api standard paragraphs rotordynamic tutorial: lateral ...api standard paragraphs rotordynamic...

API Standard Paragraphs Rotordynamic Tutorial: Lateral Critical Speeds, Unbalance Response, Stability, Train Torsionals, and Rotor Balancing

API TECHNICAL REPORT 684-1FIRST EDITION, NOVEMBER 2019

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Special Notes

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Copyright © 2019 American Petroleum Institute


Foreword

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Suggested revisions are invited and should be submitted to the Standards Department, API, 200 MassachusettsAvenue, Suite 1100, Washington, DC 20001, [email protected].

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Contents

1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-11.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-11.3 Standard Paragraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-21.4 Definitions and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-31.5 Fundamental Concepts of Rotating Equipment Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-31.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-9

2 Lateral Rotordynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-12.2 Rotor-bearing System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-12.3 Rotor Modeling Methods and Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-32.4 Support Stiffness Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-152.5 Journal Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-312.6 Seal Types and Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-592.7 Elements of a Standard Rotordynamics Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-772.8 Machinery Specific Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-932.9 API Unbalance Response Verification Testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1292.10 AMB Modeling and Analysis Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-139

3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-13.2 Rotor/Bearing System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-53.3 Journal Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-73.4 Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-333.5 Excitation Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-483.6 Support Stiffness Effects on Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-583.7 Experience Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-613.8 Machinery Specific Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-683.9 Solving Stability Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-863.10 Identifying Fluid Induced Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-933.11 Stability Testing of Machinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-983.12 Standard Paragraph Sections for Stability Analysis SP6.8.5 – SP6.8.6 . . . . . . . . . . . . . . . . . . . . . . . .3-1053.13 Active Magnetic Bearings and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-120

4 Torsional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-14.1 General Modeling Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-14.2 Machinery Specific Modeling Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-174.3 Reciprocating Machinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-354.4 Torsional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-464.5 Torsional Excitation Sources in Rotating Machinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-664.6 Fatigue Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-764.7 Contents of a torsional report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-864.8 Testing to Determine Torsional Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-924.9 Torsional–Lateral Vibration Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-994.10 Variable Frequency Drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-101

5 Balancing of Machinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-15.2 Terms and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-1

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Contents

5.3 Fundamentals of Low-speed Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-35.4 Low-speed Balancing Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-115.5 Low-speed Balancing Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-165.6 Operating-speed Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-245.7 Keys and Keyways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-335.8 Residual Unbalance Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-335.9 Check Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-345.10 Field Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-34

6 Standard Paragraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-16.2 Paragraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-1

Figures1-1 Simple Mass-spring-damper System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-41-2 Amplitude Ratio Versus Excitation Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-61-3 Phase Angle Versus Excitation Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-71-4 Response of a Spring-mass System to Transient (Stable). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-71-5 Response of a Spring-mass System to Transient (Unstable) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-81-6 Jeffcott Form for Rotor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-81-7 Simplified Model of a Beam-type Rotating Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-101-8 Simplified Model of a Beam-type Rotating Machine with Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-101-9 Spring-mass-damper Model of Beam Type Rotating Machine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-101-10 Synchronous Response of Beam Type Machine for Various Shaft Stiffness Values . . . . . . . . . . . . . . .1-112-1 Overview of Lateral Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-22-2 Schematic of a Lumped Parameter Rotor Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-42-3 3D Finite Element Model of a Complex Geometry Rotating Component . . . . . . . . . . . . . . . . . . . . . . . . . .2-62-4 Elastic Modulus vs. Temperature [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-82-5 Rotor Model Cross-section of an Eight Stage 12 MW (16,000 HP) Steam Turbine . . . . . . . . . . . . . . . . . .2-92-6 Turboexpander with Curvic Coupling Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-112-7 Turbocompressor with Rabbet and Curvic Coupling Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-112-8 Modeling of Curvic Coupling Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-112-9 Train Lateral Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-122-10 Train Lateral Guideline Diagram (Wjnl = Static Bearing Reaction) [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-122-11 Train Lateral Modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-132-12 Equivalent Coupling Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-142-13 Steam Turbine Support Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-172-14 Journal Bearing Fluid Film and Flexible Support Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-182-15 Dynamic Stiffness Analysis Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-202-16 Exhaust End Dynamic Compliance Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-202-17 Single Degree of Freedom Flexible Support Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-212-18 Steam End Test Stand Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-252-19 Exhaust End Test Stand Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-252-20 Exhaust End Constant Stiffness Support Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-272-21 Steam End Dynamic Compliance Support Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-272-22 Steam End Analytical Results, Dynamic Compliance Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-282-23 Measured and Predicted Unbalance Response for Experimental Test Rig [5] . . . . . . . . . . . . . . . . . . . .2-292-24 Journal Bearing Hydrodynamic Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-312-25 Two Axial Groove Bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-322-26 Spring Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-33


Contents

2-27 Journal Bearing Stiffness and Damping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-342-28 Pressure Dam Bearing [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-352-29 Pressure Dam Bearing—Top and Bottom Pads [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-362-30 Elliptical Bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-372-31 Offset Half Bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-372-32 Taper Land Bearing with Three Tapered Pockets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-382-33 Multi-Lobe Bearing with Three Preloaded, Offset Lobes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-382-34 Five-pad Tilting Pad Bearing Schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-412-35 Differentiating Load Between Pivots and Load Between Pads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-422-36 Zero Preloaded Pad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-432-37 Preloaded Pad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-432-38 Negative Preloaded Pad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-442-39 Stiffness and Damping vs. Preload and Bearing Clearance, 4-Pad Bearing [20] . . . . . . . . . . . . . . . . . .2-452-40 Stiffness and Damping vs. Preload and L/D Ratio, 4-Pad Bearing [20] . . . . . . . . . . . . . . . . . . . . . . . . . .2-462-41 Lund’s Data vs. Experimental [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-482-42 Jones and Martin Data vs. Experimental [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-492-43 Actual Test Stand Response, 3-Axial Groove Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-502-44 Analytically Predicted Response for Different Bearing Designs on an Axial Compressor . . . . . . . . . .2-512-45 Actual Test Stand Response, 4-Pad Tilting Pad Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-512-46 Analytically Predicted Response, Various Bearing Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-522-47 Induction Motor Test Stand Response, Tilting Pad Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-532-48 Induction Motor Analytical Response, Tilting Pad Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-532-49 Induction Motor Analytical Response, Elliptical Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-542-50 Induction Motor Test Stand Response, Elliptical Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-542-51 Oil Bushing Breakdown Seal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-602-52 Pressures Experienced by the Outer Floating Ring Seal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-612-53 Mid-Span Rotor Unbalance Response of a High-pressure Centrifugal Compressor for Different Suction

Pressures at Start-up [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-622-54 Mechanical (Contact) Shaft Seal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-632-55 Liquid-film Shaft Seal with Cylindrical Bushing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-642-56 Liquid-film Shaft Seal with Pumping Bushing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-652-57 Tilt Pad Oil Seal [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-662-58 Compressor Labyrinth Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-672-59 Typical Turbine Shaft Seal Arrangement—HP End . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-672-60 Honeycomb Damper Seal [32] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-682-61 Pocket Damper Seal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-682-62 Hole Pattern Damper Seal with Swirl Brakes [25] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-682-63 Swirl Brake Used in High-pressure Compressors [26] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-692-64 Shunt Hole System with Honeycomb Division Wall Seal Used in High-pressure Compressors [30] . .2-692-65 Measured Natural Frequency Showing the Increase of the Shaft’s First Bending Mode with Pressure [31] . . . . .2-712-66 Labyrinth Seal Bulk Flow Control Volume Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-712-67 Segmented-ring Shaft Seal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-732-68 Self-acting Dry Gas Seal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-742-69 Undamped Critical Speed Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-792-70 Mode Shape Examples for Soft and Stiff Bearings Relative to Shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-802-71 Typical Regions within an Undamped Critical Speed Map for a Between-Bearing Machine . . . . . . . . .2-802-72 Typical Undamped Mode Shapes for a Between Bearing Machine with Different Values of Support

Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-822-73 Typical Bode Plot for Asymmetric System with Split Critical Speeds . . . . . . . . . . . . . . . . . . . . . . . . . . .2-85


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2-74 Example Compressor with Probes Rotated to True Horizontal and Vertical . . . . . . . . . . . . . . . . . . . . . .2-862-75 Evaluating Amplification Factors (AFs) from Speed-amplitude Bode Plots . . . . . . . . . . . . . . . . . . . . . .2-882-76 Rotor Response Shape Plots in 2D and 3D Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-902-77 Motion of an Stable System Undergoing Free Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-912-78 Motion of an Unstable System Undergoing Free Oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-912-79 Integral Shafts Made from a Single Forging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-942-80 Built-up Rotor with Shrunk-on Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-942-81 Steam Turbine Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-952-82 Typical Resultant Bearing Load Vector Including Partial Admission Steam Forces. . . . . . . . . . . . . . . .2-962-83 Sequence of Admission for Different Turbines’ Partial Arc Segments . . . . . . . . . . . . . . . . . . . . . . . . . .2-972-84 Resolution of Partial Admission Forces into Journal Bearing Reactions . . . . . . . . . . . . . . . . . . . . . . . .2-982-85 Induction Motor Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1002-86 Salient Pole Synchronous Motor Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1002-87 Salient Pole Synchronous Motor Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1012-88 Massive Pole Synchronous Motor Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1022-89 Core Magnetic Field Distribution and Cross Section Within a Four Pole Induction Motor [2] . . . . . .2-1022-90 Gear Set Showing Rotation Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1052-91 Gear Force Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1062-92 Gear Load Angles at Partial and Full Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1062-93 Accumulated Pitch Error Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1072-94 FCC Expander Critical Speed Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1092-95 FCC Expander Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1112-96 FCC Expander Rotor-bearing Support Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1122-97 Axial Compressor Rotor Construction: Disc-on-shaft Shrink Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1132-98 Axial Compressor Rotor Construction: Stacked Discs with Tie Bolts . . . . . . . . . . . . . . . . . . . . . . . . . .2-1142-99 Axial Compressor Rotor Construction: Drum Rotor with Studs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1142-100 Axial Compressor Rotor Construction: Drum Rotor with Tie Bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1152-101 Typical Multi-stage Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1162-102 Soft Support Undamped Mode Shapes—Multi-stage Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1172-103 Stiff Support Undamped Mode Shapes—Multi-stage Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1182-104 Typical Unbalance Distributions for Multi-stage Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1182-105 Unbalance Response of 1st and 3rd Critical Speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1192-106 Rotor Response Shape @ 4500 rpm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1192-107 Unbalance Response of 2nd Critical Speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1202-108 Rotor Response Shape @ 12,800 rpm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1202-109 Typical Overhung Compressor 2-1212-110 Overhung Compressor Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1222-111 Overhung Compressor Undamped Mode Shapes (Impeller on Right) . . . . . . . . . . . . . . . . . . . . . . . . .2-1232-112 Undamped Critical Speed Map—Overhung Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1242-113 Impeller Unbalance Response—Overhung Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1252-114 Rotor Response Shape at 4,300 rpm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1252-116 Integrally Geared Compressor Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1262-115 Coupling Unbalance Response—Overhung Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1262-117 View of Integrally Geared Compressor Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1272-118 Typical Pinion Rotors from an Integrally Geared Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1282-119 Pinion Rotor Model—Integrally Geared Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1282-120 Typical Rigid and Flexible Body Mode Shapes & Unbalances Used to Excite Each . . . . . . . . . . . . . 2-1292-121 Baseline Vibration Reading (Graphically) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1332-122 Readings After the Addition of the Unbalance Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-133


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2-123 Influence of the Unbalance Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1342-124 Bode Plot for Eight-Stage Compressor [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1352-125 Unbalance Weight Influence (.... Predicted ___Test) [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1352-126 Bode Plot for Three-Stage Compressor [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1362-127 Unbalance Weight Influence (…. Predicted ___Test) [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1362-128 Bode Plot of Example #3 [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1372-129 Unbalance Weight Influence [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1382-130 AMB System 2-1392-131 AMB System Radial Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1402-132 SISO Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1432-133 SISO Control Transfer Function Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1432-134 MIMO Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1442-135 Transfer Function Matrix for MIMO Control (General Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1442-136 Typical Control System Transfer Function Bode Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1452-137 Typical Free-Free AMB Map (Nmc = 30,000 RPM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1472-138 AMB Force Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1492-139 Typical AMB Allowable Force Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1502-140 Closed Loop Transfer Function Measurement Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1522-141 Example AMB Closed Loop Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1532-142 Open Loop Transfer Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1532-143Typical Open Loop Transfer Function Measurement for Most Controlled Systems . . . . . . . . . . . . . .2-1542-144 Practical “Pseudo Open Loop” Transfer Function Measurement for AMB System . . . . . . . . . . . . . .2-1542-145 Example Pseudo Open Loop Transfer Function 2-1552-146 Closed Loop Transfer Function Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1572-147 Common Auxiliary Bearing Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1602-148 Typical Auxiliary Bearing Mount System [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1612-149 Typical Horizontal Rotor Landing [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1632-150 Vertical Rotor Landing Orbit [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1633-1 Definition of Log Dec Based on Rate of Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-23-2 Stability Analysis Flow Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-43-3 Fixed Geometry Bearing Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-83-4 High-Speed, Lightly Loaded, Unstable Bearing [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-93-5 Low-Speed, Heavily Loaded, Stable Bearing [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-103-6 Bearing-Induced Shaft Whip and Oil Whirl [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-113-7 Frequency Spectrum, Power Turbine Test, 3-axial Groove Bearings [1] . . . . . . . . . . . . . . . . . . . . . . . . .3-123-8 Rotor Bearing System Stability, Power Turbine N = 5000 rpm [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-133-9 Frequency Spectrum, Power Turbine Test, Double Pocket Bearings [1] . . . . . . . . . . . . . . . . . . . . . . . . .3-143-10 Frequency Dependent Stiffness and Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-163-11 Full Coefficient vs. Synchronous Reduced Tilting Pad Bearing Stability Sensitivity . . . . . . . . . . . . . .3-173-12 Waterfall Showing Self-Excited Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-183-13 High-speed Balance Vacuum Pit Oil Atomization Resulting in Subsynchronous Vibration [3] . . . . . .3-203-14 Single Housing Orifice Design Resulting in Subsynchronous Vibration [3] . . . . . . . . . . . . . . . . . . . . . .3-213-15 Spray Bar Evacuated Housing Design [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-213-16 Button Spray Design Resulting in Subsynchronous Vibration [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-223-17 Squeeze Film Damper Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-233-18 Typical End Seal Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-243-19 Axial Pressure Profiles of Various Damper Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-253-20 Squeeze Film Damper Coefficients vs. Eccentricity Ratio: Short Bearing Theory (Cavitated) [8] . . . .3-273-21 Idealization of Bearing-damper-support Characteristics [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-28


Contents

3-22 O-ring Supported Squeeze Film Damper Schematic [2,3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-293-23 Mechanical Arc Spring Supported Squeeze Damper [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-303-24 Squeeze Film Damper Stability Map [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-313-25 Re-excitation of Rotor First Critical from Oil Seal Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-343-26 Rotor Tracking Instability from Low-pressure Oil Seal Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-353-27 Rotor Tracking Instability from Distorted Oil Seal Lip Contact Area . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-353-28 Typical Configuration for a Back-to-Back Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-383-29 Typical Configuration for an In-line Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-383-30 Typical Configuration for the Last Stage of a Series Flow Compressor Showing the Impeller Eye Seal,

the Inter Stage Seal and a Typical Balance Piston Seal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-393-31 Shunt Line Schematic to Reduce Entry Swirl [8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-413-32 Typical Swirl Brake Schematic to Reduce Entry Swirl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-423-33 Compressor on Full Load Test: With Inert Gas Showing no Instability at Rotor First Critical Frequency

[13], with Process Gas Showing Instability from Balance Piston Excitation [5,13] . . . . . . . . . . . . . . . .3-423-34 Measured and Predicted Rotordynamic Stability vs. Discharge Pressure for a High Pressure Centrifugal

Compressor [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-453-35 Comparison of Honeycomb and Hole Pattern Seals [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-463-36 Various Hole Pattern Surface Areas Relative to Honeycomb [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-463-37 Example Plot of Predicted Rotor Stability as a Function of Damper Seal Clearance Taper [11] . . . . .3-473-38 Blade Forces Due To Centerline Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-503-39 Shrink Fit Internal Friction and Shaft Material Hysteresis Destabilizing Force [1] . . . . . . . . . . . . . . . . .3-533-40 Dry Friction Rub Backward Whirl Excitation [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-543-41 Entrapped Fluid Cross-coupled Force [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-553-42 Differential Heating at Bearing Journal for Synchronous Forward Whirl . . . . . . . . . . . . . . . . . . . . . . . .3-563-43 Predicted and Measured Stability Threshold with and without Bearing Support Models . . . . . . . . . . .3-593-44 Stiffness Ratio vs. Log Dec for Two Example Centrifugal Compressors . . . . . . . . . . . . . . . . . . . . . . . . 3-603-45 % Critical Damping vs. Log Dec for Two Example Centrifugal Compressors . . . . . . . . . . . . . . . . . . . .3-603-46 Sood’s General Rotor Stability Criteria [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-623-47 Sood/Fulton Empirical Stability Criteria [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-623-48 Kirk’s Compressor Design Map [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-633-49 API Level I Screening Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-643-50 Memmott’s Compressor Experience Plot—Flexibility Ratio vs. Average Gas Density [8, 9] . . . . . . . .3-653-51 Memmott's Compressor Experience Plot—Pressure Parameter vs. Flexibility Ratio [8,9] . . . . . . . . . .3-663-52 Memmott's Compressor Experience Plot—Bearing Span/Impeller Bore vs. Average Gas Density [8,9] . . .3-663-53 Memmott's Compressor Experience Plot—Pressure Parameter vs. Bearing Span/Impeller Bore [13,14] . . .3-673-54 Steam Turbine Leakage Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-693-55 Typical Resultant Bearing Load Vector Including Partial Arc Admission Steam Forces . . . . . . . . . . . .3-703-56 FCC Expander Critical Speed Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-723-57 FCC Expander Cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-743-58 FCC Expander Rotor-bearing-support Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-753-59 Axial Compressor Rotor Construction: Disk-on-shaft Shrink Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-763-60 Axial Compressor Rotor Construction: Stacked Disks with Tie Bolts . . . . . . . . . . . . . . . . . . . . . . . . . . .3-773-61 Axial Compressor Rotor Construction: Drum Rotor with Studs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-783-62 Axial Compressor Rotor Construction: Drum Rotor with Tie Bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-793-63 High-speed Gearbox Pressure Dam Pinion Bearing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-813-64 High-speed Gearbox Pressure Dam Bull Gear Bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-813-65 Typical Multi-stage High-pressure Centrifugal Compressor Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-823-66 Overhung Compressor Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-833-67 Pinion Rotors from an Integrally Geared Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-84


Contents

3-68 Forces Exerted on a Whirling Shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-873-69 Mode Shapes for Various Support/Rotor Stiffnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-883-70 Half Spectrum Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-953-71 Forward Precession Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-963-72 Reverse Precession Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-973-73 Full Spectrum—Reverse Precession of 0.5X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-973-74 Shaft Centerline Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-983-75 Measured vs. Identified Frequency Response Functions during a Stability Test [7] . . . . . . . . . . . . . .3-1013-76 Time Transient Waveform from Blocking Excitation during a Stability Test [7] . . . . . . . . . . . . . . . . . .3-1023-77 Estimating Stability from Time Transient Data Using Mechanical Log Decrement [12] . . . . . . . . . . .3-103SP-9 Level I Screening Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-1103-78 Process Compressor Cross-section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-1133-79 Gas Injection Compressor Cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-1133-80 Process Compressor Stability Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-1173-81 Gas Injection Compressor Stability Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-1173-82 Process and Gas Injection Compressors on Experience Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-1183-83 AMB System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-1203-84 AMB System Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-1213-85 Stability Analysis/Log Dec Requirements for AMB Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-1243-86 Example of Sensitivity Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1253-87 Sensitivity Function Measurement Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-1263-88 Sensitivity Transfer Function That Fails to Meet the Acceptance Criterion . . . . . . . . . . . . . . . . . . . . .3-1284-1 Typical Coupling Stiffness Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24-2 Side View of a Typical Motor/Gear/Compressor Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-24-3 Modeling a Typical Motor/Gear/Compressor Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-34-4 Schematic Lumped Parameter Model for the Motor/Gear/Compressor Train . . . . . . . . . . . . . . . . . . . . . .4-34-5 Side View of a Typical Steam Turbine Driven Compressor Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-44-6 Modeling a Typical Steam Turbine Drive Compressor Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-44-7 Schematic Lumped Parameter Model for the Steam Turbine Driven Compressor Train . . . . . . . . . . . . 4-54-8 A Typical Twin-pinion Integrally Geared Centrifugal Compressor That Should be Modeled as a Branched

System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-74-9 Effective Penetration of a Smaller Diameter Shaft Section into a Larger Diameter Shaft Section . . . . .4-84-10 45 Degree Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-94-11 Examples of Shrunk on Sleeves With and Without Relieved Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-104-12 Point of Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-104-13 A Reduced Moment Gear Coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-114-14 A Marine-style Diaphragm Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-124-15 A Reduced Moment Disc Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-124-16 Coupling Spacer Torsional Stiffness Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-134-17 Typical Nonlinear Stiffness vs. Torque Characteristic for an Elastomeric Coupling . . . . . . . . . . . . . . .4-144-18 An Elastomeric Hybrid Coupling (w/Disc Type). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-154-19 Temperature-dependent Shear Modulus Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-164-20 Cross-sectional View of a Parallel Shaft Speed Increaser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-184-21 Torsional Model of a Parallel Shaft Speed Increaser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-194-22 Torsional Model of a Gear Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-204-23 Typical Geometry of a Webbed Motor Rotor and Cross-Section Under the Windings . . . . . . . . . . . . . 4-224-24 Torsional Stiffness Data for Six-Webbed Shaft with Varying Web Thickness [8] . . . . . . . . . . . . . . . . . .4-234-25 Torsional Stiffness and Damping Coefficients of a 930 kW Eight-pole Induction Motor with rated

Operation Condition [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-25


Contents

4-26 HVSP Schematic [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-284-27 View of Typical Screw Compressor Rotor Pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-294-28 Typical System Model of a Dry Screw Compressor Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-304-29 Typical System Model of a Flooded Compressor Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-314-30 API 672 Integrally Geared Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-324-31 API 672 Integrally Geared Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-334-32 Typical Mode Shapes for an Integrally Geared Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-334-33 Portion of a Typical Crankshaft Throw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-364-34 Finite Element Models Used to Calculate Torsional Stiffness of Crankshaft Sections . . . . . . . . . . . . .4-374-35 Typical Reciprocating Train Response Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-414-36 Viscous Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-434-37 Campbell Diagram for a Motor-Gear-Compressor System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-484-38 Campbell Diagram for a Steam Turbine Driven Compressor System . . . . . . . . . . . . . . . . . . . . . . . . . . .4-494-39 Torsional Mode Shapes for a Typical Motor-Gear-Compressor Train . . . . . . . . . . . . . . . . . . . . . . . . . . .4-504-40 Torsional Mode Shapes for a Typical Motor-Gear-Compressor Train (Continued) . . . . . . . . . . . . . . . .4-514-41 Campbell Diagram for a Motor-Gear-Compressor Train After Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . .4-554-42 Torsional Mode Shapes for a Typical Steam Turbine-Driven Compressor Train . . . . . . . . . . . . . . . . . .4-564-43 3rd Torsional Natural Frequency of a Motor-Gear-Compressor System . . . . . . . . . . . . . . . . . . . . . . . . .4-594-44 1st Torsional Natural Frequency of a Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-604-45 A Typical Magnification Factor Plot of a Torsional Steady-State Response Analysis . . . . . . . . . . . . . .4-614-46 Transient Torsional Motor Fault Analysis Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-634-47 Speed Torque Curve for a Synchronous Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-644-48 Plot of Synchronous Motor Transient Response Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-654-49 Transient Torque Associated with a Generator Line-to-line Short Circuit . . . . . . . . . . . . . . . . . . . . . . . .4-744-50 Transient Torque Associated with Generator Synchronization Error of 45 Degrees . . . . . . . . . . . . . . .4-754-51 Shaft Operating Stress as a Function of Shaft Operating Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-774-52 Transient Torsional Simulation of a Synchronous Motor-Driven Compressor Train . . . . . . . . . . . . . . .4-834-53 Typical Speed-Torque Curve for a Synchronous Motor with Laminated Pole Construction . . . . . . . .4-844-54 Typical Speed-Torque Curve for a Synchronous Motor with Solid Pole Construction . . . . . . . . . . . . .4-854-55 Displacement Measurement Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-964-56 Torsional Measurements Using a Toothed Wheel and Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-974-57 Torsional Vibration Measurements with a Bar Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-974-58 Overview Schematic of a VFD System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-1034-59 Simplified Representation of a VFDs Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-1044-60 Six-pulse Rectifier and the Voltage Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-1054-61 Six-pulse Load Commutated Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-1074-62 Two-level Voltage Source Inverter Drive with Diode Rectifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-1084-63 Campbell Diagram Showing VFD Harmonic Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1124-64 Campbell Diagram Showing VFD Inter-Harmonic Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-1134-65 Bode Plot of Torque Transfer Functions for a Selection of Drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-1184-66 Block Diagram of Drive System Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-1184-67 VSI and the Compressor Drive Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-1205-1 Unbalance Expressed as the Product of Weight and Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-55-2 Static Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-65-3 Dynamic Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-65-4 Combination of Static and Dynamic Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-65-5 Unbalance Distribution Resolved into Static and Dynamic Components . . . . . . . . . . . . . . . . . . . . . . . . 5-85-6 Permissible Residual Specific Unbalance Based on Balance Quality Grade G and Service Speed, n (ISO

1940-1:2003) [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-10


Contents

5-7 Comparison of ISO and API Balance Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-115-8 Unbalance Versus Speed for API Limits and Balance Machine Limit (Calculated at W = 1 lbm). . . . . .5-125-9 Applicable Speed Ranges for Hard-bearing and Soft-bearing Balancing Machines [1] . . . . . . . . . . . .5-125-10 Typical Soft Bearing Balancing Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-135-11 Low-speed Balance Machine Employing Direct-end Drive Wrap-Around Belt Drive . . . . . . . . . . . . . . 5-155-12 Low-speed Balance Machine with Wrap-Around Belt Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-155-13 Fit Eccentricity Related Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-205-14 Unbalance Correction to Fit Eccentricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-215-15 Initial Reading of the Index Balancing Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-225-16 Indexed Component Relative to the Shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-225-17 Vector Representation of and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-235-18 Results of Adding and Subtracting Vectors and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-245-19 Top Loading High-speed Balance Bunker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-265-20 Close-up of Bearing Pedestal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-265-21 Bunker Drive Shaft (Shown in Red) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-275-22 Hydraulic Pedestal Stiffeners Highlighted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-28SP-6 Unbalance Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-4SP-7 Rotor Response Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-7SP-8 Definition of Speed Range for Probe Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-8SP-9 Level I Screening Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-121.C-1Undamped Critical Speed Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-251.C-2Level I Stability Sensitivity Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-271.C-3Stability Experience Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-271.C-4Geometry Definitions for Tilt Pad Bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-291.C-5Preloaded Pad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-291.D-1Typical Campbell Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-32

Tables2-1 Computer Model Generated for the Eight-Stage Steam Turbine Rotor . . . . . . . . . . . . . . . . . . . . . . . . . .2-102-2 Bearing, Support and Equivalent Characteristics, 13.1 kN Steam Turbine . . . . . . . . . . . . . . . . . . . . . . .2-232-3 Bearing, Support and Equivalent Characteristics, 22.5 kN Steam Turbine . . . . . . . . . . . . . . . . . . . . . . .2-232-4 General Behavior and Requirements of Oil and Gas Annular Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-592-5 Summary of the Results of the Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1382-6 Example Axial Damped Natural Frequency Analysis Results (Amplification Factors > 2.5) . . . . . . . 2-1583-1 Formula for Squeeze Film Damper Stiffness and Damping Coefficients . . . . . . . . . . . . . . . . . . . . . . . .3-263-2 Decreasing or Eliminating Excitation Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-893-3 Increasing Effective Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-903-4 Level I Stability Results for Process and Gas Injection Compressor . . . . . . . . . . . . . . . . . . . . . . . . . .3-1143-5 Minimum Log Decrement for Gas Injection Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-1193-6 Peak Sensitivity Function Zone Limits [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-1274-1 Penetration Factors for Selected Shaft Step Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-85-1 Relationship Between Machine Pedestal Stiffness, Rotor Weight, and Operating Speed . . . . . . . . . .5-321.C-1Tilt Pad Bearing Dimensions and Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-28

Worksheets3-1 Modified Alford’s Force—Process Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-1153-2 Modified Alford’s Force—Gas Injection Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-116

R11 R12

R11 R12


1-1

API Standard Paragraphs Rotordynamic Tutorial: Lateral Critical Speeds, Unbalance Response, Stability, Train Torsionals, and Rotor Balancing

1 Scope

1.1 Introduction

This document is intended to describe, discuss, and clarify the API Standard Paragraphs (SP) Section 6.8 whichoutlines the complete lateral and torsional rotordynamics and rotor balancing acceptance program designed by API toensure equipment mechanical reliability. Background material on the fundamentals of these subjects (includingterminology) along with rotor modeling utilized in this analysis is presented for those unfamiliar with the subject.

The standard paragraphs are introduced with references to the appropriate background material to enhance theunderstanding. This information is intended to be a primary source of information for this complex subject and isoffered as an introduction to the major aspects of rotating equipment vibrations that are addressed during a typicallateral dynamics analysis. It is not intended to be a comprehensive guideline on the execution of rotordynamicsanalyses but is intended to:

a) provide guidance on the requirements for analysis;

b) aid in the interpretation of rotordynamics reports;

c) provide guidance in judging the acceptability of results presented.

1.2 Organization

The document is divided into six sections:

1. Overview

2. Lateral Dynamic Analysis

3. Stability Analysis

4. Torsional Analysis

5. Balancing of Machinery

6. Standard Paragraphs

The individual sections have been prepared in a stand alone manner. As a result, necessary material may berepeated in a succeeding section to provide sufficient clarity to the discussion.

Sections two through four have a parallel organization:

— Modeling criteria

— Analysis techniques and results

— Machine specific considerations

— Testing

— Applications and examples


1-2 API TECHNICAL REPORT 684-1

1.3 Standard Paragraphs

In order to aid turbomachinery purchasers, the American Petroleum Institute’s Subcommittee on MechanicalEquipment has produced a series of specifications that define mechanical acceptance criteria for new rotatingequipment. Experience accumulated by turbomachinery purchasers over the past ten years indicates that if the APIstandards are properly applied, the user can be reasonably assured that the installed unit is fundamentally reliableand will, barring problems with the installation and operator misuse, provide acceptable service over its design life.

An integral component of these individual equipment specifications is contained in the API Standard Paragraphs,those specifications that are generally applicable to all types of rotating equipment. The criteria associated with lateraland torsional rotordynamics and balancing have been categorized as standard paragraphs. In rotating equipmentspecifications published by API (for example, API Standard 617—Axial and Centrifugal Compressors and Expander-compressors for Petroleum, Chemical and Gas Industry Services) there is a section on rotordynamics and balancing.The backbone of those sections is the standard paragraphs augmented by additional information that is applicableonly to the type of unit considered in the standard.

The Dynamics Standard Paragraphs originated in the Centrifugal Compressor Standard, API 617. A timeline for thefirst through eighth editions with major highlights is [1]:

— 1st Edition June 1958—dynamics section added. A critical speed separation margin defined and a balance/vibration criterion established. Vibration limit in operating speed range for mechanical test is a step function.

— 2nd Edition April 1963—torsional analysis added as an option.

— 3rd Edition October 1973—balance criteria and separation margin for laterals below operating speed rangeadjusted, rotor response required when specified, torsional analysis required on motor driven and geared units,vibration limit in operating speed range changes from step function to (12,000/MCOS)0.5 or 2.0 mils whichever isless and vibration limit added at trip speed.

— 4th Edition November 1979—AF (amplification factor) limited when passing through lateral critical speeds, lateralanalysis required including a rotor response and transient torsional analysis for all synchronous motor-drivenunits.

— 5th Edition April 1988—removed limits on AF while passing through lateral critical speeds, acceptance definedbasis rotor response analysis, lateral analysis report requirements spelled out in detail, operating seal clearancecriteria added, separation margin for laterals tied to amplification factor, amplification factor less than 2.5acceptable in any operating case, a dynamics shop test included, and balance criteria modified.

— 6th Edition February 1995—lateral paragraphs reorganized and criteria set for comparison with shop testevaluation and transient torsional required with variable speed motors, and vibration limit in operating speedrange changes to (12,000/maximum continuous speed)0.5 or 1.0 mils whichever is less (had been 2.0 mils).

— 7th Edition July 2002—formula for allowable separation margins for lateral criticals changes, unbalanceresponse test criteria clarified, stability analysis requirement included, and entire section updated.

— Proposed 8th Edition—active magnetic bearing specifications expanded including stability criteria, inclusion ofdamper seals into lateral analysis, high-speed balance specifications revamped, requirements for lateral andtorsional reports added and influence of VFD, generators and motor faults and excitations on torsional analysis.

The complete text of the Dynamic Section of the Standard Paragraphs is included at the end of the document.


API STANDARD PARAGRAPHS ROTORDYNAMIC TUTORIAL: LATERAL CRITICAL SPEEDS, UNBALANCE RESPONSE, STABILITY, TRA N TORSIONALS, AND ROTOR BALANCING 1-3

1.4 Definitions and References

Definitions are incorporated into each section of the document. Due to very large number of references employed, theyare identified at the end of each relevant section.

1.5 Fundamental Concepts of Rotating Equipment Vibrations

In order to understand the results of a rotordynamics design analysis, it is necessary to first gain an appreciation for thephysical behavior of vibratory systems. Begin by noting that all real physical systems/structures (such as buildings,bridges, and trusses) possess natural frequencies. Just as a tuning fork has a specific frequency at which it will vibratemost violently when struck, a rotor has specific frequencies at which it will tend to vibrate during operation. Eachresonance is essentially comprised of two associated quantities: the frequency of the resonance and the associateddeflections of the structure during vibration at the resonance frequency. Resonances are often called “modes of vibration”or “modes of motion,” and the structural deformation associated with a resonance is termed “mode shape.”

The modes of vibration are important only if there is a source of energy to excite them, like a blow to a tuning fork. Thenatural frequencies of rotating systems are particularly important because all rotating elements possess finite amounts ofunbalance that excite the rotor at the shaft rotation frequency (synchronous frequency) and its multiples. When thesynchronous rotor frequency equals the frequency of a rotor natural frequency, the system operates in a state ofresonance, and the rotor’s response is amplified if the resonance is not critically damped. The unbalance forces in arotating system can also excite the natural frequencies of nonrotating elements, including bearing housings, supports,foundations, piping, and the like.

Although unbalance is the excitation mechanism of greatest concern in a rotordynamics analysis, unbalance is only oneof many possible lateral loading mechanisms. Lateral forces can be applied to rotors by the following sources: impelleraerodynamic loadings, misaligned couplings and bearings, rubs between rotating and stationary components, and so on.A more detailed list of rotor excitation mechanisms of particular interest is found in the API Standard Paragraphs, 6.8.1.1.This subject is discussed in detail in 3.5, as well as scattered in the appropriate sections of this document.

The vibration behavior of a rotor can be described with the aid of a simple physical model. Assume that a rotor-bearingsystem is analogous to the simple mass-spring-damper system presented in Figure 1-1.

From physics, the governing equation of motion for this system can be written as Equation 1-1 and Equation 1-2:

ma + cv + kx = F(t) (1-1)

(1-2)

where

m is the mass of the block, kg (lbm);

a = is the acceleration, m/s2 (in./s2);

c is the viscous damping coefficient, N-s/m (lbf-s/in.);

v = is the velocity, m/s (in./s);

k is the stiffness of the elastic element, N/m (lbf/in.);

x is the displacement of the block, m (in.);

F(t) is the force applied to the block (time-dependent function), N (lbf).

mꞏꞏ cꞏ kx+ + F t =

ꞏꞏ

ꞏ



Figure 1-1—Simple Mass-spring-damper System

g (gravitationalacceleration)

M (mass)[=Weight/g]

c (damping)

k (stiffness)

F (force)

x (displacement)

v (velocity)

a (acceleration)

9.88 m/s2

kg

N-s/mm

N/mm

N

m

mm/s

g

386.4 in./s2

lbm

lbf-s/in.

lbf/in.

lbf

mils

in./s

g

- NA -

2.2046

0.17513

0.17513

4.4482

25.4

25.4

1.0

US-to-SlConversion1

US Customary Units(FLT System of Units)

Sl Units(MLT System of Units)Quantity

Table of Typical Units and Conversions

Masselement

(m)

F(t)

x, v, a

Stiffnesselement

(k)

Dampingelement

(c)

NOTE 1 Multiply the quantities listed above (in US Customary Units shown) by the US to SI conversion factors to obtain the quantity in the SI units listed in the table.

NOTE 2 NA = not applicable.

NOTE 3 Common Unitss = secondsg = acceleration

US Customary Units in. = inches mil = 1.0 x 10-3 in. lbf = pound (force) lbm = pound (mass)

SI Units mm = millimeters m = micrometers kg = kilograms N = Newtons = kg-m/s2



In this example, the displacement response of the block to the applied force is counteracted by the block’s mass andthe support’s stiffness and damping characteristics. The undamped natural frequency of this system is calculated bydetermining the eigenvalue of the second order homogeneous ordinary differential equation (F = 0) for the casewhere the damping term is neglected (c = 0) as seen in Equation 1-3.

(1-3)

where

is the undamped natural frequency, rad/s.

Since real, physical systems include damping, this needs to be included in the analysis. The damped naturalfrequency of the homogeneous system (F = 0) is defined in Equation 1-4.

(1-4)

where

d is the damped natural frequency, rad/s.

If the system was excited (hit by a hammer), this damped natural frequency is the frequency of vibration that would beseen as the system responds. Note that the damped natural frequency of the system, d, is equal to the undampednatural frequency of the system only when system damping is negligible. This observation underscores the fact thatan undamped critical speed analysis should, in general, not be used to define the critical speeds of a rotatingmachine.

If the exciting force is sinusoidal, i.e.,

F(t) = A sin (t) (1-5)

The response will be:

x(t) = B sin (t + ) (1-6)

where

is the amplitude ratio;

is the phase angle, rad.

In the case of a rotor with unbalance, the unbalance force is defined in Equation 1-7:

FUB = meu2 (1-7)

where

m is the mass of rotor, kg (lbm);

eu is the mass eccentricity, m (in.);

is the rotational speed, rad/s;

FUB is the unbalance force, N (lbf).

km---=

dkm---- c

2m-------

2

–=

BA---



This result is called “forced response analysis” and is analogous to the unbalance response analysis performed inrotordynamics studies. The amplitude ratio depends upon frequency of the excitation and the damping in the system.Figure 1-2 shows the amplitude ratio versus the excitation frequency. Maximum amplitude ratio is seen where theexcitation frequency equals the natural frequency of the system. Amplitude ratio also increases as dampingdecreases with the amplitude becoming infinite at zero damping (a situation that is not physically practical).

There is a phase difference between the excitation and response. This phase difference is a function of damping andreaches 90 degrees at the natural frequency. Figure 1-3 shows the phase angle versus excitation frequency.

If a transient rather than a sinusoidal excitation excites the system, the actual response normally looks like that shownin Figure 1-4.

In this case, the response is at the natural frequency. It decays with time based on the amount of damping. Thisresponse is called “stable.” In Figure 1-5, the response grows with time. This response is called “unstable.”

While the simple, single degree of freedom system described above is useful for examining the general concepts ofvibration theory, this system is clearly not representative of a turbomachine. Recognize that the early development ofrotor dynamics analysis took place in the late 19th to early 20th Centuries and calculations could only be performedwith pencil and paper. Thus, it was required to utilize sound, simplifying assumptions to permit solutions to beperformed. In 1869, Lord Rankine published an analysis that concluded operation of a rotating shaft above its firstresonant speed was impossible. This reflected the common feeling of many of the learned mechanics experts of theday. However, in 1889, Gustav Laval built a steam turbine which he operation above the first resonance.

It was apparent that a more accurate representation of a rotating assembly was required. Henry Jeffcott (undercharter of the British Royal Society) performed an analysis with the results published in 1919. In that analytical work,he proposed the model as shown in Figure 1-6. It is a two degree of freedom model with all mass concentrated in acentral disk. The disk is mounted on a massless flexible shaft and It is connected to ground by rigid supports. Thismodel may be analyzed and solved to indicate the presence of the first undamped critical speed. This form of modelis known as the Laval (Europe) or Jeffcott (outside Europe) model.

Figure 1-2—Amplitude Ratio Versus Excitation Frequency

6

5

4

3

2

1

01000 2000

Rotation speed (rpm)

Am

ptu

de

Decreasing damping

3000 4000 5000



Figure 1-3—Phase Angle Versus Excitation Frequency

Figure 1-4—Response of a Spring-mass System to Transient (Stable)

Rotation speed (rpm)

Decreasing damping

Pha

se a

nge

180

150

120

90

60

30

01000 2000 3000 4000 5000

-1.5

-1.0

-0.5

0.0

0.5

1.0

0 1 2 3 4 5 6 7 8 9 10

1.5

Eigenvalue of a viscously damped system:

s = p + iwdp = damping exponent

wd = frequency of oscillation{

-ept

ept

xxo

= Real (est)

x x o

envelope (p < 0)displacement

Time (seconds)

(Dm

ens

ones

s)



Figure 1-5—Response of a Spring-mass System to Transient (Unstable)

Figure 1-6—Jeffcott Form for Rotor Model

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

10 2 3 4 5 6 7 8 9 10Time (seconds)

envelope (p > 0)displacement


s = p + iwd

wd = frequency of oscillation p = damping exponent{

ept

-ept

estxxo = Real ( )

x x o(D

men

son

ess)

Unbalanced disk

Elastic shaft

Rigid supports

I

I / 2



The Laval-Jeffcott Model is used extensively to gain insight into the potential rotor dynamic performance of rotors.While it is able to provide simple, qualitative results, more complete and comprehensive models were required. Thenext level of complexity added was to replace the rigid shaft supports with springs as shown in Figure 1-7. Then,damping was added Figure 1-8 and Figure 1-9. As computing capabilities began to be available, more comprehensive,accurate models were developed. These are described in considerably detail in the rest of this document.

In order to consider the effect of the combined shaft and bearing stiffness, we need to review the effective combinedstiffness as shown in Figure 1-7. As shown, the effective stiffness, Keq, is the result of the individual stiffnesses added.The equation that describes this is:

(1-8)

(1-9)

(1-10)

where

Kshaft is the shaft stiffness, N/m (lbf/in.);

E is the Young’s modulus for shaft material, N/m2 (lbf/in.2);

L is the shaft section length, m (in.);

I is the area moment of inertia, m4 (in.4);

D is the shaft diameter, m (in.).

This equation indicates that the stiffness of the combined shaft-bearing system will be less than the stiffness of thesingle most flexible element. In this example, the shaft is the single most flexible element (Kshaft = 8756.5 N/mm or50,000 lbf/in.). According to Equation 1-8, the effective stiffness of the combined shaft-bearing spring system is only7297.7 N/mm (41,670 lbf/in.).

To carry this analysis one step higher in complexity, consider the sketch of a rotordynamic system displayed in Figure1-8. Figure 1-9 shows the model with masses, springs, and dampers.

Note that this system is identical to the system just discussed except that viscous damping elements have beenadded to the bearing model. All oil film bearings generate significant viscous damping forces. Figure 1-10 displays thecalculated response of the disk to a harmonic load acting at the disk for various values of shaft stiffness.

Note that as the shaft stiffness decreases, the peak response frequency decreases while the amplitude of the peakresponse and the sharpness of the peak both increase. These observations are understood by noting that thedecrease in shaft stiffness decreases the relative deflection of the shaft in the bearings and diminishes the magnitudeof the damping forces provided by the bearings. Thus, one may conclude that the effect of damping provided by thebearings is maximized when the shaft stiffness is large relative to the bearing stiffness.

These general concepts of vibrations will be demonstrated in considerably more detail in the sections that follow.

1.6 References

[1] Pettinato, B.C., Kocur, J.A. and Swanson, E.E. (2011), “Evolution and Trend of API 617 CompressorRotordynamic Criteria”, Turbomachinery Society of Japan: Turbomachinery, 39, pp 292-303.

Keq2

2Kshaft

------------ 1

Kbrg

--------- +

--------------------------------------=

Kshaft48EI

L3------------=

ID4

64---------=



Figure 1-7—Simplified Model of a Beam-type Rotating Machine

Figure 1-8—Simplified Model of a Beam-type Rotating Machine with Damping

Figure 1-9—Spring-mass-damper Model of Beam Type Rotating Machine

kbrg = 21,900 N/mm(125,000 lbf/in.)

kbrg = 21,900 N/mm

kshaft

(125,000 lbf/in.)Disk wt = 2220 N

(500 lbf)

cbrgcbrg

kbrg

wdisk = 2220 N (500 lbf)kbrg = 21,900 N/mm (125,000 lbf/in.)cbrg = 4.4 N-s/mm (25 lbf-s/in.)

kshaft

wdisk

Dynamic System Schematic

1/2 kshaft1/2 kshaft

kbrg kbrg

cbrg cbrg

mdisk



Figure 1-10—Synchronous Response of Beam Type Machine for Various Shaft Stiffness Values

0 1000

250

200

150

100

50

0

10

8

6

4

2

02000

Speed (rpm)

3000 4000 5000

Vbr

aton

am

ptu

de (μ

m p

-p)

Vbr

aton

am

ptu

de (m

s p-

p)

kshaft

kshaft = 35,000 N/mm (200,000 lbf/in.)

kshaft = 8750 N/mm (50,000 lbf/in.)

Increasing shaft stiffness

Calculated Responses for Several Values of Shaft Stiffness

System Schematic

cbrgcbrg

kbrg

wdisk = 2220 N (500 lbf)kbrg = 21,900 N/mm (125,000 lbf/in.)cbrg = 4.4 N-s/mm (25 lbf-s/in.)

kshaftwdisk


2-1

SECTION 2—LATERAL ROTORDYNAMICS

2.1 Introduction

The mechanical reliability of rotating equipment depends heavily upon decisions made by both the purchaser andvendor prior to equipment manufacture. Units that are designed using the appropriate application of sophisticatedcomputer-aided engineering methods will be less problematic than units designed without the benefit of suchanalysis. Even with performance and mechanical acceptance tests prior to delivery and installation, the discovery ofdesign-related problems during these tests may compromise the planned cost of the unit and/or its delivery schedule.

For this reason, rotordynamics analysis tools have been developed and are continuing to evolve. In the late 19thcentury, there was a widespread belief that machines could not be operated above the first critical speed and designsfocused on machines that operated below that critical. Once it was proven that operation above the first critical waspossible, it was still difficult to execute the designs due to the complexity of the analysis and the limits ofcomputational power. In the mid-20th century, critical breakthroughs were made in the analytical tools. Once this wascoupled with the explosion of computational power, analysis methodology began to experience significant growth.Today, very comprehensive analyses may be performed on computers and analysts are continuing to develop moreaccurate and complex modeling techniques. In addition, enhanced measurement capabilities are providing theopportunity to improve this modeling work.

Today, these analysis tools and procedures have become a standard fundamental design tool for a class ofturbomachinery that includes centrifugal and axial compressors, centrifugal pumps, steam turbines, gas turbines,electric motors, expander turbines, and gears. Application of this technology requires communication between thepurchaser and manufacturer to be very effective. This document has been prepared to facilitate this communicationby providing the means to achieve a common understanding or platform upon which to hold meaningful discussion.This document identifies the analysis requirements as well as providing an aid to interpret and understand results ofthat analysis.

The general class of machines to which this document applies are primarily custom designed, i.e. while they belong toa machine class, the specific design features like bearing span, rotor weight, operating speed, etc. fall within designranges selected for the specific application. This is different from “standard design” turbomachinery, e.g. aircraft gasturbines which are complex but which have many identical machines built. Most of the principles identified in thisdocument apply to those machines, but the details have not been directed at that class of machine.

2.2 Rotor-bearing System Modeling

Modeling is the single most important process in performing any engineering analysis of a physical system. If themodel does not accurately simulate the proposed design, the sophisticated analysis and evaluation of the design willdo little good. The steps taken to model rotating equipment are listed in sequence below.

a) Generate a mass-elastic lateral model of the unit’s rotating assembly.

b) Calculate the static bearing reactions (including miscellaneous static load mechanisms such as gear loading,loads resulting from partial arc steam admission, and so forth).

c) Calculate the linearized fluid film bearing coefficients and squeeze film damper behavior. Determine the dynamicsupport stiffness.

d) Calculate the linearized floating ring oil seal coefficients (if present).

e) Calculate all other excitation mechanisms (such as aerodynamic effects and labyrinth seal effects).

The flow chart in Figure 2-1 outlines the generalized procedure for the lateral analysis. These steps will be discussedin greater detail in this section.



Figure 2-1—Overview of Lateral Analysis Procedure

Rotor model

Bearinganalysis

Supportmodel

Undamped critical speed analysis

Unbalance response analysis Unbalance

distributions

Bode plots

Oil Seal analysis

Separationmargins

adequate?

Probe vibration levels in

OSR< AVL

Rub Check acceptable

?

Calculate vibrations at probes and

critical clearances

Scaled deflection

shapes

REDESIGN

Pass

Fail

Fail

Fail

ACCEPTABLEDESIGN

Pass

Pass

AVL = Vibration limit in operating speed range (OSR)



2.3 Rotor Modeling Methods and Considerations

2.3.1 General

This section will describe some of the methods that have been successfully employed over a period of many years tomodel the important elements of a rotor assembly. The reader is cautioned that tie-bolt rotors are potentially subject togreater modeling complications than solid shaft rotor designs. For example, the rotor bending stiffness characteristicsmay be related to the tie-bolt stretch. The nonlinear axial face friction forces between rotor segments may becomesignificant if the segments move relative to each other during rotor operation. Finally, thin shell rotors may simply notbe adequately represented by direct application of cylindrical beam elements. In such cases, either a sophisticatedfinite element analysis of the rotating element or a modal test benchmark may be necessary to build or verify accurateprediction of results.

An accurate model of a rotor system has adequate representation of the rotor’s mass and elastic (mass-inertia-stiffness) properties. For the purpose of performing a basic rotordynamics design audit, two simple building blocks arejoined together to form a complete model of the rotating assembly. These elements are the shaft lumped mass-inertiaelements and the disk lumped mass-inertia elements. Shaft elements contribute mass, inertia and stiffness to theglobal model, whereas disk elements contribute mass and inertia only. More complicated element types can be usedat the cost of introducing complexity to the model. In general, however, most turbomachinery can be adequatelymodeled using the lumped mass-inertia shaft and disk elements presented in this tutorial.

Once the type of elements to be used in the analysis has been established, it simply remains for the engineer topartition or discretize the subject rotor’s geometry using a sufficient number of the selected elements. Schematics of arotor and its associated lumped parameter model are displayed in Figure 2-2. Some general constraints must beplaced on the use of the lumped mass-inertia shaft elements, however, to ensure that accurate rotor models emergefrom the process. Clearly, if too few elements are used, the resulting model may not possess sufficient resolution toaccurately capture some of the detailed mass-elastic properties of the rotating assembly. If a large number ofelements are used to model the rotor, then numerical problems may result. A secondary benefit of minimizing thenumber of elements used to produce a rotor model is a reduction of the amount of time needed by the engineer togenerate data files and for the computer to perform calculations.

2.3.2 Division of Rotor into Discrete Sections

The modeling process starts with the analyst’s dividing the rotor into a series of stations connected by shaft elementsthat begin and end at step changes in the outside diameter (OD) or inside diameter (ID). The stations are also chosento coincide with the axial locations of the concentrated mass or shrunk on members, such as impellers, turbine disks,thrust collars, balance pistons, spacer sleeves, couplings, dry gas seals, etc. Additionally, stations are located at thejournal bearing and seal centerlines, radial probe locations, and at the ends of the rotor.

The number of stations needed to represent the physical rotor depends on: the number and shape of modes to beexamined [1], and the number to adequately approximate a continuous body with a discretized model. The followingsimple guidelines are proposed for modeling a long uniform shaft segment.

a) The length to diameter ratio of any section should not exceed 1.0 (less than or equal to 0.6 is preferred [2]).

b) The length to diameter ratio of any section should not be much less than 0.10.

The first guideline is proposed to ensure that the model possesses sufficient resolution to permit accurate calculationof the undamped critical speeds. The number of critical speeds calculated should be sufficient that the nextundamped critical speed above maximum continuous speed or trip is identified. The second guideline is proposed toensure that large differences in L/D values are avoided as this practice may generate numerical calculation problems.When a large L/D difference exists, precision errors accumulate during the numerical manipulation of the shaftelement stiffnesses. It should be noted that some transfer matrix based programs are capable of modeling well



Figure 2-2—Schematic of a Lumped Parameter Rotor Model

1

1

1 2 10

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

Length ofsectionNo. 5

Section nos.

Externallumpedinertia forcoupling Bearing

centerlineBearing

centerline

BearinglocationStation No. 4

External lumped inertiainput, Station No. 7Ext. W7 = impeller weight.Ext. Ip7 = impeller polar moment of inertia.Ext. IT7 = impeller transverse moment of inertia.

OD ofSection No. 5

12119876543

L(5)Station nos.

Station no.

BearinglocationStation No. 12

Rotor Model

Parameter

Length

Diameter

Weight

Moment of inertia

Bending stiffness

Damping

SI Units

mm

mm

N

N-mm

N/mm

N-s/mm

US Units

in.

in.

lbf

lbm-in.

lbf/in.

lbf-s/in.

Typical Units for Input

22

M , I , IN TN PN

S1K SN-1K

Mi = ith station lumped mass.ITi = ith station lumped transverse moment of inertia.Ipi = ith station lumped polar moment of inertia.Ksi = ith shaft stiffness section bending stiffness.

M1, IT1, Ip1

Notes:



outside these guidelines without any significant degradation of model accuracy. In general, the guidelines provide asafe modeling procedure for those cases where sufficient program benchmarking has not been performed.

Whenever the analyst is unsure of how to model a given feature in the rotating element, he or she may alwaysproceed by determining the sensitivity of calculated results to various ways of modeling the feature in question. Forexample, if one strictly adheres to the two modeling guidelines proposed above, a short circumferential groovemachined into the shaft cannot be modeled. Such grooves are often found on compressor shafts to locate split ringsat the ends of the aerodynamic assembly and to lock thrust collars onto the shaft. Such design features can beignored when analysis indicates that decreasing the diameter of the entire element encompassing the groove doesnot affect the critical speeds or the associated mode shapes. When a given geometric feature possesses a stronginfluence on calculated results, the designer must examine the possibility that the rotor’s design may befundamentally flawed.

On those occasions when the analyst has difficulty modeling a rotating assembly because the rotor geometry cannotbe readily described using rudimentary shaft elements, then an equivalent model can be formulated from moresophisticated analysis. For example, the bending characteristics of a stub shaft bolted to the second stage impeller onan overhung gas pipeline compressor have been determined using a finite element analysis of the shaft and impellersections. The finite element mesh is displayed in Figure 2-3. Note that the large counter-bored bolt holes dramaticallydecrease that stub shaft’s lateral bending stiffness. Once the static bending analysis of the component isaccomplished, an equivalent lumped parameter beam-type model of the type used in rotordynamics analysis can beformulated that possesses identical bending stiffnesses at the lumped mass and inertia locations.

2.3.3 Addition of External Masses and Inertial Loadings

Components that are shrunk on turbomachinery shafts (impellers, sleeves, thrust collars, and so on) affect thebending stiffness of the rotating element to varying degrees. The amount of shrink fit and contact length determine theamount of contribution to the bending stiffness of the rotor. The model used to predict the unit’s critical speeds mayhave to be refined according to data collected during mechanical testing of the actual machine if the critical speedsdiffer by more than 5 %.

Components shrunk or fitted onto the shaft affect the mass and inertia characteristics of the rotating assembly, andmust be added to the model. This is most often accomplished by adding lumped masses and inertias at the centers ofgravity of the shrunk-on components. It is occasionally necessary, as in the case of motor cores, to generate detailedinertia distributions of the shrunk-on component. Most rotors will include at least several of the following additionalmasses:

a) impellers/disks/blades*,

b) couplings*,

c) balance rings*,

d) check nuts, nose cones,

e) fans*,

f) sleeves,

g) balance pistons*,

h) thrust collars*,

i) dry gas seals*.

NOTE Items marked with (*) are typically represented by mass and inertia.



Figure 2-3—3D Finite Element Model of a Complex Geometry Rotating Component

Load-bearing stub shaft

Cross-section of rotating elementwith complex geometry component

A finite element analysis (FEA) of complex geometryrotating components is used to calculate the effective bendingstiffness of the component. This bending stiffness is thenconverted into an equivalent cylindrical section that can beinput into lateral rotor dynamics analysis software.

3D Finite element model of stub shaft



In the rotor lateral model, a station located at the center of gravity of coupling hub should be specified for the couplingassociated with the distributed coupling weight and moment of inertia. Coupling vendors provide the location of thecenter of gravity of each half of the assembled coupling and the weight distribution on the drawings. When the centerof gravity of coupling hub is beyond the shaft end, model manipulation is needed to provide the same overhungmoment. One method to accomplish this is to include a “virtual” element with very low mass and very high modulus ofelasticity to locate the center of gravity.

Particular machines will have specific masses that must be added, including the following:

a) armature windings in electric motors*,

b) shrunk-on gear meshes*,

c) wet impeller mass and inertia in pumps*,

NOTE Items marked with (*) are typically represented by mass and inertia.

It is imperative that the rotor model properly account for these masses and any additional rotating masses that maybe peculiar to a particular system. Some of these rotor modeling aspects that are specific to different machine typesare discussed in Section 2.8.

2.3.4 Addition of Stiffening Due to Shrink Fits and Irregular Sections

Most rotating assemblies have nonintegral collars, sleeves, impellers, and so forth that are shrunk onto the shaftduring rotor assembly. If the amount and length of the shrink fit and the size of the shrunk-on component aresufficiently large, then the shrunk-on component must be modeled as contributing to the shaft stiffness. The vendormust determine the importance of shrink fits for particular cases. Often, this can be accomplished only by experiencewith units of similar type. Smalley et al. [3] provide an empirical method based on static deflections. A modal test of avertically or horizontally hung rotor will give some indication of the stiffening effect of shrunk-on components, but suchmeasurements will likely exaggerate such effects because the fits will tend to be relieved as a result of centrifugalgrowth at normal operating speeds.

In some cases, a shaft segment consists of a series of short grooves and steps. Such segments are often found onturbine shafts at main labyrinth packing locations. Since the change in the diameter of the shaft segmentencompassing the grooves does not affect the rotor critical speeds and the associated mode shapes, the steppingsegment is usually simplified as one element using the average shaft diameter to evaluate the mass and bendingstiffness of the stepping shaft segment.

Noncircular rotor cross-sections are common in the midspan areas of electric motors and generators. These electricalmachines frequently possess integral or welded-on arms in the midspan area to support the rotor core. Thesestructures add significant stiffening to the rotor midspan. This contribution to the lateral bending stiffness of therotating assembly must be accounted for, as it is incorrect to model the stiffness of motor rotors using the base shaftonly. Older steam turbines of built-up construction may also possess noncircular midspan rotor cross-sections.Additional details on these machinery-specific modeling aspects are given in Section 2.8.

2.3.5 Location of Bearings, Seals, and Radial Probes

It is well understood that bearings and seals can dramatically alter the vibration behavior of a rotating machine. Eachfluid film support bearing and floating ring oil seal is typically represented using a set of eight linearized dynamiccoefficients. The linearized models of the bearings and oil seals are assumed to act at the centerlines of theassociated bearing and sealing lands.



A dry gas seal only adds lumped external mass and inertia to the shaft station located at the center of gravity of thedry gas seal’s rotating assembly. The principle stiffness and damping of gas seals are relatively small, and are notincluded in the rotordynamics model for lateral critical speed and unbalance response analyses.

Other important, tight clearance seal locations within a machine should also have their locations included asadditional rotor model stations. The unbalance response analysis must ensure these locations are not easilysusceptible to rubbing. In the case of abradable, floating ring and compliant seals, the vibration amplitude at theselocations should be calculated, but it may be allowable for the level of the amplitude to exceed the minimum runningclearances in these locations. The linearized models of seals are assumed to act at the centerlines of the associatedsealing lands.

The axial locations of the radial bearing probes’ centerlines should be included as additional stations in the rotormodel, as this is where the vibration of the rotating machine will be measured.

2.3.6 Determination of Material Properties

The material properties required to generate the rotor model are density, Young’s modulus of elasticity, shearmodulus and Poisson’s Ratio.

Manufacturers must determine the importance of temperature on shaft material properties for particular applications.For high-temperature cases, vendors should be aware of the influence of temperature on the elastic modulus of shaftmaterials in the rotor lateral models. Sometimes temperature variation exists along the shaft, resulting in the elasticmodulus changing along the axial length. The shafts of most rotating machinery are made of carbon and low-alloysteels, such as AISI 4340 and ASTM A470. The properties of carbon and low-alloy steels change considerably overthe temperature range from 0 °F to 650 °F. Figure 2-4 shows the trends of the changes in elastic modulus withtemperature for several low-alloy steels, including AISI 4340, which are determined during “static” tensile loading anddynamic loading [4].

Figure 2-4—Elastic Modulus vs. Temperature [4]

225

200

175

150

125

100

750 100 200 300 400

Testing Temperature, °C

Testing Temperature, °F

500 600

30

25

20

15

700 800

200 400 600 800 1000 1200 1400

Eas

tc M

odu

us, G

Pa

Eas

tc M

odu

us, 1

06 ks

Line 2 Cr 1 Mo14

610 (H11)60143404340 dynamicSA 517 F604Range 0 9 % Cr



2.3.7 Rotor Model Example

Results of the complete modeling process are displayed for an eight-stage 12-megawatt (16,000-horsepower) steamturbine rotor. A larger version of this drawing was used to describe shaft geometry. The measured rotor weight wasused to check the results of the modeling process. The resulting tabular description of the model is presented in Table2-1. Note that the translational and rotational inertias shown in this table are formed by the sum of externally appliedinertias (from turbine blades and disks) and shaft inertias calculated for each of the shaft sections. A cross-section ofthe rotor model is displayed in Figure 2-5.

2.3.8 Built-up Rotors

Some rotors are constructed of axially segmented sections, which are stacked and bolted together. Theturboexpander in Figure 2-6 and the turbocompressor in Figure 2-7 are examples of built-up rotors. The segments areradially located by either rabbet or spline fits and are held together axially by through bolts. When properly designed,the joints are very stiff, and can be approximated as being an integral piece of metal.

Rotor modeling of a curvic coupling joint is illustrated in Figure 2-8. The hatched areas represent the elements used tomodel the rotor stiffness, while the unhatched areas are modeled as masses and inertias only. These methods havebeen shown to be sufficiently accurate when the rotor is properly designed, and assembled with appropriatepreloading of the stacked components. While a more precise joint model can be obtained by using finite elementmethods or by verifying through modal analysis testing, such methods are typically unnecessary.

2.3.9 Train Lateral

2.3.9.1 General

Most machinery trains consist of a series of two-bearing rotors connected by spacer-type, flexible couplings as shownin Figure 2-9. Individual units are typically isolated from each other dynamically by the flexible coupling such that theycan be treated as separate units. A lateral rotordynamic analysis of the entire coupled system (train lateral analysis) istherefore rarely required. There are, however, unique circumstances in which the train lateral analysis should beconsidered.

1) The coupling spacer natural frequency is within the region shown in Figure 2-10 as ‘train lateralrecommended’.

2) The coupling is not of the flexible spacer type. These situations would include hard coupled turbomachinery,typically consisting of a single fixed joint, as well as piloted and unpiloted spline couplings.

Figure 2-5—Rotor Model Cross-section of an Eight Stage 12 MW (16,000 HP) Steam Turbine

0

0 500 1000 1500 2000 2500

25 50 75 100Rotor axial length (in.)

Rotor axial length (mm)

1

5

1015 20 25 30 35

40

Bearng 1

Bearng 2



Table 2-1—Computer Model Generated for the Eight-Stage Steam Turbine Rotor

StationNo.

AxialLocation

(in.)Wt.

(lbm)Length

(in.)

ShaftOD(in.)

ShaftID

(in.)I

(in.4)

Polar Inertia, IP

(lbm-in.2)

Transverse Inertia, IT(lbm-in.2)

Ex10-6

(lbf/in.2)

1 0.00 1.762 1.500 3.25 0 5.48 2.321 1.491 28.7

2 1.50 3.523 1.500 3.25 0 5.48 4.652 2.991 28.7

3 3.00 3.523 1.500 3.25 0 5.48 4.652 2.991 28.7

4 4.50 23.786 1.300 12.35 0 1140.08 421.909 214.386 28.7

5 5.80 47.029 1.200 13.69 0 1724.18 1005.320 508.760 28.7

6 7.00 29.793 1.723 5.00 0 30.68 600.712 304.538 28.7

7 8.72 9.579 1.723 5.00 0 30.68 29.941 17.336 28.7

8 10.45 9.579 1.723 5.00 0 30.68 29.941 17.336 28.7

9 12.17 13.195 2.100 6.00 0 63.62 52.789 30.671 28.7

10 14.27 17.970 2.390 6.00 0 63.62 80.869 48.077 28.7

11 16.66 17.007 1.190 7.50 0 155.32 95.364 53.119 28.7

12 17.85 12.185 1.010 6.50 0 87.62 77.377 39.974 28.7

13 18.86 43.105 4.260 9.00 0 322.06 413.466 265.154 28.7

14 23.12 117.719 5.680 10.00 0 490.87 906.536 563.578 28.7

15 28.80 109.175 4.910 10.00 0 490.87 1364.690 901.689 28.7

16 33.71 109.172 4.910 10.00 0 490.87 1364.650 901.648 28.7

17 38.62 515.401 2.500 15.00 0 2485.05 1402.150 811.401 28.7

18 41.12 515.404 2.500 15.00 0 2485.05 2629.720 1316.590 28.7

19 43.62 59.702 3.570 10.00 0 490.87 1215.930 650.785 28.7

20 47.19 138.463 1.250 11.25 0 786.28 9424.440 4816.280 28.7

21 48.44 127.568 2.590 10.00 0 490.87 9288.250 4722.120 28.7

22 51.03 157.523 1.570 11.57 0 879.64 12322.200 6252.180 28.7

23 52.60 158.524 2.680 10.00 0 490.87 12334.700 6260.170 28.7

24 55.28 162.368 1.560 11.56 0 876.60 13086.000 6636.280 28.7

25 56.84 165.258 2.940 10.00 0 490.87 13122.200 6660.060 28.7

26 59.78 170.604 1.570 11.57 0 879.64 14030.000 7115.410 28.7

27 61.35 171.937 3.060 10.00 0 490.87 14046.700 7126.750 28.7

28 64.41 174.432 1.570 11.57 0 879.64 14517.900 7362.870 28.7

29 65.98 177.100 3.300 10.00 0 490.87 14551.300 7386.300 28.7

30 69.28 177.100 1.570 11.57 0 879.64 14551.300 7386.290 28.7

31 70.85 170.763 2.730 10.00 0 490.87 14472.100 7332.250 28.7

32 73.58 197.509 1.880 11.88 0 977.77 17066.500 8649.220 28.7

33 75.46 211.795 4.015 10.00 0 490.87 17245.100 8779.610 28.7

34 79.47 146.058 4.785 10.00 0 490.87 1115.940 677.903 28.7

35 84.26 32.353 2.240 9.00 0 322.06 423.397 220.752 28.7

36 86.50 27.875 1.640 6.50 0 87.62 244.917 132.617 28.7

37 88.14 15.406 1.640 6.50 0 87.62 81.369 44.136 28.7

38 89.78 19.540 2.520 6.50 0 87.62 103.196 59.591 28.7

39 92.30 24.801 2.760 6.50 0 87.62 130.976 79.978 28.7

40 95.06 111.748 2.000 21.62 0 10724.00 314.579 165.750 28.7

41 97.06 77.957 0.000 21.62 0 0.00 4554.850 2292.050 28.7

________ _______

4475.292 97.059

Bearing Reactions: 2190.92 lbm at Station 10

2284.38 lbm at Station 39



Figure 2-6—Turboexpander with Curvic Coupling Fits

Figure 2-7—Turbocompressor with Rabbet and Curvic Coupling Fits

Figure 2-8—Modeling of Curvic Coupling Joints

Brg. Brg.

1st



Note that Figure 2-10 shows an increase in the recommended coupling critical speed ratio (coupling spacer criticaldivided by the train maximum continuous speed) from 2.0 to 3.0 for coupling half weights that approach the journalload. This reflects the increased influence of the coupling on train dynamics as the coupling weight approaches thejournal static reaction. For most petrochemical turbomachinery applications, however, the coupling weight is a smallpercentage of the rotor weight and a ratio 2.0 is recommended. For ratios less than 2.5, a simply supported tubecalculation for the coupling critical speed is not sufficient.

For trains with more than two bodies, a partial train lateral analysis consisting only of the bodies meeting the specifiedcriteria is acceptable, since other units in the train, connected by couplings with adequate spacer natural frequencies,would be sufficiently isolated and have little effect on the rotordynamic characteristics of the remaining train.

Figure 2-9—Train Lateral Model

Figure 2-10—Train Lateral Guideline Diagram (Wjnl = Static Bearing Reaction) [5]

Train lateral unnecessary

Train lateral recommended

0 0.2

W1/2Cplg / Wjnl

Ncr

(spa

cer) /

Nm

cos

0.4 0.6 0.8 1

6

5

4

3

2

1

0



Figure 2-11 displays the first four mode shapes of a three body string representing the first critical speed of each bodyand the second critical of the gas turbine. The first unit is a gas turbine supported by bearings 1, 2, and 3. The secondunit is a compressor supported by bearings 4 and 5. The third unit is a motor supported on bearings 6, 7, and 8. Thegas turbine and compressor are joined by a coupling with spacer natural frequency 1.67 times running speed. Thecompressor and motor are joined by a coupling with spacer natural frequency 10.28 times running speed.

The mode shapes indicate that the coupling between the gas turbine and compressor has significant modaldisplacement. The seventh mode (2680 cpm) is the bending mode of this coupling in combination with the shaft ends.In addition, there is some modal participation between the compressor and gas turbine at the 6th mode (2155 cpm),which must be traversed. By contrast, the motor is well isolated from the compressor, and by the criteria listed, apartial train lateral analysis of the turbine and compressor is sufficient.

Figure 2-11—Train Lateral Modes

1.00

0.50

0.00

0.50

1.000.00

Rel

atve

Am

pltu

de

10.00

Brg

CL1

, Pro

be 1

Brg

CL2

, Pro

be 2

Brg

CL3

, Pro

be 3

Brg

CL4

Brg

CL5

Brg

CL6

Brg

CL7

Brg

CL8

20.00 30.00 40.00 50.00Axial Length, in. *101

60.00 70.00 80.00 90.00 100.00

1.00

0.50

0.00

0.50

1.000.00

Rel

atve

Am

pltu

de

10.00

Brg

CL1

, Pro

be 1

Brg

CL2

, Pro

be 2

Brg

CL3

, Pro

be 3

Brg

CL4

Brg

CL5

Brg

CL6

Brg

CL7

Brg

CL8

20.00 30.00 40.00 50.00Axial Length, in. *101

60.00 70.00 80.00 90.00 100.00

Symbol No. 1Critical speed 1904









The lateral train analysis of a geared train, if performed can be divided into two subsystems:

1) Driver – Coupling – Gear

2) Pinion – Coupling – Driven Machine(s)

The bearing analyses of the gear and pinion are normally at the rated conditions.

Prior to performing a train lateral rotordynamic analysis, each of the individual unit rotordynamic analyses should becompleted. The modeling of the train consists of coupling the unit models together through the coupling(s). Thecoupling half weight properties, as used in the unit model, are no longer relevant and the entire coupling including thehubs, flexible joints, and spacer are modeled instead.

2.3.9.2 Coupling Models

For flexible couplings, the hubs are assumed to be integral with the shaft ends while the flexible joints are modeled aseither angular and radial springs or equivalent beam elements. The bending stiffness is to be supplied by the couplingvendor or calculated [6,7]. The spacer is modeled as a hollow shaft (tube) using the techniques described in Section2.3.2. A model using equivalent beam elements is illustrated in Figure 2-12.

Gear type couplings are modeled the same as flexible couplings, except that the joints are assumed to have nobending moment transfer and can be modeled with either zero or very low bending stiffness.

Solid flange couplings do not correspond to API 671 standards. For these couplings, the coupling hubs are assumedintegral to the shaft end, and the flange portions are modeled as an equivalent beam element or bending spring.These data are either supplied by the coupling vendor, derived by finite element analysis or calculated based oneither analytical or empirical methods [7,8,9,10].

Spline type couplings are typically modeled with separate radial and torsional stiffnesses at the joints [11].

2.3.9.3 Static Bearing Loads

The static bearing loads of the string are handled by superposition. That is, the load at each bearing would be thesame as the uncoupled case for a multi-span system where each span is supported by two bearings. For multi-bearing and rigid coupled systems, the bearing loads should be determined by including static sag and misalignmenteffects.

Figure 2-12—Equivalent Coupling Model



2.3.10 References

[1] Childs, D. W., 1993, Turbomachinery Rotordynamics, John Wiley & Sons Inc., pp. 126–127.

[2] Kirk, R. G., Baheti, S., and Ramesh, K., 1995, “Modeling of Rotor Shafting for Lower Mode Accuracy:Influence of Section L/D,” Proceedings of the 1995 Design Engineering Technical Conferences, Vol. 3, PartB., ASME, pp. 957–965.

[3] Smalley, A. J., Pantermuehl, P.J., Hollingsworth, J. R., and Camatti, M., 2002, “How Interference Fits Stiffenthe Flexible Rotors of Centrifugal Compressors,” Proceedings of the IFToMM Sixth International Conferenceon Rotor Dynamics, Sydney, Australia, Vol 2, pp. 928–935.

[4] American Society for Metals, 1978, Metals Handbook, 9th Edition, Vol. 1, pp. 641.

[5] Kirk, R. G., Mondy, R. E., and Murphy, R. C., 1984, “Theory and Guidelines to Proper Coupling Design forRotor Dynamics Considerations”, Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol 106,No. 1, pp. 129–138.

[6] Rothfuss, N. B., “Design and Application of Flexible Diaphragm Couplings to Industrial-Marine Gas Turbines”,ASME paper 73-GT-75.

[7] Johnson, S. G., Rothfuss, N. B., 1977, “Contoured Flexible Diaphragm Couplings”, International Conferenceon Flexible Couplings, Paper D1.

[8] Bannister, R. H., 1980, “Methods for Modeling Flanged and Curvic Couplings for Dynamic Analysis ofComplex Rotor Constructions”, Journal of Mechanical Design, Vol. 102, pp. 130–139.

[9] Young, Warren C., 1989, Formulas for Stress and Strain, 6th edition, McGraw-Hill Book Company, pp. 434–435.

[10] Tondl, A., 1965, Some Problems of Rotordynamics, Chapman & Hall, London.

[11] Marmol, R. A., Smalley, A. J. and Tecza, J. A., 1980, “Spline Coupling Induced Nonsynchronous RotorVibrations”, Journal of Mechanical Design, Vol. 102, pp. 168–176.

2.4 Support Stiffness Effects

2.4.1 Introduction

The accurate modeling of the effective bearing stiffness and damping coefficients acting upon the rotor is required toaccurately predict the lateral rotordynamics. An important consideration in predicting these effective coefficients is theinfluence of the pedestal support stiffness and damping characteristics. These support characteristics act in serieswith the bearing characteristics and the combined effective characteristics, i.e. bearing stiffness and dampingproperties acting on the rotating shaft, may be quite different from either of these [1-6,16]. The analysis of machinevibration response based on rigid bearing supports usually predicts critical speeds that are substantially higher thanactual values [1-4]. The dynamic support stiffness, combined with bearing stiffness and damping, produces effectivestiffness and damping coefficients that are generally lower than those for the bearing alone, resulting in lower criticalspeeds than would occur if the supports were rigid.

For oil film bearing supported machines, Nicholas and Barrett [3] found that for the four rotors analyzed, neglectingsupport flexibility resulted in predicted first critical errors that range from 14 % to 21 % high and second critical errorsthat range from 40 % to 88 % high. In addition, the dynamic support flexibility produces effective damping coefficientsthat are lower than the bearing damping coefficients alone, resulting in larger amplification factors than would occur ifthe supports were rigid.



Some high-speed machines operate on ball bearings. These machines include micro-turbines (in applications for oiland gas) and gas generators and power turbines, among others. Ball bearings are usually very stiff compared to fluidfilm bearings and can be regarded as rigid connections in most cases. This makes substructure representationimportant in these cases. In addition, a roller bearing supported rotor may rely on structural damping to control rotorvibration. In this case, accounting for support dynamic characteristics is required to accurately predict rotordynamicperformances.

Also, for active magnetic bearing (AMB) supported rotors, the effects of support flexibility might need to be consideredand included in the rotordynamic model. The issues are discussed in more detail in the AMB section of this tutorial.

In addition to the bearing type, a machine’s sensitivity to bearing supports dynamic characteristics is dependent onthe rotor properties, machine construction and excitation mechanisms. For this reason, different requirements are setby API standards in terms of support dynamic effects. For instance, in the API 617 standard for axial and centrifugalcompressors, it is required to account for supports’ dynamic stiffness and damping in the rotordynamic analysis, whenthe ratio between the bearings supports and the bearing stiffness is equal or lower than 3.5. Furthermore, it requiresavoidance of support resonances within machine operating speed range. In the API 546 standard for brushless,synchronous machines, some stricter requirements are imposed for separation margins between frame andfoundation system natural frequencies relative to electrical line frequency, operating speed and its multiples. Theseare given to avoid structural modes’ excitation within electrical machinery.

Since rotating machinery is designed, marketed and sold, for the most part, based on analytical predictions, anaccurate method of easily incorporating the support flexibility effect into rotordynamic analyses is of paramountimportance. Therefore, industrial practices have advanced to include the effects of support flexibility into rotordynamicanalyses. A basic and typical method is to model the supports with stiffness and damping coefficients which areconstant over the entire speed range [3,7]. In most cases, the support stiffness is based on static deflections of thebearing pedestal (experimentally and/or analytically calculated). While this approach can be successfully utilized topredict both the location and amplification of rotor critical speeds [3], it will not show more than a single support orfoundation resonance.

Detailed models of support structures have been incorporated into rotordynamic analyses in an effort to predict thesupport-rotor resonance interactions. The usual approach has been to use a modal model from a finite elementanalysis of the structure. Li and Gunter [8] use a component mode synthesis technique whereas Queitzsch [9] usesanalytical frequency response functions (FRF) to represent the supporting structure. These methods have beenproven successful, but they may be time-consuming.

The method proposed by Nicholas, Whalen and Franklin [1] utilizes experimental FRF data to represent the bearingsupport structure. The experimental data are determined from modal analysis techniques where the response of thestructure to a known force is measured. The resulting FRF data are plotted as a function of frequency. If the units ofthe measured FRF function are displacement divided by the force, the resulting data are called “dynamic compliance”[10].

The application of experimental FRF data to rotordynamic analyses has been discussed previously [1,2,4,5,11,16,17].One of the biggest advantages of this method is that the support mass is included implicitly in the FRF data along withthe support stiffness [1]. The FRF data can easily be incorporated into the rotordynamic support model used inReferences [1,2,3,5,7,15,16,17], either as a constant dynamic stiffness over a narrow speed range or as a frequencydependent dynamic stiffness over the entire speed range [1].

The modal analysis technique used in determining the dynamic compliance data is detailed herein and in Reference[1,15]. The data can then be employed in a forced response rotordynamic analysis, using various levels of flexiblesupport model sophistication. Example results are compared to actual test stand speed amplitude plots from a steamturbine running on the test stand with a known midspan unbalance.



2.4.2 Representation of Support Dynamic Characteristics

A typical outline drawing of a steam turbine case is shown in Figure 2-13. The steam end bearing is housed in abearing case that is supported by a flex plate to allow for axial thermal expansion. The exhaust end bearing case issupported within the exhaust casing which sits on two sets of thick horizontal plates with gussets for added stiffness.These plates along with the flex plate are attached to the baseplate.

Figure 2-13—Steam Turbine Support Schematic

SteamInlet

Ste

amIn

et

Side view

Steamendbearingcase

Steamendbearingcase

Flexplate

Flexplate

View from steam end

Exh

aust

Exhaustendbearingcase



A model for this complex support is illustrated in Figure 2-14. The first level of flexibility is the bearing fluid film whichis represented by eight principal (xx,yy) and cross-coupled (xy,yx) stiffness and damping coefficients. For tilting padbearings, the second level of flexibility is the pad [6] and the pad pivot [12]. This effect may be accounted for in thetilting pad bearing analysis [3,6,11,12]. The next level of flexibility is everything past the pad pivot. This may includethe bearing case, the supporting plates, the baseplate, the columns and the foundation. In some instances, in additionto the direct stiffness and damping components, the support can be characterized by a cross-coupling effect amongorthogonal axes on the same support, or even crosstalk between different supports through machine or supportstructure.

Each component of the dynamic system (rotor, bearings, supports and foundation) is characterized by its owndynamics, thereby influencing, to some extent, the dynamics of the overall system. Thus, to determine the dynamicperformances of the entire system, the dynamics characteristics of each single component needs to be considered.Usually, this results in combining the dynamic properties of each component into a single equivalent dynamic systemfor rotordynamic analyses.

Several methods exist to combine the effects of supports dynamics into rotordynamic analyses, each of themresulting in a different level of complexity and accuracy on vibration performance prediction. The modelingapproaches can be distinguished based on the level of dynamic coupling within and between pedestals (direct termsonly, with cross-coupling, with crosstalk between pedestals) as well as by the approach used to analytically combinethe rotor/bearing and support system dynamics. Examples of system integration approaches include the use of fullsupport matrices (K, C, M), state space representation, modal methods, transfer functions, and single degree offreedom (SDOF) approximations.

Figure 2-14—Journal Bearing Fluid Film and Flexible Support Model

Csyx

Csxy

Cbyx

Cbxy

Ksyx

Ksxy

Kbyx

Kbxy

Kbxx

Cbxx

Y

X

Support

Ksxx

Csxx

Kbyy Cbyy

Ksyy Csyy

ms

Shaftm



Application of each method is dependent on the origin of the support stiffness dynamic properties (i.e. numerical orexperimental) and on the level of accuracy required by the analysis and the machine’s sensitivity to the supports’dynamics.

2.4.3 Dynamic Support Stiffness Characterization

In general, the support flexibility is dynamic. That is, the support stiffness and damping properties are a function of theapplied excitation frequency, .

Predicting the support stiffness and damping characteristics is not always straightforward. Although it is possible tomake reasonable assumptions about the dynamic characteristics for some support configurations, most situationsrequire the characteristics to be predicted from finite element analyses of the support structure or to be determinedfrom test data. Further complicating the situation is the fact that the support characteristics may vary with the vibrationfrequency of the rotor.

When experimentally determining the support FRF, several classical and advanced techniques are available andimplemented in commercial instrumentation and identification software. The identification can be conducted in difficultenvironmental conditions or even in operation [10].

The block diagram of the test system used by Nicholas et al. [1] to determine the stiffness and damping properties ofan actual bearing support is illustrated in Figure 2-15. An impact hammer is used to excite the bearing case at thebearing centerline. An internal load cell registers the force imparted on the bearing case by the hammer. Mounted onthe case at the bearing centerline is an accelerometer that senses the case motion that results from the impact force.The force and acceleration signals are used as input to a frequency or spectral analyzer. The acceleration is doubleintegrated and the resulting displacement is divided by the force from the impact hammer. Using these signals over aspecified frequency range, the dynamic compliance FRF, (), is calculated and is a complex quantity, containingboth amplitude and phase information. The inverse of the dynamic compliance is the dynamic stiffness, Kdyn():

Kdyn() = (2-1)

Equivalent support stiffness and damping can be found from:

ks() = Re (Kdyn()) (2-2)

cs() = (2-3)

These calculations can be used to determine the support characteristics at each vibration frequency being analyzedand combined with the bearing coefficients at each frequency in order to perform rotordynamic calculations.

By measuring the pedestal response orthogonal to the applied excitation force, but on the same pedestal, the cross-coupled compliance is obtained experimentally. Conversely, when the excitation force and the measured vibrationresponse are located at different pedestals, crosstalk FRFs can be obtained.

An example of a compliance FRF plot is shown in Figure 2-16 for a steam turbine case. The exhaust end verticalcompliance results from a vertically mounted accelerometer sensing vertical acceleration from a vertical excitation(principal compliance). Two different excitation sources are shown: an impact hammer and an electromagnetic exciteror shaker. Note that for frequencies below 200 Hz (12,000 cpm), both excitation sources give very nearly identicalresults. The impact hammer offers the advantage of being significantly quicker to set up and conduct the actualdynamic compliance testing.

The compliance plot in Figure 2-16 plots the magnitude of the complex compliance FRF.

1 ------------

Im Kdyn

------------------------------



Figure 2-15—Dynamic Stiffness Analysis Diagram

Figure 2-16—Exhaust End Dynamic Compliance Plots

F

X

Impacthammer

AccelerometerLoad

cell

X/F = Dynamic compliance

F/X = Dynamic Stiffness

Frequencyanalyzer

Signalconditioner

Bearingcase

X

IE-6

IE-7

IE-8

IE-9

0 100 200Frequency, Hz

300

Shaker

Com

pan

ce,

n./b

Impact

Exhaust end vertical



When the pedestal dynamics is represented by a SDOF spring-mass-damper system, as discussed in [1,10], themagnitude of the dynamic compliance is:

(2-4)

where

F is the applied force in N (lbf);

X is the resulting displacement in meters (in.).

Inverting, the magnitude of the dynamic stiffness is obtained:

(2-5)

Thus, for small support damping, the dynamic stiffness’ magnitude becomes:

(2-6)

Note that since |KD()| is an absolute value, the magnitude can never be negative.

2.4.4 SDOF Representation of Support Dynamic Effects

A simplified support model for a rotor supported on tilting pad journal bearings is shown in Figure 2-17. The singlesupport mass with two degrees of freedom model illustrated in Figure 2-14 is reduced to two single degrees offreedom (SDOF) support spring-mass-damper systems in the horizontal X and vertical Y directions. Unlike the modelin Figure 2-14, the SDOF models discussed from here on assume that the support has no cross-coupling. The X andY direction equations are uncoupled since the cross-coupled terms are zero, or near zero, for tilting pad bearings. Forillustrative purposes, only the Y direction is considered in Figure 2-17, but an identical system also exists in the Xdirection.

Figure 2-17—Single Degree of Freedom Flexible Support Model

XF----- 1

Ks ms2–

2Cs 2+

----------------------------------------------------------= =

KD FX----- Ks ms

2– 2

Cs 2+= =

KD Ks ms2– =

Ks

Kb

Cs

Cby1

ms

m

f

yShaft

Tilt padbearing

Support

EquivalentSupport

KeqCeq

m

f

y

Shaft



The Y displacement shown in Figure 2-17 is the absolute rotor response; Y1 is the support or pedestal response andY–Y1 is the relative rotor response. Since most vibration probes are mounted on the bearing case to monitor shaftmotion, it is the relative response that is of primary importance for correlation purposes.

From Figure 2-17, the bearing stiffness and damping are combined with the support mass, stiffness and damping toyield an equivalent support model. In this model, the bearing stiffness and damping, Kb and Cb, are functions of theshaft rotational speed, . The equivalent support properties are also speed dependent while the support stiffness anddamping, Ks and Cs, may be constant or speed dependent. The combination of pedestal and bearing properties for tiltpad bearing with no cross-coupling terms, both on bearing and support, yield the equivalent stiffness and dampingproperties in horizontal (x) direction [1,3]:

(2-7)

(2-8)

and similarly for the vertical (y) direction:

(2-9)

(2-10)

In equations 2-7 through 2-10, and . Note that when using experimental FRFdata, it is not necessary to calculate the support mass ( ) explicitly, since is the real part of this function.

Equations and modeling techniques outlined by Barrett et al. [4] or by Vázquez et al. [5] can easily consider supportcross-coupling in the horizontal and vertical directions, as well as crosstalk from one support to another.

Examples of using the SDOF model and equations 2-7 through 2-10 are summarized in the tables below for twosteam turbines operating on tilting pad journal bearings [3]. Table 2-2 is for a 13.1 kN steam turbine running at 4291rpm. Table 2-3 is for a 22.5 kN steam turbine running at 5833 rpm. Since the bearing case flexibility is modeled, thesupport mass is assumed to be the approximate bearing case mass of 227 kg for both turbines. A small amount ofsupport damping is assumed.

From Table 2-2, the equivalent stiffness values are 8 % to 36 % lower while the equivalent damping values are62 % to 74 % lower compared to the original bearing characteristics. From Table 2-3, the equivalent stiffnessvalues are 2 % to 33 % higher while the equivalent damping values are 71 % to 77 % lower compared to theoriginal bearing characteristics. Typically, support flexibility has a much more predominate effect on dampingcompared to stiffness. Table 2-3 actually shows an increase in equivalent stiffness while the equivalent dampingdecreases by more than 70 %.

Keqxx

K̂sxxKbxx

K̂sxxKbxx

+ 2 KbxxC2

sxxK̂sxx

C2bxx

+ +

K̂sxxKbxx

+ 2 2 Csxx

Cbxx+ 2+

----------------------------------------------------------------------------------------------------------------------=

Ceqxx

K2bxx

CsxxK̂

2sxxCbxx

2CsxxCbxx

CsxxCbxx

+ + +

K̂sxxKbxx

+ 2 2 Csxx

Cbxx+ 2+

-----------------------------------------------------------------------------------------------------------------=

Keqyy

K̂syyKbyy

K̂syyKbyy

+ 2 KbyyC2

syyK̂syy

C2byy

+ +

K̂syyKbyy

+ 2 2 Csyy

Cbyy+ 2+

----------------------------------------------------------------------------------------------------------------------=

Ceqyy

K2byy

CsyyK̂2

syyCbyy

2CsyyCbyy

CsyyCbyy

+ + +

K̂syyKbyy

+ 2 2 Csyy

Cbyy+ 2+

-----------------------------------------------------------------------------------------------------------------=

K̂sxxKsxx

msx2–= K̂syy

Ksyymsy

2–=

msx,msy K̂sxx,yy



2.4.5 SDOF Representation of Support Dynamic Effects Including Bearing Cross-Coupling

When stiffness and damping cross-coupling exists in the bearing oil film, e.g. when a fixed geometry bearing ispresent, the equivalent stiffness and damping equations become much more difficult since the X and Y directionequations of motion are coupled by the cross terms. Details of equivalent stiffness and damping equations includingbearings cross-coupling terms are discussed in [15].

Defining the following additional terms:

Kbxy, Kbyx is the bearing stiffness cross-coupling;

Cbxy, Cbyx is the bearing damping cross-coupling;

A = Kbxx + iwCbxx

B = Kbxy + iwCbxy

D = Kbyx + iwCbyx

E = Kbyy + iwCbyy

G = Ksxx – msx2 + iCsxx

H = Ksyy – msy2 + iCsyy

R = (A + G) (E + H) – B D

Table 2-2—Bearing, Support and Equivalent Characteristics, 13.1 kN Steam Turbine

Parameter Bearing Support Support/Bearing Equivalent

Kxx (kN/m) 124,861 350,240 2.81 114,879

Kyy (kN/m) 225,905 350,240 1.55 145,700

Cxx (kN-s/m) 327 56.4 0.17 125

Cyy (kN-s/m) 434 56.4 0.13 112

Table 2-3—Bearing, Support and Equivalent Characteristics, 22.5 kN Steam Turbine

Parameter Bearing Support Support/Bearing Equivalent

Kxx (kN/m) 158,308 525,360 3.32 210,144

Kyy (kN/m) 224,154 525,360 2.34 227,656

Cxx (kN-s/m) 599 69.0 0.12 172

Cyy (kN-s/m) 687 69.0 0.10 161



Assuming the support has no cross-coupling, the equivalent stiffness and damping characteristics of the combinedbearing support system become (based on work by Lloyd Barrett):

(2-11)

(2-12)

Unfortunately, it would be very cumbersome to separate the real and imaginary parts from equations 2-11 and 2-12 and solve for the equivalent properties by hand. However, they can easily be separated and solved for using a computer code or a commercial math program.

For zero bearing cross-coupling, B = D = 0, and equations 2-11 and 2-12 reduce to:

(2-13)

(2-14)

Now, the real and imaginary parts of equations 2-13 and 2-14 can be easily separated, resulting in equations 2-7through 2-10.

2.4.6 Forced Response Correlation

A practical case where SDOF support model representation was successfully applied to predict the forced responseof a real machine is reported by Nicholas et al. [1]. The machine investigated consisted of a nine stage 1097 kg (2418lbm) steam turbine rotor operating on 5-pad tilting pad bearings with 127 and 102 mm (5.0 and 4.0 in.) diameterjournals on the exhaust and steam ends, respectively. The rotor tested was run up to a trip speed of 6150 rpm with16,560 g-mm (23 oz-in.) of unbalance placed at the center wheel rim.

The resulting speed-amplitude plot is shown in Figure 2-18 for the steam end probes (see Reference [1] for theexhaust end probe plots). The probes, which are mounted on the bearing case, are clocked 45° from top-dead-centerand are referred to as “right probe” and “left probe.” From Figure 2-18, both probes exhibit split or dual first criticalspeed peaks. Split critical speeds often indicate strong support interaction with the rotor-bearing system. Figure 2-19shows a phase-amplitude plot for the exhaust end right probe. The inner loop at 2872 rpm also indicates strongsupport interaction.

iCeqXXKeqXX

G 1G E H+

R----------------------–

=+

iCeqXYKeqXY

BGHR

------------=+

iCeqYXKeqYX

DGHR

-------------=+

iCeqYYKeqYY

H 1H A G+

R-----------------------–

=+

iCeqXXKeqXX

A GA G+-------------=+

iCeqYYKeqYY

E HE H+-------------=+



Figure 2-18—Steam End Test Stand Response

Figure 2-19—Exhaust End Test Stand Response

4

3

2

1

00

Ms,

PK

-PK

1700

2700

3200

1 2 3 4Speed, RPM x 1000

5 6

Right

7

4

3

2

1

00

Ms,

PK

-PK

1750

2500

3000

1 2 3 4 5 6

Left

7

3150

2712

27612807

28242838

2872

30903141

3175

3193322632573289

3372357136393687

37193751

40814214

42654729

59975904

2695270

< < <

ROTN

0

90

Polar Plot—Exhaust End Right ProbeFull Scale Amp = 2 Mils, PK PK Amp per Div = 0.1 Mils, PK PK

180

26802664

263126162538

25062440236322502154

17061538

547



In an effort to accurately predict the actual turbine response plot of Figure 2-18, inclusion of the dynamic supportflexibility was required. The simplest flexible support model that can be employed is to use the identical spring-mass-damper support system over the entire speed range, for both bearing cases and for both the horizontal and verticaldirections [1,2,3]. Values for the spring-mass-damper system can be calculated from the compliance FRF plots. A plotfor the exhaust end bearing case, horizontal direction, is shown in Figure 2-20. From Equation 2-6, with small supportdamping, the dynamic stiffness contains not only the support stiffness, Ks, but also the support mass, ms. However, Ksand ms need not be determined explicitly, as the value for Kd may be used directly in the equivalent support equations[1].

The dynamic stiffness, Kd, for the exhaust end horizontal location is picked off of Figure 2-20 at 3000 cpm (near thecritical speed in question). The dynamic stiffness for the steam end horizontal, steam end vertical and exhaust endvertical locations may be determined in the same manner. Thus, the constant stiffness model is employed by usingthe actual Kd values from each of the four dynamic compliance plots. While it is not necessary, for simplicity, a single,average value for the four locations may be used. The average support stiffness value is 1.5x106 lbf/in. (262,680 kN/m). The average bearing stiffness is 135,200 kN/m, resulting in an average support-to-bearing stiffness ratio of 1.94.

This type of constant stiffness model has been successful in accurately predicting the location and amplification of thefirst and second critical speeds [2,3]. Different models should be used for each critical in question as the dynamiccompliance can be significantly different in the vicinity of the first critical speed compared to the second or the thirdcritical speeds. Thus, separate forced response runs should be made with the different SDOF support models tolocate each critical speed.

In an attempt to predict the split critical peak frequencies of Figure 2-18, a more sophisticated model is devised wheremultiple SDOF spring-mass-damper systems are used to represent the supports over the entire speed range. Themodel approximations are illustrated in Figure 2-21 on the steam end horizontal compliance FRF curve. As indicatedin the figure, the dynamic compliance curve is approximated as a series of straight lines. The dynamic stiffness alongwith the frequency is tabulated for all points where the straight lines intersect. These data are then used as flexiblesupport input parameters in the forced response computer program.

Thus, a different SDOF spring-mass-damper support system is used for every speed increment in the responseprogram. Linear interpolation is used for all speeds between the input speeds.

The relative support-to-shaft forced response plot for the steam end probes using this dynamic compliance model isshown in Figure 2-22 (see Reference [1] for the exhaust end probe plots). Figure 2-22 predicts a split critical for theleft probe at 2925 cpm and 3100 cpm, which correlates very closely to the actual values of 3000 cpm and 3150 cpmfrom Figure 2-18. The predicted right probe split critical peaks are at 2700 cpm and 3225 cpm, with a supportresonance peak at 3675 cpm. Actual right probe split critical peaks are at 2700 cpm and 3200 cpm. The predicted3675 cpm support resonance is more evident on the right exhaust end probe, where the actual support resonancespeed is 3600 cpm [1].

For the example presented, modeling each bearing support as two SDOF systems and utilizing impact hammercompliance FRF data produced excellent analytical forced response correlation with actual test stand results. Usingthe constant stiffness model, the location of the first critical speed was accurately predicted. However, the split criticalpeaks were not evident. Using many SDOF spring-mass-damper systems over the operating speed range (dynamiccompliance model) resulted not only in an accurate first critical speed prediction, but the split critical peaks were alsoevident along with one of the support resonances.

An extensive experimental and numerical activity to investigate support flexibility effects on the forced rotor responseand predictability with alternative support modeling methods was performed by Vázquez et al. [5,15–17]. Theexperimental apparatus consisted of a flexible rotor test rig with several flexible support configurations and fluid filmjournal bearings. For the specific test configuration in [5], the support stiffness to bearing stiffness ratio was rangingfrom 0.13 to 0.17 in horizontal direction and from 1.3 to 3.7 in vertical direction. In addition, the crosstalk dynamiccompliance was in the same order of magnitude as the direct dynamic compliance. Vázquez et al. [17] show the



Figure 2-20—Exhaust End Constant Stiffness Support Model

Figure 2-21—Steam End Dynamic Compliance Support Model

IE-5

IE-6

IE-7

IE-820 60 100 140

Frequency, Hz

3000

CP

M

Com

pan

ce,

n./b

1.7E6 lb/in.

Exhaust end horizontal

IE-5

IE-6

IE-7

IE-80 20 60 140100

Frequency, Hz

Com

pan

ce,

n./b

Dynamiccompliancemodel

Steam end horizontal



experimental results and predictions with 15 different support stiffness and damping conditions and a different set offixed geometry bearings.

Figure 2-23a shows the response near the center of shaft for an unbalance distribution selected to excite the firstcritical speed of the rotor. In the case of the SDOF support model, the predicted response shows the first and thirdcritical speeds. Figure 2-23b shows the response near the quarter span for a coupled unbalance distribution selectedto excite the second critical speed of the shaft. A comparison between measured and predicted unbalance responsein terms of resonance frequency and peak response, represented in Figure 2-23, for this specific case indicated thelarge influence of the crosstalk of the supports on the unbalance response and poor correlation with the single masssupport model results. The best correlation between experimental and predicted unbalance response results wasobtained when a complete dynamic model of the supports, including crosstalk and cross-coupling effects, wasintroduced into the simulation using transfer functions.

In conclusion, the accuracy of the SDOF based support model depends on the dynamic characteristics of thesupport, the bearing dynamic properties, and the rotor geometry. The support modeling approach to be used in therotordynamic analysis should be devised on a case by case basis depending on the support-to-bearing stiffness ratio,the cross-coupled vs direct dynamic support stiffness and the required level of accuracy of the analysis. The requiredanalysis accuracy depends on the margin of the predicted vibration amplitude plus the critical speed separationmargin as they compare to acceptance values.

At times, a complete representation of the support dynamics in the rotor model will produce more accurate resultscompared to SDOF models. Conversely, in cases where the support dynamics produce several resonances close toeach other or close to the rotor’s critical speed, irregular forced response curves may result, making it even moredifficult to identify the rotor critical speeds and amplification factors. In this case, one of the SDOF support dynamicstiffness representation methods mentioned above may be a more effective method for interpreting the forcedresponse results.

Figure 2-22—Steam End Analytical Results, Dynamic Compliance Model

2.5

2.0

1.5

Am

ptu

de (m

s)

Rotor Speed (RPM x 10-3)1 2 3 4 5

1.0

0.5

0.0

Probes clocked 45 degrees

Relative ResponseSteam End Probes

DynamicComplianceSupportModel

X (Left) ProbeY (Right) Probe

Unbalance = oz-in.Exh Stm

230

2925

3100

2700

3225

3675



Figure 2-23—Measured and Predicted Unbalance Response for Experimental Test Rig [5]

35

30

25

20

15

10

5

01500 2000 2500 3000 3500 4000

25

20

15

10

5

01500 2000 2500 3000 3500 4000

Mag

ntu

tde

(μm

0-p

)M

agn

tutd

e (μ

m 0

-p)

SDOF model1st critical

SDOF model3rd critical

No

cros

s-ta

k

No cross-talk

Exp.

Fullmodel

SDOF model2nd critical

No cross-talk

Exp.

Full model

Pred. SDOF model

Experimental dataPred. full model

Pred. no cross-talk

Pred. no cross-coupling

Pred. SDOF model

Experimental dataPred. full model

Pred. no cross-talk

Pred. no cross-coupling

Speed (rpm)

Speed (rpm)

a. Response near the center disk to an excitation at mid-span

b. Response near the right disk to a coupled excitation at quarter span

No cross-coupling



2.4.7 References

[1] Nicholas, J. C., Whalen, J. K. and Franklin, S. D., 1986, “Improving Critical Speed Calculations Using FlexibleBearing Support FRF Compliance Data,” Proceedings of the Fifteenth Turbomachinery Symposium, TexasA&M University, pp. 69–78.

[2] Nicholas, J. C., 1989, “Operating Turbomachinery on or Near the Second Critical Speed in Accordance withAPI Specifications,” Proceedings of the Eighteenth Turbomachinery Symposium, Texas A&M University, pp.47–54.

[3] Nicholas, J. C., and Barrett, L. E., 1986, “The Effect of Bearing Support Flexibility on Critical SpeedPrediction,” ASLE Transactions, 29 (3), pp. 329–338.

[4] Barrett, L. E., Nicholas, J. C. and Dhar, D., 1986, “The Dynamic Analysis of Rotor-Bearing Systems UsingExperimental Bearing Support Compliance Data,” Proceedings of the 4th International Modal AnalysisConference, Union College, Schenectady, New York, II, pp. 1531–1535.

[5] Vázquez, J. A., Barrett, L. E. and Flack, R. D., 2001, “Including the Effects of Flexible Bearing Supports inRotating Machinery,” International Journal of Rotating Machinery, 7 (4), pp. 223–236.

[6] Lund, J. W., and Pedersen, L. B., 1986, “The Influence of Pad Flexibility on the Dynamic Coefficients of aTilting Pad Journal Bearing,” ASME Journal of Tribology, 109 (1), pp. 65–70.

[7] Kirk, R. G., and Gunter, E. J., 1972, “The Effect of Support Flexibility and Damping on the SynchronousResponse of a Single Mass Flexible Rotor,” ASME Journal of Engineering for Industry, 94 (1), pp. 221–232.

[8] Li, D. F., and Gunter, E. J., 1982, “Component Mode Synthesis of Large Rotor Systems,” ASME Journal ofEngineering for Power, 104 (2), pp. 552–560.

[9] Queitzsch, G. K., 1985, Forced Response Analysis of Multi-Level Rotor Systems with Substructure, Ph.D.Dissertation, University of Virginia.

[10] Ewins, D. J., 1984, Modal Testing: Theory and Practice, Letchworth, Hertfordshire, England, ResearchStudies Press.

[11] Caruso, W. J., Gans, B. E. and Catlow, W. G., 1982, “Application of Recent Rotor Dynamics Developments toMechanical Drive Turbines,” Proceedings of the Eleventh Turbomachinery Symposium, Texas A&MUniversity, pp.1–17.

[12] Nicholas, J. C., and Wygant, K. D., 1995, “Tilting Pad Journal Bearing Pivot Design for High Load Applications,”Proceedings of the Twenty-Fourth Turbomachinery Symposium, Texas A&M University, pp. 33–47.

[13] Moore, J. J., Vannini, G., Camatti, M., Bianchi, P., 2006, “Rotordynamic Analysis of A Large Industrial Turbo-Compressor Including Finite Element Substructure Modeling,” ASME Paper GT2006-90481.

[14] De Santiago, O., and Abraham, E., 2008, “Rotordynamic Analysis of A Power Turbine Including SupportFlexibility Effects,” ASME Paper GT2008-50900.

[15] Vázquez, J. A., 1999, Using Transfer Functions to Model Flexible Supports and Casings of RotatingMachinery, Ph.D. Dissertation, University of Virginia.

[16] Vázquez, J. A., Barrett, L. E. and Flack, R. D., 2002, “Flexible Bearing Supports, Using Experimental Data,”Transactions of ASME, Journal of Engineering for Gas Turbine and Power, Vol 124, pp. 369–374.

[17] Vázquez, J. A., Barrett, L. E. and Flack, R. D., 2001, “A Rotor on Flexible Bearing Supports, Stability andUnbalance Response,” Transactions of ASME, Journal of Vibrations and Acoustics, Vol 123, pp. 137–144.



2.5 Journal Bearings

2.5.1 Introduction

Fluid film journal bearings play a key role in determining the rotordynamic characteristics of rotating machinery. Theirstiffness and damping properties influence the location and severity of the rotor’s critical speeds and the stabilitycharacteristics of the rotor-bearing system. Because of this key role, bearing modeling is critical for accuraterotordynamic predictions.

Furthermore, accurate bearing modeling is essential in determining the bearing’s oil flow requirements as well as itsoperating temperatures. Designing a bearing to perform dynamically is not good enough, as it is also necessary thatthe bearing operate at reasonable temperatures for prolonged life.

Determining the static or steady state (temperature and oil flow) and dynamic (stiffness and damping) properties of afluid film bearing is difficult. The governing hydrodynamic pressure equation, Reynold’s equation, is a second ordernonlinear differential equation which must be solved using numerical methods such as the finite difference or the finiteelement methods. However, computer codes are readily available that perform these calculations along with, in someadvanced codes, solving the energy equation and the elasticity equation (see 2.5.4.1). Additionally, computationalresults have been compared to experimental results with a fair degree of accuracy (see 2.5.4.9).

In the following sections, many different journal bearing designs are discussed along with some of the key parametersavailable to the bearing designer that have a profound effect on rotordynamics. Several popular fixed geometrysleeve bearing designs are included (see 2.5.3). Tilting pad journal bearings are addressed in detail includingcomprehensive discussions of many of the important tilting pad bearing geometric variables that are available to thebearing designer for improved rotordynamic behavior (see 2.5.4.2).

Figure 2-24—Journal Bearing Hydrodynamic Film

++

W

Y

X

P

Minimum filmMaximum filmtemperature

Bearing

Hydrodynamicpressure profile

Convergingoil wedge

Divergentcavitated film

Maximum pressure

Oj

Ob

Line ofcenters

Fx = O Fy = W

For Equilibrium



2.5.2 Stiffness and Damping Properties

Figure 2-24 and Figure 2-25 display exaggerated views of an operating fixed geometry journal bearing. Note that theoperating position of the journal is located below the bearing centerline. In most of the top half of the bearing, the oilfilm is cavitated because the thickness of the oil film is divergent (increases in the direction of shaft rotation) in thisarea. In most of the bottom half of the bearing, the film thickness converges (decreases in the direction of shaftrotation) so a pressurized oil film wedge forms to support the rotating journal. Note that for the journal to attain asteady state equilibrium position in the bearing, the oil film forces (integrated pressures) must balance in both thehorizontal and vertical directions. For this reason, for fixed geometry journal bearings, the rotating shaft does notdisplace solely in the vertical downward (–Y) direction but also displaces sideways in the positive horizontal (+X)direction. The steady state operating position is most often measured by analyzing the DC gap of local eddy currentdisplacement probes.

The displacement of the journal from the center of the bearing is a nonlinear function of the applied load and journalspeed. Thus, the bearing’s oil film stiffness coefficients (rate change of force with displacement), Kxx, Kxy, Kyx and Kyy,are also nonlinear functions of the journal’s equilibrium position. The bearing’s damping coefficients (rate change offorce with velocity) generated by the bearing are similarly nonlinear functions of the journal’s equilibrium position.Thus, the bearing’s damping coefficients, Cxx, Cxy, Cyx and Cyy, are not solely dependent on the bearing geometry, butalso on the applied bearing load and the rotational speed of the journal. The differences in the bearing dynamiccharacteristics with bearing type and applied load can make a great difference in the lateral rotordynamiccharacteristics of a given machine.

In order to establish the influence that bearings have on the dynamic characteristics of a rotor system, the linearizedbearing dynamic coefficients are calculated based on small perturbations in displacement and velocity from thejournal’s equilibrium position. Many effects must be considered including the effect of heat generation due to fluidshearing in the film, fluid turbulence, variation in oil supply temperatures, and so on.

Figure 2-25—Two Axial Groove Bearing

X

YVertical

Divergingoil film

(cavitation)

Oil feedgroove

Horizontal

Min film

Convergingoil film

Oil out

Oil inlet

Hot/coldoil mix

Max film

xob

oj

Line ofcenters

Shaftrotation

W



Linearized stiffness is examined in Figure 2-26 using a simple spring-mass system. Perturbing the mass by a small+X displacement results in a restoring force in the –X direction. If the spring stiffness, K, is linear, then K = –F/x for allvalues of the displacement, x. If the spring stiffness is nonlinear, then K = –F/x only holds for small values of x since Fwas initially determined via a small +X displacement.

Expanding this concept to a fluid film bearing becomes somewhat more complicated. From Figure 2-27, a small +Xdisplacement or perturbation not only results in a –X restoring force called “direct horizontal stiffness,” Kxx, but it alsocreates a force in the vertical Y direction resulting in a cross-coupled stiffness, Kyx. The net modeling result is a set ofeight (four stiffness and four damping) linearized hydrodynamic bearing coefficients. They are linear as since it isassumed that the stiffness value remains proportional to the displacement regardless of the magnitude of thedisplacement. In truth, these forces are nonlinear. Thus, the resulting dynamic coefficients are said to be good for“small displacements about equilibrium.” In other words, these linearized coefficients are valid for small vibrationalmotion about the steady state operating position. For large journal displacements/vibration above 80 % of the bearingclearance, the bearing’s nonlinear effects would become critical to consider. The effect of nonlinearity below thispercentage is dependent on bearing design and eccentricity.

2.5.3 Fixed Geometry Journal Bearings

2.5.3.1 Introduction

Fixed geometry or sleeve bearings have the undesirable property of creating an excitation force that can drive therotor unstable by creating a subsynchronous vibration. This phenomenon usually occurs at relatively high rotorspeeds and/or light bearing loads, or, more generally, at high Sommerfeld Numbers (see 3.3.2.1). The problem is thatsleeve bearings (i.e., all journal bearings excluding tilting pad bearings) support a resultant load with a displacementthat is not directly in line with the resultant load vector but at some angle with rotation from the load vector. This angle

Figure 2-26—Spring Stiffness

UnstretchedSpring Position

Mass Perturbatedby a X Displacement

F = Restoring spring force (lbs)

X = Displacement or spring stretch (in.)

K = Spring stiffness (lbs/in.)

K = – lbs/in.

K = –

Linear spring(F proportional to X)

A displacement in the +X directionresults in a force in the –Xdirection (a restoring force)K = –

FX

DFDX

dFdX

K

F

X

M

M



can approach 90° for light loads and high speed, resulting in high cross-coupling forces that may drive the rotorunstable. This subject is discussed in detail in 3.3.2.

2.5.3.2 Axial Groove Bearings

Axial groove bearings, sometimes called “plain sleeve bearings,” have a cylindrical bore with typically two to four axialoil feed grooves. Figure 2-25 illustrates a 2-axial groove design. These bearings are very popular in relatively low-speed equipment. For a given bearing load magnitude and orientation, the stability characteristics of axial groovebearings are primarily controlled by the bearing clearance. Tight clearances produce higher instability thresholds buttight clearance bearings present other problems that make them undesirable. For example, as clearance decreases,the bearing’s operating oil temperature increases. Furthermore, Babbitt wear during repeated start-ups will increasethe bearing’s clearance, thereby, degrading stability. In fact, many bearing induced instabilities in the field are causedby bearing clearances that have increased due to wear from oil contamination, repeated starts or slow-rolling withboundary lubrication.

One method of increasing the instability threshold speed of a 2-axial groove bearing is to place a circumferential oilrelief track or groove in the bottom half of the bearing (see the bottom pad in Figure 2-28 and Figure 2-29, Nicholasand Allaire [1]). The relief track removes some of the bearing’s load carrying capacity, thereby, forcing the bearing tooperate at a higher eccentricity ratio. While adding a circumferential groove to the lower half of a 2-axial groovebearing is a popular field fix for a bearing induced instability, care must be taken before this is attempted. This fixshould not be attempted when the original bearing unit load is above 100 psi without performing additional bearinganalysis.

Figure 2-27—Journal Bearing Stiffness and Damping

+

+ +

X

Oj Oj

Ob

Fy

Fx

X

Bearingsurface

Journal atequilibrium

Journalperturbedat + X

Y Kxx, Kxy, Kyx, KyyCxx, Cxy, Cyx, Cyy

Kxx = – FxX

A +X Displacement results in a –X force (restoring force)

Kyx = – FyX

A +X Displacement results in a –Y force (cross-coupled force)

Oyy = – FyY

A +Y Velocity results in a –Y force (restoring force)

xx,yy = Principle coefficientsxy,yx = Cross-coupled coefficients



2.5.3.3 Pressure Dam Bearings

Because of the limitations of grooving the lower half of an axial groove bearing to increase stability, another fixed-bore, anti-whirl bearing design is desirable that is easily manufactured, relatively insensitive to design tolerances, andavailable for quick retrofits in existing 2-axial groove bearing inserts. The pressure dam bearing falls into this category(Figure 2-28 and Figure 2-29). The details of the surface inside the pocket are of secondary importance, since theside lands hold the flow and the pressure. The hydrodynamic load created by the pocket provides the increasedmargin of stability for step bearings compared to plain bearings. Finally, the tolerance on the pocket depth is not ascritical as lobe clearance tolerances for multi-lobe bearings.

Pressure dam or step journal bearings have long been used to improve the stability of turbomachinery asreplacements for plain journal or axial groove bearings. In many cases, these bearings provide a quick andinexpensive fix for machines operating at high speeds near or above the stability threshold. For example, a plaincylindrical axial groove bearing can easily be removed from a machine displaying subsynchronous vibration. Milling astep in the top pad of the proper size and location may be all that is necessary to eliminate the stability problem. Thisis much less expensive and faster than installing tilting pad bearings that may require a change in the bearinghousing.

Most pressure dam bearings have two oil supply grooves located in the horizontal plane as shown in Figure 2-28. Fora downward directed load (negative y-direction) corresponding to a portion of the rotor weight, a pocket is cut in theupper half of the bearing with the end of the pocket (the step or dam) located in the second quadrant for counter-clockwise shaft rotation. The pocket has side lands to hold the pressure and flow as shown in Figure 2-29. Acircumferential relief groove or track is sometimes grooved in the bottom half of the bearing as illustrated in Figure 2-28 and Figure 2-29. Both of these effects (dam and relief track) combine to increase the operating eccentricity of thebearing compared to a plain cylindrical bearing.

At high Sommerfeld numbers (light loads and/or high speeds), the axial groove journal bearing’s eccentricity ratioapproaches zero and the journal runs centered in the bearing. This condition leads to unstable operation. However,the pressure dam bearing’s eccentricity either approaches some minimum value or increases as the Sommerfeld

Figure 2-28—Pressure Dam Bearing [1]

X

Oilsupply

Oilsupply

Pocket

Y

Relieftrack

Step

w

R

R e

c

cd

qs

ob oj

j



number increases due to the effect of the stepped pocket. At high speeds and/or light loads, the step creates aloading that maintains a minimum operating eccentricity. That is, as speed is increased, the bearing eccentricity doesnot approach zero as it would for axial groove bearings. The eccentricity approaches some minimum value or mayeven increase with increasing speed due to the step loading. Thus, a properly designed step bearing would operateat a moderate eccentricity ratio even at high Sommerfeld numbers. This condition helps to stabilize the pressure dambearing in the high Sommerfeld number range.

Many rotors used in the rotating equipment industry operate on vendor installed or retrofitted pressure dam bearings.The most common applications are steam turbines and gear boxes. In high-speed gear boxes, the gear loading mayvary by several orders of magnitude from minimum to full power. This large variance in load is often accompanied bya change in load direction. A pressure dam bearing example in gearbox application may be found in 3.8.5.

While the pressure dam or step journal bearings will not solve all rotordynamic instability problems, it remains anextremely effective, low cost anti-whirl bearing. If near optimum clearance ratios and step locations are used [1],many oil whirl instabilities may be eliminated with pressure dam bearings.

2.5.3.4 Elliptical Bearings

A popular field fix for an unstable sleeve bearing is to make the bore elliptical by placing a 1.0, 2.0, or even a 3.0 milshim at top-dead-center between the bearing insert and housing. This has the effect of crushing the insert andreducing the vertical bearing clearance, thereby, increasing the bearing’s instability threshold speed. Note that if thereis 1.0 mils of looseness between the existing insert and the bearing housing, a 2.0 mil shim is necessary for 1.0 milsof insert of crush.

Figure 2-29—Pressure Dam Bearing—Top and Bottom Pads [1]

Ld = Ld / L

Ld

Side land

LPocket

CLSte

pSide land

Oil Supply Grooves Oil Supply GroovesTop Pad

Lt = Lt / L

Lt LRelief track

CL

Bottom Pad

2 1



Two axial groove bearings are also machined elliptical and supplied in new equipment often with their horizontalclearance equal to twice the vertical clearance. An elliptical or “lemon bore” bearing is illustrated in Figure 2-30. It isessentially a 2-axial groove bearing with a slightly tighter vertical clearance and a more open horizontal clearance.This type of bearing is very popular in Europe and is the standard sleeve bearing design of at least one majorEuropean compressor manufacturer. It is also used in the United States with at least one large domestic motor-generator manufacturer utilizing this design. Making a pressure dam bearing elliptical for improved stabilitycharacteristics is also possible, Mehta et al. [2].

2.5.3.5 Offset Half Bearings

Another method to stabilize a 2-axial groove bearing is to offset the bore, usually at the horizontal split. An offset halfbearing is shown in Figure 2-31. This bearing is also very popular in Europe. Offsetting a pressure dam bearing isalso possible for improved stability, Mehta and Singh [3].

2.5.3.6 Taper Land Bearings

Taper land bearings (Figure 2-32) are also very popular sleeve bearings that are successful in increasing theinstability threshold speed compared to cylindrical sleeve bearings. The taper land bearing has side lands similar to apressure dam and, thus, is a pocket bearing. Care must be taken when using this design in heavy load applicationsas load capacity may be a problem, Nicholas and Kirk [4]. Taper land bearings are very frequently utilized in small,light rotors operating at high speeds such as small turbo-expanders, cryogenic expanders and turbochargers.

2.5.3.7 Multi-Lobe Bearings

Multi-lobe bearings, Flack and Lanes [5,6], do not have side lands. The elliptical bearing is a 2-lobe, multi-lobebearing. The lobes are always preloaded, since a multi-lobe bearing with zero preload is simply an axial groovebearing. Note that for fluid film journal bearings, preload relates the geometric ratio of bore radius to pad radius, not toan actual preload force. Preloading each lobe provides a converging film, thus, producing a hydrodynamic forceregardless of bearing load or journal operating position. Preload is discussed in detail in 2.5.4.5. The lobes can also

Figure 2-30—Elliptical Bearing Figure 2-31—Offset Half Bearing

X

Y

w

X

Y

w

d

d



be offset with the minimum lobe clearance located at some angle with rotation from the center of the lobe as in Figure2-33.

Measurement of the dynamic characteristics of a highly preloaded three-lobe bearing were presented by Taylor et al.[7]. Pettinato et al. [8, 9] performed testing on the same bearing and determined a lack of frequency dependence forthis fixed geometry test bearing, which is theoretically expected. For additional information, Swanson and Kirk [10]provide a good survey on published experimental data for fixed geometry bearings.

2.5.3.8 References

[1] Nicholas, J. C. and Allaire, P. E., 1980, “Analysis of Step Journal Bearings—Finite Length, Stability,” ASLETransactions, 23 (2), pp. 197–207.

[2] Mehta, N. P., Singh, A., Gupta, B. K., 1981, “Stability of Finite Elliptical Pressure Dam Bearings with RotorFlexibility Effects”, ASLE Transactions, 24 (2), pp. 269–275.

[3] Mehta, N. P. and Singh, A., 1986, “Stability Analysis of Finite Offset-Halves Pressure Dam Bearings”, ASMEJournal of Tribology, 108 (2), pp. 270–274.

[4] Nicholas, J. C. and Kirk R. G., 1981, “Theory and Application of Multi-Pocket Bearings for Optimum TurborotorStability,” ASLE Transactions, 24 (2), pp. 269–275.

[5] Flack, R. D. and Lanes, R. F., 1982, “Effects of Three-Lobe Bearing Geometries on Rigid-Rotor Stability”,ASLE Transactions, 25 (2), pp. 221–228.

[6] Lanes, R. F. and Flack, R. D., 1982, “Effects of Three-Lobe Bearing Geometries on Flexible Rotor Stability”,ASLE Transactions, 25 (3), pp. 377–385.

Figure 2-32—Taper Land Bearing with Three Tapered Pockets

Figure 2-33—Multi-Lobe Bearing with Three Preloaded, Offset Lobes

X

Y

w

X

Y

w



[7] Taylor, D.V., Kostrzewsky, G.J., Flack, R.D., and Barrett, L.E., 1995, “Measured Performance of a HighlyLoaded Three-Lobe Journal Bearing-Part II: Dynamic Characteristics,” Tribology Transactions, 38, (3), 707–713.

[8] Pettinato, B., and Flack, R.D., 2001, “Test Results for a Highly Preloaded Three-Lobe Journal Bearing- Effectof Load Orientation on Static and Dynamic Characteristics,” Lubrication Engineering, Journal of the Society ofTribologists and Lubrication Engineers, Vol. 57, No. 9, 23–30.

[9] Pettinato, B., Flack, R.D., and Barrett, L.E., 2001, “Effects of Excitation Frequency and Orbit Magnitude on theDynamic Characteristics of a Highly Preloaded Three-Lobe Journal Bearing,” Tribology Transactions, Vol. 44,No. 4, 575–582.

[10] Swanson, E. E., and Kirk, R. G., 1997, “Survey of Experimental Data for Fixed Geometry HydrodynamicJournal Bearings,” ASME Journal of Tribology, 119, pp.704–710.

2.5.4 Tilting Pad Journal Bearings

2.5.4.1 Historical Perspective

The state-of-the-art in tilting pad journal bearing design and analysis has advanced tremendously starting with thelandmark paper by Jorgen Lund in 1964 [1]. Lund’s paper, Spring and Damping Coefficients for the Tilting-PadJournal Bearing, was the first major published document that contained tilting pad journal bearing stiffness anddamping coefficients. Furthermore, his paper presented the innovative analytical methodology that Lund used todetermine these dynamic characteristics. His analytical procedure is commonly known as “Lund’s pad assemblymethod.”

Prior to 1964, tilting pad journal bearing studies consisted of steady-state analyses which were limited to determiningload capacity and power loss. For many years, the only analysis available was detailed in a 1953 paper by Boyd andRaimondi [2]. A pivoted flat slider on a flat runner was used to determine results which roughly approximated thespecial case of the bearing assembled clearance equal to the tilting pad machined-in clearance (i.e. zero padpreload). With no knowledge of the dynamic characteristics, they concluded that tilting pad bearings offer “No strikingadvantages over plain journal bearings….” A second paper by the same authors in 1962 expanded their analysis inan attempt to include preloaded pads by approximating the actual pad radius of curvature by adding a crown to thepivoted slider [3].

Lund’s pad assembly method calculates the stiffness and damping contribution of each individual pad of a tilting padjournal bearing by considering each pad as a partial arc bearing. As Lund states, “A summation over all pads resultsin the combined spring and damping coefficients for the complete tilting-pad journal bearing” [1]. The inertia of the padwas included in the analysis. Lund utilized the finite difference method for the partial arc Reynolds equation solution.Stiffness and damping design curves were presented for assembled tilting pad journal bearings with 4, 5, 6, and 12centrally pivoted pads.

Orcutt [4] extended the work of Lund with the inclusion of turbulence in a paper published in 1967. Using Lund’s padassembly method, a 4-pad tilting pad journal bearing was analyzed and design plots presented.

Nicholas, Gunter and Allaire [5], employing the finite element method to determine the single pad dynamic data,utilized Lund’s pad assembly method to present stiffness and damping design curves for assembled 5-pad tilting padbearings of varying pad preloads, pivot offsets and pivot load orientations. Nicholas, Gunter and Barrett [6] used thedata in [5] to show the effects of pad preload, pivot offset and pivot loading on the stability of an 11 stage centrifugalcompressor.

Jones and Martin [7] also utilized the pad assembly method along with the finite difference method to produce steadystate and dynamic properties of 5-pad centrally pivoted bearings including the effects of turbulence. Furthermore,while using an isoviscous solution for the partial arc single pad analysis, their model allowed for different temperatures



on each pad. They compared their results to experimental data and to the analytical results from [5] and from Shapiroand Colsher [8].

In Reference [8], the authors discuss the fact that, since the pads of a tilting pad journal bearing tilt, each pad adds adegree of freedom to the journal bearing system. Thus, for a 5-pad bearing, there are 7 degrees of freedom (the x,yjournal motion and the 5 tilt modes of the pads). This results in a 7 x 7 stiffness and a 7 x 7 damping matrix. These 7 x7 matrices may be reduced to standard 2 x 2 matrices by assuming a pad excitation frequency. The design curvespresented in Lund’s 1964 paper [1] are based on synchronous frequency as does the reduced data in References [4-7]. Reference [8] presents two sets of full 7 x 7 stiffness and 7 x 7 damping matrices for a 5-pad bearing with zero and50 % pad preload along with the reduced 2 x 2 data for a synchronous frequency, called “synchronously reducedcoefficients.” The authors also present the equations for reducing full stiffness and damping matrices that include padtilt degrees of freedom to reduced 2 x 2 matrices.

Other authors [9-12] have investigated the frequency dependency of the reduced tilting pad bearing characteristics [9,12]. This subject is addressed in detail in 3.3.3.1.

Also starting in the late 1980s, more advanced tilting pad journal bearing codes were developed which did not treatthe pads as independent partial arc bearings. Instead, the steady state and dynamic operating characteristics weredetermined with a global, fully assembled analysis [13-18].

The first example of this development was presented by Knight and Barrett [13]. They solved Reynolds equationusing the finite element method for a fully assembled tilting pad bearing assuming a parabolic axial pressure profile.Their methodology includes the solution of a first order energy equation with constant axial and approximate radialtemperature profiles. The authors found the journal equilibrium position by iterating on the imposed load, leadingedge boundary conditions, journal temperature, pad rotation angles and the coupled pressure (Reynolds) and energyequations.

Branagan [14] used similar axial pressure and temperature approximations but included pad and pivot elasticityeffects when solving for the dynamic properties of a fully assembled tilting pad journal bearing including pad tilt. Otherauthors followed in the 1990s solving the Reynolds equation with the energy equation and the elasticity equation forthe assembled bearing [15-18]. All three equations are iteratively coupled. For example, Kim, Palazzolo and Gadangi[17] check the convergence on the pad tilt angles, journal eccentricity, shaft temperature, fluid film temperature, padtemperatures, pad deformations, and drain temperature at each iterative step.

A more detailed discussion concerning the historical development of tilting pad journal bearing analytical tools may befound in Nicholas [19].

2.5.4.2 Geometric Properties

One advantage of tilting pad bearings is the many design parameters that are available for variation [6, 20, 21]. Theseincluding pad load orientation, pivot offset, pad preload and pad axial length.

2.5.4.3 Pad Orientation

First, consider pivot or pad load orientation. The load between pivot configuration is shown in Figure 2-34. Directingthe resultant loading between pivots provides more symmetric stiffness and damping coefficients, Nicholas and Kirk[20]. Symmetric support properties provide circular orbits whereas asymmetric supports cause elliptical orbits.Circular orbits are preferable for synchronous response attenuation since, in general, their vibration amplitudes aresmaller going through a critical compared to the major axis of an elliptical orbit. However, the asymmetry associatedwith the load on pivot configuration often provides superior stability characteristics.

If the pivot offset is 50 % (see next paragraph for a more detailed description), then the pivot is located at the middleof the pad’s arc length and no complications arise with respect to the load orientation’s reference frame. However, forhigher pivot offsets, it becomes important to differentiate whether the loading orientation is relative to the pivots or to



the pads. Figure 2-35 shows the difference between a load between pivots and load between pads configurations fora 60 % offset pivoted bearing. It is important to consider such orientation differences when performing bearingcalculations.

2.5.4.4 Pad Pivot Offset

Another tilting pad parameter available to the bearing designer is pad pivot offset. Referring to Figure 2-34, the padpivot offset is defined as

(2-15)

For centrally pivoted pads, = 0.5 (50 % offset). Typical offset pivot values range from = 0.55 to = 0.6 (55 to 60 %offset).

Offset pivots are very popular with thrust bearings as offsetting the pivot increases the operating film thicknessthereby decreasing the operating temperature (i.e. offset pivots increase bearing load capacity). For tilt pad journalbearings, offset pivots also increase load capacity [5,21] which results in lower journal bearing operatingtemperatures. Also, offset pivots increase bearing stiffness, especially Kxx (horizontal), compared to centrally pivotedpads [5]. Finally, offsetting the pad pivot makes the tilting pad journal bearing unidirectional with respect to rotation.

Figure 2-34—Five-pad Tilting Pad Bearing Schematic

p

-----=

Pivot

X

+

+

Y

Loadbetween

pivots

Ob

p

Oj

Wj

R

R +

c' b



2.5.4.5 Pad Preload

An important tilting pad bearing parameter available to the bearing designer is tilting pad bearing preload [6,19-21]. Ageometric parameter, not an actual load, tilting pad bearing preload is defined as:

(2-16)

For zero preload, the pad machined-in clearance (cp) equals the assembled bearing clearance (cb). When the bearingand journal centers coincide, the journal-to-pad radial clearance at any circumferential location along the pad isconstant and equal to the bearing radial clearance (Figure 2-36).

For a preloaded pad, the pad clearance is greater than the bearing clearance (cp > cb). Typical preload values rangefrom 0.0 to 0.6 (0 % to 60 %). When a pad is preloaded (m > 0), a converging film section exists even if the journalruns centered in the bearing (Figure 2-37). Thus, the pad will continue to produce hydrodynamic forces as the bearingload approaches zero. A negative preloaded pad is illustrated in Figure 2-38.

The biggest advantage of reducing the tilting pad preload to near zero is illustrated in Figure 2-39. For this tilt padbearing example, as preload decreases, bearing damping increases while bearing stiffness remains approximatelyconstant. Both of these trends help in increasing the bearing’s effective damping. This trend generally holds for amajority of turbomachinery applications.

Effective damping is a measure of how much bearing damping is effective in shaft vibration suppression. As effectivedamping increases, shaft vibration decreases. Bearing stiffness has a big influence on the amount of effectivedamping that a bearing produces. Normally, as bearing damping increases, bearing stiffness also increases. Thistrend can also be seen from Figure 2-39. As bearing assembled clearance decreases for a constant preload, bearingstiffness and damping both increase. Even though bearing damping increases, the effective damping decreasesbecause the corresponding increase in bearing stiffness makes the bearing damping less effective. The increasedbearing stiffness restricts the shaft from moving in the bearing, thereby reducing the effectiveness of the oil filmproduced damping.

Figure 2-35—Differentiating Load Between Pivots and Load Between Pads

rotationShaft

rotationShaft

Split

line Split

line

Load between pivots Load between pads

m 1cb

cp---- –=



Figure 2-36—Zero Preloaded Pad

Figure 2-37—Preloaded Pad

Journal

Tiltingpad

Rb

Rp

Oj Op

R

cb

Rb = R = cb Rp = Pad radius of curvature

Preload = m = 0Rp = Rb

Journal

Tiltingpad

Rb

Rp

Oj

Ob

Op

R

cb

cp – cb

Rb R + cb Rp R + cp

cb Assembled bearing clearancecp Pad clearance

Preload m 1 – (cb/cp) Typical m 0.0 to 0.6



Since the decreasing pad preload, increasing effective damping trend is a typical characteristic of many rotor-bearingsystems, the temptation to decrease tilting pad preload to near zero to improve machine stability is strong. However,there are several major disadvantages to low preload pads. First, there may be a drastic decrease in horizontalstiffness and damping (Kxx and Cxx) as the pad preload becomes negative. This trend is shown analytically anddiscussed in detail in Nicholas [21] for a centrifugal compressor on 5-pad tilting pad bearings with on pivot loadingwhere the negative preload results from a rather generous machining tolerance range. If zero preload is defined bynominal dimensions, the tolerance range on the journal diameter, pad radius of curvature and assembled bearingclearance can produce a negative preload.

The second problem with light preload is the loss of damping when the top pads become unloaded. Top unloadedpads also flutter since there does not exist a tilt angle at which the pad can seek equilibrium. Fluttering pads maycause rotor vibration and excessive pivot wear [21].

A third problem is the frequency dependent effect. As preload deceases, and as the whirl frequency decreases fromsynchronous, damping decreases and stiffness increases [9,19].

2.5.4.6 Bearing Length-to-Diameter Ratio

Another powerful design parameter available to the tilting pad bearing designer is pad length-to-diameter, L/D, ratio.An example where increasing the pad L/D ratio increases bearing damping but decreases bearing stiffness is shownin Figure 2-40. Again, both changes contribute to the increase in effective damping.

The effect of the L/D parameter on journal bearing stiffness and damping is related to the Sommerfeld number ofinterest and the stiffness and damping trend may be different (stiffness increasing or decreasing with L/D) dependingon it [38]. Generally, for low Sommerfeld number (S < 0.05) bearings, the L/D increase leads to lowering the stiffness.However, for high Sommerfeld (S > 0.05) number bearings, the L/D increase leads to increasing the stiffness.

Of course, it is usually more practical to increase the pad length as opposed to decreasing the journal diameter. Forthis reason, longer pad lengths have become more popular with bearing designers. Whereas older pad designstypically had L/D near 0.5, the range of L/D considered in modern designs is broader ranging up to L/D = 1.0.

Figure 2-38—Negative Preloaded Pad

Journal

Tiltingpad

RbRp

Oj

Op

R

cb

cb – cp

Rb R + cb Rp R + cp

cb Assembled bearing clearancecp Pad clearanceFor negative preload, cp < cb



2.5.4.7 Pivot Design

As long as the pad geometry is defined and the journal bearing operating conditions are specified, the stiffness anddamping coefficients of the oil film can be evaluated with a proper numerical tool. The overall journal bearing stiffnessand damping coefficients may be affected by the pad pivot design because the pivot’s stiffness can be comparablewith the oil film’s stiffness.

Figure 2-39—Stiffness and Damping vs. Preload and Bearing Clearance, 4-Pad Bearing [20]

10.0

1.0

0.10.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Preload, M

17.5

1.75

0.175

C (

b-s/

n. x

10-

4 )

K (

b/n.

x 1

0-6 )

C (N

-s/cm x 10

-4) K

(N/cm

x 10-6)

0.0762 mm (3.0 mils)

0.1016 mm (4.0 mils)

0.1270 mm (5.0 mils)

Tolerance range

Axial compressor #1

Wr = 21,079 N (4739 lbs)

N = 5500 RPM

4-pad between bearings

Kxx = Kyy

Cxx = Cyy

cb



The most common pivot designs are the following:

— line contact, often called “rocker-back”;

— elliptical (spherical), often called “button”;

— ball and socket.

Even if they can be practically realized according to different mechanical arrangements, the basic principle whichdifferentiates them is the type of contact between the pad and the bearing housing surfaces. Depending on thespecific type of contact, the relevant stiffness can be evaluated using finite element analysis or closely approximated

Figure 2-40—Stiffness and Damping vs. Preload and L/D Ratio, 4-Pad Bearing [20]

10.0

1.0

0.10.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Preload, M

17.5

1.75

0.175

C (

b-s/

n. x

10-

4 )

K (

b/n.

x 1

0-6 )

C (N

-s/cm x 10

-4) K

(N/cm

x 10-6)

0.5

0.6

0.7

0.8

L /D

Axial compressor #2

Wr = 72,102 N (16,210 lbs)

N = 3600 RPM

4-pad between bearings

Kxx = Kyy

Cxx = Cyy



by Hertzian theory. References [40,41] provide detailed descriptions of the Hertzian analytical formulas necessary toevaluate the pivot stiffness and how it affects the overall bearing dynamics. Pivot flexibility must be considered foraccurate determination of critical speeds and vibration amplitudes in machines whose bearing oil film stiffness valueshave the same order magnitude as the pivot stiffness. Large machines with heavy rotor weights and highly loadedapplications, such as gearboxes, are typical instances where pivot flexibility should be modeled.

The selection of a pivot design for a certain application is typically governed by the bearing manufacturer’s, or theOEM’s experience. Some general design considerations include:

— axial alignment capability: elliptical pivot and ball and socket types are preferable;

— high radial loads: line contact and ball and socket types are preferable.

Finally, another type of pivot also exists in modern turbomachinery, the flexible pivot. While not as commonly applied,this design’s pad is physically connected to the bearing housing through a flexible web-type pivot element where thepad, pivot, and housing are machined out of single piece of metal using EDM. Such an arrangement offers thegeneral advantages of reduced manufacturing tolerances on the bearing’s clearances and preload, and a reducedrisk for pad fluttering or spragging (leading edge in contact with the shaft) during operation [38].

2.5.4.8 Summary

As with many journal bearing design trends, a design change that results in improved synchronous responseattenuation in the operating speed range at the bearing probes may adversely affect stability performance and raisesynchronous vibration at the midspan at the first critical speed. This occurs because the first aspect is generallyimproved with an increase of stiffness of the bearing, while the second and third aspects are improved with theopposite modification. In the same manner, a journal bearing design change to improve stability and decreasemidspan response may often increase synchronous response vibration at the probe locations. Thus, whencontemplating a bearing redesign, synchronous response and stability must both be considered.

2.5.4.9 Experimental Verification

The earliest comparison of analytical and experimental tilting pad journal bearing dynamic coefficients was made byLund [1]. Lund compares his theoretical data to some experimental data from Hagg and Stankey [22]. His comparisonplot for a load between pivot 4-pad bearing is included here as Figure 2-41. It should be noted that Lund’s coordinatesystem is defined as: y-axis horizontal to the right, x-axis vertically downward with counter-clockwise rotation. Anotherearly comparison may be found in Jones and Martin [7]. They compared their analytical data to the experimental datain Yamauchi and Someya [23]. The comparison plot is illustrated in Figure 2-42. Both of these figures showreasonable correlation.

Since then, many other researchers have conducted tilting pad journal bearing experimental testing. For example,eccentricity measurements are presented in Tripp and Murphy [24] for a 5-pad tilting pad bearing showing, for the firsttime, a slight attitude angle with rotation from bottom dead center.

Other examples include Brockwell et al. [25]. The authors measured the stiffness and damping characteristics for a76 mm diameter (3.0 in.) 5-pad tilting pad bearing and compared his results to theoretical calculations. They concludethat, while “… there is reasonable agreement between theory and experiment …”, the theoretical model“… overestimates the damping characteristics of the bearing” even with the inclusion of pad pivot stiffness.

Dmochowski and Brockwell [26] also considered a 76 mm diameter (3.0 in.), 5-pad tilting pad bearing. Their datainclude an uncertainty analysis based on the work of Kostrzewsky and Flack [27,28]. Dmochowski and Brockwellconclude that the “… uncertainty analysis of the measured coefficients has shown that the error associated with thestiffness coefficients is in the range ±5 % to ±11 %, while that for the damping coefficients is in the range ±5 % to±17 %.”



Wygant et al. [29] present measured dynamic coefficients for a 70 mm diameter (2.75 in.), 5-pad tilting pad bearingwith on-pivot loading. The authors present stiffness and damping coefficients for their test bearing with “rocker-back”and “spherical seated” pad pivots. Their data also include an uncertainty analysis. “Uncertainty levels ranged from8 % to 42 % for Kxx, from 5 % to 45 % for Kyy, from 9 % to 28 % for Cxx from 6 % to 82 % for Cyy …”

Pettinato and De Choudhury [30] tested a 127 mm diameter (5.0 in.) 5-pad tilting pad bearing with between-pivotloading. The authors present stiffness and damping coefficients including an uncertainty analysis for their test bearingwith “key-seat” and “spherical seated” pad pivots. Theoretical data are also included and compared to the test data.

Steady state tilting pad bearing test data are also available [31-34]. Simmons et al. [31] present temperature data fora 200 mm (7.87 in.) 5-pad tilting pad bearing for unit loads up to 600 psi and surface velocities up to 345 ft/s forcentrally pivoted pads. Offset pivoted pads are considered in [32] where temperature comparisons are made showinga significant pad operating temperature reduction for offset pivots.

Brockwell et al. [33] present pad operating temperature data for a 152 mm (6.0 in.), 5-pad tilting pad bearing,comparing offset pivoted to centrally pivoted pads and LOP to LBP for unit loads up to 318 psi and surface velocitiesup to 279 ft/s. The authors concluded that “In terms of bearing operating temperatures, there does seem to be anadvantage in switching from LOP to LBP. This advantage improves at higher shaft speeds.” (Note that LOP is load onpivot and LBP is load between pivots.) They also conclude that “In all test conditions, the offset pivot bearing rancooler than the center pivot bearing. In some cases the difference was as much as 20 °C.”

Figure 2-41—Lund’s Data vs. Experimental [1]

10.08.0

6.0

4.0

2.0

1.00.8

0.6

0.4

0.2

0.1

00

S =

0.1 0.2 0.4 0.6 1.0 2.0 4.0 6.0 10.0

Test, Ref. 4

Theory

Kxx

Kxx

wCxxwCxx

mNDLW

RC

2( )

,C

Kxx

WC

wC

xxW



DeCamillo and Brockwell [34] tested a 6.0-in., 5-pad tilting pad bearing comparing “… the effects of pivot offset, oilflow, load orientation, method of lubrication, and oil discharge configuration.” Unit bearing loads ranged up to 320 psiwith surface velocities up to 393 ft/s. Most of their data were for 288 ft/s, however. Among their conclusions is that“Reducing flow rate to the offset pivot bearing by 50 % reduced power losses 10 % to 20 %.”

Childs [39] shows a summary of the most recent testing experience in the field of tilting pad journal bearings.Kulhanek and Childs [42] present test data for a 4.0” rocker-back tilting pad bearing for both center and offset pivot forspeeds up to 16,000 rpm and unit loads up to 450 psi.

2.5.5 Journal Bearing Retrofits

A powerful design tool available to bearing and rotordynamic designers concerns the stiffness and dampingasymmetry of sleeve bearings such as axial groove, pressure dam, elliptical or multi-lobe bearings. Theseasymmetric properties often result in split first critical speeds [20,35]. That is, since the horizontal stiffness anddamping are much softer than the vertical, a horizontal first critical speed may appear several hundred to severalthousand revolutions per minute lower than the first vertical critical speed.

Tilting pad bearings, however, produce more symmetric bearing properties especially when loaded between pivots. Infact, the stiffness and damping values for a 4-pad tilting pad bearing loaded between pivots are exactly equal [20].

Figure 2-42—Jones and Martin Data vs. Experimental [7]

5 x 107

107

106

1000 2000

StiffnessDamping

Stiffness (driven end)Stiffness (free end)Damping (driven end)Damping (free end)

Experimentalresults fromYamauchi& Someya

Theoretical predictionsby Jones & Martin

3000Speed (rev/min)

4000 5000

5 x 105

105

104

Stf

fnes

s –

K (N

/m)

Dam

png

– B

(Ns/

m)



This symmetry often results in a single, un-split critical that is located approximately midway between the sleevebearings split peaks [20,35].

As an example, consider the design and retrofit application of a 4-pad tilting pad bearing for a relatively heavy (16,210lb), low-speed (3600 rpm) axial compressor [20]. The compressor was originally designed to operate on plain,cylindrical, 3-axial groove sleeve bearings with a unit load of Lu = 179 psi. The actual test stand results for thecompressor with 3-axial groove bearings are shown in Figure 2-43. A peak response is evident at 3750 rpm which isunacceptably close to the 3600 rpm operating speed.

The high unit load on the 3-axial groove bearings produces bearing properties that are extremely asymmetric. Thisasymmetry results in two distinct peaks for the first critical speed. The lower, horizontal peak is nonresponsive inFigure 2-43 and the peak at 3750 rpm is the higher, vertical first critical. This is illustrated in Figure 2-44 where ananalytical response curve is shown for the axial compressor with the original 3-axial groove bearings. The 3-axialgroove predicted response shows two peaks at 2000 and 3500 rpm. Close examination of the mode shapes andcritical speed map indicates that both peaks are first mode criticals due to the asymmetry in the axial groove bearings.

The major advantage of a 4-pad tilting pad bearing is the symmetric stiffness and damping properties that result whenloaded between pivots [20]. With symmetric dynamic characteristics, the split first mode no longer exists. It isreplaced by a single peak located approximately midway between the sleeve bearing split peaks. Figure 2-44 alsoshows the predicted response for the axial compressor with a proposed retrofit: a 4-pad tilting pad bearing design withload between pivots. The 4-pad tilting pad bearing with between pivot loading is symmetric and results in only onepeak at 2800 rpm.

Test results for the compressor operating on 4-pad tilting pad bearings with between pivot loading is shown in Figure2-45. The critical is now located between 2850 rpm and 3000 rpm, 16.7 % to 20.8 % below operating speed.

Figure 2-46 illustrates a similar trend for a smaller axial compressor operating at 5500 rpm, comparing three differenttilting pad bearing designs to the original 3-axial groove bearing with split first critical peaks.

Figure 2-43—Actual Test Stand Response, 3-Axial Groove Bearings

1.0

0.5

0.0

1.0

0.5

0.00 1 2 3 4 5

2.54

0.0

2.54

0.0

Speed, RPM x 10–3

Pea

k-P

eak

Am

ptu

de, M

s

Pea

k-P

eak

Am

ptu

de, c

m x

103

1 PerRevolution

Filtered

Overall

Unfiltered



Figure 2-44—Analytically Predicted Response for Different Bearing Designs on an Axial Compressor

Figure 2-45—Actual Test Stand Response, 4-Pad Tilting Pad Bearings

0.5

0.4

0.3

0.2

0.1

0.0

1.27

1.02

0.76

0.51

0.25

0.001 2 3 4 5

Speed, RPM x 10–3

Am

ptu

de, M

s

Am

ptu

de, c

m x

103Axial compressor #2

Wr = 72,102 N (16,210 lbs)N = 3600 RPM

3-axial groove

4-pad between

0

1

2000 2500 3000 3500

Speed (rpm)

Thrust vertical

Plain horizontal

Plain vertical

Thrust horizontal

Mils



In another example, a two-pole 5000 HP induction motor was designed and built as a rigid shaft machine [35]. That is,it was to operate with the first critical speed located at least 20 % above the synchronous operating speed of 3600rpm. Thus, the first critical speed had to be above 4320 rpm. The motors relevant mass-elastic rotordynamicproperties include a rotor weight of 4430 lb, a bearing span of 69.4 in. and a journal diameter of 5.5 in.

The resulting test stand response plot is illustrated in Figure 2-47 for the induction motor operating on the original 4-pad tilting pad bearings. Clearly, the first critical speed, N1, is located at 3900 rpm with an associated amplificationfactor of A1 = 5.7. With 10 oz-in. of unbalance placed in-phase at each fan inboard of the bearings, the resultingvibration is 2.5 mils peak-to-peak (pk-pk).

The corresponding analytical response plot is shown in Figure 2-48 with 10 oz-in. of unbalance in-phase at each fan.The resulting predicted first critical speed is 3900 rpm with an amplification factor of 6.4 and 2.6 mils of pk-pkvibration. Unfortunately, the initial analysis assumed an unreasonably high support stiffness (see Section 2.4). Thus,the first critical was initially erroneously predicted above 4320 rpm.

Numerous failed attempts were made to raise the first critical speed by stiffening the bearing support. For example,stiffeners were added to the bearing brackets and the motor was moved around the test floor seeking the least flexiblelocation.

After these and other design efforts were exhausted, short of a complete rotor redesign, an analysis was conductedreplacing the original tilting pad bearings with elliptical sleeve bearings. The resulting analytical response plot isshown in Figure 2-49. Again, with 10 oz-in. of unbalance at the fan locations, two distinct first criticals are evident. Thelower or horizontal first critical, Nh, is located at 2350 rpm with Ah = 2.2 and with 1.6 mils of pk-pk vibration. Thehigher, or vertical, first critical, Nv, is at 4550 rpm with Av = 3.8. Both criticals now meet the required separation margin.

The resulting test stand plot for the motor with elliptical bearings and with 10 oz-in. of in-phase unbalance at the fansis shown in Figure 2-50. The vibration level is 1.6 mils pk-pk with Nh = 2740 rpm and Ah = 1.5. Compared to thepredicted results, the vibration magnitude and amplification factor are quite close, but the actual location of the criticalis 390 rpm higher than predicted. The motors were commissioned with the elliptical bearings and have beenoperating free of vibration problems since 1993.

Figure 2-46—Analytically Predicted Response, Various Bearing Designs

0.5

0.4

0.3

0.2

0.1

0.0

1.27

1.02

0.76

0.51

0.25

0.001 2 3 4 5 6 7 8 9

Speed, RPM x 10 3

Am

ptu

de, M

s

Am

ptu

de, c

m x

103

Axial compressor #3Wr = 31,661 N (7118 lbs)N = 5500 RPM

3-axial groove

5-pad between

5-pad on

4-pad between



Figure 2-47—Induction Motor Test Stand Response, Tilting Pad Bearings

Figure 2-48—Induction Motor Analytical Response, Tilting Pad Bearings

90

180

270

360

101.6

76.2

50.8

25.4

0.00 1000 2000 3000 4000

63.5 μm(2.5 mils)

μm

5000

Pha

se, D

eg

4

3

2

1

0

PK

-PK

Am

ptu

de, M

s

Rotor Speed (RPM)

N1 = 3900 RPMA1 = 5.7

4

3

2

1

0

101.6721

OperatingRPM = 3600

10.0 10.0Unbalance = oz-in.

721

76.2

50.8

25.4

0.01000 2000 3000 4000

Rotor Speed, RPM5000 6000

PK

-PK

Am

ptu

de, M

s

g-cm μm

N1 = 3900 RPMA1 = 6.4

66 μm(2.6 mils)



Figure 2-49—Induction Motor Analytical Response, Elliptical Bearings

Figure 2-50—Induction Motor Test Stand Response, Elliptical Bearings

2.0

2.5

1.5

1.0

0.5

0.0

721

OperatingRPM = 3600

10.0 10.0Unbalance = oz-in.

721

76.2

50.8

25.4

0.01000 2000 3000 4000

Rotor Speed, RPM5000 6000

PK

-PK

Am

ptu

de, M

s

g-cm

μm

Nv = 4500 RPMAv = 3.8

Nh = 2350 RPMAh = 2.2

40.6 μm(1.6 mils)

Majoraxis

Minoraxis

360

180

0

38.1

25.4

12.7

0.00 1000 2000 3000 4000

40.6 μm(1.6 mils) μm

5000

Pha

se, D

eg

2.0

1.5

1.0

0.5

0.0

PK

-PK

Am

ptu

de, M

s

Rotor Speed (RPM)

Nh = 2740 RPMAh = 1.5



2.5.6 Viscosity and Viscosity Units

Although kinematic viscosity is more commonly used in fluid mechanics, dynamic or absolute viscosity is the keyviscosity for hydrodynamic lubrication. Probably the least used unit of dynamic or absolute viscosity in defininglubricants is the reyn, named in honor of Sir Osborne Reynolds. The reyn, however, is a key viscosity unit as it is therequired unit for the dynamic pressure equation (Reynolds equation). A reyn is defined as [36]:

(2-17)

Another viscosity unit is the poise, named in honor of the physician, Doctor Poiseuille [36]:

(2-18)

where 1 lbf = 448,000 dynes. Likewise, a centipoise is equal to 10-2 poise:

(2-19)

The relationship between a reyn and a centipoise is:

(2-20)

In the metric system, the dynamic viscosity unit is the pascal-second:

(2-21)

The relationships between a pascal-second to a reyn and to a centipoise are:

(2-22)

(2-23)

The kinematic viscosity is the dynamic viscosity divided by the lubricant density. Convenient to measure, thekinematic viscosity is a common parameter given for lube oil characteristics. The metric unit for kinematic viscosity isthe stoke which is defined as:

(2-24)

Also, a centistoke is equal to 10-2 stokes:

(2-25)

Lubricants can be defined by a viscosity unit called “Saybolt Seconds Universal,” SSU. This unit is simply the time inseconds required for the lubricant to empty out of a cup in a Saybolt viscometer through a capillary opening. For highflow times (high viscosity lubricants), the Saybolt viscosity is proportional to the kinematic viscosity. For low viscosity

rlbf s–

in.2-------------- reyn= =

pdyne s–

cm2--------------------- poise= =

cp p 10 2– centipoise= =

r cp 1.45 7–=

paN s–

m2------------ Pa s–= =

pa r 6.8943=

pa cp 10 3–=

vscm2

s--------- stoke= =

vcs vs 10 2– mm2

s----------- centistoke= = =



lubricants (values below 70 centistokes, 325 SSU), turbulence and other effects influence the efflux time making therelationship nonlinear [37].

An approximate equation relating centistokes to Saybolt viscosity is:

(2-26)

A detailed discussion of this equation and its limitations may be found in Reference [37].

Most lubricant viscosity is now defined by their ISO viscosity grade. ISO viscosity grades (VG) 32 and 46 are the mostcommon oil types being used for the turbomachinery related to this tutorial. Generally, the oil type is specified by theend user who has the global view of the turbomachinery train and performs this choice aiming to reach the bestcommonality among all the involved machines. This means that the oil selection is not part of the rotordynamic designoptimization, but is just an input for the design.

2.5.7 Final Notes for Rotordynamic Analysis

For the purpose of verifying a machine’s rotordynamic design, the journal bearing parameters which need to be takeninto account are all the parameters described in previous paragraphs (load configuration, offset, clearance, preload,L/D ratio, pad arc length, pivot design, etc.). Furthermore, operating conditions such as static load, rotational speedrange as well as oil type, inlet temperature, inlet oil flow and viscosity must also be input for any basic rotordynamicsimulation.

For a robust design, the simulation should include the possible variation of these parameters and, specifically, themanufacturing tolerances which affect the clearance and preload values. Moreover, the inlet oil temperature willchange in a specified range to be considered. Finally, the properly combined effects of tolerances and oil temperaturewill produce the extremes in the journal bearing coefficients; in most of the cases, the minimum clearance, maximumpreload, minimum oil inlet temperature combination will represent the maximum stiffness case, and the maximumclearance, minimum preload, maximum oil inlet temperature will represent the minimum stiffness case.

2.5.8 References

[1] Lund, J. W., 1964, “Spring and Damping Coefficients for the Tilting-Pad Journal Bearing,” ASLE Transactions,7, pp. 342–352.

[2] Boyd, J., and Raimondi, A. A., 1953, “An Analysis of the Pivoted-Pad Journal Bearing,” MechanicalEngineering, 75, No. 5, pp. 380–386.

[3] Boyd, J., and Raimondi, A. A., 1962, “Clearance Considerations in Pivoted Pad Journal Bearings,” ASLETransactions, 5, No. 2, pp. 418–426.

[4] Orcutt, F. K., 1967, “The Steady-State and Dynamic Characteristics of the Tilting-Pad Journal Bearing inLaminar and Turbulent Flow Regimes,” ASME Journal of Lubrication Technology, 89, No. 3, pp. 392–404.

[5] Nicholas, J. C., Gunter, E. J., and Allaire, P. E., 1979, “Stiffness and Damping Coefficients for the Five PadTilting Pad Bearing,” ASLE Transactions, 22, No. 2, pp. 112–124.

[6] Nicholas, J. C., Gunter, E. J., and Barrett, L. E., 1978, “The Influence of Tilting Pad Bearing Characteristics onthe Stability of High Speed Rotor-Bearing Systems,” Topics in Fluid Film Bearing and Rotor Bearing SystemDesign and Optimization, an ASME publication, pp. 55–78.

vcs 0.22 ssu 180ssu

---------–=



[7] Jones, G. J., and Martin, F. A., 1979, “Geometry Effects in Tilting-Pad Journal Bearings,” ASLE Transactions,22, No. 3, pp. 227–244.

[8] Shapiro, W. and Colsher, R., 1977, “Dynamic Characteristics of Fluid Film Bearings,” Proceedings of the SixthTurbomachinery Symposium, Texas A&M University, pp. 39–-53.

[9] Parsell, J. K., Allaire, P. E. and Barrett, L. E., 1983, “Frequency Effects in Tilting-Pad Journal Bearing DynamicCoefficients,” ASLE Transactions, 26, pp. 222–227.

[10] Barrett, L. E., Allaire, P. E. and Wilson, B. W., 1988, “The Eigenvalue Dependence of Reduced Tilting PadBearing Stiffness and Damping Coefficients,” ASLE Transactions, 31, pp. 411–419.

[11] Ha, H. C., and Yang, S. H., 1999, “Excitation Frequency Effects on the Stiffness and Damping Coefficients ofa Five-Pad Tilting Pad Journal Bearing,” ASME Journal of Tribology, 121 (3), pp. 517–522.

[12] Wygant, K. D., 2001, “The Influence of Negative Preload and Nonsynchronous Excitation on the Performanceof Tilting Pad Journal Bearings,” Ph.D. Dissertation, University of Virginia, Charlottesville, Virginia.

[13] Knight, J. D., and Barrett, L. E., 1988, “Analysis of Tilting Pad Journal Bearing with Heat Transfer Effects,”ASME Journal of Tribology, 110, No. 1, 128–133.

[14] Branagan, L. A., 1988, “Thermal Analysis of Fixed and Tilting Pad Journal Bearings Including Cross-FilmViscosity Variations and Deformation,” Ph.D. Dissertation, University of Virginia, Charlottesville, Virginia.

[15] Ettles, C. M., 1992, “The Analysis of Pivoted Pad Journal Bearing Assemblies Considering ThermoelasticDeformation and Heat Transfer Effects,” STLE Tribology Transactions, 35, No. 1, pp. 156–162.

[16] Parkins, D. W. and Horner, D., 1993, “Tilting Pad Journal Bearings—Measured and Predicted StiffnessCoefficients,” STLE Tribology Transactions, 36, No. 3, pp.359–366.

[17] Kim, J., Palazzolo, A., and Gadangi, R., 1995, “Dynamic Characteristics of TEHD Tilt Pad Journal BearingSimulation Including Multiple Mode Pad Flexibility Model,” ASME Journal of Vibration and Acoustics, 117, No.1, pp. 123–135.

[18] Fillon, M., Desbordes, H., Frene, J., and Wai, C. C. H., 1996, “A Global Approach of Thermal Effects IncludingPad Deformation in Tilting-Pad Journal Bearings Submitted to Unbalance Load,” ASME Journal of Tribology,118, No. 1, pp. 169–174.

[19] Nicholas, J. C., 2003, “Lund’s Pad Assembly Method for Tilting Pad Journal Bearings,” Special Issue: TheContributions of Jørgen W. Lund to Rotor Dynamics, ASME Journal of Vibration and Acoustics, Vol. 125, No.4, pp. 448–454.

[20] Nicholas, J. C. and Kirk R. G., 1982, “Four Pad Tilting Pad Bearing Design and Application for Multi-StageAxial Compressors,” ASME Journal of Lubrication Technology, 104, No. 4, pp. 523–532.

[21] Nicholas, J. C., 1994, “Tilting Pad Bearing Design,” Proceedings of the Twenty-Third TurbomachinerySymposium, Texas A&M University, College Station, Texas, pp. 179–194.

[22] Hagg, A. C. and Stankey, G. O., 1958, “Elastic and Damping Properties of Oil-Film Journal Bearings forApplications to Unbalance Vibration Calculations,” ASME Journal of Applied Mechanics, 25, No. 1, pp. 141–143.

[23] Yamauchi, S. and Someya, T., 1977, “Balancing of a Flexible Rotor Supported by Special Tilting-PadBearings,” CIMAC Twelfth International Congress on Combustion Engines, Tokyo, Japan.



[24] Tripp, H. and Murphy, B., 1984, “Eccentricity Measurements on a Tilting-Pad Bearing,” ASLE Transactions,28, pp. 217–224.

[25] Brockwell, K., Kleinbub, D. and Dmochowski, W., 1990, “Measurement and Calculation of the DynamicOperating Characteristics of the Five Shoe Tilting Pad Journal Bearing,” STLE Tribology Transactions, 33 (4),pp. 481–492.

[26] Dmochowski, W. and Brockwell, K., 1995, “Dynamic Testing of the Tilting Pad Journal Bearing,” STLETribology Transactions, 38 (2), pp. 261–268.

[27] Kostrzewsky, G. J. and Flack, R. D., 1990, “Accuracy Evaluation of Experimentally Derived DynamicCoefficients of Fluid Film Bearings Part I: Development of Method,” STLE Tribology Transactions, 33 (1), pp.105–114.

[28] Kostrzewsky, G. J. and Flack, R. D., 1990, “Accuracy Evaluation of Experimentally Derived DynamicCoefficients of Fluid Film Bearings Part II: Case Studies,” STLE Tribology Transactions, 33 (1), pp. 115–121.

[29] Wygant, K. D., Barrett, L. E. and Flack, R. D., 1999, “Influence of Pad Pivot Friction on Tilting-Pad JournalBearing Measurements—Part II: Dynamic Coefficients,” STLE Tribology Transactions, 42 (1), pp. 250–256.

[30] Pettinato, B. and De Choudhury, P., 1999, “Test Results of Key and Spherical Pivot Five-Shoe Tilt Pad JournalBearings—Part II: Dynamic Measurements,” STLE Tribology Transactions, 42 (3), pp. 675–680.

[31] Simmons, J. E. L. and Dixon, S. J., 1994, “Effect of Load Direction, Preload, Clearance Ratio, and Oil Flow onthe Performance of a 200 mm Journal Pad Bearing,” STLE Tribology Transactions, 37 (2), pp. 227–236.

[32] Simmons, J. E. L. and Lawrence, C. D., 1996, “Performance Experiments with a 200 mm, Offset Pivot JournalPad Bearing,” STLE Tribology Transactions, 39 (4), pp. 969–973.

[33] Brockwell, K., DeCamillo, S. and Dmochowski, W., 2001, “Measured Temperature Characteristics of 152 mmDiameter Pivoted Shoe Journal Bearings with Flooded Lubrication,” STLE Tribology Transactions, 44 (4), pp.543–550.

[34] DeCamillo, S. and Brockwell, K., 2001, “A Study of Parameters that Affect Pivoted Shoe Journal BearingPerformance in High-Speed Turbomachinery,” Proceedings of the Thirtieth Turbomachinery Symposium,Texas A&M University, pp. 9–22.

[35] Nicholas, J. C. and Moll, R. W., 1993, “Shifting Critical Speeds Out of the Operating Range by Changing fromTilting Pad to Sleeve Bearings,” Proceedings of the Twenty-Second Turbomachinery Symposium, Texas A&MUniversity, pp. 25–32.

[36] Fuller, D. D., 1984, Theory and Practice of Lubrication for Engineers, John Wiley & Sons, New York.

[37] Wilcock, D. F. and Booser, E. R., 1957, Bearing Design and Application, McGraw Hill Book Company, NewYork.

[38] Zeidan, F., 2011, “Fluid Film Bearings Fundamentals”, Proceedings of 1st Middle East TurbomachinerySymposium, Texas A&M University, February.

[39] Childs, D., 2010, “Tilting-Pad Bearings: Measured Frequency Characteristics of their RotordynamicCoefficients”, Proceedings of the 8th IFToMM International Conference on Rotor Dynamics, Seoul, Korea, pp.1–8.



[40] Kirk, R. G. and Reedy, S. W., 1988, "Evaluation Of Pivot Stiffness For Typical Tilting-Pad Journal BearingDesigns,” Journal Of Vibration, Acoustics, Stress, And Reliability In Design, Vol. 110, Pp. 165–171, April.

[41] Nicholas, J. C., and Wygant, K. W., 1995, “Tilting Pad Journal Bearing Pivot Design for High LoadApplications,” Proceedings of the Twenty-Fourth Turbomachinery Symposium, Texas A&M University, pp. 33–47.

[42] Kulhanek, C., and Childs, D., 2011, “Measured Static and Rotordynamic Coefficient Results for a Rocker-Pivot, Tilting-Pad Bearing with 50 and 60 % Offsets,” Proceedings of ASME Turbo Expo 2011, No. GT2011-45209, Vancouver, Canada.

2.6 Seal Types and Modeling

2.6.1 Introduction

For convenience in rotordynamic modeling, seals can be categorized into two groups: liquid and gas. The effect theseseals have on the rotordynamic performance differs according to the type of seal and analysis being considered. Theimpact on turbomachine stability of all seals is significant. Whether positive or negative, nearly every seal needs to beaccounted for when performing a stability analysis. How the seals are modeled may vary from CFD analyses toempirical relations. We can summarize the general behavior and requirements of oil and gas seals in Table 2-4.Section 3.4 covers in detail the role that seals play in rotordynamic stability aspects.

For unbalance response analysis, the API requirement is to include oil seals only. Oil is specified since the use ofliquid film seals with fluids other than oil is rare in the equipment discussed in this tutorial. The distinction madebetween oil and other liquid seals, such as water, is a result of the viscosity of the sealing liquid. In journal bearingsand oil film seals, the film forces produced by the oil viscosity supports the journal load and creates the dynamiccoefficients. The fluid dynamics are modeled using Reynolds equation. For low viscosity fluids (e.g. gas, water, etc.),annular seals under large pressure gradients are highly turbulent and require a different analytical approach.

The unbalance response analysis involves forces and vibratory motion at the synchronous frequency. At thisfrequency, the oil seal’s damping improves the rotor response. However, the destabilizing aspects of these seals atsubsynchronous frequencies are one factor that has led to their replacement with dry gas seals. The trade-off is anincrease in synchronous response; the magnitude of which is dependent on the rotor/bearing design.

Toothed labyrinth seals (this includes impeller eye labyrinths, balance piston labyrinths, etc.) are basically neutral intheir behavior for unbalance response due to several factors. First, their principal stiffness terms are normally severalorders of magnitude less than the bending stiffness of the shaft or the rotor support stiffness. With lower principalstiffness terms, they have little influence on the location of the critical speeds. However, damper seals can have apronounced effect on the location of the critical speeds as discussed below in the gas annular seal section (seeSection 2.6.4).

Table 2-4 —General Behavior and Requirements of Oil and Gas Annular Seals

Oil Seals Gas Annular Seals

Inclusion in Unbalance Response Analysis API requirement Not required but may be needed for accurate analyses

Inclusion in Stability Analysis API requirement Effects need to be accounted for

Strong Effective Stiffness (K-m2) Terms Geometry and model dependent No (see exceptions in 2.6.6)

Strong Effective Damping (C-k/) Terms Yes (can be positive or negative) Yes (can be positive or negative)

Positive Effective Damping at Synchronous Whirl Frequency Yes Geometry dependent



2.6.2 Oil Seals

One major application of oil seals is the pressure containment in centrifugal compressors. It is in these applicationsthat the seals play a major role in the dynamic behavior of the rotor/bearing system. Oil seals are designed to keepprocess fluids from discharging into the atmosphere by placing a barrier between the process gas and theatmosphere. A typical single breakdown liquid-film shaft seal with cylindrical bushings is shown in Figure 2-51. Adiagram of the outer sealing ring is presented in Figure 2-52, showing key dimensions and the general oil pressuredistribution.

Typically, only the outer pressure reducing rings (sweet oil rings) are included in the rotordynamic calculations. With asmall clearance and small differential pressure, the inner ring (sour oil ring) is assumed to float and keep nearlyconcentric with the rotating shaft. In this condition, the dynamic characteristics produced by the inner seal ring arelow.

The axial force on the outer ring is obtained from integrating the pressure distribution on the radial faces. This forcedetermines the contact pressure at the lapped sealing face on the left side of the ring, Figure 2-52. The coefficient offriction between the lapped face and the housing produces a radial holding force that inhibits the outer seal ring frommoving. The magnitude of the axial force can be affected by relocating the dimensions, Do and Di. For low-pressureapplications, the lapped sealing face outer diameter Do is typically high. However, for high-pressure applications, thelapped seal face Do is typically low. Since a certain lapped surface area is required for sealing and to preventdistortion of the face, the axial force can only be minimized and not eliminated. A low axial force on the outer ringcould induce fretting on the face and even allow the outer seal ring to whirl. Other considerations in the design of theouter ring are the length of the lands and the clearances that will maintain a certain leakage rate and limits on thetemperature.

Figure 2-51—Oil Bushing Breakdown Seal

Babbited bore of seal Inner seal ring

Outer seal ring

Processgas atpressure Pp

Innerdrain

Rotor shaft

Lapped sealingface

Oil supply atpressure Ps

Antirotationpin

Springs for initialseating of sealson lapped faces

Seal cartridge

Lapped sealing face

Outer drain atpressure Pambient



During eccentric operation of the seal ring, the oil film between the ring and shaft will produce a restoring force similarto that found in journal bearings. In bearings, the force supports the shaft load. In oil seals, equilibrium is achievedwhen the restoring force is equal to or is less than the holding force. In this “locked” position, dynamic coefficients ofthe ring can be obtained and applied to the rotor/bearing system. Highly eccentric operation of the ring has beenshown to be detrimental to both the rotor stability and unbalance response. Allaire et al. [1] describe a case historyinvolving the balancing of a seal’s pressure distribution, modification of the seal restoring force and the subsequentimpact on the rotor’s dynamic behavior.

For synchronous response, locking of the seal position may occur when the machinery is started up. The lockedposition is determined by the pressure distribution axially and the oil film characteristics. While modifying the lappedsealing face diameters can affect the holding force, so can varying the suction pressure during start-up. For example,the damped response of the hydrogen recycle compressor rotor, Allaire et al. [1], to a general unbalance distributionhas been calculated for various start-up sealing pressures. The mid-span response of this compressor through thefirst critical speed is displayed in Figure 2-53 for three start-up sealing pressures. Note that as the sealing pressureincreases, the influence of the oil seals increases due to an increase in the damping generated by the seals. Ingeneral, the damping provided by oil seals tends to reduce the amplification associated with the fundamental criticaland to raise the frequency of the critical speed.

Although the effect of oil seals on unbalance rotor response characteristics is extremely positive, oil seals may provequite detrimental to the unit’s rotor stability at operating speed and cause large amplitude subsynchronous vibrations.Much design effort has focused on minimizing the destabilizing effect of the seals. Some oil seals are designed tominimize axial forces; the sealing lands are frequently grooved in order to diminish hydrodynamic load capacity; andvarious centering mechanisms are used to reduce the eccentricity of the ring, relative to the shaft during operation.One OEM combined a tilt-pad bearing with the floating outer ring oil seal to reduce the seal eccentricity, reduce thecross-coupled stiffness of the outer ring and add additional direct stiffness and damping from the tilt pads in the sealassembly [2]. The influence of oil seals on stability will be discussed in Section 3.4.

Figure 2-52—Pressures Experienced by the Outer Floating Ring Seal

Anti-rotation pin

Ps

Ps

Ls

Ps

Pambient

Ps

DO

DI

D

PsPambient

DDIDOLs

seal oil supply pressureambient pressure diameter of sealing landinner diameter of lapped sealing faceouter diameter of lapped sealing facelength of sealing land

== ====

Notes:



Lastly, the determination of the oil film characteristics of the locked ring is dependent on many factors: the sealingpressure and temperature, eccentricity of the locked ring, geometric data, and lubricant properties. The pressure fieldof the oil film in the seals is described using similar differential equations employed to describe the oil film pressures inhydrodynamic journal bearings. The principal difference between the numerical methods used to analyze bearingsand seals lies in part due to the pressure differential across the axial length of the seal. Additionally, turbulence andfluid inertia may not be negligible especially in water film seal. Baheti and Kirk [3] present a finite-element method offloating ring seals using a thermo-hydrodynamic solution that takes into account many of these factors. These factorsinclude:

d) journal diameter;

e) shaft speed;

f) sealing pressure and temperature;

g) axial length of seal ring;

h) number of grooves;

i) radial seal clearance;

j) inner and outer diameter of sealing face or lip;

k) coefficient of friction between ring and housing;

l) oil viscosity, density, and thermal properties.

Figure 2-53—Mid-Span Rotor Unbalance Response of a High-pressure Centrifugal Compressor for Different Suction Pressures at Start-up [1]

0 1000

125

100

75

50

25

0

5

4

3

2

1

02000

Speed (r/min)

Low Psuction start-upHigh Psuction start-upRecommended Psuction start-up

3000 4000 5000

Vbr

aton

am

ptu

de (μ

m p

-p)

Vbr

aton

am

ptu

de (m

s p-

p)



Various configurations of oil seals are presented in Figure 2-54, Figure 2-55, Figure 2-56, and Figure 2-57. Somehave fixed breakdown rings, others have a floating outer ring and some have multiple floating rings. All have an oilfilm between the rotating assembly and stationary components that affect the rotordynamic characteristics ofcentrifugal compressors in the manner described above. Figure 2-57 shows a tilt pad oil seal, where tilt pad oil sealsreplaced a “no-load” oil film ring seal to eliminate a subsynchronous vibration problem.

Reedy and Kirk [4] show the importance of determining the locked position by stating that cross-coupled stiffness canincrease by tenfold with increasing eccentricity. A quasi-static method for estimating the lock-up eccentricity waspresented by Allaire et al. [5]. Static measurements of these holding forces were made by Kirk and Browne [6].

2.6.3 Turbulent Liquid Annular Seal Modeling & Testing

The modeling of liquid film seals got a significant start with Black [7,8]. In this work, the influence of high axial flowseals with Reynolds numbers greater than 2000 on the rotordynamic behavior of pumps was explained. Lightviscosity liquids, large axial pressure gradients and large clearance to radius ratio characterized these seals. Thiswork brought into attention how dramatically different seal flow was from conventional circular journal bearing flow.

Figure 2-54—Mechanical (Contact) Shaft Seal

Internalgas pressure

Clean oil inPressurebreakdownsleeve

Oil out

Contaminatedoil out

Atmosphere

Running face

Rotating seatStationary seat

Carbon ring



The bulk of the work that followed examined various aspects of modeling turbulent flow. Hirs’ bulk flow model [9] wasapplied to the turbulent flow seals by Childs [10,11]. Hirs’ theory assumes the friction factor is a function of Reynoldsnumber only. This was found restrictive when analyzing seals with rough surfaces. Moody’s equation, which assumesthat friction factor is a function of both Reynolds number and relative roughness, provided closer agreement withexperimental results [12,13].

Dietzen and Nordmann [14] applied fully three-dimensional flow methods using a k-ε turbulence model. Turbulencemodels applied to CFD codes attempt to model the turbulence flow (Hirs’ model treats the turbulence effects as achange in viscosity.) Large centered whirl amplitudes were included by Tam et al. [15]. In both cases, the additionalcomputational effort produced marginal benefits over bulk flow models. Other research included variable properties[16] and fluid inertia effects [17].

Figure 2-55—Liquid-film Shaft Seal with Cylindrical Bushing

Clean oil in

Inner bushing Outer bushing

Shaft sleeve

Atmosphere

Oil out(sweet sidedrain)

Contaminatedoil out(sour side drain)




Experimental results for liquid annular seals were largely obtained by Childs at Texas A&M. Childs et al. measureddamper seals with different surface roughness [18] and tapered seals [19]. Grooved seals [20,21] and hole-patterndamper seals [22] were also tested.

As noted earlier, annular seals with low viscosity fluids tend to be highly turbulent requiring a specialized analyticalapproach. In some cases, these solutions can be applied to the lower Reynolds number axial flow found in oil sealsbut may not include the lock-up analysis that is unique with floating ring seals. Kirk and Miller [23] developed anapproach based on Reynolds lubrication equation including the locking mechanism. This approach was extended toinclude a thermal heat balance [24] and later to include multiple floating rings [4].

Figure 2-56—Liquid-film Shaft Seal with Pumping Bushing

Shaft sleeve

Clean oilrecirculation

Clean oil in


Pumping area

Oil out

Atmosphere

ContaminatedOil out

Innerbushing

Outerbushing



2.6.4 Gas Annular Seals

Gas annular seals are used throughout the centrifugal compressor to separate regions of high and low pressure andminimize internal leakage. These seals can have a large effect on the rotordynamic stability of the compressor. Theeffects of the seals are a part of a Level 2 stability analysis (discussed in Section 3). Labyrinth seals consist ofcircumferential teeth either on the rotor or stator (or both) and create a tortuous path for the leakage flow therebyminimizing leakage. In general, labyrinth seals are treated as a second order effect in unbalance response analysis.For synchronous response, principal stiffness and effective damping terms are usually small in comparison to oil filmseals and journal bearings. However, damper type seals, such as hole pattern and honeycomb seals, can producesignificant stiffness and damping terms. In these situations, an accounting of their effect would be needed for anaccurate response analysis. These exceptions are discussed later in this section.

Damper seals are used at seal locations with high-pressure differential such as balance piston and center divisionwall seal locations to maximize their beneficial effect on stability. Some examples of gas annular seals found inturbomachinery are illustrated in Figure 2-58 through Figure 2-64. A hole pattern damper seal with swirl brakes isshown in Figure 2-62. A swirl brake is a row of small vanes added to the upstream side of a labyrinth seal to block theswirling flow from entering the seal thereby reducing the cross-coupled stiffness in the seal. Figure 2-63 from Mooreand Hill [26] shows an example of a swirl brake used on an impeller eye seal location. These authors described aCFD method used to design the swirl brakes. Shunt hole systems have been used since the 1970s to take gas fromthe last stage diffuser and send it near to the upstream side of division wall or balance piston seals to block theswirling flow from entering the seal. The usage of shunt hole systems has been described in many papers [27-30].Such a system, including a damper honeycomb seal at the division wall, is shown in Figure 2-64 from Memmott [30].Further discussion for the stability aspects of gas annular seals is provided in Section 3.4.

Damper seals, however, can have a significant effect on unbalance response levels and the location of the firstnatural frequency if there is a significant pressure differential or density across the seal. Even though these gasannular seals demonstrate the capability of changing the location and behavior of the rotor first natural frequencywhile the compressor is running at the design speed, one important factor still exists; a pressure drop is required toproduce the dynamic coefficients. Most flexible rotors will have their first critical speed around 50 % ±15 % of designspeed. Assuming that discharge pressure (and pressure gradient) is developed as the square of speed, at 50 % of

Figure 2-57—Tilt Pad Oil Seal [2]

Tiltingpad

Lapped sealingface

Outer ringInner ring



Figure 2-58—Compressor Labyrinth Seals

Figure 2-59—Typical Turbine Shaft Seal Arrangement—HP End

Impeller eye seal

Balance piston

Inter-stage shaft seal

Seal leakage direction



Figure 2-60—Honeycomb Damper Seal [32]

Figure 2-61—Pocket Damper Seal

Figure 2-62—Hole Pattern Damper Seal with Swirl Brakes [25]

Cell size

Cell depth

Fluidpreswirl

Shaft

Clearance

Honeycombhousing



Figure 2-63—Swirl Brake Used in High-pressure Compressors [26]

Figure 2-64—Shunt Hole System with Honeycomb Division Wall Seal Used in High-pressure Compressors [30]

Shunt hole

HP sideLP side

Honeycomb div. wall sealwith shunt holes



the design speed, only one quarter of the design pressure gradient has been achieved. At this level, gas sealsdynamics will be greatly reduced accordingly and thus, their effect on the rotordynamics of the shaft. In cases wherethe natural frequency is pushed up into the operating speed range, the damping from the seal typically causes theamplification factor to be less than 2.5, allowing operation on the critical speed.

Memmott [30] reported on a 7 stage, back-to-back compressor that underwent modification to eliminate asubsynchronous vibration problem due to unacceptable vibration amplitude at the first critical speed frequency at71 % of running speed. The division wall toothed labyrinth (with shunt holes) was replaced with a honeycomb sealand the shunts were covered by the honeycomb (without shunt). There still was a subsynchronous vibration problemdue to unacceptable levels of amplitude at the first critical speed frequency. The frequency of the subsynchronousvibration increased to 94 % of running speed indicating that the location of the first critical speed natural frequencywas affected. Subsequent inclusion of the shunt holes and redesign of the honeycomb eliminated thesubsynchronous vibration (see Figure 2-64 for the final configuration). Damper seals can benefit from reduction ofinlet swirl from anti-swirl devices. Bidault and Bauman [31] show the increase of the first natural frequency withpressure with a hole pattern seal, as shown in Figure 2-65.

2.6.5 Annular Gas Seal Modeling

Section 3.4.2 presents background on the analysis and testing of labyrinth seals in the literature. Some of thatinformation is repeated here. In addition, Childs [32] provides an excellent overview of modeling techniques,analytical methods and testing results for various types of gas seals from smooth to interlocking labyrinths. A briefsummary of Childs’ chapter dealing with gas seals is included here.

Labyrinth seals have been modeled using bulk flow models with various control volumes. The different control volumemodels are illustrated in Figure 2-66. A single volume was investigated by Childs and Scharrer [33] and extended toeccentric shaft positions by Kurohashi et al. [34] and stepped labyrinths by Scharrer [35]. A two-control volume modelwas proposed by Wyssman et al. [36]. The model has one volume in the cavity between teeth and one covering thethrough-flow region. A three-control volume model has been studied by Nordmann and Weiser [37]. Work by Thorat[38] has introduced real gas properties to the predictions. Other relevant papers are provided in [39-42]. In addition,others have investigated the use of Navier-Stokes CFD models to analyze gas labyrinth seals [43-44] including holepattern seals [45].

The research summarized in Childs [32] provided the first measurements of the stiffness and damping coefficients.While the results gave the first comprehensive basis for comparison against predictions, Childs and Ramsey [46]revealed the importance of testing at or near the application conditions. Ramsey’s test rig was extended toward thisgoal by Childs and Scharrer [47] and then again by Elrod et al. [48]. Teeth-on-stator, teeth-on-rotor, and interlockingconfigurations were tested.

Wagner and Steff [49] developed a test rig supported by magnetic bearings. The intent of the rig was to furtherexpand the existing knowledge database to geometries and gas conditions matching industrial applications, namely,pressure differential, size and speed. The magnetic bearings made it possible to set the eccentricity and superimposecirculatory movement independent of speed. The rig permitted the inlet pressure and the backpressure to be setindependently of one another. Pressures of 70 bar (1000 psi) were possible at surface speeds up to 157 m/s (515fps). The testing of Picardo and Childs [50] present test data for a 20 tooth, teeth-on-stator, labyrinth seal at pressuresalso up to 70 bar (1000 psi).

Vannini et al. [58] presented results for a long labyrinth seal from a high-pressure test rig working with active magneticbearings and reaching 380 bar as a maximum pressure. A rotational speed of 10 krpm was needed to achieve 0.8inlet preswirl ratio. Nitrogen was the test gas and the main findings were:

— Laby Seal behavior (stiffness and damping coefficients) was shown to be linear with inlet pressure and constantpressure ratio (about 2).

— Frequency dependence of cross-coupled stiffness and damping coefficients was low or negligible.



Figure 2-65—Measured Natural Frequency Showing the Increase of the Shaft’s First Bending Mode with Pressure [31]

Figure 2-66—Labyrinth Seal Bulk Flow Control Volume Approaches

0 100 200 300 400 500 600 700Discharge pressure (bar)

220

200

180

160

140

120

100

80

Nat

ura

freq

uenc

y (H

z)

105 % Speed 100 % Speed

Stator

Rotor

I

Single-control volume

Stator

Rotor

I

II

Two-control volumeStator

Rotor

I

II

III

Three-control volume



— Frequency dependence of direct stiffness was more pronounced with the trend towards negative values. Themaximum absolute values were one order lower than the typical bearing stiffness.

Finally, predictions for honeycomb and hole pattern damper seals also utilize bulk flow analysis using a friction factorto model the roughened surface, Ha and Childs [51]. The friction factor was defined using a Blasius or Moody-equation. The weakness of these models stems from origin of the friction factors. Literature has shown that thesemust be obtained experimentally to provide reasonable predictions of the dynamic behavior of these damper seals.

Kleynhans and Childs [52] developed a two-control volume isothermal bulk flow model for damper seals. This modelpredicts frequency dependent rotordynamic coefficients due to the interaction of an acoustic frequency with thehoneycomb cavity gap. Experimental measurements of dynamic impedance for honeycomb seals by Dawson andChilds [53] confirm the frequency dependent behavior for these types of seals. Hole-pattern seals show similarfrequency dependent nature for rotordynamic coefficients as observed for honeycomb seals. Experimentalmeasurements by Yu and Childs [54], Holt and Childs [55], Childs and Wade [56] confirm the frequency dependentbehavior of these seals.

Experimental results have confirmed the beneficial aspects of the damper seal [53-56]. These advantages includegood stability when used with swirl brakes or shunt holes and ruggedness. Manufacturing costs and times can bereduced when a hole-pattern configuration is employed, instead of a honeycomb, without loss in dynamic behavior,Holt and Childs [55]. Data from full load, full pressure shop tests of a high-pressure centrifugal compressor with a holepattern seal and a smooth division wall seal indicate principal stiffness and damping terms of the hole pattern are onthe same order of magnitude as the smooth seal and significantly higher than that of a labyrinth seal. The smooth sealmeasured 50 % greater leakage than the hole pattern seal [57].

2.6.6 Segmented Ring Shaft Seals and Gas Lubricated Face Seals

The segmented-ring shaft seal and the gas lubricated face seal (dry gas seal) basically have negligible rotor dynamicstiffness and damping coefficients. The segmented-ring seal is frequently applied as a barrier seal separating thebearing lube oil from the gas seal. Their contributions to the unbalance response and stability analyses are addedmass to the rotor model (if sleeves are used in conjunction with the segmented-ring seal). Examples of each areshown in Figure 2-67 and Figure 2-68.

2.6.7 References

[1] Allaire, P. E., Stroh, C. G., Flack, R. D., Kocur, Jr., J. A. and Barrett, L. E., 1987, “Subsynchronous VibrationProblem and Solution in Multistage Centrifugal Compressor,” Proceedings of the 16th TurbomachinerySymposium, Texas A&M University, pp. 65–73.

[2] Memmott, E. A., 2004, "The Stability of Centrifugal Compressors by Applications of Tilt Pad Seals,"Proceedings of the 8th International Conference on Vibrations in Rotating Machinery, IMechE, Swansea,pp.81–90, September 7–9.

[3] Baheti, S. K. and Kirk, R. G., 1994, “Thermo-Hydrodynamic Solution of Floating Ring Seals for High PressureCompressors Using the Finite-Element Method,” STLE Tribology Transactions, 37, No. 2, pp. 336–346.

[4] Reedy, S. W. and Kirk, R. G., 1992, “Advanced Analysis of Multi-Ring Liquid Seals,” ASME Journal ofVibration and Acoustics, 114, pp. 42–46.

[5] Allaire, P. E., Kocur, J. A. and Stroh, C. G., 1986, “Oil Seal Effects and Subsynchronous Vibrations in HighSpeed Compressors,” Proceedings of the Rotordynamics Instability Problems in High-PerformanceTurbomachinery, NASA Publ. 2409, pp. 205–223.

[6] Kirk, R. G. and Browne, D. B., 1990, “Experimental Evaluation of Holding Forces in Floating Ring Seals,”Proceedings of 3rd International Conference on Rotordynamics, IFToMM, Lyon, France, pp. 319–323.



[7] Black, H. F., 1969, “Effects of Hydraulic Forces on Annular Pressure Seals on the Vibration of CentrifugalPump Rotors, “Journal of Mechanical Engineering Science,” 11, No. 2, pp. 206–213.

[8] Black, H. F. and Jenssen, D. N., 1971, “Effects of High Pressure Ring Seals on Pump Rotor Vibrations,”ASME Paper 71-WA/FF-38.

[9] Hirs, G. G., 1973, “A Bulk Flow Theory for Turbulence in Lubricating Films,” ASME Journal of LubricationTechnology, 95, No. 2, pp. 137–146.

[10] Childs, D. W., 1983, “Dynamic Analysis of Turbulent Annular Seals Based on Hirs Lubrication Equation,”ASME Journal of Lubrication Technology, 105, pp. 429–436.

Figure 2-67—Segmented-ring Shaft Seal

Internalgaspressure

Atmosphere

Ports may beadded forsealing

Scavengingport may beadded forvacuumapplication



[11] Childs, D. W., 1983, “Finite Length Solutions for Rotordynamic Coefficients of Turbulent Annular Seals,”ASME Journal of Lubrication Technology, 105, pp. 437–444.

[12] Nelson, C. C. and Nguyen, D. T., 1987, “Comparison of Hirs’ Equation with Moody’s Equation for DeterminingRotordynamic Coefficients of Annular Pressure Seals,” ASME Journal of Tribology, 109, pp. 144–148.

[13] Nelson, C. C. and Nguyen, D. T., 1988, “Analysis of Eccentric Annular Incompressible Seals: Part 1 – A NewSolution Using Fast Fourier Transforms for Determining Hydrodynamic Forces,” ASME Journal of Tribology,Vol. 110, pp. 361–366.

[14] Dietzen, F. J. and Nordmann, R., 1988, “A Three-Dimensional Finite-Difference Method for Calculating theDynamic Coefficients of Seals,” Proceedings of the Rotordynamics Instability Problems in High-PerformanceTurbomachinery, NASA Publ. 3026, pp. 211–228.

[15] Tam, L. T., et al., 1988, “Numerical and Analytical Study of Fluid Dynamics Forces in Seals and Bearings,”ASME Journal of Vibration, Acoustics, Stress and Reliability Design, pp. 315–325.

[16] San Andres, L. A., 1991, “Analysis of Variable Fluid Properties, Turbulent Annular Seals,” ASME Journal ofTribology, 113, pp. 694–702.

[17] Simon, J. and Frene, J., 1990, “Rotordynamic Coefficients of Turbulent Annular Misaligned Seals,”Proceedings of the 3rd International Symposium on Transport Phenomena and Dynamics of RotatingMachinery (ISROMAC-3), Vol. 2: Dynamics, pp. 124–143.

Figure 2-68—Self-acting Dry Gas Seal

Innerlaby

Primaryseal

Secodaryback-up seal

Separation seal

Intermediatelabyrinth

Bearing sideProcess side

Separation gas

Secondary seal gas leakage

Secondary seal gas supply

Primary seal gas leakage

Primary seal gas supply



[18] Childs, D. W. and Kim C.-H., 1985, “Analysis and Testing for Rotordynamic Coefficients of Turbulent AnnularSeals with Different, Directionally-Homogenous Surface Roughness Treatment for Rotor and StatorElements,” ASME Journal of Tribology, 107, No. 3, pp. 296–306.

[19] Childs, D. W. and Dressman, J. B., 1985, “Convergent-Tapered Annular Seals: Analysis and Testing forRotordynamic Coefficients,” ASME Journal of Tribology, 107, No. 3, pp. 307–317.

[20] Childs, D. W. and Kim C.-H., 1986, “Testing for Rotordynamic Coefficients and Leakage: CircumferentiallyGrooved Seals,” Proceedings of the Second IFToMM International Conference on Rotordynamics, Tokyo,Japan, pp. 609–618.

[21] Childs, D. W., Nolan, S. and Kilgore, J., 1990, “Test Results for Turbulent Annular Seals Using Smooth Rotorsand Helically-Grooved Stators,” ASME Journal of Tribology, 112, pp. 254–258.

[22] Childs, D. W., Nolan, S. and Kilgore, J., 1990, “Additional Test Results for Round Hole-Pattern Damper Seals:Leakage, Friction Factors and Rotordynamic Force Coefficients,” ASME Journal of Tribology, 112, pp. 365–371.

[23] Kirk, R. G. and Miller, W. H., 1979, “The Influence of High Pressure Oil Seals on Turbo-Rotor Stability,” ASLETransactions, 22, No. 1, pp. 14–24.

[24] Kirk, R. G. and Nicholas, J. C., 1980, “Analysis of High Pressure Oil Seals for Optimum TurbocompressorDynamic Performance,” Proceeding of the 2nd International Conference on Vibrations in Rotating Machinery,IMechE, Cambridge, pp. 125–134.

[25] Memmott, E. A., 2011, “Stability of Centrifugal Compressors by Applications of Damper Seals,” ASME,Proceedings of ASME Turbo Expo 2011, Power for Land, Sea and Air, Vancouver, Canada, June 6–10,GT2011-45634.

[26] Moore, J. J., Hill, D. L., 2000, “Design of Swirl Brakes for High Pressure Centrifugal Compressors Using CFDTechniques,” Proceedings of the 8th International Symposium of Transport Phenomena and Dynamics ofRotating Machinery (ISROMAC-8), March 26–30, 2000, Honolulu, Hawaii, pp. 1124–1132.

[27] Memmott, E. A., 1992, “Stability of Centrifugal Compressors by Applications of Tilt Pad Seals, DamperBearings, and Shunt Holes,” Proceedings of the 5th International Conference on Vibrations in RotatingMachinery, IMechE, Bath, pp. 99–106, September 7–10.

[28] Gelin, A., Pugnet, J-M., Bolusset, D, and Friez, P., 1996, “Experience in Full Load Testing Natural GasCentrifugal Compressors for Rotordynamics Improvements,” Proceedings of ASME Turbo Expo 1996, Powerfor Land, Sea and Air, Birmingham, UK, 96-GT-378.

[29] Camatti, M., Vannini, G., Fulton, J. W., and Hopenwasser, F., 2003, “Instability of a High Pressure CompressorEquipped with Honeycomb Seals,” Proceedings of the Thirty Second Turbomachinery Symposium, TexasA&M University, pp. 39–48, September 8–11.

[30] Memmott, E. A., 1994, “Stability of a High Pressure Centrifugal Compressor Through Application of ShuntHoles and a Honeycomb Labyrinth,” Presented at the 13th Machinery Dynamics Seminar, CMVA, Toronto,Canada, September 12–13.

[31] Bidaut, Y., Baumann, U., 2009, “Rotordynamic Stability of a 9500 psi Reinjection Centrifugal CompressorEquipped with a Hole Pattern Seal – Measurement vs. Prediction taking into Account the OperationalBoundary Conditions,” Proceedings of the Thirty Eighth Turbomachinery Symposium, Texas A&M University,pp. 251–260, September 14–17.



[32] Childs, D. W., 1993, Turbomachinery Rotordynamics: Phenomena, Modeling, and Analysis, John Wiley &Sons, Inc., New York.

[33] Childs, D. and Scharrer, J., 1986, “An Iwatsubo-Based Solution for Labyrinth Seals: Comparison toExperimental Results,” ASME Journal of Engineering for Gas Turbines and Power, 108, pp. 325–331.

[34] Kurohasi, M., Inoue, Y., Abe, T. and Fujikawa, T., 1980, “Spring and Damping Coefficients of the LabyrinthSeals,” Proceedings of the 2nd International Conference on Vibrations in Rotating Machinery, IMechE,Cambridge, England, pp. 215–222.

[35] Scharrer, J., 1988, “Rotordynamic Coefficients for Stepped Labyrinth Gas Seals,” ASME Journal of Tribology,111, pp. 101–107.

[36] Wyssman, H., Pham, T. and Jenny, R., 1984, “Prediction of Stiffness and Damping Coefficients for CentrifugalCompressor Labyrinth Seals,” ASME Journal of Engineering for Gas Turbines and Power, 106, pp. 920–926.

[37] Nordmann, R. and Weiser, H., 1990, “Evaluation of Rotordynamic Coefficients of Look-Through Labyrinths byMeans of a Three Volume Bulk Model,” Rotordynamic Instability Problems in High-PerformanceTurbomachinery, NASA CP-3122, pp. 141–157.

[38] Thorat, M., 2010, “Impact of Rotor Surface Velocity, Leakage Models, and Real Gas Properties onRotordynamic Force Predictions of Gas Labyrinth Seals,” M.S. Thesis, Texas A&M University.

[39] Benckert, H. and Wachter, J., 1980, “Flow Induced Spring Coefficients of Labyrinth Seals for Application toRotordynamics,” NASA CP-2133, pp. 189–212.

[40] Iwatsubo, T., Matooka, N. and Kawai, R., 1982, "Spring and Damping Coefficients of the Labyrinth Seal,"NASA CP-2250, pp. 205–222.

[41] Kirk, R. G., 1985, “Evaluation of Aerodynamic Instability Mechanisms for Centrifugal Compressors,”Proceedings of the Design Engineering Vibration Conference, 85-DET-147, Cincinnati, Ohio, Sept. 10–13.

[42] Kirk, R. G., 1990, “A Method for Calculating Labyrinth Seal Inlet Swirl Velocity,” ASME Journal of Vibration andAcoustics, 112 (3), pp. 380–383.

[43] Kwanka, K., Sobotzik, J., and Nordmann, R., 2000, “Dynamic Coefficients Of Labyrinth Gas Seals: AComparison Of Experimental Results And Numerical Calculations,” Proceedings of the ASME InternationalGas Turbine and Aeroengine Congress and Exposition, May 8–11, Munich, Germany.

[44] Moore, J. J., 2003, “Three-Dimensional CFD Rotordynamic Analysis of Gas Labyrinth Seals,” ASME Journalof Vibrations and Acoustics, 125, October, pp. 427–433.

[45] Chochua, G., and Soulas, T. A., 2007, “Numerical Modeling of Rotordynamic Coefficients for DeliberatelyRoughened Stator Gas Annular Seals,” ASME Journal of Tribology, 129, pp. 424–429.

[46] Childs, D. W. and Ramsey, C., 1990, “Seal-Rotordynamic-Coefficient Test Results for a Model SSME ATD-HPFTP Turbine Interstage Seal With and Without a Swirl Brake”, Rotordynamic Instability Problems in High-Performance Turbomachinery, NASA CP-3122, pp. 179–190.

[47] Childs, D. W. and Scharrer, J. K., 1986, “Experimental Rotordynamic Coefficient Results for Teeth-on-Rotorand Teeth-on-Stator Labyrinth Gas Seals”, ASME 86-GT-12.



[48] Elrod, D. A., Pelletti, J. M. and Childs, D. W., 1995, “Theory Versus Experiment for the RotordynamicCoefficients of an Interlocking Labyrinth Gas Seal,” Proceedings of the ASME International Gas Turbine andAeroengine Congress and Exposition, 95-GT-432, Houston, Texas, June 5–8.

[49] Wagner, N. G. and Steff, K., 1996, “Dynamic Labyrinth Coefficients From a High-Pressure Full-Scale Test RigUsing Magnetic Bearings,” Rotordynamic Instability Problems in High-Performance Turbomachinery, NASACP-3344, pp. 95–111.

[50] Picardo, A. and Childs, D., Rotordynamic coefficients for a Teeth-on-Stator Labyrinth Seals at 70 bar SupplyPressures Measurements Versus Theory and Comparisons to a Honeycomb Seal, ASME Journal ofEngineering for Gas Turbines and Power, 127, October, pp. 843–855.

[51] Ha, T. W., and Childs, D.W., 1994, “Annular Honeycomb-Stator Turbulent Gas Seal Analysis Using a NewFriction-Factor Model Based on Flat Plate Tests,” ASME Journal of Tribology, 116, pp. 352–360.

[52] Kleynhans, G. F., and Childs, D. W., 1996, “The Acoustic Influence of Cell Depth on the RotordynamicCharacteristics of Smooth-Rotor/Honeycomb-Stator Annular Gas Seals,” ASME Journal of Engineering forGas Turbines and Power, 119, pp. 949–956.

[53] Dawson, P. and Childs, D., 2002, “Measurements versus Predictions for the Dynamic Impedance of AnnularGas Seals-Part II: Smooth and Honeycomb Geometries”, ASME Journal of Engineering for Gas Turbines andPower, 124, pp. 963–970.

[54] Yu, Z., and Childs, D. W., 1998, “A Comparison of Experimental Rotordynamic Coefficients and LeakageCharacteristics Between Hole-Pattern Gas Damper Seals and a Honeycomb Seal,” ASME Journal ofEngineering for Gas Turbines and Power, 120, pp. 778–783.

[55] Holt, C. G., and Childs, D. W., 2002, “Theory Versus Experiment for the Rotordynamic Impedances ofTwo Hole-Pattern-Stator Gas Annual Seals,” ASME Journal of Tribology, 124, pp. 137–143.

[56] Childs, D. and Wade, J., 2004, “Rotordynamic-Coefficient and Leakage Characteristics for Hole-Pattern-Stator Annular Gas Seals-Measurements versus Predictions”, ASME Journal of Tribology, 126, pp. 326–333.

[57] Moore, J. J., Soulas, T. S., 2003, “Damper Seal Comparison in a High-Pressure Re-Injection CentrifugalCompressor During Full-Load, Full-Pressure Factory Testing Using Direct Rotordynamic StabilityMeasurement,” Proceedings of the DETC ‘03 ASME 2003 Design Engineering Technical Conference,Chicago, IL, Sept. 2–6.

[58] Vannini, G., Calicchio V., Cioncolini S., Tedone F., 2011, “Development of a High Pressure Rotordynamic TestRig for Centrifugal Compressors Internal Seals Characterization”, Proceedings of the 40th TurbomachinerySymposium, Texas A&M University.

2.7 Elements of a Standard Rotordynamics Analysis

2.7.1 Introduction

The purpose of a standard rotordynamic analysis and design audit is to enable an engineer to characterize the lateraldynamics design characteristics of a given design. While analysis of some rotating equipment may require analysisspecific to the unit, a general method has emerged for performing the standard lateral analysis. With modeling of theindividual components (rotor, bearings, seals, etc.) completed and described in the previous sections, the standardlateral analysis is composed of three parts: (a) undamped critical speed analysis, (b) damped unbalance responseanalysis, and (c) stability (damped eigenvalue) analysis.



The analysis of the lateral dynamics of rotating shafts began in 1869 [1] when Rankine identified the existence ofcritical speeds. Dunkerley [2] showed that the problem of calculating these critical speeds was equivalent to theproblem of finding the natural frequencies of the shaft, treating it as a flexible beam on simple supports (rigidbearings). Based on Rankine’s and Dunkerley’s conclusions about the destructive nature of operating near or abovethe first critical speed, manufacturers focused on simply calculating the first critical speed and designed theirmachines to operate well below it [3]. This design approach was eventually found to be overly conservative throughthe experimental steam turbine work of De Laval and the landmark analysis of Jeffcott [4].

Possessing no knowledge of the bearings’ stiffness and damping characteristics, for many years, rotordynamicanalysis focused on accurately modeling the rotor’s mass/elastic properties for critical speed prediction assumingrigid bearings. Without the computational power available today, approximating the first critical speed of a shaft withvariable cross-section was a serious challenge. Several methods gained popularity. These included Rayleigh’squotient where the first critical speed was estimated by assuming the first mode shape, and Stodola’s graphicaltechnique [5] which was more widely used. These were replaced by the transfer matrix method of Myklestad [6] andProhl [7], which was a variation of Holzer’s technique for torsional dynamic analysis. This method allowed the mass/elastic properties of a variable cross-section rotor to be more accurately represented. It also made use of the growingaccuracy of computers [8]. For almost 20 years after Prohl’s publication, most machines were designed using thismethod to predict the critical speeds assuming rigid bearings with no oil film flexibility or damping.

With a greater availability of computational power in the early 1970’s, manufacturers began applying the latest fluidfilm bearing models for the stiffness and damping characteristics, negating the need to assume rigid supports. Thisallowed for improved estimates of the critical speeds based on an undamped critical speed analysis. At the sametime, manufacturers began performing unbalanced response analysis as a standard machine design calculation. Inthis analysis, the predominant technique was the Lund and Orcutt algorithm [9] which expanded the transfer matrixmethod and allowed for actual vibration levels to be predicted along with the critical speeds and amplification factors.This algorithm also incorporated the full dynamic (stiffness and damping) characteristics of the machine’s bearings.

Lund [10] also provided industry with a stability analysis algorithm in 1974. Also based on the transfer matrix method,this analysis technique was first used to compute the basic stability condition of the rotor including the journalbearings, oil seals and damper bearings. Studies were also made of the sensitivity of the log dec to arbitrary amountsof cross-coupling at the midspan. The stability analysis also made a great improvement over the undamped criticalspeed map, as it could be used to produce plots of natural frequency and log dec versus speed in order to obtainimproved estimates of the location of the peak unbalance response speeds and their amplification factors. Themethod yielded mode shapes, which provided a good guide to the location and types of unbalances to be used in theunbalance response analysis to excite particular modes. It wasn't until the early 1980’s that programs were availableto calculate the stiffness and damping coefficients of the labyrinths to include in the stability analysis. See 3.1.1 for acomplete historical review of stability analysis.

Today, a complete dynamic picture of the machine can be obtained using advanced bearing/seal codes along withforced response and stability analysis codes.

2.7.2 Undamped Critical Speed Analysis

Serving as the primary analysis technique for many years, the undamped critical speed analysis is still performedtoday for the preliminary estimation of critical speeds and mode shape characteristics. It is deemed preliminarybecause this analysis is based on the bearing oil film stiffness only. It also excludes forces such as unbalance or otheritems that contribute to the machine’s actual dynamic behavior such as the labyrinth seals. However, the undampedcritical speed analysis can be of great value in rapidly assessing the general dynamic behavior of the machine.

The critical speeds and their associated mode shapes are most influenced by the support (bearing and pedestalstructure) stiffness magnitudes, the support locations, and the rotor’s mass and stiffness properties. Aimed at definingthis influence for the specific machine, the undamped critical speed analysis applies a varying amount of stiffness atthe support locations to the rotor model. The critical speeds and mode shapes of the system are then calculated foreach support stiffness level. The primary result is the undamped critical speed map, for which a typical map is shown



in Figure 2-69. This plot typically presents the first four undamped, forward-whirling modes as a function of totalbearing/support stiffness. Both the abscissa (support stiffness) and ordinate (frequency of the critical speed) axes ofthe critical speed map are generally log scales.

The important relationship that governs the overall characteristics of the critical speed map is the one between theshaft’s stiffness and the support stiffness. Illustrated in Figure 2-70 and Figure 2-71, the natural frequencies with zerosupport stiffness, or free-free condition, are governed by the shaft stiffness, with the first two natural frequencies beingzero due to rigid body movement. When the stiffness of the bearings are small relative to the shaft’s bending stiffness,the stiffness of the bearings and the rotor mass and mass moments of inertia govern the frequency of the unit’s lowestcritical speed. The mode shapes affirm this behavior with little bending of the shaft. Conversely, when the bearingsare much stiffer than the bending stiffness of the shaft, the frequency of the unit’s undamped critical speeds willprincipally be governed by the mass and mass moments of inertia and bending stiffness of the rotor (refer to Equation1-8). Under this condition, the bearing supports become node points and a high degree of shaft bending occurs in themode shape. This mode shape behavior is shown in Figure 2-70 and Figure 2-71.

The undamped critical speed map summarizes these relationships for the particular rotor being analyzed. Focusingour attention on the first two critical speed lines as displayed in Figure 2-69 and Figure 2-71, there lies a region of

Figure 2-69—Undamped Critical Speed Map

Low Pressure Charge Gas ServiceBetween Bearing CompressorRotor weight: 14,650 lbs

Bearing span: 151.5 in.Rotor midspan diameter: 17 in.

5 Pad tilting pad bearingsLoad on padL/D = 0.79

104

103

102104 105 106

Support Stiffness (lbf/in.)

Crt

ca S

peed

(cpm

)

107108

4th Critical speed

3rd Critical speed

2nd Critical speed

1st Critical speed

Operating speed range

Kxx

Kyy



Figure 2-70—Mode Shape Examples for Soft and Stiff Bearings Relative to Shaft

Figure 2-71—Typical Regions within an Undamped Critical Speed Map for a Between-Bearing Machine

First mode

Second mode

Third mode

Stiff bearings/flexible rotorBearing stiffness >> shaft stiffness

Soft bearings/stiff rotorBearing stiffness << shaft stiffness

10 2

10 3

10 4

10 5

10 410 3 10 5

Support stiffness (N/mm)

Support stiffness (lbf/in.)

10 6

10 4 10 5 10 6 10 7

Crt

ca s

peed

(cpm

)

Flexible rotor modes

Operatingspeed

Range of bearing stiffnessStiff rotor modes

Low a.f. High a.f.

KXX KYY



bearing stiffness where the slope of these two lines is positive and approximately constant. This area is called the“stiff rotor part of the critical speed map” because the bending stiffness of the rotor is appreciably greater or the sameorder of magnitude as the bearing stiffness. On the right hand side of the undamped critical speed maps lies an areawhere the critical speeds do not change with increasing bearing stiffness. Referred to as the stiff bearing part of thecritical speed map, the curves have reached asymptotic values because the shaft stiffness is dominating the systemstiffness (refer to Equation 1-8). The asymptotic values of the critical speed lines are often referred to as the rigidbearing critical speeds. Since both are governed by the shaft properties, the “rigid bearing criticals” and the “free-freecriticals” are sometimes measured experimentally to verify and improve the mathematical model of the rotor [11,12].

With the critical speed map defined, the final step in the analysis is to define what the actual support characteristicsare in order to estimate the critical speeds. Using the results from bearing and support analysis techniques, speed-dependent total bearing/support principal stiffnesses (Kxx, Kyy) are cross-plotted on the map like that displayed inFigure 2-69. Speeds where the support coefficient curves intersect the critical speed curves are the potential criticalspeeds of the system. For highly asymmetric bearings, like the one shown in this example, these intersections maybe at substantially different speeds for the same mode indicating a critical speed associated with the horizontaldirection and one for the vertical direction, i.e. a “split critical.” Note that, in beam-type machines (no overhung wheelsor stages), the difference in bearing static loads often does not vary significantly from end to end. Consequently, thecalculated bearing coefficients often are nearly equal on both ends of the unit. For this reason, only one set of abearing’s coefficients typically appears on a critical speed map. For machines where the bearings’ static loads differsubstantially, the undamped critical speed map and the overall undamped analysis are more difficult to interpret. Seethe later sections of this tutorial for how to handle specific machines which fall into this category. The discussions herefocus on between bearing machines.

Cross-plotting the calculated bearing coefficients on undamped critical speed maps allows one to infer the generaldamped unbalance response characteristics of a rotor-bearing system. If the bearing stiffness intersects a criticalspeed line in the stiff rotor part of the undamped critical speed map, then the amplification factor associated with thecritical will typically be less than 8, and the rotor’s response to unbalance during operation near the critical speed willbe well damped. For the compressor example map shown in Figure 2-69, the horizontal stiffness intersects in the stiffrotor part of the critical speed map. If, however, the bearing/support stiffnesses intersect a critical speed line near theflat part of the critical speed line, the amplification factor associated with the critical speed will typically be greater than8. The rotor’s response to unbalance will be highly amplified during operation near the critical speed. Referring to ourexample map in Figure 2-69, this compressor would be expected to have a large amplification factor in the verticaldirection.

The relationship between the undamped critical speed map and the results of other lateral dynamics analysis, suchas the damped unbalance response analysis, may be better understood if the undamped mode shapes associatedwith the first few undamped critical speeds are examined for the soft and stiff bearing cases (see Figure 2-70 andFigure 2-71). Note that, in the case of soft bearings (relative to shaft bending stiffness), shaft deflections are smallrelative to the bearing deflections. The damping generated by the bearings will be used to attenuate rotor vibrationscaused by potential rotor exciting forces such as unbalance. On the other hand, when the bearings are stiff relative tothe shaft bending stiffness, the shaft deflections are large relative to the bearing deflections. In this case, even if thebearing damping coefficients are large, the damping forces provided by the bearings will be small because the rotormotion at the bearings is small. Thus, rotor vibrations caused by unbalance and other forces will be highly amplified atcritical speeds if the bearings are much stiffer than the rotor bending stiffness.

Mode shapes associated with the undamped lateral natural frequencies should be calculated as part of theundamped critical speed analysis and presented as shown in Figure 2-72. This figure shows the typical mode shapesfor a between bearing rotor. Other machine types will have fundamentally different mode shapes and the readershould consult the machine specific consideration sections which follow. Mode shapes are usually calculated usingbearing principle stiffnesses which bracket the predicted bearing stiffness range. These plots display the rotor’s



Figure 2-72—Typical Undamped Mode Shapes for a Between Bearing Machine with Different Values of Support Stiffness

Charge gas compressor

Charge gas compressor

Bearing stiffness = 1.0e6 lb/in.

Bearing stiffness =5.0e6 lb/in.

Critical speeds (cpm):Mode 1 = 1475.29Mode 2 = 3654.37Mode 3 = 6064.07Mode 4 = 8609.70

Critical speeds (cpm):Mode 1 = 1803.37Mode 2 = 5519.65Mode 3 = 7922.24Mode 4 = 11701.10

Re

atve

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ent (

Dm

)R

eat

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1.2

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0.2

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-0.2

-0.4

-0.6

-0.8

-1

-1.2

1.2

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

-1.2

0 50 100 150 200Length (in.)

0 50 100 150 200Length (in.)

22 2 2 2

2

2

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3

3

3

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3 3 3 3 3 3 3 33 3 3

3

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g =

5.0e

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11

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11

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44

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2

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normalized free (unforced) rotor deflections associated with the lateral undamped natural frequencies. Theundamped mode shapes are useful for the following reasons:

a) The undamped mode shapes are planar or two dimensional; undamped mode shapes do not possess thecomplex three dimensional bending experienced by the rotor due the presence of damping.

b) The undamped mode shapes provide an approximate indication of the relative displacements that the shaftundergoes when the rotor operates in the vicinity of the associated critical speed.

c) An undamped mode shape gives an indication of what unbalance distribution will be necessary to excite theassociated critical speed. This is information is vital for determining the unbalance locations that are needed forthe damped unbalance response analysis where unbalance must be specified at certain locations along the rotor.

d) Undamped mode shapes can be used to determine unbalance response and stability through modal analysistechniques. Using the undamped mode shapes, the rotor system can be simplified (known as model reduction [8])allowing for quicker computations of response and stability. When these modal analysis techniques are employed,most often the rotor’s undamped free-free rigid and bending mode shapes are utilized where no bearings orsupports are present.

2.7.3 Damped Unbalanced Response Analysis

With the undamped critical speed analysis providing some knowledge of the critical speeds and their sensitivity, itdoes not provide any information regarding the actual vibrational amplitudes expected. This is accomplished byconducting an unbalance response analysis with damping included. Its objective is to determine whether the machinewill meet the required separation margins and vibration limits. The API standard paragraphs establish the followingunbalance response performance requirements for the machine:

a) adequate separation margin between critical speeds and operating speeds;

b) the probe vibration limit is not exceeded within the specified operating speed range even with twice the maximumallowable residual unbalance present;

c) no rubbing will occur even if the rotor’s balance state degrades to the probe vibration limit.

The unbalance response analysis determines whether or not these performance requirements are achieved.

Given the criticality of this analysis, an accurate model is imperative and this accuracy is obviously dependent uponthe level of detail incorporated. The model must incorporate the speed/frequency dependence of the bearings’, seals’and supports’ dynamic properties for the machine’s entire speed range, including trip speed. Furthermore, asdiscussed earlier, these properties are highly dependent on clearances, loads, and other parameters likely to varythroughout a machine’s life span. The analysis must cover the wide spectrum of these possible component variationsin order to bracket the expected range of critical speeds and vibration amplitudes.

With the machine’s governing dynamics defined by the mass, stiffness and damping properties of the components inthe model, the rotor unbalance forces must also be defined to determine the predicted vibrational amplitudes. Theresponse at any location depends on both the amount of unbalance as well as its axial and phase distribution alongthe rotor. The amount of unbalance to be applied analytically is prescribed by the API standard paragraphs. Thisamount is based on the unbalance allowable under API following final balancing, with the design intent being that therotor should not exceed probe vibration limits with twice this level of unbalance present. The distribution of thisunbalance becomes just as important as its amount. This is due to the fact that to excite a natural frequency of anysystem, the forces must not be at node points of the natural frequency’s mode shape. For example, unbalance at themidspan of the compressor in Figure 2-72 would excite the first mode, but would not excite the second modebecause the midspan is a node point for the second mode. Examining the second mode shape, it will be excited byany out-of-phase unbalances at opposite ends of the rotor. Likewise, to excite this example compressor’s third mode,



Figure 2-72 indicates that any unbalance present at the rotor’s left end would accomplish this. For most betweenbearing machines, unbalance at the shaft ends where coupling hubs and thrust collars typically exist are effective inexciting the third mode, a mode usually having high deflections at the shaft ends.

The actual unbalance distribution along a rotor following balancing is unknown. Therefore, to ensure that the vibrationlimits are not exceeded and the critical speeds are accurately identified, different distributions must be applied whichmost adversely excite the critical speeds of concern. While the API standard paragraphs prescribe some appropriategeneric distributions, the mode shapes from the undamped critical speed analysis can provide better informationtoward the selection of unbalance distributions for a specific machine.

Much of the unbalance response analysis performed today is based on the landmark work of Lund and Orcutt [7] whoprovided the first comprehensive algorithm for conducting this type of analysis. While the analysis techniques haveremained essentially the same, significant advancements have been made in modeling various components such asbearings and seals. These advancements have helped to improve the overall accuracy of the predictions. Eventhough an accurate model is vital, several other factors must also be considered when performing the analysis. Theseinclude the following:

a) Probe orientation and mounting: For accurate comparison with test data, the predicted response at probelocations should account for the probe angular orientation. Furthermore, when probes are mounted on flexiblepedestals, it may be necessary to calculate the absolute shaft displacement as well as its displacement relative tothe flexibly mounted probes.

b) Speed step: The response is calculated at specified speed increments. Too large of a speed increment can give acritical speed the appearance of being well damped with low peak response and low amplification factor.

As required by the API standard paragraphs, the main products of the unbalance response analysis are the Boderesponse plots, critical speed locations including their amplification factors and separation margins, and rotordeflection shapes at these critical speeds. The Bode plots and rotor deflection shapes are the primary outputs used todetermine whether the machine meets the unbalance response performance requirements.

Displaying the calculated vibration amplitude and phase resulting from a particular unbalance distribution as afunction of operating speed, Bode plots are normally provided only for probe locations. Each plot should clearlyindicate the identifying parameters used to create the plot, i.e. response location, unbalance distribution, bearingcondition, etc.

Sample Bode plots are shown in Figure 2-73, displaying the X and Y response at a probe location and its adjacentjournal bearing of the example compressor. For this example, the level of unbalance is the API allowable residualunbalance applied at the rotor’s midspan, in a static unbalance configuration. (Note: The applied unbalance requiredby the Standard Paragraphs is twice the allowable unbalance or 8W/N.) This distribution was chosen to excite the firstcritical speed as suggested by the undamped mode shapes in Figure 2-72. Comparing the probe to the bearingresponse, it is clear that the probes “see” more vibration than the bearing and that the unbalance distribution hasindeed excited some sort of resonance. Examining the bearing Bode plot, it appears that multiple resonances arepresent at 1600 rpm, 1850 rpm, and 1900 rpm. This situation highlights the difficulties in interpreting Bode plots thatsometimes occur. Since the undamped critical speed map (Figure 2-69) indicated very asymmetric load on padbearings, a split critical speed can be expected.

To verify whether these three peaks are actually a split critical, it is often helpful to analytically rotate the probes from45° and 135° to observe the true horizontal and vertical response. Figure 2-74 displays the magnitude Bode plots forthe true horizontal and vertical (virtual probes at 0° and 90°) response. A split critical is clearly present with thehorizontal direction experiencing a well damped resonance at 1700 rpm and then peaking again as part of the poorlydamped critical at 1850 rpm which is predominantly in the vertical direction. Thus, the peaks at 1850 rpm and 1900rpm are the same resonance. This figure illustrates the influence that probe orientation has on the presented results.One must always keep in mind that the X and Y response are spatially fixed while observing an orbit that candramatically change in shape from circular to very elliptical.



Cha

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p

Res

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e Lo

cato

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tato

n 10

Pro

bes

X-P

robe

Ang

le =

45

degr

ees

Y-P

robe

Ang

le =

135

deg

rees

Unb

alan

ce D

strb

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Sta

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29 M

dspa

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45 o

z-n

at 0

00°

017

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015

012

5

01

007

5

005

002

5 0

Displacement Magnitude (Mils 0p)

010

0020

0030

00S

peed

(rpm

)

1

900

00A

mp y

= 1

378

SM

A =

50

6S

MR =

15

6

1

850

00A

mp x

= 9

04S

MA =

51

9S

MR =

14

75

4000

5000

6000

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135 90 45 0

-45

-90

-135

-180

Phase Angle (degrees)

010

0020

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peed

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)40

0050

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3850 rpm

4704 rpm

X Y

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Bea

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Pro

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= 1

35 d

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peed

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)

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160018

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4000

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4704 rpm

X Y

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0020

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peed

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)40

0050

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Fig

ure

2-73

—Ty

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lot

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Sp

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Figure 2-74—Example Compressor with Probes Rotated to True Horizontal and Vertical

Charge Gas CompressorBearings: Min. Clearance, Max. Preload, Min. Oil Supply Temp.

Response LocationStation 10: ProbesX Probe Angle = 0 degreesY Probe Angle = 90 degrees

Unbalance DistributionStation 29: Midspan12.45 oz in. at 0.00°

0.175

0.15

0.125

0.1

0.075

0.05

0.025

0

Dsp

acem

ent M

agn

tude

(Ms

0-p)

0 1000 2000 3000Speed (rpm)

1850.00Ampy = 15.76SMA = 51.9SMR = 15.8

1700.00Ampx = 2.94SMA = 55.8SMR = 5.2

1900.00Ampx = 5.45SMA = 50.6SMR = 12.7

4000 5000 6000

3850

rpm

4704

rpm

X

Y

Charge Gas CompressorBearings: Min. Clearance, Max. Preload, Min. Oil Supply Temp.

Response LocationStation 11: Bearing CenterlineX Probe Angle = 0 degreesY Probe Angle = 90 degrees

Unbalance DistributionStation 29: Midspan12.45 oz in. at 0.00°

0.175

0.15

0.125

0.1

0.075

0.05

0.025

0

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0-p)

0 1000 2000 3000Speed (rpm)

1700

1850

1900

4000 5000 6000

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4704

rpm

X

Y



The various unbalance distributions are meant to excite the critical speeds of concern for their accuratedetermination. For the example compressor, the midspan distribution excites a split critical speed with the highlyamplified critical speed at 1900 rpm being of primary concern. Relative to the minimum operating speed, this criticalhas a separation margin of roughly 50 %. To determine whether this margin is adequate, the amplification factor mustbe calculated as outlined in Figure 2-75.

The amplification factor is a simplified measure of the amount of damping available to a natural frequency. Whenconsidering the amplification factor for use in determining the separation margin, several considerations must be keptin mind. First, not only will the amplification factors vary with bearing clearance and other variations, it will also varydepending on the probe response location and the unbalance distribution. For a given unbalance distribution andmodel, an amplification factor calculated at one end of the machine may be different than the one calculated based onthe other end. Second, the amplification factor should be determined based on the individual probe responses. Usingthe major axis to calculate the amplification factor can result in liberal values (low amplification factor, high damping),especially when significant support asymmetry exists creating split critical speeds.

For this particular machine, the highly amplified first critical meets the separation margin requirements, yet, extensiveshop balancing effort was made to help ensure that this critical speed was not excited in actual operation. Asemphasized earlier, to ensure overall acceptability of the design, all of the calculated output must be duplicated forexpected changes in components’ operation, i.e. bearing clearance variation, etc.

Continuing to examine this midspan unbalance distribution, the analysis must next address whether the vibrationamplitude design objectives are achieved. First, one must verify that, within the operating speed range Nma to Nmc,the vibration limit at the probes will not be exceeded when twice the allowable residual unbalance is present. Thisquestion can easily be answered by utilizing the fact that this unbalance response analysis is linear. Recalling that theunbalance applied at the midspan of the example compressor was only the API allowable, the response with twicethe allowable unbalance is simply twice the response already calculated. More specifically, taking the maximum probeamplitude within the operating speed range from Figure 2-73:

(2-27)

Therefore, with twice the allowable unbalance (2 12.45 oz-in.) at the midspan, the maximum response at the probesin the operating speed range is 0.152 mils p-p, well below the vibration limit Lv of 1.0 mils p-p. One should notice thatthe probe vibration limit is based upon the maximum continuous speed. A proper analysis would perform a similarverification at the other probe axial locations.

The second vibration limit requirement ensures that rubbing at critical clearance locations in the range zero to tripspeed will not occur even if the vibration limit is reached at the bearing probes in the operating speed range Nma toNmc. Once again utilizing the analysis’ linearity, the unbalance level required to cause the probes to reach thevibration limit can be determined. Here, the unbalance sensitivity for the particular unbalance distribution is applied.Using the Bode plot response in the operating speed range (Figure 2-73), the midspan unbalance needed to achievethe vibration limit is calculated accordingly.

(2-28)

Y 1U 0.076 mils p-p=

Y 2U 2 Y 1U 0.152= mils p-p=

Lv Nmc 4704=1.0= mils p-p

S0.076 mils p-p

12.45 oz-in.-------------------------------------- 0.0061

mils p-poz-in.

---------------------= =

Unbalance necessary to reach Lv

Uvibelimit

Lv

S----- 1.0

0.0061----------------- 163.8 oz-in.== =



Figure 2-75—Evaluating Amplification Factors (AFs) from Speed-amplitude Bode Plots



This amount of unbalance is 163.8/12.45 = 13.15 times the allowable residual unbalance. API limits the amount ofunbalance to 12 times the allowable residual unbalance (or a scale factor of 6 twice the allowable residualunbalance) for the consideration of rubbing at close clearance locations. So, in this case 12 12.45 = 149.4 oz-in.would be used. API also requires that at least two times the allowable residual unbalance must be used representinga scale factor of at least one. As previously stated, a similar calculation would need to be performed at the other probelocation(s) to determine the minimum amount of unbalance necessary to reach the vibration limit within the operatingrange. (Note: In practice the probes are not considered separately. All of the machine’s probes are used to determinethe probe with the highest amplitude in the speed range.) The unbalance sensitivities determined are only applicablefor a particular unbalance distribution and speed. Therefore, the sensitivities would need to be calculated for eachunbalance distribution. As a side note, the unbalance sensitivities are equivalent to influence coefficients measuredduring balancing procedures.

With this calculated level of unbalance applied, the response at all the critical clearance locations must be determinedfor the individual speeds where rubbing could occur during startup, shutdown or overspeed trip. The results aretypically displayed as a table or by using a rotor deflection plot at the speeds of concern, such as the first criticalspeed and maximum continuous speed.

For the example compressor, Figure 2-76 presents the rotor deflection plot with the rotor at 1850 rpm in both two andthree-dimensional views. As with a Bode plot, the identifying parameters to create the plot should be shown. Thethree-dimensional plot illustrates the orbital response at the individual stations across the entire rotor. As concludedearlier, this critical speed at 1850 rpm is dominated by response in the vertical direction and Figure 2-76 reaffirms this.Here, one can also see that the response at the bearings is not indicative of the rest of the rotor. In fact, relative to therest of the rotor, the bearings are close to being node points having almost no amplitude, causing their damping to beineffective, and resulting in the highly amplified critical speed seen in the Bode plot. Comparing these levels to theavailable clearances along the rotor, the designer can determine whether this unbalance distribution causes vibrationamplitudes to exceed the 75 % clearance limit established by the API standard paragraphs.

This entire process is repeated for each unbalance distribution as well as for the expected changes in thecomponents during the operational life of the machine, such as bearing clearances or load variation, oil sealpressures, etc. When investigating such variations, it is important to maintain the same unbalance distributions foraccurate comparison.

Poor unbalance response performance can be exhibited in many different ways. A highly amplified first critical maycause significant damage to tight clearance areas during startup or shutdown. Increased bearing clearance orvariations in oil supply temperature may reduce a separation margin. The machine may be prone to fouling problemsat locations that have high unbalance sensitivities with respect to the probes. While each machine design is unique,solving poor unbalance response performance, typically, involves optimizing the rotor characteristics with its supportstructure characteristics. Introducing more damping is not always the solution. Section 2.5.5 discusses how journalbearing design can be used to improve response performance. References [13,14] are also important fundamentalpapers examining this rotordynamic design problem.

2.7.4 Stability Analysis

Current technology identifies the damped eigenvalue, evaluated at the rotor’s operating speed, as the principalmeasure of rotor stability. Each damped eigenvalue is a complex number of the form s = p ± iωd where p is thedamping exponent, ωd is the frequency of oscillation, and i = square root (–1). The effect of the sign of the dampingexponent on the motion of the rotor is presented in Figure 2-77 and Figure 2-78. As previously noted, if the dampingexponent of an eigenvalue is negative, then the rotor vibrations associated with this mode will be stable (envelope ofvibrations decreases with time). If the damping exponent of an eigenvalue is positive, then the rotor vibrationsassociated with the mode will be unstable (envelope of vibrations increases with time) and a frequency componentmatching ωd will be expected to appear in the machine’s response spectrum.



Figure 2-76—Rotor Response Shape Plots in 2D and 3D Form

Speed = 1850 rpm

X X X X X X X X X X X X X X X XX X

Y

Y

Y

Y

Y

Y

Y

YY Y Y Y

Y

Y

Y

Y

Y

Speed = 1850 rpm

Length (in.)

Length (in.)

-20

50

100

150

200

Unbalance distributionStation 29: midspan163.8 oz-in. @ 0.0°

-10 0 10 20

X amplitude (mils)

-200

-10

0

10

20

Y am

ptu

de (m

s)

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ent (

ms

0-p)

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10

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180

135

90

45

Phas

e (d

egre

es)

-45

-90

-135

-180

Major axisX magnitudeY magnitudeY phase

XY

Charge Gas CompressorBearings: min. clearance, max. preload, min. oil supply temp.

Pro

be

Pro

beB

earn

g C

L

Bea

rng

CL

163.8 oz-in.0.0°

0



Figure 2-77—Motion of an Stable System Undergoing Free Oscillations

Figure 2-78—Motion of an Unstable System Undergoing Free Oscillations

-1.5

-1.0

-0.5

0.0

0.5

1.0

0 1 2 3 4 5 6 7 8 9 10

1.5


s = p + iwdp = damping exponent

wd = frequency of oscillation{

-ept

ept

xxo

= Real (est)

x x o

envelope (p < 0)displacement

Time (seconds)

(Dm

ens

ones

s)

-4.0

-3.0

-2.0

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0.0

1.0

2.0

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10 2 3 4 5 6 7 8 9 10Time (seconds)

Envelope (p > 0)Displacement


s p + iwd

wd Frequency of oscillation p Damping exponent{

ept

-ept

estxxo

Real ( )

x x o(D

men

son

ess)



Although the real part of the complex eigenvalue is the direct result of rotor stability calculations, API specificationsevaluate rotor stability using a derived quantity called “log decrement.” The log decrement, , is calculated as follows:

(2-29)

The log decrement is a measure of how quickly the free vibrations experienced by the rotor system decay. When thelog decrement is positive, the system is stable. Conversely, when the log decrement is negative, the system isunstable. The log decrement has proven to be a useful measure of rotor stability because it is a nondimensionalquantity allowing easy comparison of different machines and may be interpreted using general design rules.

A damped eigenvalue analysis can be used to estimate the location of the critical speeds and their amplificationfactors [10,15,16]. However, an unbalance response analysis cannot determine the stability characteristics of amachine. Some qualitative relationships do exist between the two analyses, namely, a first critical speed with a highamplification factor may suggest a sensitive machine in terms of stability. Crudely, for low log decs, the amplificationfactor is approximately /. However, only a stability analysis can confirm such an estimation.

The API standards lacked a uniform guideline for evaluating stability until the 7th Edition of API 617 was released in2002. Experience with stability problems, improvements in modeling capabilities, and consensus around appropriatemethodologies, however, resulted in the stability requirements defined in the API standard paragraphs which wereincorporated into the 7th Edition of API 617. These requirements are discussed in Section 3 which contains anextensive state of the art tutorial on stability.

2.7.5 References

[1] Rankine, W.J.M., 1869, “On the Centrifugal Force of Rotating Shafts,” The Engineer, London, 27, pp. 249.

[2] Dunkerley, S., 1895, “On the Whirling and Vibration of Shafts,” Proceedings of the Royal Society, London,Series A, 185, pp. 279–360.

[3] Gunter, E.J., 1966, Dynamic Stability of Rotor-Bearing Systems, NASA SP-113.

[4] Jeffcott, H.H., 1919, “The Lateral Vibration of Loaded Shafts in the Neighborhood of a Whirling Speed—TheEffect of Want of Balance,” Philosophical Magazine, Series 6, 37, pp. 304–314.

[5] Stodola, A., 1910, Dampf und Gasturbinen, Springer, Berlin, 1910.

[6] Myklestad, N.O., 1944, “A New Method for Calculating Natural Modes of Uncoupled Bending Vibrations ofAirplane Wings and Other Beams,” Journal of Aeronautical Sciences, 11, pp. 153–162.

[7] Prohl, M.A., 1945, “A General Method for Calculating Critical Speeds of Flexible Rotors,” Journal of AppliedMechanics, Trans. ASME, 12(3), pp. A142–A148.

[8] Nelson, H.D., 1994, “Modeling, Analysis and Computation in Rotordynamics: A Historical Perspective,”Proceedings of the IFToMM Fourth International Conference on Rotor Dynamics, Chicago, pp. 171–177.

[9] Lund, J.W. and Orcutt, F.K., 1967 “Calculations and Experiments on the Unbalance Response of a FlexibleRotor,” ASME Journal of Engineering for Industry, pp. 785–796.

[10] Lund, J.W., 1974, “Stability and Damped Critical Speeds of a Flexible Rotor in Fluid-Film Bearings,” ASMEJournal of Engineering for Industry, 96(2), pp. 509–517.

2p–d

-------------=



[11] Vance, J.M., Murphy, B.T., and Tripp, H.A., 1987 “Critical Speeds of Turbomachinery: Computer PredictionsVs. Experimental Measurements,—Part I: The Rotor Mass-Elastic Model,” ASME Journal of Vibration,Acoustics, Stress and Reliability in Design, 109(1), pp. 1–7.

[12] Vázquez, J.A. and Barrett, L.E., 1998 “Comparison Between the Calculated and Measured Free-Free Modesfor a Flexible Rotor,” ASME Paper 98-GT-51.

[13] Barrett, L.E., Gunter, E.J., and Allaire, P.E., 1978 “Optimum Bearing and Support Damping for UnbalanceResponse and Stability of Rotating Machinery,” ASME Journal of Engineering for Power, 100, pp. 89–94.

[14] Nicholas, J.C., 1989 “Operating Turbomachinery On or Near the Second Critical Speed in Accordance withAPI Specifications,” Proceedings of the Eighteenth Turbomachinery Symposium, Texas A&M University, pp.47–54.

[15] Kirk, R. G., 1980, "Stability and Damped Critical Speeds: How to Calculate and Interpret the Results," CAGITechnical Digest, Vol. 12, No. 2, pp. 1–14.

[16] Memmott, E. A., 2003, “Usage of the Lund Rotordynamic Programs in the Analysis of CentrifugalCompressors,” Special Issue: The Contributions of Jørgen W. Lund to Rotor Dynamics, ASME Journal ofVibration and Acoustics, Vol. 125, No. 4, pp. 500–506.

2.8 Machinery Specific Considerations

2.8.1 Steam Turbines


The rotordynamic characteristics of steam turbines may require special consideration of the rotor construction,bearing support stiffness, partial admission forces, and thermal effects.

2.8.1.2 Rotor Construction

Steam turbine shafting is typically manufactured from low alloy steel forgings. Rotors consist of a shaft and a series ofdisks, which may be either integral parts of the shaft (Figure 2-79) or shrunk fit onto the shaft (Figure 2-80). Theeffective stiffness diameter at the wheel stages can require special consideration. High temperature applicationsrequire further material consideration to resist corrosion rate and degradation mechanisms, such as creep. Mostmanufacturers further specify that solid rotor shaft forgings are to be heat indicated. Heat indication is a test intendedto demonstrate that the forging is not susceptible to deflections caused by thermally induced stresses fromdiscontinuities, changes in grain orientation, or other section flaws. Such rotor deflection would result in rotor bowduring operation, which would be observed as changing unbalance.

2.8.1.3 Bearing Supports

A typical outline drawing of a steam turbine case is shown in Figure 2-81. The steam end bearing is housed in abearing case that is supported by either a flex plate or a sliding support to permit free casing and rotor axial thermalexpansion. The exhaust end bearing case is fitted to the exhaust casing, which may be arranged with horizontalplates and gussets for added support stiffness. These bearing support structures are attached to the baseplate. Thedynamic characteristics of the bearing support structure at each end of the machine are largely a function of design,and could have an appreciable effect on rotor response [1]. These support effects are discussed in Section 2.4. Thesupport stiffness coefficients can be either estimated based on experience [2], measured [3], or calculated by finiteelement analysis [4]. The amount of support damping is based on the manufacturer’s experience with similar



Figure 2-79—Integral Shafts Made from a Single Forging

Figure 2-80—Built-up Rotor with Shrunk-on Disks



equipment, or it is measured. The amount of support damping will typically be in the range of 1 % to 10 % of thecritical damping Ccr which is given by the formula:

2-30

2.8.1.4 Partial Admission Forces

Partial admission forces are a factor since they affect rotor position and bearing loading causing changes in stiffnessand damping characteristics. An analysis methodology and example are presented by Caruso et al. [1].

It has long been recognized that the entire control valve opening sequence and the effect of partial admissiondiaphragm stages must be considered in a rotor response analysis. This is especially so in cases where partialadmission forces are large relative to the rotor weight. The resultant effect can be one in which the rotor is loaded intoa sector of the bearing where the dynamic characteristics are significantly different from what they would be due togravity load alone (Figure 2-82). Consequently, changes in turbine load can yield significantly different operatingvibration amplitudes [1]. There are two sources of partial admission force. The primary source is the inlet andextraction control stages (if included), which can have a wide range of admission arcs and hence loading conditionsdependent on the operating point. Another source is partial admission diaphragm stages that are occasionally usedfor flow-path efficiency considerations.

The resultant forces imposed on the rotor are of two types: a tangential component derived from the stage torquereaction, and an axial thrust from the pressure drop across the blades. The axial thrust is orientated at the centroid ofthe admission arc and can be resolved into radial force couples at the bearings. The axial thrust forces are usuallysmall compared to the tangential torque reactions and are typically neglected. A possible situation where they might

Figure 2-81—Steam Turbine Outline

Steam endbearing

Exhaust endbearing

1011121314

1516

17

18

19

20

21

22

10

23

24

25

26 27

9

8

7

65

4

3

2

1

Ccr 2 MK 2Mn= =



need to be accounted for, however, would be at the inlet control stage with only one or two valves open yielding asmall admission arc. Since the pressure drop across the control stage is typically quite high, the axial thrust on thefirst row of blades would also be high and the location of the centroid at a large radius. As the admission arcincreases, the radius to the centroid reduces, as does the resultant bearing loading. Figure 2-83 shows different arcsegments that provide steam in accordance with the valve opening sequence. A typical force diagram illustrating howto resolve these forces into journal bearing reactions is shown in Figure 2-84. In this figure, stage torque reactionstypical of an inlet and extraction control stage are represented, given by the subscripts ‘i’ and ‘e’, respectively.

2.8.1.5 Thermal Effects

The steam inlet end bearing will tend to run hotter than the steam exhaust end bearing. When modeling the bearings,there may be a need to account for a loss of clearance (from the cold or assembly dimensions) and to account for thejournal thermal boundary conditions of the journal. In addition, temperature can also affect the modulus of elasticity ofthe rotor.

2.8.1.6 Analysis Methodology

The analysis methodology and acceptance criteria for the damped response analysis are outlined in Section 2.7.Special considerations for steam turbines have been discussed in the preceding sections. These include thefollowing:

a) consideration of the effective stiffness diameter at the wheel stages;

b) bearing support effects;

c) external bearing forces, such as from partial arc forces, as appropriate;

d) thermal effects on the bearings and the rotor modulus of elasticity.

Figure 2-82—Typical Resultant Bearing Load Vector Including Partial Admission Steam Forces

Rotor weightResultant load vector

Steam force

Bearin

g pad

Bearing ShellBearing Shell



2.8.1.7 References

[1] Caruso, W. J., Gans, B. E. and Catlow, W. G., 1982, “Application of Recent Rotor Dynamics Developments toMechanical Drive Turbines,” Proceedings of the Eleventh Turbomachinery Symposium, Texas A&MUniversity, pp. 1–17.

[2] Edney, S. L. and Lucas, G. M., 2000, “Designing High Performance Steam Turbines with Rotordynamics as aPrime Consideration,” Proceedings of the Twenty-Ninth Turbomachinery Symposium, Texas A&M University,pp. 205–224.


[4] Stephenson, R.W. and Rouch, K.E., 1992, “Generating Matrices of the Foundation Structure of a RotorSystem from Test Data,” Journal of Sound and Vibration, Vol. 154, pp. 467–484.

[5] Nicholas, J. C. and Wygant, K. D., 1995, “Tilting Pad Journal Bearing Pivot Design for High LoadApplications”, Proceedings of the Twenty Fourth Turbomachinery Symposium, Texas A&M University, pp. 33–47.

Figure 2-83—Sequence of Admission for Different Turbines’ Partial Arc Segments



Nomenclature

Ccr is the critical damping (lbf-sec/in.);

Fex is the exhaust end bearing partial admission force reaction X direction (lbf);

Fey is the exhaust end bearing partial admission force reaction Y direction (lbf);

Fix is the inlet end bearing partial admission force reaction X direction (lbf);

Fiy is the inlet end bearing partial admission force reaction Y direction (lbf);

L is the bearing span (in.);

Le is the length from inlet end bearing to extraction control stage (in.);

Li is the length from inlet end bearing to inlet control stage (in.);

Oe is the origin extraction control stage;

Oi is the origin inlet control stage;

Ve is the extraction control stage partial admission force vector (lbf);

Figure 2-84—Resolution of Partial Admission Forces into Journal Bearing Reactions

LLe

Li

Fiy

FixOi

Vi

Ve

e

i

Fey

Oe Fex



Vi is the inlet control stage partial admission force vector (lbf);

θe is the angle from X direction to extraction control stage force vector (deg);

θi is the angle from X direction to inlet control stage force vector (deg).

2.8.2 Electric Motors and Generators


The rotordynamic characteristics of electric motors and generators may require some special consideration of therotor construction, bearing support stiffness, and electromagnetic effects.

2.8.2.2 Types of Motor Rotors

Most alternating current machines used for mechanical drive in the petroleum industry can be broadly classified aseither synchronous or asynchronous (so-called induction). The stator construction for these motor types is generallysimilar; however, rotor construction can differ significantly.

2.8.2.2.1 Induction Motor Rotors

Rotors of electric machines are almost always fabricated. The center core region of induction rotors, such as thatshown in Figure 2-85, is composed of electrical conductors, electrical steel sheets, magnetic materials, and/or otherstructural members. The multitude of parts is fitted together with different types of joining methods. In addition, therotor might be impregnated with resin or other insulating material penetrating effectively to small gaps. One of theapplied methods is called “vacuum pressure impregnation” (VPI). The stiffness of the shaft is usually well known, butthe stiffening effect of this core region is one of the main uncertainties in the rotordynamic modeling of electricmachines.

2.8.2.2.2 Synchronous Motor Rotors

The synchronous motor rotor requires an exciter to produce excitation current resulting in a magnetic field to generatemechanical torque. This exciter is either mounted on the main rotor, or is a separately coupled unit with anotherbearing set. The complete assembly of synchronous motor along with exciter system may have a three bearingsystem.

The majority of alternating current synchronous motors used in petroleum industry either have salient pole or massivesteel rotor construction. In a salient pole rotor, the solid steel shaft is machined to create pole legs, where-in copperwindings are wrapped along with an iron pole shoe to create an electric pole. Examples of salient pole rotorassemblies are given in Figure 2-86 and Figure 2-87. The rotor assembly also consists of many other parts to securethe copper windings from flying off due to centrifugal forces generated during rotation. Salient pole rotor constructionis more practical for four and higher pole motors. For a massive pole rotor construction, such as the example shownin Figure 2-88, a solid steel shaft is milled to create longitudinal slots to insert electrical copper windings. Massive polerotor construction is usually used for two poles synchronous motors and for high-speed applications.

2.8.2.3 Electromechanical Interaction

Electrical machines typically consist of a rotor housed within a stator separated by an annular air gap as shown inFigure 2-89. Electromagnetic fields in the air gap separation produce torque on the rotor, and also influence the lateraldynamics of the rotor as well due to attractive magnetic forces [1,2]. This effect is traditionally referred to aselectromagnetically induced negative stiffness. In addition, there will be unbalanced magnetic pull (UMP) since therotor is never perfectly symmetric with respect to its properties, manufacture, or motion. These interaction effects are



most significant in high power, cage induction motors with a flexible shaft, large bearing span and small air-gaplength. The traditional approach to calculate the value of negative spring stiffness is to apply the following formula:

(2-31)

where d and l are the outer diameter and the overall length of rotor core, respectively, is the flux-density amplitudeof the fundamental component, is the radial air-gap distance, the constant p = 0.5 for two-pole machines and p =1 for other machines, and the constant sw ≤ 1 describing the effects of a) electromagnetic damping of the rotor cage,and b) parallel paths in the stator windings in synchronous whirling motion [4-6]. These electromagnetic forces are

Figure 2-85—Induction Motor Rotor

Figure 2-86—Salient Pole Synchronous Motor Rotor

ke p sw dlB̂p2

40--------------=

B̂p



distributed around the rotor core as shown in Figure 2-89. The effect of electromechanical interaction on criticalspeeds varies depending on other design choices. In addition, the spring effect is dependent on operationalparameters and whirling frequency. The machine vendors can usually provide the data needed for evaluation ofelectromagnetic effects.

This rotordynamic model demonstrates electromagnetic forces acting on the rotor due to uneven air gap generatedduring manufacturing as well as stack up tolerances for different components. Electromagnetic forces, if ignored ornot accounted accurately during rotordynamic calculations, can lead to significant error in predicted values incomparison to measurements.

2.8.2.4 Thermal Bow

Some small amount of change in vibration will always occur because of the movement of the rotor components whenthe rotor is heated from cold to the running condition [3]. This phenomenon is referred as rotor thermal sensitivity orthermal unbalance or thermal bow. This thermal bow can be caused by an uneven temperature distributioncircumferentially around the rotor, or by axial forces which are not distributed uniformly in the circumferential direction.The main reason for the later cause is the large difference in coefficient of thermal expansion between the copper (oraluminum) winding and the steel components of the rotor.

The rotordynamic effects of thermal bow are similar to shaft or rotor bow. However, the thermal bow depends on theloading condition of the machines and thus the rotordynamic behavior is a function of operation condition and recentoperation history.

Figure 2-87—Salient Pole Synchronous Motor Rotor



Figure 2-88—Massive Pole Synchronous Motor Rotor

Figure 2-89—Core Magnetic Field Distribution and Cross Section Within a Four Pole Induction Motor [2]



2.8.2.5 Modeling

2.8.2.5.1 Modeling of Induction Motors

Rotordynamic modeling of induction motors has only a few machinery specific considerations [5-8]. The mostimportant is related to the modeling of core region. When developing the mass-elastic model for an induction motor,the mass and stiffness of the core must be considered. As noted in Section 2.8.2.2.1, the stiffness of the shaft isusually well known, but the stiffening effect of this core region is one of the main uncertainties in the rotordynamicmodeling of electric machines. The purely analytical methods cannot usually predict this stiffening effect accurately.Improper modeling of the rotor core stiffness can lead to significant error in critical speed prediction. The stiffeningeffect can be estimated by experimental methods, and the motor vendors can usually provide this information. Fromexperimental modal analysis and rotor dynamics calculations, rotor core stiffness is usually 3 % to 6 % of the Young’smodulus of steel, keeping the same inertia and mass [9-11].

2.8.2.5.2 Modeling of Synchronous Motors and Generators

When developing the mass-elastic model for a synchronous electric motor or generator, the mass of the windingsmust be considered. However, the stiffness contribution of the windings is negligible and should not be considered.Since the rotor cross-section will usually be slotted for the windings, determining the equivalent stiffness diameter ofthe rotor center section can be challenging. One approximate method is to determine the equivalent mass diameterand use that as the stiffness diameter. Often, the motor or generator manufacturer can provide a total rotor weight.This is helpful in determining the equivalent mass diameter. The mass and inertia of items attached to the motor rotor,such as coupling(s), fan(s) and the exciter should be included in the rotor model. Finally, the motor box supportstiffness must be included for accurate critical speed prediction (see Section 2.4 for more details on flexible supportmodeling).

2.8.2.5.3 Asymmetrical Rotor Modeling

In the case of two-pole motors and generators, which have an asymmetrical geometry, an equivalent symmetricmodel is normally sufficient for the purpose of rotordynamic modeling. A precise analysis can only be performed bytime-transient method making use of the exact rotor geometry.

2.8.2.5.4 Rotor Model Validation

The rotor model can be validated by measuring the natural modes and frequencies of a vertically or horizontally hungrotor. This can be particularly beneficial for induction motor designs, which have uncertainty relating to the corestiffness. However, the centrifugal loads and thermal effects may change the rotor characteristics under operatingconditions.

2.8.2.6 Support Stiffness

The bearing support stiffness of electrical machines is an important parameter for rotordynamic calculations. Usually,the horizontal support stiffness is lower than the vertical stiffness. The motor manufacturer can usually provide thestiffness values including all the components included to the electrical machine, i.e. bearing housing, gable, mountingmembers, etc. The situation is more difficult when the machine is installed on a flexible foundation. In this case, themotor “rigid body” modes are coupled with the rotordynamics of the system.

In the case of vertical machines, the motor is usually mounted on top of the driven equipment. The machinerydynamics can be heavily affected by the cantilevered support structure such that the rotordynamics of these verticalmachines cannot be analyzed by simply applying support stiffness values for the bearings. The lowest natural modeof the system is the so-called reed mode [12]. The frequency of this reed mode is the same order of magnitude as therotational speed of the machine. The structural reed frequency should not be confused with the lateral critical speedof the rotor. The lateral critical speeds are usually much higher than the rotational speed corresponding to this reedfrequency.



2.8.2.7 References

[1] Holopainen, T.P., 2009, “Simple electromagnetic force model for industrial rotordynamic analyses of electricalmachines,” Proceedings of the 22nd Biennial Conference on Mechanical Vibration and Noise, ASME, SanDiego, California, USA, August, 9 pages.

[2] Holopainen, T.P., and Arkkio, A., 2008, “Electromechanical interaction in rotordynamics of electricalmachines—an overview.” Proceedings of the Ninth International Conference on Vibrations in RotatingMachinery, IMechE, Exeter, UK,. Vol. 1, pp. 423–436.

[3] Zawoysky, R.J., and Genovese, W.M., 2001, “Generator rotor thermal sensitivity – theory and experience.”GE Library Reference: GER-3809.

[4] Seinsch, H.-O., 1992, Oberfelderscheinungen in Drehfeldmaschinen, Teubner-Verlag, Stuttgart.

[5] Neto, R.R., Bogh, D.L., and Flammia, M., 2008, “Some experiences on rigid and flexible rotors in inductionmotors driving critical equipment in petroleum and chemical plants.” IEEE Transactions on IndustryApplications, Vol. 44, No. 3, pp. 923–931.

[6] Werner, U., 2008, “A mathematical model for lateral rotor dynamic analysis of soft mounted asynchronousmachines. Journal of Applied Mathematics and Mechanics, Vol. 88, No. 11, pp. 910–924.

[7] Schuisky, W., 1972, “Magnetic pull in electrical machines due to the eccentricity of the rotor,” ElectricalResearch Association Trans., 295, pp. 391–399.

[8] Früchtenicht, J., Jordan, H., Seinsch, H.-O., 1982, “Exzentrizitätsfelder als Ursache von Laufinstabilitäten beiAsynchronmaschinen,“ Archiv für Elektrotechnik 65, pp. 271-281 and pp. 283–292

[9] Singhal, S., 2010, “Selection of Optimized Balancing Planes To Attain Low Vibrations In Large High SpeedInduction Motors”, ASME International Mechanical Engineering Congress and Exposition.

[10] Singhal S., Singh, K. V., and Hyder, A., 2011, “Effect of Laminated Core on Rotor Mode Shape of Large HighSpeed Induction Motor”, Proceedings of the IEEE International Electric Machines and Drives Conference,Niagara Falls, May.

[11] Seo, Y., Lee, J., Lee, S., 2010, “Modeling and Analysis of Rotor with Laminated Core in Electric Machine”,Proceedings of the 8th IFToMM International Conference on Rotor Dynamics, Seoul, Korea, pp. 186–192.

[12] Gaylord, F.D., 1989, “Solving reed frequency vibration problems,” Hydrocarbon Processing, January, pp. 50–54.

2.8.3 Gearboxes

Gearboxes, like that shown in Figure 2-90, have unique characteristics that must be considered when performing arotordynamic study.

Torque is transmitted from one shaft to another through radial forces acting between the gear teeth. These toothforces are directly proportional to the torque. The bearings supporting the shafts have a very significant change inload as a result of these tooth forces (as shown in Figure 2-91). As a result, the stiffness and damping of the bearingsalso have significant changes. Therefore, transmitted torque must be considered in rotordynamics studies ofgearboxes. Usually, the analysis is performed at 10 %, 50 %, and 100 % of full torque, but it is common to alsoperform the calculations at the expected test stand torque.



The high radial loads on gearbox bearings reduce the tendency for bearing instability at normal operating conditions.However, it is important to select the shaft offset and shaft rotation to load the bearings in the most favorable way. Thecombination of the rotor weight and the direction of load from the gear teeth can increase or reduce the bearing loads.

Operating at reduced torque may cause the normally uploaded radial bearing to be loaded down (as shown in Figure2-92). In general, for horizontal offset gearboxes, downloading the gear bearings and uploading the pinion bearings ispreferable. When there is insufficient torque to prevent bearing instability, such as during start up or on a test stand,the use of a stability feature such as a pressure dam or tilt pad bearing is necessary.

Unlike most machines, lateral vibrations and torsional vibrations are related in gearboxes. Torque variations result inradial load variations. Therefore, it is possible for torsional and radial vibrations to interact. If unexpected radialvibrations occur, consider the possibility of a torsional and radial vibration interaction.

Gearbox rotors are usually relatively stiff; therefore, support stiffness can have a significant impact on lateralrotordynamics. Most pinions are designed to operate below their first lateral critical speed with a mode shape that hasthe highest amplitude at the coupling. This can make gearboxes sensitive to coupling unbalance and misalignment.However, most gear rotors operate far enough below the first lateral critical speed to be considered stiff shafts, sospecial balancing procedures are seldom necessary.

Manufacturing tolerances for any gear and pinion teeth lead to minute variations in tooth spacing. The errors result ina slight increase in rotational speed followed by a slight decrease at a first order frequency. This feature, known as“accumulated pitch error,” can result in a substantial excitation in addition to any unbalance forces. Any ellipticalshaping of the gear and/or pinion can lead to second order frequencies. Usually higher orders of accumulated pitcherror are a result of wear caused by continued vibration. Figure 2-93 shows an accumulated pitch error chart from agear tooth checker showing a wear pattern at the tenth order.

Figure 2-90—Gear Set Showing Rotation Directions



Figure 2-91—Gear Force Schematic

Figure 2-92—Gear Load Angles at Partial and Full Load

Turbine Generator

Gear load

Higher Transmitted Power

Resultant load

Weight loadGear load

Lower Transmitted Power

Resultant load

Weight loadLoad load

Gear load

Full Transmitted Power

Resultant load

Weight load



Most analysis is concerned with shaft speed or low order multiples of shaft speed. However, gearboxes are unique inthat the gear teeth enter and leave engagement at a frequency equal to the rotor speed times the number of teeth.This frequency is called the “tooth mesh frequency.” For instance, a pinion with thirty teeth will have a thirtieth ordermeshing frequency. This frequency is higher than normally evaluated during a rotordynamics study. The toothmeshing frequency is not purely sinusoidal and higher order harmonics are typically present. Resonance in gearblanks, casings or support structures at tooth mesh frequency are rare but do occur and can occasionally causeproblems.

Another problem that requires consideration is excitation of higher order modes, such as the 4th mode of the pinion.This can come from high multiples of running speed such as 7x and 8x running speeds [2-4].

It is possible to have an excitation because of the “assembly frequency.” The assembly frequency is the tooth meshfrequency divided by a common prime number. As an example, a gearset with 9 teeth in the pinion and 15 teeth in the

Figure 2-93—Accumulated Pitch Error Chart

Accumulatedpitch error

LH RH

0.001



gear have a common prime number of three. This would result in a third order excitation. If a gearset is ran and thenreassembled with the teeth in a different combination, there is often substantial excitation at the assembly frequency.This example is taken from Winterton [1]. A “hunting tooth design”, required by API, by definition does not have acommon prime number so the assembly frequency is the same as the tooth passing frequency. A hunting tooth isusually supplied in a modern design.

The following formula is a common method used to calculate principal and cross-couple stiffness’ from gear meshwhen conducting a coupled gear-pinion lateral vibration analysis. Analysts do not typically include the gear meshstiffness in lateral vibration analyses.

(2-32a)

(2-32b)

(2-32c)

(2-32d)

(2-33)

where

FW is the net facewidth of gear, mm (in.);

β is the helix angle, deg;

γ = (A α) + B;

α is the normal pressure angle, deg;

A = 1 for downloaded rotors; or

A = –1 for uploaded rotors;

B = 90° for CW rotation looking into coupling end; or

B = 270° for CCW rotation looking into coupling end;

C = 12.057 (1.75);

K is the stiffness, N/m (lbf/in.).

2.8.3.1 References

[1] Winterton, J. G., 1991, “Identification of Gear Generated Spectra,” Orbit Magazine, 2nd Quarter, June, BentlyNevada Corp.

[2] Memmott, E. A., 2005, “Should Pinions of Gear Sets be Designed to Avoid Critical Speeds at Eight TimesRunning Speed?,” Case History, Presented at the Thirty-Fourth Turbomachinery Symposium, Texas A&MUniversity, December.

Kxx K' cos2 =

Kxy K' sin cos =

Kyx Kxy=

Kyy K' sin2 =

K' C FW cos2 106=



[3] Memmott, E. A., 2006, “Case Histories Of High Pinion Vibration When Eight Times Running Speed CoincidesWith The Pinion’s Fourth Natural Frequency,” CMVA, Proceedings of the 24th Machinery Dynamics Seminar,Montreal, October 25–27.

[4] Marin, M., 2009, “Practical Uses of Advanced Rotordynamics Tools to Ensure Trouble Free Operation of aGear Box,” Case History, Presented at the Thirty-Eighth Turbomachinery Symposium, Texas A&M University,September.

2.8.4 Power Turbines and FCC Power Recovery Expanders

The following discussion applies to overhung one or two stage power turbines and Fluid Catalytic Cracking (FCC)power recovery expanders. This equipment is subjected to adverse operating conditions; inlet temperatures canexceed 1200 °F and, in the case of FCC expanders, the working gas can be laced with solid particles or contain highlevels of corrosive compounds. For the purpose of this discussion, it is assumed that fluid film journal bearings, notrolling element bearings, support the rotating element.

2.8.4.1 Rotor Construction and Support Flexibility

Typically, this type of equipment appears to have an extremely stiff rotor, as the first undamped rigid bearingsynchronous forward critical speed of the rotor is generally above the operating speed range. A typical undampedforward synchronous critical speed map for a FCC expander rotor is illustrated in Figure 2-94.

Figure 2-94—FCC Expander Critical Speed Map

Undamped Critical SpeedSingle Stage Overhung FCC PRT Expander

5 Shoe Load-between-pads Tilt Pad Brg s/Kavg, diaphragm coupling

1.0E+05

1.0E+05

1.0E+04

1.0E+04

1.0E+03

1.0E+021.0E+06 1.0E+07 1.0E+08


Crt

ca s

peed

(RP

M)

Operating speed = 4635 RPM

1st Critical speed2nd Critical speed3rd Critical speedKxxKyyDxx = Sqrt (Kxx2 + omega2 x Cxx2)Dyy = Sqrt (Kyy2 + omega2 x Cyy2)



A flexible bearing support structure having stiffness less than 3.5 times the journal bearing stiffness can be typical ofsome expander designs. In such cases, the effect of the bearing support structure should be included in therotordynamic model. A percentage of critical damping is used, which is usually in the range of 1 % to 10 % of criticaldamping. The percentage used is based on the manufacturers experience with similar equipment. Critical dampingCcr is given by the formula:

(2-34)

Because of the flexible bearing support structure, the first critical speed of the rotor-bearing support system could beboth well below the first undamped rigid bearing critical speed of the rotor and below the operating speed range. Thisfirst critical speed is a system mode, principally determined by the mass of the rotor in series with the hot end journalbearing and the bearing support.

Proper modeling of the gyroscopic effects is extremely important in determining the dynamics of the rotor due to thelarge overhung mass and inertia [1]. The locations of the modes are heavily influenced by the diameter of the rotornear the hot end, the journal bearing stiffness, the stiffness of the bearing support, and the mass and polar andtransverse moment of inertia of the overhung mass.

Also because of the large overhung mass on the hot end as compared to the mass of the coupling on the oppositeend, the center-of-gravity will be somewhat close to the disc(s) end bearing, which can create a tendency for the rotorto tip during assembly/disassembly. Designers may attempt to shift the center-of-gravity without the coupling installedinboard between bearings to ensure a downward load on the coupling end bearing. This tends to prevent the rotorfrom tipping during assembly/disassembly. Such features are applied only when practical to all aspects of the designincluding rotordynamic effects.

2.8.4.2 Bearing Considerations

A low resultant load on the coupling end bearing may make that end of the rotor sensitive to unbalance. Coupling endunbalance sensitivity may be reduced by decreasing coupling end bearing assembly clearance, increasing tilting padbearing preload, or using offset tilt pads. If fixed geometry bearings are used, the lightly loaded coupling end bearingmay be susceptible to oil whirl. A case history of a sleeve bearing induced instability for an overhung power turbine issummarized in 3.3.2.4.

Because of sleeve bearing induced instability problems, many manufacturers have installed or retro-fitted stabilizingbearing designs such as pressure dam or multi-pocket sleeve bearings [2], offset half bearings, or tilt pad bearings.This has been done on both the hot and the cold end. In general, such retrofits have eliminated subsynchronousvibrations but have not eliminated misalignment problems that are common to “hot” equipment. Rotors withmisalignment issues and bearings with signs of edge loading can benefit from tilting pad journal bearings with axialmisalignment capability.

2.8.4.3 Thermal Effects

Since inlet gas temperatures to expanders can exceed 1200 °F, thermal management is particularly important. Steaminjection is used as a method for cooling the disk and limiting the amount of heat that travels down the rotor to thebearing journal disk end bearing journal. When modeling the disk end bearing, there may be a need to account for aloss of clearance (from the cold or assembly dimensions) and to account for the journal thermal boundary conditionsof the journal. In addition, temperature can also affect the modulus of elasticity of the rotor.

2.8.4.4 Example and Experience

Figure 2-95 illustrates a cross-section of a FCC power recovery expander. Note that the coupling end bearing is wellsupported in the radial direction by a pedestal box-structure, while the disk end bearing is only supported by a box-likestructure that is cantilevered from the pedestal.

Ccr 2 MK 2Mn= =



For the expander displayed in Figure 2-95, the static stiffness of the coupling end bearing support is greater than3.0 106 lbf/in. while the static stiffness measured for the disk end bearing support is much less than 1.0 106 lbf/in.The low stiffness associated with the disk end bearing support reduces the effectiveness of the disk end journalbearing’s damping capability [3]. This, in turn, reduces the ability of the disk end bearing to suppress rotor vibrationsand to promote stable rotor operation. For this reason, among others, expander designers have increased thediameter of the disk end journal in an effort to reduce disk end journal bearing unit loading and thus, disk end journalbearing stiffness. Also, the bearing diameter on the disc(s) end has been increased in power turbines in order to raisethe first rotor mode above the operating speed range.

Any lateral model of an overhung power turbine or FCC expander must incorporate accurate descriptions of thebearing support’s dynamic properties. Such properties may be calculated or measured, but should always betempered by experience. Figure 2-96 contains a schematic of the resulting rotor-bearing support model of thisequipment. Inclusion of the flexible support data in lateral models can dramatically lower calculated critical speeds [3]and damped natural frequencies for these types of units (see 2.4 and 3.6).

It is not an easy matter to measure or calculate the dynamic stiffness and damping properties of the bearing supportsfor the following reasons.

1) The exhaust casing may influence support properties.

2) Support properties may be influenced by temperature.

Figure 2-95—FCC Expander Cross-Section



Given the unknowns in the support data, any lateral analysis of these overhung machines should incorporatepreviously accumulated experience. Specifically, the calculated critical speeds should be compared to thosemeasured for similar units, if available.

2.8.4.5 Analysis and Methodology

The analysis methodology and acceptance criteria for the damped response analysis are outlined in Section 2.7.Special considerations for expanders have been discussed in the preceding sections. These include the following.

a) Rotor construction. The overhung disk requires consideration of the polar and transverse moments of inertia andmodeling of the gyroscopic effects.

b) The support flexibility under the disk end journal bearing will often require modeling due to low flexibility inducedby the cantilevered support.

c) Bearing considerations. Coupling end bearings are lightly loaded and may have high sensitivity to couplingunbalance.

d) Thermal effects on both the disk end bearing and the rotor modulus of elasticity may need to be considered.

2.8.4.6 References

[1] Green, R. B., 1948, “Gyroscopic Effects of the Critical Speeds of Flexible Rotors,” Journal of AppliedMechanics, December, pp. 369–376.

[2] Nicholas, J. C., 1985, “Stability, Load Capacity, Stiffness and Damping Advantages of the Double PocketJournal Bearing,” ASME Journal of Tribology, 107 (1), pp. 53–58.


Figure 2-96—FCC Expander Rotor-bearing Support Model

Disk bearingcenterline

Coupling bearingcenterline

Kbrg

KbrgCbrg Cbrg

Bearing “Coffin”



2.8.5 Axial Compressors

The rotor mass-elastic characteristics of axial compressors depend on the method of rotor construction and bladeattachment. The four typical methods of rotor constructions are disc-on-shaft shrink fit (similar to most centrifugalcompressors), stacked disk with through tie bolts, drum rotors with studs or tie bolts, and solid rotors. Rotors with tiebolts often require modal testing to tune the model for accuracy. Four typical axial compressor rotor constructionexamples are illustrated in Figure 2-97, Figure 2-98, Figure 2-99, Figure 2-100.

Because of their complex construction features, the modeling of axial compressor rotors demands significantattention. Tie rod and stud torque levels along with stacked disk surface interactions greatly affect the stiffnessproperties of the rotor. In general, axial compressor rotors are more rigid than other process turbomachinery becauseof their large mid-span diameters and their hollow construction. For a solid cylinder and a hollow cylinder of equalcross-sectional area, the hollow cylinder will have higher lateral bending stiffness.

Furthermore, the support characteristics (pedestals) of axial compressors could be of the same order of magnitude asthe fluid film characteristics of the journal bearings in which the rotor is supported. Hence, this should be accountedfor in the rotordynamic analysis as required, Nicholas and Kirk [1] (see 2.4 and 3.6).

The destabilizing aerodynamic cross-coupling stiffness in an axial compressor stage can be estimated from eitherseal calculations or from Alford’s equation which is discussed in detail and given as Equation 3-3 in 3.5.1. Per API617, the stability characteristics of an axial compressor due to aerodynamic cross-coupling forces should beevaluated using the cross-coupling coefficients calculated from Alford’s equation. Stability characteristics of axialcompressor should also be evaluated if supported by fixed geometry bearings [1], as these bearings can induce anexcitation force that may drive the axial compressor rotor unstable (see 3.3.2). In general, stability is not a problemassociated with axial compressors.

Figure 2-97—Axial Compressor Rotor Construction: Disc-on-shaft Shrink Fit

View “Z”View “A”

Blade Spacing

Balanceplug holes “A”

“B”

“Z”

Directionof flow

Rotation

View “B”Blade Root

and Lock Piece



Figure 2-98—Axial Compressor Rotor Construction: Stacked Discs with Tie Bolts

Figure 2-99—Axial Compressor Rotor Construction: Drum Rotor with Studs

Tiebolts


Blade Spacing

Balanceplug holes

“A”

“B”

“Z”

Directionof flow

Rotation


and Lock Piece

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 2 3 4 5 6 7 8 9 10 11

Tiebolts

View “Z”

View “A”Blade Spacing

Balanceplug holes

Holes forlifting stubshaft

“A”

“Z”

Directionof flow

Rotation

Blade Root



2.8.5.1 References

[1] Nicholas, J. C. and Kirk, R. G., 1982, “Four Pad Tilting Pad Bearing Design and Application for Multi-StageAxial Compressors,” ASME Journal of Lubrication Technology, 104 (4), pp. 523–532.

2.8.6 Centrifugal Compressors


Centrifugal compressors present equipment designers with challenges not typically found in other rotating equipment.They arise from such factors as the aerodynamic characteristics of the centrifugal impeller and the containment of theprocess pressure within the case. These challenges impact the dynamic behavior of the rotor/bearing system byrestricting the design flexibility. As an example, axial flow machinery (turbines and compressors) derives efficiencybenefits from locating the aerodynamic components, blades and vanes, at larger diameters in the flow path. Incontrast, efficiency gains with centrifugal impellers are obtained from using smaller inlet eye diameters. Therotordynamics impact of this factor is that centrifugal compressors will tend to operate more in the flexible shaft regionthan axial flow machinery.

Another challenge involves containing the higher pressures levels developed within a centrifugal compressor in thecase. These levels in combination with the high rotating speeds magnify the effects of sealing components. The sealdynamic behavior impacts both unbalance response and stability. This increases the complexity of both analyses.The development of the API lateral analysis specifications for centrifugal compressors in comparison to other rotatingequipment mirrors this complexity. Weaver [1] presented an early attempt to define the critical dimensions governingthe dynamic behavior of multi-stage and overhung compressors.

Figure 2-100—Axial Compressor Rotor Construction: Drum Rotor with Tie Bolts

Tiebolts


Blades, Spacers,and Lock Piece


“Z”

Directionof flow

Rotation

Tangential T-SlotBlade Root



Special considerations of centrifugal compressors for the undamped critical speed and unbalance response analyseswill be presented in the following sections. Section 3.8.6 discusses the factors/components that require carefulexamination to obtain accurate stability predictions.

2.8.6.2 Multi-stage Compressors

Modeling of multi-stage compressor rotors, such as that shown in Figure 2-101, does not normally require specialconsideration. Past construction trends have used solid shafts with impellers, thrust collars and couplings applied byinterference fits. Basic FEA and lumped mass techniques have proven more than adequate in modeling these rotors.A few manufacturers use a tie-bolt or built-up construction technique more reminiscent of gas turbines or axialcompressors. These require special attention to the shaft diameter that is used to determine the shaft bendingstiffness (see 2.3).

Additionally, support stiffness considerations beyond the bearing are not normally required. Due to the pressurecontainment requirements, the case construction is normally very stiff in relation to the bearings. With lower thermalgrowth than steam or gas turbines, soft and/or separate housing supports are also not usually required. Thiscombination of a stiff case and bearing housings produces support stiffness in excess of 3.5 times the bearingstiffness and can be treated as infinite without a great loss of accuracy.

The unique characteristic of multi-stage compressors is the seals and their contribution to the dynamic behavior of therotor/bearing system. These include the impeller eye, balance piston, section inlet and the casing end seals. While asignificant portion of the attention has been paid to their impact on the rotor stability (see 3.4), the impact of liquid filmseals on the synchronous behavior of compressors can be just as great (see 2.6.2). Accurate models of oil film sealsare necessary to ensure reliable critical speed predictions.

Typical mode shapes of multi-stage compressors from an undamped analysis follow the classical examples. Figure 2-102 and Figure 2-103 illustrate the soft and rigid support modes from a typical multi-stage compressor. The bounceand rocking rigid body mode shapes can be seen, as well as, the classic bending mode shapes one might expectfrom a beam with central weight addition. The shaft bending stiffness, between bearing weight and bearingproperties, will determine the location and shape of the modes during operation.

It is worth noting that for the last two modes pictured in both regions, the coupling weight determines which shaft endwill possess the maximum displacement at the lower (3rd) mode. In this example, the coupling is on the right end ofthe shaft. Thus, this end of the shaft will have the maximum displacement at the lower mode. For drive-throughcompressors (couplings at both ends), the overhung moment of the shaft end will determine the maximum shaft enddisplacement at the lower mode.

Typical unbalance weight distributions used to excite the first four modes in the unbalance response are shown onFigure 2-104. The magnitudes of the weights are sized according to the rotor weight, journal reaction and overhungweight. While the primary mode of interest is listed under the unbalance distribution, it should be noted that excitationof higher modes using these distributions is possible.

Figure 2-101—Typical Multi-stage Compressor



For drive-through compressors, an additional unbalance may be used to excite the fourth mode. Higher modes maybe excited by using a single unbalance weight at each end or by placing two unbalances in-phase and out-of-phasewith each other. In the first case, dynamics of each end may be identified separately. In the second, a higher level ofresponse may be achieved for an insensitive system. The advantages of each need to be weighed to select theoptimum unbalance configuration.

Due to trade-offs of aerodynamic efficiency versus rotordynamic behavior, the first mode in a multi-stage compressorwill normally have significant bending and an amplification factor exceeding 2.5. Figure 2-105 through Figure 2-108plot the unbalance response characteristics (Bode plots and operating deflection shapes) of the first two unbalanceresponse peaks. The compressor operates between these modes. For a large sized compressor with long bearingspan, the second mode may well have AF ≥ 2.5 and the compressor will likely operate between the first and secondmodes.

2.8.6.3 Overhung Compressors

The overhung gyroscopic effects and weight of the single stage impeller and coupling govern the dynamics of theoverhung compressor (see Figure 2-109). In contrast to multi-stage compressors, overhung compressors have shortbearing spans and relatively stiff shafts. As a result, the first two modes are characterized by rigid body motion. Thefrequency of the first mode is determined by the overhung mass moment of the impeller, impeller polar andtransverse inertia and impeller bearing properties. The second mode is influenced by the same mass properties of thecoupling and coupling end bearing.

The magnitude of the overhung moment also increases the likelihood of experiencing another dynamic phenomenonreferred to as synchronous thermal instability. Synchronous thermal instability or “Morton’s Effect” (see 3.5.2.5)occurs when the overhung dynamics couples with asymmetric heating of the journal. Simply stated, synchronousvibration causes nonuniform heating of the shaft under the bearing. This will lead to a thermal bow of the shaft end,producing unbalance. If the overhung dynamics exist that magnify the journal response to that bow or unbalance,than an unstable system results. An excellent reference to this dynamic effect can be found at de Jongh [2]. “Morton’s

Figure 2-102—Soft Support Undamped Mode Shapes—Multi-stage Compressor

1

0 100

Natural Frequency: 727

–1

1

0 100


–1

1

0 100


–1

1

0 100


–1



Figure 2-103—Stiff Support Undamped Mode Shapes—Multi-stage Compressor

Figure 2-104—Typical Unbalance Distributions for Multi-stage Compressors

1

0 100


–1

1

0 100

Natural Frequency: 16,986

–1

1

0 100


–1

1

0 100


–1

Excites the 1st Mode

Excites the 3rd Mode

Excites the 2nd Mode

Non Drive-Thru

or

Drive-Thru

Excites the 1st Mode

Excites the 3rd Mode Excites the 4th Mode

Excites the 3rd Mode Excites the 4th Mode

Excites the 2nd Mode



effect” has also been shown to afflict integrally geared and drive-through compressors as well as double-ended driveturbines. Note, that there are no API analysis requirements for this behavior. However, API vibration limits may beexceeded during testing.

Figure 2-105—Unbalance Response of 1st and 3rd Critical Speeds

Figure 2-106—Rotor Response Shape @ 4500 rpm

32W/N Mid-Span UnbalanceMinimum Bearing Clearance

Rotor Speed (rpm x 10-3)

1.20

0.80

0.40

0.000 2 4 6 8 10 12 14 16 18 20

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

150

180

120

210

240

270300

330

360

30

60

902.50

2.00

1.50

1.00

0.50Y

X

X

2.50

2.00

1.50

1.00

0 50

0.501.00

1.502.00

2.50

10 020.00

30.0040.00

50.0060.00

70.00

80.00

90.00Z

100.00

Z

A

A



Figure 2-107—Unbalance Response of 2nd Critical Speed

Figure 2-108—Rotor Response Shape @ 12,800 rpm

32W/N Journals OP UnbalanceMinimum Bearing Clearance

Rotor Speed (rpm x 10 3)

1.20

0.80

0.40

0.000 2 4 6 8 10 12 14 16 18 20

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

80.0070.00

60.0050.00

40.0030 00

20 0010.00

150

180

120

210

240

270300

330

360

30

60

900.50

0.40

0.300.200.10

A

Z

Y

0.50

0.50

0.40

0.40

0.30

0.30

0.20

0.20

0.10

0.10

X

A

Z

Z

100.00

90.00



While the rotor support dynamics in between-bearing compressors are similar end-to-end, in overhung compressors,they are very different. At the coupling end, journal loads can approach zero and, in some cases, actually be loadedupwards. The impeller end bearing may also be subjected to loads other than gravity loading of the shaft. Someresearchers have reported volute loading at levels significant enough to affect the impeller end bearing dynamiccharacteristics. A commonly used equation to compute the volute loading is the Stepanoff equation [3]:

(2-35)

where

Fr is the radial force, N (lbf);

Kr is the nondimensional factor;

P2 is the discharge pressure, Pa (psia);

D is the impeller outer diameter, m (in.);

L is the active impeller length, m (in.) (includes shroud surface and impeller exit).

The support structure behind the bearings also varies. A typical case is shown on Figure 2-110. A simple beamsupports the coupling end bearing while the impeller end bearing is nested within the case. The combination of thetwo produces support stiffness that can differ by a factor of three or greater.

Figure 2-111 illustrates the undamped modes of an overhung compressor with soft and stiff supports. As expected,the lower rigid and bending modes show maximum displacement at the impeller end of the shaft. These weregenerated with equal support stiffness at the bearing locations.

Figure 2-109—Typical Overhung Compressor

Fr KrP2DL=



Overhung rotors, especially single overhung rotors, have bearing loading characteristics that are significantly differentfrom typical multistage compressors or steam turbines. The impeller end bearing normally has a larger load than thecoupling end bearing. Also, it is not unusual for the coupling end bearing to have a negative, upward, load. Theimpeller end bearing is normally larger than coupling end bearing due to the difference in loading. If similar bearingdesign parameters are used for both bearings, that is L/D ratio, preload range, assembled bearing clearance to shaftdiameter ratios, pivot locations, etc. then it is not uncommon for the two bearings to have significantly different supportstiffnesses. When the two bearing support stiffnesses are significantly different, the usefulness of a typical undampedcritical speed map becomes questionable. The curves of the first, second, and other critical speeds are typicallycalculated based on the assumption that the two bearing stiffnesses are equal. Some manufacturers may achieveapproximately equal bearing stiffnesses with a design modification to the smaller bearing to increase its supportstiffness, so that a conventional undamped critical speed calculation can be used.

If the bearing stiffnesses are significantly different, some manufacturers have used analytical methods that aredifferent from those used to calculate critical speeds based on multiple values of identical support stiffness for bothbearings. One approach is to ratio the stiffness used at the journal locations to the stiffness calculated at somereference speed. For example, the undamped map plotted on Figure 2-112 was generated with a stiffness ratio of 2:1for the impeller end bearing versus the coupling end bearing. This represents the approximate ratio of verticalstiffness near the operating speed. Obviously, the shortcoming of this approach is that the map is truly valid only in theregion where the stiffness ratio applies. Another approach is to use an iterative method to solve for critical speeds.This approach starts the calculation by assuming support stiffnesses for both bearings and solves for the critical

Figure 2-110—Overhung Compressor Assembly



Figure 2-111—Overhung Compressor Undamped Mode Shapes (Impeller on Right)

1

0 100


–1

1

0 100


–1

1

0 100


–1

1

0 100


–1

1

0 100


–1

1

0 100


–1

1

0 100


–1

a) Soft Support Mode Shapes

b) Stiff Support Mode Shapes



speed. The critical speed is then used to calculate the bearing stiffnesses. If the assumed stiffness used to start thecalculation does not agree with final calculated stiffness, the stiffness values are revised and the process repeateduntil the bearing stiffness at the calculated critical agree with the bearing stiffnesses assumption.

Figure 2-113, Figure 2-114, and Figure 2-115 display the unbalance response and rotor shape for the first two cases.Notice that a peak response is not identified for the coupling excitation case. The second mode does occur above theanalysis range.

As expected, the peak response of the first mode in Figure 2-113 is very broad with a low amplification factor. This ischaracteristic of an excitation of a rigid body mode. The response shape can be seen on Figure 2-114 and shows littlebending over the length of the rotor. The coupling unbalance was unable to excite a peak in the response range asshown in Figure 2-115.

2.8.6.4 Integrally Geared Compressors

The integrally geared compressor is composed of two rotor types: the bull gear, which can be treated in the same wayas in typical gear sets, and the pinion rotor. See Figure 2-116 and Figure 2-117 for views of typical integrally gearedcompressor arrangements. Dynamically, the pinion rotor behaves somewhat the same as overhung compressors, i.e.the overhung mass and polar and transverse moment of inertia partially controls the behavior. A differing feature of

Figure 2-112—Undamped Critical Speed Map—Overhung Compressor

106

105

104

103

102

104 105 106

Bearing Stiffness 107 108

Operating Speed

KH KV

(lbf/in.)

Freq

uenc

y (c

pm)



Figure 2-113—Impeller Unbalance Response—Overhung Compressor

Figure 2-114—Rotor Response Shape at 4,300 rpm

32W/N Impeller UnbalanceMinimum Bearing Clearance


1.20

0.80

0.40

0.000 1 2 3 4 5 6 7 8 9 10

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

150

180

120

210

240

270300

330

360

30

60

901.000.80

0.500.400.20

A

Z

Y

1.00

1.00

0.80

0.80

0.60

0.60

0.40

0.40

0.20

0.20

X

A

ZZ

10 00

20.00

30 00

40.00

50.00

60.00



the pinion rotor is the gear loading at the shaft center. This is taken into account during the calculation of the bearingcharacteristics as an additional load.

Figure 2-115—Coupling Unbalance Response—Overhung Compressors

Figure 2-116—Integrally Geared Compressor Cross-Section

32W/N Coupling UnbalanceMinimum Bearing Clearance


1.20

0.80

0.40

0.000 1 2 3 4 5 6 7 8 9 10



With integrally geared compressors, the complication of unequal journal loading experienced in overhungcompressors is not as significant for two reasons. First, identical bearings are normally used at both ends. Second,the gear loading at the rotor center is typically greater than the rotor weight. This evens the journal load distribution inpinions with two impellers of different weights or even just one impeller. Figure 2-118 and Figure 2-119 present typicalpinion rotors along with a model for an integrally geared compressor pinion rotor with two stages (impellers at bothends).

Typical undamped modes shapes are shown on Figure 2-120. An unbalance distribution to excite each mode is alsoshown. While the undamped map would look very similar to that of an overhung compressor, assumptions concerningthe gear load are needed. The gear load is required to compute the bearing coefficients. The complications arise fromthe fact that the gear load does not behave in a similar fashion from one application to another. One assumptionfrequently made is that the torque (and thus the gear load) will vary as the square of the speed reaching full load at fullspeed. However, integrally geared compressors equipped with variable inlet guide vanes (VIGV) are able to unloadthe compressor during starting sequences.

Another complication arises at maximum continuous speed. The VIGV also enable the operating conditions to bechanged independently of speed. Thus, whatever assumptions were applied to the load versus speed function mayno longer be valid at Nmc. Additionally, integrally geared compressors are frequently used with multiple processstreams. Often the sequencing of those streams is critical for the plant process resulting in loading that may varysignificantly (and not uniformly) with both speed and time. For these reasons, the validity and usefulness ofundamped critical speed maps are questioned. (It should be noted that the same assumptions are required to performthe unbalance response analysis. As before, these assumptions define the regions where the analysis is valid.)

Figure 2-117—View of Integrally Geared Compressor Assembly



Figure 2-118—Typical Pinion Rotors from an Integrally Geared Compressor

Figure 2-119—Pinion Rotor Model—Integrally Geared Compressor

(a) With thrust bearing

(b) With thrust collar

203.0 203.0524.0

12 3 4 5 6

7 8 9

101112 13

14 15 16

17 1819

20

21 2223

24 25 26 2728 293031



2.8.6.5 References

[1] Weaver, F. L., 1972, “Rotor Design and Vibration Response,” Proceedings of the 1st TurbomachinerySymposium, Texas A&M University, pp. 142–147.

[2] de Jongh, F. M. and van der Hoeven, P., 1998, “Application of a Heat Barrier Sleeve to Prevent SynchronousRotor Instability,” Proceedings of the 27th Turbomachinery Symposium, Texas A&M University, pp. 17–26.

[3] Moore, J. J. and Flathers, M. B., 1998, “Aerodynamically Induced Radial Forces in a Centrifugal GasCompressor: Part 1 Experimental Measurement,” Journal of Engineering for Gas Turbines and Power, Vol.120, April, pp. 383–390.

2.9 API Unbalance Response Verification Testing

2.9.1 Introduction

The primary goal of API testing is to verify the unit design by observing the dynamic behavior. Performance testingdetermines the aerodynamic/thermodynamic behavior (within the specifications of ASME PTC 10 Type I or II testing.)The compressor head and efficiency are calculated from measurements of the fluid pressure, temperature and flowrate. Inference of the compressor performance is not needed as it is measured directly during Type I or with minimalextrapolation with Type II. Accuracy of the analytic predictions is determined directly as the performance is measured.

Figure 2-120—Typical Rigid and Flexible Body Mode Shapes & Unbalances Used to Excite Each

U

U1

U1 U1

U2

U2

U2

U



Ideally, the same direct measurement would be made to determine the machine’s rotordynamic behavior. However,the freedom to completely measure the vibratory behavior of the rotor/bearing system is not available on the test stand.Certain parameters can be measured (journal response, bearing temperatures, etc.) while others are impractical, ifnot, impossible (vibration at impeller eye labyrinth, for example). Probe (bearing) vibration levels reveal only a smallpart of the dynamic behavior of the rotor. In many situations, of greater concern is the vibration level at close clearancelocations throughout the rotor. Rubbing at these locations can detrimentally affect the operability of the rotor.

Rotordynamic predictions easily permit the determination of unbalance response at any selected rotor location.However, during the mechanical testing, some key dynamic behavioral traits (vibration levels at the close clearancelocations) will be inferred from the rotordynamic prediction and the probe measurements close to the bearings. Thus,to verify the acceptability of the unit design, the testing program will also need to determine the accuracy of therotordynamic predictions, not just monitor the probe readings. To determine the accuracy of the unbalance responsecalculations, the unbalance response verification test was developed.

Basically, the unbalance verification test selects an unbalance weight and rotor placement location. The proberesponse due to that weight is measured during the test. This measured unbalance response is compared against therotordynamic predictions for the same unbalance weight and placement. If the probe predictions match themeasurements, then the rotordynamic method/predictions are determined to be accurate. Confidence in using thatmethod to then determine vibration levels at points not measured during testing is increased. If the measured doesnot agree with the predicted, then corrections to the analytical model are needed to gain that confidence.

The restrictions and complexities faced during verification testing are summarized below. The list is not intended to beall encompassing. Different machine types and test setups may present more possibilities and be less restrictive thanothers.

a) Residual unbalances and forces exist — Unbalance response analysis predicts the rotor response to a knownunbalance or distribution of unbalances. However, real life rotors will possess an unknown residual unbalancedistribution and are subjected to varying levels of dynamic forces. While these factors are minimized by carefulbalancing, precise alignment, and reduced aerodynamic forces during mechanical testing, they do exist and canraise the vibrations levels to a significant percentage of the limit. Efforts are required to identify the response duesolely to the applied unbalance verification weight.

b) Few choices exist for the applied verification weight location — In many instances, the possible locations in whicha test unbalance weight may be applied is limited to one, the coupling. (Some steam turbines, generators andmotors have the potential for other locations if discussed during the design phase of the equipment.) A couplingapplied trial weight has many advantages and disadvantages. It is a poor location to excite the first mode but maybe better suited to excite higher order modes. However, isolation of the coupling response may prove difficult dueto the decreased sensitivity of the rotor design.

c) Applied trial weight is normally limited in size — Analytically, it is possible to apply unlimited amounts of unbalanceto the rotor. This is not possible during testing due to safety concerns and physical limitations. In some cases,attempting to raise the vibration levels to the test limit employing coupling unbalance weights may require anunbalance force application in excess of the rotor weight. Stress considerations and physical space may provethis impractical and unsafe. The verification test and analytical prediction need to be coordinated to ensure asuccessful test.

d) Risks associated with elevating vibration levels — The API 617 6th Edition testing method relied on raising thevibration levels measured at the probe locations at the operating speed nearest to the critical of concern to the APIlimit. The rotordynamics analysis was then used to determine if internal clearances were exceeded. Thismethodology overly relies on an accurate rotordynamic model, when verifying the model’s accuracy was theoriginal point of the test. Any inaccuracies in the analysis could result in damage during the shop testing atelevated levels, leading to repairs, retesting and delayed shipments.



e) Limited measurement capabilities — Without substantial efforts, vibrations levels at the probe locations (usuallysituated near the journals) are the only response measurements made of the rotor. Additionally, taking thesemeasurements only at the operating speed nearest the critical speed of concern greatly limits the ability tocompare the test results to the analytical predictions. It should be noted that some manufacturers offer to performverification tests in a high-speed balancing facility. This permits application of the unbalance weights at mid-span,quarter-span and other combinations not possible on the test stand in the case. Additionally, mid-span vibrationreadings are possible. (Note: The high-speed balance facility will have different support characteristics that needto be addressed in the rotordynamic model used to compare against the measurements.)

The verification test seeks to confirm the response prediction for one unbalance distribution or placement. Theresponse predictions share a common rotor/bearing model, which form the core of the unbalance response analysis.The various response analyses apply different unbalance forces to this common model. Confirmation of the accuracyof one response prediction is a verification of the model, and thus, all other response predictions using that model.

2.9.2 Qualitative Aspects of the Verification Test

While the verification test is based on quantitative comparisons (i.e. the ability to isolate the effects of the unbalanceweight), all qualitative aspects cannot be removed. These arise due to several factors, multiple probe readings,instrumentation sampling differences and operating condition changes to name a few. A brief description of thesefactors is contained below.

a) Multiple probe readings — As anyone who has run a test can confirm, each of the vibration probes used tomonitor the shaft vibrations will produce slightly different results. Some of the differences are expected due tobearing loading and geometry. Others cannot be easily explained. (This was primarily the reason behind removingthe calculated amplification factor as one of the comparison values from API 617 6th to 7th Edition.)

b) Instrumentation sampling differences — To identify the effects of the unbalance weight, a baseline reading of themachine is necessary. The response includes the effects of the unknown residual unbalance and other forces inthe machine. After adding the unbalance weight, the data (usually in the form of a Bode plot) are taken again. Theeffect of the weight is obtained from a vector subtraction of the two data sets. However, slight differences in thedata points in speed will produce small errors in the vector subtraction. For the most part, these are insignificantand do not affect the results. In the areas of critical speeds where larger phase changes occur over smaller speedranges, the errors may result in changes to the “measured” values.

c) Operating condition changes — During a “normal” verification test, the final X-Y data for the Bode plot are takenfollowing the 4-hour mechanical run. Following this, the machine is shutdown to add the unbalance weight andbrought back up to speed to reach steady state. Some conditions may not return to the same levels reached at theend of the four-hour run. In those situations, the response at the critical speeds (the most sensitive area) will varyfor reasons other than the addition of the unbalance weight. Since these cannot be isolated, they will be attributedto the unbalance weight during the vector subtraction.

d) Multiple plane readings — Most API equipment has vibration probes near both journal bearings locations. Duringverification tests using the coupling as the weight location, the probe readings at the opposite end of the shaft maysee a relatively small influence. The influence can be masked by other factors such as background noise. Thiscan lead to mistakenly attributing the response to the placement of the unbalance weight.

Differences between the predicted and measured influence of the trial weight should be discussed prior toproclaiming the test a failure. These “qualitative” aspects of the test may alter the results but not affect the underlyingintegrity of the predictions.

2.9.3 Pre-testing Preparations

For an effective verification test, the rotordynamic analysis of the equipment should be examined at the design stage.Areas of concern would include critical speed locations, amplification factor, and unbalance sensitivity. Beyond



configuration and design changes, the verification test can be used to satisfy these concerns by confirming therotordynamic analysis.

The verification test can help identify important vibration performance behaviors such as the following.

a) High amplification factors through the first critical speed — This is usually accompanied by high ratios of mid-spanto journal vibration. Careful balancing cannot always hide high amplification factors. A high amplification factorimplies a more sensitive rotor and is a function of the damping and rotor stiffness (not the amount of unbalancepresent.) Normally, high amplification factors can be seen on the baseline data taken during the mechanical test.Verification testing performed in an at-speed bunker may permit more flexibility in weight placement and centerspan vibration readings but also include different rotor support properties. These trade-offs should be discussed todetermine the feasibility of this option.

b) High unbalance sensitivity of the equipment — This is easily measured by the verification weight applied to therotor. However, care in selecting the size of the weight should be exercised. Always applying (or asking to apply)the maximum weight may prove damaging to the rotor internals. High sensitivity implies that the influence ofsmaller weights can be properly identified and will satisfy API requirements.

c) Critical speed locations of the lower mode — Location of the first critical speed in most turbomachinery is easilyidentifiable during the mechanical test. The exception comes with stiff rotors whose first mode is heavily damped.In this case, it is unlikely that location of the first mode will be enhanced during a verification test without additionalefforts. Of course, the location of heavily damped modes is also less important since they are heavily damped.Higher order modes infringing on the Nmc may be excited by the verification test even with unbalance weights atthe coupling. Should verification of the predictions not be possible with a coupling weight (due to the reasons inthe previous section), alternate means of performing the test should be discussed.

2.9.4 Identifying the Unbalance Weight Influence

The cornerstone of the API unbalance response verification test is the identification of the unbalance weightinfluence. This permits direct comparison of the predicted to the measured response without the residual influences.This process is demonstrated both graphically and vectorially for a single speed and probe and is easily expanded fora speed range and multiple probes.

To illustrate the method, a simple example is presented. During the baseline run (the X-Y Bode plot data), Rotor A hasreadings for a given probe and speed as shown on Figure 2-121. The synchronous (1x) vibration, V1, is a result of theresidual unbalance, Ur, left in the rotor. (For the purpose of this explanation external or residual forces will be ignored.)

Expressed as a vector, the synchronous vibration and unbalance are:

(2-36)

(2-37)

After adding the test unbalance weight, Ut, to Rotor A, the synchronous vibration reading, V2, is taken at the samespeed and probe location. Figure 2-122 presents the measurements before and after the addition of the unbalanceweight.

Since the only change to Rotor A was the addition of the unbalance weight, Ut, the change in vibration from point 1 topoint 2 must be the influence of the weight. The weight’s vibration influence, Vt, is shown on Figure 2-123.

V1 1.2 mils @ 40=

U1 Ur=



At point 2, following the addition of the weight, the total vibration and unbalance are:

(2-38)

(2-39)

The vector math to compute the effect of the unbalance weight is simply:

(2-40)

(2-41)

or:

Figure 2-121—Baseline Vibration Reading (Graphically)

Figure 2-122—Readings After the Addition of the Unbalance Weight

Y

Vibration reading

X

V1 = 1.2 mils @ 40°

Y

Unbalance level

X

Ur = Unknown

Y

Vibration reading

X

V1 = 1.2 mils @ 40°Ut = 10 gr-in.

V2 = 2.0 mils @ 300°

Y

Unbalance level

X

Ur = Unknown

V2 V1 Vt+=

U2 Ur Ut+=

U2 1– U2 U1– Ur Ut+ Ur Ut=–= =

V2 1– V2 V1– V1 Vt+ V1 Vt=–= =

Vt 2.5 mils @ 272=



Notice that the residual unbalance magnitude and location does not need to be known to identify the influence of theunbalance weight. This method is automated by several commercially available software packages to perform thevector math for a data set encompassing the operating range. Furthermore, it is not necessary to compensate(subtract) for runout from the baseline and test synchronous vibration readings since the runout effects areautomatically eliminated during the calculations.

2.9.5 Verification Test Examples

The identification method was presented in the previous section for a simple example of a single probe and speed.Three examples of actual verification tests are presented next. The first two examples contain vibration data from tworotors representing flexible and rigid shaft designs. These examples are taken from Nicholas et al. [1]. This referencealso contains excellent examples of the process applied to steam turbines. The third illustrates the influences thatforce qualitative decisions on the accuracy of the test. (Note: While not discussed here, the analyses indicated thatvibration levels would not exceed 75 % of the internal clearances for all examples.)

The first rotor is an eight-stage compressor with a Nmc (11,470 rpm) to critical speed ratio of 3.1. Figure 2-124 plotsthe baseline vibration (solid) and results with the unbalance weight added to the rotor (dotted) at the coupling(vibration data are taken from the coupling end probe.) The first critical speed is clearly identified by the peak in bothresponse curves. The shaded region represents the range of critical speed predicted by the analysis for minimum andmaximum bearing clearances. As expected with flexible rotors, the first critical speed frequency is not sensitive to thechange in bearing stiffness (the bearing stiffness changes as a result of the clearance range specified for the journalbearings).

Taking the two data sets obtained for the eight-stage compressor and performing the vector subtraction yields thesolid curve on Figure 2-125. Overlaid upon the test data is the predicted response at minimum and maximum bearingclearance. For the most part, the test data lie within the predicted range. The response at Nmc is within the rangepredicted by the analysis.

The verification test was also performed on a three-stage compressor operating at 13,500 rpm. As required, thevibration data were taken following the four hour run (baseline) and after the unbalance weight was added to thecoupling. Figure 2-126 plots the vibration data taken from the coupling end probe. The first critical is clearly evident(10,000 rpm) even though it is well damped. Since the rotor has only three stages, the shaft bending to support

Figure 2-123—Influence of the Unbalance Weight

Y

Vibration reading

X

V1 = 1.2 mils @ 40°

Vt = V2 – V1

Ut = 10 gr-in.

V2 = 2.0 mils @ 300°

Y

Unbalance level

X

Ur = Unknown



stiffness ratio is significantly larger than with the eight-stage compressor. This rotor is characterized by modes moreclosely resembling rigid body motion. Rigid body modes are more sensitive in frequency to changes in the bearing/support stiffness. This can be seen by the larger shaded region predicted for the location of the first mode over thebearing clearance range.

Computing the influence of the unbalance weight and overlaying these data with the predicted response yields Figure2-127. The rotor appears to be more sensitive than predicted with the measured response 20 % greater thanpredicted at Nmc and a curve shape that exaggerates the predicted plot. The amplitude levels are mostly bounded bythe unbalance response predictions for minimum and maximum clearance. The variations witnessed are in the 0.05mil range. Additionally, the predicted response is taken at the extremes of the bearing clearance range. Differentcombinations of pad curvature and pad placement, while still within the predicted range of response may have adifferent curve shape depending on the stiffness and damping properties calculated.

The final example details some of the influences that complicate the process of verifying the rotordynamic analysis.The vibration response of a seven-stage compressor before and after application of the unbalance weight is shownon Figure 2-128. In this case, both the run up and rundown are included in the Bode plot. Differences between the twoarise from changes in the steady state conditions that are reached at Nmc (7150 rpm) before the speed sweeps begin.These may include bearing temperature differences, both oil and pad, among others.

Figure 2-124—Bode Plot for Eight-Stage Compressor [1]

Figure 2-125—Unbalance Weight Influence (.... Predicted ___Test) [1]

0 2 4 6 8 10 123700 rpm

180

270

360

90

180

Pha

se L

ag:

15 d

eg/d

v

0 2 4 6Speed: 0.5 krpm/div

8 10 12

1.0

0.5

0.0

Am

ptu

de:

0.05

m p

p/d

v

Predicted rangeof 1st critical Baseline

Unbalanceweight

Point: Coupling end vert. 45° left 1X uncomp 1.31 294°


8 10 12

0.15

0.10

0.05

0.00

Am

ptu

de:

0.01

m p

p/d

v



Note that runout is not removed from the vibration data in this case. Ideally, slow roll readings would not changeduring the mechanical testing. Since the vector subtraction removes all aspects of the vibration data that are or shouldbe constant from one run to another, the runout would therefore be eliminated in the process. Figure 2-129 plots theinfluence of the unbalance weight as determined from the data in Figure 2-128. At the lower speeds, the influence isnearly zero, as one would expect. This indicates that the runout was unchanged between runs and successfullyremoved.

Figure 2-129 identifies three areas in the plot that require further discussion. First is the region around the 1st criticalspeed, region 1. As noted earlier, the region surrounding the critical speeds is more sensitive to minor changes in therotor/bearing system. Thus, any differences in support stiffness, rotor balance condition, etc., between runs areemphasized in this region. This can be seen on Figure 2-129. While the analysis predicts that the 1st critical is notexcited by the weight, a peak nonetheless exists. This is due to the minor changes being emphasized by the 1stcritical speed.

The differences in the runup and rundown were noted on Figure 2-128. Region 2 on Figure 2-129 indicates that thesedifferences are not uniform from run to run. From the baseline run to the unbalance weight run, the system changesbetween the speed sweeps varied roughly 0.03 mils, not a significant amount but nearly 50 % of the predictedresponse to the unbalance weight at the lower speeds. The mean of the two measured curves nearly equals thepredicted response indicating the behavior is as predicted.

Figure 2-126—Bode Plot for Three-Stage Compressor [1]

Figure 2-127—Unbalance Weight Influence (…. Predicted ___Test) [1]

0 2 4 6 8 10 12

90

180

270

360

90

Pha

se L

ag:

15 d

eg/d

v

20 4 6Speed: 0.5 krpm/div

8 10 12

0.15

0.10

0.05

0.00

Am

ptu

de:

0.01

m p

p/d

v Predicted rangeof 1st criticalBaseline

Verification


8 10 12

0.10

0.08

0.06

0.04

0.02

0.00

Am

ptu

de:

0.00

5 m

pp/

dv



Finally, thermal changes as the system stabilizes at Nmc can be seen at Region 3. While time dependent vibrationsare not normally plotted on a Bode plot, they can influence the results depending on the rate of acceleration to Nmc.For slower ramp rates, time dependency can creep into the response levels. However, the analytical predictionrepresents a steady state condition. From the plotted results, it appears that the baseline run included some of theseeffects which translated in to the weight influence on Figure 2-129 at Nmc. Since the speed changed only slightly, therise is nearly vertical identifying it as a time dependent effect. In the worst case where the speed is varying slowly, thismay appear as a rise to a response peak.

These examples emphasize the need to ensure that the test conditions/procedures focus on obtaining accurateconsistent measurements. Table 2-5 summarizes the predicted versus measured unbalance response characteristicsfor the three examples. While assessment of critical speed location is a relatively straightforward comparison,unbalance sensitivity can be much more difficult which highlights the importance of consistent operating conditionsduring baseline and test weight vibrations.

2.9.6 References

[1] Nicholas, J. C., Edney, S. L., Kocur, J. A., and Hustak, J. F., 1997, “Subtracting Residual Unbalance forImproved Test Stand Vibration Correlation,” Proceedings of the 26th Turbomachinery Symposium, Texas A&MUniversity, pp. 7–18.

Figure 2-128—Bode Plot of Example #3 [1]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 1000 2000 3000 4000Speed (rpm)

Dsp

acem

ent (

ms

p-p)

5000 6000 7000 80

Predicted Range of 1st Critical

Baselilne

Verification



Figure 2-129—Unbalance Weight Influence [1]

Table 2-5—Summary of the Results of the Tests

Example 8-stage Compressor 3-Stage Compressor 6-Stage Compressor

Predicted 1st Critical Speed (rpm) 3700 to 3750 8600 to 10,800 3050 to 3150

Measured 1st Critical Speed (rpm) 3700 10,000 3050

Predicted Verification Response @ Nmc (mils p-p)

0.11 to 0.16 0.04 to 0.085 0.11 to 0.135

Measured Verification Response

@ Nmc (mils p-p)0.16 0.09 to 0.105 0.085 to 0.125

Discussions

API verification test requirements met. Good agreement between predictions and test results.

Most API verification test requirements met. Measured data may indicate a greater sensitivity to unbalance than predicted. However, short span rotor is already insensitive so indicated change has no influence. Additionally, response close to predicted levels.

API verification test requirements met. Influence response indicates excitation of 1st critical, transient effects at Nmc, and differences between run up and rundown plots. All of which is not attributable to the unbalance weight.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

Dsp

acem

ent (

ms

p-p)

6000 7000 80001000 2000 3000 4000 5000Speed (rpm)

Predicted

Test

1

23



2.10 AMB Modeling and Analysis Issues

2.10.1 Introduction

Rotordynamic modeling and analysis of machinery with active magnetic bearings (AMBs) has some importantdifferences compared to fluid film or rolling element bearing machines. This section discusses these differences andthe related issues in the context of lateral rotordynamics and response. This section also discusses rotordynamicissues related to the auxiliary bearing system. Issues more directly related to stability and stability analysis arediscussed in Section 3.

The fundamental difference for AMBs, relative to other bearing systems, is that AMB systems have a feedbackcontrol system which determines much of the overall rotordynamic performance. A portion of the feedback controlsystem for a typical rotor and AMB system is shown in Figure 2-130. The major subcomponents are:

— rotor assembly with AMB radial bearing sleeves and axial thrust disk;

— radial and axial electromagnetic actuators;

— radial and axial position sensors and associated signal conditioning (typically also used for machinerymonitoring);

— control system including signal processing (typically implemented digitally);

— power amplifiers;

— auxiliary bearing system.

Figure 2-130—AMB System

Auxiliarybearings

Radialpositionsensor

Radialcontrolsystem

Radialmagneticactuator

Signalconditioner

Poweramplifier(s) Axial actuator

and shaft disk



From a functional perspective, these subcomponents control shaft position as follows.

— The position sensor system measures rotor motion, and sends signals to the control system proportional to theshaft position.

— The control system generates an output signal which tries to move the shaft back to the desired location(generally the center of the clearance space).

— The power amplifiers convert this signal into a high power voltage/current. The power amplifier supplies theenergy used to control the rotor’s static and dynamic displacement.

— The electromagnetic actuators convert the voltage/current into a strong, attractive magnetic field that acts on therotor sleeve/thrust disk to pull the rotor towards the desired location.

Under normal design conditions, the auxiliary bearing system is inactive. The rotor operates without contact within asmall clearance space between the auxiliary bearing and the corresponding rotor surface. The auxiliary bearing(s)only come into contact with the rotor when the AMB system is de-energized or overloaded.

In a two bearing machine, there are five control axes—two radial axes at each of the two actuators, and one axialaxis. The radial axes are usually oriented at plus and minus 45 degrees from vertical. These are frequently referred toas the “V” and “W” axes as shown in Figure 2-131. The direction of rotation relative to the V-W axes varies for differentmachines. Some integrally driven machines (motor directly coupled to the compressor or pump) have three radialbearings, for a total of seven control axes. More detailed discussions of AMB systems can be found in the referencelist below.

Figure 2-131—AMB System Radial Axes

Axis 1WAxis 1V

Bearing 1

Bearing 2

Axis 2WAxis 2V



2.10.1.1 References

[1] ISO 14839-1, 2004, “Mechanical Vibration—Vibration of Rotating Machinery Equipped With Active MagneticBearings - Part 1: Vocabulary,” No. 14839-1, International Organization for Standardization.

[2] ISO 14839-2, 2004, “Mechanical Vibration—Vibration of Rotating Machinery Equipped with Active MagneticBearings, Part 2: Evaluation of Vibration,” No. 14839-2, International Organization for Standardization.

[3] ISO 14839-3, 2006, “Mechanical Vibration—Vibration of Rotating Machinery Equipped with Active MagneticBearings, Part 3: Evaluation of Stability Margin,” No. 14839-3, International Organization for Standardization.

[4] ISO 14839-4, 2012, “Mechanical Vibration—Vibration of Rotating Machinery Equipped with Active MagneticBearings, Part 4: Technical Guidelines,” No. 14839-4, International Organization for Standardization.

[5] Schweitzer, G., and Maslen, E. H., editors, 2009, Magnetic Bearings: Theory, Design and Application toRotating Machinery, Springer, Dordrecht.

2.10.2 Rotor Modeling Considerations

The general guidelines for rotor modeling presented elsewhere in this tutorial apply to rotors with AMB systems.However, there are also some AMB specific modeling practices that should be followed.

2.10.2.1 Rotor Stations

In general, a single actuator station is assigned at the centerline of each radial magnetic bearing’s stator poles. Thatcenterline point can change slightly with thermal expansion of the shaft/housing. If the aspect ratio of bearing’s lengthto diameter ratio exceeds a factor of 3, more than one station should be considered.

Likewise, a station is assigned at the centerline of each AMB radial position sensor. The radial position sensors,generally, are not located at the axial centerline of the actuator. The effect of this offset (also known as“noncolocation”) is extremely important to include in the overall rotordynamic model. Note that this centerline canchange slightly with thermal expansion of the shaft/housing. If two position sensors per actuator are combined to forma ‘virtual’ sensor location, it is frequently reasonable to place a single station at this ‘virtual’ sensor location. However,use of a single station will not allow the effects of any shaft or housing dynamics between the sensors to be evaluated.Thus, this approximation should be used with some caution.

Stations should also be assigned at the auxiliary bearing centerlines and the centerline of the axial magnetic bearingthrust disk.

2.10.2.2 AMB Sleeves and Thrust Runner

Mass and inertia properties of the AMB sleeve(s) need to be included in the rotor model. This could mean multipleassemblies for radial AMB rotor lamination sleeves, axial thrust disk, auxiliary bearing landing sleeves and positionsensor targets, or integrated sleeves that combine one or more of these features.

Shaft stiffening effects from the sleeve(s) should also be considered. In some cases, the radial rotating assembly is arelatively long sleeve. This sleeve can add enough bending stiffness to affect the higher shaft modes. If the rotorlaminations are directly mounted on the shaft, no added stiffness should be included for them. A modal (impact) testmay be required to accurately determine any stiffening effects.

AMB thrust runners can be much larger in diameter than for typical fluid film thrust bearings. They are also relativelythin. Thus, disk flexibility and attachment flexibility may have to be included to obtain an accurate model [1]. Thisflexibility can be estimated via finite element analysis or from modal testing.



2.10.2.3 AMB Rotor/Stator Clearance Spaces

The radial clearance between the rotor and stator for the AMB system is small and, in many machines, is filled withthe process fluid. This fluid-filled annular gap can give rise to appreciable fluid-structural interaction effects [2]. Theseeffects can be modeled using linearized added mass, stiffness and damping coefficients at the annular gap location.Thus, for high density compressors, and especially for pumps, additional dynamic coefficients may need to beincluded in the model to account for these effects.

2.10.2.4 Negative Magnetic Stiffnesses

Attractive magnets, such as those used in AMB systems, inherently have the characteristic that the attractive forceincreases as the rotor-actuator gap decreases (assuming constant current). This inherent characteristic is usuallymodeled as a negative stiffness that is obtained by linearizing the equations relating actuator geometry, materials,force, gap and current. This negative stiffness at each actuator location is one of the reasons that all practical AMBsystems must have a feedback control system. Without the control system, the rotor would simply be pulled over tothe backup bearings by this negative stiffness effect [3].

This negative stiffness is generally not included in the AMB transfer function model provided by the AMB vendor. Inthis case, it is necessary to include a separate bearing-like, constant, negative stiffness at the centerline of eachactuator location.

Likewise, many AMB supported machines include integral motor drives that may have significant negative magneticstiffnesses. In this case, it is necessary to also include a separate bearing-like, constant, negative stiffness at eachmotor location [4]. Depending on the length of the motor, a single negative stiffness at the motor centerline may besufficient. For long motor rotors though, multiple stiffnesses along the length of the motor may be required.

2.10.2.5 Supports and Housing Dynamics

From a lateral response perspective, the same considerations as with any rotordynamic model apply. As is discussedat greater length in Section 3, a more detailed housing/structural model may be required for accurate stability analysiswith an AMB system due to the possibility of interaction between the AMB control system and higher order vibrationmodes. Thus, the support and housing dynamic model for an AMB system might be much more detailed than for afluid-film or rolling element bearing supported rotor.

2.10.2.6 References

[1] Díaz, S., De Santiago, O., Solórzano, V., 2012, “Rotordynamic Modeling of Centrifugal Compressor Rotors forUse with Active Magnetic Bearings,” Proceedings of the XIII Latin American Turbomachinery Congress andExposition, Paper Number CELT-019-2012, March 12–15, Querétaro, Mexico.

[2] Fritz, R.J., 1970, “The Effects of an Annular Fluid on the Vibrations of a Long Rotor, Part I—Theory,” ASMEJournal of Basic Engineering, Vol. 92, pp. 923–929.

[3] Schweitzer, G., and Maslen, E. H., editors, 2009, Magnetic Bearings: Theory, Design and Application toRotating Machinery, Springer.

[4] Holopainen, T.P., and Arkkio, A., 2008, “Electromechanical interaction in rotordynamics of electricalmachines—an overview.” Proceedings of the Ninth International Conference on Vibrations in RotatingMachinery, IMechE, Exeter, UK,. Vol. 1, pp. 423–436.

2.10.3 AMB Control System Considerations

The dynamics of an AMB supported machine are largely governed by the AMB control system. This sectiondiscusses some of the special modeling concerns.



2.10.3.1 Control System Model

AMB developers use a variety of models and tools to develop the control system for a machine. The models usuallyare quite detailed, including separate dynamic models for the sensors, anti-aliasing filters, digital control algorithm,calculation delays, power amplifier dynamics and other special feature such as flux feedback. These models arefrequency dependent, and may also depend to some extent on the machine operating conditions. Many of the detailsof these models are considered proprietary, intellectual property of each vendor.

In an attempt to avoid requiring vendors to disclose as much proprietary information, the standard paragraphs werewritten to allow AMB vendors to supply a set of overall displacement to force transfer functions. These transferfunctions hide much of the proprietary detail, but allow independent rotordynamic analyses to be performed.Obviously, if only the overall transfer functions are provided, it is not be possible to perform a detailed audit/analysis ofthe AMB control system. Optionally, the vendor might supply the equivalent set of state-space system matrices.

The number of transfer functions required varies depending on the control system implementation. At one extreme, isa design where exactly one sensor is used to generate the control feedback for exactly one actuator. This is typicallythe case for the axial (thrust) axis. In this case there is one transfer function per axis. This is also known as “single-input, single-output” (SISO) or decentralized control.

For a two radial bearing machine with no coupling between the axes, SISO control would require exactly fourcontrollers for the radial bearings as shown in Figure 2-132. If we described this control system as a matrix of transferfunctions, all of the off-diagonal terms would be zero as in Figure 2-133.

Figure 2-132—SISO Control

Figure 2-133—SISO Control Transfer Function Matrix

h11X1v

Sensor signal (X)

F1v

h22X1w F1w

h44X2w F2w

h33X2v F2v

F1v

F1w

F2v

F2w h11 0 0 0

0 h22 0 0

0 0 h33 0

0 0 0 h44 X1v

X1w

X2v

X2w

=



At the other extreme, each sensor has some impact on the feedback signal to each actuator. In this case, a full matrixof transfer functions from each radial position sensor location to each radial actuator location is required to accuratelymodel the AMB system. This is shown in Figure 2-134. Thus, in the case of the lateral dynamics for a two radialbearing (four axis) system, there would be 16 different transfer functions. This is also known as “multi-input, multi-output” (MIMO) control. This is the more general case.

If we wrote the most general MIMO control system as a matrix of transfer functions, all of the terms would be presentas shown in Figure 2-135.

It is important to note that the distinction between SISO and MIMO only applies to the control system. Even forindependent SISO controllers, the dynamics of the rotor will couple the axes to some extent. Thus, the overall systemis coupled. Therefore, even if the system has SISO controllers, the overall system is inherently MIMO due to therotordynamics.

As an example, consider the two radial bearing machine shown previously in Figure 2-130 and Figure 2-131. If adynamic force is applied by a bearing actuator at one end of a rotor (V1 axis), there will be a displacement seen at theposition sensor at the other end of the rotor (V2) for most excitation frequencies. Likewise, if the rotor is spinning, thegyroscopic effects will also result in a displacement being seen by the other two position sensors (W1 and W2) for aforce being applied at the V1 actuator location.

Figure 2-134—MIMO Control

Figure 2-135—Transfer Function Matrix for MIMO Control (General Case)

hij

X1v

Sensor signal (X) Command signal (F)

F1v

X1w F1w

X2w F2w

X2v F2v

F1v

F1w

F2v

F2w h11 h12 h13 h14

h21 h22 h23 h24

h31 h32 h33 h34

h41 h42 h43 h44 X1v

X1w

X2v

X2w

=



An example of a transfer function written in the s-domain is shown below in Equation 2-42. In this equation, ai and biare the equation coefficients, and s is the complex frequency. Note that there usually are fewer terms in the numeratorthan in the denominator for practical systems.

(2-42)

A Bode plot of a typical control system transfer function is shown in Figure 2-136. The gain or magnitude units willvary depending on which inputs and outputs are used. For example, a digital control system might actually “measure”volts/volts which is conceptually a scaled representation of the physical force/displacement relationship.

There is no simple, general way to accurately re-write a general AMB control system transfer function into a formatthat is compatible with the coefficients used for conventional bearings. Thus, some rotordynamic codes may not beappropriate for doing analysis of AMB supported machinery.

Some early works proposed treating a table of frequency dependent stiffness and damping characteristics as if theywere speed dependent bearing parameters to model an AMB system. This approach was fairly easy to handle withthe rotordynamic codes in existence at that time. However, this approach is not sufficient to allow all of the analysesrequired in the current API specifications to be performed [1]. Therefore, it is no longer considered to be an

Figure 2-136—Typical Control System Transfer Function Bode Plot

01

12

2

01

12

2

...

...

asasasa

bsbsbsb

ntDisplaceme

Forceh

mm

nn

ij

VDC-XE Compensator Transfer Function

Ga

n, d

B

20

0

-20

-40

-60101

101

102

102

103

103

Frequency (Hz)

Frequency (Hz)

Pha

se, d

eg

180

90

0

-90

-180



appropriate approach for analysis of AMB supported machinery. Any code used for AMB supported machinery mustbe capable of representing the full frequency dependence of the AMB in the system dynamic model.

Two good references for further mathematical details on implementing an AMB control system plus rotor model forrotordynamic analysis are [2] and [3].

2.10.3.2 Control System Complications

There are several issues that can complicate the control system model and the overall analysis.

The first is gain scheduling. In some cases, the control system parameters change with operating speed. Oneexample where this may be required is highly gyroscopic rotors. In these cases, the analysis must also consider thesevariations. Frequently, the parameters only change at a few specific speeds, meaning it is possible to do the analysisover several discrete speed intervals.

A second issue is nonlinear behavior. Generally, AMB control systems are designed to be fairly linear for "small"orbits. However, in the case of large orbits, large unbalances, or unusual control system implementations, nonlinearbehavior may need to be considered as part of an adequate rotordynamic analysis. However, this is pretty unusual.

A third issue is the sampling and calculation delays inherent in a digital control system. This delay can be importantfor high frequency behavior. The transfer function supplied by the AMB vendor should include any such delays. If not,then a Padé approximation [3] would be required.

2.10.3.3 Unbalance Force Rejection Control

One of the unique capabilities of AMB systems is the ability to reduce or eliminate transmitted synchronous dynamicforces or motion due to unbalance. There are two main synchronous open loop control approaches:

— minimize the transmitted AMB force;

— minimize the rotor synchronous motion.

If there no separation margin or clearance concerns, the typical approach is to minimize the transmitted AMB force.

For a rotordynamic audit to meet API specifications, it was decided that the most conservative approach for mostmachinery would be to require that the system meet all requirements with unbalance force rejection control inactive.Thus, the rotordynamic control system model should not include any force cancellation algorithms for the purpose ofevaluating whether the system meets API specifications.

2.10.3.4 References

[1] Swanson, E.E., Maslen, E.H., Li, G., and Cloud, C.H., 2008, “Rotordynamic Design Audits of AMB SupportedMachinery,” Proceedings of the Thirty-Seventh Turbomachinery Symposium, Texas A&M University.

[2] Nise, N.S., 2004, Control System Engineering, 4th Edition, John Wiley.

[3] Schweitzer, G., and Maslen, E.H., editors, 2009, Magnetic Bearings: Theory, Design and Application toRotating Machinery, Springer.

2.10.4 Standard Lateral Rotordynamics Analysis for AMB

Once an appropriate model has been created, the basic elements of a lateral rotordynamic analysis for an AMBsupported system are much the same as for a fluid-film or rolling element bearing system. However, there are some



important additions and changes to account for the different characteristics of an AMB system. These are discussedin this section. Stability and stability robustness are discussed in Section 3.

2.10.4.1 Free-Free Map

The rotordynamics report for an AMB system is required to include a “free-free” map. As shown in Figure 2-137, themap presents the variation in the rotor natural frequencies as a function of speed with zero support stiffness (i.e. therotor is modeled as a free body in free space with free-free boundary conditions). Some rotordynamic analysis codeswill encounter numerical problems if the stiffness is exactly zero. In this case, it may be necessary to add a soft springsupport to the rotor. The stiffnesses should be selected such that the rigid body mode frequencies are at least twoorders of magnitude smaller than the first shaft bending mode.

This free-free map is not the same as the undamped critical speed map, which presents the variation in criticalspeeds versus support stiffness. For the evaluation of an AMB system, the free-free map provides a number of usefulpieces of information.

— It can help give some idea of whether the control system design task needs to deal with free-free modes in ornear the operating speed range.

— It can help give some idea of whether there will be free-free modes near or below the running speed range thatcould affect the auxiliary bearing performance during a drop event.

— It shows whether the natural frequencies change significantly with speed due to strong gyroscopic effects.

The free-free mode shapes are also very helpful in determining where the nodes are located relative the actuatorsand sensors. A node between the sensor and actuator means that there will be a 180 degree phase shift betweenthem, which complicates the control system design.

Likewise, a lateral Campbell diagram generated using the AMB system characteristics may be very useful inunderstanding the overall system dynamics.

Figure 2-137—Typical Free-Free AMB Map (Nmc = 30,000 RPM)

Rotordynamic Free-Free Natural Frequency Map

VDC-XE Rotor2000

1800

1600

1400

1200

1000

800

600

400

200

00 10,000

Nat

ura

freq

uenc

y, H

z

20,000 30,000 40,000 50,000Rotor speed, rpm



The conventional undamped critical speed map is still required. The free-free map does not replace it.

2.10.4.2 Unbalance Response Limits

The traditional unbalance response and test stand displacement limits were developed through long experience withrolling element and fluid film bearings. These limits were modified for AMB systems [1-3]. For AMB systems, the limitis given by the smaller of Equation 2-43 or 0.3 times the minimum diametral close clearance. These limits assume acentered rotor. If the rotor operates well off center (fairly unusual), the limits may need to be reconsidered.

In SI units (m peak-peak):

(2-43a)

In US Customary units (mils peak-peak):

(2-43b)

This limit is intended to balance the desire to be very conservative with regards to unbalance response amplitudes,with the unique characteristics of AMB systems.

There are two issues driving the need to modify the proven limits. The primary issue is the finite force capacity of AMBsystems as discussed below in Section 2.10.6. An AMB control system designer can usually make the bearing stiffenough to meet the traditional API displacement limits if required. However, forcing the shaft to have a very small orbituses up more of the system's finite force capacity. This leaves less force capacity available for controllingaerodynamic flow forces and transients such as surge, etc.

The other issue is that many AMB systems are tuned to be dynamically softer at operating speed relative to the otherbearing types. Thus, the transmitted forces for a given response amplitude are lower. This means less stress onmachinery components, and hence less potential for fatigue problems [2]. Likewise, the minimum seal and otherinternal clearances in an AMB system are frequently larger than for other bearing systems. This is done to ensure thatthe auxiliary bearing system clearance is the minimum clearance. This approach ensures that no other componentswill come into contact with the rotor in the event of a delevitation or overload event.

2.10.4.3 References

[1] Alban, T., et al., 2009, “Mechanical and Performance Testing Method of an Integrated High-Speed MotorCompressor,” Proceedings of the Thirty-Eighth Turbomachinery Symposium, Texas A&M University.

[2] Jumonville, J., 2010, “Tutorial on Cryogenic Turboexpanders,” Proceedings of the Thirty-NinthTurbomachinery Symposium, Texas A&M University.

[3] Swanson, E.E., Maslen, E.H., Li, G., and Cloud, C.H., 2008, “Rotordynamic Design Audits of AMB SupportedMachinery,” Proceedings of the Thirty-Seventh Turbomachinery Symposium, Texas A&M University.

2.10.5 Mechanical Run Test Limits

The limit on synchronous response amplitudes discussed above is also used for the mechanical running test. This isa change from the traditional vibration limit requirements, and is intended to account for the typically softer dynamicstiffnesses of an AMB system and the issues discussed in the previous section. In addition, it is also required thatboth sub and super-synchronous amplitudes be checked. This accounts for the more complex dynamics of AMBsystems.

Avl 3 25.4 12,000Nmc

------------------ =

Avl 3 12,000Nmc

------------------ =



2.10.6 Bandwidth, Dynamic Force Limits

One significant difference between AMB’s and fluid-film or rolling element bearings is load capacity. Fluid-film androlling element bearings can generally handle a short term overload well in excess of the rated load capacity(although there may be some reduction in bearing life). Practical AMB systems cannot. There is no overload capacitybeyond what is designed into the system by the AMB vendor. The five main limiting factors are [1]:

— magnetic saturation;

— amplifier current;

— amplifier voltage/actuator inductance;

— thermal considerations;

— eddy current effects (especially in thrust bearings).

The usual relationship among these is shown in Figure 2-138. The maximum possible force an actuator couldproduce is set by magnetic saturation and the actuator geometry. It is not possible to exceed this limit.

The peak transient force capability is an upper bound set by magnetic saturation or the maximum amplifier current,whichever is smaller. Given a particular power amplifier, actuator geometry and materials, there is an absolute upperlimit to how much magnetic force will be generated.

The static load capacity, which is the maximum constant load that can be supported for an unlimited period of time, isdetermined by the smaller of the maximum power amplifier current, or the actuator coil temperature limits. Themaximum amplifier current is set by the amplifier components and design. The coil temperature limit is related to howwell the actuator configuration can dissipate heat. As more and more current is pushed through the actuator coils,

Figure 2-138—AMB Force Limitations

Available dynamic force (continuous)

Peak force (transient)

Frequency

Forc

e

Maximum Actuator Force per Axis

Material saturation limit

Amplifier current limit

Coil temperature limit

Voltage (slew rate) limit



they will get hotter due to the resistance of the wires. At some point, the temperature will exceed the insulationtemperature limit of the wires.

The dynamic load capacity is frequency dependent. AMB actuators have significant inductance. From basic electricaltheory, as drive frequency goes up, inductance increasingly resists changes in current flow. Thus, for the samedynamic force (current), an increasingly higher voltage is required. At some point this voltage will exceed themaximum available amplifier voltage, and the available dynamic capacity is reduced as shown in Figure 2-138. Thisissue is generally referred to as “slew rate” limiting. It depends on parameters such as power amplifier properties andbearing air gap. The frequency where slew-rate becomes the limiting factor is sometimes referred to as the “kneefrequency.” This point could be above or below running speed.

One other factor that can limit the available dynamic force is eddy currents. Eddy currents are electrical currents thatare induced in the magnetic material by changing magnetic fields. Especially in the case of thrust actuators, they canact to reduce the available dynamic force for control at higher frequencies. Eddy currents are rarely an issue in radialactuators except on “canned” type radial AMBs where reduction on actuator bandwidth should be accounted duringthe design phase.

All of these effects are primarily concerns at high frequencies and/or for large dynamic forces.

Due to the difficulty in modeling all of the relationships within the bearing that govern these effects, the APIspecifications place the burden of dealing with these issues on the AMB vendor. The vendor is required to provide abearing operating envelope as a function of frequency to be used for audit and validation purposes. A typicalenvelope might look like Figure 2-139. The standards indicate that the envelope must be conservative, with a factor ofsafety greater than or equal to 1.5.

From the audit perspective, a comparison of the predicted forces (from unbalance or other sources) versus thisdynamic force envelope allows for independent verification that the expected forces are within an acceptable range. It

Figure 2-139—Typical AMB Allowable Force Envelope

Allowable (FOS = 1.5)Actuator limit

XMB36A Maximum Actuator Force per Axis (Safety Factor = 1.5)

Frequency (Hz)0 500 1000 1500 2000

6000

5000

4000

3000

2000

1000

0

Forc

e (b

f)

Nmc



should be noted that this envelope is an absolute upper bound for all forces acting at the same time. Thus, if multiplestrong excitations are expected at different forcing frequencies, the overall combination needs to considered. Anexample might be unbalance forces in the presence of a strong low frequency aerodynamic force. Discussion of theexpected forces with the AMB vendor would be recommended in such cases.

2.10.6.1 References

[1] Alban, T., et al., 2009, “Mechanical and Performance Testing Method of an Integrated High-Speed MotorCompressor,” Proceedings of the Thirty-Eighth Turbomachinery Symposium, Texas A&M University.

2.10.7 Open and Closed Loop Transfer Functions

The AMB specifications mention three other transfer functions: open loop, closed loop, and sensitivity. These transferfunctions come from feedback control systems analysis and design concepts [1,2]. They can provide significantinsight into the behavior of the overall AMB/rotor system. They allow for some very useful approaches to modelvalidation and performance verification which may be much easier to perform than an unbalance response test formany AMB machines. The sensitivity function is also related to system stability, and is discussed in Section 3.Transfer functions are discussed at length in most control texts ([1], for example), ISO 14839-3 [2], and in [3].

These transfer functions are all measured using the AMB system to excite the system, and measure the response. Inmany cases, the digital AMB control system will include the ability to make the required calculations and performthese measurements as a user function. These transfer function measurements are almost always performed as partof commissioning the AMB system. Some amount of caution does need to be taken, since these measurementsrequire that an excitation force be applied to the shaft by the AMB system.

2.10.7.1 Full versus Axis-by-Axis Transfer Function Measurements

There are two basic approaches to measuring AMB system transfer functions. The first is to measure the transferfunctions for each controlled axis separately. This is the “axis-by-axis” approach. For a typical two bearing machinewith four radial axes and one axial axis, a total of five transfer functions would be measured, one for each controlledaxis. In the case where the control systems for each axis are completely separate (i.e. SISO control), this approach isappealing.

The other approach is to measure the transfer functions from each actuator to each sensor. This is the “all” or “full”approach. Since a typical two bearing AMB machine has four radial control axes (two at each bearing), for a total offour sensor signals, and four force signals, there are 16 distinct possible radial transfer functions. Likewise, for a threebearing machine, there are 36 distinct possible radial transfer functions. Many of the transfer functions betweendifferent axes may be quite small. Generally, there is no significant coupling between the radial and axial axes, so theoptional full measurement described in the standard is for the lateral (radial bearing) axes only.

The standard provides for both “axis-by-axis” or “all” for measurement of the optional open and closed loop transferfunctions. The axis-by-axis measurement is adequate for many machines, and limits the amount of data that must beacquired, processed, reviewed and approved. The shortcoming of this axis-by-axis approach, is that it does notprovide any insight into coupling between the different AMB system axes.

This coupling can be important to consider. In the case of a MIMO controller, the controller inherently couples at leastsome of the axes. Even for independent SISO controllers, the dynamics of the rotor will couple the lateral axes tosome extent. To put this another way, even if the system has SISO controllers the lateral system is inherently MIMOdue to the rotor dynamics. A force applied at one location on the rotor almost always causes the entire rotor to moveand have displacements at all sensor locations (although perhaps not at all frequencies).

Requiring all of the transfer functions to be measured might be appropriate for highly gyroscopic machines or othermachines that are expected to have substantial cross-axis coupling (use of tilt-translate control, for example), or



machines that are outside of the OEM and end user experience base. The additional measurements would also beuseful in cases where there are difficulties in validating the analytical model.

2.10.7.2 The Closed Loop Transfer Function

The closed loop transfer function (CLTF) is the ratio of output response to input excitation signal for an activelycontrolled system, including the effects of the feedback loop. The excitation and measurement locations are specifiedin the AMB specifications, and are as shown in Figure 2-140. A Bode plot of an example CLTF is shown in Figure 2-141. The CLTF measurement can optionally be used for model validation instead of an unbalance response test. Thisis discussed more fully in Section 2.10.8. This transfer function is also discussed at length in ISO 14839-3 [2].

Conceptually, this transfer function has some similarities to the traditional unbalance influence coefficient plot, butthere are some important differences. Both provide the ratio of machine response to a particular excitation versusexcitation frequency. The CLTF is the transfer function (ratio) of rotor position from rotor excitation by the AMB, whilethe influence coefficients are the ratio of rotor position response to rotor excitation by rotating unbalance. Theexcitation frequency for the unbalance response is typically shaft speed, thus gyroscopic effects are present. Thefrequency for the CLTF is independent of shaft speed. Indeed, the measurements can be performed both at zerospeed and with the machine running. A zero speed measurement will generally be a higher quality measurement,since it will not be contaminated by effects such as unbalance, aerodynamic noise, etc. On the other hand,measurements with the machine running are required to see the effects of gyroscopic effects, aerodynamic cross-coupling, etc.

Many modern digital AMB control systems are capable of measuring the CLTF without any external instrumentation,since all of the signals are generally available in digital form. The CLTF can also be calculated analytically.

2.10.7.3 Open Loop Transfer Functions

An open loop transfer function (OLTF) is extremely useful with regards to characterizing a system with a feedbackcontrol system. Conceptually, this transfer function is nothing more than the ratio of the compensator (control system)output signal to the input signal as shown in Figure 2-142.

In the case of an AMB system, however, there is a complication. Without the feedback control system active, thesystem is inherently unstable. The shaft would immediately be pulled over to the auxiliary bearings. Thus, it is notpossible to cut the feedback loop for an AMB system. The control system must be active. This situation is quitedifferent from the typical control system, where measurements can be made without all of the control system loopsactive.

This seemingly minor point is quite important with regards to the true open loop transfer function. Typically, an openloop transfer function is measured or calculated by disconnecting/ opening each feedback loop individually and

Figure 2-140—Closed Loop Transfer Function Measurement Locations

PlantExcitation

(Exc) Compensator

Current orforce

command(Cmd)



measuring or calculating the open loop response as the ratio of Cmd/Exc as shown in Figure 2-143. The goal of thismeasurement is to look at the behavior of the “Plant” (the rotor) and the “Compensator” (AMB system) only.

For an AMB system, a “pseudo open loop” transfer function is measured. The pseudo open loop transfer functionmeasurement is generally implemented in an AMB system as shown in Figure 2-144. This pseudo open loop transferfunction is extremely valuable for diagnostic or model tuning purposes. In particular, it provides a quick way to identifythe frequencies of the rotor free-free modes. It also provides a quick check that the general control system isfunctioning correctly.

This transfer function can easily be calculated analytically. So it can also be used for model tuning, especially for therotor model. Due to the similarity with the true open loop transfer function, many people working with AMB systemsrefer to this transfer function as the “open loop” transfer function. This can be a source of confusion in some cases.

Figure 2-141—Example AMB Closed Loop Transfer Function

Figure 2-142—Open Loop Transfer Function

VDC-XE Closed Loop Transfer Function

Frequency (Hz)

Frequency (Hz)

Ga

n, d

BP

hase

, deg

101

101

102

102

103

103

20

0

-20

-40

-60

-80

180

90

0

-90

-180

OutputPlantExcitation Compensator



An example of a pseudo OLTF is shown in Figure 2-145. This transfer function and the differences relative to a trueopen loop transfer function are discussed at length in references [2] and [3].

This pseudo open loop transfer function is not quite the same as the true open loop transfer function. There are twomajor issues that need to be considered:

— The coupling with the other control axes will generally increase the apparent damping of the free-free shaftmodes. Thus, instead of almost no damping, as would be measured with a true OLTF or rap test, there modescould appear to have noticeable amounts of damping. The frequencies, however, are generally correct.

— The coupling with the other control axes will generally affect the rigid body modes. In a true open loopmeasurement, these modes would appear at 0 Hz. However, with the other control loops active and the couplingdue to the shaft, these are likely to appear higher in frequency. This can lead to some confusion if only free-freemodes were expected.

Many modern digital AMB control systems are capable of measuring the pseudo OLTF without any externalinstrumentation, since all of the signals are generally available in digital form.

2.10.7.4 References

[1] Nise, N.S., 2004, Control System Engineering, 4th Edition, John Wiley.

[2] ISO 14839-3, 2006, “Mechanical Vibration—Vibration of Rotating Machinery Equipped with Active MagneticBearings, Part 3: Evaluation of Stability Margin,” No. 14939-3, International Organization for Standardization.

Figure 2-143—Typical Open Loop Transfer Function Measurement for Most Controlled Systems

Figure 2-144—Practical “Pseudo Open Loop” Transfer Function Measurement for AMB System

Plant

Open/disconnectfeedback loop

Excitation(Exc) Compensator

Current orforce

command(Cmd)

Plant

V1

PseudoOLTF =CmdV1

Excitation(Exc)

Compensator

Current orforce

command(Cmd)



[3] Schweitzer, G. and Maslen, E.H., editors, 2009, Magnetic Bearings: Theory, Design and Application toRotating Machinery, Springer.

2.10.8 Model Verification Testing

2.10.8.1 Overview

AMB systems inherently have the ability to both measure shaft motion, as well as apply excitation forces to the rotor.Thus, many rotordynamic performance measurements are more easily facilitated by AMB equipped machines ascompared to fluid film bearing equipped machines.

The AMB system allows measurements to be made with the machine running or with the machine not rotating.Calculation results for both the zero speed and running cases can be made available to compare with the measuredresults. A good correlation between the zero speed calculations and measurements will minimize the risks of anyproblems occurring when the actual running tests are performed. This is in contrast to the situation with a fluid filmbearing machine, where the machine must be operating to make meaningful measurements since unbalance istypically the only practical source of excitation force.

The specifications include an option to replace the traditional unbalance response model verification test with a CLTFmodel verification test. This test compares the measured CLTF to the predicted CLTF. This approach is especially

Figure 2-145—Example Pseudo Open Loop Transfer Function

VDC-XE Closed Loop Transfer FunctionG

an,

dB

Pha

se, d

eg

Frequency (Hz)

Frequency (Hz)

20

0

-20

-40

-60101

101

102

102

103

103

180

90

0

-90

-180



useful in the case of a sealed machine, where there is no convenient coupling hub on which to place a test weight. Inthese machines, the traditional unbalance test would require opening the machine casing.

2.10.8.2 Unbalance Response Tests

A traditional unbalance response test can still be used as a rotordynamic performance verification measurement.With an AMB machine, however, both shaft position and the AMB currents should be monitored. The AMB bearingcurrent signals provide a measurement that is proportional to the actual dynamic loading of the AMB. In most systemswith linearized actuator characteristics, the dynamic force will be directly proportional to the dynamic current over theuseful range of bearing loading. Technically though, the dynamic load is dependent on both the bearing currents andthe bearing air gap, but if the vibration level is a small percentage of the magnetic bearing gap (less than 10% of theair gap), then the dynamic component of the current measurement is a reasonable approximation of the dynamicbearing load.

The dynamic bearing currents (force) can be compared to the allowable bearing dynamic force envelope limits toensure that adequate reserve margin is maintained.

2.10.8.3 Model Verification Example

The results of one axis of a CLTF model verification test are shown in Figure 2-146. These measurements were madeon one radial axis of a small, high-speed machine. As can be seen, the lower shaft modes are extremely heavilydamped in both analysis and measurement. The frequency and amplification factor of the first shaft bending modenear 700 Hz are well predicted. There is some difference for the second mode near 1700 Hz. The machine has amaximum operating speed of 36,000 rpm. API 617’s Annex E requires that “the frequency of radial resonance peaksfrom the closed loop transfer function up to 1.25 Nmc shall not deviate from the corresponding frequency predictedby the analysis by more than ±5 %, and the measured peak amplitudes must not be greater than 1.0 times, nor lessthan 0.5 times the predicted amplitudes.” Thus, the match is acceptable. This measurement would be repeated on theremaining axes. If all are acceptable, then the model is validated.

2.10.9 Axial Rotordynamic Analysis


In a fully AMB supported rotor, there is also an axial AMB system. Thus, it is necessary to perform an axial analysis.An axial analysis is not typically performed for most fluid-film or rotating element bearing machinery. The mainconcerns for the axial analysis are stability and stability robustness, which are discussed in Section 3. However, thespecifications also require a damped natural frequency/modeshape analysis, which will be discussed in this section.

2.10.9.2 Axial Rotor Modeling Considerations

Generally, at least a partial train model is required to accurately model the axial dynamics. There have been caseswhere the dynamic behavior of a “flexible” disk-pack coupling were significant, so it is not a safe practice to assumethat a flexible coupling decouples the various bodies in the train for the axial analysis.

An adequate axial model can generally be obtained by considering each body in a train to be a rigid mass, similar to asimplified torsional analysis. Some other considerations include:

— Coupling disk pack stiffnesses should be included in the model if flexible coupling are used.

— Rigid couplings may result in only one rigid body mode that needs to be considered.

— Axial stiffness and damping of dry gas seal assemblies may need to be included.



— Axial stiffnesses due to magnetic centering forces from motors or generators, as well as the radial AMB systemmay need to be included.

— Axial stiffnesses due to aerodynamic centering forces may need to be included.

— The effect of eddy currents on the dynamic characteristics of the axial actuator may need to be considered. Ifsignificant, these effects should be included in the displacement to force transfer function provided by the AMBvendor.

— In some cases, accurate models have required additional degrees of freedom at thin, large diameter disks (thrustdisks, impellers, etc.) to account for disk flexibility effects from one or two nodal diameter modes of the disk.

— It may be necessary to consider the axial dynamics of the casing if there are casing modes in the control systembandwidth. The specification requires that the basis for the structural model be provided. In particular, if thestructure is assumed to be rigid, the basis for this assumption should be documented.

Figure 2-146—Closed Loop Transfer Function Model Validation

Frequency (Hz)101 102 103

Gai

n, d

B

-60

-40

-20

0

20VDC-XE Closed Loop Prediction/Measurement Comparison

AnalysisMeasured

Frequency (Hz)101 102 103

Phas

e, d

eg

-180

-90

0

90

180



2.10.9.3 Axial AMB Control System Considerations

Most of the comments related to radial control system models also apply for the axial dynamic model. In particular, thefull dynamics of the AMB plant (sensor, controller, power amplifiers and actuator) must be included in the analysismodel.

2.10.9.4 Damped Natural Frequency/Modeshape Analysis

The specifications require that the axial damped natural frequencies (eigenvalues) for all modes with an amplificationfactor greater than 2.5 be calculated. Given that a simplified, lumped mass model is being used for the axial system,this would be expected to be a fairly small list of modes, so no bandwidth limits are specified. From a practicalperspective, all of the axial modes would be calculated, since it is generally not possible to know in advance whichones would have an amplification factor less than 2.5.

The reporting requirements indicate that the modeshapes also need to be plotted or at least described in the report. Atypical report table for a two body system with a flexible coupling is shown in Table 2-6.

There are no explicit acceptance criteria for the axial modes in the standard. However, the axial sensitivity functionmust fall within zone B or better. Implicitly, this indicates that the modes must be stable, since the sensitivity functionanalysis is meaningless for an unstable system. It is also noted that modes which are dominated by motion within aflexible coupling are excluded, since the AMB system generally has little or no control authority over these modes.

2.10.10 As Installed Analysis

One very unusual feature of the AMB specifications is the requirement to perform a final set of analyses using the as-tuned parameters following initial field commissioning. The final rotordynamics report is not considered complete untilthis step is performed. At the time this requirement was developed, it was recognized that it will cause someadministrative complications. However, given that many AMB systems currently require at least some field tuning, itwas felt that it is crucial that the final rotordynamic report include consideration of the implications of this tuning.

As currently written, there are no acceptance requirements related to the results of this analysis. It is intended to bepurely informative for all parties. Obviously though, it is certainly desirable that the as-tuned system meet all of thestandard acceptance criteria.

2.10.11 Auxiliary Bearings

2.10.11.1 Introduction/Overview

It is expected that all API machinery using AMB’s will have an auxiliary bearing (also called “touchdown bearings,”“backup bearings,” or “catcher bearings”). These bearings are typically located adjacent to the radial actuators asshown previously in Figure 2-130. As described in the standard, these bearings are purely machinery protectiondevices. In many cases, they will have a very limited operational life. In addition, the rotordynamic performance of themachine when operating on the auxiliary bearings may be quite unacceptable by the criteria used to evaluate normaloperation. However, it must still be evaluated in so far as possible, to ensure that the auxiliary bearings can fulfill theirrequired function.

Table 2-6—Example Axial Damped Natural Frequency Analysis Results (Amplification Factors > 2.5)

Mode Frequency (Hz) Amplification Factor Description

1 80 2.5 Bodies Out of Phase

2 310 50 Spacer Mode



The goal of this section is to provide some background on the unique requirements for auxiliary bearings, evaluationof the rotordynamic performance through analysis and modeling, as well verifying the performance through droptesting.

It should be noted that there was no proven, industry standard approach for modeling and analysis of auxiliarybearings when the AMB specific standards were developed in 2009–2010. Therefore, the standards require that theauxiliary bearing performance be evaluated, but do not provide specific guidance for how the evaluation is to beperformed. The intent was to allow OEMs a great deal of latitude with regards to how to meet machinery operationalgoals, while hopefully encouraging the development of a consensus approach. The results of the evaluation areincluded as part of the overall rotordynamics report to allow comparison with future experience.

2.10.11.2 Auxiliary bearing function

The function of the auxiliary bearing system is to provide additional shaft support for three conditions:

— rotor not levitated by AMB system;

— AMB system failure;

— AMB system overload.

For each condition, the auxiliary bearing system must ensure that there is no rotor-stator contact at any closeclearance locations other than the auxiliary bearings (with the possible exception of abradable/compliant seals). Thefollowing paragraphs provide more detail for each of the scenarios in which the auxiliary bearing system is necessary.

The standard requires that the auxiliary bearings be designed for a minimum of two drops with a complete coastdown,and an agreed upon number of momentary contacts due to transient overload during operation, without requiringreplacement. Although infinite life would clearly be desired, this will probably be beyond the state of the art for quitesome time. The intent of the standards is to encourage a realistic discussion among all of the parties to ensure thatthe machine is capable of performing its intended function.

2.10.11.2.1 Rotor support when not levitated

Any time the AMB system is powered down for maintenance, inspection, or simply because it is not in use, theauxiliary bearing system must support the weight of the rotor. Thus, even a fault tolerant AMB system capable ofsupporting all possible operating loads will still have some sort of “auxiliary bearing” surface to support the rotor whenthe AMB system is inactive. For some machines, the auxiliary bearing must also support the rotor duringtransportation or installation. In this case, consideration must also be given to the relevant loading, which may belarger, be oriented differently, and/or have significant components due to vibration. Likewise, seismic events mayhave to be considered for some installations.

2.10.11.2.2 Rotor protection in the event of magnetic bearing failure

Although AMB system reliability is historically high, it is still possible that a component of the system can fail, bedamaged by electrical transients (lightning strike, for example), or physically damaged by some outside source(interconnecting cabling accidentally cut, for example). In these situations, the auxiliary bearing system must providethe necessary shaft support at the operating condition of the machine. This includes supporting the weight of therotor, the dynamic loads due to unbalance, and the radial and thrust loads associated with the process (i.e. side loadsand thrust loads). The auxiliary bearing system must provide this support instantaneously upon the event of a failure.It is typical to initiate an emergency stop in the event of AMB failure, so the auxiliary bearing system will only berequired to provide support long enough to bring the rotor safely to rest. This failure scenario is often referred to as a“landing” event, since the rotor essentially lands on the auxiliary bearings.



2.10.11.2.3 Rotor protection in the event of magnetic bearing overload

Because AMB systems have a limited load capacity, there can be overload conditions in which the AMB systemcannot provide the necessary support and additional bearing capacity is required. Typical examples are unexpectedprocess upsets that generate forces not considered during the design process, or bearings that cannot be practicallysized large enough to handle all possible operating conditions. In this scenario, the auxiliary bearing system mustprovide the extra load capacity required to prevent contact between the stator and rotor. Unlike the landing eventdescribed above, in this scenario the AMB system is still active, so the auxiliary bearing system only has to providepartial load support. Operation in a load sharing mode may continue until normal operation is restored, or brief loadsharing immediately followed by a trip and delevitation event where auxiliary bearings carry the entire load.

2.10.11.3 Auxiliary Bearing Types

Almost all API relevant AMB supported machinery use either rolling element bearings or solid lubricated bushings forthe auxiliary bearings. A possible exception is pumps, which might use process lubricated bearings or seal(s) thatdouble as auxiliary bearing(s).

Figure 2-147 provides sketches of typical arrangements. As shown in the sketches, it is most common for theauxiliary bearings to be located outboard of the AMB components. The choice of bearing type is a function ofapplication requirements and AMB vendor experience base. In each case, there are radial and axial clearancesbetween the bearing surfaces and the shaft during normal operation.

Typically, rolling element auxiliary bearings are either deep groove (radial only) or a duplex pair of angular contact(radial and thrust) bearings. Frequently, ceramic balls are used. Considerable care is required in the bearinggeometry and lubricant selection because of the combination of very high operating speeds and relatively largebearing diameter that is typically required. The operating DN values are often well beyond what would be acceptablefor continuous operation. The bearing race/ball/cage acceleration rates when the shaft contacts the bearing race arealso quite high. Special cages or cageless (full ball complement) designs may be required. The choice of materials forthe race and shaft, along with any lubrication or coating is also important.

A variety of solid lubricated bushing designs have been used. The bearings may or may not have segmented pads.The radial bearing may or may not be separate from the thrust bearing. Considerable care is required in selection ofbushing geometry, materials (bearing and shaft), lubrication, coatings and mounting due to the very high surfacespeeds involved.

Figure 2-147—Common Auxiliary Bearing Configurations

Compliantmounting

Radialmagneticbearing

Bushingtype

auxiliarybearing

RadialmagneticbearingCompliant

mounting

Rollingelementauxiliarybearing



2.10.11.4 Bearing Mounting Stiffness and Damping

The auxiliary bearings are almost always mounted in a support system that provides stiffness and damping (API 617requires this for compressors). The dynamic characteristics of this mount largely control the overall rotordynamicbehavior of the system during a delevitation or overload event. Figure 2-148 shows a typical arrangement.

Tradeoffs between competing requirements are generally required during auxiliary bearing mount design. Adequatedamping is required to reduce the dynamic loads on the auxiliary bearings during rotor impacts and to suppress rotorwhirl motion within the auxiliary bearing clearances. However, as with squeeze film dampers, too much dampingwould be undesirable, since it could prevent the mount from moving. This would again lead to very large whirlamplitudes and forces. Stiffness has two opposite effects: high stiffness will reduce rotor vibration amplitudes.However, high stiffness also tends to increase the rotor natural frequencies on auxiliary bearings, triggering higherfrequency rotor whirl and larger auxiliary bearings loads. Very high stiffness can also prevent the damping from beingeffective.

The mount stiffness characteristics are typically nonlinear for large loads due to the presence of a hard stop whichlimits maximum mount motion Additional nonlinearities due to friction damping, nonlinear springs, etc. are alsofrequently present.

Figure 2-148—Typical Auxiliary Bearing Mount System [4]

Structure

Damper ribbon

Angular contact bearing

Shaft Z

Y

1

2

3



2.10.11.5 Vertical Versus Horizontal Machinery

By their nature, AMB systems allow for operation in any orientation. The choice of the orientation has some influenceon the design of the auxiliary bearing system and the system rotordynamic characteristics.

For the horizontal application, there is almost always adequate radial load so that the rotor tends to sit at the bottom ofthe auxiliary bearing. This radial load is primarily the weight of the rotor, although additional side loads may have to beaccounted for as well. Two common examples are suction and discharge nozzles for compressors and pumps. Thesecan apply significant radial loads that increase or decrease the bearing load. Converging seals can also have verylarge centering forces. In the case where the auxiliary bearing can be unloaded by side forces, there is typically anincreased likelihood of large amplitude whirl.

The axial or thrust load will generally be related only to the fluid forces for horizontal machines. Depending on thedesign of the machine and the secondary flow management scheme, these fluid forces can be quite large, but theywill generally decay quickly with shaft speed. So, for the design of horizontal machines, gravity and fluid forces are themain drivers in auxiliary bearing selection with regard to load capacity. Schmied and Pradetto [1] present typical whirlresults for a horizontal machine which experiences a few circular orbits prior to settling down into the bottom of theauxiliary bearing race for the remainder of the landing event (Figure 2-149).

In vertical applications, the radial bearings are not loaded by the shaft weight. Therefore, the design load for the radialbearings is more difficult to assess. Because there is no gravity load to bias the rotor to any one lateral direction, therotor is more likely to whirl within the auxiliary bearing clearance. There are now two important terms which determinethe loads sustained by the auxiliary bearings; the radius of the whirl (i.e. the orbit of the shaft) and the frequency of thewhirl. Clearly, the rotor begins the landing event spinning at the operating speed, but it will establish a secondfrequency of rotation, in which the unit whirls about the center of the bearing clearance, while still spinning. Anexcellent example of this phenomenon is provided by McMullen et al. [2] and shown in Figure 2-150. In this case,there is clearly an orbit of the rotor around the circumference of the auxiliary bearing. There is also a synchronouscomponent of whirl which is due to rotor unbalance.

Depending on the whirl frequency (and the whirl radius), the radial load due to whirl can be large. Therefore, as adesign philosophy, it is important to minimize the whirl frequency of the vertical rotor while supported on auxiliarybearings. Prediction of the whirl frequency continues to be an active research subject, but is generally closely relatedto the support stiffness of the auxiliary bearing system. Research in flywheel applications has demonstrated that thesupport stiffness is the determining factor for these machines, whether that support stiffness is dominated by bearingcomponents or by the structure itself [3]. For oil and gas applications, the support stiffness may not be the onlydetermining factor. Ransom et al. [4] found that typical aerodynamic forces related to labyrinth seals and impellershrouds can drive the whirl frequency up. This is due to the presence of cross-coupled stiffness, which converts radialmotion into a forward whirl force.

For a vertical machine, the thrust bearing will experience the direct gravity load of the rotor in the event of delevitation.As mentioned above, fluid forces developed within the machine can be significant as compared to the rotor weight.Eventually though, the auxiliary bearing will carry the entire rotor weight as the fluid forces decay with shaft speed.

2.10.11.5.1 References

[1] Schmied, J. and Pradetto, J.C., 1992, “Behavior of a One Ton Rotor Being Dropped into Auxiliary Bearings,”Proceedings of the Third International Symposium on Magnetic Bearings.

[2] McMullen, P., Vuong, V., and Hawkins, L., 2006, “Flywheel Energy Storage System with AMB’s and HybridBackup Bearings,” Proceedings of the Tenth International Symposium on Magnetic Bearings.

[3] Caprio, M.T. et al., 2004, “Spin Commissioning and Drop Test of a 130 kW-hr Composite Flywheel,”Proceedings of the Ninth International Symposium on Magnetic Bearings.



Figure 2-149—Typical Horizontal Rotor Landing [1]

Figure 2-150—Vertical Rotor Landing Orbit [2]

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[4] Ransom, D., Masala, A., Moore, J.J., Vannini, G., Camatti, M., 2008, “Numerical and Experimental Simulationof a Vertical High Speed Motorcompressor Rotor Drop onto Catcher Bearings,” Proceedings of the 11thInternational Symposium on Magnetic Bearings, Nara, Japan, August 26-29.

2.10.11.6 Auxiliary Bearing Life Considerations

Auxiliary bearing systems are almost always very life limited systems. Thus, the key question we would really like toanswer from the rotordynamics/bearing analysis is, “how many drops will be possible?” The answer to this question isa complex combination of:

— rotor weight and operating speed;

— auxiliary bearing design;

— rotordynamic characteristics;

— Whether process fluids are present at the auxiliary bearing location during normal operation, machine upset, orstorage, and the composition of these fluids and thermodynamic conditions;

— potential for contaminates in the bearing compartment (rust, unexpected process fluids, etc.);

— thermodynamic conditions (temperature, pressure, cooling air) at the auxiliary bearing location.

Unfortunately, the current state of the art is not adequate to allow a reliable prediction of bearing life in the generalcase using readily available tools. As the statistics and experience become more widely known, statistical analysis,such as Weibull analysis, to quantify life data and predict component reliability may become practical.

From the rotordynamics perspective, a key to maximizing auxiliary bearing life is to avoid large amplitude dynamicforces due to unbalance and rotordynamic effects. Analytical predictions of dynamic loads of one or even two ordersof magnitude larger than the static load have been presented in the literature [1]. Clearly, these large loads can havea very detrimental effect on bearing life, if not leading directly to failure. Adequate control of unbalance state, andappropriate rotordynamic characteristics of both the rotor and the bearing/mount system are clearly required. There isalso interaction with the length of the contact or coastdown time.

The capability to meet service life, reliability, and observability requirements may involve tradeoffs in the design ofauxiliary bearing systems. The space envelope may grow, resulting in compromises in the rotordynamics of themachine, both when supported on magnetic bearings and when supported on auxiliary bearings. Compromises inrotordynamics may become a vicious circle where the rotor vibration amplitude and loads imparted to the auxiliarybearings increase, accompanied by increases in the bearing temperatures as well. Rotor-bearing system stabilitycould even be compromised in the extreme case. Thus, the design of an adequate auxiliary bearing is challenging.


[1] Kirk, R.G. 1999, “Evaluation of AMB Turbomachinery Auxiliary Bearings.” Journal of Vibrations and Acoustics,121, 2, pp. 156–162.

2.10.11.7 Rotordynamic Analysis and Modeling

There are two main concerns with regard to the rotordynamics of a rotor running on auxiliary bearings:

— excessive loading during the landing and stable whirl events;

— rubbing of shaft seals during landing and whirl events.



The full prediction of the landing event requires a transient, nonlinear analysis due to the time dependent nature of theevent and the nonlinear stiffness of the auxiliary bearing component. The whirl response analysis can be performedusing the same transient nonlinear analysis, or a simplified simulation using a steady state linear model with bearingcoefficients that have been linearized to the stable whirl condition can be used.

2.10.11.7.1 Transient Nonlinear Analysis of Landing Events

A transient nonlinear analysis of the landing event may be appropriate for two reasons. First, the initial landing eventis transient in nature. At the time of AMB de-levitation, the rotor must drop from the magnetic center to the radially (oraxially) offset location of the auxiliary bearing system. Following the initial impact, there will typically be several cyclesof relatively large amplitude vibration as the rotor comes to a new stable condition. The nonlinearity associated withthis transient scenario is primarily due to the nonlinear properties of the impacts which occur during the landing.Models that also include thermal effects are unusual. However, thermal effects can be quite significant, especially withregards to auxiliary bearing life.

The main model and analysis inputs to such a transient nonlinear analysis include finite element model of the rotor,orientation of gravity vector, dynamic coefficients for seals and aerodynamic components, nonlinear model of auxiliarybearings, nonlinear model of the auxiliary bearing mount, dynamic model of supports/casing (especially if casing isrelatively flexible), and rotor initial conditions. The nonlinear bearing and mount models can range from a fairly simplecontact friction plus contact model (i.e. Hertzian contact, coefficient of restitution, etc.) and/or as a nonlinear load-deflection curve, up to a very complex model that includes multiple contacts, detailed accounting of rolling elements,cage dynamics, and lubrication effects (see Figure 2-148). Component and/or subcomponent testing may also berequired [3-12]. The main outputs from a drop simulation are the peak displacement of the rotor relative to the statorat all critical locations, and the peak loads experienced by the auxiliary bearings.

It should kept in mind that the results of any transient calculation need to be benchmarked against experimental data.There are many application specific parameters that generally need to be tuned to achieve adequate predictions [8].These types of analyses should not be thought of as a standard rotordynamic calculation.

2.10.11.7.2 Steady State Analysis of Whirl During Coastdown

Another important consideration in the design of an auxiliary bearing system is the steady whirl response during thecoast-down event. While this can be obtained from a transient, nonlinear analysis, it can also be estimated from afairly standard steady state analysis if the whirl orbits are assumed to be “small” [2]. This simpler analysis may betterserve early system design tasks. Such an analysis will not predict the maximum response due to the initial drop, but itcan be used to predict the whirl frequency and the rotor operating deflected shape during the coast down, subject tothe assumption that orbits remain small. Obviously, this analysis would be less appropriate for a rotor that is driven tovery large amplitude whirl when passing through a critical speed.

The inputs to such an analysis are similar to those required for a typical rotordynamic design analysis and includefinite element model of the rotor, dynamic coefficients for seals and aerodynamic components, linearized coefficientsfor the auxiliary bearings, possibly a casing/support system dynamic model, and rotor initial conditions. Typicaloutputs include the operating deflected shape of the rotor and the rotor whirl frequency. From these two values, thebearing loads and the rotor-stator clearances can be estimated. Tuning parameters for such a design include theradial clearance, location and support stiffness of the auxiliary bearings.

All of the normal considerations for rotordynamic modeling apply for operation on auxiliary bearings. In particular, forcompressors and turbines, consideration needs to be given to effects such as seal cross-coupling. Aerodynamiccross-coupling can increase or decrease the likelihood of damaging whirl while operating on the auxiliary bearings,depending on whether the whirl is forward or backward.



2.10.11.7.3 Interpretation of Rotordynamic Analyses and Auxiliary Bearing Performance

Ultimately, there are two basic questions that we would like the auxiliary bearing analysis to answer. The first iswhether the auxiliary bearing design can fulfill the machinery protective function. The second question is whether theauxiliary bearing life will be adequate.

In principle, the first question is a rotordynamic performance question that can be answered though the analysistechniques outlined above. In practice though, the modeling of this nonlinear system support is quite challenging.There is also the difficulty of assigning a realistic worst case unbalance distribution. Since this is a very nonlinearsystem, an analytical prediction of an acceptable drop for one set of parameters does not guarantee acceptableperformance for all parameter combinations. Thus, the experience seems to be that the transient analysis can providegeneral guidance, but not precise answers.

Transient (and even linear static) analyses can be used as screening tools to confirm that the proposed auxiliarybearing system can at least provide the protective function under limiting assumptions.

The bearing life question is very difficult to address analytically. It is certainly strongly influenced by the rotordynamiccharacteristics of the system. But, it is also strongly influenced by the overall system design, rotor coast down time,static and dynamic loading, thermal environment, etc.

This tutorial is not intended to be an auxiliary bearing design guide. However, from a rotordynamic perspective, somesuggestions that have been made for improving auxiliary bearing life include [1,2]:

— operating below the first free-free critical speed;

— maintaining a well-balanced rotor (preferably stack balanced);

— ensuring that the damped mount structure dynamic characteristics are well matched to the rotor system;

— testing.


[1] Kirk, R.G. 1999, “Evaluation of AMB Turbomachinery Auxiliary Bearings.' Journal of Vibrations and Acoustics,121, 2, pp. 156–162.

[2] Swanson, E.E., Maslen, E.H., Li, G., and Cloud, C.H., 2008, “Rotordynamic Design Audits of AMB SupportedMachinery”, Proceedings of the Thirty-Seventh Turbomachinery Symposium, Texas A&M University.

[3] Kirk, R.G., Raju, K.V.S., Ramesh, K., 1997, “Modeling of AMB Turbo-Machinery for Transient Analysis”,Proceedings of MAG ‘97, Alexandria, USA, pp. 139–153.

[4] Swanson, E. E., Kirk, R. G., 1995, “AMB Rotor Drop Initial Transient on Ball and Solid Bearings,” Proceedingsof MAG ‘95, Alexandria, USA, pp. 227–235.

[5] Orth, M., Erb, R. and Nordmann, R., 2000, “Investigations of the Behavior of a Magnetically Suspended RotorDuring Contact with Retainer Bearings”, Proceedings of 7th International Symposium on Magnetic Bearings,August 23-25, 33–38, Zürich, Switzerland.

[6] Cole, M. O. T., Keogh, P. S., Burrows, C. R., 2002, “Predictions on the dynamic behavior of a rolling elementauxiliary bearing for rotor/AMB systems,” Proceedings of the 8th International Symposium on MagneticBearings, August 26-28, Mito, Japan, pp. 501–506.



[7] Keogh, P.S., Seow,Y.H., Cole, M.O.T., 2005, “Characteristics of a Magnetically Levitated Flexible Rotor Whenin Contact with One or More Auxiliary Bearings”, Proceedings of ASME Turbo Expo, Paper GT-2005-68583,June 6–9, 2005, Reno-Tahoe, Nevada, USA.

[8] Ransom, D., Masala, A., Moore, J., Vannini, G., Camatti, M., 2009, “Numerical and experimental simulation ofa vertical high speed motorcompressor rotor drop onto catcher bearings,” Journal of System Design andDynamics, Vol. 3, No. 4, pp. 596–606, [DOI: 10.1299/jsdd.3.596].

[9] Hawkins, L., Filatov, A., Imani, S., Prosser, D., 2006, “Test results and analytical predictions for rotor droptesting of an AMB Expander/Generator,” Proceedings of ASME Turbo Expo, Paper GT2006-90283, May 8–11, Barcelona, Spain.

[10] Kärkkäinen, A., Helfert, M., Aeschlimann, B., Mikkola, A., 2008, “Dynamic analysis of rotor system withmisaligned retainer bearings,” Journal of Tribology, Vol. 130, pp. 1-10. (021102-1 – 021102-10)

[11] Sun, G., Palazzolo A. B., Provenza, A., Montague, G., 2004, “Detailed ball bearing model for magneticsuspension auxiliary service,” Journal of Sound and Vibration, Vol. 269, pp. 933–963.

[12] Sun, G., 2006, “Rotor drop and following thermal growth simulations using detailed auxiliary bearing anddamper models,” Journal of Sound and Vibration, Vol. 289, pp. 334–359.

2.10.11.8 Auxiliary Bearing Testing

Rotor drop testing onto auxiliary bearings is typically performed to validate the auxiliary bearing life and rotordynamicperformance. This type of testing is typically performed at the end of the unit’s factory acceptance test at the OEM’sworkshop.

It should be noted that any drop testing will consume some portion of the auxiliary bearing life. Thus, test goals,bearing inspection procedures and bearing refurbishment/replacement should be discussed with the AMB vendorprior to testing. Upon completion of rotor drop testing, it is generally advisable to inspect the auxiliary bearings androtor prior to further unit operation. Damaged or worn-out components would need to be replaced with newcomponents. In the case of identical machines, very limited testing might be conducted after the design has beenvalidated on the initial machine. This limited testing might not require machine disassembly for bearing inspection/replacement.

The standard also mentions testing as part of “the basis of expecting the auxiliary bearing system to meet the designrequirements.” Although it is not explicitly stated, this paragraph is intended to allow component level tests of anauxiliary bearing design to be used in meeting the other requirements. For example, it may be very difficult toanalytically predict mount damping from a friction damper, but fairly straightforward to do a component level test.

2.10.11.8.1 Options and Test Strategies

When a rotor drop test is specified, a comprehensive planning discussion between the customer, machinery OEMand AMB vendor regarding test strategies and expectations should take place prior to the tests.

Two alternative types of tests can be performed to assess rotor and auxiliary bearings performances in the case of adelevitation event. These are referred as “Transient ‘proof’ testing” or “Full drop testing” depending on the extent ofthe rotor delevitation time.

2.10.11.8.2 Transient “Proof” Testing

This test is typically performed as a lower risk preliminary test, or when the test stand configuration does not allowreproducing a rotor coast-down time equivalent to what the machine would experience during field operation. For thistest, the machine is delevitated for a limited amount of time, then relevitated, all while running at some typical



operating speed or speed range. The goal would generally be to examine some aspects of rotor dynamic behavior. Inthe case of test stand limitations, one option might be to attempt to achieve auxiliary bearing temperature riseequivalent to what would be generated in the case of full drop down with the more rapid deceleration of a fully loadedmachine.

A proposed test strategy for turboexpander-compressors equipped with AMBs is a rotor delevitation period of threeseconds, followed by relevitation. The drop test would be performed from unit trip speed and with all control axessimultaneously shut down. This test sequence might be repeated several times before auxiliary bearing inspectionand final replacement.

Upon completion of rotor drop testing, it is generally advisable to inspect the auxiliary bearings and rotor prior tofurther unit operation. Damaged or worn-out components would need to be replaced with new components.

2.10.11.8.3 Full Drop Testing

A full drop test would be performed from agreed upon operating conditions (speed, power, pressures, etc) at machinetrip speed down to rotor standstill condition: this test might be appropriate when the final acceptance testconfiguration is able to guarantee drop conditions and rotor deceleration similar to the ones that might occur duringfield operation.

A rotor drop test is typically aimed to simulate the worst case failure scenario represented by complete failure of themain AMB system and simultaneous shut-down of all control axes. In certain cases, a partial axis delevitation test canbe performed to reproduce process related rotor drop events or investigate specific dynamic behavior.

Because of the risk for auxiliary bearings and machine integrity associated with the rotor drop onto auxiliary bearings,special attention must be paid to rotor drop conditions identification, procedure preparation, as well as measurementcollection and handling.

To prevent catastrophic failures of the auxiliary bearings during the test, bearings and rotor failure modes and damageindicators should be clearly identified and investigated before each drop test. Typical information to be leveragedduring the test to identify ongoing damage mechanisms are:

— auxiliary bearings clearance measurements;

— auxiliary bearings temperature measurements;

— auxiliary bearings and supports vibration signature;

— rotor transfer function measurements;

— modification on machine performances.

To identify auxiliary bearing degradation after each test, the drop conditions (i.e. speed, gas pressure, etc.) shouldgenerally be maintained constant for each drop and the results compared to the previous test to identify abruptmodification or unusual trends.

Numerical results on rotor drop dynamics predictions, auxiliary bearing loads, and thermal regimes should also beused as references to determine when excessive stress conditions are being sustained by the auxiliary bearings androtor parts during the test.

Upon completion of rotor drop testing, it is generally advisable to inspect the auxiliary bearings and rotor prior tofurther unit operation. Damaged or worn-out components would need to be replaced with new components.



2.10.11.8.4 Facility Requirements and Instrumentation

Rotor drop tests are typically performed at OEM testing facilities in a protected and safe area. Emergency machineprotection systems to recover the rotor in case of unexpected performances during the landing test event should beidentified and implemented. Due to the fast dynamics of the drop event, automated relevitation of the rotor is usuallypreferable to manually operated command [1].

In the case of transient “proof” testing, pilot transient drop at low speed or partial axis delevitation tests may beadvisable to verify that the protection and relevitation systems are correctly implemented and operating. When a fulldrop test is going to be performed, a pilot machine coast-down test from identified drop conditions is typicallyperformed before starting the drop test, in order to verify that the rotor deceleration trend is in accordance withpredicted values.

In addition to the standard AMB displacement probes, special instrumentation may be required to capture the rotordrop dynamics and perform the condition monitoring activities necessary to verify the integrity of the auxiliarybearings after each drop test. These may include temperature probes, accelerometers or acoustic devices located onauxiliary bearings supports. Additional dynamic probes to identify fast transients of process parameters inside themachine may be required. Fast data acquisition systems (>20 kHz sampling rate) are typically used to capture thedrop dynamics and dedicated post-processing and verification tools are used to verify proper performances andhealth of the auxiliary bearings.


[1] Hawkins, L., McMullen, P., Vuong, V., 2007, “Development and Testing of the Backup Bearing System for anAMB Energy Storage Flywheel,” Proceedings of the ASME Turbo Expo, Paper GT2007-28290.

2.10.12 Machinery Specific Considerations

Each machine type has slightly different issues, both for the AMB system and the auxiliary bearing system. Thissection covers some of the more common issues from both perspectives.

2.10.12.1 Turbine Versus Motor Driven Machinery and Auxiliary Bearings

In general, quick coast downs are preferred for maximizing the life of the auxiliary bearing components, since life isrelated to both the load on the bearing system as well as the number of cycles. The shorter the shutdown transient is,the fewer the cycles accumulated in a single emergency stop. Very rapid passage through critical speeds can meanless time for vibrations to build-up to large amplitudes. Motor driven compressors can typically be started and stoppedin very short periods of time. Turbine driven machines, typically, have longer coast-down times.

2.10.12.2 Integrally Motor Driven Machinery

As with any motor or generator, there is the potential for a large negative magnetic stiffness due to electromagneticeffects. These machines may also have more than two bearings, with rigid couplings or a single shaft. Thesemachines require a full train analysis of the entire rotating assembly.

2.10.12.3 Corrosive Fluids (Sour Gas)

For corrosive fluids, such as those in sour-gas applications, the AMB and auxiliary bearing components must beprotected against the corrosive exposure. For the AMB system, this is usually done by “canning” the stationarycomponents. For a canned bearing, the stator components (actuator and sensor) are completely encased in acorrosion proof container. The AMB system (actuator and sensor in particular) dynamic model must consider theeffects of this corrosion proof liner. It is very difficult, however, to can the auxiliary bearings, since they must contactthe rotor. Thus, they must survive in the corrosive environment. This means very careful selection of materials,lubricants and coatings for both the bearings and the compliant bearing mount system.



2.10.12.4 Multiphase Fluid and Pump Applications

Multiphase machinery and pumps can have significant additional load due to fluid effects. The possibility of addedmass effects and cross-coupling from close clearance, annular spaces for flooded machines must also beconsidered.

2.10.13 AMB Data Requirements for Lateral/Axial Reports and Data for Independent Analysis

There are a few AMB specific items that are added to the standard lateral report for an AMB supported machine.These items include the following.

— General actuator and rotor component dimensions, along with the actuator coordinate system (i.e., pole faceorientations). The intent being that a basic sanity check of actuator sizing versus expected load is possible.Although not explicitly stated, the nominal rotor-stator air gap would be expected to be included.

— A plot of the allowable force envelope and what factor of safety was assumed.

— Plots of the AMB system transfer functions used for the rotordynamic analysis, including any significant cross-coupling.

— The auxiliary bearing gap(s) and any speed dependency.

— The free-free map.

— The bearing forces relative to the allowable force per the specified force envelope.

— The closed loop transfer function if the optional closed loop model verification test is specified.

— The results of the axial natural frequency analysis, including either mode-shapes and/or a brief discussion ofeach mode.

— The as-installed analysis results using the controller coefficients from the initial field commissioning (final report).

There are also a few items more nearly related to stability analysis that are described in Section 3. The standard alsoincludes a list of the minimum additional information that is to be provided if an independent analysis is planned. Thisincludes the following.

— Axial locations and angular orientations of sensors and actuators.

— Coefficients for “adequate,” AMB displacement to force transfer functions, including all relevant dynamic effects,as well as any negative stiffness not accounted for in the coefficients. Any significant coefficient variations due togain scheduling or other effects also need to be described. These are to be provided in physical coordinates, sothat they can be directly used in a rotordynamic model. Transformed coordinates, such as “tilt-translate,” do notmeet this requirement. The substitution of first-order state-space matrices for transfer functions is allowed if allparties agree, since this may be more appropriate. The intent of this data requirement is to ensure that theindependent model will be able to accurately predict the rotordynamic behavior of the system (assuming all of theother system elements are accurately modeled). If there are significant nonlinearities or other unusualcharacteristics to the control system, it is expected that the data provided would allow someone with areasonably good AMB background to model them appropriately.

— Data for the allowable force envelope in tabular form, as well as a brief discussion of what issues wereconsidered.



— Enough basic information about the system to allow the reasonableness of the dynamic force envelope to beconfirmed. This requirement was intentionally left somewhat vague, since it was not clear what specificinformation would be most appropriate. There should be enough information to perform a check on the static loadcapacity, as well as the basic slew rate limit(s).

— Enough information on the auxiliary bearing system to allow at least a static sag analysis and examination of theapproximate undamped critical speed map when supported by the auxiliary bearings (for a horizontal rotor).


3-1

SECTION 3—STABILITY ANALYSIS

3.1 Introduction

3.1.1 Historical Perspective

A stability analysis is also referred to as a damped natural frequency or damped eigenvalue analysis. For a rotor-bearing-seal system, a stability analysis leads to a multitude of solutions corresponding to each of the system’sdamped natural frequencies. The solutions to the equations of motion are complex with real (growth factor or stabilityindicator) and imaginary (damped natural frequency or instability frequency) parts called eigenvalues. The associatedmode shapes are called eigenvectors.

Measuring the rate of decay of free oscillations (real part of the eigenvalue) is a convenient way to determine theamount of damping present in the system. Greater damping values produce faster rates of decay. For stabilityanalysis of rotating equipment, the logarithmic decrement (log dec) is a common measure of damping present andthus, the stability. The log dec is defined as the natural logarithm of the ratio of any two successive amplitudes, Figure3-1. The expression for the log dec is then:

(3-1)

For stable systems with a positive rate of decay, the log dec is positive. For unstable systems with a negative rate ofdecay (growth rate), the log dec is negative. From a damping point of view, stable systems with positive log decvalues contain sufficient damping to overcome the excitation, resulting in a negative decay rate. Conversely, unstablesystems with negative log dec values do not contain sufficient damping to overcome the excitation, resulting in apositive decay rate.

The evaluation of dynamic stability has become an essential element for rotating machinery design analysis. Thecurrent capability to compute the damped eigenvalues of complex rotor-bearing-seal systems was motivated bymajor field problems in the 1960s and early 1970s wherein the major vibration characteristic was self-excitedsubsynchronous vibration. Two famous and classic centrifugal compressor instability cases from the early 1970s arereferred to as Kaybob, Smith [1] and Ekofisk, Booth [2] (see Figure 3-48).

The ability to compute the eigenvalues of a six degree of freedom system (i.e. a six rotor station model) wasconsidered a great design tool in the 1960s and early 1970s. Attempts to find higher order system eigenvalues (i.e.rotor models with more than six stations) proved very difficult and time consuming. The first attempt at thedevelopment of a stability computer code was by Jorgen Lund [3] in 1965. This code, however, was not easy to useand it was said that only if you knew the approximate answer would it converge on the solution. Also of note from themid-1960s are two classic publications by Gunter where basic stability methodology [4] and internal friction excitation[5] are discussed.

The work of Ruhl and Booker [6] published in 1972 presented both finite element and transfer matrix techniques forrotor systems employing a solver known as Muller’s method. For lightly damped systems, their transfer matrixsolution analysis worked fine but for more heavily damped structures, such as turbomachinery supported on fluid filmbearings, the program’s analysis methodology produced incorrect and false modes.

Jorgen Lund presented his landmark paper on rotor bearing stability in 1973. Lund’s paper [7], published first in 1974and again in 1987, not only gives a detailed transfer matrix solution procedure, but also describes how the resultsmay be presented to study machine design parameters. The most basic result of a damped eigenvalue or dampednatural frequency analysis is the stability prediction from the real part of the eigenvalues. It is usually the lowest mode,which corresponds to the rotor’s first fundamental natural frequency that is “re-excited” causing a subsynchronous

X1X2------

ln=



vibration and rotor instability. Coincidently, the beauty of the transfer matrix solution is that it is able to search for thefirst 6 to 10 modes very efficiently, finding the lowest frequency modes first.

Ruhl’s transfer matrix analysis was updated to include flexible supports by Bansal and Kirk [8] in 1976, replacingMuller’s solver with a Cauchy-Rieman condition finite difference algorithm plus a Newton-Raphson search solutionspecified by Kirk [9]. This was essentially the same solution as used by Lund [7] but with minor differences. Bothprocedures work but will occasionally skip modes. This problem is more likely to occur when asymmetric flexiblesupports are included at the bearing locations. Other transfer matrix computer programs have been developed basedon Lund’s original analysis such as Barrett and Gunter [10].

More recent computer codes are based on a finite element solution that is successful in extracting all of the correctmodes. These finite element code authors include Nelson [11], Rouch [12], Chen [13], Edney [14], and Ramesh [15].One disadvantage of the finite element analysis is that the problem size increases dramatically with the number ofelements used to model the rotor. Consequently, the run times with the finite element analysis are longer than with thetransfer matrix analysis depending on the solution technique employed. Generally, however, the numericaltechniques used with the finite element analysis are quite robust and do not miss any roots. Some methods requirethat all of the roots are found extracting the highest eigenvalue (and hence natural frequency) first and ending with thelowest, which is usually the only mode of interest. The run times associated with these methods, therefore, areconsiderably longer than those used with the transfer matrix analysis. Another approach employs a substitutiontechnique to reduce the second order set of equations to first order form in which the efficient QR algorithm can beused. Other methods used to reduce run times involve extracting the lowest eigenvalues first, either by solving theinverse problem or using determinant search techniques.

3.1.2 Controversial Issues

There are two main areas of a rotor-bearing-seal stability analysis that remain controversial. The first is what shouldbe used for the tilting pad journal bearing stiffness and damping coefficients, which are a function of the pad tiltfrequency. The frequency in question is the synchronous or once per shaft revolution frequency versus thesubsynchronous damped natural frequency or instability frequency. The use of both synchronous andsubsynchronous characteristics has been commonly applied over the years and both methods have been successfulin predicting and solving complicated turbomachinery stability problems. This issue is further discussed in 3.3.3.1.

Figure 3-1—Definition of Log Dec Based on Rate of Decay

X1X2

X

t



The second area of controversy concerns the labyrinth seals and aerodynamic cross-coupling. For years, it wasbelieved that the major destabilizing driving force was an aerodynamic excitation produced by impeller or bladeinteraction with the stator. Empirical equations by Alford and Wachel were developed and used extensively to predictthese destabilizing aerodynamic cross-coupling forces. Recently, labyrinth seal code development has shown that thelabyrinth seals produce a major destabilizing excitation. This has lead to the belief by some rotordynamicsts that thelabyrinth seals and balance piston produce the major destabilizing excitation in centrifugal compressors. Othersmaintain that impeller aerodynamic excitation is a significant driving force for compressor instability. These topics arecovered in detail in 3.4.2, 3.5.1, and 3.8.6.

3.1.3 Stability Specifications—Standard Paragraphs

The stability specification is segmented into two parts: a simplified Level 1 analysis and a detailed Level 2 analysis(see flow chart, Figure 3-2). The Level 1 analysis is meant to be a screening process in which a quick and simpleanalysis can be conducted to filter out machines that are well away from the instability threshold. Level 1, therefore,utilizes a modified Alford’s equation to estimate the destabilizing forces. Level 2, however, requires that the dynamicproperties of labyrinth seals be included implicitly through the use of an appropriate labyrinth seal code.

Currently the stability specification has continued the use of synchronous tilting pad bearing characteristics eventhough frequency dependency may significantly influence the rotor's predicted stability characteristics. While furtherexperimental work is desired, consideration will be given in the next revision of the stability specifications to requirethe use of nonsynchronous, frequency dependent tilting pad coefficients for stability analyses. This is discussed indetail in 3.3.3.1.

NOTE It is recognized that the stability specification is a work in progress and that revisions will be made as the state-of-the-artin rotor, bearing, and seal dynamics progresses.

The following tutorial sections basically outline the critical components of a stability analysis. These are grouped andpresented in a flow chart of a typical process as depicted in Figure 3-2. The procedure follows the specification setforth in 6.8.5 and 6.8.6 of the SP R22. For analysis of rotors with potential field related stability vibration problems, aLevel 2 analysis should be used, as outlined in 6.8.6, even if the rotor passed the Level 1 analysis, as described in6.8.5.



Figure 3-2—Stability Analysis Flow Chart

Rotormodel

Bearinganalysis

Oil sealanalysis

SFDanalysis

Level Istability analysis

Evaluate vs.screening

criteria

AcceptabledesignPass

Level 2stability analysis

Fail

Excitationapproximation

Labyrinth seals anddamper seals

Evaluate vs.stability criteria

Aerodynamics and 2nd order effects

Pass

Fail

RedesignYes

Mutualacceptability agreement

No



3.1.4 References

[1] Smith, K. J., 1975, “An Operational History of Fractional Frequency Whirl,” Proceedings of the FourthTurbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp.115–125.

[2] Booth, D., 1975, “Phillips’ Landmark Injection Project,” Petroleum Engineering, pp.105–109.

[3] Lund, J. W., 1965, “Rotor Bearing Dynamics Design Technology, Part V,” AFAPL-TR-65-45, Aero PropulsionLaboratory, Wright-Patterson Air Force Base, Dayton, Ohio.

[4] Gunter, E. J., 1966, Dynamic Stability of Rotor-Bearing Systems, NASA SP-113.

[5] Gunter, E. J., 1967, “The Influence of Internal Friction on the Stability of High Speed Rotors,” ASME Journal ofEngineering for Industry, Series B, 89 (4), pp. 683–688.

[6] Ruhl, R. L. and Booker, J. F., 1972, “A Finite Element Model for Distributed Parameter Turborotor Systems,”ASME Journal of Engineering for Industry, Series B, 94 (1), pp. 126–132.

[7] Lund, J. W., 1974, “Stability and Damped Critical Speeds of a Flexible Rotor in Fluid Film Bearings,” ASMEJournal of Engineering for Industry, 96 (2), pp. 509–517.

[8] Bansal, P. N. and R. G. Kirk, 1975, “Stability and Damped Critical Speeds of Rotor-Bearing Systems,” ASMEJournal of Engineering for Industry, Series B, 97 (4), pp. 1325–1332.

[9] Kirk, R. G., 1980, “Stability and Damped Critical Speeds: How to Calculate and Interpret the Results,” CAGITechnical Digest, 12 (2).

[10] Barrett, L. E., Gunter, E. J. and Allaire, P. E., 1976, “The Stability of Rotor-Bearing Systems Using LinearTransfer Functions—A Manual for Computer Program ROTSTB,” Report No. UVA/643092/MAE81/124,School of Engineering and Applied Science, University of Virginia, Charlottesville, Virginia.

[11] Nelson, H. D. and McVaugh, J. M., 1975, “The Dynamics of Rotor-Bearing Systems using Finite Elements,”ASME Journal of Engineering for Industry, 98 (2), pp. 593–600.

[12] Rouch, K. E. and Kao, J. S., 1979, “A Tapered Beam Finite Element for Rotor Dynamics Analysis,” ASMEJournal of Sound and Vibration, 66, pp. 119–140.

[13] Chen, W. J., 1996, “Instability Threshold and Stability Boundaries of Rotor-Bearing Systems,” ASME Journalof Engineering for Gas Turbines and Power, 118, pp.115–121.

[14] Edney, S. L., Fox, C. H. J. and Williams, E. J., 1990, “Tapered Timoshenko Finite Elements for RotorDynamics Analysis,” Journal of Sound and Vibration, 137 (3).

[15] Ramesh, K. and Kirk, R. G., 1993, “Stability and Response of Rotors Supported on Active Magnetic Bearings,”Proceedings of the 14th ASME Vibrations and Noise Conference, DE- 60, Vibration of Rotating Systems, pp.289–296.

3.2 Rotor/Bearing System Modeling

3.2.1 Baseline Stability

Baseline stability refers to a stability analysis performed with only the effects of the rotor’s support system included inthe stability model. The support system may include any of the following as applicable; the journal bearings (see 2.5and 3.3), the oil seals (see 3.4.1), the squeeze film dampers (see 3.3.4) and the support or casing flexibility (see 2.4and 3.6). The mechanical rotor model is identical to that used in the undamped critical speed analysis or synchronousresponse analysis (without the defined unbalance).

Fixed geometry bearings are modeled with eight coefficients. Four coefficients represent the direct stiffness anddamping while the remaining four represent the cross-coupled stiffness and damping. Tilting pad bearings will have



additional frequency dependent coefficients as described in 3.3.3.1. The tilting pad bearing may be modeled in eitherof three methods. The bearing may be represented by 5N+4 stiffness and 5N+4 damping coefficients referred to as“full” or “pad dynamic” coefficients where N = number of tilting pads. Alternatively, the bearing is often modeled usingthe standard four stiffness and four damping coefficients that have been “reduced” from the full pad dynamiccoefficients using the synchronous frequency (synchronously reduced coefficients) or reduced using the instability ordamped natural frequency (frequency dependent coefficients). For a more detailed explanation, refer to 3.3.3.1.

It should be recognized that the method chosen to represent tilt pad bearings will affect the results of the stabilityanalysis. In some cases, the difference between an analysis with frequency dependent coefficients andsynchronously reduced coefficients may be quite large (see 3.3.3.1). Any resulting stability requirement must considerwhich tilting pad journal bearing modeling method is employed and the “pass/fail” criteria adjusted accordingly. Ineither case, the bearing coefficients should be calculated at the extremes of the manufacturing tolerances that resultin minimum and maximum values of clearance and the corresponding maximum and minimum values of preload forfixed lobe or tilting pad bearings. In addition, the normal operating range of oil inlet temperature should be considered.In some cases, it may prove beneficial to include the permissive-to-start oil inlet temperature. Stability calculationsshould be made at such extremes.

Another important modeling consideration is the pedestal or support stiffness and damping properties. These supportcharacteristics, acting in series with the bearing’s oil film stiffness and damping coefficients, can have a profoundeffect on the resulting rotor system stability parameters. The support flexibility includes everything past the bearing’soil film; the bearing case, the bearing pedestal, the machine casing, the base plate, the supporting columns and thefoundation. When tilting pad journal bearings are used, the flexibility of the tilting pads and the pad pivots must also bemodeled. In general, the support flexibility is dynamic. That is, the support stiffness and damping properties are afunction of the applied vibration frequency. Refer to 3.6 for a more detailed discussion.

Any external forces, such as gear reaction components, shall be included when the bearing coefficients arecalculated. In the event that several operating conditions are specified for the machinery, these must also beevaluated to insure that the extremes in the stability analysis have been identified. Any factor that is a variable in thespecified operating condition that influences the bearing dynamic coefficients should be considered to insure that theminimum and maximum log decrement of the rotor has been determined.

3.2.2 Analysis with Additional Cross-coupling

The baseline model shall be modified to include one or more additional “bearing or support” locations to represent thedestabilizing effects of components such as internal close clearance seals (e.g. labyrinth seals) or other identifiedareas within the machinery that will destabilize the rotor. The “bearing” may represent the combination of thedestabilizing forces at one rotor location, such as mid-span for a conventional centrifugal compressor or steamturbine, or the center of gravity of an impeller for an overhung design.

The combined destabilizing effects will be represented by a “bearing” with an equivalent or combined cross-coupledstiffness. For an axial turbine or axial compressor the combined equivalent cross-coupled stiffness is represented bythe Alford equation (see 3.5.1.2). For a centrifugal compressor, the combined equivalent cross-coupled stiffness isrepresented by the modified Alford equation (see 3.5.1.3). The resulting total excitation value is then generally placedat the rotor centerline for stability calculations. Other methods to represent the destabilizing effects include usingthese equations on a stage-by-stage basis and then placing each calculated excitation at the specific stage locationsalong the rotor. Alternately, each calculated stage excitation value may be weighed by a factor determined from thenormalized rotor amplitude of the mode in question and then the total weighed value placed at the rotor center.

In cases where the destabilizing effects are quantified more directly, using the results of specialized codes to analyzeseals, the destabilizing effects may be located at each component that has been analyzed. In such cases, thedestabilizing effects may be represented with the eight coefficients obtained from the component analysis or with anequivalent cross-coupled stiffness. Regardless of which method is used to determine the destabilizing mechanism,the excitation must be evaluated at the conditions that will result in the maximum and minimum log decrement of therotor.



3.3 Journal Bearings

3.3.1 General

The stability of a turbomachine or any dynamic system is significantly influenced by the amount of damping present.In most applications, the bearings provide the majority of damping in the rotor system, their design is crucial formaintaining adequate stability.

The following subsections describe some of the special aspects of fixed geometry bearings, tilting pad bearings andsqueeze film dampers with regards to stability. Identified early as a major influence on rotor stability (Newkirk [1]),bearing characteristics can now be predicted fairly accurately in order to avoid such phenomena as oil whirl and shaftwhip.

While stability improvements may be obtained by reducing the destabilizing forces of particular components (seals,bearings, etc.), the journal bearings, squeeze-film dampers, and damper seals along with the rotor govern themachine’s ability to withstand and dissipate such forces, i.e. the machine’s stability robustness. One element inobtaining stability robustness is the optimization of the journal bearing characteristics relative to the shaft stiffness.The degree of asymmetry between the bearing’s horizontal and vertical characteristics can also greatly impact thesystem’s stability [2,3,4]. Since there are many design variables available that determine a journal bearing’s stiffnessand damping properties, proper design may be attainable (Nicholas et al. [3]). In some cases, however, the bearingcharacteristics cannot be optimized adequately, especially for high pressures and high densities, making squeeze filmdampers and/or damper seals a necessity.

3.3.1.1 References

[1] Newkirk, B. L. and Taylor, H. D., 1925, “Shaft Whipping Due to Oil Action in Journal Bearing,” General ElectricReview, 28, pp. 559–568.

[2] Gunter, E. J., 1966, “Dynamic Stability of Rotor-Bearing Systems,” NASA SP-113, pp. 153–157.

[3] Nicholas, J. C., Gunter, E. J. and Barrett, L. E., 1978, “The Influence of Tilting Pad Bearing Characteristics onthe Stability of High-Speed Rotor Bearing Systems,” Report No. UVA/643092/MAE81/141, School ofEngineering and Applied Science, University of Virginia, Charlottesville, Virginia, pp. 30-32. Also in Topics inFluid Film Bearing and Rotor Bearing System Design and Optimization, an ASME publication, April 1978. pp.55–78.

[4] Wohlrab, R., 1976, “Einflub der Lagerung auf die Laufstabilitat einfacher Rotoren mit Spalterregung,”Konstruktion, 28, pp. 473–478.

3.3.2 Fixed Geometry Journal Bearing Stability

3.3.2.1 Unstable Journal Bearings

Fixed geometry or sleeve bearings create an excitation force that can drive the rotor unstable by creating asubsynchronous vibration. This phenomena usually occurs at relatively high rotor speeds and/or light bearing loads,or, more generally, at high Sommerfeld Numbers.

The Sommerfeld Number, S, is defined below:

(3-2)SNLD60W

---------------- DCd------- 2

=



where

is the average fluid viscosity, Pa-s (lb-s/in.2);

N is the rotor speed, rpm;

L is the bearing length, m (in.);

D is the bearing diameter, m (in.);

W is the bearing load, N (lbf);

Cd is the bearing diametral clearance, m (in.).

The problem is that sleeve bearings (i.e. all journal bearings excluding tilting pad bearings) support a resultant loadwith a displacement that is not directly in line with the resultant load vector but at some angle with rotation from theload vector. This angle can approach 90° for light loads and high speed. The specific case of a vertically downwardgravity load is illustrated in Figure 3-3 for a two axial groove bearing. Now, the sleeve bearing supports this verticallydownward load with a displacement that is not directly downward but at some angle with rotation from bottom deadcenter. This angle is defined as the attitude angle, as shown in Figure 3-3.

Figure 3-4 illustrates the hydrodynamic pressure distribution for a high-speed, lightly loaded, unstable journal bearing(Nicholas [1]). Note that the bearing eccentricity ratio, = e/c (Figure 3-3), is very small and the attitude angle, ,approaches 90°. In this manner, a light –Y direction load is supported by a +X displacement. This occurs since theload is so light, the resulting pressure profile becomes very small with very little change from the maximum film to theminimum film locations. For equilibrium, the summation of all vertical components of the hydrodynamic forces times

Figure 3-3—Fixed Geometry Bearing Schematic

Eccentricity ratio

Attitude angle,

X

Oil atviscosity

Y

W

D

R

e

R + c

ob

oj

= e/c,



the area must be equal and opposite to the external load, W. Likewise, the sum of all horizontal forces must be zero.This can only occur for attitude angles that approach 90°.

Since a downward load is supported by a horizontal displacement, any downward force perturbation will result in ahorizontal displacement that will result in a horizontal force that in turn produces a vertical displacement, etc. Thus,the bearing generates unstable cross-coupling forces that actually drive the rotor and cause it to vibrate at afrequency that is normally in the range of 40 % to 50 % of running speed.

3.3.2.2 Stable Journal Bearings

Figure 3-5 [1] illustrates a relatively low speed, heavily loaded (i.e. low Sommerfeld Number), stable journal bearing.Note that the bearing eccentricity ratio, , is very large and the attitude angle, , approaches 0°. In this manner, aheavy –Y direction load is supported by a –Y displacement. This occurs since the load is so heavy, the resultingpressure profile becomes very large with very large gradients from the maximum film to the minimum film locations.From a force summation, Fx = 0 and Fy = W. This can only occur for attitude angles that approach 0°. Since adownward load is now supported by a vertical displacement, cross-coupling forces are at a minimum and the bearingis stable.

3.3.2.3 Oil Whirl and Shaft Whip

Sleeve journal bearing induced oil whirl and shaft whip are illustrated in Figure 3-6 [1]. The 1x or synchronousvibration line is clearly indicated on the plot. Oil whirl, caused by the destabilizing cross-coupling forces produced byhigh-speed, lightly loaded sleeve bearings (i.e. high Sommerfeld Number, Section 3.3.2.1), manifests itself asapproximately a 50 % of running speed frequency (shaft vibrates approximately once per every two shaft revolutions).This can be seen in Figure 3-6 at speeds below about 7500 rpm (below twice the rotors first critical).

Above 7500 rpm, the instability frequency locks onto the rotor’s first fundamental natural frequency, which is at about3800 cpm. This re-excitation of the rotor’s first natural frequency is sleeve bearing induced shaft whip that shows up

Figure 3-4—High-Speed, Lightly Loaded, Unstable Bearing [1]

+ +

W

Light Y loadsupported by

+X displacement

Light load/high speedhigh cross coupling

unstable

Y

XOj

Ob



as a vibration component that is below 50 % of running speed and occurs at speeds that are above twice the rotor’sfirst critical.

Usually, sleeve bearings are designed not to go unstable until the rotor speed exceeds twice the rotor’s first criticalspeed. Thus, an approximate 0.5x is a rare occurrence and bearing induced instabilities usually show up as shaftwhip at frequencies less than 50 % of synchronous speed.

Axial groove bearings have a cylindrical bore with typically 2- to 4-axial oil feed grooves. These bearings are verypopular in relatively low-speed equipment. For a given bearing load magnitude and orientation, the stabilitycharacteristics of axial groove bearings are primarily controlled by the bearing clearance. Tight clearances producehigher instability thresholds but tight clearance bearings present other problems that make them undesirable. Forexample, as clearance decreases, the bearing’s operating oil temperature increases. Furthermore, babbitt wearduring repeated start-ups will increase the bearing’s clearance thereby degrading stability. In fact, many bearinginduced instabilities in the field are caused by bearing clearances that have increased due to wear from oilcontamination, repeated starts or slow-rolling with boundary lubrication.

Because of these limitations, other fixed-bore bearing designs have evolved to counteract some of the poor stabilitycharacteristics of axial groove bearings. Some anti-whirl sleeve bearing examples include pressure dam bearings[2,3], offset half bearings (Mehta and Singh [4]), and multi-lobe bearings (Lanes and Flack [5]). These bearingdesigns have been successful in increasing the instability threshold speed compared to axial groove bearings [1–5].

Figure 3-5—Low-Speed, Heavily Loaded, Stable Bearing [1]

+

+

W

Heavy Y loadsupported byY displacement

Heavy load/low speedlow cross coupling

stable

Y

X

Oj

Ob



3.3.2.4 Double Pocket Bearing Gas Turbine Application

A frequency spectrum is shown in Figure 3-7, Nicholas [1,6] for a large overhung power turbine operating on test with3 axial groove bearings. A large subsynchronous component is evident at 40 Hz (0.033 mm, 1.3 mils peak-to-peak at5000 rpm) predominately in the horizontal direction. The amplitude of the 40 Hz component exceeded 6.0 mils soonafter this signature was recorded. Also indicated on Figure 3-7 are the synchronous power turbine component labeled“1X” and the synchronous gas generator component labeled “1X GG.”

The 40 Hz instability is 48 % of synchronous speed at 5000 rpm and is thought to be caused by a combination of oilwhirl and aerodynamic excitations from the turbine blades. In an attempt to improve the turbine’s stabilitycharacteristics, a double pocket bearing is considered [1,6].

Results from a full rotor-bearing system stability analysis are presented in Figure 3-8. From Figure 3-8, the logarithmicdecrement is plotted against aerodynamic cross-coupling, Q, a destabilizing excitation placed at the turbine disklocation. The original design 3-axial groove bearings place the rotor in the unstable region of the map with a logdecrement value of –0.25 for low cross-coupling levels. The damped natural frequency is 2389 cpm (39.8 Hz) at a

Figure 3-6—Bearing-Induced Shaft Whip and Oil Whirl [1]

0 1 2 3 4 5

0.5X 1X

2X

3X

6Frequency (RPM x 1000)

Mac

hne

Spe

ed (E

vent

s/M

n x

1000

)

7 8 9 10 11 12

12

11

10

9

8

7

6

5

4

3

2

Sha

ft w

hp

Machine naturalfrequency = 3800

Oil

whi

rl

Sync

hron

ous



cross-coupling level of 1.0 x 104 lb/in. This corresponds to the 40 Hz instability illustrated in Figure 3-7. The doublepocket bearing designs, however, are well into the stable area with log decrement values under 0.75.

Figure 3-9 shows the turbine’s frequency spectrum operating on the modified double pocket bearings. The 40 Hzsubsynchronous component is suppressed to an amplitude of 0.25 mils at 5000 rpm. This level is bounded and wellwithin the customer specifications.

Figure 3-7—Frequency Spectrum, Power Turbine Test, 3-axial Groove Bearings [1]

0

1X

1XGG

5000RPM

4500RPM

40 Hz

0.0127 mm(0.5 mils)

50 100f, Hz

150 200

3 Axial Groove



Figure 3-8—Rotor Bearing System Stability, Power Turbine N = 5000 rpm [1]

Stable

Doublepocket

3 Axialgroove

Unstable

0

105

104

103

105

104

103

Q,

b/n.

0.5 1.00.51.0

Log Decrement

N/c

m

c = 0°

c = 15°



3.3.2.5 References

[1] Nicholas, J. C., 1996, “Hydrodynamic Journal Bearings—Types, Characteristics and Applications,” MiniCourse Notes, 20th Annual Meeting, The Vibration Institute, Willowbrook, Illinois, pp. 79–100.


[3] Nicholas, J. C., 1986, “Stabilizing Turbomachinery with Pressure Dam Bearings,” Encyclopedia of FluidMechanics, 2, Gulf Publishing Co.

[4] Mehta, N. P. and Singh, A., 1986, “Stability Analysis of Finite Offset-Halves Pressure Dam Bearings”, ASMEJournal of Tribology, 108 (2), pp. 270–274.

[5] Lanes, R. F. and Flack, R. D., 1982, “Effects of Three-Lobe Bearing Geometries on Flexible Rotor Stability”,ASLE Transactions, 25 (3), pp. 377–385.


3.3.3 Tilting Pad Journal Bearings

Even though they are costlier than fixed geometry bearings, tilting pad bearings have gained popularity because oftheir superior stability performance. Unlike fixed geometry bearings, tilt pad bearings generate very little destabilizingcross-coupled stiffness regardless of geometry, speed, load or operating eccentricity. However, rotors supported ontilting pad bearings are still susceptible to instabilities due to other components within the machine such as labyrinthseals, impellers, etc., which can generate destabilizing forces.

3.3.3.1 Tilting Pad Bearing Frequency Dependency

3.3.3.1.1 General

The low destabilizing nature of tilting pad bearings is a direct result of the pad’s ability to rotate or pivot (Lund [1]).These pad rotations also result in the bearing’s stiffness and damping characteristics becoming frequency dependent,

Figure 3-9—Frequency Spectrum, Power Turbine Test, Double Pocket Bearings [1]

0

1X

1XGG

5000RPM

4500RPM

40 Hz0.0127 mm(0.5 mils)

50 100f, Hz

150 200

Double Pocket



in addition to their dependence on geometry, load, speed, etc. [2,3]. This additional dependence on vibrationalfrequency is important with respect to stability because instabilities almost invariably occur at vibrational frequenciesother than shaft speed, particularly, at subsynchronous frequencies, as discussed for fixed geometry bearings in3.3.2.

— Fixed Geometry Journal Bearing:

K & C = f (geometry, oil, materials, load, shaft speed)

— Tilting Pad Journal Bearing:

K & C = f (geometry, oil, materials, load, shaft speed, vibrational frequency)

Because only 2 degrees of freedom (shaft lateral motion) are predominant, fixed geometry bearing’s dynamiccharacteristics can be represented by 4 stiffness and 4 damping coefficients as described in 2.5.2. However, a tiltingpad bearing, with N pads has N+2 degrees of freedom, and requires 5N+4 stiffness and 5N+4 damping coefficients,called “full” or “pad dynamic” coefficients, to define its dynamic properties. For a 5 pad bearing, this means 29stiffness and 29 damping coefficients determine its dynamic characteristics [2,4].

This large number of coefficients introduces increased mathematical and computational complexity into therotordynamic analyses. Furthermore, it is difficult to physically interpret their meaning, and their impact on the rotorsystem. To alleviate these problems, a method can be used to reduce these coefficients into the more common 8stiffness and damping coefficients. This reduction requires a vibrational frequency to be assigned to the pads, thus,establishing a frequency dependence upon the 8 dynamic coefficients. When the vibrational frequency is set at theshaft rotational speed, these 8 reduced coefficients for the tilting pad bearing are designated “synchronously reduced”bearing coefficients [1–5].

Figure 3-10 illustrates the importance of vibrational frequency on the reduced stiffness and damping characteristics oftilting pad bearings. The whirl ratio represents the ratio of shaft (and pad) vibrational frequency to shaft rotationalfrequency. Synchronously reduced coefficients are associated with a whirl ratio of 1.0. Figure 3-10 shows that whenthe shaft is vibrating subsynchronously (for instance, near its first natural frequency, whirl ratio of about 0.4), thepredicted stiffness, K, and damping, C, properties that the bearing imposes on the rotor are altered. Typical of aturbomachine’s bearing design, this particular bearing increases the equivalent stiffness and decreases theequivalent damping available as the shaft vibrates subsynchronously (whirl ratio <1). Generally, such stiffness anddamping trends degrade a machine’s stability characteristics.

The extent of the predicted frequency dependence is numerically a function of several parameters; preload, pivotoffset and Sommerfeld Number (see 3.3.2.1). Increasing preload and pivot offset decreases the amount of frequencydependence, i.e. reduces the difference between the synchronously reduced and the nonsynchronously reducedcoefficients. Frequency effects also decrease as the Sommerfeld Number diminishes due to higher loading or lowerspeed [2–4].

Up until the late 1990s, no experimental work had been conducted to investigate the nonsynchronous characteristicsof tilting pad bearings’ stiffness and damping properties. Early published experiments [5,6] correlated reasonably wellwith predicted trends.

Childs [7] published the results of a large number of experimental investigations examining a variety of differentdesign configurations such as number of pads, load orientation, pivot offset, pivot type and preload. Theseexperiments have focused on measuring the test bearing's dynamic stiffness characteristics (H(w)) versus whirlfrequency for various speeds and loads. The 2x2 complex valued, dynamic stiffness matrix H(w) relates the shaftdisplacements to forces created by the bearing according to:

(3-3)Fx

Fy Hxx Hxy

Hyx Hyy

X

Y

H X

Y

= =



where is the whirl frequency, Fx and Fy are the bearing forces, and X and Y are the shaft displacements. For aparticular whirl frequency, the stiffness and damping coefficients of interest, i.e. the eight coefficients predicted byconventional fluid film bearing codes, are related to the complex dynamic stiffness by the following:

(3-4)

With respect to the frequency dependency of stiffness characteristics, Childs [7] experiments have correlated wellwith the predicted trends. Specifically, tilting pad bearing stiffness shows significant frequency dependency for somebearing designs and operating conditions, and not for others. However, the experimental results from [7] for dampinghave not shown compelling evidence of any frequency dependency. Other researchers have found similar trends(Delgado et al. [8]).

It is important to note that many of these recent experimental efforts have generated a separate debate on what typeof model is most appropriate to represent any frequency dependency that does exist. In this debate, one modelattributes any frequency dependency to the added mass effects of the lubricant film [7,8,9]. The other model arguesthat the nonsynchronous characteristics are primarily governed by the full coefficients that are derived from the pad’sability to tilt (Cloud et al. [10], Diamond et al. [11]). Additional research will have to be conducted to settle this debateand identify which is the dominant phenomenon. An alternate approach is to ensure the bearing design has minimalfrequency dependence (increasing preload and offset).

Current trends in petrochemical machinery design are leading toward bearing designs running at higher speeds. Inaddition, low preload bearings are also a popular choice because of their improved stability performance. For both(lower preloads and higher speeds), the frequency dependency is increased, leading toward bearings that providehigher equivalent stiffness and lower equivalent damping at subsynchronous frequencies. These effects translate intoa reduction in predicted stability levels compared to those predicted using synchronously reduced coefficients.

Figure 3-10—Frequency Dependent Stiffness and Damping

Kyy

Kxx

Cyy

Cxx

2.5E+06

Tilting Pad Bearing Analysis

2.0E+06

1.5E+06

1.0E+06

0.5E+06

1st N

atur

a fr

eque

ncy

Syn

chro

nous

y re

duce

d

Dam

png

(b-

s/n.

)

Whirl Ratio0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Stf

fnes

s (b

/n.)

2000 5-pad, load between pads, L/D = 0.43Bore diametral clearance = 4 milsPivot offset = 0.5ISO VG 32 110 °F inlet temperaturePreload = 0.529Journal diameter = 3.496 in.Vertical load = 497 lbsPad arc length = 60 degreesShaft speed = 10,569 rpmSommerfeld no. 1.1

1800

1600

1400

1200

1000

800

K i j, Re H i j=

C i j,1---Im H

i j=



Figure 3-11 reveals these prediction differences for a particular industrial compressor. At the bearing’s maximumpreload, the compressor exhibits its worst stability characteristics. Furthermore, one can see how influential thebearing’s frequency dependence can be in decreasing the predicted base log decrement (i.e. log decrement for Kxy =0). For the maximum preload in the tolerance range, m = 0.367, the predicted log decrement is 0.28 usingsynchronous coefficients and 0.17 using full coefficients. For the minimum preload in the tolerance range, m = 0.178,the predicted log decrement is 0.38 using synchronous coefficients and 0.23 using full coefficients.

Because of limited computational resources, early stability analysis techniques did not incorporate the 5N + 4 fullstiffness and 5N + 4 full damping coefficients of tilting pad bearings. Stability levels were determined using the 8synchronously reduced coefficients. As shown in Figure 3-11, such an approach can yield stability predictions, whichcan be significantly different from an analysis incorporating the full set of nonreduced coefficients [12,13].

However, many manufacturers still incorporate synchronously reduced coefficients for their tilting pad bearingproperties, Kocur et al. [14]. In fact, most of the confidence factors surrounding destabilizing force models (such asAlford’s, Modified Alford's or Wachel’s equation) and historically acceptable stability levels (e.g. 0.1 for stableoperation) are based on many years of design experience using synchronously reduced tilting pad bearingcharacteristics for stability calculations.

Furthermore, using synchronous tilting pad bearing characteristics in stability predictions may be physically justifiedsince, prior to an instability, the shaft (and pads) vibrate synchronously (1x) as illustrated in Figure 3-12 (Gunter et al.[15]). Thus, the destabilizing forces that increase with speed and finally produce a subsynchronous component atabout 10,500 rpm, are counteracted by a rotor-bearing system that is vibrating synchronously. Even at speeds above

Figure 3-11—Full Coefficient vs. Synchronous Reduced Tilting Pad Bearing Stability Sensitivity

UNSTABLE

0 10,000 20,000 30,000Kxy at Midspan (lb/in.)

40,000 50,000

0.4

0.3

0.2

0.1

0

–0.1

Loga

rthm

c D

ecre

men

t

Coker Wet Gas Compressor7-Stage In-Line

MCOS = 10,569 rpmRotor weight = 991 lbsRotor L/D = 11.5

Bearing span = 71.8 in.Midspan diameter = 6.24 in.Critical speed ratio = 2.86

Bearings: 5-pad load between padL/D = 0.43 center offset

Preload = 0.367 (Somm = 1.2) full coeff.Preload = 0.367 (Somm = 1.2) synch redPreload = 0.176 (Somm = 0.8) full coeff.Preload = 0.176 (Somm = 0.8) synch red



12,500 rpm where a strong subsynchronous vibration component is evident, an equally strong synchronouscomponent is also present causing the tilting pads to vibrate both synchronously and subsynchronously.

However, a recent investigation where stability thresholds were measured on a tilting pad bearing supported rotorsystem, has provided some compelling evidence that the synchronously reduced representation does not accuratelypredict thresholds because of the synchronous vibration’s dominance at the onset of stability. Stability thresholds andlevels were consistently more accurately predicted by the full coefficient representation [10]. A similar conclusion wasfound for a compressor examined as part of an API survey [14].

There is some evidence that frequency dependency can exist (depending on the bearing design and operation) andthat it may significantly influence the rotor's stability characteristics. While further experimental work is desired,consideration will be given in the next revision of the SPs to require the use of nonsynchronous, frequency dependentcoefficients for stability analyses. Currently, the stability specification requires the use of synchronous tilting padbearing characteristics in both Level 1 and 2 analyses. The corresponding acceptance criteria in both the Level 1 andthe Level 2 analyses were developed specifically for the use of synchronous coefficients.

Figure 3-12—Waterfall Showing Self-Excited Instability

0

16

14

12

10

8

6

4

2

02 4 6 8

Frequency Spectrum f x 10 3 (CPM)

Frequency Spectrum vs. SpeedNonsynchronous Whirl

10 12 14 16

Rot

or S

peed

RP

M x

10-

3

N = 13,500 Design speed

1N2N




[1] Lund, J. W., 1964, “Spring and Damping Coefficients for the Tilting-Pad Journal Bearing,” ASLE Transactions,7, pp. 342–352.

[2] Parsell, J. K., Allaire, P. E. and Barrett, L. E., 1983, “Frequency Effects in Tilting-Pad Journal Bearing DynamicCoefficients,” ASLE Transactions, 26, pp. 222–227.

[3] White, M. F. and Chan, S. H., 1992, “The Subsynchronous Dynamic Behavior of Tilting-Pad JournalBearings,” ASME Journal of Tribology, 114, pp. 167–173.

[4] Barrett, L. E., Allaire, P. E. and Wilson, B. W., 1988, “The Eigenvalue Dependence of Reduced Tilting PadBearing Stiffness and Damping Coefficients,” ASLE Transactions, 31, pp. 411–419.

[5] Ha, H. C., and Yang, S. H., 1999, “Excitation Frequency Effects on the Stiffness and Damping Coefficients ofa Five-Pad Tilting Pad Journal Bearing,” ASME Journal of Tribology, 121 (3), pp. 517–522.

[6] Wygant, K. D., 2001, “The Influence of Negative Preload and Nonsynchronous Excitation on the Performanceof Tilting Pad Journal Bearings,” Ph.D. Dissertation, University of Virginia, Charlottesville, Virginia.

[7] Childs, D., 2010, “Tilting-Pad Bearings: Measured Frequency Characteristics of their RotordynamicCoefficients,” Proceedings of the 8th IFToMM International Conference on Rotor Dynamics, Seoul, Korea.

[8] Delgado, A., Vannini G., Ertas, B., Naldi, L., Drexel, M., 2011, “Identification And Prediction Of ForceCoefficients In A Five-Pad And Four-Pad Tilting Pad Bearing For Load-On-Pad And Load-Between-PadConfigurations,” ASME Journal of Engineering for Gas Turbines and Power.

[9] Kulhanek, C. D. and Childs, D. W., 2011, “Measured Static and Rotordynamic Coefficient Results for aRocker-Pivot, Tilting-Pad Bearing with 50 and 60% Offsets,” Proceedings of ASME Turbo Expo 2011: Powerfor Land, Sea and Air, Vancouver, British Columbia, GT2011-45209.

[10] Cloud, C. H., Maslen, E. H., Barrett, L. E., 2010, “Influence of Tilting Pad Journal Bearing Models on RotorStability Estimation,” Proceedings of the 8th IFToMM International Conference on Rotor Dynamics, Seoul,Korea, pp. 94–102.

[11] Diamond, T. W., Younan, A. A., Allaire, P. E. and Nicholas, J. C., 2010, “Modal Frequency Response of aFour-Pad Bearing with Finite Pivot Stiffness and Different Pad Preloads,” Proceedings of ASME Turbo Expo2010: Power for Land, Sea and Air, Glasgow, UK, GT2010-23609.

[12] Brockett, T. S. and Barrett, L. E., 1993, “Exact Dynamic Reduction of Tilting-Pad Bearing Models for StabilityAnalyses,” STLE Tribology Transactions, 36, pp. 581–588.

[13] Chan, D. S. H. and White, M. F., 1996, “Stability Thresholds of Rotor Systems Supported on Tilting-PadJournal Bearings,” I.Mech.E., pp. 235–257.

[14] Kocur, J. A., Nicholas, J. C., and Lee, C. C., 2007, “Surveying Tilting Pad Journal Bearings and Gas LabyrinthSeal Coefficients and Their Effect on Rotor Stability,” Proceedings of the Thirty-Sixth TurbomachinerySymposium, Texas A&M University, pp. 1–10.

[15] Gunter, E. J., Barrett, L. E. and Allaire, P. E., 1975, “Design and Application of Squeeze Film Dampers forTurbomachinery Stabilization,” Proceedings of the Fourth Turbomachinery Symposium, TurbomachineryLaboratory, Texas A&M University, College Station, Texas, pp. 127–141.

3.3.3.2 Lubrication Starved Tilting Pad Bearings

In 1991, Tanaka [1] introduced the concept of removing the end seals from tilting pad journal bearings. He found thatevacuating the bearing housing cavity produced significantly lower pad operating temperatures compared to theconventional flooded housing designs. This concept has been developed further by Nicholas [2,3] with theintroduction of spray bars and wide open housing drains as illustrated in Figure 3-13.



While the Figure 3-13 design has been successful in reducing pad operating temperatures from 10% to 15% [2], caremust be taken when implementing the design. The wide open end seals and housing drains result in low housingcavity pressures, often below 1.0 psig. Conversely, conventional flooded housing designs have typical housingpressures that range from 5 psig to 15 psig for a 20 psig oil inlet. These low housing pressures can lead to oilstarvation and subsynchronous vibration if improperly applied [3].

Figure 3-13 illustrates one misapplication example. Open end seals and open housing drains will not work in a high-speed balance vacuum. Even though the inlet oil is introduced to the housing through a spray bar at around 20 psig,the housing cannot maintain a positive pressure and the oil immediately atomizes, resulting in oil starvation. Thismanifests itself as a subsynchronous rotor vibration [3].

The solution is to use dummy end seals with a reasonable clearance and to temporally block the open housing drains.Dummy end seal clearances of around twice the bearing clearance will produce a reasonable housing pressure ofaround 10 psig. In this case, the inlet oil exiting the spray bars will not atomize even in a high-speed balance vacuumbunker.

Another example of the misapplication of the evacuated housing design is shown in Figure 3-14. In this case, the oil isintroduced with a single housing hole between each set of tilting pads at the pad’s axial centerline. The pad length todiameter ratio is 1.0 resulting in a relatively long pad. The designer, desiring lower pad operating temperatures, usedopen end seals that resulted in a relatively low housing pressure of 1.3 psig. This low housing pressure coupled withthe long pad caused oil starvation and a subsynchronous rotor vibration. The spray bar design shown in Figure 3-15distributes the oil along the full axial length of the pad and was successful in eliminating the subsynchronous rotorvibration [3].

A final example of the misapplication of the evacuated housing design is shown in Figure 3-16. In this case, the oil isintroduced with a mushroom orifice spray between each set of tilting pads at the pad’s axial centerline. Again, the padlength to diameter ratio is 1.0 resulting in a relatively long pad. As before, lower pad operating temperatures weredesired prompting the designer to used open end seals. Clearly from Figure 3-16, no oil is directly sprayed toward thepad’s axial ends resulting in a 1.9-in. “oil free” zone at both pad ends. This again caused oil starvation and asubsynchronous rotor vibration. A spray bar similar to the one illustrated in Figure 3-15 again solved the problem [3].

Figure 3-13—High-speed Balance Vacuum Pit Oil Atomization Resulting in Subsynchronous Vibration [3]

10.00 in. PadL/D = 1.00

Open housingdrains

Open end sealCd = 500 mils dia.

• Oil atomization• Starvation• Subsynchronous vibration



While it may appear that these tilting pad bearings caused a rotor instability, it was the misapplication of theevacuated housing design that caused the subsynchronous vibration. This phenomenon cannot be predicted orprevented by performing a rotor bearing stability analysis.

Figure 3-14—Single Housing Orifice Design Resulting in Subsynchronous Vibration [3]

Figure 3-15—Spray Bar Evacuated Housing Design [3]

Low housingpressure1.3 psig

Single orificein housing

Long L/D = 1.0 pad• Open end seals• Low housing pressure• Long L/D = 1.0• Oil starvation• Subsynchronous vibration

Open end sealCd = 50 mils dia.



3.3.3.3 References

[1] Tanaka, M., 1991, “Thermohydrodynamic Performance of a Tilting Pad Journal Bearing with Spot Lubrication,”ASME Journal of Tribology, 113 (3), pp. 615–619.

[2] Nicholas, J. C., 1994, “Tilting Pad Bearing Design,” Proceedings of the Twenty-Third TurbomachinerySymposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 179–194.

[3] Nicholas, J. C., 2003, “Tilting Pad Journal Bearings with Spray Bar Blockers and By-Pass Cooling for HighSpeed, High Load Applications," Proceedings of the Thirty-Second Turbomachinery Symposium,Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 27–38.

3.3.4 Squeeze Film Dampers

3.3.4.1 General

It is widely known from early investigations on rotordynamic instability that the use of flexible damped supports mayprevent instability or alter the speed at which instability occurs. Installing a squeeze film damper in series with theradial bearings is the easiest and perhaps most common method of adding a flexible damped support. In addition toenhancing stability, they are also used to reduce first critical speed peak response amplitudes at the mid-span ofhighly flexible rotors. Early in their use, squeeze film dampers were sometimes regarded as a last resort solutionapplied to problem machinery. Since then they have gained acceptance not only as a credible fix in problemmachinery but also as a design tool routinely used in new equipment.

Squeeze film dampers are widely applied in aircraft jet engines to supplement the otherwise negligible dampinginherent in the rolling element bearings used in this type of machinery. Although their main purpose is to reduce rotorresponse amplitudes, they provide some improvement in the stability of the rotor system. Squeeze film dampers arealso used in a variety of land-based machinery operating on oil film bearings. Of these applications, they are morefrequently used in high-pressure and high-density centrifugal compressors to provide stable operation, although they

Figure 3-16—Button Spray Design Resulting in Subsynchronous Vibration [3]

7.480

3.670 1.905

Oil starvation at pad ends-subsynchronous vibration

Oil freeregion



are also used to reduce the critical speed response characteristics of machines with highly flexible rotors. A briefhistory of the development and early applications of squeeze film dampers along with a general discussion ontheoretical models and design considerations is given in Zeidan et al. [1]. Many case histories concerning theapplications of squeeze film dampers are given in Memmott [2,3].

Although tilting pad bearings are routinely used to avoid oil whirl instability problems, machines with flexible rotorssupported on tilting pad journal bearings may exhibit instability if there exists an external destabilizing force ofmagnitude that exceeds the restoring damping of the bearing. This instability may be avoided by either eliminating thedestabilizing mechanism, raising the rotor’s first damped natural frequency, or by adding sufficient external dampingto raise the stability threshold. In cases where it is not possible to eliminate the instability mechanism or raise thenatural frequency, the only practical solution is to add external damping. One of the key features in the successfuldesign of a squeeze film damper, therefore, is the introduction of flexibility as well as damping at the bearing supportstructure. Allowing the bearing housing or cage to move within the squeeze film damper's oil film increases thebearing’s ability to suppress rotor vibration. Other benefits include lower transmitted forces and increased bearing life,particularly in the case of machinery operating above the first critical speed.

3.3.4.2 Design Considerations

A schematic of a squeeze film damper bearing is given in Figure 3-17. The inner bearing element is a conventionalhydrodynamic tilting pad journal bearing that is supported and centered in a fixed housing. A small radial clearance isrequired between the housing and inner element to provide a cavity for an oil film. Unlike with a conventional journalbearing, the inner element is free to float within this cavity and not rotate. In most designs, oil is supplied to this cavityand the inner bearing via a central annular groove machined into either the housing or the bearing shell. O-ring sealsare typically used in centrifugal compressor and steam turbine applications to enclose this cavity to minimize axialthrough flow and increase damping. The most common end seal arrangement incorporates a circumferentialelastomeric o-ring as illustrated in Figure 3-18.

Figure 3-17—Squeeze Film Damper Schematic

Shaft

Bearing

Squeeze film

End seals(o-rings)

Outer shellor case

Circumferential oil groove

Oilinlet



Other designs more typically used with rolling element bearings include circumferential piston rings or side mountedo-rings, also depicted in Figure 3-18. The outside diameter of the inner element forms the damper journal, which isprevented from spinning by a loose anti-rotation pin. This feature allows the inner element (or damper journal) to whirlbut not spin in a precession motion, thus squeezing the oil in the clearance cavity that in turn generates an oil filmpressure and hence a damping force. Since the inner element cannot rotate, there is no net fluid rotation in the cavityto develop a destabilizing cross-coupled stiffness, only direct coefficients as a result of the orbital motion of the innerelement. For optimum damping, a centering device of the inner element is often incorporated into the design to keepthe damper centered within the cavity. On machines with lightweight rotors, the o-ring end seals provide enoughsupport to center the inner bearing (Leader et al. [4]). Machines with heavy rotors, however, present a uniquechallenge requiring a mechanical spring element that can support a large weight and yet retain some inherentflexibility (Kuzdal and Hustak [5], Edney and Nicholas [6]).

3.3.4.3 Stiffness and Damping Coefficients

Gunter et al. [7] originally proposed the theory on which many squeeze film damper analyses are based. A squeezefilm damper is essentially a plain journal bearing in which radial motion only is allowed. As with conventional bearings,the stiffness and damping coefficients may be derived from a solution of the incompressible fluid Reynolds’ equation,but for a nonrotating journal assuming laminar flow, and a circular whirl orbit. Typical axial pressure profiles of fourdifferent combinations of damper end seal and oil feed are illustrated in Figure 3-19. Cases 3 and 4 are moretraditional arrangements with a center feed groove. Without end seals, Case 3, the damper is equivalent to a plainjournal bearing with two lands of equal length L/2. The film pressure reduces to atmosphere at the oil groove anddamper ends creating two separate pressure profiles equivalent to two parallel dampers of length L/2. Consequentlythe maximum pressure in each side of the damper is reduced by a factor of four and the force on each side by a factorof eight. The net effect is a reduction in hydrodynamic force, and hence both bearing coefficients, by a factor of four.With end seals, Case 4, the pressure profile is equivalent to that of a plain land without an oil groove and without endseals of overall length L. The values for total force, stiffness and damping, therefore, are also the same. Most practicaldampers are designed with some type of end seal to reduce the axial through flow and increase damping. Axialpressure profiles for a damper with a central feed hole, Cases 1 and 2, are other options although much less popular.

For unrestricted end flow (Cases 2, 3, and 4) Reynolds’ equation is solved using boundary conditions appropriate tothe short or Ocvirk bearing theory. Without end flow (Case 1) the boundary conditions are those referred to as thelong or Sommerfeld bearing theory.

Figure 3-18—Typical End Seal Arrangements

Shaft

Radial o-ring Seal

Shaft

Side o-ring Seal

Shaft

Piston Ring Seal



Another factor in the determination of the stiffness and damping coefficients is whether the oil film is cavitated or not.By virtue of their design, squeeze film dampers may be prone to two types of cavitation: vapor and gaseous. Thedamper may be susceptible to cavitation if the oil inlet pressure is low and the damper operates highly eccentric in thedamper housing resulting in high peak pressures. The damper will be less susceptible to cavitation if the oil inletpressure is high and the damper operates nearly centered in the damper housing resulting in low peak pressures. Inthe solution of Reynolds’ equation, cavitation is represented as a pi-film model, which in the case of vapor cavitation isa reasonable assumption. With gaseous cavitation, however, a compressible fluid model assuming properties of amixture of oil and air would be more appropriate, although the random nature of the air entrapment may not yield anysignificant improvement in the results. Diaz and San Andres [9] provide a more detailed discussion on this topic,along with a review of the many conceptual approaches that have been considered. These authors also conclude thatair ingestion leading to a bubbly mixture can substantially affect the dynamic performance of a squeeze film damper[10]. A sensible design procedure is to include features in the damper design that would preclude cavitation.

Despite the concern expressed in the preceding section, most squeeze film damper analyses employ the stiffnessand damping coefficients derived from Reynolds’ equation. Expressions for the direct stiffness and damping based onthe work of Gunter et al. [7] are given in Table 3-1 for a damper precessing in steady state circular motion. Formulaeare given for both the short and long bearing theory solutions, and for both a cavitated (pi-film) and uncavitated (fullfilm) model are given.

Practical experience has shown that damper bearings with small L/D ratios (<0.5) that can be more reasonablymodeled using the short bearing theory yield predictions that correlate well with actual damper performance.

Figure 3-19—Axial Pressure Profiles of Various Damper Arrangements

L

L

L L

L2

L2

CaseNo.

Closed endscenter hole feed

Open endscenter hole feed

Open endscenter groove feed

Closed endscenter groove feed

Long journal bearingtheory

Short journal bearingtheory



Equivalent to Case 2

1

2

3

4

Sealing and FeedingConfiguration

PressureProfile

Spring and DampingConstants



For the short bearing model with cavitation, the stiffness and damping coefficients expressed in dimensionless formare plotted against eccentricity ratio in Figure 3-20. Both coefficients clearly increase sharply at values of eccentricityratio above 0.6 tending to infinity at a ratio of 1.0. Since a damper’s effectiveness degrades significantly witheccentricity, it is important for optimum performance that the damper is positioned and remains well centered in theclearance cavity. Similar trends are observed with the coefficients of the long bearing model.

3.3.4.4 Support Model Representation

In the support model, the squeeze film damper is included in series with the bearing oil film and supportcharacteristics as idealized in Figure 3-21. Kd and Cd are the direct stiffness and damping coefficients of the squeezefilm damper and Kds is the stiffness of the centering device, which is included in parallel with the damper coefficients.For mechanical centering, Kds is equal to the o-ring stiffness plus the mechanical spring stiffness. With o-ring onlycentered dampers, Kds is equal to the o-ring stiffness. A method for calculating equivalent support values is outlined inNicholas et al. [11]. A good rule of thumb is that the damper stiffness should be less than one half of the oil filmstiffness in order for the damper to be effective.

3.3.4.5 Centering Devices

The importance of statically centering a squeeze film damper is described in Kuzdzal and Hustak [5]. The authorspresent results for a bottom resting damper vs. o-ring supported and mechanical spring supported dampers atvarious damper eccentricities. Regardless of the centering device employed, its stiffness value must be carefullydetermined. Theoretical calculations are often approximate and should be used with caution. Leader et al. [4] presenta method for measuring the stiffness of an o-ring and show how to use the Nicholas et al. [11] formulas. They alsodescribe a method for installing the o-rings eccentrically to keep the inner element centered under the static weight ofthe rotor. A mechanical centering device [5,6] and a method for measuring stiffness is presented in [6]. Withmechanical elements, hooks, bolted joints or other means of fixing the centering device in the housing might introducesome additional flexibility thus reducing the overall stiffness, but this flexibility can be avoided by proper design of theconnection. Test fixtures for measuring stiffness must insure that all such effects are properly accounted for. This isbest accomplished by testing the centering device in its housing. A schematic showing a typical o-ring centereddamper is illustrated in Figure 3-22 [2,3], whereas a mechanical arc spring centered damper is shown in Figure 3-23[12]. Similar photos may also be found in [5].

Particularly in the case of mechanical centering devices used to support heavy rotors, a fatigue calculation should beperformed. The mean stress is the maximum stress with the centering device statically loaded. For the alternating

Table 3-1—Formula for Squeeze Film Damper Stiffness and Damping Coefficients

Film ModelShort Bearing Theory Long Bearing Theory

Stiffness: Kd Damping: Cd Stiffness: Kd Damping: Cd

Cavitated: pi-film

Uncavitated: full film 0 0

2RL3

cr3

1 2–

2---------------------------- RL

3

2cr3

1 2–

3 2-------------------------------------- 24R

3L

cr3

2 2+ 1 2

–

--------------------------------------------12R

3L

cr3

2 2+ 1 2

– 1 2

-------------------------------------------------------

RL3

cr3

1 2–

3 2-----------------------------------

24R3L

cr3

2 2+ 1 2

– 1 2

-------------------------------------------------------



stress, a worst case value should be assumed equal to the load that would deflect the centering device beyond thestatically loaded condition an amount equal to the damper radial clearance.

3.3.4.6 Design Procedure

There is no single procedure that would adequately cover the design of a squeeze film damper for every application.Most designs and analysis procedures have evolved through the practical experience of the individual ormanufacturer applying them. As a practical guide, however, the following general procedure is suggested.

a) Identify the purpose of the squeeze film damper: improved stability characteristics or reduced synchronousresponse at rotor mid-span at the first critical speed.

b) Perform rotordynamic stability and response calculations to determine the optimum damping required for stability,reduced synchronous response at Nc1 and acceptable synchronous response at Nmc. There may be a need fortrade-offs between these items. In most cases a Level I stability analysis will be sufficient. For a reference onoptimizing the log dec see Lund [13]. For a reference on optimizing the log dec and the stability threshold seeMemmott [14] and Memmott and Ramesh [15]. A method of optimizing the synchronous response may be foundin [15].

c) Choose the damper design considering the available space and configuration of the machine. Determine the typeof centering device, end seal preference (usually o-rings for centrifugal compressors and steam turbines), and thelubricant supply arrangement.

d) Calculate the squeeze film damper stiffness and damping coefficients using the appropriate equations from Table3-1 or, if available, a computer code. For the minimum clearance damper, use an eccentricity ratio based on themaximum allowable absolute eccentricity for the damper along with minimum oil inlet temperature. For themaximum clearance damper, assume the damper is centered and use the maximum oil inlet temperature.

e) Add the squeeze film damper coefficients with the stiffness of the centering device. These coefficients are inparallel.

Figure 3-20—Squeeze Film Damper Coefficients vs. Eccentricity Ratio: Short Bearing Theory (Cavitated) [8]

0 0.2 0.4 0.6 0.8 1.0Eccentricity

LsfC3

RμL3N

orm

aze

d da

mp

ngN

orm

aze

d st

ffnes

s

5

10

15

20

BsfC3

RμL3



f) Combine the resultant coefficients from step e with the bearing oil film coefficients. Include the support values asappropriate. These coefficients are in series. Combine minimum clearance damper values with minimumclearance bearing oil film values and combine maximum clearance damper values with maximum clearancebearing oil film values. Some rotordyamics computer codes may do these combinations automatically.

g) Input the resultant coefficients into an appropriate rotordynamics computer program to predict stability orsynchronous response.

A useful design tool for optimizing damper stability performance is a stability map as shown in Figure 3-24 [7]. In thisexample, the growth factor (real part of the eigenvalue) is plotted as a function of support damping for several damperstiffness values. Clearly, a near optimum squeeze film damper stiffness and damping range can be extracted from themap. For this case, a stiffness range of 50,000 to 100,000 lb/in. and a damping range of 500 to 1000 lb-sec/in. wouldbe near optimum. Design efforts to obtain these values can then be attempted by varying damper clearance and axiallength.

Figure 3-21—Idealization of Bearing-damper-support Characteristics [6]

Ks Cs

Kds Kd Cd

Kb Cb

Md

Ms

M

Bearing

Damper

Support

Shaft

Spring



Figure 3-22—O-ring Supported Squeeze Film Damper Schematic [2,3]

O-rings

Housing

Clearancearea

Bearingcage

Tiltingpad



3.3.4.7 General Comments

A squeeze film damper is an excellent design tool that can be used to improve the vibration characteristics of rotorsby adding damping to either stabilize otherwise unstable machinery or reduce synchronous vibration amplitudes whilepassing through critical speeds. Research is continuing into the development of more advanced squeeze film modelsin an attempt to include the more detailed effects of fluid dynamic cavitation, two phase flows, and end seals ondamper performance. Nevertheless, provided oil film cavitation is not excessive and the L/D ratio small (<0.5),squeeze film dampers can be adequately modeled and successfully applied using the incompressible fluid Reynolds’equation based on the short bearing model.

It must be recognized that with a damper the resultant damping will always be less than the tilting pad bearingdamping. The overall stiffness, therefore, must also be lowered to allow rotor motion at the bearings in order to makethe available damping more effective. Even so, the stiffness of the centering device and the clearance of the dampercavity must be chosen carefully to insure that the supports are not under or over-damped. Over damped supportsmay lockup and significantly reduce the effectiveness of the damping. This condition can be avoided by not specifyingtoo small a damper clearance and by insuring that the damper is well centered within the cavity. The latter is ofparticular concern since the stiffness and damping coefficients are highly nonlinear functions of eccentricity, sharplyincreasing at values of eccentricity ratio above 0.6.

Figure 3-23—Mechanical Arc Spring Supported Squeeze Damper [12]



3.3.4.8 References

[1] Zeidan, F. Y., San Andres, L. and Vance, J. M., 1996, “Design and Application of Squeeze Film Dampers inRotating Machinery,” Proceedings of the Twenty-Fifth Turbomachinery Symposium, TurbomachineryLaboratory, Texas A&M University, College Station, Texas, pp. 169–188.

[2] Memmott, E. A., 1990, “Tilt Pad Seal and Damper Bearing Applications to High Speed and High DensityCentrifugal Compressors,” IFToMM, Proceedings of the 3rd International Conference on Rotordynamics,Lyon, pp. 585–590.

[3] Memmott, E. A., 1992, “Stability of Centrifugal Compressors by Applications of Tilt Pad Seals, DamperBearings, and Shunt Holes,” IMechE, 5th International Conference on Vibrations in Rotating Machinery, Bath,pp. 99–106.

[4] Leader, M. E., Whalen, J. K., Hess, T. D. and Grey, G. G., 1995, “The Design and Application of a SqueezeFilm Damper Bearing to a Flexible Steam Turbine Rotor,” Proceedings of the Twenty-Fourth TurbomachinerySymposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 49–57.

[5] Kuzdzal, M. J. and Hustak, J. F., 1996, “Squeeze Film Damper Bearing Experimental vs. Analytical Results forVarious Damper Configurations,” Proceedings of the Twenty-Fifth Turbomachinery Symposium,Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 57–70.

[6] Edney, S. L. and Nicholas, J. C., 1999, “Retrofitting a Large Steam Turbine with a Mechanically CenteredSqueeze Film Damper Bearing,” Proceedings of the Twenty-Eighth Turbomachinery Symposium,Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 29–40.

Figure 3-24—Squeeze Film Damper Stability Map [7]

50

0

–50

–100

–150

10 50 100 200 500 1000 2000 5000 10,000

Support Damping (lb-sec/in.)

Rea

Par

t—R

oot o

f Cha

ract

erst

c E

quat

on

Q = 20,000 lb/in.N = 10,000 RPM

Unstable

Stable

Rotor Characteristics

Bearing Characteristics

W2 = 675 lb

Wj = 312 lb

W2 = 15 lb

KS = 500,000

KS = 250,000

KS = 100,000

KS = 50,000

Kxx = 1.287 x 106 lb/in.

KYY = 1.428 x 106 lb/in.

Cxx = 1200 lb sec/in.

CYY = 1290 lb sec/in.

KS = 2.8 x 105 lb/in.

CS = 1.0 x lb sec/in.

CI = 0.0



[7] Gunter, E. J., Barrett, L. E. and Allaire, P. E., 1975, “Design and Application of Squeeze Film Dampers forTurbomachinery Stabilization,” Proceedings of the Fourth Turbomachinery Symposium, TurbomachineryLaboratory, Texas A&M University, College Station, Texas, pp. 127–141.

[8] Ehrich, F.F., 1992, Handbook of Rotordynamics, McGraw Hill.

[9] Diaz, S. E. and San Andres, L. A., 1997, “Forced Response of SFDs Operating with a Bubbly (Air/Oil)Mixture,” Texas A&M University, Turbomachinery Research Consortium Progress Report, TRC-SFD-1-97.

[10] Diaz, S. E. and San Andres, L. A., 1999, “Reduction of the Dynamic Load Capacity in a Squeeze Film DamperOperating with a Bubbly Lubricant,” ASME Journal of Gas Turbines and Power, 121, pp. 703–709.

[11] Nicholas, J. C., Whalen, J. K. and Franklin, S. D., 1986, “Improving Critical Speed Calculations Using FlexibleBearing Support FRF Compliance Data,” Proceedings of the Fifteenth Turbomachinery Symposium,Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 69–78.

[12] Nicholas, J. C., Edney, S. L., Matthews, T. and Varela, F. J. M., 2001, “Eliminating a Rub Induced Start-UpVibration Problem in an Ethylene Drive Steam Turbine,” Proceedings of the Thirtieth TurbomachinerySymposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 65–78.

[13] Lund, J. W., 1974, “Stability and Damped Critical Speeds of a Flexible Rotor in Fluid-Film Bearings,” Trans.ASME, Journal of Engineering for Industry, pp. 509–517.

[14] Memmott, E. A., 2003, “Usage of the Lund Rotordynamic Programs in the Analysis of CentrifugalCompressors,” Special Issue: The Contributions of Jørgen W. Lund to Rotor Dynamics, ASME Journal ofVibration and Acoustics, Vol. 125, No. 4, pp. 500–506.

[15] Memmott, E. A. & Ramesh, K., 2008, “Application of Squeeze-film Dampers with a Centrifugal Compressor,”IMechE, 9th International Conference on Vibrations in Rotating Machinery, Exeter, pp. 787–798.

Nomenclature

cr is the damper radial clearance, m (in.);

Cb is the bearing direct damping, N-s/m (lbf-s/in.);

Cd is the squeeze film direct damping, N-s/m (lbf-s/in.);

Cs is the support direct damping, N-s/m (lbf-s/in.);

D is the damper diameter, m (in.);

Kb is the bearing direct stiffness, N/m (lbf/in.);

Kd is the squeeze film direct stiffness, N/m (lbf/in.);

Kds is the damper centering device spring stiffness, N/m (lbf/in.);

Ks is the support direct stiffness, N/m (lbf/in.);

L is the damper axial length, m (in.);

M is the journal mass, kg (lbm);

Md is the damper mass, kg (lbm);

Ms is the support mass, kg (lbm);

R is the damper radius, m (in.);

is the damper eccentricity ratio, dim;

is the oil viscosity, Pa-s (lbf-s/in.2);

is the whirl frequency, rad/s.



3.4 Seals

3.4.1 Oil Seals

3.4.1.1 General

The analysis to determine the influence of oil seals on the dynamic stability and unbalance response of the rotor isessential for modern centrifugal compressors. The influence on unbalance response was discussed in 2.6.2. Thecross-section of a common oil bushing seal was shown in Figure 2-51. The outer seal ring is of major concern due tothe high axial force that can lock the otherwise floating outer ring to the compressor casing. A typical pressuredistribution for an outer ring was shown in Figure 2-52. Early oil seal analyses [1,2] used a constant operating oil filmtemperature obtained by an approximate temperature flow balance. An improved linear axial temperature distributionwas used in the later analyses [3,4]. More recent publications [5,6] included the solution of the energy equation for abetter estimation of the temperature distribution in an oil seal ring where the finite element technique was used toperform the thermo-hydrodynamic analysis. The influence of the circumferential grooves, tapered bore and axialgrooves on the seal characteristics and the dynamic stability of the compressor rotor can also be estimated by thefinite element method discussed in these more recent publications.

The two dimensional pressure and temperature distributions in the seal can be determined by solving the Reynoldsand the energy equations respectively. The domain is made up of a thin fluid film between the rotor and the seal. Thisproblem is not axi-symmetric because the fluid film around the rotor is not of constant thickness. Consequently, thegoverning equations must be solved over the whole domain.

These governing equations, Reynolds equation for pressure and the energy equations for temperature, are coupledand nonlinear as the viscosity depends on both pressure and temperature while the density and specific heat dependon temperature. The Reynolds and the energy equations can be solved using a finite element iterative technique toobtain the pressure and temperature distributions in a seal ring. A detailed explanation of the finite element iterativetechnique is given in Baheti [7]. The hydrodynamic forces are obtained by integrating the pressure distribution axiallyand circumferentially. The stiffness and damping coefficients of the seal are determined using a perturbationtechnique.

The majority of centrifugal compressor design evaluations for oil seals are currently conducted using a simplifiedanalysis as outlined in Kirk [2]. Multiple rings and circumferential grooving is found in modern high-pressurecompressor end seals. The eight stiffness and damping coefficients for estimated design point operating eccentricityratio are used in standard damped stability analysis evaluation. The computed average film temperature is used forthe calculation at each speed point.

It is important to note that the influence of a high-pressure oil seal should not be evaluated using 180° cavitatedbearing coefficients. An uncavitated 360° is more appropriate for all but the lowest pressure applications. Importantconsiderations for stability include the diameter and length of the seal land region, the operating clearance and thelocation and width of the casing sealing lip contact area. More extensive design guidelines are given in Kirk [2].

An example of frequency content for locked oil ring instability is shown in Figure 3-25 where the rotor is showing astrong subsynchronous excitation of its first natural frequency. Solutions to this type instability are dedicated toreducing the cross-coupling and focus on either reducing this directly (larger clearances, circumferential grooves) orby improving the bushing’s centering capability (soft starts, pressure balancing the bushing, tapering, reducingbushing weight). This is the type of oil seal induced rotor instability that can be predicted and prevented with adamped stability analysis that includes the oil seals.

The other major type of instability produced by oil seals is a low order tracking instability shown in Figures 3-26 and 3-27. Here the ring was not properly seated to the case and was driving the rotor at a frequency proportional torotational speed and not at a system critical speed frequency. The solution to low order tracking instability could be abetter surface finish on the sealing lip, reduced distortion of the casing end wall, an increased pressure drop across



the sealing lip, or increased sealing lip face diameters to prevent the ring from freely moving. This type of oil seal ringinstability cannot be predicted or prevented with a damped natural frequency stability analysis.

Kirk [2] presents an excellent description of the pressure distribution on both the inner and outer rings. A case study isdescribed where grooving is used to eliminate the destabilizing aspects of the outer ring. While this achieved thedesired results, it was also noted that the accompanying decrease in damping increased the unbalance response ofthe first critical speed. Grooving greatly reduces the hydrodynamic effects of the oil film with the resulting decrease incross-coupled stiffness. The rule of thumb is that the cross-coupled stiffness and direct damping vary with the lengthcubed. Thus, adding two grooves reduces the land length by a factor of 3, but triples the number of lands, so the netreduction in coefficients is 1/9th, Allaire et al [8]. Memmott [9] shows an example where multiple grooves and loweringof the seal shaft diameter reduced but did not eliminate a subsynchronous vibration problem on the mechanical testwith a ring type oil seal. There was no subsynchronous vibration when a tilt pad seal and other features (damperbearings and shunt holes) were applied in the field before start-up in the latter case.

Childs et al. [10] and [11] tested smooth annular oil seals and three centrally grooved seals with three depths. Themeasurements show the direct stiffness near zero up to an eccentricity of 0.3 after which a significant direct stiffnessis measured. The addition of grooves reduces the cross-coupled stiffness, but the amount is a function of rotationspeed. Significant added mass coefficients are obtained from dynamic tests indicating fluid inertia effects, which

Figure 3-25—Re-excitation of Rotor First Critical from Oil Seal Excitation

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Figure 3-26—Rotor Tracking Instability from Low-pressure Oil Seal Test

Figure 3-27—Rotor Tracking Instability from Distorted Oil Seal Lip Contact Area

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increase with increasing groove depth. The results show that the groove does not isolate the dynamic characteristicsof the two seal lands. The analysis code of Semanate and San Andres [12] was compared to these experimentalresults. The code does not predict the measured increase in direct stiffness at higher eccentricities and under predictsthe cross-coupled stiffness and direct damping (though the trends with eccentricity are similar). The work of Delgadoand San Andres [13] include the fluid dynamics of the groove in the model and greatly improve the predictionscompared to experiment including the inertia coefficients.

Cerwinske et al. [14] also presents a case study describing the influence of high-pressure bushing seals on therotordynamic characteristics of a centrifugal compressor. An analysis of the ring to determine the locked position ofthe ring and its effect on the stability is described. As with Kirk [2], the author states the importance of analyzing notonly the stability but the change in the unbalance response of the rotor as well. One of the few papers documentingring seal monitoring during compressor mechanical shop testing is discussed in Emerick [15] for one particular type ofseal design. It is very difficult to get probes into the high-pressure cartridge. Tests conducted in a static seal test rig toevaluate effective friction factors are discussed in Kirk and Browne [16].

3.4.1.2 References

[1] Kirk, R. G. and Miller, W. H., 1979, “The Influence of High Pressure Oil Seals on Turbo-Rotor Stability,” ASLETransactions, 22 (1), pp. 14–24.

[2] Kirk, R. G., 1986, “Oil Seal Dynamics: Considerations for Analysis of Centrifugal Compressors,” Proceedingsof the 15th Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station,Texas, pp. 25–34.

[3] Reedy, S. W. and R. G. Kirk, 1992, “Advanced Analysis of Multi-Ring Liquid Seals,” ASME Journal of Vibrationand Acoustics, 114 (1), pp. 42–46.

[4] Kirk, R. G. and S. W. Reedy, 1990, “Analysis of Thermal Gradient Effects in Oil Ring Seals,” STLE TribologyTransactions, 33 (3), pp. 425–435.

[5] Baheti, S. K. and Kirk, R. G., 1994, “Thermo-Hydrodynamic Solution of Floating Ring Seals for High PressureCompressors Using Finite-Element Method,” STLE Tribology Transactions, 37 (2), pp. 336–346.

[6] Baheti, S. K. and Kirk, R. G., 1995, “Finite Element Thermo-Hydrodynamic Analysis of a CircumferentiallyGrooved Floating Oil Ring Seal,” STLE Tribology Transactions, 38 (1), pp. 86–96.

[7] Baheti, S. K., 1995, “Non-linear Finite Element Thermo-Hydrodynamic Analysis of Oil Ring Seals Used inHigh Pressure Centrifugal Compressors,” Ph.D. Dissertation, Virginia Polytechnic Institute and StateUniversity, Blacksburg, Virginia.

[8] Allaire, P. E., Stroh, C. G., Flack, R. D., Kocur, Jr., J. A. and Barrett, L. E., 1987, “Subsynchronous VibrationProblem and Solution in Multistage Centrifugal Compressor,” Proceedings of the 16th TurbomachinerySymposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 65–73.

[9] Memmott, E. A., 1996, “Stability of an Offshore Natural Gas Centrifugal Compressor,” CMVA, 15th MachineryDynamics Seminar, Banff, pp. 11-19, October 7–9.

[10] Childs, D. W. Rodriguez, L. E., Cullotta, V., Al-Ghasem, A., and Graviss, M.,2006, “Rotordynamic-Coefficientsand Static (Equilibrium Loci and Leakage) Characteristics for Short, Laminar-Flow Annular Seals,” J. Tribol.,128 (2), pp.378–387.

[11] Childs, D.W., Graviss, M., and Rodriquez, L.E., 2007, “The Influence of Groove Size on the Static andRotordynamic Characteristics of Short, Laminar-Flow Annular Seals,” ASME J. Tribology, 129 (2), pp. 398–406.

[12] Semanate, J. and San Andrés, L., 1993, “Analysis of Multi-Land High Pressure Oil Seals,” TribologyTransactions, 36, No. 4, pp. 661–669.



[13] Delgado, A., and San Andrés, L., 2010, “A Model for Improved Prediction of Force Coefficients in GroovedSqueeze Film Dampers and Oil Seal Rings,” Journal of Tribology, 132, No. 3, 032202.

[14] Cerwinske, T. J., Nelson, W. E. and Salamone, D. J., 1986, “Effects of High Pressure Oil Seals On theRotordynamic Response of a Centrifugal Compressor,” Proceedings of the 15th Turbomachinery Symposium,Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 35–51.

[15] Emerick, M., F., 1982, “Vibration and Destabilizing Effects of Floating Ring Seals in Compressors,” NASA CP-2250, pp. 187–204.

[16] Kirk, R. G. and D. B. Browne, 1990, “Experimental Evaluation of Holding Forces in Floating Ring Seals,”Proceedings of the IFTOMM International Conference, Lyon, France, pp. 319–323.

3.4.2 Labyrinth Seals

3.4.2.1 General

The labyrinth seal has drawn considerable attention in rotordynamics concerning rotor stability. Since the mid-1970s,many articles have recognized the importance of the labyrinth seal on rotor stability in a variety of rotating equipment:centrifugal compressors, aircraft gas turbines and cryogenic rocket turbopumps.

A major goal for turbomachinery designers is to reduce internal labyrinth seal leakages, resulting in less flow acrossthese devices. This would contribute to increased overall machine efficiency and throughput capacity. The desire toreduce clearances of internal labyrinth seals could often be a compromise between low leakage and rotor systemstability.

For a back-to-back centrifugal compressor configuration, Figure 3-28, there is a long labyrinth balance piston (drum)seal at mid-span. This balances and reduces thrust loading of the rotor, which in turn reduces the size of the thrustbearing. A large pressure drop across the balance piston and the interaction of the gas flow in the small sealclearance spaces could produce considerable destabilizing excitation forces. Since these forces are produced at ornear rotor mid-span, they coincide with maximum rotor deflection at the first critical response of the rotor-bearingsystem; the effects on rotor stability could be significant. The mid-span location of the balance piston seal could be agood or bad influence, depending on how well the effective damping of the device is designed. Location of a seal neara nodal region limits its overall effectiveness for providing good direct damping. For damper seals with shunt holesand/or swirl brakes, the mid-span location could be an effective stabilizing influence in the rotor-bearing system.Adversely, the mid-span location for a balance piston seal with poor damping and high cross-coupling could beunfavorable and possibly contribute to rotor instability.

For an in-line centrifugal compressor configuration, Figure 3-29, the balance piston (drum) seal is located away fromthe mid-span, immediately past the final stage impeller. At first glance, this seal might be viewed as having less of aninfluence on rotor stability compared to a seal at mid-span location. However, for in-line compressors, the balancepiston seal takes the full pressure drop across the compressor, as compared to half the pressure drop in acomparable back-to-back compressor. The balance piston seal is typically longer and larger in diameter for an in-linecompressor. This larger diameter produces a higher surface speed for this configuration. With an in-line compressor,there might be more available direct damping from a damper seal due to the larger pressure drop. However, the endlocation of the balance piston seal is not as effective (as mid-span) for allowing this direct damping to help thecompressor. Cross-coupling could still contribute to rotor instability, even with the end location of the seal. Regardlessof compressor configuration, shunt holes and/or swirl brakes should be considered. There is no general conclusionthat could be made as to which type of compressor configuration, back-to-back or in-line, would have the betterchance for stable operation.

In addition to the balance piston seal, a compressor with covered impeller stages also has seals at each of theimpeller suction eyes. These are commonly known as impeller suction eye seals. The pressure rise across eachimpeller stage causes a corresponding pressure drop at these locations, which is lower than the pressure drop acrossa balance piston in a multi-stage compressor. Although the damping and cross-coupling contributions of a single



suction eye seal are typically less than that of a balance piston seal, the overall contribution of suction eye sealsbecomes increasingly significant for multiple stages at numerous locations along the rotor. Similar to a balance pistonseal located at mid-span, suction eye seals near the centermost part of the rotor become more influential. Theireffects should be considered by their pressure drops at each stage and location.

Centrifugal compressors also have interstage shaft seals, located from the diaphragms between impeller stages.These seals operate near the shaft O.D. or the O.D. of shaft sleeves. They are located at numerous locations alongthe rotor, similar to suction eye seals. There is not a significant pressure drop across these seals in comparison toother seals previously described. These shaft seals have smaller diameters than both balance piston and suction eyeseals, giving them a lower surface speed as well. Also, gases entering shaft seals have lost significant circulation frompassing through stationary return bends. Given all this, it is not practical to use shunt holes, swirl brakes, or elaboratedamper type seals at these locations. In most applications, shaft seals have minimal effects on rotor stability, so someturbomachinery designers may or may not include them in stability analyses.

In the late 1960s and early 1970s, an analytical calculation method to predict compressor shaft seal influence ondynamic stability did not exist. A method for computing multi-mass flexible rotor system stability was also unavailable.In the early to mid-1970s, computer analyses capable of predicting damped eigenvalues, and hence stability, weredeveloped using both transfer matrix and finite element methods (see 3.1.1). By the late 1970s and 1980s, a largepercentage of analytical rotordynamic research was developed for oil and labyrinth seals. The analysis of floating oilring seals (see 3.4.1), turbulent pump seals and gas labyrinth seals [1–6] were the key areas of development.

Figure 3-28—Typical Configuration for a Back-to-Back Compressor

Figure 3-29—Typical Configuration for an In-line Compressor

Balance drum

Balance Drum



Computer codes are now readily available for most turbomachinery designers to predict labyrinth seal stiffness anddamping effects.

Figure 3-30 shows a typical configuration for the last stage of a centrifugal compressor depicting an impeller eye seal,an inter stage shaft seal, and a typical balance piston seal configuration (Kirk [6]). The balance piston leakage flowsdown the backside of the impeller and across the balance piston labyrinth seal. The flow path between the impellerand the adjacent diaphragm of this particular example shows a converging section. Typically, the swirl velocityassociated with the gas exit angle from the impeller blades is approximately 60 % of impeller tip speed. This velocitywill vary with the impeller’s blade exit angle and with the amount of volume passing through the impeller. When theflow passes through this balance piston seal and the flow is not deswirled by a device such as shunt holes or swirlbrakes, there is potentially a large cross-coupling force generated, which may drive the rotor system unstable. Asimilar effect results from the impeller eye seal leakage that flows between the cover disc of the impeller and theadjacent diaphragm. The inlet swirl ratio is defined as the ratio of the average swirl velocity divided by the rotorsurface speed. For an impeller eye or balance piston labyrinth seal with no swirl brakes or shunt-injection, this is onthe order of 0.6 to 0.8 for many applications. With swirl brakes or shunt holes, the inlet swirl ratio is typically assumedto be from 0.0 to 0.3.

A flow is also created across the inter stage shaft seal due to the differential pressure across the diaphragm betweenstages. The pressure differential is created by the conversion of velocity head from the preceding impeller into staticpressure through the diaphragm. Inter stage shaft seal leakage flows from the region upstream of an impeller entrythrough the inter stage shaft seal up through the passage formed by the previous impeller’s hub disc and adjacentdiaphragm. The flow feeding these seals contain low swirl (15 % to 20 % of the surface velocity of the seal) since it isfed following the deswirling cascade in the return channel. Therefore, these seals have minimal effect on stability formost applications. There have been reported cases where these seals can be largely destabilizing during off design(high flow) operation. In these instances, the pressure gradient is reversed, due to stationary flow path losses causinghigh swirl flow to enter the seals (Baumann [7]).

Figure 3-30—Typical Configuration for the Last Stage of a Series Flow Compressor Showing the Impeller Eye Seal, the Inter Stage Seal and a Typical Balance Piston Seal

Impeller eye seal

Balance piston

Inter-stage shaft seal

Impeller eye seal and balance piston leakage direction



All labyrinth seals have a forward driving mechanism for entry swirls in the range of 20 % and higher. Althoughunlikely, lower entry swirl ratios or negative entry swirl can drive the system in backward whirl instability. The labyrinthseals are a likely source of forward driving subsynchronous excitation for rotating machinery [4,5].

One solution to labyrinth excitation is larger clearances, but this lowers efficiency. A larger chamber size or toothvolume is also a design factor to reduce the cross-coupled stiffness. Tooth spacing can be increased to reduce jeteffect leakage, but longer spacing produces a larger net force on the rotor for a given pressure field. Therefore, of allthe options to reduce a labyrinth seal’s destabilizing nature, the most attractive becomes controlling the entry swirl.Often, this may not be enough to suppress an instability and other seal types may be required such as a honeycombor a hole pattern seal (see 3.4.3).

Theoretical analyses have shown that toothed annular seals can develop relatively high cross-coupled stiffness andlow direct damping values that in turn give rise to excessive destabilizing forces exerted on the rotor. Predictions oflabyrinth seals are somewhat more difficult compared to a smooth plain annular seal. However, bulk flow models, aswell as CFD predictions, have shown improved results. The availability of high-pressure test data has permittedvalidation of these codes under realistic operating conditions.

A common approach to control the destabilizing influences arising in compressor seals is entry swirl reduction. Theinlet swirl in balance piston or division wall seals can be eliminated by use of shunt lines internal to the stationarycomponents that produce a buffer flow at the balance piston entrance to block the swirl from entering the seal asshown in Figure 3-31 (Kanki et al. [8]) and as shown in Figure 2-64 (Memmott [9]). When the compressor operates athigh flows, especially when diffuser vanes are used, the shunt differential pressure decreases thereby decreasing theeffectiveness of the shunt line to block or reduce entry swirl. Other methods of inlet swirl reduction throughaerodynamic or mechanical flow turning, commonly referred to as swirl brakes have also been utilized with near zeroswirl at the seal entrance (Figure 3-32), Childs and Ramsey [10], Moore and Hill [11] (Figure 2-63), and Memmott [12](Figure 2-62). Swirl brakes are commonly used upstream of impeller suction eye seals and division wall or balancepiston seals.

Figure 3-33 shows a compressor spectrum plot for (a) inert gas and (b) process gas during full load test runs Kirk [5]and Kirk and Simpson [13]. The increased molecular weight of the process gas results in a labyrinth seal inducedcompressor instability. The instability in (b), caused by the balance piston labyrinth seal, produces a forward drivingforce that excites the rotor’s first critical speed. This type of instability can be predicted and prevented with a dampedcritical frequency stability analysis that carefully includes the labyrinth seals and other excitation sources.

With the importance of labyrinth seal behavior on rotor stability known and demonstrated, testing efforts ranconcurrently with the analytical development. Benckert [1,14,15] gave the first published results of the stiffnesscoefficients of labyrinth gas seals. Teeth-on-stator seals were tested for interlocking, smooth, and stepped rotorconfigurations. Much of the data were for nonrotating seals at low pressures. Damping coefficients were not obtainedsince only static pressure measurements of the individual chambers were made. Kwanka et al. [16] as well asThieleke and Stetter [17] have also carried out similar efforts.

Wright [18] tested various configurations of a single cavity seal. While limited in its application, the results did provideinsight into the effects of pressure drop, clearance divergence or convergence and forward or backward whirl of aseal. Leong and Brown [19] investigated various teeth-on-stator configurations for variations of pressure, rotor speed,geometry and inlet tangential velocity. Again, damping was not considered.

On the basis of the results from Benckert [15] and recalculations made for various stable compressors, it wasdetermined that stiffness-only measurements were incomplete. Damping needed to be considered to accuratelydetermine the dynamic behavior of labyrinth gas seals.

The research summarized in Childs [20] provided the first measurements of the stiffness and damping coefficients.While the results gave the first comprehensive basis for comparison against predictions [10] revealed the importanceof testing at or near the application conditions. The test rig was extended toward this goal by Childs and Scharrer [21]and then again by Elrod et al. [22]. Teeth-on-stator, teeth-on-rotor, and interlocking configurations were tested.



Wagner and Steff [23] developed a test rig supported by magnetic bearings. The intent of the rig was to furtherexpand the existing knowledge database to geometries and gas conditions matching industrial applications, namely,pressure differential, size and speed. The magnetic bearings made it possible to set the eccentricity and superimposecirculatory movement independent of speed. The rig permitted the inlet pressure and the backpressure to be setindependently of one another. Pressures of 70 bar were possible at surface speeds up to 157 m/s.

Experimental results for long toothed labyrinths have measured significant negative principal stiffness but the bulk-flow codes over-predicted the negative stiffness at higher rotation speeds (Picardo and Childs [24]). Vannini et al. [25]utilize a magnetic bearing test rig and measures labyrinth seal coefficients at speeds up to 10,000 rpm and pressuresup to 200 bar (with capability of going to 350 bar in the future). These results represent the highest-pressure datapublished to date. The authors show agreement within 15 % to the prediction code of Thorat and Childs [26].

Figure 3-31—Shunt Line Schematic to Reduce Entry Swirl [8]

Labyrinth(balance piston)

Directionof rotation

Swirl cancelling port

Compressedgas

Impeller

Swirl control gas



Figure 3-32—Typical Swirl Brake Schematic to Reduce Entry Swirl

Figure 3-33—Compressor on Full Load Test: With Inert Gas Showing no Instability at Rotor First Critical Frequency [13], with Process Gas Showing Instability from Balance Piston Excitation [5,13]

Entrance flow tolabyrinth seal

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3.4.2.2 References

[1] Benckert, H. and Wachter, J., 1980, “Flow Induced Spring Coefficients of Labyrinth Seals for Application toRotordynamics,” NASA CP-2133, pp. 189–212.

[2] Iwatsubo, T., Matooka, N. and Kawai, R., 1982, “Spring and Damping Coefficients of the Labyrinth Seal,”NASA CP-2250, pp. 205–222.

[3] Childs, D. W. and Scharrer, J. K., 1986, “An Iwatsubo-Based Solution for Labyrinth Seals: Comparison toExperimental Results,” ASME Journal of Engineering for Gas Turbines and Power, 108 (2), pp. 325–331.

[4] Kirk, R. G., 1988, “Evaluation of Aerodynamic Instability Mechanisms for Centrifugal Compressors - Part I:Current Theory,” ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, 110 (2), pp. 201–206.

[5] Kirk, R. G., 1988, “Evaluation of Aerodynamic Instability Mechanisms for Centrifugal Compressors - Part II:Advanced Analysis,” ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, 110 (2), pp. 207–212.

[6] Kirk, R. G., 1990, “A Method for Calculating Labyrinth Seal Inlet Swirl Velocity,” ASME Journal of Vibration andAcoustics, 112 (3), pp. 380–383.

[7] Baumann, U., 1999, “Rotordynamic Stability Tests on High-Pressure Radial Compressors,” Proceedings ofthe Twenty-Eighth Turbomachinery Symposium, Turbomachinery Laboratory, Department of MechanicalEngineering, Texas A&M University, College Station, Texas, September, pp. 115-122.

[8] Kanki, H., Katayama, K., Morii, S., Mouri, Y., Umemura, S., Ozawa, U. and Oda, T., 1988, “High StabilityDesign for New Centrifugal Compressor,” Rotordynamic Instability Problems in High-PerformanceTurbomachinery, NASA CP-3026, pp. 445–459.

[9] Memmott, E. A., 1994, “Stability of a High Pressure Centrifugal Compressor Through Application of ShuntHoles and a Honeycomb Labyrinth,” Presented at the 13th Machinery Dynamics Seminar, CMVA, Toronto,Canada, September 12-13.

[10] Childs, D. W., and Ramsey, C., 1991, “Seal Rotordynamic-Coefficient Test Results for a Model SSME ATD-HPFTP Turbine Interstage Seal With and Without a Swirl Brake,” ASME Journal of Tribology, 113, pp. 113–203.

[11] Moore, J. J., and Hill, D. L., 2000, “Design of Swirl Brakes for High Pressure Centrifugal Compressors UsingCFD Techniques,” Proceedings of the Eighth International Symposium of Transport Phenomena andDynamics of Rotating Machinery (ISROMAC-8), March 26-30, Honolulu, Hawaii, pp. 1124–1132.

[12] Memmott, E. A., 2011, “Stability of Centrifugal Compressors by Applications of Damper Seals,” ASME,Proceedings of ASME Turbo Expo 2011, Power for Land, Sea and Air, Vancouver, Canada, June 6-10,GT2011-45634.

[13] Kirk, R. G. and Simpson, M., 1985, “Full Load Shop Testing of 18,000 HP Gas Turbine Driven CentrifugalCompressor for Offshore Platform Service: Evaluation of Rotordynamics Performance,” Instability in RotatingMachinery, NASA CP-2409, pp. 1–13.

[14] Benckert, H. and Wachter, J., 1979, “Investigations on the Mass Flow and the Flow Induced Forces inContactless Seals of Turbomachines,” Proceedings of the Sixth Conference on Fluid Machinery, ScientificSociety of Mechanical Engineers, Akadémiaki Kikado, Budapest, pp. 57–66.

[15] Benckert, H., 1980, “Stršmungsbedingte Federkennwerte in Labyrinthdichtungen,” Doctoral Dissertation,UniversitŠt Stuttgart.

[16] Kwanka, K., Ortinger, W. and Steckel, J., 1993, “Calculation and Measurement of the Influence of FlowParameters on Rotordynamic Coefficients in Labyrinth Seals,” Rotordynamic Instability Problems in High-Performance Turbomachinery, NASA CP-3239, pp. 209–218.



[17] Thieleke, G. and Stetter, H., 1990, “Experimental Investigations of Exciting Forces Caused by Flow inLabyrinth Seals,” Rotordynamic Instability Problems in High-Performance Turbomachinery, NASA CP-3122,pp. 109–134.

[18] Wright, D. V., 1983, “Labyrinth Seal Forces on Whirling Rotor,” Rotordynamic Instability, Proceedings of theASME Applied Mechanics, Bioengineering, and Fluids Engineering Conference, June 20-22, Houston, Texas,pp. 19–31.

[19] Leong, Y. and Brown, D., 1984, “Experimental Investigation of Lateral Forces Induced by Flow Through ModelLabyrinth Seals,” Rotordynamic Instability Problems in High-Performance Turbomachinery, NASA CP-2388,pp. 187–210.

[20] Childs, D. W., 1993, Turbomachinery Rotordynamics: Phenomena, Modeling, and Analysis, John Wiley &Sons, Inc., New York.

[21] Childs, D. W. and Scharrer, J. K., 1986, “Experimental Rotordynamic Coefficient Results for Teeth-on-Rotorand Teeth-on-Stator Labyrinth Gas Seals,” ASME 86-GT-12.

[22] Elrod, D. A., Pelletti, J. M. and Childs, D. W., 1995, “Theory Versus Experiment for the RotordynamicCoefficients of an Interlocking Labyrinth Gas Seal,” ASME paper 95-GT-432, presented at the InternationalGas Turbine and Aeroengine Congress and Exposition, Houston, Texas, June 5–8.


[24] Picardo, A. and Childs, D., “Rotordynamic coefficients for a Teeth-on-Stator Labyrinth Seals at 70 bar SupplyPressures Measurements Versus Theory and Comparisons to a Honeycomb Seal,” ASME J. of Gas Turbines,October 2005, 127, pp. 843–855.

[25] Vannini, G., Cioncolini, S., Calicchio, V., Tedone, F., 2011, “Development of a Ultra High PressureRotordynamic Test Rig for Centrifugal Compressors Internal Seals Characterization”, Proceedings of theFortieth Turbomachinery Symposium, September 12-15, 2011, Houston, Texas.

[26] Thorat, M., Childs, D., 2009, “Predicted rotordynamic behavior of a labyrinth seal as rotor surface velocityapproaches Mach 1”, Paper GT2009-59256, Proceedings of ASME International Gas Turbine Institute TurboExpo, Orlando, USA 2009.

3.4.3 Damper Seals

3.4.3.1 General

Historically, as operating conditions across annular compressor shaft seals have become more extreme (higherpressure differential, higher gas density, and increased speed), a corresponding decrease in rotor stability has beenmanifested. Oil and labyrinth seals have been the mainstay of the industry for many years and have been discussedabove. Oil seals have been largely replaced by dry gas seals due in part to stability and environmental concerns. Insome cases an alternative to labyrinth seals is required to obtain stable operation. Damper seals, such as hole patternand honeycomb seals, provide substantial damping thereby increasing rotor stability as described in 2.6 and arewidely utilized. Hole pattern seals are typically constructed of similar materials as toothed labyrinth seals and made byCNC milling. Honeycomb seals have traditional been two piece brazed designs using superalloy honeycombmaterial. More recent methods have employed EDM techniques with one-piece aluminum designs.

While honeycomb seals have been used for years as an abradable seal running against labyrinth seals in gasturbines, their use as a damper seal in compressors started in the 1990s (though some compressors were equippedwith honeycomb seals since the 1960s). In order to achieve damping, a smooth rotor is required. Figure 3-34(adapted from [1]) measure the rotor stability as a function of discharge pressure and demonstrate the effectivenessof a hole pattern seal that actually increases the logarithmic decrement as pressure is increased. This is opposite tothe behavior of traditional compressors with toothed labyrinth seals.



Honeycomb seals usually replace the tooth-on-stator labyrinth design with a metallic material insert with individualcells shaped in a honeycomb pattern. The hole pattern seals features are machined into the seal insert, makingretrofit of toothed labyrinth seals simpler and faster. A honeycomb seal is illustrated in Figure 2-60 from Childs [2].Cell size and depth along with overall seal length and clearance are parameters that can be varied to optimize thedynamic characteristics of the honeycomb seal. Research indicates that these dynamic characteristics are stronglyfrequency dependent due to the acoustic frequencies of the cells (Kleynhans and Childs [3]).

Typically the honeycomb material cannot support the high axial forces by itself and is thus contained as an insert in acarrier. Only very limited contact with the rotor can be tolerated before the honeycomb is damaged. If enough cells aredamaged, the effectiveness of the seal may be reduced. Additionally, honeycomb cells are susceptible to fouling (i.e.filling in of the honeycomb cells). However, Moore and Soulas [1] demonstrate that the damping is preserved, but theleakage increases when fouling occurs. Therefore, rotor stability will be maintained, but compressor performance willsuffer.

Figure 3-35 from Yu and Childs [4] shows a honeycomb seal compared to a hole pattern seal that uses a smooth rotorrunning against a stator surface filled with radial holes or dimples. Hole diameter, depth, spacing, clearance and seallength are optimization parameters. This seal can be manufactured from a solid piece and thus be more robust, beingable to withstand some contact with the rotor without a significant decay of damping properties. Figure 3-36 [4],shows a relative comparison of surface area for 3 different hole pattern geometries and a honeycomb pattern.

Implementing damper seals into a rotordynamic analysis and into the hardware design requires consideration ofseveral parameters and should be performed by experienced design and rotordynamic experts (Memmott [5]). It hasbeen found that it is prudent to use some kind of deswirling device with the honeycomb seal and with the hole patternseal. See [6, 7, and 8], where there were unacceptable levels of subsynchronous vibration at the first naturalfrequency with honeycomb balance piston or division wall seals without a shunt system and in each case a shunt

Figure 3-34—Measured and Predicted Rotordynamic Stability vs. Discharge Pressure for a High Pressure Centrifugal Compressor [1]

Test

LogDec

0

1

2

3

0 500 1000

Discharge Pressure (psia)

1500 2000 2500 3000 3500

Prediction



system was eventually used and the compressors were stable. Some of the compressors were of the back-to-backconfiguration and some were of the in-line configuration.

Experiences have shown that large dynamic coefficients are possible with these seals. Principal stiffness, bothpositive [6] and negative [8-11], can be created depending on the selection of the clearance profile. The referencedpapers [8-11] have shown that a divergent damper seal (honeycomb or hole pattern) can produce a negative directstiffness. This can lower the first natural frequency by reducing the overall support stiffness that reduces the effectivedamping of the system thereby leading to instability. Figure 3-37 (adapted from [11]) shows the effect of clearancetaper on the rotor stability. As the seal clearance becomes divergent, the calculated log dec first increases thensuddenly decreases becoming unstable. The seal clearance profile can be affected by the deformation of the sealand its carrier due to pressure and thermal effects. A finite element analysis of the seal mounting structure should beperformed in compressors that exhibit this sensitivity, such as high pressure re-injection compressors, including

Figure 3-35—Comparison of Honeycomb and Hole Pattern Seals [4]

Figure 3-36—Various Hole Pattern Surface Areas Relative to Honeycomb [4]

Hole-Pattern

Honeycomb

Honeycomb



assembly clearances and tolerances. A convergent tapered bore can be machined into the seal to give the desiredprofile during operation.

3.4.3.2 References

[1] Moore, J.J., Soulas, T.S., 2003, “Damper Seal Comparison in a High-Pressure Re-Injection CentrifugalCompressor During Full-Load, Full-Pressure Factory Testing Using Direct Rotordynamic StabilityMeasurement,” Proceedings of the DETC ‘03 ASME 2003 Design Engineering Technical Conference,Chicago, IL, Sept. 2-6, 2003.

[2] Childs, D. W., 1993, Turbomachinery Rotordynamics Phenomena, Modeling, and Analysis, John Wiley &Sons, Inc., New York.

[3] Kleynhans, G. and Childs, D. W., 1997, “Acoustic Influence of Cell Depth on the Rotordynamic Characteristicsof Smooth-Rotor/Honeycomb-Stator Annular Gas Seals,” NASA CP-3344, pp. 49–76.

[4] Yu, Z. and Childs, D. W., 1998, “Comparison of Experimental Rotordynamic Coefficients and LeakageCharacteristics Between Hole-Pattern Gas Damper Seals and a Honeycomb Seal,” ASME Journal ofEngineering for Gas Turbines and Power, 120 (4), pp. 778–783.

[5] Memmott, E. A., 1999, “Stability Analysis and Testing of a Train of Centrifugal Compressors for High PressureGas Injection,” ASME Journal of Engineering for Gas Turbines and Power, 121 (3), pp. 509–514.

[6] Memmott, E. A., 1994, “Stability of a High Pressure Centrifugal Compressor Through Application of ShuntHoles and a Honeycomb Labyrinth,” Presented at the 13th Machinery Dynamics Seminar, CMVA, Toronto,Canada, September 12–13.

[7] Gelin, A., Pugnet, J-M., Bolusset, D, and Friez, P., 1996, “Experience in Full Load Testing Natural GasCentrifugal Compressors for Rotordynamics Improvements,” ASME, Proceedings of ASME Turbo Expo 1996,Power for Land, Sea and Air, Birmingham, UK, 96-GT-378.

[8] Camatti, M., Vannini, G., Fulton, J. W., and Hopenwasser, F., 2003, “Instability of a High Pressure CompressorEquipped with Honeycomb Seals,” Proceedings of the Thirty Second Turbomachinery Symposium,

Figure 3-37—Example Plot of Predicted Rotor Stability as a Function of Damper Seal Clearance Taper [11]

Divergent Convergent-0.3 -0.2 -0.1 0 0

0

0.5

1

0.5

1

1.5

2

2.5

0.1 0.2

Seal Taper (mm)

Log

Dec



Turbomachinery Laboratory, Department of Mechanical Engineering, Texas A&M University, College Station,Texas, pp. 39-48, September 8–11.

[9] Kocur, J. A., and Hayles, G. C., 2004, “Low Frequency Instability in a Process Compressor,” Proceedings ofthe Thirty Third Turbomachinery Symposium, Turbomachinery Laboratory, Department of MechanicalEngineering, Texas A&M University, College Station, Texas, pp. 25–32, September 20-23.

[10] Eldridge, T. M. and Soulas, T. A., 2005, “Mechanism and Impact of Damper Seal Divergence on theRotordynamics of Centrifugal Compressors,” ASME, Proceedings of GT2005, ASME Turbo Expo 2005:Power for Land, Sea and Air, Reno-Tahoe, Nevada, USA, June 6-9, GT2005-69104.

[11] Moore, J.J., Camatti, M., Smalley, A.J., Vannini, G.V., Vermin, L.L., 2006, “Investigation of a RotordynamicInstability in a High Pressure Centrifugal Compressor Due to Damper Seal Clearance Divergence,” 7thInternational Conference on Rotor Dynamics, Sept 25-28, 2006, Vienna, Austria.

3.4.4 Dry Gas Seals

3.4.4.1 General

One of the characteristics of dry gas seals in compression applications is the neutral dynamic behavior of the seal.While oil seals have been shown to possess large magnitudes of damping and cross-coupled stiffness (see 3.4.1),dry gas seals values are several orders of magnitude less. In fact, it is common practice to only include the addedmass of the dry gas seal to the stability rotor model. Since they are treated as “neutral,” dry gas seal retrofits caneither correct existing stability problems, Atkins and Perez [1], or create problems, Kocur et al. [2], depending on thebehavior of the seals they are replacing.

3.4.4.2 References

[1] Atkins, K. E. and Perez, R. X., 1988, “Influence of Gas Seals on Rotor Stability of a High Speed HydrogenRecycle Compressor,” Proceedings of the Seventeenth Turbomachinery Symposium, TurbomachineryLaboratory, Texas A&M University, College Station, Texas, pp. 9–18.

[2] Kocur, Jr., J. A., Platt, Jr., J. P. and Shabi, L. G., 1987, “Retrofit of Gas Lubricated Face Seals in a CentrifugalCompressor,” Proceedings of the Sixteenth Turbomachinery Symposium, Turbomachinery Laboratory, TexasA&M University, College Station, Texas, pp. 75–83.

3.5 Excitation Sources

3.5.1 Aerodynamic Cross-coupling

3.5.1.1 General

The stability analysis of turbomachinery requires the examination of all forces arising from fluid/structure interactions.Many of these interaction forces are modeled as “aerodynamic cross-coupling.” Labyrinth seals are an excellentexample of this force interaction, possibly resulting in significant destabilizing cross-coupling forces. Numerousresearch papers on the instability of labyrinth seals have been published. Other force interaction examples includeimpellers-stator interactions on centrifugal compressors and turbine blades.

Aerodynamic cross-coupling, as defined in this context, refers to all rotor/stator interaction forces. In addition tolabyrinth seal induced cross-coupling, other rotor/stator interaction forces may arise from such phenomena asimpeller or blade tip leakage flows, secondary impeller flows, and shroud/stator cavity flows (Ehrich and Childs [1]).Due to the complex geometry involved, the transient nature of the interaction, and the difficulty in developing testapparatus, adequate design tools describing these forces are relatively unavailable. While significant work continueson this subject, the end result is most often a CFD (computational fluid dynamics) model of these rotor/statorinteraction forces that requires specialized knowledge of the software package and/or significant computational andpersonnel resources. Note, the definition of the aerodynamic cross-coupling term is not widely agreed in the Industry.



Some treat it as an all-inclusive global term, whereas for others it is a term that has to be added on top of the labyrinthseal coefficients since it only accounts for the stator/rotor fluid interactions.

3.5.1.2 Axial Stages

One of the earliest attempts to quantify the fluid/structure interactions as a design tool was performed by Alford [2].Alford established a theory and mathematical model to describe the aerodynamic cross-coupling produced in axialflow stages due to eccentric operation. As described by Alford, forces are produced normal to the shaft deflection dueto the circumferential variation in the blade tip clearance. The normal direction of the force to deflection is the natureof cross-coupled terms, i.e. a displacement in the ±x-direction produces a force in the ±y-direction. The converse isalso true.

Figure 3-38 presents the forces found in an axial stage from either a turbine or compressor. The torque forceproduced by an eccentric turbine stage is unbalanced due to the increased efficiency of blades with decreased tipclearance (solid arrows). Thus, a turbine stage with rotation and deflection shown in Figure 3-38 will produce a netvertical force in response to the horizontal displacement. In axial compressors, the inlet face has a uniform pressure.Additionally, the stage will discharge into a region of equal static pressure. Therefore, the pressure rise for everyblade must be the same in any given stage. For eccentric operation, the increased tip clearance will lower theefficiency of blades in that region. These blades will require more work in the form of larger input forces to generatethe same pressure rise (dashed arrows). As in the turbine, a net vertical force is produced. In both cases, theresultant force is normal to the displacement and tends to cause the rotor to whirl in the direction of rotation.

Alford accordingly developed a simple equation for the cross-coupled stiffness as a function of the stage torque (T),blade length (Ht), pitch diameter (Dt) and change in efficiency coefficient (Bt).

(3-5)

The equation provided designers with a simple relation to determine the expected aerodynamic cross-coupling foraxial flow stages. Alford’s equation has remained one of the few mathematical design tools available to calculate thedestabilizing forces in turbomachinery. Vance and Laudadio [3] attempted to provide some experimental backgroundfor Alford’s equation. While they were able to confirm the existence of the force, qualitative conclusions were mainlyreached.

3.5.1.3 Centrifugal Stages

Numerous attempts to develop a similar simple relationship for centrifugal impellers have been made. While CFDapproaches have yielded some success, none have provided the necessary ease of use that rotordynamicists needat the design stage. To date, most design tools for centrifugal turbomachinery represent all destabilizing forces foundbetween the casing end seals. Specifically, aerodynamic cross-coupling, impeller eye and hub seals and caselabyrinth seals are treated as one “global” destabilizing mechanism. Since this global destabilizing coefficient isintended to include all excitation sources (labyrinth seals, stator/rotor fluid interactions, etc.), it is a “black box”modeling approach.

A popular approach was developed empirically by Wachel [4]. The relation was developed using Alford’s equation asstarting point and applying it to compressor case histories where the stability (or instability) was known. It was foundthat several additional parameters (density ratio and molecular weight) were needed to adequately estimate this“global” destabilizing force. However, as designers gained experience applying Wachel’s relationship to a largerdatabase, the equation was thought to erroneously predict the destabilizing force for molecular weight gases muchdifferent than a molecular weight of 30. To remedy this, the “modified Alford’s” equation was developed, Memmott [5].

qi

TBtDtHt------------

HP BtC

DtHtN-----------------------= =



As with Wachel’s equation, the density ratio is added but the molecular weight is removed. As defined, the “modifiedAlford’s” equation takes the form:

(3-6)

Comparing the “modified Alford’s” equation to Wachel’s, we find that the two equations agree for gases with amolecular weight of 30. For lighter molecular weights, the “modified Alford’s” force predicts a stronger destabilizingforce. For heavier molecular weights, a smaller destabilizing force is predicted. These differences were based onexperience with a larger database of compressors [5].

Benckert and Wachter [6] have also developed a relationship that has been applied to approximate this “global”destabilizing force. Used with more regularity in Europe, the equation was developed explicitly for labyrinth sealsforces. However, it has been shown to sufficiently represent all destabilizing terms within centrifugal compressorapplications.

Kirk [7,8] compared theory against the actual operating experience of the compressor stability for severalapplications. Alford’s and Benckert’s equations were used to approximate the “global” destabilizing effects in the

Figure 3-38—Blade Forces Due To Centerline Displacement

F

YResultant force

Solid arrows = turbine forcesDashed arrows = axial compressor forces

e

w

XF2

F3

F1

F4

H

D

qi

HP BcC

DcHcN-----------------------

ds------

=



compressors. These were measured against two analytical methods developed solely to predict the labyrinth sealdynamic behavior [7,8] and Childs and Scharrer [9]. While a more in-depth discussion on the destabilizing effect oflabyrinth seals is included in 3.4.2, the major findings in [7,8] are emphasized here.

In comparing the estimated equations to the actual operating experience, Kirk [7,8] concluded that Alford’s equationprovided a reliable predictor of the actual operating stability. Benckert’s equation over-predicted the destabilizingforces by two to three times. More importantly was that either Kirk’s or Childs’ labyrinth seal coefficient predictionnearly equaled the destabilizing force predicted by Alford’s equation. The conclusion that was drawn then is that thelabyrinth seals play a dominant role in the instability of centrifugal compressors and may actually reflect the truesource of the “global” destabilizing effects predicted by either Alford’s or Benckert’s equations.

Due to the complexity of the flows in a centrifugal stage, CFD would seem to be the most accurate way to develop therequired force coefficients. Progress has been made by Moore et al. [11]. They applied a proper CFD technique to thevolume enclosed between the impeller shroud/hub and the relevant statoric parts with the purpose to extract the forcecoefficients for the stability analysis. They started analyzing a specific geometry and afterwards they made the resultsmore general using a nondimensional approach. They proposed a formula where the cross-coupled coefficient isdepending by the impeller geometry (namely the radial extension of the shroud), the operating conditions (namely, theperipheral speed, the ratio between actual and design flow coefficient and the gas density) and a constant factor thatis related to a certain impeller flow coefficient range.

Finally, it is worth noting that some researchers in the industry agree that the aerodynamic forces may not besignificant in comparison to the labyrinth seal forces with respect to the destabilizing cross-coupling in centrifugalcompressor applications [7,8]. Others have concluded that the aerodynamic forces represented by the modifiedAlford's equation for centrifugal compressors is independent from the labyrinth seal forces with respect to thedestabilizing cross-coupling in centrifugal compressor applications [5,10].

3.5.1.4 References

[1] Ehrich, F. F. and Childs, D. W., 1984, “Self-Excited Vibration in High-Performance Turbomachinery,”Mechanical Engineering, May, pp. 66–79.

[2] Alford, J. S., 1965, “Protecting Turbomachinery from Self-Excited Rotor Whirl,” ASME Journal of Engineeringfor Power, 87 (4), pp. 333–344.

[3] Vance, J. M. and Laudadio, F. J., 1984, “Experimental Measurement of Alford’s Force in Axial FlowTurbomachinery,” ASME Journal of Engineering for Gas Turbines and Power, 106 (3), pp. 585–590.

[4] Wachel, J. C. and von Nimitz, W. W., 1981, “Ensuring the Reliability of Offshore Gas Compression Systems,”Journal of Petroleum Technology, pp. 2252–2260.

[5] Memmott, E. A., 2000, “Empirical Estimation of a Load Related Cross-Coupled Stiffness and The LateralStability Of Centrifugal Compressors,” CMVA, Proceedings of the Eighteenth Machinery Dynamics Seminar,Halifax, April 26-28, 2000, pp. 9–20.

[6] Benckert, H. and Wachter, J., 1980, “Flow Induced Spring Coefficients of Labyrinth Seals for Application inRotordynamics,” NASA CP-2123.

[7] Kirk, R. G., 1988, “Evaluation of Aerodynamic Instability Mechanisms for Centrifugal Compressors—Part I:Current Theory,” ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, 110 (2), pp. 201–206.

[8] Kirk, R. G., 1988, “Evaluation of Aerodynamic Instability Mechanisms for Centrifugal Compressors—Part II:Advanced Analysis,” ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, 110 (2), pp. 207–212.

[9] Childs, D. W. and Scharrer, J. K., 1986, “An Iwatsubo-Based Solution for Labyrinth Seals: Comparison toExperimental Results,” ASME Journal of Engineering for Gas Turbines and Power, 108 (2), pp. 325–331.



[10] Memmott, E. A., 2000, “The Lateral Stability Analysis of a Large Centrifugal Compressor in Propane Serviceat an LNG Plant,” IMechE, Proceedings of the 7th International Conference on Vibrations in RotatingMachinery, Nottingham, England, pp. 187–198.

[11] Moore, J., Ransom, D., Viana, F., 2007, “Rotordynamic Force Prediction Of Centrifugal Compressor ImpellersUsing Computational Fluid Dynamics”, ASME Turbo Expo 2007 Proceedings.

3.5.2 Other Excitation Sources

3.5.2.1 Internal Friction, Shrink Fits and Shaft Material Hysteresis

The internal friction generated by rotor component shrink fits or in the rotor material itself can generate a destabilizingforce that may drive the rotor unstable at subsynchronous frequencies that are generally greater that 50% ofsynchronous rotor speed, Ehrich [1]. This instability, often referred to as hysteretic whirl, was the first self-excited rotorinstability phenomena identified, Newkirk [2] and Kimball [3]. The internal friction that is generated internally in therotor material is sometimes referred to as material hysteresis or internal damping.

Referring to Figure 3-39 [1], as the rotor deflects, the rotor’s neutral strain axis is normal or perpendicular to thedeflected direction. Usually, the neutral stress axis would be coincident with the neutral strain axis resulting in anelastic restoring force that is parallel and opposite to the rotor displacement. However, material hysteresis or internalfriction in a rotating shaft causes a phase shift in the neutral stress axis. This leads to a hysteretic lag angle where theneutral stress axis “lags” the neutral strain axis by some angle against rotation. Now, the restoring force is not paralleland opposite to the rotor displacement. In fact, the neutral stress axis phase lag results in a component of therestoring force that is perpendicular to the rotor displacement (and hence destabilizing) and in the direction of rotation.If this force is large enough, forward whirl will be induced, resulting in a subsynchronous rotor vibration.

Gunter [4], Ehrich [5], and Vance and Ying [6] have shown that internal friction from rotor component shrink fits cancause substantial destabilizing forces. Wheel, sleeve, or other shaft component shrink fit axial lengths should notexceed the shaft diameter and, in most cases, should be smaller depending on rotor geometry and configuration.Unfortunately, a method of analytically quantifying this internal friction induced destabilizing force is not readilyavailable.

3.5.2.2 Dry Friction Rubs

A dry friction rub is experienced when a rotating surface contacts a stationary surface without lubrication. Examplesinclude lubrication-starved journal bearing rubs (see 3.3.3.2), labyrinth seal rubs and impeller-to-stator rubs.

Figure 3-40 illustrates a rub phenomenon [1]. As the rotating part hits the stationary part, a tangential friction forceresults that is in a direction opposite of shaft rotation. This results in a backward whirl where the rotor whirls orprecesses in the opposite direction of shaft rotation. Rubs can produce a rotor vibration that is exactly 50 % ofsynchronous rotor speed, Goggin [7]. Depending on the location of the rotor’s critical speeds, rubs can also producerotor vibration frequencies at 1/n of synchronous rotor speed with n = 2, 3, 4, 5, 6, etc., Lee and Allaire [8].

In terms of a damped natural frequency analysis, a rub induced subsynchronous vibration is not technically an“instability”. Since the vibration can be subsynchronous and appear suddenly with rapid growth, it has been labeledby researchers as “rub induced instability.” However, rubs cannot be predicted or prevented by performing a rotordamped natural frequency stability analysis. Rubs are more appropriately classified and analyzed as a transientresponse problem [8].

3.5.2.3 Entrapped Fluids

Entrapped fluids in hollow rotors or coupling spacer tubes, Kirk et al. [9], Wolf [10], and Saito et al. [11], can alsoinduce a subsynchronous rotor vibration. This phenomenon is illustrated in Figure 3-41 and described in detail in [1].At some rotor deflection, the entrapped fluid is flung out radially in the direction of shaft deflection. The rotating



surface of the cavity drags the fluid in the direction of rotation. This results in a phase “lead” angle for the fluid’scentrifugal force. The entrapped fluid force has a component that is perpendicular to shaft displacement, in thedirection of shaft rotation. Thus, forward whirl may result at frequencies that range from roughly 80 % to 95 % ofsynchronous speed [9].

Entrapped fluid induced subsynchronous vibration should not be treated as instability of the rotor-bearing system.The vibration, while subsynchronous, will not grow unbounded. It cannot be predicted or prevented by performing arotor damped natural frequency stability analysis. It is more appropriately classified as a subsynchronous forcedresponse problem where the forcing function, the entrapped fluid, excites the rotor at a frequency that is less thanonce per shaft revolution.

3.5.2.4 Rotating Stall

Rotating stall is an aerodynamic phenomenon that occurs in vaneless diffusers and, less commonly, impellers ofcentrifugal compressors. The periodic unsteady flow pattern consists of low and high pressure regions that rotate inthe direction of the impeller rotation at speeds. For diffuser stall, the excitation frequency typically ranges from 5 % to20 % of shaft speed. For impeller stall, the excitation frequency will range from 50 % to 90 % of shaft speed. Both will

Figure 3-39—Shrink Fit Internal Friction and Shaft Material Hysteresis Destabilizing Force [1]

Rotatio

n

Cen

trfu

gafo

rce

Neutral axis

of stress

Destabilizingforce component

Hysteretic lag

Neutral axisof strain

Whirl

Force resultant

of flexural stress

Undeflectedshaft axis

Eas

tc re

stor

ngfo

rce

com

pone

ntExternaldamping



track with shaft speed. Rotating diffuser stall is usually encountered near the surge line of a compressor. However,poor diffuser design practices can place stall well within the operating range.

Basically, diffuser stall occurs when the radial velocity component at the impeller discharge is insufficient to allow thegas to escape the diffuser. In this instance, one or more pockets of recirculating gas form and proceed to rotate withinthe diffuser. The stall pocket(s) reduce the effective diffuser area thereby increasing radial velocity of the gas near thepocket. Further reduction of the radial component would lead the compressor into surge or a total collapse of the flowattempting to leave the diffuser.

Figure 3-40—Dry Friction Rub Backward Whirl Excitation [1]

Rotation

Cen

trfu

ga fo

rce

External damping

Tota

lco

ntra

ct fo

rce


Eas

tc re

stor

ngfo

rce

Whirl

Nor

ma

cont

act f

orce

Frictionangle

Contactcoulomb friction

force



Stall has an adverse effect on performance witnessed as drops in the efficiency and head curves usually near surge.Additionally, the rotating pressure regions can be a source of mechanical excitation of the impeller blades and shaft.Single pocket stall will have the largest impact on shaft vibration.

Some have erroneously classified this as rotor instability since the vibration is subsynchronous. Rotating stall is aforced response phenomenon with the forcing function being the rotating cells of gas. Design guidelines to avoid stallin centrifugal compressors can be found in Aungier [12].

Figure 3-41—Entrapped Fluid Cross-coupled Force [1]

Rota

tion

Cen

trfu

gafo

rce

on ro

tor

Trappedfluidsubject toviscousdrag indirection ofrotation

Destabilizingforce componentof trapped fluid

Viscousdrag angle

Whirl


Eas

tc re

stor

ngfo

rce

Cen

trifu

gal f

orce

of tr

appe

d flu

id

Externaldamping



3.5.2.5 Synchronous Thermal Instability “Morton’s Effect”

Research has shown that rotors supported in fluid film bearings will exhibit a nonuniform temperature distributionaround the bearing journals circumference under synchronous vibration (Keogh and Morton [13]). The thermal effectarises from the viscous shearing of the oil film during synchronous vibratory motion. Therefore, the phenomenonoccurs only at 1X vibration. A specific point on the rotor will always be on the outside of the orbit (the high spot) andwill therefore be closer to the bearing wall, Figure 3-42. This surface will have a smaller film thickness averaged overthe period of one orbit than the opposite side of the shaft.

At the journal locations, the oil film’s viscous shear will perform more work at the minimum clearance location. Thedifference in work over the circumference of the shaft produces a hot and cold spot on the journal. The temperaturedifference has been shown in test conditions to exceed 7 °C over the circumference (de Jongh and van der Hoeven[14]), thereby producing a thermal bend in the shaft end nearest the journal location. Larger shaft orbits producegreater temperatures differences and, thus, greater shaft bending.

The excitation source of this phenomenon is the shaft unbalance that is produced by the thermally induced shaftbending. If the rotor system is sensitive to shaft end unbalance and the phase angle between the residual unbalanceand the thermal bow is in a certain range, the thermal bend will result in an increased vibration level (and larger orbit).The larger orbit produces a greater thermal differential, which results in a greater bend or bow. In this case, thesystem is unstable and the synchronous vibrations can grow unbounded until the system becomes nonlinear or afailure occurs in one of the components. For other relative phase angles, the synchronous vibrations can slowly spiralout to a large value, then slowly spiral back to a small value in a very repeatable way.

Figure 3-42—Differential Heating at Bearing Journal for Synchronous Forward Whirl

V = 0

Cold spot

V = 0

h1

h2

V = R

V = RHot spot

Orbit



While synchronous thermal instability fits the classical definition of an unstable system, rotor stability codes arecurrently not used to predict its existence. As outlined in [14], a combination of an unbalance response code andempirically derived data is used to test for the stability condition of the rotor/bearing system. Additionally, thisphenomenon occurs only at synchronous frequencies. Verification tools have been developed in order to computeboth the circumferential thermal distribution in the shaft and the associated unbalance to be accounted for in therotordynamic analysis (Kirk [15] and Childs [16]).

The above discussion concerns the condition of the shafting outboard of the bearing. The majority of field casesreported in the literature are for overhung rotor locations and the previous design tools have been restricted to theseconditions. One example of a mid-span rotor excitation was reported by Larsson [17,18] and a rotor design tool forboth overhang and mid-span excitation has now been developed by Guo and Kirk [19,20].

3.5.2.6 References

[1] Ehrich, F. F., 1992, Handbook of Rotordynamics, McGraw Hill.

[2] Newkirk, B. L., 1924, “Shaft Whipping,” General Electric Review, 27 (3), pp. 169–178.

[3] Kimball, A. L., 1924, “Internal Friction Theory of Shaft Whirling,” General Electric Review, 27 (4), pp. 244–251.

[4] Gunter, E. J., 1967, “The Influence of Internal Friction on the Stability of High Speed Rotors,” ASME Journal ofEngineering for Industry, Series B, 89 (4), pp. 683–688.

[5] Ehrich, F. F., 1964, “Shaft Whirl Induced by Rotor Internal Damping,” Journal of Applied Mechanics, 23 (1), pp.109–115.

[6] Vance, J. M. and Ying, D., 2001, “Effect of Interference Fits on Threshold Speeds of Rotordynamic Instability,”International Symposium on Stability Control of Rotating Machinery, August 20-24, 2001, South Lake Tahoe,California.

[7] Goggin, D. G., 1982, “Field Experiences with Rub Induced Instabilities in Turbomachinery,” RotordynamicInstability Problems in High-Performance Turbomachinery, NASA CP-2250, pp. 20–32.

[8] Lee, C. C. and Allaire, P. E., 1999, “Nonlinear Transient Analysis of Point Rub Effects in a Flexible Rotor,”Journal of Applied Mechanics and Engineering, 4.

[9] Kirk, R. G., Mondy, R. E. and Murphy, R. C., 1984, “Theory and Guidelines to Proper Coupling Design forRotor Dynamic Considerations,” ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, 106(1), pp. 129–138.

[10] Wolf, J. A., 1968, “Whirl Dynamics of a Rotor Partially Filled with Liquid,” ASME Journal of Applied Mechanics,4, pp. 676–682.

[11] Saito, S. and Someya, T., 1979, “Self-Excited Vibration of a Rotating Hollow Shaft Partially Filled with Liquid,”ASME Paper No. 79-DET-62.

[12] Aungier, R. H., 2000, Centrifugal Compressors, A Strategy for Aerodynamic Design and Analysis, ASMEPress, New York.

[13] Keogh, P. S. and Morton, P. G., 1994, “The Dynamic Nature of Rotor Thermal Bending Due to UnsteadyLubricant Shearing Within a Bearing,” Proceedings of the Royal Society, London, England, A445, pp. 273–290.

[14] de Jongh, F. M. and van der Hoeven, P., 1998, “Application of a Heat Barrier Sleeve to Prevent SynchronousRotor Instability,” Proceedings of the Twenty-Seventh Turbomachinery Symposium, TurbomachineryLaboratory, Texas A&M University, College Station, Texas, pp. 17–26.

[15] Kirk, R. G., Guo, Z. and Balbahadur, 2003, “Synchronous Thermal Instability Prediction for Overhung Rotors”,Proceedings of the Thirty-Second Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&MUniversity, College Station, Texas, pp. 121–136.



[16] Childs, D. and Saha, R., 2011, “A New, Iterative, Synchronous-Response Algorithm For Analyzing The MortonEffect”, ASME Turbo Expo, Vancouver, Canada.

[17] Larsson, B., 1999a, “Journal Asymmetric Heating - Part I: Nonstationary Bow,” Journal of Tribology, Vol. 121,pp. 157–163.

[18] Larsson, B., 1999b, “Journal Asymmetric Heating—Part II: Alteration of Rotor Dynamic Properties,” Journal ofTribology, Vol. 121, pp. 164–168.

[19] Guo, Z. and Kirk, R. G., 2011, “Morton Effect Induced Synchronous Instability in Mid-span Rotor–BearingSystems, Part 1: Mechanism Study,” ASME Journal of Vibration and Sound, 133(6).

[20] Guo, Z. and Kirk, R. G., 2011, “Morton Effect Induced Synchronous Instability in Mid-span Rotor–BearingSystems, Part 2: Models and Simulations,” ASME Journal of Vibration and Sound, 133(6).

3.6 Support Stiffness Effects on Stability

3.6.1 General

In order to accurately predict rotor stability performance, accurate predictions of the effective bearing stiffness anddamping coefficients are a necessity. An important consideration in predicting these effective coefficients is theinfluence of the pedestal support stiffness and damping characteristics. These support characteristics act in serieswith the bearing oil film characteristics and the combined effective characteristics may be considerable lower than theoil film stiffness and damping properties. A detailed discussion of support stiffness may be found in 2.4.

In general, flexible supports provide an additional spring stiffness in series with the bearing oil film stiffness but withessentially zero support damping. This reduces the amount of effective damping provided by the bearing oil filmresulting in lower stability thresholds compared to rigid supports. The effective stiffness coefficients are also generallylower than those for the bearing alone resulting in lower damped natural frequencies compared to rigid supports. Thisreduction in effective stiffness and effective damping due to support flexibility generally reduces the stability of rotorswhere the instability mechanism is due to seals, balance pistons or impeller flows.

However, with squeeze film dampers, the additional flexibility from the damper oil film is beneficial. This is due to thelarge amount of additional damping provided by the damper oil film.

It may, however, increase the instability onset speed when the bearings are the primary source of instability (i.e. fixedgeometry bearing induced rotor instability discussed in 3.3.2). As an example, Vàzquez and Barrett [1] used a flexiblerotor test rig with flexible supports and fixed geometry bearings to investigate support flexibility effects on the rotorstability.

Figure 3-43 shows the predicted and measured instability onset speeds for the test rotor. The bearings are fixedgeometry multi-lobe bearings and are the source of instability in this rotor. The rotor is predicted to be unstable above5200 rpm if the bearing supports are considered rigid. A simple single mass flexible bearing support model (see 2.4.4)over-predicts the instability onset speed by a considerable margin. However, the predicted instability onset speedusing a more complex transfer function support model [1], closely predicts the measured instability onset speed of therotor.

Numerical examples of support flexibility effects on rotors operating on tilting pad journal bearings are illustrated with2 example centrifugal compressors. The 1089 kg “low-speed” compressor operates at 5600 rpm. The 232 kg “high-speed” compressor operates at 12,130 rpm. The average bearing stiffness for the low-speed compressor at 5600 rpmis 152,350 kN/m. The average bearing stiffness for the high-speed compressor at 12,130 rpm is 90,400 kN/m. Thesevalues already include the effects of pad pivot flexibility.

Figure 3-44 shows the effects of reducing the support stiffness on the log dec values for both compressors. Log dec isplotted against the stiffness ratio, Kratio, which is defined as the support stiffness divided by the average bearingstiffness. The support damping is set to zero for these calculations. As anticipated, as the support stiffness decreases,Kratio decreases and the compressor’s log dec values decrease. This effect is quite pronounced for stiffness ratiosbelow 5.0. API states that for stiffness ratios above 3.5, support stiffness may be neglected. This API limit line at 3.5 isalso shown on Figure 3.44.



Figure 3-45 shows the effects of reducing the support damping on log dec values for both compressors. Log dec isplotted against the % critical damping. Physically, critical damping, Cc, is the amount of damping necessary for anegligible increase in vibration passing through the critical speed. It is defined mathematically as:

Cc = 2.0 x (Ks x Ws/g)1/2

Cc = Critical Damping, N-s/m (lbf-s/in.)

Ks = Support Stiffness, N/m (lbf/in.)

Ws = Support Weight, N (lbf)

g = Acceleration of Gravity = 9.80665 m/s2 (386.1 in./s2)

Returning to Figure 3-45, the support stiffness is set equal to 525,360 kN/m for these calculations. As the supportdamping decreases, %Cc decreases and the compressor’s log dec value decreases. This effect is not as pronouncedas the support stiffness effects shown in Figure 3-44.

Reasonable %Cc values are in the range of 2 % to 10 %. The support weight is the weight of the component that thesupport stiffness represents. For example, if the bearing case is used in the model for support stiffness, then thebearing case weight should be used as the support weight.

Figure 3-43—Predicted and Measured Stability Threshold with and without Bearing Support Models

9.0E+06 1.0E+06 1.1E+06 1.2E+06 1.3E+06 1.4E+06

13,000

12,000

11,000

10,000

9000

8000

7000

6000

5000

4000

Inst

abty

Thr

esho

d (r

pm)

Assuming rigid supports

Horizontal Support Stiffness (N/m)

Lower bound ( = 0.01)PredictedHigher bound ( = –0.01)MeasuredSingle mass support



Figure 3-44—Stiffness Ratio vs. Log Dec for Two Example Centrifugal Compressors

Figure 3-45—% Critical Damping vs. Log Dec for Two Example Centrifugal Compressors

High-speed CCLow-speed CCAPI limit

0

0.00

0.05

0.10

0.15

0.20

0.25

0.30

-0.05

-0.1010

Stiffness Ratio, Kratio

Log

Dec

2015 255

% Critical Camping

Log

Dec

High-speed CCLow-speed CC

00.00

0.05

0.10

0.15

0.20

0.25

0.30

10 20155



3.6.2 References

[1] Vàzquez, J. A. and Barrett, L. E., 1999, “Transfer Function Representation of Flexible Supports and Casingsof Rotating Machinery,” Proceedings of the Seventeenth International Modal Analysis Conference, February8–11, Kissimmee, Florida, pp.1328–1334.

3.7 Experience Plots

3.7.1 General

In the 60s and early 70s, the need for high pressure gas reinjection machines was created by the natural gasproduction techniques in use. This represented an extension of the design capabilities and operating experience ofrotating equipment manufacturers and users. For the most part, mechanical and aerodynamic knowledge existed topermit the construction of these higher pressure compressors. What was lacking was the ability to accurately predictthe dynamic behavior of these rotors. This led to several highly published failures resulting from self-excitedsubsynchronous vibrations, i.e. instability (see 3.1.1).

Analytical methods were soon developed to calculate the damped eigenvalues and eigenvectors of rotatingequipment supported by journal bearings, oil seals, and squeeze-film dampers. Basic stability calculations becamepossible coupled with the development of the computer. However, the impact on rotor stability of two key componentsstill lagged behind, namely, labyrinth (gas) seals and impellers.

Experience plots were created to provide an initial screening criterion in the design or acceptance process. The plotswere based on existing experience concerning both operating machinery and analytical capabilities. The plotsinvolved a mix of calculated factors like the critical speed ratio, bearing span/impeller bore, and process variables likethe gas density, discharge pressure and case lift. These plots were intended to alert designers and users to rotors thatmay be susceptible to instability.

Sood [1] presented an experience plot in 1979 that was in use as a general design criterion for the rotor stabilityevaluation of high pressure compressors, Figure 3-46. His curve plotted the rotor flexibility ratio versus the averagegas density. Rotor flexibility ratio was defined as the ratio of maximum continuous speed to the first critical speed onrigid supports. A low flexibility ratio might indicate a more rigid rotor. His experience had shown that compressors witha low flexibility ratio are less susceptible to subsynchronous vibration in high-density applications. It was also notedthat compressors in lower density applications could be operated with higher flexibility ratios. No numbers wereincluded with the plot. However a low flexibility ratio could just indicate that the rotor is not running very fast relative toits first critical speed, yet the rotor might be less rigid.

Fulton [2], incorporating his company’s compressor experience and published data from unstable rotors, put somenumerical values to Sood’s curve. Plotting three known cases with instability, he drew a “Typical Threshold Line,”Figure 3-47. A “Worst Case Threshold Line” was developed using an unstable case with the lowest combination offlexibility ratio and average gas density that was known at the time by Fulton for Sood’s coordinates. Fultonrecommended that proposals for compressors above “the worst case line be required to include prices for an optionalfull speed, full gas density test with rated differential pressure across the compressor. The final decision to test couldthen be made later when the information from complete analytic studies is available.” Fulton also said that “The ‘worstcase’ line given should be a useful rule-of-thumb for indicating a threshold of concern for subsynchronous instability insimilar industrial centrifugal compressors.” He is not saying that if the compressor is above the worst case line then itwill be unstable. The Fulton plot was developed before dry gas seals were used and was developed fromcompressors with oil-film ring seals.

Kirk [3] proposed an empirical plot based on the pressure parameter, defined as discharge pressure multiplied by thecase lift (differential pressure across the compressor), vs. the flexibility ratio. See Figure 3-48 where he added theexperience points from the case history discussed in [4]). The pressure parameter was used to represent the workbeing done by the compressor (differential pressure) and the discharge pressure both thought to be importantparameters in determining the rotor stability. The slope and relative placement of the lines forming the boundaries



Figure 3-46—Sood’s General Rotor Stability Criteria [1]

Figure 3-47—Sood/Fulton Empirical Stability Criteria [2]

1

1.5

2

2.5

3

1 10 100 1000

Roto

r Fle

xibi

lity

Ratio

, R

Average Gas Density (kg/m3)

Safe Region

Unstable Region

=

1

Average Gas Density

Rotor Flexibility Ratio Maximum Continuous Speed First Critical Speed on Rigid Supports

Roto

r Fex

bty

Rato



were located using the Ekofisk and Kaybob compressors (see 3.1.1). Operating to the left of the “acceptable” line wasachieved through use of standard design features. Advanced features such as stabilized oil seal designs andstabilized labyrinth seal and balance piston designs (e.g. with the use of shunt holes or swirl brakes) were needed foroperation to the right of the “unacceptable” line. A gray zone between the “acceptable” and “unacceptable” linesindicated the zone where advanced features may be needed. Fulton [2] added a “worst case threshold line,” whichwas to the left of the Kirk “acceptable” line, to Kirk’s plot. Kirk [4] shows a sample of his company’s experience at thetime on his plot and on this had an experience limit for the pressure parameter for his company at that time. Thisconcept of showing an experience limit, no matter what the CSR, was very farseeing. There are now manycompressors that are above that experience limit that he showed.

Memmott [5,6,7], for compressors with tilt pad seals, plotted the flexibility ratio using the first critical speed with theinclusion of the tilt pad seals vs. the average gas density. He showed a representative sample of his manufacturer’sexperience up to the time of each paper. The critical speed was either from rotor response calculations or test andfield experience. He recommended using this plot when tilt pad seals are used, instead of the plot with the flexibilityratio with the rigid bearing first critical speed vs. average gas density. He showed a significant reduction in theflexibility ratio by the use of the tilt pad seals, i.e. the first critical speed with tilt pad seals could be significantly abovethe rigid bearing first critical speed. Plots as were given in Memmott [5,6,7] show that with the use of the rigid bearingfirst critical speed there are many points well over Fulton’s “typical” lines, but with the use of the first critical speed withthe tilt pad seals all points are under or just on Fulton’s “typical” line. Memmott [6] also gave a version of the Kirk plotusing the flexibility ratio with the tilt pad seals and this showed everything below Kirk’s “acceptable” line.

The 7th Edition of API Std 617 (2002) adopted a modification of the Fulton plot to use as an experience plot to dostability screening. An end-user had adopted a cut-off average gas density above which there was more strictscreening criteria. Some end-users and manufacturers had limits on the flexibility ratio irrespective of the average gas

Figure 3-48—Kirk’s Compressor Design Map [4]

Pre

ssur

e P

aram

eter

, P2D

P +

100

0(b2

/n.4

)

P2DP/1000 vs. N/NCRP2 = Discharge pressure (PSIA)DP = Pressure rise (PSI)N = Compressor operating speed (RPM)NCR = Compressor rigid bearing critical (RPM)

105

5

2

5

2

5

2

104

103

102

Ekofisk final(stable)

Ekofisk original(unstable)

Unacceptable

Kaybob original(unstable)

Kaybob final(stable)

Acceptable

1.0 2.0Critical Speed Ratio, N/NCR (DIM.)

3.0 4.0

Limit w/o megaswirl

Pressure parameterversus

speed ratio

Full load test

Inert gas condition



density. On the basis of several examples and these concerns, API Std 617 adopted a plot as is shown in Figure 3-49with a Region A, for which the analytical screening criteria is not severe, and a Region B, for which the analyticalscreening criteria is severe.

Memmott [8,9] showed experience plots with the use of the flexibility ratio with the rigid bearing first critical speed, forboth with the average gas density and with the pressure parameter. The plots in [8] contained a representativesample of his manufacturers’ experience to the time of publication with dry gas or toothed labyrinth casing end seals.Of the seven case histories of instability described in [8], four were for compressors with oil seals, one compressorhad used both oil and dry gas seals, one had toothed labyrinth casing end seals, and the other had dry gas seals.Each instability case history shown in [8] and [9] was solved by some combination or just one of tilt pad seals, damperbearings, shunt holes, and damper seals. There were no modifications to the rotors.

See the redrawn plot Figure 3-50 from [8]. Some of the case histories from both [8] and [9] are shown on Figure 3-50as the large symbols. On this plot of CSR vs. average gas density he showed the API regions A and B and Fulton’s“worst case” and “typical” lines. There are quite a few compressors that are above the Fulton “typical” line. On the plotof pressure parameter vs. CSR he defined a Region A and Region B and showed the Kirk “unacceptable” and“acceptable” lines and the “worst case” line that Fulton had drawn on the Kirk plot. See the redrawn plot Figure 3-51from [8], with the same case histories as in Figure 3-50. The case history with the diamond marker may have helpeddefine the Kirk “acceptable” line. Nothing appears above the Kirk “unacceptable” line.

A case history is shown on those plots, by a large square symbol. It is an in-line CO2 compressor with 167.7 bar(2432 psia) discharge pressure. During the field start-up there were unacceptable levels of subynchronous vibrationat the first natural frequency. This was with rigid support tilt pad bearings and no deswirling devices at the toothedlabyrinth balance piston seal. Squeeze-film dampers were installed in series with the tilt pad journals and shunt holeswere applied before the balance piston seal. The compressor then was stable. This case history showed that both theFulton and Kirk plots need to have a cut off average gas density or pressure parameter above which stricter analyticalcriteria need to apply no matter what the flexibility ratio. On the Fulton plot it is below the “worst case” line. On the Kirkplot it is well below both the Kirk “acceptable” line and the Fulton “worst case line”.

Figure 3-49—API Level I Screening Plot

Average Gas Density, ave, kg/m3 (lbf/ft3)

1(0.625)

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

10(6.25)

Region A

Region B

100(62.5)

API 617 Stability Criteria

Rot

or f

exb

ty ra

to, C

SR

1000(625)



Memmott [8,9] gave a plot of bearing span/impeller bore vs. average gas density. On this he defined a Region A andRegion B and showed that there can be a completely different evaluation of the potential for instability by use of thebearing span/impeller bore instead of the flexibility ratio. See an updated plot Figure 3-52 from [8] with the same casehistories as in Figure 3-50 and Figure 3-51. The example CO2 compressor discussed above appeared to be arelatively benign application on the Fulton and Kirk plots, yet it is on the edge of the experience envelope on Figure 3-52. It is a relatively slow running compressor with relatively long bearing span. The same comment may be madeabout some of the other examples on the plots Figure 3-50, Figure 3-51, and Figure 3-52.

Camatti et al. [10] gave their manufacturer’s experience on the plot of flexibility ratio with the rigid bearing first criticalspeed vs. average gas density. They showed a boundary experience line and a “warning” region, and a “stable”region, which was somewhat close to the API Region A.

Bidaut et al. [11] gave their manufacturer’s experience with high-pressure compressors on the plot of flexibility ratiovs. average gas density. Bidaut and Baumann [12] gave this plot again, now showing which compressors had toothedlabyrinth balance piston seals and which had hole pattern seals. The trend for higher densities and flexibility ratioswas to use hole pattern seals.

In Memmott [13,14], his experience plots are updated and are for compressors with dry gas or toothed labyrinthcasing end seals. Memmott [13,14] also added plots of pressure parameter vs. bearing span/impeller bore. SeeFigure 3-53, with the same data points and experience case histories are as shown as in Figure 3-50, Figure 3-51,and Figure 3-52. On this plot he also defined a Region A and Region B. The example CO2 compressor discussedabove is not as benign an application on Figure 3-53 as it appears to be in Figure 3-51. In [13,14], the points on thevarious experience plots are indicated whether or not the compressors have damper seals (honeycomb or holepattern) and whether or not the compressors have squeeze film dampers in series with the journal bearings. In totaltwelve case histories are discussed in [13,14] and it is also seen that these damper seals have significantly extendedthe experience envelope of centrifugal compressors in high density and high-pressure applications.

Figure 3-50—Memmott’s Compressor Experience Plot—Flexibility Ratio vs. Average Gas Density [8, 9]

With toothed laby or dry gas end sealsWith low pressure oil sealsWith oil or dry gas end seals

With toothed laby or dry gas end sealsFulton typical lineFulton worst case lineAPI border line between A & B regionsExample CO2 comp. with TL end seals

Region B

Region A

0.11.0

1.5

2.0

2.5

3.0

3.5

1.0 10.0Average Gas Density (lbm/ft3)

CS

R

100.0



Figure 3-51—Memmott's Compressor Experience Plot—Pressure Parameter vs. Flexibility Ratio [8,9]

Figure 3-52—Memmott's Compressor Experience Plot—Bearing Span/Impeller Bore vs. Average Gas Density [8,9]

With toothed laby or dry gas end sealsWith low pressure oil sealsWith oil or dry gas end sealsWith toothed laby or dry gas end sealsKirk Donald unacceptable lineKirk Donald acceptable lineFulton worst case lineBorder line between A & B regionsExample CO2 comp. with TL end seals

Region A

Region B

CSR1.0

10

100

1000

10,000

100,000

2.0 3.01.5 2.5 3.5

P2

x D

eta

P (p

s2 )/

1000

With toothed laby or dry gas end seals

With low pressure oil seals

With oil or dry gas end seals

With toothed laby or dry gas end seals

Border line between A & B regions

Example CO2 comp. with TL end seals

Region A

0.16

7

8

9

10

11

12

13

14

15

16

1.0Average Gas Density (lbm/ft3)

Bea

rng

Spa

n/Im

peer

Bor

e

10.0 100.0

Region B



3.7.2 References

[1] Sood, V. K., 1979, “Design and Full Load Testing of a High Pressure Centrifugal Natural Gas InjectionCompressor,” Proceedings of the Eighth Turbomachinery Symposium, Turbomachinery Laboratory, TexasA&M University, College Station, Texas, pp. 35–42.

[2] Fulton, J. W., 1984, “Full Load Testing in the Platform Module Prior to Two-Out: A Case History ofSubsynchronous Instability,” Rotordynamic Instability Problems in High-Performance Turbomachinery, NASACP-2338y, pp. 1–16.

[3] Kirk, R. G. and Donald, G. H., 1983, “Design Criteria for Improved Stability of Centrifugal Compressors,” AMDVol. 55, ASME, pp. 59–71, June).

[4] Kirk, R. G. and Simpson, M., 1985, “Full Load Shop Testing of 18,000 HP Gas Turbine Driven CentrifugalCompressor for Offshore Platform Service: Evaluation of Rotor Dynamics Performance,” Instability in RotatingMachinery, NASA CP-2409, pp. 1–13.

[5] Memmott, E. A., 1992, “Stability of Centrifugal Compressors by Applications of Tilt Pad Seals, DamperBearings, and Shunt Holes,” IMechE, 5th International Conference on Vibrations in Rotating Machinery, Bath,pp. 99–106, Sept. 7-10.

[6] Memmott, E. A., 1998-9, “Stability Analysis and Testing of a Train of Centrifugal Compressors for HighPressure Gas Injection,” ASME Turbo Expo ‘98, Stockholm, June 2-5, Journal of Engineering for GasTurbines and Power, July 1999, Vol. 121, pp. 509–514.

[7] Memmott, E. A., 2004, “Rotordynamic Stability Of The High Pressure Gas Injection Compressors At TheNorth Slope,” CMVA, Proceedings of the 22nd Machinery Dynamics Seminar, Ottawa, October 27-29.

[8] Memmott, E. A., 2002, “Lateral Rotordynamic Stability Criteria for Centrifugal Compressors,” CMVA,Proceedings of the 20th Machinery Dynamics Seminar, Quebec City, pp. 6.23–6.32, October 21-23.

Figure 3-53—Memmott's Compressor Experience Plot—Pressure Parameter vs. Bearing Span/Impeller Bore [13,14]

Region A

Region B

106 7 8 9 10 11

Bearing Span/Impeller Bore

P2

x D

eta

P (p

s2 )/

1000

12 13 14 15 16

100

1000

10,000

100,000With toothed laby or dry gas end sealsWith low pressure oil seals

With oil or dry gas end sealsWith toothed laby or dry gas end sealsBorder line between A & B regions

Example CO2 comp. with TL end seals



[9] Memmott, E. A., 2002, “API 684 2nd Edition—Section 3.0—Stability Analysis,” API, Presentation to SOME,API 67th Fall Refining Meeting, Dallas, TX, Sept. 30–October 2.

[10] Camatti, M., Vannini, G., Fulton, J. W., and Hopenwasser, F., 2003, “Instability of a High Pressure CompressorEquipped with Honeycomb Seals,” Proceedings of the Thirty Second Turbomachinery Symposium,Turbomachinery Laboratory, Department of Mechanical Engineering, Texas A&M University, College Station,Texas, pp. 39–48, September 8-11.

[11] Bidaut, Y., Baumann, U., and Al-Harthy, S. M. H., 2009, “Rotordynamic Stability Of A 9500 Psi ReinjectionCentrifugal Compressor Equipped With A Hole Pattern Seal-Measurement Versus Prediction Taking IntoAccount The Operational Boundary Conditions,” Proc. of the Thirty-Eighth Turbomachinery Symposium,Turbomachinery Laboratory, Department of Mechanical Engineering, Texas A&M University, College Station,Texas.

[12] Bidaut, Y. and Baumann, U., 2010, “Improving the Design of a Compressor Casing with the Help of FiniteElement Analysis to Ensure the Stability of High Pressure Centrifugal Compressor Equipped with a HolePattern Seal,” ASME, Proceedings of ASME Turbo Expo 2010, Power for Land, Sea and Air, Glasgow,Scotland, UK, June 14-18, GT2010-222185.

[13] Memmott, E. A., 2010, “Application of Squeeze-film Dampers to Centrifugal Compressors,” CMVA,Proceedings of the 28th Machinery Dynamics Seminar, Universite de Laval, Ville Quebec, Canada, October27-29.

[14] Memmott, E. A., 2011, “Stability of Centrifugal Compressors by Applications of Damper Seals,” ASME,Proceedings of ASME Turbo Expo 2011, Power for Land, Sea and Air, Vancouver, Canada, June 6-10,GT2011-45634.

3.8 Machinery Specific Considerations

The same rotor model used for unbalanced response analysis is normally sufficient for use in the stability model.However, bearings and seals can require different modeling as compared to the models used for unbalancedresponse analysis due to such items as frequency dependence. Machinery specific consideration is primarilyconcerned with the use of predicted cross-coupling forces unique to each type of machine along with relatingexperience applicable to each type of machine.


3.8.1.1 General

Early observations of instability problems in steam turbines reported a load dependent phenomenon. Pollman andTermeuhlen [1] described several cases in which they classified the instability as “steam whirl.” Some of these cases,however, turned out to be oil whirl caused by the unloading of fixed arc hydrodynamic bearings due to partial arcadmission forces. Other cases clearly were not caused by unloading of the bearings but were a genuine instability.Greathead and Bostow [2] described a load dependent instability in which the unit could be satisfactorily operated at90 % power, but would whirl subsynchronously at its first natural frequency at higher powers. In steam turbines, theseobservations led to the characterization of an “onset power level” of instability contrary to the more widely known“onset speed” of instability associated with fixed arc hydrodynamic bearings.

Steam turbines, in general, are much less prone to instability compared to other classes of turbomachinery such ashigh-pressure centrifugal compressors. Operation on fixed geometry hydrodynamic bearings is a general concern forany turbomachine. Aside from bearing related issues, those turbines that might be susceptible to instability typicallyoperate at high rotational speeds with high inlet steam pressures, and/or develop very high stage horsepower. Thedestabilizing forces that drive these instabilities are caused by steam leakage through eccentric peripheralclearances, as illustrated in Figure 3-54, Greathead and Bostow [2]. These forces are often referred to asaerodynamic since they are the result of fluid forces acting on the rotor. In steam turbines, the two sources of concernare labyrinth shaft seals and blade tip clearance leakage. Partial arc admission forces are a factor also since theyaffect rotor position and bearing loading causing changes in stiffness and damping characteristics. An instability



problem and solution on a 3600 rpm generator drive steam turbine operating on tilting pad bearings is presented inArmstrong and Perricone [3]. An analysis methodology and example is presented in Edney and Lucas [4].

3.8.1.2 Shaft Seals

In steam turbines, labyrinth shaft seals are the primary destabilizing source. The magnitude of the destabilizing force,and hence the likelihood of unstable operation, increases with pressure, speed, and rotor-seal eccentricity. Due to thelarge pressure drop and number of seals used, the labyrinths located in the high-pressure end gland of the machinetypically generate the highest destabilizing forces. On the other hand, the seals located near the maximum modalamplitude can have a greater influence on the rotor stability.

Brush seals have been increasingly applied to steam turbine applications. These seals have the advantage of being acompliant type seal, which tends to have good stability characteristics. Childs [5] have performed tests indicating thatthese seals have low cross-coupling forces.

3.8.1.3 Blade Tip Clearance Leakage

A secondary aerodynamic source is from tip clearance leakage of the rotating blades in eccentric operation. Thisexcitation arises due to the variation in blade efficiency associated with a change in tip clearance, which was firstpostulated by Thomas [6] in connection with instability problems on steam turbines. Alford [7] later identified the samemechanism associated with instability problems on aircraft gas turbines. In steam turbines these forces aresometimes neglected due to their relatively small magnitude compared to labyrinth seal forces. The cross-coupledstiffness is approximated from Alford’s equation, which is given in 3.5.1.2 as Equation 3-5. For steam turbines, the BTin Equation 3-5 is an empirical adjustment factor, usually in the range from 1.0 to 3.

The clearance excitation force is purely destabilizing without any direct stiffness or damping. For further information,the experimental work of Urlichs [8] on unshrouded, and Leie and Thomas [9] on shrouded tip leakage effects aresuggested. Although these researchers have measured values for BT of up to 5, experience has shown that forpractical purposes a value of 1.5 yields more appropriate results.

3.8.1.4 Partial Arc Admission Forces

It has long been recognized that the entire control valve opening sequence and the effect of partial arc admissiondiaphragm stages must be considered in a rotor unbalance response analysis. This is especially so in cases wherepartial arc admission forces are large relative to the rotor weight. The resultant effect can be one in which the rotor is

Figure 3-54—Steam Turbine Leakage Path

Rotating blade tip leakage

Interstage diaphragmgland leakage

Steam inlet

HP rearbearing

HP inlet endgland leakage

HP exhaust endgland leakage

Steamexhaust

HP frontbearing



loaded into a sector of the bearing where the dynamic characteristics are significantly different from what they wouldbe due to gravity load alone in Figure 3-55. Similarly, the stability of the rotor system can be affected. In the case of amachine with small stability margin, the change in bearing loading associated with a change in admission arc couldalter the oil film stiffness and damping characteristics sufficiently to cause the machine to go unstable. Modeling of thepartial arc admission forces is discussed in 2.8.1.4 along with a typical force diagram. The same model(s) used in theunbalanced response analysis should be used in the stability analysis.

3.8.1.5 References

[1] Pollman, E. and Termuehlen, H., 1975, “Turbine Rotor Vibrations Excited by Steam Forces,” ASME Paper 75-WA/Pwr-11.

[2] Greathead, S. and Bostow, P., 1976, “Investigations into Load Dependent Vibrations of the High PressureRotor on Large Turbo-Generators,” Proceedings of the Institution of Mechanical Engineers Conference onVibrations in Rotating Machinery, Cambridge, England, pp. 279–286.

[3] Armstrong, J. and Perricone, F., 1996, “Turbine Instability Solution—Honeycomb Seals,” Proceedings of theTwenty-Fifth Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, CollegeStation, Texas, pp. 47–56.

[4] Edney, S. L. and Lucas, G. M., 2000, “Designing High Performance Steam Turbines with Rotordynamics as aPrime Consideration,” Proceedings of the Twenty-Ninth Turbomachinery Symposium, TurbomachineryLaboratory, Texas A&M University, College Station, Texas, pp. 205–224.

[5] Childs, D. W., 1993, Turbomachinery Rotordynamics: Phenomena, Modeling, and Analysis, WileyInterscience, New York, Wiley-Interscience, pp 341–343.

Figure 3-55—Typical Resultant Bearing Load Vector Including Partial Arc Admission Steam Forces

Rotor weight

Resultant load vector

Steam force

Bearin

g pad

Bearing Shell



[6] Thomas, H., 1958, “Unstable Oscillations of Turbine Rotors Due to Steam Leakage in the Clearance of theSealing Glands and the Buckets,” Bulletin Scientifique, A.J.M. 71, pp 1039–1063.

[7] Alford, J. S., 1965, “Protecting Turbomachinery from Self-Excited Rotor Whirl,” ASME Journal of Engineeringfor Power, 87 (4), pp. 333–344.

[8] Urlichs, K., 1976, “Die Spaltstromung bei Thermischen Turbo-Machinen als Ursache fur die EnstehungSchwingungsanfacher Querkrafte,” (The gapflow in thermal turbo-machines as the cause for the lateralvibration forces), Ingenieur-Archiv, 45, pp 193–208.

[9] Leie, B. and Thomas, H., 1980, “Self-Excited Rotor Whirl due to Tip-Seal Leakage Forces,” RotordynamicInstability Problems in High-Performance Turbomachinery, NASA CP-2133, pp 303–316.

3.8.2 Electric Motors and Generators

3.8.2.1 General

Electric motors and generators do not contain significant cross-coupling force generating mechanisms such as oilseals or labyrinth seals. (Exceptions to this include hermetically sealed and liquid-filled motors). However, if theyoperate on fixed geometry bearings, a stability analysis should be conducted since these bearings can producecross-coupling forces that may be strong enough to drive the motor and/or generator rotor unstable (see 3.3.2).

The electromagnetic fields in the air gap of an electric machine generate the unbalance magnetic pull (UMP) withcross-coupling terms (Holopainen [1]). Due to the inherent dimensioning requirements the UMP is usually low and thecross-coupling terms do not have any effect on system stability. The reported cases of stability problems are very few[2,3,4]. In general, the cross-coupling effects are largest in high-power induction motors with a flexible shaft, largebearing span and small air-gap length.

High-speed electric motors operating well above the grid line supply frequency are more susceptible to instabilityproblems. In these machines, the bearing solution is usually based on the multi-lobe or tilting-pad designs. Very often,a high-speed electric motor is closely integrated to a centrifugal compressor or other driven machine. The couplingbetween the motor and driven machine is usually rigid, and the number of bearings is two or more. In these cases, thestability control requires the inclusion of all the system components including the electric motor.

In many canned applications, the electric machine is supported on active magnetic bearings, which are discussed indetail in 3.13.

3.8.2.2 References

[1] Holopainen, T.P. 2009. Simple electromagnetic force model for industrial rotordynamic analyses of electricalmachines. In: The 22nd Biennial Conference on Mechanical Vibration and Noise. San Diego, California, USA,August 30–September 2, 2009. New York: The American Society of Mechanical Engineers. 9 p.

[2] Kellenberger, W. 1966. “Der magnetische Zug in Turbogenerator-Rotoren als Ursache einer Instabilität desmechanischen Laufes” (The magnetic pull in turbo-generator rotors as a cause of instability of the mechanicalrotation), Archiv für Electrotechnik, Vol. 50, No. 4, pp. 253–265.

[3] Früchtenicht, J., Jordan, H. & Seinsch, H. O. 1982a. Exzentrizitätsfelder als Ursache von Laufinstabilitäten beiAsynchronmaschinen, Teil 1: Elektromagnetische Federzahl und elektromagnetische Dämpfungskonstante”(Eccentricity as a cause of rotation instability in asynchronous machines, Part 1: Electromagnetic spring andelectromagnetic damping coefficient). Archiv für Electrotechnik, Vol. 65, pp. 271–281.

[4] Früchtenicht, J., Jordan, H. & Seinsch, H. O. 1982b. “Exzentrizitätsfelder als Ursache von Laufinstabilitätenbei Asynchronmaschinen, Teil 2: Selbsterregte Biegeeigenschwingungen des Läufers”, (Eccentricity as acause of rotation instability in asynchronous machines, Part 2: Self-excited flexural vibrations of the rotor),Archiv für Electrotechnik, Vol. 65, pp. 283–292.



3.8.3 Power Turbines and FCC Power Recovery Expanders

3.8.3.1 General

The following discussion applies to overhung one or two stage power turbines and Fluid Catalytic Cracking (FCC)power recovery expanders. This equipment is subjected to adverse operating conditions: inlet temperatures exceed1200°F and, in the case of FCC expanders, the working gas can be laced with solid particles or contain high levels ofcorrosive compounds. Typically, this type of equipment possesses an extremely stiff rotor as the first undamped rigidbearing critical speed is generally above operating speed. A typical undamped critical speed map for an FCCexpander rotor is illustrated in Figure 3-56. Since the rotating element is much stiffer than most ordinary compressorsor steam turbines, the journal bearings and bearing support structures largely determine the critical speed andstability characteristics of these types of units. For the purpose of this discussion, it is assumed that the rotatingelement is supported by fluid film journal bearings.

Assuming that the rotating element is “stiff” relative to the journal bearings and bearing support stiffnesses, the lateralbehavior of overhung power turbines or FCC expanders is governed by the mass distribution of the rotating elementand the frequency dependent stiffness and damping characteristics of the bearings and bearing supports. The massdistribution of the rotating element is important because of its influence on the applied bearing loads. In particular,designers try to keep the center-of-gravity of the entire rotating element as far from the disk end bearing as possible(i.e. toward the coupling end bearing) in order to maintain load on the coupling end bearing. Keeping a net downwardload on the coupling end bearing in a two stage overhung unit may require a massive centerbody section (large shaftsection between journal bearings) resulting in a relatively stiff rotor compared to the journal bearings. If fixed geometrybearings are used, the lightly loaded coupling end bearing may be more susceptible to oil whirl. A case history of asleeve bearing induced instability for an overhung power turbine is summarized in 3.3.2.4.

Figure 3-56—FCC Expander Critical Speed Map

Undamped Critical SpeedSingle Stage Overhung FCC PRT Expander

5 Shoe Load-between-pads Tilt Pad Brg s/Kavg, diaphragm coupling1.0E+05

1.0E+05

1.0E+04

1.0E+04

1.0E+03

1.0E+021.0E+06 1.0E+07 1.0E+08


Crt

ca s

peed

(RP

M)

Operating speed = 4635 RPM

1st Critical speed2nd Critical speed3rd Critical speedKxxKyyDxx = Sqrt (Kxx2 + omega2 x Cxx2)Dyy = Sqrt (Kyy2 + omega2 x Cyy2)



Many manufacturers have installed or retro-fitted stabilizing bearing designs such as pressure dam or multi-pocketsleeve bearings in favor of two or three axial groove sleeve bearings because of fixed geometry bearing inducedinstability problems (Nicholas [1]). In general, such retrofits have eliminated subsynchronous vibrations but have noteliminated misalignment problems that are common to “hot” equipment. For this reason, new overhung units aresupported by tilting pad journal bearings.

Unlike most “cold” units such as centrifugal compressors, the bearing support structure must be capable ofaccommodating the thermal growths associated with “hot” operation. This generally means the supporting structure islight and/or flexible relative to the rotating assembly. Figure 3-57 illustrates a cross-section of an FCC power recoveryexpander. Note that the coupling end bearing is well supported in the radial direction by a pedestal box-structure whilethe disk end bearing is only supported by a box-like structure that is cantilevered from the pedestal.

For the expander design displayed in Figure 3-57, the static stiffness of the coupling end bearing support is greaterthan 3.0 x 106 lbf/in while the static stiffness measured for the disk end bearing support is much less than 1.0 x 106 lbf/in. The low stiffness associated with the disk end bearing support reduces the effectiveness of the disk end journalbearing’s damping capability (Nicholas et al. [2]). This, in turn, reduces the ability of the disk end bearing to suppressrotor vibrations and to promote stable rotor operation. For this reason, careful design of the disk end cantileveredsupport for purpose of maximizing the support stiffness is necessary.

It is not an easy matter to measure or calculate the dynamic stiffness and damping properties of the bearing supportsfor the following reasons:

1) The exhaust casing may influence support properties.

2) Temperature may influence the support properties through changes in the material modulus.

The dynamic properties of the bearing support system may have a heavy influence on the rotordynamics, particularlyon older machines. The dynamic properties of the bearing support system may be calculated or measured but shouldalways be tempered by experience. One approach to determining acceptability would be by performing an analysisusing worst case parameters. If such analysis is unacceptable, then a more detailed model could be determinedbased on either further analysis or by benchmarking to similar units. Figure 3-58 contains a schematic of the resultingrotor-bearing-support model of this equipment. Inclusion of the flexible support data in lateral models can dramaticallylower calculated critical speeds [2] and damped natural frequencies for these types of units (see 3.6). Manufacturer’sexperience with similar equipment will also help in providing an accurate measure of lateral critical speed and stabilityperformance.

3.8.3.2 References


[2] Nicholas, J. C., Whalen, J. K. and Franklin, S. D., 1986, “Improving Critical Speed Calculations Using FlexibleBearing Support FRF Compliance Data,” Proceedings of the Fifteenth Turbomachinery Symposium,Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 69–78.

3.8.4 Axial Compressors

3.8.4.1 General

The rotor mass-elastic characteristics of axial compressors depend on the method of rotor construction and bladeattachment. The four typical methods of rotor constructions are disc-on-shaft shrink fit (similar to most centrifugalcompressors), stacked disk with through tie bolts, drum rotors with studs or tie bolts, and solid rotors. Four typicalaxial compressor rotor construction examples are illustrated in Figure 3-59, Figure 3-60, Figure 3-61, and Figure 3-62.


3-74A

PI T

EC

HN

ICA

L RE

PO

RT 6

84-1

Figure 3-57—FCC Expander Cross-section


AP

I STA

ND

AR

D P

AR

AG

RA

PH

S RO

TO

RD

YN

AM

IC T

UT

OR

IAL: L

AT

ER

AL C

RIT

ICA

L SP

EE

DS, U

NB

AL

AN

CE R

ES

PO

NS

E, S

TAB

ILIT

Y, TR

AIN

TO

RS

ION

AL

S, AN

D R

OT

OR

BA

LA

NC

ING

3-75

The modeling of axial compressor rotors demands significant attention due to their complex construction features. Tie rod and stud torque levels along withstacked disk surface interactions greatly affect the stiffness properties of the rotor. In general, axial compressor rotors are more rigid than other processturbomachinery because of their large mid-span diameters and their hollow construction. For a solid cylinder and a hollow cylinder of equal cross-sectionalarea, the hollow cylinder will have higher lateral bending stiffness.

Furthermore, the support characteristics (pedestals) of axial compressors could be of the same order of magnitude as the fluid film characteristics of thejournal bearings in which the rotor is supported. Hence, this should be accounted for in the rotordynamic analysis (Nicholas and Kirk [1]) (see 3.6).

Figure 3-58—FCC Expander Rotor-bearing-support Model

Disk bearingcenterline

Coupling bearingcenterline

Kbrg

KbrgCbrg Cbrg

Bearing “Coffin”


3-76A

PI T

EC

HN

ICA

L RE

PO

RT 6

84-1

Figure 3-59—Axial Compressor Rotor Construction: Disk-on-shaft Shrink Fit


Blade Spacing


“B”

“Z”

Directionof flow

Rotation


and Lock Piece


AP

I STA

ND

AR

D P

AR

AG

RA

PH

S RO

TO

RD

YN

AM

IC T

UT

OR

IAL: L

AT

ER

AL C

RIT

ICA

L SP

EE

DS, U

NB

AL

AN

CE R

ES

PO

NS

E, S

TAB

ILIT

Y, TR

AIN

TO

RS

ION

AL

S, AN

D R

OT

OR

BA

LA

NC

ING

3-77

Figure 3-60—Axial Compressor Rotor Construction: Stacked Disks with Tie Bolts

Tiebolts


Blade Spacing

Balanceplug holes

“A”

“B”

“Z”

Directionof flow

Rotation


and Lock Piece

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15



The destabilizing aerodynamic cross-coupling stiffness in an axial compressor stage can be estimated from Alford’sequation that is discussed in detail and given as Equation 3-5 in 3.5.1.2. The stability characteristics of an axialcompressor due to aerodynamic cross-coupling forces should be evaluated using the cross-coupling coefficientscalculated from Alford’s equation.

Figure 3-61—Axial Compressor Rotor Construction: Drum Rotor with Studs

1 2 3 4 5 6 7 8 9 10 11

Tiebolts

View “Z”

View “A”Blade Spacing

Balanceplug holes

Holes forlifting stubshaft

“A”

“Z”

Directionof flow

Rotation

Blade Root



3.8.4.2 References

[1] Nicholas, J. C. and Kirk, R. G., 1982, “Four Pad Tilting Pad Bearing Design and Application for Multi-StageAxial Compressors,” ASME Journal of Lubrication Technology, 104 (4), pp. 523–532.

3.8.5 Gearboxes

3.8.5.1 General

The primary source of gearbox instability is fixed geometry bearings under light load conditions.

Figure 3-62—Axial Compressor Rotor Construction: Drum Rotor with Tie Bolts

Tiebolts


Blades, Spacers,and Lock Piece


“Z”

Directionof flow

Rotation

Tangential T-SlotBlade Root



3.8.5.2 Pinions with Tilt Pad Bearings

High-speed pinions with tilting pad bearings will not go unstable, as there are no destabilizing mechanisms. The onlyexception may be if the tilting pad bearing’s oil inlet and discharge flow geometries starve the bearing of oil. Oilstarvation can result in a subsynchronous rotor vibration. This phenomenon is described in detail in 3.3.3.2.Furthermore, Figures 3-14, 3-15, and 3-16 from 3.3.3.2 all concern high-speed gearbox pinion bearings.

3.8.5.3 Gear Force Influence on Bearing Load

Bearing stability is a concern with gearboxes due to manner in which load vectors change the bearings can becomelightly loaded or unloaded. The magnitude and direction of gear reaction load is directly proportional to the powertransmitted through the gear box. Events that change the power transmission directly influence the gear forces.Gearboxes may be found with “up or down mesh” gear sets. With down mesh sets, the pinion is always loadeddownward from zero to full unit load. However, the bull gear transitions from a gravity downward load to upward loadat rated torque. With up mesh sets, the bull gear bearings are always loaded downward from zero to full load whilethe pinion bearing load transitions from downward to upward. There are stability concerns in both cases when sleeve(fixed geometry) bearings are present.

Instability problems can occur at partial gear loads with sleeve bearings. The bull gear and pinion bearings must besized to handle the full gear load. Thus, the bearings are oversized for the partial load condition. As described in3.3.2, lightly loaded sleeve bearings are susceptible to oil whirl instability. This typically occurs on the bull gear withdown mesh gear sets with partial unit loading.

For this reason, a full stability analysis should be conducted on the pinion and bull gear at light or partial loadconditions to ensure stable operation. A stabilized sleeve bearing design, such as a pressure dam bearing (Nicholasand Allaire [1]) may help overcome lightly loaded sleeve bearing induced oil whirl instabilities.

An example of pressure dam bearings designed for gearbox application is shown in Figure 3-63 and Figure 3-64(Nicholas [2]). Figure 3-63 shows a pressure dam bearing design for the high-speed pinion. Since the resultantexternal gear force, Wg, is directed downward, the pressure dam is placed in the upper half. This ensures that thebearing’s load capacity is at a maximum for 100 % load. Note that for the 100 % load case, both the load vector andthe minimum film thickness are located in the lower half. For the 25 % load case, although the load vector is located inthe lower half, the minimum film thickness is located at the horizontal split. The difference between load and minimumfilm thickness is an indication of bearing instability. - angle approaching 90 indicates large cross-coupling forcesand thus, increased tendency toward oil whirl. The pressure dam helps to load the bearing at partial gear loadconditions minimizing the - angle, thereby improving the sleeve bearing’s stability performance.

Figure 3-64 shows the pressure dam bearing design for the bull gear. Now, the gear force is directed upward and thepressure dam is placed in the lower half. This ensures that the bearing’s load capacity is at a maximum for 100 %load. Note that the 25 % and 100 % load vectors as well as the minimum film thickness for both cases are all locatedin the upper half.

The above example is for a “down mesh” configuration where the bull gear “drives” the pinion downward due to thetangential gear force. The pinion is also pushed away from the bull gear due to the gear separating force. This resultsin the resultant load shown in Figure 3-63 at 18° (25 % load condition) and 19° (100 % load condition) with rotationfrom bottom dead center, which is approximately the gear pressure angle.



Figure 3-63—High-speed Gearbox Pressure Dam Pinion Bearing

Figure 3-64—High-speed Gearbox Pressure Dam Bull Gear Bearing

0.011 in. Pocket135°

25 % Loadminimumfilm location

100 % Load

Rb = 1.7535 in.

WjWg25%

Load

18°

100% Load

r = 19°

= 89°

= 67°

25 % Load

135°

25 % Load

Minimumfilm location

0.020 in.Pocket

100 % Load100 % Load

Rb = 3.006 in.

r = 135°

Wg

Wj

Wg

Gravity load

r = 157°

= -127°

= -145°



3.8.5.4 References



3.8.6 Centrifugal Compressors

3.8.6.1 Multi-Stage Compressors

The multi-stage configuration covers perhaps the largest variety of compression equipment, from large rotors incracked gas service to high compression equipment in re-injection service, Figure 3-65. This equipment category alsocontains the highest risk in terms of instability, and therefore has received the most attention. The stability analysisprocess for multi-stage compressors is well defined.

The factors affecting stability in multi-stage compressors are numerous. A list of some of those is included below.

a) The ratio of bearing span to shaft diameter (slenderness ratio)—This is used extensively to approximate the shaftstiffness or degree of bending of the first critical speed (Memmott [1]).

b) Power requirement of the compressor—There is a correlation between the magnitude of the instabilitymechanisms and the gas power requirements (Wachel and Von Nimitz [2]).

c) Pressure rise across the compressor and final discharge pressure—Large pressure differentials and highdischarge pressures increase the dynamic coefficients of labyrinth seals (Elrod et al. [3], Wyssmann et al. [4], Kirkand Donald [5], and Memmott [1]).

d) Suction pressure of the compressor—Determines the sealing pressure of the compressor. Higher suctionpressures equate to higher differentials across the main seals.

e) Shaft speed—Taken on its own, higher shaft speed will increase the ratio of operating speed to the first criticalspeed and, thus, the likelihood of instability (Wachel [6]).

f) Bearing configuration—More of a concern on older designs with fixed geometry bearings. The application of tiltpad bearings has eliminated the bearing as a major source of cross-coupling and possible whirl (see 3.3.2 and3.3.3) (Allaire and Flack [7]).

g) Casing end seal type—As with tilt pad bearings, the dry gas seal, as a substitute for oil seals, has greatly reducedthe instability drivers in newer designs. Dry gas seals are considered dynamically neutral (Atkins and Perez [8])(see 3.4.4), and therefore enable easier and more reliable rotordynamic prediction. Oil seals, on the other handare a possible source of both cross-coupling stiffness and damping. As such, oil seals can have either a positive

Figure 3-65—Typical Multi-stage High-pressure Centrifugal Compressor Rotor



or negative effect on rotordynamic stability, and therefore add uncertainty to the prediction (see 3.4.1). Theinclusion of tilt pads in the outer ring of the oil seals can drastically lower their associated cross-coupled stiffness(Memmott [9]).

h) Compressor configuration (back-to-back vs. straight-through)—The compressor configuration is usually dictatedby the performance requirements (e.g. inter-stage cooling). There may be some advantage in terms of rotorstability of one configuration over the other (Kirk [10,11]), but it also has been shown that either configuration canhave instability problems (Memmott [9]).

i) Gas molecular weight—Another parameter that experience has shown is directly related to the level ofdestabilizing forces present in a centrifugal compressor. Heavier molecular weight gases tend to increase thepossibility of instability (Kirk and Simpson [12]).

j) Gas density—Related to the discharge pressure and molecular weight, it is not surprising that higher gas densitiesare associated with an increased likelihood of instability (Wagner and Steff [13]). This factor is also found as theabscissa on numerous stability experience charts (Sood [14], Fulton [15], and Memmott [1,9,16]). However, thetrend is the opposite with deswirled hole pattern seals [1,20] as stability increases with increasing gas density.

k) Internal Seals—Labyrinth seals are considered to be a primary source of destabilizing forces within a compressor.Labyrinth seal destabilizing forces are heavily dependent on inlet pre-swirl entering into the seal; hence, anti-swirldevices, such as swirl brakes [17,18,19,20] and shunts [1,9,16,20] have been used to reduce cross-coupling.Damper type seals such as honeycomb [16], hole pattern type [1,20], and pocket damper seals (Richards et al.[21]) have also been used successfully, usually in conjunction with the use of anti-swirl devices. Of course, theselection of an internal seal is a balance between rotordynamic characteristics and leakage for compressorperformance.

3.8.6.2 Overhung Compressors

Stability is not normally a concern with this type of centrifugal compressor, Figure 3-66, due to the relatively highbending stiffness of the shaft. There are, however, high-speed and power applications where stability should beexamined. In these instances, the anticipated cross-coupling should be applied at the overhung impeller center-of-gravity (CG). In the case of multiple overhung impellers, the placement should coincide with the combined CG of theimpellers. Bearing type has considerable effect on overhung compressor stability. Operation on fixed geometryhydrodynamic bearings is a general concern for any turbomachine. Particular attention must be paid to the bearingloading at the coupling end of the machine, which can be lightly loaded due to the overhung weight of the impeller.

Figure 3-66—Overhung Compressor Rotor



In overhung compressors, the critical speed prediction is heavily dependent on the impeller mass, inertia and CGlocation, which must be accurately defined. Minor errors in these properties are magnified when dealing withoverhung dynamics as compared to multi-stage beam type compressors.


Integrally geared compressors differ from overhung compressors and must take into consideration the influences ofboth the destabilizing influences of compressors and gas seals and also the influence of the gear loading on thebearings. The existence of the gear mesh at the center of the shaft (or pinion) generates radial forces that should betaken into consideration for the bearing loads. As discussed in 3.8.5 the gear forces depend on the power transmittedthrough the gear and the rotational speed. Therefore the range of operating conditions must be considered asinfluences to the bearing load. The radial load is treated as a vector addition to the rotor weight at the bearings.Variable inlet guide vanes make it possible to change the gas load at a constant speed. The varying loads maychange the dynamic behavior of the radial bearing, thus, influencing the rotor stability. The gear loads producedduring a reduced pressure start should be analyzed as these may represent the least stable condition. The range ofgear loads should be considered when performing the stability analysis.

The destabilizing influences as discussed in 3.8.6.1 are present since overhung impellers at one or both ends of theshaft are present. These distinguishing features can be seen on Figure 3-67.

The possibility of having impellers at both ends of the shaft does add a small complication to the Level 1 analysis.Each impeller should be treated as a separate compressor. The implication is that an anticipated cross-coupling willbe calculated and applied to each impeller for the Level 1 analysis. This is the only instance where more than oneanticipated cross-coupling is applied in a Level 1 analysis. In the case of having a blind end pinion (one impeller only),the anticipated cross-coupling is calculated and applied at the location of the individual impeller. The stability analysisthen proceeds as described in 3.8.6.1.

Figure 3-67—Pinion Rotors from an Integrally Geared Compressor



3.8.6.4 References

[1] Memmott, E. A., 2011, “Stability of Centrifugal Compressors by Applications of Damper Seals,” Proceedingsof ASME Turbo Expo 2011, Power for Land, Sea and Air, Vancouver, Canada, June 6-10, GT2011-45634.

[2] Wachel, J. C. and von Nimitz, W. W., 1981, “Ensuring the Reliability of Offshore Gas Compression Systems,”Journal of Petroleum Technology, pp. 2252–2260.

[3] Elrod, D. A., Pelletti, J. M. and Childs, D. W., 1995, “Theory Versus Experiment for the RotordynamicCoefficients of an Interlocking Labyrinth Gas Seal,” ASME paper 95-GT-432, presented at the InternationalGas Turbine and Aeroengine Congress and Exposition, Houston, Texas, June 5–8.

[4] Wyssmann, H. R., Pham, T. C. and Jenny, R. J., 1984, “Prediction of Stiffness and Damping Coefficients forCentrifugal Compressor Labyrinth Seals,” ASME Journal of Engineering for Gas Turbines and Power, 106, pp.920–926.

[5] Kirk, R. G. and Donald, G. H., 1983, “Design Criteria for Improved Stability of Centrifugal Compressors,” AMDVol. 55, ASME, pp. 59–71.

[6] Wachel, J. C., 1982, “Rotordynamic Instability Field Problems,” Rotordynamic Instability Problems in High-Performance Turbomachinery, NASA CP-2250, pp. 1–19.

[7] Allaire, P. E. and Flack, R. D., 1982, “Design of Journal Bearings for Rotating Machinery,” Proceedings of theTenth Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station,Texas, pp. 25–46.

[8] Atkins, K. E. and Perez, R. X., 1988, “Influence of Gas Seals on Rotor Stability of a High Speed HydrogenRecycle Compressor,” Proceedings of the Seventeenth Turbomachinery Symposium, TurbomachineryLaboratory, Texas A&M University, College Station, Texas, pp. 9–18.


[10] Kirk, R. G., 1988, “Evaluation of Aerodynamic Instability Mechanisms for Centrifugal Compressors Part I:Current Theory,” ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, 110 (2), pp. 201–206.

[11] Kirk, R. G., 1988, “Evaluation of Aerodynamic Instability Mechanisms for Centrifugal Compressors--Part II:Advanced Analysis,” ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, 110 (2), pp. 207–212.

[12] Kirk, R. G. and Simpson, M., 1985, “Full Load Shop Testing of 18,000 HP Gas Turbine Driven CentrifugalCompressor for Offshore Platform Service: Evaluation of Rotordynamics Performance,” Instability in RotatingMachinery, NASA CP-2409, pp. 1–13.


[14] Sood, V. K., 1979, “Design and Full Load Testing of a High Pressure Centrifugal Natural Gas InjectionCompressor,” Proceedings of the Eighth Turbomachinery Symposium, Turbomachinery Laboratory, TexasA&M University, College Station, Texas, pp. 35–42.



[15] Fulton, J. W., 1984, “Full Load Testing in the Platform Module Prior to Tow-Out: A Case History ofSubsynchronous Instability,” Rotordynamic Instability Problems in High-Performance Turbomachinery, NASACP-2338, pp. 1–6.

[16] Memmott, E. A., 1999, “Stability Analysis and Testing of a Train of Centrifugal Compressors for High PressureGas Injection,” ASME Journal of Engineering for Gas Turbines and Power, 121 (3), pp. 509–514.

[17] Benchert, H., and Wachter, J., 1980, “Flow Induced Spring Coefficients of Labyrinth Seals for Application inRotordynamics,” Rotordynamic Instability Problems in High Performance Turbomachinery, NASA CP2133,pp. 189–212.

[18] Baumann, U., 1999, “Rotordynamic Stability Tests on High-Pressure Radial Compressors”, Proceedings ofthe Twenty-Eighth Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, CollegeStation, Texas, pp. 115–122.

[19] Wagner, N. G., 2001, “Reliable Rotor Dynamic Design of High-Pressure Compressors Based on Test RigData,” ASME Journal of Engineering for Gas Turbines and Power, 123, pp. 849–856.

[20] Moore, J. J., Walker, S. T., and Kuzdzal, M. J., 2002, “Rotordynamic Stability Measurement During Full-LoadFull-Pressure Testing of a 6000 PSI Re-injection Centrifugal Compressor,” Proceedings of the Thirty FirstTurbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas,September

[21] Richards, R. L., Vance, J. M., Paquette, D. J., and Zeidan, F. Y., 1995, “Using a Damper Seal to EliminateSubsynchronous Vibrations in Three Back-to-Back Compressors,” Proceedings of the Twenty-FourthTurbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp.59–71.

3.9 Solving Stability Problems

3.9.1 General

Rotordynamic instability is perhaps the most damaging and disruptive vibration problem experienced in the field.While less frequent than other sources of vibration, the potential for machine damage and extended downtime ismuch greater. This is due in large part to the nature of an instability problem. The onset of the subsynchronousvibration is sudden and virtually unbounded. Furthermore, rotor instability is normally a re-excitation of the rotor’s 1stcritical speed with a peak deflection near the mid-span for multi-stage rotors. This is coupled with a tangential forcethat encourages whirling. This is not the case with unbalance excitation of the 2nd critical, the 1st critical or, in fact,any critical speed. With large peak internal deflections and little control of the vibratory motion, the likelihood ofinstability related damage to the rotor and stator components is greatly increased.

The lengthy disruption to operations caused by rotordynamic instability stems from three sources: the consequentialdamage, an inability to operate in conjunction with the instability, and the nature of the solution. As previouslymentioned, instability has a high probability of causing internal flow path damage and loss of performance. Repairsmay involve extended shutdowns to remove the aero bundle for replacement of parts not normally spared.

Most sources of vibration permit continued operation under all but extreme situations. Unfortunately, it is rare thatinstability vibration levels are not damaging to the rotor. Most cases involve immediate shutdown and protractedoutings (Doyle [1]).

Finally, solutions to instability problems tend to be more complicated involving reconfiguration of the rotor-bearingsystem to some extent. These can range from bearing parametric changes to shaft modifications. The solution willalmost always entail the replacement of some part (i.e. new bearings, modified labyrinth seals) and may includeredesigning and/or re-machining of the internal configuration (i.e. bearing span reduction, shrink fit modification, etc.).



This is very costly and time consuming in contrast to field balancing, realignment, or elimination of mechanicallooseness that can be performed immediately to remedy other types of vibration problems. An excellent exampleillustrating the difficulty of solving an instability problem may be found in Fowlie and Miles [2] and Smith [3]. Bothpapers describe the classic Kaybob instability problem from the early 1970s and the costly, time consuming solution.

These solutions can be grouped into one of two categories: decreasing or eliminating the excitation force andincreasing the effective damping. If the instability problem is modeled as a sum of the tangential forces inducing whirlat any particular natural frequency (see Figure 3-68 for a graphical description of these forces), then the rationalebehind this grouping becomes obvious. Describing the tangential force as:

(3-7)

In this expression, the destabilizing forces contained in the rotor-bearing system are represented by the modal sum ofk. The second term, C, represents the effective damping at frequency that is available to counteract the tendencyto whirl. Viewed in this sense, the tangential whirling force, Ft, may be reduced by decreasing k and/or by increasingC.

Before proceeding, a brief definition of the effective bearing damping is presented as used in the context of thissection. Effective damping is defined as the percentage of damping provided by the rotor supports available to controlthe particular critical speed in question. For beam compressors, this mode is usually the first and control of the mid-span is critical. For a mode behaving as a rigid body, (Kshaft >> Ksupport), the percentage of bearing damping availableto control the motion is nearly 100 % (top mode shape in Figure 3-69). Conversely, for modes behaving as a flexiblebody, (Kshaft << Ksupport), the bearings become nodal and the percentage of bearing damping available to control the

Figure 3-68—Forces Exerted on a Whirling Shaft

FtA----- k C– =

Y-axis

Whirl orbit

Shaft

A

X-axis

Fr (K – M 2) x A

Ft (k – C ) x A



vibratory motion is nearly 0 % (bottom mode shape in Figure 3-69). Keep in mind that motion at the bearings isneeded for the bearing damping terms to generate a suppressing force. In the case of rigid bearings, no motion existsat the bearing location and therefore, no bearing damping force is created. Thus, the options to increase the effectivedamping are to stiffen the shaft, soften the support system through incorporation of a damper bearing or increase theoverall effective damping present. If oil-film ring seals are used then tilt pad oil-film seals have been used to increasethe overall effective damping. Another option to increase the overall effective damping is to use a damper seal at thebalance piston location.

Table 3-2 and Table 3-3 give some guidelines for performing either task of eliminating the excitation or increasing theeffective damping. However, instability mechanisms and predictor tools are complicated and greatly intertwined withother dynamic effects within the turbomachine. These guidelines are not intended as a substitute for thorough fieldinvestigation or a detailed stability analysis. The most common practices are shown in boldface type.

Figure 3-69—Mode Shapes for Various Support/Rotor Stiffnesses

Soft support/rigid shaft mode shape

Stiff support/flexible shaft mode shape



Table 3-2—Decreasing or Eliminating Excitation Forces

Component Modifications Benefits Drawbacks References

Oil Seals (see 3.4.1)

Increase Clearance

Dynamic properties decrease with increasing clearance.

Increases oil leakage, decreases principal stiffness. Decreases the centering capability of the seal ring.

Emerick [4]

Circumferential Grooves

Grooving separates the seals into individual units.

Decreases the centering capability of the seal ring. Decreases principal stiffness. Increases oil leakage.

Allaire [5]

Pressure Balance Seal

Reduces the force that locks up floating seal ring eccentrically.

May require extensive seal modifications, i.e. face lip change (reduce axial force) or seal face coating (reduce friction force).

Allaire [5]

Tilt Pad Seal Adding tilt pads to the outer bushing reduces the eccentricity and increases the effective damping. May increase the critical speed.

Heat load and minor cost increase. May increase the critical speed.

Memmott [6]

Reduced Suction Pressure Start

Enables floating seal rings to center during start transients.

May require piping or process control modifications that may not be practical and economical.

Gas Seal Retrofit Replaces oil seals with a dynamically neutral component.

Cost and machining. Removal of the oil seals may also increase the response of critical speeds to synchronous and nonsynchronous forces.

Kocur [7]

Labyrinth Seals (see 3.4.2)

Clearance Changes

There is no clear trend on how clearance affects the stability.

Leakage is directly proportional to clearance.

Childs [8] and Kirk [9]

Number of teeth No clear trends. All dynamic coefficients increase with greater number of teeth (and thus length).

Leakage is inversely proportional to number of teeth.

Iwatsubo [10] andScharrer [11]

Increase Tooth Height

Reduces all dynamic coefficients. Increased height will reduce tooth strength and rub durability.

Wyssmann [12]

Decrease swirl Reduces tangential force (can change sign for smaller values or against rotation pre-swirl). Normally achieved with swirl brakes or shunt injection.

Increased machining complexity and space requirements. May influence performance if labyrinth seal is shortened or if number of teeth is reduced.

Wagner [13], Memmott [6]

Journal Bearings (see 3.3)

Fixed to Tilting Pad

Tilting pad bearings have been shown to have little or no destabilizing effect. Removal of fixed pad configurations eliminates a large source of cross-coupling and potential bearing whirl.

Fixed geometry bearings may have more damping for control of synchronous vibrations. Care needs to be taken to avoid creating an unbalance response problem.

Kludt [14],Jackson [15],Herbage [16].

Configuration or clearance modifications.

There are considerable differences in the stability characteristics of the various fixed pad bearings, more so than in varying tilting pad bearing characteristics.

As before, bearing changes will also affect the synchronous behavior of the rotor-bearing system.

Nicholas [17]

Kxy1

clearance3---------------------------

Kxy length3



Shrink Fits (see 3.5.2.1)

Reduce (remove material from the fit axial center or reduce fit length) or eliminate (integral wheels) fits.

Eliminate sources of internal friction that has been shown to cause rotordynamic instabilities.

Keyways may be needed to offset the loss in fit contact area. Machining cost and balance complexities are added.

Kimbal [18], Robertson [19]

Aero Forces (see 3.5.1)

None Minimal positive benefits. Due to the large impact on efficiency, stage changes (tip clearance, etc.) are avoided at all costs. Solutions usually involve increasing the effective damping.

Addition of swirl brakes to long axial passages

Reduces the cross-coupling produced by rotor/stator interaction of the swirling fluid

Cost and machining

Partial Arc Admission(see 3.8.1.4)

Reschedule the valve opening sequence.

Partial arc admission will alter the bearing loading. Inappropriate sequencing can unload bearings causing whirl or instability (especially in the case of fixed pad designs).

May require modifications to valve system.

Caruso [20]

Table 3-3—Increasing Effective Damping


Squeeze Film Dampers (see 3.3.4)

Incorporate into bearing design.

Squeeze film dampers are intentionally designed to be soft supported. This increases the effective damping of the rotor-bearing system.

Increased bearing complexity. Soft supports also affect the synchronous behavior of the system.

Gunter [21], Memmott [6], and Kuzdzal [22]

Damper Seals (see 3.4.3)

Honeycomb or hole pattern retrofit.

Provides high levels of damping to produce a stabilizing tangential force. Tangential force can be further increased using deswirl devices (see 3.4.3).

Reconfiguration of existing design. Rub intolerance and pocket fouling with honeycomb seals. Possible complexities due to large dynamic behavior.

Sorokes [23], Zeidan [24], and Moore [25]

Pocket damper seal retrofit.

Provides positive damping and low levels of cross-coupling to produce a stabilizing tangential force.

Reconfiguration of existing design. Leakage may increase.

Richards [26], Li [27], Butras [28]

Rotor Decrease bearing span.

Increases the bending stiffness of the shaft and, thus, the effective damping provided by the bearings.

Major redesign of the turbomachine. Manufacturers usually reduce bearing span for cost and dynamic considerations to practical limits.

Smith [3]

Increase shaft diameter between bearings.

Increases shaft bending stiffness. Aerodynamic efficiency normally suffers as shaft diameter is increased.

Smith [3]

Table 3-2—Decreasing or Eliminating Excitation Forces (Continued)




3.9.2 References

[1] Doyle, H. E., 1980, “Field Experience with Rotordynamic Instability in High-Performance Turbomachinery,”Rotordynamic Instability Problems in High-Performance Turbomachinery, NASA CP-2133, pp. 3–13.

[2] Fowlie, D.W. and Miles, D.D., 1975, “Vibration Problems with High Pressure Centrifugal Compressors,” ASME75-Pet-28.

[3] Smith, W. W., 1975, “An Operational History of Fractional Frequency Whirl,” Proceedings of the FourthTurbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp.115–125.

[4] Emerick, M. F., 1982, “Vibration and Destabilizing Effects of Floating Ring Seals in Compressors,”Rotordynamic Instability Problems in High-Performance Turbomachinery, NASA CP-2250, pp. 187–204.

[5] Allaire, P. E. and Kocur, J. A., 1985, “Oil Seal Effects and Subsynchronous Vibrations in High-SpeedCompressors,” Rotordynamic Instability Problems in High-Performance Turbomachinery, NASA CP-2409, pp.205–223.


[7] Kocur, J. A., Platt, J. P. and Shabi, L. G., 1987, “A Retrofit of Gas Lubricated Face Seals in a CentrifugalCompressor,” Proceedings of the Sixteenth Turbomachinery Symposium, Turbomachinery Laboratory, TexasA&M University, College Station, Texas, pp. 65–74.

[8] Childs, D. W. and Scharrer, J. K., 1987, “Theory Versus Experiment for the Rotordynamic Coefficients ofLabyrinth Gas Seals: Part II—A Comparison to Experiment,” Rotating Machinery Dynamics, presented at the1987 ASME Design Technology Conferences—Eleventh Biennial Conference on Mechanical Vibration andNoise, Boston, Massachusetts, September, 2, pp. 427–434.

[9] Kirk, R. G., 1986, “Labyrinth Seal Analysis for Centrifugal Compressor Design Theory and Practice,”Proceedings of the International Conference on Rotordynamics, Tokyo, September 14–17.

Changing components from interference fits to integral.

Increases shaft bending stiffness. Assembly may become complicated. Increased cost of shaft forging.

Tilting Pad Geometry

Bearing optimization studies:

• L/D (+)

• Clearance (+)

• Preload (–)

• # of pads

• Load direction

• Offset (–)

Intent is to optimize stability through geometry changes in the tilting pad bearing. The parameter changes that lead to higher stability levels are:

(+) Increasing

(–) Decreasing

NOTE: These are general trends only. They are not absolute!

Limited ability to change stability levels through pad changes. Space, design practice and synchronous response considerations may limit possibilities.

Nicholas [29]

Table 3-3—Increasing Effective Damping




[10] Iwatsubo, T., Fukumoto, K. and Mochida, H., 1993, “An Experimental Study of Dynamic Characteristics ofLabyrinth Seal,” Rotordynamic Instability Problems in High-Performance Turbomachinery, NASA CP-3239,pp. 219–237.

[11] Scharrer, J. K., 1988, “Rotordynamic Coefficients for Stepped Labyrinth Gas Seals,” Rotordynamic InstabilityProblems in High-Performance Turbomachinery, NASA CP-3026, pp. 177–195.

[12] Wyssmann, H. R., Pham, T. C. and Jenny, R. J., 1984, “Prediction of Stiffness and Damping Coefficients forCentrifugal Compressor Labyrinth Seals,” Journal of Engineering for Gas Turbines and Power, Transactions ofthe ASME, 106, pp. 920–926.

[13] Wagner, N. G., 2001, “Reliable Rotor dynamic Design of High-Pressure Compressors Based on Test RigData,” ASME Journal of Engineering for Gas Turbines and Power, Indianapolis, 123, pp. 849–856.

[14] Kludt, F. H., and Salamone, D. J., 1983, “Rotor Dynamic Modification of an Eight Stage Compressor forSafety/Reliability Improvement,” Proceedings of the Twelfth Turbomachinery Symposium, TurbomachineryLaboratory, Texas A&M University, College Station, Texas, pp. 81–96.

[15] Jackson, C., 1985, “Radial and Thrust Bearing Practices with Case Histories,” Proceedings of the FourteenthTurbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp.73–85.

[16] Herbage, B. S., 1993, “Bearing Upgrade Reduces Vibration and Improves Unit Load,” Power Engineering, 97(7), pp 41–43.


[18] Kimball, A. L., 1924, “Internal Friction Theory of Shaft Whirling,” General Electric Review, Vol. 27, No. 4, April,pp. 244–251.

[19] Robertson, D., 1935, “Hysteretic Influences on the Whirling of Rotors,” IMechE Proceedings, Vol. 131, pp.513-537.

[20] Caruso, W. J., Gans, B. E. and Catlow, W. G., 1982, “Application of Recent Rotor Dynamics Developments toMechanical Drive Turbines,” Proceedings of the Eleventh Turbomachinery Symposium, TurbomachineryLaboratory, Texas A&M University, College Station, Texas, pp. 1–17.

[21] Gunter, E. J., Allaire, P. E. and Barrett, L. E., 1975, “Design and Application of Squeeze Film Dampers forTurbomachinery Stabilization,” Proceedings of the Fourth Turbomachinery Symposium, TurbomachineryLaboratory, Texas A&M University, College Station, Texas, pp. 127–142.

[22] Kuzdzal, M. J. and Hustak, J. F., 1996, “Squeeze Film Damper Bearing Experimental Versus AnalyticalResults for Various Damper Configurations,” Proceedings of the Twenty-Fifth Turbomachinery Symposium,Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 57–70.

[23] Sorokes, J. M., Kuzdzal, M. J., Sandberg, M. R. and Colby, G. M., 1994, “Recent Experiences in Full LoadPressure Shop Testing of a High Pressure Gas Injection Centrifugal Compressor,” Proceedings of the Twenty-Third Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station,Texas, pp. 3–18.

[24] Zeidan, F., Perez, R. and Stephenson, E. M., 1993, “The Use of Honeycomb Seals in Stabilizing TwoCentrifugal Compressors,” Proceedings of the Twenty Second Turbomachinery Symposium, TurbomachineryLaboratory, Texas A&M University, College Station, Texas pp. 3–15.



[25] Moore, J. J., Walker, S. T., and Kuzdzal, M. J., 2002, “Rotordynamic Stability Measurement During Full-LoadFull-Pressure Testing of a 6000 PSI Re-injection Centrifugal Compressor,” Proceedings of the Thirty FirstTurbomachinery Symposium, Turbomachinery Laboratory, Department of Mechanical Engineering, TexasA&M University, College Station, Texas, September.

[26] Richards, R. L., Vance, J. M., Paquette, D. J., and Zeidan, F. Y., 1995, “Using a Damper Seal to EliminateSubsynchronous Vibrations in Three Back-to-Back Compressors,” Proceedings of the Twenty-FourthTurbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp.59–71.

[27] Li, J., Kushner, F. and DeChoudhury, P., 2000, “Gas Damper Seal Test Results, Theoretical Correlation andApplications in Design of High-Pressure Compressors,” Proceedings of the Twenty-Ninth TurbomachinerySymposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 55–64.

[28] Ertas, B. H., Delgado, A., and Vannini, G., 2011, “Rotordynamic Force Coefficients for Three Types of AnnularGas Seals with Inlet Preswirl and High Differential Pressure Ratio,” Proceedings of ASME Turbo Expo 2011:Power for Land, Sea and Air, GT2011.

[29] Nicholas, J. C. and Kirk, R. G., 1979, “Selection and Design of Tilting Pad and Fixed Lobe Journal Bearings forOptimum Turborotor Dynamics,” Proceedings of the Eighth Turbomachinery Symposium, TurbomachineryLaboratory, Texas A&M University, College Station, Texas, pp. 43–58.

Nomenclature

C is the labyrinth seal direct damping, N-s/m (lbf-s/in.);

A is the precession orbit, m (in.);

Fr is the radial force component, N (lbf);

Ft is the tangential force component, N (lbf);

k is the cross-coupled stiffness coefficient, N/m (lbf/in.);

K is the direct stiffness coefficient, N/m (lbf/in.);

M is the direct mass coefficient, N/m (lbf/in.);

is the whirl frequency, rad/s;

is the shaft speed, rad/s (rpm).

3.10 Identifying Fluid Induced Instabilities

3.10.1 General

The mechanisms causing fluid induced instability are not limited to oil lubricated bearings. It can occur when any fluid(e.g. oil, steam, process gas, etc.) is enclosed within a small clearance area (typically on the order of m or mils),between two body surfaces forming a cylinder within a cylinder, one of which is rotating and dragging the enclosedfluid into circumferential rotation. As discussed extensively in other sections of this tutorial, these fluid motions cangenerate significant levels of destabilizing cross-coupled stiffness. When this fluid generated cross-coupling issufficient to overcome the beneficial damping in the rotor system, the resulting unstable vibrations are designated asa fluid induced instability.



If a fluid induced instability is suspected to be present on a rotating machine, the data (vibration and process) must beproperly reviewed to confirm its presence. The characteristics of the vibration that must be determined, as well as thetypes of data that can be used, are the following.

— Frequency of the vibration:

— half or full FFT spectrums.

— Direction of the vibration’s precession (forward or backward whirling with respect to rotation):

— full FFT spectrums;

— orbits (filtered at the frequency of concern may be necessary);

— time based waveform (filtered at the frequency of concern may be necessary).

— The vibration’s dependence on speed and machine operating condition:

— process data (inlet and exit pressures and temperatures, flow rates, fluid properties, etc.);

— cascade FFT plot versus speed or time.

— The shaft’s static operating position when the vibration occurs:

— shaft centerline trends;

— DC proximity probe gaps.

Vibration frequency alone is insufficient confirmation of rotor instability. Fluid induced instabilities will have asubsynchronous vibration frequency typically in the range of 0.3X to 0.5X (e.g. 30 % to 50 % of synchronous speed).However, there are a number of other malfunctions that can occur with a subsynchronous vibration in the samefrequency range. For example, certain types of rubs can produce subsynchronous vibrations at 0.333X or 0.50X.Frequency alone cannot quantify the root cause of the vibration. Consider the following example of a half spectrumplot as shown in Figure 3-70. The half spectrum clearly indicates a subsynchronous vibration at 0.5X.

Direction of precession is an important vibration characteristic to determine because fluid induced instabilities areprimarily associated with forward whirling rotor modes. If the orbit precession is reverse, the source of thesubsynchronous vibration is likely not due to a fluid induced instability. However, recent research on axialcompressors shows that fluid forces can excite backward modes (Ehrich et al. [1]). Furthermore, backward modeinstabilities can be experienced by machines supported on active magnetic bearings, where stability is a concern forall modes, forward and backward, within the controller’s bandwidth.

The terms “forward” and “reverse” precession refer to the direction of the shaft centerline motion (orbits) that areclockwise (CW) or counterclockwise (CCW) relative to the direction of shaft rotation. If the orbit precession is in thesame direction as the shaft rotation, it is referred to as a “forward” precession orbit. Conversely, if the orbit precessionis opposite to the direction of the shaft rotation, it is referred to as “reverse” or “backward” precession. Precessionaldirection is an important diagnostic consideration in determining which malfunction may be present on a machine.

There are several ways to determine the direction of shaft precession. The two most common techniques are thetime-base presentation of the vibration waveforms or the use of a full spectrum plot.

Figure 3-71 and Figure 3-72 are typical time base presentations that can be used to differentiate forward or reverseprecession. Figure 3-71 indicates a forward precession orbit. The individual time-base waveforms are evaluatedbased on their peak amplitude occurrence relative to real time. For the example shown in Figure 3-71, the shaftrotation is CCW. Based upon the probe orientation, the horizontal probe will see the peak vibration first (t0) followed



by the vertical probe at time t0+1. Figure 3-72 indicates reverse precession vibration. The vertical probe detects thepeak vibration at t0 followed by the horizontal transducer at time t0+1. For clarity, the time based presentations shouldbe filtered at the subsynchronous frequency.

A diagnostic technique referred to as a “full spectrum” can also be utilized to determine the precession of an orbit. Anexample of reverse precession shaft centerline motion is shown in Figure 3-73. The full spectrum plot presents thevibration components in the frequency domain. It identifies the forward and reverse components on the same plot. Ifthe magnitude (amplitude) of a reverse frequency component is larger than the amplitude of the forward componentat the same frequency, the shaft centerline motion at that frequency is a reverse precession. If the reverse andforward spectrum components at the same frequency have the same amplitude, it means that the orbit is a straightline. Any deviation from this condition means the orbit is elliptic with forward/backward precession direction(depending on the magnitude of the prevailing component).

Knowledge of the rotor’s static operating position is another important piece of diagnostic information for identifyingfluid induced instabilities. Changes in the rotor’s static operating position with speed, time or operating condition cangreatly alter the dynamic characteristics of the journal bearings, which are some of the most important componentsgoverning a machine’s stability.

Figure 3-74 presents a typical shaft centerline plot showing the operating position as a function of time. The circle inFigure 3-74 represents the available diametral clearance of the bearing. The plotted points and curve define the shaftcenterline position within the bearing as a function of time. This could also be plotted as a function of rotational speedduring a startup or shutdown. For this example, the shaft rotation is counterclockwise (CCW). The shaft can be seento rise in the bearing to its operating position near the bearing’s geometric center under steady state conditions; i.e. avery low eccentricity ratio. Operation at such low eccentricity is particularly a concern for fixed geometry bearings (see3.3.2).

The presence or absence of a fluid induced instability is often highly dependent on the operating conditions of themachine. Such dependence on operating condition means that process data are an important element for diagnosingfluid induced instability. Process variables are typically stored on a different data collection system than vibration dataand can be overlooked. Correlations between vibration characteristics (frequency, amplitude, centerline position, etc.)

Figure 3-70—Half Spectrum Plot

Half Spectrum Plot10

5

Am

ptu

de (M

s P

P)

0.5X 1X



and operating conditions can help distinguish fluid instabilities from other vibration phenomena. Using process andoperating data in conjunction with cascade FFT plots can help determine such correlations.

Figure 3-6 shows an example of a cascade plot in which oil whirl and shaft whip are occurring. Oil whirl and shaft whipare classical examples of fluid-induced instability that are self-excited rotor vibrations caused by interaction betweenthe fluid and the rotor. Fluid-induced instabilities are conventionally separated into two regimes called “whirl” and“whip.” Whirl has a frequency proportional to shaft rotational speed, approximately one-half (see 3.3.2.3).

With an increase in rotational speed, whirl transforms into whip as the instability frequency approaches the firstnatural frequency of the rotor system (at higher rotational speeds, fluid whirl and whip of the higher modes may alsooccur). The vibration frequency of whip will remain essentially at the 1st natural frequency even with an increase inrotor speed.

Whirl instability has been seen in machinery that employ fixed geometry journal bearings or oil-film ring seals. Lookingat all classes of machinery, the predominant fluid induced instability that generates high amplitudes is shaft whip.

In some cases, structural or torsional resonances (in gears) can cause vibration components at fixed frequencies.Depending on the particular relation between speed and the vibration frequency, these can appear in the samefrequency range as fluid induced instability if examining a single FFT. In both situations, differentiating between whipor whirl or eliminating other sources of the subsynchronous vibration, a cascade plot of the vibration can prove to beuseful.

Diagnosis of rotating machinery vibrations is a complex subject and needs a good understanding of relevantmeasurement techniques and data post-processing algorithms. Bently and Hatch [2] and Eisenmann and Eisenmann[3] present a general overview of both these topics together with practical examples.

Figure 3-71—Forward Precession Vibration

Vert

Vert

Timebase Presentation—Forward Precessiont0 1

t0

Am

ptu

de

CCW Rotation

Horiz

Time



Figure 3-72—Reverse Precession Vibration

Figure 3-73—Full Spectrum—Reverse Precession of 0.5X

Vert

Vert

Timebase Presentation—Reverse Precession

t0+1t0A

mp

tude

Am

ptu

de

CCW Rotation

Horiz

Time

Full Spectrum Plot

10

5Am

ptu

de, M

s P

P

–1X –0.5X 0

Rev. Vib. Components Fwd. Vib. Components

0.5X 1X



3.10.2 References

[1] Ehrich, F.F. et al., 2001, “Unsteady Flow and Whirl Inducing Forces in Axial-Flow Compressors: Part II—Analysis,” Journal of Engineering for Gas Turbines and Power, ASME, Vol. 123, July, pp. 446–452.

[2] Bently, D. with Hatch, C.T., 2002, Fundamentals of Rotating Machinery Diagnostics, Bently PressurizedCompany Press.

[3] Eisenmann, Sr., R.C., and Eisenmann, Jr., R.C., 1997, Machinery Malfunction Diagnosis and Correction:Vibration Analysis and Troubleshooting for the Process Industries, Prentice-Hall, Inc.

3.11 Stability Testing of Machinery

3.11.1 General

Traditionally, the only method of assessing a machine’s actual stability has been through shop testing at full pressure,load and speed. For compressors, this usually means an expensive ASME PTC 10 Type I compressor test at full gasdensity conditions (in relation to Type II testing). Such testing can only identify the presence or absence ofnonsynchronous vibrations without verifying the machine’s actual log dec and stability margin, leaving unansweredquestions about the modeling accuracy and risk level of a particular machine.

Figure 3-74—Shaft Centerline Plot

Up

15

10

5

0

1 mil/div –5 0 5

07:25:34

22:37:2722:38:08

22:40:4122:43:07

22:46:1322:50:3722:52:54

22:57:57

22:57:57

23:03:1123:08:43

23:12:12



By quantitatively measuring a machine’s stability level (i.e. to measure a machine’s log decrement), stability testing ismeant to help resolve these unanswered questions and avoid problematic vibrations before a machine reaches thefield. Initially conducted as part of some research and development efforts [1,2], stability testing is becoming moreprevalent for assessing the instability risks of machines in extreme services [3–6], as well as for relatively simpleapplications as part of shop acceptance testing (Pettinato et al. [7]). Such testing is ideally performed in conjunctionwith PTC 10 Type I test conditions. However, as a cost effective alternative, it can also be conducted duringmechanical run testing, other performance tests, or within high-speed balancing facilities (Atkins and Perez [8]).These testing alternatives possess varying degrees of model validation because of the presence and absence ofdifferent components and effects. For example, stability testing during a mechanical run test does not include anysignificant annular seal dynamics, but does help to validate the combined rotor-bearing-pedestal system model.

Stability testing’s objective is directly analogous to unbalance response verification testing (see 2.9). Specifically,these objectives are to measure and quantify a machine’s log dec in order to assess the accuracy of its rotordynamicmodel’s predictions. Confidence in the accuracy of the rotordynamic model is vital because this model must be reliedupon for predicting the machine’s stability for operating conditions that can often not be tested.

Fundamentally, rotor stability measurements fall within the field of operational modal testing or, more specifically,experimental modal parameter identification (Cloud [9]). Here, the modal parameter that is trying to be identified isusually the log decrement of the first forward whirling mode. The modal testing and controls communities usuallyexpress a mode’s stability level in terms of its modal damping ratio, . Log decrement and modal damping ratio aredirectly related by the following equations:

(3-8)

(3-9)

The modal damping of interest for stability is that available within the operating-speed range, which can besignificantly different (and typically lower) than the modal damping present when running at lower speeds or whenpassing through a critical speed. The damping of the first critical speed, as measured by amplification factor on aBode plot during a machine’s startup or shutdown, should not be used as a measurement of the log dec in theoperating range.

Like many modal testing efforts, it is necessary to excite the modes of interest using some external forcingmechanism. For a typical machine, this means injecting a sufficient amount of nonsynchronous energy to excite thefirst forward whirling natural frequency while the rotor is spinning at some higher speed, like its maximum continuousspeed. Enough energy must be applied to register meaningful results, but a great deal of caution must be exercised inorder to avoid applying too strong of an excitation force that might result in damage to the machine. Since eachmachine is different, the required magnitude of the excitation force will be different for each case.

Some of the devices and mechanisms that can produce the type of nonsynchronous excitation required aremechanical impact [1,2], seismic shakers [8,10], and some type of aerodynamic or process flow variations [11,12].Mechanical impact to an operating rotor is not recommended and is usually not practical due to space and safetylimitations. Seismic shaker energy must be applied to a stationary object like a bearing housing and a fairly large forceis sometimes required to impart sufficient energy to the rotor. An active magnetic actuator, temporarily mounted onthe machine to directly excite the shaft, is currently the most popular choice [3–7]. It is important that the addition ofany device chosen minimizes the alteration of the machine’s rotordynamics, in particular the first forward mode’sstability level and natural frequency. Any components added to the rotor assembly for testing, such as a magnetic

2

1 2

–

------------------ 2 1«=

2

42

+

--------------------------

2------

1«=



actuator’s lamination sleeve, must be included in the modeling correlations. A minimal amount of mass should beadded to the rotor system to minimize impact on the system dynamic behavior.

Active magnetic actuators are popular and effective because of their flexibility in controlling the excitation’smagnitude, frequency content and directional capabilities. Directional capability implies the actuator’s ability to applyforces with different orbital nature from a straight line to circular in either the direction of rotation (forward whirling) oropposite to it (backward whirling) [9]. These capabilities allow the excitation to be selected that will maximize themode of interest’s signal to noise ratio, an important objective for any modal testing effort.

The type of excitation signal that is applied depends on the capabilities of the device as well as what modal parameterestimation technique is chosen to extract the log decrement from the measurement data. As in the modal testing andcontrols communities, stepped sine or swept sine excitations to measure frequency response functions (FRFs) arethe most popular choices. Blocking excitation has also been shown to be effective when modal parameter estimationtechniques in the time domain are employed [7,10].

Some example frequency and time domain data from stability tests are shown in Figure 3-75 and Figure 3-76. Sixdifferent FRFs are presented in Figure 3-75, one for each probe. These are compliance FRFs whose magnitude is interms of displacement per unit force. This highlights the fact that, for stability measurements based on FRF data, theexcitation force should be fully characterized1, i.e. the injected force’s magnitude versus frequency characteristicsshould be measured [13].

The coupling end probes in Figure 3-75 clearly show peak response in the frequency region where the first forwardwhirling mode has been excited. Near 132 Hz, the FRF data are corrupted by the synchronous vibrations and shouldnot be used in the modal parameter estimation process.

Time transient data from a stability test incorporating blocking excitation are shown in Figure 3-76. The blockingexcitation has been tuned to excite the rotor’s first forward whirling natural frequency that is subsynchronous. Oncethe blocking excitation is turned off, the vibration decays slowly back to steady state levels being dominated bysynchronous vibrations. This transient decay region is the primary data of interest for the log decrement estimationprocess.

As mentioned earlier, the type of test data collected is usually selected based on the modal parameter estimationtechnique that is chosen to extract the log decrement measurement. For frequency domain FRF data, amplificationfactor (Q) where:

(3-10)

or phase slope where:

(3-11)

1 Force calibration can be difficult under real testing conditions.

Qr

HP---------------- 1

2-----

---

1«=

d d

1

rdd-------

wr

-------------------------- 180

--------



Figure 3-75—Measured vs. Identified Frequency Response Functions during a Stability Test [7]

400

0.01

0.02

Shaker probes

Thrust end Brg probes

Cplg end Brg probes

Frequency (Hz)

Mag

ntu

de (m

s/b f)

Mag

ntu

de (m

s/b f)

Mag

ntu

de (m

s/b f)

0

0.02

0.04

Xp MeasuredYp MeasuredXp ModelYp Model

0

0.01

0.02

60 80 100 120 140 160

40 60 80 100 120 140 160

40 60 80 100 120 140 160



and, for time domain data, mechanical log decrement ( ) where:

(3-12)

have historically been the popular modal parameter estimation techniques being used for stability tests. See Figure 3-77 for an example. Like all single degree of freedom (SDOF) techniques, their popularity stems from their simplicitywith little computational effort. The SDOF designation refers to their fundamental assumption that the responsearound a resonance is dominated by only one mode of vibration, where other modes’ contribution is assumed to beconstant around the resonance, but not necessarily negligible (Ewins [13]).

Research has revealed that the accuracy of SDOF modal parameter estimation techniques can be unreliable forsome rotor systems. This is primarily due to the interaction of the first forward mode with its sister first backwardmode, which both can have similar shapes and frequencies (Cloud et al. [14]).

The problem of close modal proximity can be easily resolved using multiple degree of freedom (MDOF) techniquesthat are already available from the modal analysis, controls and speech processing communities. These MDOFestimation techniques allow multiple modes to be identified at the same time, meaning the first forward mode’sstability can be more accurately distinguished and identified. Even though they are more computationally intensivethan their SDOF counterparts, such MDOF techniques have been successfully applied for rotor stability testing[1,7,14]. Since there are numerous MDOF estimation techniques available, one must be careful to ensure that theestimation technique chosen handles rotor systems’ unique modal characteristics and testing requirements [1,9].

Figure 3-76—Time Transient Waveform from Blocking Excitation during a Stability Test [7]

Time

Blocking turned OFF

Dsp

acem

ent

m

m1

ncycle---------------ln

y 0 y ncycle -----------------------

=



Careful consideration must be given to all aspects of a machine’s environment and operating conditions whenperforming a stability test. As described in the previous sections, the mechanical tolerance variations and condition ofthe machine parts, especially journal bearings, squeeze film dampers and annular seals, are very important indetermining the machine’s stability level. Synchronous vibration excitations like excessive unbalance andmisalignment may affect the test results substantially. During testing, oil temperatures and all process variables mustbe carefully controlled and the machine should be thermally stable.

As with any shop testing, it must be recognized that results from a test stand may differ significantly from the installedfield results due to differences in foundation and piping configurations as well as process conditions. As long as suchdifferences are recognized and the test stand environment and operating conditions are accurately considered,stability testing can be effective for minimizing the risk of field instability problems.

3.11.2 References

[1] Nordmann, R., 1984, “Identification of Modal Parameters of an Elastic Rotor With Oil Film Bearings,” ASMEJournal of Vibration, Acoustics, Stress, and Reliability in Design, 106, pp. 107–112.

[2] Wohlrab, R., 1975, Experimentelle Ermittlung spaltstromungsbedingter Krafte an Turbinenstufen und derenEinflub auf die Laufstabilitat einfacher Rotoren, Ph. D. Dissertation, Technical University of Munich, Germany.

[3] Baumann, U., 1999, “Rotordynamic Stability Tests on High-Pressure Radial Compressors,” Proceedings ofthe Twenty-Eighth Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, CollegeStation, Texas, pp. 115–122.

[4] Moore, J. J., Walker, S. T. and Kuzdzal, M. J., 2002, “Rotordynamic Stability Measurement During Full-Load,Full-Pressure Testing of a 6000 psi Re-Injection Centrifugal Compressor,” Proceedings of the Thirty-FirstTurbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp.29–38.

Figure 3-77—Estimating Stability from Time Transient Data Using Mechanical Log Decrement [12]

Logarithmic Decrement

Probe Data:

Location: Suction, Vert

Orientation: 135°

Sensitivity: 200 mV/mil

Runout < 0.25 mils

Shaft Speed 5614 RPM

= ln (A1/A2) = ln (3.2/2.25) = 0.35

Time Scale = 62.5 mSec/Division

1.0 Mil

Resonate Excitation of Axial AirCompressor Induced by Surge



[5] Bidaut, Y., Baumann, U., and Al-Harthy, S. M. H., 2009, “Rotordynamic Stability of a 9500 psi ReinjectionCentrifugal Compressor Equipped with a Hole Pattern Seal—Measurement versus Prediction Taking intoAccount the Operational Boundary Conditions,” Proceedings of the Thirty-Eighth TurbomachinerySymposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 251–259.

[6] Sorokes, J. M., Soulas, T. A., Koch, J. M., and Gilarranz, J. L., 2009, “Full-Scale Aerodynamic andRotordynamic Testing for Large Centrifugal Compressors,” Proceedings of the Thirty-Eighth TurbomachinerySymposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 71–79.

[7] Pettinato, B. C., Cloud, C. H., and Campos, R. S., 2010, “Shop Acceptance Testing of CompressorRotordynamic Stability and Theoretical Correlation”, Proceedings of the Thirty-Ninth TurbomachinerySymposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 31–42.

[8] Atkins, K. E. and Perez, R. X., 1992, “Assessing Rotor Stability Using Practical Test Procedures,” Proceedingsof the Twenty-First Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, CollegeStation, Texas, pp. 151–159.

[9] Cloud, C. H., 2007, Stability of Rotors Supported by Tilting Pad Journal Bearings, Ph.D. Dissertation,University of Virginia.

[10] Kanki, H., Fujii, H., Hizume, A., Ichimura, T. and Yamamoto, T., 1986, “Solving Non-Synchronous VibrationProblems of Large Rotating Machineries By Exciting Test in Actual Operating Condition,” Proceedings of theIFToMM International Conference on Rotordynamics, pp. 221–225.

[11] Jackson, C., 1981, “Four Compressor Trains of a Large Ethylene Plant—Design Audit, Testing andCommissioning,” Proceedings of the Tenth Turbomachinery Symposium, Turbomachinery Laboratory, TexasA&M University, College Station, Texas, pp. 4–9.

[12] Jackson, C. and Leader, M. E., 1983, “Design, Testing and Commissioning of a Synchronous Motor-Gear-Axial Compressor,” Proceedings of the Twelfth Turbomachinery Symposium, Turbomachinery Laboratory,Texas A&M University, College Station, Texas, pp. 97–111.

[13] Ewins, D. J., 2000, Modal Testing: Theory, Practice and Application, Research Studies Press Ltd., 2nd Edition.

[14] Cloud, C. H., Maslen, E. H., and Barrett, L. E., 2009, “Damping Ratio Estimation Techniques for RotordynamicStability Measurements,” ASME Journal of Engineering for Gas Turbines and Power, Vol. 131, No. 1, pp.012504.



Nomenclature

y[0] is the amplitude before decay begins;

y[ncycle] is the amplitude of the nth cycle;

ln is the natural logarithm;

ncycle is the number of decay cycles;

is the mechanical logarithmic decrement;

Q is the amplification factor;

ωr is the peak response frequency;

is the half power point bandwidth;

is the phase angle (degrees);

is the modal damping ratio.

3.12 Standard Paragraph Sections for Stability Analysis SP6.8.5 – SP6.8.6

The philosophy and detailed explanation of the stability paragraphs are presented in this appendix. Application of theparagraphs to two types of centrifugal compressors is included as an example following these sections.

3.12.1 Philosophy

The task of creating specifications for stability was undertaken to reflect the general philosophy behind the standardrotordynamic paragraphs. In a manner similar to the unbalance response calculations, a procedure was formulatedthat explicitly outlines the method, tools and equations, while at the same time, serving as a simplified screening tool.The screening tool employs a simplified but conservative approach to efficiently serve as a first stage filter andprovide the structure and commonality to permit database construction. To reflect the state-of-the-art that exists, alongwith the experience that has developed at the different vendors, a more involved analysis is employed to predict theactual operating conditions of the rotor. This analysis is expected to more closely follow the vendor’s development oftools through innovation and experience. From this desire, a two-tiered procedure was created. This is representedby the Level I and Level II analyses.

3.12.1.1 Level I

The Level I analysis was created with two specific purposes in mind. First, a common methodology was defined.Second, a simplified but conservative screening criterion was created to identify individual rotors and classes ofapplications that require a more involved analysis.

With the lack of an existing standard approach to rotor stability, a uniform methodology is needed. As described in3.1, uncertainty remains in several key areas of rotor/bearing stability due to the complexity of the problem. Over theyears, experience and development efforts have created predictor tools that vary from vendor to vendor to addressthese unknowns. The tools are closely guarded and reflect the different design philosophies, construction techniques,and application experiences of each vendor. These differences also influence the stability analysis by varying the typeof excitations to include, which tools to employ and what stability level is acceptable. In response, the Level I analysiswas developed.

m

HP



The Level I analysis was intended to create a common basis for comparison. Similar to the unbalance responseanalysis, a procedure was developed that specifies what is to be included, how to include it and how to measure theresults. Thus, purchasers could compare new equipment proposals from different vendors on a common basis.Additionally, this methodology could be used to generate a database for existing equipment of both the purchaser andvendor with minimized effort.

The second purpose that the Level I analysis serves is that of a screening tool. The tool, as defined, is conservativeand straightforward. The intent is not to accurately represent the stability condition of the rotor/bearing system but toprovide a step that can efficiently and effectively identify rotors or groups of applications that require a more detailedanalysis. This is an attempt to balance the desire (by the purchaser) to have a stability analysis performed for everyturbomachine and the contention (by the vendor) that some machines never need a stability analysis let alone adetailed study. As an example, high-pressure gas re-injection compressors will always undergo a stability analysis bythe thorough vendor whether required or not. That same vendor might, as common practice, not perform a stabilityanalysis for rotors that operate below their first natural frequency, except for machines with fixed geometry bearingsand oil-film ring seals. Thus, a screening tool was developed that is simply applied, straightforward and conservative.

An initial analysis is done on all rotors operating above the first mode and those operating below the first mode thathave fixed geometry bearings or oil film ring seals. This identifies those rotors that require a more in-depth study ofthe dynamic behavior. Some applications that are characterized by high discharge gas densities or operate wellabove the first critical speed have stricter criteria to meet if they are to avoid an additional (Level II) analysis. Inbetween the extremes, a method based on the experiences of several vendors and purchasers was formulated.While acknowledged as not completely representative of the dynamic behavior, it is nonetheless capable ofidentifying those rotors requiring a Level II analysis.

3.12.1.2 Level II

For those rotors identified in the Level I analysis as requiring a more in-depth investigation, the Level II analysis wascreated. The Level II analysis more accurately represents the dynamic behavior of the rotor at the maximumoperating conditions. As previously mentioned, some unknowns remain pertaining to the instability drivers withinturbomachines. The various vendors have addressed these unknowns in different manners. Rather than select an“optimum” approach, especially considering the lack of experimental data, it was left to the individual vendor todetermine the best method to analyze the rotor. This takes advantage of the vendor’s experience with both theanalytical tools and operating experience of their own equipment. However, there are requirements with respect to thegeometry for the modeling of toothed labyrinths and damper seals.

The Level II analysis describes the operating condition that should be used for the analysis, the sources of dynamiceffects that should be considered for inclusion in the analysis, and the final acceptable stability level. Properconsideration of these factors may require levels of effort not justifiable for all rotors. By specifying the mechanismsand not the tools, the Level II analysis is open-ended permitting future developments to be applied without changes tothe specification.

3.12.2 Stability Standard Paragraph Discussion

A detailed discussion of the individual API Standard Paragraphs (SP) relating to rotordynamic stability will bepresented. The following format is employed for this discussion: each of the paragraphs are individually reproduced insequence, followed by commentary designed to illustrate or clarify the material presented in the paragraph. Thestability paragraphs are displayed in boldface type with the numbering scheme intact. Comments immediately followin each paragraph in normal type.

SP6.8.5 Level 1 Stability Analysis



SP6.8.5.1 A stability analysis shall be performed on all centrifugal or axial compressors, turbines and/or radial flowrotors that meet the following:

a) Those rotors whose maximum continuous speed is greater than the first undamped critical speed on rigidsupports, FCSR, in accordance with SP6.8.2.3.

b) Those rotors with fixed geometry bearings or oil film ring seals.

The stability analysis shall be calculated at the API defined maximum continuous speed.

NOTE Level I analysis was developed to fulfill two purposes: first, it provides an initial screening to identify rotors that do notrequire a more detailed study. The approach as developed is conservative and not intended as an indication of an unstable rotor.Second, the Level I analysis specifies a standardized procedure applied to all manufacturers similar to that found in SP6.8.2.

To avoid confusion, the first critical speed is defined using the undamped critical speed analysis of SP6.8.2.3. This isconsistent with most empirical stability criteria that use the undamped critical speed on rigid supports as onemeasurement. Maximum continuous speed was selected for the Level I analysis since this should represent the leaststable speed in the operating-speed range. The reference to a specific speed facilitates the creation of a standardizedprocedure.

Historical data and experience have indicated that instability in subcritical machinery (operating below the firstundamped critical speed on rigid supports) is rare in centrifugal compressor, steam turbine and axial and/or radial flowrotors (excluding pumps). However, it has happened. De Santiago and Memmott [1] where instability happened withan overhung compressor with a fixed geometry bearing that was combined with an oil film ring seal.

It bears emphasizing again that the Level I analysis is a screening tool. It is fully expected that some rotors,regardless of their design, will not satisfy the screening criteria and require a Level II analysis. This should not beviewed as a deficiency in the design. Additionally, it is possible to have a more conservative design requiring a LevelII analysis than one needing only a Level I analysis for the same application. The intent of the Level I analysis is toidentify rotors that have the possibility of being more sensitive to destabilizing forces (for example, high Critical SpeedRatio [CSR]) or having greater destabilizing force levels (for example, high gas density).

SP6.8.5.2 The model used in the Level I analysis shall include the items listed in SP6.8.2.4.

To maintain consistency with the damped unbalance response, the analysis shall include the same items required forthat analysis.

SP6.8.5.3 When tilt pad journal bearings are used, the analysis shall be performed with synchronous tilt padcoefficients.

In API Std 617 8th Edition in Annex C of Part 1, it is allowed to use synchronous or frequency dependent coefficientsin the Level II stability analysis.

The committee recognizes that there is a question as to which bearing coefficients to use for tilt pad bearings for astability analysis: synchronous or frequency dependent (see 3.3.3.1). The intention of the committee is to considermoving toward frequency dependent coefficients corresponding to the 9th Edition of API Std 617.

SP6.8.5.4 For rotors that have quantifiable external radial loading (e.g. integrally geared compressors), the stabilityanalysis shall also include the external loads associated with the operating conditions defined in SP6.8.5.5. For somerotors, the unloaded (or minimal load condition) may represent the worst stability case and shall be considered.

Recognizing that external loading effects the dynamic behavior of the journal bearings, and thus the stability, theywere included in the Level I analysis. In some instances, (e.g. the pinion shaft in a gearbox) the unloaded case may



represent the worst stability case (see 3.8.5). This should be discussed between the purchaser and vendor. For theLevel II analysis, the loads at the rated condition are extrapolated to the maximum continuous speed as necessary.

SP6.8.5.5 The anticipated cross-coupling, QA, present in the rotor is defined by the following procedures:

a) For centrifugal compressors:

The parameters in Equation 7 shall be determined based on the machine conditions at normal operating point unlessanother operating point is agreed upon.

(SP-7)

Equation 7 is calculated for each impeller of the rotor. QA is equal to the sum of qa for all impellers

b) For axial flow rotors

(SP-8)

Equation 8 is calculated for each stage of the rotor. QA is equal to the sum of qa for all stages.

The anticipated cross-coupling is derived from Alford’s force for steam turbines and axial flow rotors, Equation (8),and a modified form of the same equation for centrifugal compressors, Equation (7). Section 3.5 describes thedevelopment of each equation. These equations are calculated for each individual stage. In steam turbines and axialflow rotors, this is consistent with the development of Alford’s force. For centrifugal compressors, this was determinedto yield the most representative application of the equation across a wide range of configurations.

SP6.8.5.6 An analysis shall be performed with a varying amount of cross-coupling introduced at the rotor mid-spanfor between bearing rotors or at the center of gravity of the stage or impeller for single overhung rotors. For doubleoverhung rotors, the cross-coupling shall be placed at each stage or impeller concurrently and should reflect the ratioof the anticipated cross-coupling (qA, calculated for each impeller or stage).

This is part of the conservatism built into the Level I analysis. The location that most adversely affects the stability wasselected. This location should normally coincide with the peak modal displacement. For between bearing rotors withthe typical first bending mode, this corresponds to the mid-span. For overhung rotors, this corresponds to the shaftend with the turbine stage or impeller that is typically the largest overhung moment. The cross-coupling is applied tothe c.g. of the overhung component.

By varying the amount of cross-coupling applied, a sensitivity study is created. The study contains two importantpieces of information. First, at zero applied cross-coupling, the log decrement of the rotor/support system is identified.Second, the amount of cross-coupling needed to zero the log decrement, Q0, (produce an unstable rotor) iscalculated, SP6.8.5.7. The ratio between Q0 and the anticipated cross-coupling determines the rotor’s sensitivity andsafety margin.

While a single amount of cross-coupling is applied to between bearing and single overhung rotors, two values areused for double overhung rotors. The values used should reflect the ratio of anticipated cross-coupling. For example,a double overhung compressor rotor is calculated to have an anticipated cross-coupling of 10,000 lbf/in. for the firstimpeller and 15,000 lbf/in. for the second impeller. Therefore, in applying the two cross-couplings to the rotor for thisanalysis, the amounts will be varied according to the ratio of qA (i.e. if 1000 lbf/in. is applied to impeller #1, 1500 lbf/in.should be applied concurrently to the second impeller; for 20,000 lbf/in. applied to the first impeller, 30,000 lbf/in.should be applied to the second impeller, etc.).

qa

HP BcCDcHcNr

-----------------------d

s----- =

qa

HP BtCDtHtNr

----------------------=



SP6.8.5.7 The applied cross-coupling shall extend from zero to the minimum of the following.

a) A level equal to 10 times the anticipated cross-coupling, QA.

b) The amount of the applied cross-coupling required to produce a zero log decrement, Q0. This value can bereached by extrapolation or linear interpolation between two adjacent points on the curve.

This dictates that the sensitivity study either shows the point that the log decrement reaches zero or that it applies atleast ten times the anticipated cross-coupling. While the first objective is obvious, the second is used to indicate thatthe rotor is basically insensitive to cross-coupling for its given application. Stated another way, the analysis wouldstate that the rotor is still stable even after applying ten times the amount of cross-coupling that is expected to occurfor the application. In this case, the rotor is considered to be insensitive to the levels of anticipated cross-coupling.

A graphical presentation of the results of the sensitivity study is to be provided. The plot (see Annex 1.C Figure 1.C-2)shows the anticipated cross-coupling, QA, corresponding log decrement, A, and the cross-coupling needed todestabilize the rotor, Q0 (if less than ten times QA). These values are taken as the minimum A at the anticipatedcross-coupling, QA and the minimum Q0 derived from the minimum or maximum bearing clearance. (Note: A and Q0may be taken from different clearance conditions, i.e. one from minimum and the other from maximum bearingclearance.)

Several pieces of information are contained in the graph. Two of which are discussed here. First, the slope of thecurve is an indication of the sensitivity of the rotor to destabilizing forces. For example, the benefit of having A = 0.5 ifQ0 is only 10 % greater than QA is questionable. In this case, minor deviations in the destabilizing forces, whetherfrom changes in the operating conditions or uncertainties in the analysis, would be sufficient to cause the rotor tobecome unstable. Second, the stability of the rotor/support system is shown at the left-hand side of the plot with noexcitation present. In most situations, this will be the highest level of stability achieved. The situation where this maynot be true is with the incorporation of honeycomb, hole pattern or pocket balance pistons and seals. See 3.4.3 forfurther information.

SP6.8.5.8 Level I Screening Criteria:


If any of the following criteria apply, a Level II stability analysis shall be performed:

i. Q0 / QA < 2.0.

A Level II analysis is necessary if the cross-coupling needed to destabilize the compressor is less than two times theanticipated amount. It was felt that an adequate safety margin (expressed as the ratio Q0/QA) did not exist to coveruncertainties in the estimation equation and analytical predictions. For safety margins less than two, a reduction inthese uncertainties is provided by a more accurate prediction of the instability mechanisms in the Level II analysis.

ii. A < 0.1.

For similar reasons as with i), a log decrement of less than 0.1 requires a Level II analysis. Once again, an inadequateseparation margin from the unstable region (negative log dec) exists that requires a more accurate prediction of theinstabilities. Additionally, the destabilizing effects of projecting the operating condition to maximum continuous speedneeds to be examined.

iii. Q0 / QA < 10 and the point defined by CSR and the average density at the normal operating point is located inRegion B of Figure SP-9.

For ratios less than 10, a further examination of the rotor susceptibility and application sensitivity is required. The rotorsusceptibility to unstable operation is represented by the CSR (see Symbols section at the end of SP6.8.6 for CSR



definition). The average gas density measures the application sensitivity. In Region B, it was determined that thecompressor is susceptible to destabilizing forces (flexible rotor), or the excitation forces themselves will be large (highgas densities) (see 3.8.6), and thus for all but the insensitive turbomachines (those with a Q0/QA 10 and a A 0.1),a Level II analysis is required. In Region A, the low levels of susceptibility and excitation may not require a moreinvolved analysis (unless Q0/QA < 2 or A < 0.1). Figure SP-9 was derived from the experience plots described in 3.7.Gas density was selected as the independent variable since it was felt to more represent the excitation levels in thecompressor.

Otherwise, the stability is acceptable and no further analyses are required.

Essentially, this includes only compressors that have a A 0.1 and Q0/QA 10 in Region B or Q0/QA 2 in Region Aof Figure SP-9. This defines a rotor that is insensitive to the anticipated levels of destabilizing forces and has a logdecrement sufficient to overcome any analytical prediction uncertainties.

b) For axial flow rotors:

If A < 0.1, a Level II stability analysis shall be performed. Otherwise, the stability is acceptable and no further

analyses are required.

For axial flow rotors, Alford’s force lumped at the center span (or stage center of gravity for overhung rotors) issufficient to conservatively represent the destabilizing forces present. In this case, a log decrement at the anticipatedlevel of cross-coupling of less than 0.1 warrants a Level II analysis for the rotor.

SP 6.8.6 Level II Stability Analysis

SP6.8.6.1 A Level II analysis, which reflects the actual dynamic forces (both stabilizing and destabilizing) of therotor, shall be performed as required by 6.8.5.8.

Figure SP-9—Level I Screening Plot

0 20(1.25)

1.0

1.5

2.0

CS

R

Region A

Region B

3.0

2.5

3.5

40(2.5)

60(3.75)

80(5.0)

100(6.25)




The Level II analysis is a more in-depth and time-consuming analysis. For those rotors deemed susceptible to orexposed to large levels of cross-coupling, an analysis shall be performed that takes advantage of the vendor’sexperience and analytical developments.

SP6.8.6.2 The Level II analysis shall include the dynamic effects from all sources that contribute to the overallstability of the rotating assembly. These dynamic effects shall replace the anticipated cross-coupling, QA. Thefollowing sources shall be considered:

a) labyrinth seals;

b) damper seals;

c) impeller/blade flow aerodynamic effects;

d) internal friction.

The anticipated cross-coupling calculated by Equation (7) or (8) is replaced with a more detailed analysis of thepossible sources of instability. A potential list of sources is presented. (Damper bearings and oil seals are alreadyincluded in the analysis). The list is not intended to be all-inclusive. Other sources are to be discussed as necessaryby the vendor.

In contrast to the Level I analysis, the vendor shall determine the most appropriate procedure to use to predict thedynamic effects of the instability drivers in their rotor. This should be based on their experience, current research andapplicability. In many instances, this will differ between the various vendors reflecting the continued uncertainties inthe knowledge base. The purchaser and user should also be aware that research and knowledge are ongoing toaccurately predict or confirm the behavior of some of these effects. While their impact on the overall rotor stability canbe shown in individual cases or experiments, a predictor tool does not exist that can be applied across a wide rangeof applications. This includes aerodynamic cross-coupling produced by centrifugal impellers, shrink fits and shafthysteresis, to name a few. For these instances, reliance upon the combined and shared knowledge of the vendor andpurchaser is required.

SP6.8.6.2.1 The vendor shall state how the sources are handled in the analysis.

NOTE It is recognized that methods may not be available at present to accurately model the destabilizing effects from allsources listed above.

SP6.8.6.3 The Level II analysis shall be calculated at Nmc.

The Level II analysis is intended to represent the worst possible case for rotor stability. With this in mind, themaximum continuous operating speed (Nmc) is selected since stability normally degrades with increasing rotor speed.

SP6.8.6.4 The operating conditions defined for the normal operating point shall be extrapolated to Nmc.

NOTE Extrapolated conditions should not fall outside the operating limits (the defined operating map) of the equipment trainsuch as horsepower, discharge pressure, etc.

Operating conditions used in the Level II analysis shall be extrapolated to Nmc.

SP6.8.6.5 The modeling requirements of Level I shall also apply.

The bearing (including SFD) and seal dynamic coefficients shall be calculated with the extremes of clearance and oilinlet temperature.

SP6.8.6.6 The dynamic coefficients of the labyrinth seals shall be calculated at minimum seal running clearance.



In the range of typical labyrinth seal clearances, the minimum clearance normally represents the configuration withthe highest destabilizing forces. For labyrinth seals used in abradable applications, this may not be valid and theclearance effects on stability should be examined.

SP6.8.6.7 When calculating the dynamic coefficients of damper seals, the running clearance profile range, which isdetermined by drawing dimensions, manufacturing tolerances and deformations in the seal, seal support and rotor,shall be included.

Research and experience have shown that damper seals (honeycomb or hole pattern) to be very sensitive to theclearance profile within the seal with respect to their dynamic coefficients. A divergent damper seal (more clearanceon the low pressure side than on the high pressure side) can lead to a negative direct stiffness, which in turn lowersthe first natural frequency. A lower natural frequency reduces the effective damping, which in turn can lead toinstability at this lower frequency.

SP6.8.6.8 The frequency and log decrement of the first forward damped mode shall be calculated progressively forthe following configurations (except for double overhung machines where the first two forward modes shall beconsidered).

The first forward mode is the mode of concern for rotor stability. Double overhung rotors will have modes associatedwith each impeller that can be excited and possibly driven unstable. Thus, the first two modes must be considered inthe analysis.

a) Rotor and support system only (basic log decrement, b).

b) Each source from 6.8.6.2 utilized in the analysis.

c) For damper seals, the dependence due to parameters defined in 6.8.6.7.

d) Complete model including all sources (final log decrement, f).

The Level II analysis should identify the relative impact of adding each item to the system. By separating the effects,the critical component affecting the rotor stability is identified. This will permit effective troubleshooting of problemsand behavior impact assessment of damage, operation changes or geometry variations. Should instability problemsoccur, it might be more advantageous to address or modify the component that is causing the greatest destabilizingforce or the component that has the most potential for stabilizing the system (like changing a toothed labyrinthbalance piston/division wall seal to a hole pattern). This analysis should help identify that component or effect.

SP6.8.6.9 Acceptance Criteria—The Level II stability analysis shall indicate that the machine, as calculated inSP6.8.6.1 through SP6.8.6.8, shall have a final log decrement, f, greater than 0.1.

The minimum acceptable level of the stability in this worst-case scenario is a log decrement greater than 0.1. At thislevel of log decrement and given the conservative approach of the worst-case scenario, sufficient margin shouldremain to guarantee stability during operation.

SP6.8.6.10 If after all practical design efforts have been exhausted to achieve the requirements of SP6.8.6.9,acceptable levels of the log decrement, f, shall be agreed upon.

If the final log decrement >0.1 cannot be reached, the vendor and purchaser shall mutually agree to an acceptablelevel based on historical data, operating experience and analysis methods of the vendor.

NOTE It should be recognized that other analysis methods and continuously updated acceptance criteria have been usedsuccessfully since the mid-1970s to evaluate rotordynamic stability. The historical data accumulated by machinery manufacturersfor successfully operated machines may conflict with the acceptance criteria of this specification. If such a conflict exists and thevendors can demonstrate that their stability analysis methods and acceptance criteria predict a stable rotor, then the vendors’criteria should be the guiding principle in the determination of acceptability.



This note highlights the fact that there remains sufficient unknowns and experience to develop standard stabilityspecifications and methodology for API equipment. As noted earlier, that even in the absence of specifications,vendors have refined their own design rules to achieve stable rotors since the mid-1970s. This specification is acombination of their experience mixed with purchaser’s requirements and does not represent a single existingmethod. Conflicts may exist between this specification and the vendor’s experience as to what is deemed acceptable.If sufficient proof is presented to the purchaser’s satisfaction, the vendor's criteria and experience can be used as thedetermining factor to assess the rotor stability.

3.12.3 Analysis Examples

3.12.3.1 General

To illustrate the analysis procedure, two compressors were selected. One is typical of process compressors. It has sixstages, 4330 HP and a rated speed of 13,600 rpm (Nmc = 14,280 rpm). The other compressor represents a gasinjection rotor. This compressor has seven stages, 6300 HP, a rated speed of 13,751 rpm (Nmc = 14,439 rpm) and anaverage density of 107 kg/m3. Both compressors were designed to maximize rotor stability and meet API Std 617specifications.

3.12.3.2 Compressor Cross-Sections

A cross-section of the process compressor rotor is shown on Figure 3-78. Future design conditions necessitated theincreased impeller spacing. To avoid lateral critical speed and stability problems, the shaft diameter was increasedwherever practical. Similar efforts were made during the design process of the injection compressor, Figure 3-79.Stability concerns over the high discharge pressure (275 bar) and gas density led to design efforts to stiffen the shaft.The larger shaft is reflected in the ratio between the impeller OD and the shaft OD.

Before proceeding with the stability analysis requirements of the standard paragraphs, some relevant points need tobe addressed concerning the Level I analysis. As discussed, efforts were made during the design phase of bothcompressors to maximize the rotor stability. Both compressors are operating without evidence of instability. A full load,full pressure test was performed on the injection compressor also looking for signs of instability. The compressor wastested from surge to carry out (stonewall) at maximum continuous speed with no subsynchronous vibrations present.These points are emphasized because this compressor will be required to undergo a Level II analysis. This reflects

Figure 3-78—Process Compressor Cross-section

Figure 3-79—Gas Injection Compressor Cross-section



the intention of the Level I analysis to require most high-density applications to undergo a Level II analysis. It does notimply that there is a deficiency in the injection compressor rotor design. To emphasize the purpose of the Level Ianalysis again, it is to identify those rotors or class of applications that require a more intensive analysis. It is notintended as a judgment of the design merits of the rotors undergoing the analysis.

3.12.3.3 Level I Analysis

Both rotors were modeled according to SP6.8.2.4. Neither contained damper bearings or oil seals, so the supportsystem was comprised entirely of tilt pad bearings. Housing stiffness in both cases was an order of magnitude higherthan the bearing stiffness and was disregarded without having an impact on the dynamic predictions. Minimum andmaximum values were used for the bearing clearance and oil inlet temperature. Synchronous coefficients were usedto model the tilt pad bearings. The Level I analysis was performed at Nmc.

NOTE The anticipated cross-coupling is derived using the normal operating speed.

To facilitate the calculation of Equation (7), a spreadsheet was used. For the process compressor, the design valuesused in the equation are shown in Worksheet 3-1. The values for the injection compressor are shown on Worksheet3-2.

Per SP6.8.5.9, a plot of the log decrement versus mid-span cross-coupled stiffness was created for both compressorsat the bearing conditions stated above. Since the Q0 value was reached before ten times the anticipated cross-coupling (per Worksheets 3-1 and 3-2) for either compressor, the x-axis of the plot was extended to include Q0 only.The graphs are shown on Figure 3-80 and Figure 3-81.

For reference, the values in Table 3-4 were obtained for each compressor.

The 1st critical speed was taken from the undamped critical speed analysis on rigid supports and determined inaccordance with SP6.8.2.3. Details of the lateral analysis were omitted for space considerations.

Table 3-4—Level I Stability Results for Process and Gas Injection Compressor

Process Compressor Gas Injection Compressor

Q0 (lbf/in) 46,700 133,300

QA (lbf/in) 12,576 49,609

A 0.44 0.55

Q0/QA 3.71 2.69

1st Critical Speed (rpm) 5994 8677

Nmc 14,280 14,440

CSR 2.38 1.66

Gas Density (kg/m3) 12.3 107



Worksheet 3-1—Modified Alford’s Force—Process Compressor

Stage by Stage Calculation of Modified Alford Predicted Aerodynamic Forces V1.0

B

D

H

N

C

PTZPTZ

DH

Kxy

Kxy

s

dxy

NHDCHPBK



Worksheet 3-2—Modified Alford’s Force—Gas Injection Compressor

Stage by Stage Calculation of Modified Alford Predicted Aerodynamic Forces V1.0

B

D

H

N

C

PTZPTZ

DH

Kxy

Kxy

s

dxy

NHDCHPBK



Figure 3-80—Process Compressor Stability Plot

Figure 3-81—Gas Injection Compressor Stability Plot

0

1.2

1.0

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

Log Dec vs. Mid Span KxyProcess Compressor

Log

Dec

QA

Q0

0 10 20 30 40 50 60 70 80 90 100

Kxy (Klbf/in.)

A

MinimumMaximum

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

0.2

Log Dec vs. Mid Span KxyGas Injection Compressor

Log

Dec

QA

Q00 20 40 60 80 100 120 140

Kxy (Klbf/in.)

A

MinimumMaximum



3.12.3.4 Level I Screening Criteria

a) Neither compressor has a Q0/QA ratio of less than two. This criterion does not then call for a Level II analysis.

b) The log decrement at the anticipated cross-coupling for the process compressor is 0.44. For the gas injectioncompressor, the log decrement is 0.55. Thus, this criterion does not require that a Level II analysis be performed.

c) The Q0/QA ratio for both compressors is between 2 and 10. Thus, the average gas density and CSR ratio need tobe examined. For the process compressor with a CSR of 2.38 and an average gas density of 12.3 kg/m3 (shownas point Process Compressor, Figure 3-82), it lies within Region A and Q0/QA 2. Thus, no further analysis isrequired. The combination of shaft stiffness and excitation levels is such that the predictions based on the Level Ianalysis are sufficient to ensure a stable rotor.

The gas injection compressor has an average gas density of 107 kg/m3. At this density, a Level II analysis is requiredregardless of the CSR because Q0/QA < 10, point Injection Compressor (relative shaft stiffness measurement). Highgas density applications have the potential for creating large destabilizing forces (see 3.5.1). Under these conditions(average gas density exceeding 60 kg/m3 and Q0/QA < 10), a Level II analysis is used to more accurately model thesedestabilizing forces.

3.12.3.5 Level II Analysis

A Level II analysis of the gas injection compressor is required due to the high gas density and the Q0/QA ratio of theapplication. Per SP6.8.6.3, the stability analysis is also performed at Nmc of the compressor. For componentssensitive to pressure differential, operating conditions near surge (representing the maximum differential attainable)may be considered.

Figure 3-82—Process and Gas Injection Compressors on Experience Plot


Ex #1

Ex #2

0 20(1.25)

40(2.5)

60(3.75)

80(5.0)

100(6.25)

MC

SR

3.5

3.0

2.5

2.0

1.5

1.0

Region A

Region B



The anticipated cross-coupling was replaced with the following excitation sources.

a) Impeller eye labyrinths—Impeller eye seals are exposed to the pressure rise developed in the impeller. Dependingon the number and compressor configuration, their collective effect may be similar in magnitude to the balancepiston. For this example, the impeller hub seals were ignored due to the small differential across each.

b) Balance piston—Large destabilizing forces were created by the balance piston. In its original configuration, thecompressor was predicted to be unstable. Swirl brakes were required to reduce the destabilizing forces in thebalance piston (see 3.4.2).

c) Aerodynamic cross-coupling—Introducing cross-coupling as predicted by one of the empirical equationsduplicates the destabilizing forces already in the model from the labyrinth seal analysis (see 3.5.1). Theaerodynamic cross-coupling was assumed to be zero for this example.

d) All remaining terms were assumed to be second order in magnitude and neglected for this example.

Per SP6.8.6.4, a table of the log decrement is developed for each stage of the analysis of the gas injectioncompressor (see Table 3-5).

The minimum log decrement at the initial step is produced by the combination of bearing clearance and oil inlettemperature within the operating range. This combination is held throughout the remainder of the analysis.

Displaying the results in this fashion highlights the balance piston as the major influence on stability. With no pre-swirlcontrol, the balance piston degrades the log decrement by 0.65. Thus, it is clear which component should beredesigned to address the deficiency in rotor stability. Adding swirl brakes to the balance piston increases the final logdecrement to an acceptable level of 0.32. This may be further improved with swirl brakes on the eye labys.

Note, in the table above, if the balance piston had been designed with swirl brakes then the step of modeling it withoutswirl brakes would be omitted. The analysis should be completed with the as-designed components. If a deficiency isfound, then efforts should be made to redesign the parts.

The table highlights the needs to perform a complete analysis by illustrating the impact of the various components onthe stability. Individual component model quality contributes to the assessment of the rotor stability.

3.12.3.6 Level II Acceptance Criteria

The gas injection compressor was predicted to have an acceptable level of stability with swirl brakes added to thebalance piston (0.322 vs. the required 0.1). This configuration is what was assembled and successfully tested. Theunit has been commissioned and is operating.

3.12.4 References

[1] De Santiago, O. & Memmott, E. A., 2007, “A Classical Sleeve Bearing Instability in an OverhungCompressor,” CMVA, Proceedings of the 25th Machinery Dynamics Seminar, St. John, NB, October 24-26.

Table 3-5—Minimum Log Decrement for Gas Injection Compressor

Configuration Minimum Log Decrement

Rotor and Bearings (Basic Log Decrement) 0.871

Rotor and Bearings + Impeller Eye Labys 0.492

Rotor and Bearings + Impeller Eye Labys +Balance Piston – 0.159

Rotor and Bearings + Impeller Eye Labys +Balance Piston w/Swirl Brake (Final Log Decrement)

0.322



3.13 Active Magnetic Bearings and Stability

3.13.1 General

Rotordynamic modeling and analysis of machinery with Active Magnetic Bearings (AMBs) has some importantdifferences compared to fluid-film or rolling element bearing machines. This section discusses these differences andthe related issues in the context of stability and stability analysis. General modeling practices and issues related toresponse analysis and auxiliary bearings are discussed in Section 2.

3.13.2 Introduction

The fundamental difference for AMBs relative to other bearing systems, is that AMB systems have a feedback controlsystem that determines much of the overall rotordynamic performance. A typical rotor and AMB system with a singleaxis of the feedback control system is shown in Figure 3-83. The major subcomponents are:

— rotor with AMB radial bearing sleeves and axial thrust disk;

— radial and thrust electromagnetic actuators;

— radial and axial position sensors and associated electronics (typically also used for machinery monitoring);

— control system (typically implemented digitally);

— power amplifiers;

— auxiliary bearing system.

Figure 3-83—AMB System

Auxiliarybearings

Radialpositionsensor

Radialcontrolsystem

Radialmagneticactuator

Signalconditioner

Poweramplifier(s) Axial actuator

and shaft disk



From a functional perspective, these subcomponents control shaft position as follows.

— The position sensor system measures rotor motion, and sends signals to the control system proportional to theshaft position.

— The control system generates an output signal that tries to move the shaft back to the desired location (generallythe center of the clearance space).

— The power amplifiers convert this signal into a high power voltage/current.

— The electromagnetic actuators convert the voltage/current into a strong magnetic field that acts on the rotorsleeve/thrust disk to pull the rotor towards the desired location.

Under normal design conditions, the auxiliary bearing system is inactive. The rotor operates without contact within asmall clearance space between the auxiliary bearing and the corresponding rotor surface. The auxiliary bearing(s)only come into contact with the rotor when the AMB system is deenergized or overloaded.

In a two bearing machine, there are five control axes - two radial axes at each of the two actuators, and one axial axis.The radial axes are usually oriented at plus and minus 45 degrees from vertical (for horizontal machines). These arefrequently referred to as the “V” and “W” axes as shown in Figure 3-84. The direction of rotation relative to the V-Waxes varies for different machines. Some integrally driven machines (motor directly coupled to the compressor orpump) have three radial bearings, for a total of seven control axes.

More detailed discussions of AMB systems can be found in the reference list [1–5].

Figure 3-84—AMB System Axes

Axis 1WAxis 1V

Bearing 1

Bearing 2

Axis 2WAxis 2V



3.13.3 AMB Stability Overview

The issue of stability is somewhat different for AMB systems versus fluid-film bearings. For a fluid-film bearingmachine, the question is whether fluid (or aerodynamic and hysteretic) cross-coupled forces caused by shaft rotationare strong enough to overcome the bearing (or bearing and damper) stiffness and damping and create an unstablemode. There is almost always a threshold speed for the onset of instability. For an AMB machine, on the other hand,the fundamental questions are related to the inherent stability of the closed loop system including the rotor, controlsystem, actuator, sensor, etc. Indeed, without a control system (i.e. open loop), the negative magnetic stiffness of theactuator would result in an unstable system.

The magnetic bearing is able to provide damping to the rotor, but if tuned incorrectly, the feedback control system canalso excite and destabilize any natural frequency of the rotor bearing system within the control system's bandwidth.There can be unstable modes at any operating speed, including when the shaft is not rotating. Thus, there is thepossibility of both subsynchronous as well as supersynchronous instability. In principle, it is possible for both forwardwhirling and backwards whirling modes to be unstable. As with fluid-film bearings, it is also possible for aerodynamic(or hysteretic) cross-coupling to cause an instability above some onset operating speed or power level.

In many cases, the identification and resolution of stability issues can be somewhat easier for AMB systems than forfluid-film bearings. There is a body of know-how from the control system community that can be applied. In mostcase, stability problems can be resolved by changes to the control system parameters, which are quite easy to updatein modern digital control systems. A substantial amount of testing and validation can be performed with the machinenot spinning, which substantially reduces the risk of machine damage. Finally, the API specifications also requiremeasurement of a sensitivity transfer function, which provides a measurement of stability robustness to parametervariations such as sensor or actuator gain changes due to temperature, loading, or off-center operation. AMBsystems also inherently include actuators capable of applying dynamic forces to the shaft, thus facilitating stabilitytesting at operating conditions.

3.13.4 Modeling Considerations

3.13.4.1 General

Since any mode within the control system bandwidth can go unstable, it is very important that the rotor model used forstability analysis be capable of accurate modeling of all high frequency modes within the control system bandwidth(i.e. the first few free-free bending modes). This will include modes that may be well above operating speed.Accurately modeling these modes can be difficult. This section highlights a few issues that may need to beconsidered.

Due to the difficulties associated with accurately modeling the higher bending modes, It may also be advisable toperform a rotor ring test to allow all of these effects to be accurately quantified. Ring tests from similar rotors may givesome insight.

3.13.4.2 Shaft Sleeve Bending Stiffness

For most rotor models with fluid-film or rolling element bearings, it is reasonable to assume that most sleeves do notadd much bending stiffness. This may not be a good assumption for higher bending modes. Thus, componentsadded to the shaft should be evaluated for stiffening effects. These shaft stiffening effects may have an importantinfluence on the natural frequencies and mode shapes of the high frequency shaft modes.

3.13.4.3 Disk Modes

Any sizable added components assembled on the shaft should also be evaluated for relevant added degrees offreedom. Relatively thin, large diameter axial magnetic bearing thrust disks, for example, may have 1D mode shapetype natural frequencies within the frequency range of the relevant shaft bending modes. Thus, disk flexibility andattachment flexibility may have to be included to obtain an accurate model [6,7]. This flexibility can be estimated via



finite element analysis or from modal testing. Other components, which may have 1D mode shape type naturalfrequencies within the frequency range of the relevant shaft bending modes, include the compressor impellers andcoupling hubs.

3.13.4.4 AMB Rotor/Stator Clearance Spaces

The radial clearance between the rotor and stator for the AMB system is small and in many machines is filled with theprocess fluid. This fluid-filled annular gap can give rise to appreciable fluid-structural interaction effects (Fritz [8]).These effects can be modeled using linearized added mass, stiffness and damping coefficients at the annular gaplocation. Thus, for high density compressors, and especially for pumps, additional dynamic coefficients may need tobe included in the model to account for these effects.

3.13.4.5 Structural/Rotor Damping

Experience has shown that it is generally appropriate to omit any inherent or structural damping that may be providedby the shaft material for the stability analysis of an AMB equipped compressors with typical construction. Unusualmachines that have long axial shrink fits in regions of high bending stress for one or more modes below operatingspeed might be an exception to this rule (the effects of hysteretic cross-coupling may have to be considered). From anumerical perspective though, it may be necessary to add a small amount modal damping to avoid these lightlydamped modes from being predicted to be unstable due to numerical and round-off effects.

3.13.4.6 Support/Stator Dynamics

It is possible for structural modes to adversely affect overall AMB system stability and even to have the AMB systemdrive a structural mode unstable. This makes it critically important to include flexible supports in the model to achieveaccurate system dynamic predictions. An AMB system can excite any structural mode that has motion at the actuatorlocation and is within the control system’s bandwidth.

Due to the uncertainty in structural models for many machines, structural modes have historically usually beensuccessfully dealt with via field tuning adjustments during initial commissioning. Since any delays at this point can bequite expensive, it is obviously desirable to include any known support/structural dynamics in the analysis model.Consideration also needs to be given to the possibility of the structural dynamics changing for different operatingconditions. Examples include test stand versus field installation, extremely high operating temperatures, dry versusliquid filled, dry versus submerged, etc. Extremely detailed analyses to predict support dynamic effects are typicallyincluded only in special cases, such as for new unique equipment design, or where very flexible support systems areused (Masala et al. [9]). One example where this was required was a vertical subsea machine that is cantilevermounted from one end (Ransom et al. [10]).

Due to the possible adverse impact of structural dynamic effects, users may want to consider building up the actualmachine skid for the mechanical test to reduce the likelihood of field retuning. This would generally only be worthwhileif the support stiffness of the test configuration is significantly different than the field installation.

3.13.4.7 Couplings

From the lateral analysis perspective, it is usually reasonable to treat each body of a multibody AMB supportedsystem with flexible couplings separately, just as is done for fluid-film or rolling element bearings. However, there arealso a group of AMB machines that use a rigid coupling between an electric motor driver and a compressor ormultiphase pump. Many of these are three radial bearing machines. These machines must have a train analysis.

For an axial stability analysis, coupling flexibility effects should be included as part of a train model. Typically, alumped-mass train model is used for this analysis.



3.13.4.8 Control System Modeling

See Section 2.10.

3.13.5 Stability and Stability Robustness Analysis Considerations

3.13.5.1 General

The specifications require both a stability analysis and a stability robustness analysis. AMB specific changes to thesetwo analyses are discussed below.

3.13.5.2 Lateral Stability Analysis

The AMB stability analysis developed for API Std 617 is quite similar to the analysis for fluid-film bearings, but with afew changes related to the wider bandwidth of AMB systems. This wider bandwidth means that AMB system canapply significant dynamic forces at frequencies well above running speed. This leads to the possibility ofsupersynchronous instability. For example, if the control system is not well designed, it is quite possible for an AMBsystem to have an instability corresponding to a bending mode well above operating speed. In this case, the instabilitywould result in the AMB system generating a high frequency force that leads to high supersynchronous vibration.

From practical control system and electrical considerations, however, the available AMB dynamic force will decayrapidly at higher frequencies. For example, most AMB systems are digitally controlled. Thus, the control systembandwidth is limited to half the sample rate. Actuator slew rate limits discussed in 2.10 impose another upper boundon the bandwidth. The control algorithm typically also limits the magnitude of high frequency forces. Thus, for thetypical industrial AMB supported machine, only a small number of bending and/or structural modes are candidates forinstability.

Rather than trying to explicitly define a universally applicable control system “bandwidth” in terms of gain and/orphase characteristics, it was felt that a reasonable approach would be to set a minimum requirement to consider atleast the modes up to 2x Nmc for typical API process machinery. This limit may need to change as experience isgained with applying these standards. It should also be noted that some engineering judgment is required to ensurethat the AMB system does not have unusually large bandwidth (i.e. slow roll-off).

For AMB systems, the Level I and II analyses are extended to include frequencies up to 2x Nmc due to the potential forunstable supersynchronous modes. For frequencies up to Nmc, the same limits apply as for fluid-film bearings. Thisincludes the Level I Q0/QA margin. Above 1.25x Nmc, the log dec for the anticipated cross-coupling must be greater orequal to zero (i.e. not be unstable). A linear decrease was specified between Nmc and 1.25x Nmc. This linear transitionis not based on a rigorous analysis, but seemed reasonable to the standards task force.

The extended criterion for logarithmic decrement is shown in Figure 3-85.

Figure 3-85—Stability Analysis/Log Dec Requirements for AMB Systems

AcceptableRegion

Frequency

0.1

125x NmcNmc 2x Nmc

Log

Dec

.



It needs to be highlighted that, when the system model includes casing and substructure dynamics, the minimum logdec requirements below 1.25x Nmc may NOT be achieved for these additional static structure dynamics, nor is itnecessary. The requirements below 1.25x Nmc are primarily intended for rotor modes.

3.13.5.3 Axial Stability Analysis

The sections relate to the AMB axial analysis do not explicitly state a minimum log dec for the axial system. However,clearly the axial system needs to be stable in a practical machine. This is also implicit in the requirement to meet theISO 14839 sensitivity transfer function limits, since these limits are only valid for a stable system.

3.13.6 ISO 14839/Sensitivity (Transfer) Function

In addition to the Level I and/or II stability analysis, the sensitivity function analysis described in the ISO AMBspecifications (ISO 14839-3) must be evaluated. The sensitivity function must also be measured as part of themechanical run test. The API standards require that the measured sensitivity function fall within specified rangespresented in the ISO standard.

This section discusses the sensitivity function and interpretation of the results. Additional information is alsopresented in [11] and [12].

3.13.6.1 Overview

The sensitivity function is the ratio of a response signal directly after an excitation point to the excitation signal. It isconceptually similar to stability margin as measured by gain and phase margin for single input-single output controlsystems, but generalized to multivariable systems. An example of a sensitivity transfer function is shown in Figure 3-86.

Typically, the sensitivity transfer function starts with a low value (less than 0 dB) at low frequency, then has one ormore peaks in the middle frequency range, then settles down to 0 dB at high frequency where the control system haslimited or no effect.

The overall peak value of the sensitivity function is a measure of stability robustness of a stable system to parametervariations such as sensor or actuator gain changes due to temperature, loading, or off-center operation. Thesensitivity function can both be calculated analytically, as well as measured on the machine using the AMB system asthe source of measurement and the source of excitation. Typically, the measurement can be made by a digital AMBcontrol system without any additional instrumentation.

Figure 3-86—Example of Sensitivity Transfer Function

101-20

-10

0

10

20

102

VDC-XE Sensitivity Transfer Function

Ga

n, d

B

103



Note that the sensitivity function analysis requires that the system first be stable. If the system is unstable, a sensitivityfunction analysis should not be performed.

3.13.6.2 Measurement/Calculation Locations

There are several possible sensitivity functions that could be measured and/or generated analytically. This issue isdiscussed at length in both ISO 14839, part 3 [11], and by Schweitzer and Maslen [12]. ISO 14839-3 indicates that thesensitivity transfer function is to be measured or calculated on an axis by axis basis in physical sensor coordinates, atthe input to the AMB controller, as shown in Figure 3-87.

It is important to note that the measurements must be made on an physical axis by axis basis. In particular, it is nottechnically correct to make the measurements and calculation in tilt/translate coordinates. The transformation fromphysical (V1, V2, W1, W2, etc.) coordinates to tilt-translate coordinates can be shown to change the shape of thesensitivity transfer function. In some cases, the limits developed by the ISO committee might not be appropriate forthis transformed system.

Making the sensitivity function measurement in transformed coordinates, however, may be quite useful from a controlsystem tuning/validation perspective.

3.13.6.3 Some Observations about the Sensitivity Transfer Function

The requirements to compute and measure the sensitivity transfer function are included in the specifications primarilyas a way to address stability robustness of an AMB system. The sensitivity function measurement is a fairly practical,well defined, way to evaluate this issue for a closed loop AMB system. How it works can be considered from at leasttwo viewpoints.

The first is to view it as the multi-input, multi-output (MIMO) equivalent of gain and phase margin as defined for single-input, single-output (SISO) control systems. For a SISO system, gain margin is a measure of how much the controlsystem gain can increase before the overall system goes unstable. Phase margin is a measure of how much thephase lag through the control system can increase before the system goes unstable. High peak sensitivity is relatedto low gain and phase margins.

The second is to note that the sensitivity function is formed by summing of the excitation signal plus the system'sresponse to that signal, than scaling this sum by dividing it by the excitation signal. If the system does not respond tothe signal at all, then the ratio is 1.0 (excitation/excitation = 1). This would be low sensitivity system. On the otherhand, if the system amplifies the excitation signal, it will have a high sensitivity, since the sum will be much larger thanthe original excitation signal.

The performance of the nominal system with a high sensitivity may be quite acceptable. The problem, however, iswhat happens if one of the system parameters changes. For example, the position sensor scale factor might increase

Figure 3-87—Sensitivity Function Measurement Locations

Compensator

V1

Sensitivity TF =V1

Exc

Excitation(Exc)

Plant(amplifier, rotor

and sensor)



due to temperature. If the system has a high sensitivity, the system response at the input to the compensator with thehigher sensor scale factor will be quite different. Thus, the overall performance of the system will also be quitedifferent, and may eventually go unstable. Conceptually, a system with a high sensitivity (gain) is more “sensitivity” tochanges in the rotor or AMB system parameters. A system that is too sensitive to rotor/AMB parameter changes isobviously less robust than one that is relatively insensitive.

Achieving an acceptable peak sensitivity always involves tradeoffs between low sensitivity and good forced(unbalance) response. A high gain, very stiff AMB may have good forced response, but high peak sensitivity.Likewise, a soft AMB may have low peak sensitivity, but poor forced response characteristics. The sensitivity transferfunction is also frequency dependent. Reducing peak sensitivity over one frequency range often will increasesensitivity over another frequency range.

Finally, AMB systems always have multiple inputs and multiple outputs. Even if the control system is not a multi-input/multi-output configuration, real AMB machinery always has some amount of coupling between the inputs and outputs,especially if the machine is rotating and gyroscopic effects are present. Thus, there is actually a matrix of sensitivityfunctions that describe the interactions between each input and each output. In some cases, the diagonal terms ofthis matrix that the API and ISO specs require to be measured may not tell the whole story about sensitivity. Thiswould be especially true in systems with strong coupling between planes, like highly gyroscopic systems, or systemswith strong end-to-end coupling like relatively rigid rotors with centralized masses. However, acceptance criteria forthese nondiagonal, cross-coupling, sensitivity functions have not been established.

It should also be noted that the sensitivity function is only interesting for a closed loop system. For a linear system thathas no feedback, the sensitivity function by definition is 1.0.

3.13.6.4 Acceptance Criterion

The peak magnitude of the sensitivity function is a measure of how robust the system is to parameter variations suchas sensor or actuator gain changes due to temperature, loading, or off-center operation. The larger the peak value is,the less margin the system has. The ISO technical committee established the limits shown in Table 3-6 as beingappropriate for industrial machinery. For API machinery, the 0 RPM mechanical run test measurements should fallwithin zone A for the radial bearings, and zone B for the axial bearing.

The example sensitivity transfer function shown in Figure 3-86 meets this requirements, since the peak value is lessthan 9.5 dB. An example of a sensitivity transfer function that does not meet this criterion is shown in Figure 3-88.

3.13.6.5 Limitations

The sensitivity function analysis is a powerful technique for evaluating the robustness of the system to feedback loop(AMB system) uncertainty. It is one of the best ways of ensuring that the AMB system is capable of handling variationswithin the AMB system. It is important to note, however, that the sensitivity function cannot replace the stabilityevaluation. The sensitivity function analysis evaluation makes the implicit assumption that the overall AMB/rotorsystem is stable. This is one reason why this evaluation is included as an additional step in the stability evaluation,rather than as a stand-alone item. It is not recommended to consider the sensitivity function results unless the stabilitycriteria are first met.

Table 3-6—Peak Sensitivity Function Zone Limits [11]

Zone Range (dB) Range (Ratio) Description

A <= 9.5 < 3 Newly commissioned machinery

B 9.5 < S <= 12 3< S <= 4 Acceptable for long-term operation

C 12 < S <= 14 4< S <= 5 Short-term operation only, not acceptable for long-term operation

D > 14 > 5 Machine damage likely



In addition, as discussed in a paper by Li, Maslen and Allaire [13] the sensitive function analysis does not reliablypredict AMB system robustness with regards to some important practical variations in the machine. In particular, itdoes not address:

— cross-coupled stiffness variations (such as aerodynamic excitation and seals);

— variations in natural frequencies due to physical changes in mass and/or stiffness (changes in mountingstructures, or converging seal clearances, for example).

The standard attempts to address the issue of cross-coupled stiffness by requiring that a representative amount ofcross-coupling be applied using the same basic approach used for the Level I and Level II stability analyses.

The possibility for variations in natural frequencies is difficult to address in general. The standard does not attempt todo so. As with all aspects of the rotordynamics evaluation, some engineering judgment is required in cases wherechanges in natural frequency may be an issue. An example might be variations in the direct stiffness coefficients of aconverging or diverging hole-pattern or honeycomb seal. Another might be large density variations in a mixed-flowpumps. In these cases, additional parametric studies would be appropriate.

Figure 3-88—Sensitivity Transfer Function That Fails to Meet the Acceptance Criterion

Frequency (Hz)10-1

-40

-30

-20

-10

0

10

20

30

40

101 102 103100

Max=25.6 dB

DE Y

DCBA

Comp End Y

0 RPM WV23Z Compressor Sensitivity FunctionsS

ens

tvty

(dB

)

10,0

00 R

PM

23,0

00 R

PM



3.13.6.6 Level I/Level II/Axial Sensitivity Function Analysis

In keeping with the spirit of the Level I stability analysis, the Level I sensitivity analysis is to be performed with twotimes the anticipated cross-coupling. This decision was not based on a rigorous analysis nor is there a great deal offield experience to support this choice. As the compressor community gains experience with the new AMBrequirements, it may be desirable to adjust the cross-coupling multiplier to achieve a reliable screening evaluation.

For the Level II sensitivity analysis, it is intended that the sensitivity function analysis be performed very similar to theLevel II stability analysis. The basic idea is that as each group of destabilizing effects are added, the stability is firstevaluated. If the system is stable, then the sensitivity function is also evaluated with this group of destabilizing effects.This should allow the relative importance of each group of destabilizing effect to be evaluated.

Although it is not explicitly stated in the standard paragraphs, it really does not make sense to evaluate the sensitivityfunction for an unstable system. Likewise, it should be noted that the only sensitivity function evaluation that is to beexamined with regards to meeting the specification, is the final calculation for the complete model.

For the axial analysis, the acceptance criteria are relaxed slightly to zone B or better. A distinction is also madebetween rotor motion dominated modes and modes dominated by coupling spacer motion. The AMB system shouldbe able to control the rotor modes. It may not have very much influence on flexible coupling spacer modes. Thus, it isnot appropriate to consider these modes. These modes also exist in non-AMB supported machinery with flexiblecouplings, and are rarely a source of problems.

3.13.6.7 Measurement during Mechanical Run Test

The specifications require measurement and evaluation of the sensitivity functions at 0 RPM prior to operating themachine for the mechanical run test. This requirement is intended to help catch any major problems prior to runningthe machine. Ensuring that the measurements fall within zone A gives some indication that there are no unexpectedhousing dynamics, and that the rest of the AMB components are as specified.

The API acceptance criterion is based on 0 RPM to avoid additional sensor noise due to rotation, unbalance forces,added seal cross-coupling, etc.

In the case of a pump or mixed-flow machine, a decision would need to be made as to whether to make themeasurements for both dry and wet impellers.

Measurement at additional operating speed(s) is appropriate in many cases. This is especially true in the case of again scheduled controller, which has coefficients that change as a function of operating speed, although these areexpected to be rare for compressors. If a full-speed, full-load test is performed, measurements at operating conditionscan give some indication of the effects of seal and other aerodynamic/fluid effects, which are not addressed by the 0rpm sensitivity function measurement. Fully assessing the impact of these effects may also require measurement ofthe logarithmic decrement of the actual modes as described elsewhere in this tutorial.

3.13.7 As Installed Analysis

The final requirement of the AMB specification is for an analysis using the final AMB parameters from the as-installedmachine (if they do not match the original design parameters). This analysis is then to be included in the finalrotordynamics report. This requirement is a significant departure from the requirements for traditional machinery. Fortraditional machinery, the rotordynamics report could probably be finalized once the machine leaves the test stand.Thus, the potential need to perform additional rotordynamics work after field startup is awkward. However, the currentexperience is that field tuning adjustments for AMB supported rotors are fairly common. Thus, it is crucial that thisadditional step be performed to ensure that the implications of any field tuning have been fully evaluated.

Field tuning during commissioning is generally a response to either unmodeled support or stator structural dynamics,or actual process conditions. Differences in unmodeled or incompletely modeled structural dynamics between the test



stand and the field installation can lead to re-tuning. AMB systems are frequently far more sensitive to supportstructure dynamics than a hydrodynamic bearing might be. In some cases, it has been suggested that it may bedesirable to build-up at least a portion of the actual machinery skid for the shop test. One example might be a shoptest setup where the machine is rigidly bolted directly to a massive foundation, while the actual installation is moreflexibly supported. Conversely, some installations are far stiffer than the shop test setup.

The effects of actual process conditions versus test stand conditions can also lead to re-tuning. The effects ofpressure or density on seals, for example may result in significantly different stiffnesses than for a mechanical runtest.

Either or both of these issues can make it desirable to fine-tune the AMB controller to obtain maximum rotor bearingsystem performance. With modern digital controllers, this fine-tuning generally just involves testing and downloadingnew sets of parameters.

The intent of this final requirement is to ensure that the final rotordynamics report includes the effects of any tuningthat was performed during initial commissioning. It is important to note that there are no acceptance criteria for thisfinal analysis. It is intended strictly for end-user and OEM reference.

Obviously, it is desirable that the final analysis demonstrate that the system meets all of the same requirements.However, it is recognized that there may be modeling limitations with regards to effects such as seals or impellerforces that are beyond the state of the art to accurately include. It is not the intent to have this final analysis involve asignificant amount of model updating and validation.

From the end user perspective, the primary question is probably whether the field tuning is predicted to significantlyreduce the stability of the machine or significantly increase the unbalance sensitivity. It also provides a moreconsistent analytical baseline for evaluating future machinery performance.

3.13.8 AMB Data Requirements for Lateral/Axial Reports and Data for Independent Analysis

There are a few AMB specific items related to stability that are added to the standard lateral report for an AMBsupported machine. These items include:

— plots of the ISO 14839 sensitivity functions with 2x the anticipated cross-coupling;

— additional modes for the Level I stability analysis;

— Level II (if performed):

— results for additional modes for the Level II stability analysis;

— table of peak sensitivity function values and frequencies by added component/group of components;

— plots of the ISO 14839 sensitivity functions for final model.

Note that these items are also required for the as-installed analysis.

3.13.9 References

[1] ISO 14839-1, (2004) “Mechanical Vibration—Vibration of Rotating Machinery Equipped With Active MagneticBearings - Part 1: Vocabulary,” No. 14839-1, International Organization for Standardization.

[2] ISO 14839-2, (2004) “Mechanical Vibration—Vibration of Rotating Machinery Equipped with Active MagneticBearings, Part 2: Evaluation of Vibration,” No. 14839-2, International Organization for Standardization.



[3] ISO 14839-3, (2006) “Mechanical Vibration—Vibration of Rotating Machinery Equipped with Active MagneticBearings, Part 3: Evaluation of Stability Margin,” No. 14839-3, International Organization for Standardization.

[4] ISO 14839-4, (2012) “Mechanical Vibration—Vibration of Rotating Machinery Equipped with Active MagneticBearings, Part 4: Technical Guidelines,” No. 14839-4, International Organization for Standardization.

[5] Schweitzer, G., Maslen, E.H., editors (2009), Magnetic Bearings: Theory, Design and Application to RotatingMachinery, Springer, Dordrecht.

[6] Díaz, S., De Santiago, O., Solórzano, V., 2012, “Rotordynamic Modeling of Centrifugal Compressor Rotors forUse with Active Magnetic Bearings,” XIII Latin American Turbomachinery Congress and Exposition, PaperNumber CELT-019-2012, March 12-15, Querétaro, Mexico.

[7] Vance, J.M., 1988, Rotordynamics of Turbomachinery, John Wiley & Sons.

[8] Fritz, R.J., 1970 “The Effects of an Annular Fluid on the Vibrations of a Long Rotor, Part I—Theory,” ASMEJournal of Basic Engineering, p. 923–929.

[9] Masala A., Vannini G., Lacour M.,. Tassel F-M, Camatti M., (2010) “Lateral Rotordynamic Analysis and Testingof a Vertical High Speed 12.5MW Motorcompressor Levitated by Active Magnetic Bearings”, Proceedings of12th International Symposium on Magnetic Bearings.

[10] Ransom, D., Masala, A., Moore, J.J., Vannini, G., Camatti, M., 2008, “Numerical and Experimental Simulationof a Vertical High Speed Motorcompressor Rotor Drop onto Catcher Bearings,” 11th International Symposiumon Magnetic Bearings, Nara, Japan, August 26–29.

[11] ISO 14839-3, (2006) “Mechanical Vibration—Vibration of Rotating Machinery Equipped with Active MagneticBearings, Part 3: Evaluation of Stability Margin,” No. 14839-3, International Organization for Standardization.

[12] Schweitzer, G., Maslen, E.H., editors (2009), Magnetic Bearings: Theory, Design and Application to RotatingMachinery, Springer, Dordrecht.

[13] Li G., Maslen E. H., and Allaire P., (2006) “A Note on ISO AMB Stability Margin.” 10th International Symposiumon Magnetic Bearings.


4-1

SECTION 4—TORSIONAL ANALYSIS

4.1 General Modeling Considerations

4.1.1 Introduction

In direct contrast to lateral modeling, which is often done on a component-by-component basis, a good torsionalmodel should account for all rotating elements within the drive train. Train modeling begins by dividing the componentshafts into discrete sections or finite elements. Locations on the rotor that contain significant inertia, such ascompressor impellers, turbine discs, bull gears, etc. are identified and lumped inertia is concentrated at theselocations. Connections between rotors, such as couplings, are identified as lumped inertia connected by anequivalent stiffness to complete the mechanical model. The model contains information regarding speed ratios of therotors so that all the system inertias and stiffnesses are properly referenced to one speed. The final data required forthe model is the identification of the material properties of the connected system.

Since torsional analysis often involves several sets of vendor prints and/or data sets, which may contain differentsystems of units, care must be exercised in the detailed modeling of a torsional system in order to ensure the requiredaccuracy. A consistent set of engineering units must be used throughout the analysis in order to avoid potential errors.Prior to analyzing a system, the analyst must have accurate drawings to model the rotors and couplings used in thesystem. These may consist of actual manufacturing drawings or an accurate summary of the rotor construction, alsoknown as a mass-elastic drawing. The mass-elastic drawing should include the following information:

a) materials used for the rotor and how the properties will change with operating temperature, if appropriate;

b) shaft section lengths and diameters and any noncircular cross-section details;

c) axial locations and polar moments of inertia for all significant inertial components;

d) details of rotor attachments such as shrunk-on or bolted-on components including lumped inertia.

The analyst should not rely on a single sketch of a rotor which tabulates inertia and stiffness. Unless the shaft stiffnessis properly calculated between major lumped inertias and for couplings, serious misrepresentations may be made ofthe torsional system model. Such modeling errors can result in significant errors in the torsional natural frequencies ofa system.

Coupling drawings are required to accurately model the system. Correct modeling of the couplings is crucial as thesecomponents have a major impact on the first mode of a directly coupled system and the first two modes of motor/gear/compressor systems. Systems with multiple bodies may have the first three or four modes influenced bycouplings between components. Coupling drawings should contain the polar mass moment of inertia of the drive anddriven halves separately (including half of the spacer in each) and the torsional stiffness of the coupling assembly. It isimportant to identify how the shaft penetration into the coupling has been treated when calculating the completecoupling stiffness (see Figure 4-1). For metallic couplings, the coupling stiffness is a single value. However, fortorsionally resilient elastomeric block couplings, the stiffness is a nonlinear function of the applied torque. The data forsuch elastomeric couplings should include a curve of the torsional stiffness versus the applied torque.

If train units are accurately modeled, the undamped torsional natural frequency analysis usually predicts actual trainnatural frequencies within a small margin of error because most equipment trains generally possess low levels ofsystem torsional damping. In other words, the damped and undamped natural frequencies are almost always nearlyidentical.

Figure 4-2, Figure 4-3, Figure 4-4, Figure 4-5, Figure 4-6, and Figure 4-7 present, in sequence, the general approachto torsional modeling for a typical motor-gear-compressor train and a turbine-compressor train. Sideview drawings ofthe two trains are presented in Figure 4-2, Figure 4-3, Figure 4-4, and Figure 4-5. Cross-sectional views of the



rotating elements for these two trains are displayed in Figure 4-3, Figure 4-4, Figure 4-5, and Figure 4-6. Finally, usinggeometric and inertia properties of the rotating elements, computer models of the trains can be assembled.Schematics of the train computer models are presented in Figure 4-4, Figure 4-5, Figure 4-6, and Figure 4-7.

Typical items which can easily be modeled by concentrated mass-elastic data exclusively are couplings, gears,impellers, turbine stages, and motor rotor attachments. Experience has shown that metallic flexible couplings aremost appropriately modeled as a single torsional spring (vendor supplied torsional stiffness) with the respective half-coupling inertias at each end.

Gears lend themselves to lumped mass modeling since the bulk of their inertia is in the gear wheels, and the shaftsclosely approximate low inertia torsional springs. Impellers and turbine stages can usually be modeled as discretelumped inertias since they typically do not contribute to the torsional stiffness of their respective shafts. The onlynormal exception to this occurs in low pressure steam turbine stages where the blades are so long that their flexibilityneeds to be accounted for in the torsional model. Motor rotor attachments such as rotor cores and brushless excitersare not easily modeled because they are typically shrunk onto the shaft over an extended length It is not obvious howthe attachment’s inertias and stiffnesses affect the motor shaft inertias and stiffnesses. (This is discussed in moredetail in Section 4.2.)

Figure 4-1—Typical Coupling Stiffness Boundary

Figure 4-2—Side View of a Typical Motor/Gear/Compressor Train

Coupling stiffness typically includes shaftsection contained within the hub.

MotorGear Compressor



Figure 4-3—Modeling a Typical Motor/Gear/Compressor Train

Figure 4-4—Schematic Lumped Parameter Model for the Motor/Gear/Compressor Train

1 3 4 52 6 7 8 9 10 11 12

1314 15 16 17 18 1920 21 22 23 24 25

Motor

Stationno.

Shaft penetration

Couplingmodel

ImpellerC.G.

Armaturecore

Motor Gear PinionLow speed

couplingHigh speed

couplingCentrifugalcompressor

DBSE

Couplingmodel

NOTE Coupling vendors typically provide WR2 and KTorsional for each coupling. The WR2 value does not include the WR2 of the coupling journal (shaft inside the coupling HUB). The KTorsional value typically assumes 1/3 shaft penetration into the coupling HUB.

LengthDiameterWR2 (Ip x gc)KTorsional

inchesincheslbf in.2

in. lbf/radian

mmmmN mm2

N mm/radian


Shrink fit

Motor

LowSpeed

Coupling

Gear(Gear and

Pinion)

HighSpeed

CouplingCentrifugal

Compressor

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Kt1

Ip1 Ip15 Ip25

Kt15 Kt24

Ipi = ith station lumped polar moment of inertia.Kti = ith shaft section torsional stiffness.

StationNo.

NOTE 1

NOTE 2 All WR2 values have been converted into polar mass moments ofinertia (Ip) at each station.

NOTE 3. All Ip and Kt values have been referenced to a single shaftrotation speed.



Figure 4-5—Side View of a Typical Steam Turbine Driven Compressor Train

Figure 4-6—Modeling a Typical Steam Turbine Drive Compressor Train

Turbine Compressor

1 3 4 52 6 7 8 9 1011

12 13 14 15 16 1718

19 20 21 22 23

Disk C.G.

Compressor

ImpellerC.G.

Couplingmodel

DBSE

StationNo.

CentrifugalcompressorCoupling

Coupling

Steam turbine

LengthDiameterWR2 (Ip x gc)KTorsional

inchesincheslbf in.2

in. lbf/radian


NOTE Coupling vendors typically provide WR2 and KTorsional for each coupling. The WR2 value does not included the WR2 of the coupling journal (shaft inside the coupling HUB). The KTorsional value typically assumes 1/3 shaft penetration into the coupling HUB.

Shrink fit



Caution must be exercised with approaches that lump the inertia and stiffness of a whole unit because inaccuraciesmay result if the shaft ends are sufficiently torsionally flexible relative to couplings. This is quite commonly found inmotors, as motor suppliers often provide only one inertia and one stiffness in their mass-elastic data. This approach isstrongly discouraged since it will result in errors in the prediction of higher order natural frequencies. It also makes theperformance of a detailed stress analysis on that particular component, which is often required (see Section 4.6),quite difficult. Accordingly, it is strongly recommended that all shaft sections be modeled using multi-station models,with the same detail that is used in lateral modeling.

This section presents some of the more important modeling problems and concerns typically encountered in torsionalvibration analysis:

a) speed referencing inertia (Ip) and stiffness (Kt),

b) shaft stepping,

c) shrink fits,

d) integral disks or hubs,

e) couplings,

f) material properties,

g) built-up rotors.

Figure 4-7—Schematic Lumped Parameter Model for the Steam Turbine Driven Compressor Train

Kt22Kt1

Ip1

Ip13

Ip23

Kt13

16 17 18 19 20 21 22 231 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Turbine

StationNo.

Coupling Centrifugal compressor

Ipi = ith station lumped polar moment of inertia.Kti = ith shaft section torsional stiffness.

NOTE 1

NOTE 2 All WR2 values have been converted into polar mass moments ofinertia (Ip) at each station.

NOTE 3 All Ip and Kt values do not require speed referencing since trainelements all spin at the same speed.



4.1.2 Speed Referencing Inertia and Stiffness

A computer code for calculating undamped train torsional natural frequencies should have the capability to analyze atrain of coupled rotors that operate at different rotational speeds (for example, an equipment train with multiple speed-changing gears). An equivalent single shaft model for a geared train is required before calculating the undampedtorsional natural frequencies and mode shapes of the system. An equivalent single shaft model requires all inertiasand stiffnesses to be referenced to a common speed, typically, the driver speed. The relationships between inertiaand stiffness of components that operate at a different speed from a selected reference speed are written as follows:

(4-1)

(4-2)

where

Ip is the polar mass moment of inertia, kg-m2 (lbm-in.2);

Kt is the torsional stiffness, N-m/rad (in.-lbf/rad);

N is the rotation speed, rpm;

a (subscript) denotes actual;

r (subscript) denotes reference.

If the software used to calculate the undamped torsional natural frequencies does not have the capability to analyze atrain whose elements are operating at different speeds, then the analyst must manually perform speed referencingusing the above relationships.

Simple gear reductions (single or multiple) represent single-branch systems, and combined with possible simplebranches (single inertia, one-degree of freedom) can be analyzed with the basic Transfer Matrix (Holzer) computercode. Multiple-branch systems of greater complexity require a more sophisticated computer analysis, typically, a finiteelement formulation. A side view drawing of a train with a single reduction gear is displayed in Figure 4-2. An exampleof a multiple branch system, a multi-stage integrally geared plant air compressor, is displayed in Figure 4-8. This unithas four overhung compressor stages driven through a single bull gear. Note that the two pinions operate at differentspeeds.

4.1.3 Shaft Stepping

When the shaft geometry, length (L) and diameter (D), are used as input, the analyst should consider the effectivepenetration of smaller diameter shaft sections into adjacent larger diameter sections. The smaller shaft, in effect,penetrates the larger by the distance defined as the penetration factor (PF). As a result, the length of the smallerdiameter shaft is effectively increased by the amount PF and the length of the larger diameter shaft is reduced by thesame amount. Figure 4-9 displays the effective length increase of the smaller diameter section as a function of thestep geometry. Allowance for penetration effects enables one to more accurately approximate the actual flexibility ofthe physical system than by simply summing the calculated flexibility of the individual shaft sections. For this type ofdiscontinuity, the effective length of a shaft which joins another of larger diameter is greater than the actual length dueto local deformation at the juncture. As shown in Figure 4-9 this penetration factor (PF) depends on the ratio of shaftdiameters.

Ipr Ipa

Na

Nr

-----

2

=

Ktr Kta

Na

Nr

------

2

=



Figure 4-8—A Typical Twin-pinion Integrally Geared Centrifugal Compressor That Should be Modeled as a Branched System

Wpinion2

Wpinion1

WDrive



Table 4-1 gives some characteristic results.

Another commonly employed method for accounting for this effect is the 45 degree rule, which is illustrated in Figure4-10. In using the rule, a 45 degree angled line is simply drawn from the point of diameter transition. This line isterminated at either the point where it intersects another 45 degree line (as shown in the figure) or at the point whereit intersects the larger diameter. As shown by the dashed line in the figure, the effective stiffness diameter in thatregion is then simply taken to be the diameter that bisects the 45 degree line. Although this methodology is not asaccurate as the use of Figure 4-9, it is quicker and normally provides sufficient accuracy.

It should be noted that both of the above methodologies only apply to the effective stiffness diameter in the region ofthe larger shaft diameter. If a computer code that permits input of different diameters for stiffness and masscalculations is being employed, the mass diameter should be simply set equal to the actual diameter of the shaft.

Figure 4-9—Effective Penetration of a Smaller Diameter Shaft Section into a Larger Diameter Shaft Section

Table 4-1—Penetration Factors for Selected Shaft Step Ratios

D2/D1 PF/D1

1.00 0

1.25 0.055

1.50 0.085

2.00 0.100

3.00 0.107

0.125

1.0

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0

DiameterRatio

D2D1

(dm

)

(dim)

Leng

th o

f Red

uced

Sec

ton

Dam

eter

, D1

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

NOTE Lambda represents a length added to the smaller diameter which will yield the actual torsional stiffness of a stepped shaft section.D2D1



NOTE Refer to Figure 4-9 for definition of parameters. Lambda ( ) in Figure 4-9 represents a length added to the smallerdiameter which will yield the actual torsional stiffness of a stepped shaft section. Penetration factor (PF) equals lambda.

4.1.4 Shrink Fits

Most rotating assemblies used in machinery trains have nonintegral collars, sleeves, and so on, that are shrunk ontothe shaft to create a rotor assembly. These shrunk-on components may or may not contribute to the torsional stiffnessof the shaft, depending on the degree of the interference fit, the length of the shrink fit and the size of the shrunk-oncomponent. Precise guidelines regarding inclusion or exclusion of the effect of the shrunk-on member on the shafttorsional stiffness are difficult to quantify; however, the following general principles apply.

a) If the fit of the shrunk-on component is relieved over most of its length, then the torsional stiffening effect isnegligible. A fit is relieved when some portion of the designed interference has been removed. Schematics of shaftsleeves with and without typical relieved fits are displayed in Figure 4-11. Sleeves and impellers often possessrelieved fits to aide the rotor assembly process and to minimize internal friction forces that contribute to rotorsystem instability. The stiffening effect of shaft sleeves and impellers with a high degree of relief (small fit length) isoften neglected.

b) If the fit of a shrunk-on component is (1) not relieved over a significant part of its length, (2) made of the samematerial as the shaft, and (3) manufactured with a shrink fit equal to or greater than 1 mil/in. of shaft diameter, thenthe effective stiffness diameter of the shaft should be assumed equal to the actual diameter under the sleeve plusthe thickness of the sleeve.

c) If the shrunk-on component has a large rotational inertia, then centrifugal force due to rotation may diminish thedegree of the shrink fit and the attendant torsional stiffening effects of the component over the rotor operatingspeed range.

d) If a shrunk-on component has a nominal fit length with an L/D greater than or equal to 1.0, the shaft is assumed tobe free to twist torsionally up to a point that Reference [1] refers to as a “point of rigidity” within the hub. Asillustrated in Figure 4-12, this fictional point is the point where the effective stiffness diameter transitions from thatof the shaft to that of the hub. The most well-known one is the 1/3-2/3 rule used by most coupling vendors (i.e. theshaft is allowed to twist freely over 1/3 of the length of its interface with the hub). However, Reference [1] providesequations for locating the point of rigidity in several other common configurations.

Figure 4-10—45 Degree Rule

Element boundaryPart boundary

45° Line



4.1.5 Integral hubs or discs

The effect of integral thrust collars or disks forged on the shaft can be determined by the method used for steppedshafts (see 4.1.3). For short collars, with an axial length less than 1/4 of the shaft diameter, the effect is negligible andthe axial length of the collar may be assumed to have an effective torsional stiffness diameter equal to the diameter ofthe shaft.

4.1.6 Couplings

Train components are normally connected by flexible couplings such as the gear, metallic flexible element, or variouselastomeric types. Of these, the flexible element dry couplings are the most popular. The flexible element drycouplings are usually the disc or diaphragm type. Sample cross-sectional drawings of the gear and flexible elementtypes of couplings are displayed in Figure 4-13, Figure 4-14, and Figure 4-15.

Figure 4-11—Examples of Shrunk on Sleeves With and Without Relieved Fits

Figure 4-12—Point of Rigidity

Shrunk-on sleeve

Sleeve with relieved fitSleeve with no relief

Fit area Fit area Fit area

Shaft

Disk

Point of rigidity

No deflectionassumed over

this length

Shaft assumed totwist freely over

this length



Normally, a vendor-supplied torsional stiffness value is input for the total coupling. Most coupling vendors include theeffects of shaft penetration (i.e. the flexibility of the shaft twisting within the coupling hub) in this value but there are afew that do not. Thus, it is important to ascertain whether or not shaft penetration effects have been included within agiven coupling stiffness value. If they have been included, then the stiffness of the portions of the shafts within thecoupling hubs should not be duplicated in the overall analysis model. With a flange connection, the stiffness value willinclude the coupling flange and the bolted connection, but will not include the stiffness of the flange to which thecoupling is bolted.

The coupling can be thought of as an assembly of torsional springs in series. For example, the reduced moment gearcoupling of Figure 4-13 can be modeled with:

a) Ka1,Ka2 = hub-to-shaft connection stiffness (for each hub includes the portion of the shaft inside the mounted huband the hub body);

b) Kb1,Kb2 = gear mesh stiffness;

c) Kc1,Kc2 = sleeve stiffness (for each sleeve includes the sleeve tube and sleeve flange);

d) Kd1,Kd2 = bolted connection stiffness;

e) Ke = spacer stiffness (includes spacer flanges and the spacer tubular section).

The total coupling stiffness, Kt, would then be found from the equation:

(4-3)

Where typical SI units for Kt are N-m/rad, and typical US Customary Units are in.-lbf/rad.

For a reduced moment flexible element dry coupling as shown in 4-15, the method of calculation is the same, exceptthat instead of the gear mesh stiffness, the flexible element stiffness and any flange connection stiffness are used.

The formulas for calculating the stiffness are relatively simple, but the assumptions used to define certain variablesmake the analysis more complex. As an example, the length to use in the formula for the tubular section of a spaceris not the overall length of the spacer, but is the overall length of the spacer minus half the thickness of each flange.

The spacer flanges are considered to be disk sections, which are circular sections through which the torque istransmitted between the outside diameter and the inside diameter of the section. When the torque transmitting pathchanges from a tube section to a disk section, as in a coupling spacer, the tubular length is taken from the center ofthe disk section. Moreover, the outside diameter used in the disk (flange) section is not the extreme outer diameter,but the bolt circle diameter. All this is shown in Figure 4-16. For detailed equations and calculation examples, refer toReferences [2] and [3].

Figure 4-13—A Reduced Moment Gear Coupling

1Kt----- 1

Ka1-------- 1

Ka2-------- 1

Kb1--------- 1

Kb2--------- 1

Kc1-------- 1

Kc2-------- 1

Kd1-------- 1

Kd2-------- 1

Ke------+ + + + + + + +=



Even more complex is the hub-to-shaft connection stiffness analysis. Refer to Reference [3] 4.4.2.3 Hub/shaftconnection which reads, “Some of the parameters that affect torsional stiffness of a hub/shaft connection include: thetype of fit (clearance, interference, keyed or nonkeyed), magnitude of torque, speed, and hub flange location.”

A simplified method commonly known as “1/3 shaft penetration” was introduced in Reference [4]. When applicable,the 1/3 shaft penetration should be included in the coupling stiffness. Other methods such as found in Reference [5] orthe vendor’s standard can also be used upon mutual agreement.

Figure 4-14—A Marine-style Diaphragm Coupling

Figure 4-15—A Reduced Moment Disc Coupling



Further, “The 1/3 shaft penetration method assumes, through experience, that the shaft is regarded as unrestrainedfor one third the length of the hub. The torque travels through only the shaft for the first one-third of the hub shaftengagement, and through only the hub... for the remaining two-thirds” [4].

Of course, the actual penetration varies with hub connection design. For typical applications of flexible element drycouplings, the 1/3 penetration model is a very good assumption, since the torsional stiffness of the flexible elementswill generally be much less than the hub/shaft connection stiffness. In gear couplings, the situation is different,because the gear mesh stiffness is much higher than the flexible element stiffness. In that case, the hub/shaftstiffness plays more of a role. Unless the gear coupling has a short shaft separation (or an otherwise very stiff spacer)the 1/3 penetration assumption is still reasonable, since longer shaft separations have a much lower spacer tubestiffness which reduce the overall coupling stiffness.

In most couplings that do not contain elastomers, the torsional stiffness is assumed to be constant, independent ofapplied torque. Tests and finite element analysis presented in Reference [2] show this assumption to be overlyoptimistic for disc couplings, as the stiffness is torque-dependent, tending to decrease as applied torque is increased.Additionally, the torsional stiffness may also be a function of the coupling axial pre-stretch. However, it is customary toignore these effects and treat the stiffness as being constant. Experience has shown these effects have not been asource of failure due to an unintended torsional interference. The coupling vendor should be consulted as to the basisof their torsional stiffness calculation.

Elastomeric couplings are typically of the shear type or compression block type. Elastomeric block couplings aresometimes used to de-couple drivers from loads and/or add torsional damping to equipment trains, especially oneswith high steady state or transient vibratory torques, such as those containing variable speed or synchronous motorsor reciprocating compressors. This damping, which varies with the different elastomeric materials, reduces theamplification factor of a resonant frequency, typically by absorbing the torsional vibration energy. This energy isconverted to heat inside the elastomer.

Elastomeric block couplings are more complex to model as the torsional stiffness is nonlinear. The torsional stiffnessof an elastomeric coupling is a function of the transmitted torque. A typical elastomeric block coupling torsionalstiffness versus torque curve and elastomeric durometer is shown in Figure 4-17. It can be seen from Figure 4-17 thatthe coupling stiffness is dependent upon the elastomer used in the coupling. The coupling manufacturer can providethe stiffness at a given torque or provide an explanation of how to calculate the stiffness for the nonlinear torquedeflection plot. In addition, for transient conditions, the coupling vendor can recommend appropriate stiffness anddamping to be used for a transient analysis. Also, if the torque supplied by the driver has a significant variation, then

Figure 4-16—Coupling Spacer Torsional Stiffness Model

T

L

D

2

tf

tf2tf



the system may have to be analyzed with the minimum and maximum torque conditions to define a range offrequencies for a given mode.

There are some other considerations which limit the use of elastomeric block couplings. An elastomeric blockcoupling will weigh much more than a corresponding flexible element dry or gear type coupling. This will affectbalance and lateral rotor dynamics. Note that a hybrid coupling, where an elastomeric coupling is mated with a moreconventional style (see Figure 4-18) can alleviate this situation, by having the heavier elastomeric portion on the shaftmore able to handle the weight.

Figure 4-17—Typical Nonlinear Stiffness vs. Torque Characteristic for an Elastomeric Coupling

0

T A

ppca

ton

Torq

ue n

%

100

90

80

70

60

50

40

30

20

10

010 20 30 40 50

X Torsional Angle in %60 70 80 90 100

80°

70°

60°

50° R

ubbe

r blo

cks



Moreover, although elastomeric couplings are considered dry nonlubricated couplings, they are more maintenanceintensive than flexible element dry couplings. The elastomeric material torsional (and other) properties are not stableover long periods of time and are affected by exposure to light, chemical exposure (e.g. ozone), degradation due toheat from operating ambient temperature, heat from torsional vibration energy dissipation, and even time. Theelastomeric materials age harden, and have a shelf life of typically 5 years or less, where the durometer can increasemore than 10 %. These changes will affect the stiffness and damping properties. The material needs to be inspectedat least once a year, or more frequently if there have been operational problems and frequent start-ups. Where asynchronous motor is involved which does not have soft start technology, the start-up vibration magnitudes are high,and heating of the elastomeric material occurs each start-up. If the start-ups are severe enough, and a sufficient timehas not elapsed from one to another, the material can begin to melt and lose its damping properties, or worse, fail.

It should be noted that although elastomeric couplings are generally considered to provide excellent performance insituations, such as synchronous motor startups, where they only have to withstand occasional transient resonances,many users have encountered problems when these couplings have been subjected to continuous (i.e. steady-state)high-amplitude oscillating torque. These situations should be discussed with the coupling vendor to avoid problems.When properly applied, these type couplings will solve a torsional problem, and operate successfully for years at atime. Care should be taken to account for the oscillating torque when selecting, and the long-term effects of materialproperty changes.

4.1.7 Material Properties

The torsional modulus, G, of the shaft material(s) varies with temperature. Figure 4-19 displays the effect oftemperature on the shear modulus. The temperature dependence of the shear modulus should be considered intorsional analysis when large temperature changes in the shafting occurs between start-up and steady-stateoperation. Temperature-dependent shear modulus occurs in machinery such as steam and gas turbines, hot gasexpanders, and some fans. Experience indicates that temperature effects can result in a several percent difference incalculated undamped torsional natural frequencies. Reference [6] is one source for the temperature-dependentmodulus of metals.

Figure 4-18—An Elastomeric Hybrid Coupling (w/Disc Type)



4.1.8 Built-up Rotors

Accurate modeling of the following shaft geometry and mechanical fits (often associated with built-up shafts) isimportant for accurate torsional natural frequency calculation:

a) tapered (solid or hollow) shaft sections;

b) splined fits;

c) curvics or serrated couplings.

4.1.9 References

[1] Nestorides, E. A. BICERA Handbook on Torsional Vibration, London, Cambridge University Press, 1958.

[2] Steiner, S., “Revisiting Torsional Stiffness of Flexible Disc Couplings,” Proceedings of the 2007 Gas MachineryResearch Council Conference, GMRC, Dallas, TX, 2007.

Figure 4-19—Temperature-dependent Shear Modulus Curve

15

10

5

00

Structural alloys handbook(static shear modulus)

Aerospace structural metals handbook(ratio determined from dynamic tensile modulusPoisson's ratio = 0.290)

-400 400Temperature °F

800 1200

She

ar M

odu

us K

SI x

102



[3] ANSI/AGMA 9004-A99, Flexible Couplings-Mass Elastic Properties and Other Characteristics.

[4] Wilson, W.K., Practical Solution of Torsional Vibration Problems, Volume 1, Frequency Calculation, New York,New York, John Wiley & Sons Inc. 1956.

[5] Calistrat, M.M. and Leaseburge, G.G., “Torsional Stiffness of Interderence Fit Connections,” ASME Paper 72-PTG-37, 1972.

[6] ASME Unfired Pressure Vessel Code Section II.

4.2 Machinery Specific Modeling Considerations

4.2.1 Introduction

The material in this section identifies methods used to model unique types of machinery having torsionalcharacteristics that require specialized techniques.

4.2.2 Gearing

Single-branch gear systems can be analyzed using either the basic Transfer Matrix (Holzer) torsional method or finiteelement method. As displayed in Equation 4-1 and Equation 4-2, formulation of an equivalent single shaft modelrequires that all inertia and stiffness be referenced to the reference speed by the square of the gear ratio. In modelingthe shaft stiffness characteristics of gears (both integral and shrink-fit gear construction), it is necessary to considerpenetration of the shaft into the gear mesh. For shrink-fit gears, 1/3rd shaft penetration should be considered. Forintegral gears, Wilson [15] should be consulted for the appropriate stiffness dimensions. The rotational inertia of thegear and pinion are normally given on the drawings supplied by the manufacturer and are typically referenced to theirown respective speeds. The torsional model of the gear should include stiffness and inertia of the gear from the inputcoupling to the centerline of the gear; the model should also include the pinion’s stiffness and inertia from thecenterline to the coupling end of the pinion. The inertia of the shaft sections opposite the coupling extensions for thegear and pinion may be lumped at the centerline of the gear (see Figure 4-20 and Figure 4-21). Alternatively, thepinion and gear may be continuously modeled out to their free ends. Finally the pinion and gear mass-elastic modelsare coupled in the gearbox through the torsional stiffness generated by the gear mesh.

A unique feature of gears is the mesh. A typical gear mesh can be represented by a spring and a damper along thepressure line which is tangent to the base circles of the gears as shown in Figure 4-22. Ozguven and Houser [1]discuss mathematical models used in gear dynamics. The mesh stiffness and damping are dependent on the loading.There is a time varying characteristic to the mesh stiffness due to variation in the number of teeth in contact as thegears rotate, which changes the effective length of the line of contact. In the case of a helical gear mesh, the changein total length of line of contact is small due to large contact ratios. As a result, mesh stiffness variations aresignificantly smaller when compared with those of spur gears. Treating the mesh stiffness as a constant is, therefore,typical and sufficient for torsional analyses of API 613 gear sets.

Most analytical models are based on contact of a single tooth of the gear with a single tooth of the pinion as describedby Nestorides [2]. Gregory et al. [3] measured steel spur gears of standard 20 degree pressure angle and shape ashaving linear meshing stiffness of 1.4 x 1010 N/m/m face-width (2 x106 lbf/in./in. face width) [3,4]. Tooth stiffnessvalues in the range of 1.5 x106 to 3.0 x106 lbf/in./in. are discussed by AGMA [5]. The gear mesh torsional stiffness canbe calculated using Equation 4-4 if the reference speed equals the pinion rotation speed, or Equation 4-5 if the gearrotation speed is used as the reference speed:

KtMESH = (0.013625) Wf E (PDp)2 cos2() (4-4)

KtMESH = (0.013625) Wf E (PDg)2 cos2() (4-5)



where

KtMESH is the mesh torsional stiffness, N-m/rad (lbf-in./rad);

Wf is the face width of mating gears, m (in.);

PDp is the pitch diameter of pinion, m (in.);

PDg is the pitch diameter of bull gear, m (in.);

is the helix angle of gear set, deg.

E is the Young’s modulus, N/m2 (lbf/in.2).

For a helix angle of 0 degrees, Equation 4-4 and Equation 4-5 reduce to a mesh stiffness of 1,635,000 lbf/in./in. facewidth.

Since the design of high speed gearing usually results in numerous teeth with the base being wide relative to thedepth of the tooth, the resulting stiffness of the teeth is usually much higher than the torsional stiffness associated withthe shafting or couplings used in the power transmission lineup. As a result, the calculated natural frequencies areoften not influenced when the stiffness of gear teeth is introduced into the torsional model. Thus, for trains having

Figure 4-20—Cross-sectional View of a Parallel Shaft Speed Increaser

Gear mesh

Plane of gear mesh centers

Extended end

Pinion

Bull gear

Blind end

1

PD

pPD

g

Wf

4 5 6 7

32

Modeling a parallel single reduction gear (see Figure 2-13 for model schematic):— bull gear model extends from coupling hub to center of gear mesh;— pinion model extends from center of gear mesh to coupling hub;— lump rotor inertia outboard of plane of gear contact at the center of gear mesh;— account for bull gear mesh penetration and stiffening.

PDp pinion pitch diameter; PDg gear pitch diameter; PDf face width of mating gears. (All dimensions in mm or inches.)

NOTE 1

NOTE 2



only a single gear mesh, the analyst may usually make the mesh infinitely stiff with little loss of accuracy. The analystshould confirm the insensitivity of the model to relative mesh rigidity. However, for trains having multiple gear meshes,such as marine propulsion systems, gear tooth flexibility may be significant and should be accounted for in the model.

In some geared systems, there are occasions where the torsional motion couples with lateral motion of the gearshafts, usually the pinion shaft. There can also be interaction with axial vibration for those systems having a singlemesh helical gear mesh. For the majority of geared systems, the standard practice of evaluating the lateral andtorsional behavior separately is sufficient. Section 4.9 provides additional discussion on the subject of coupledtorsional-lateral analyses.

4.2.3 Electric Motors


A precise torsional model of electric motors must be developed for inclusion in the train model to accurately calculatethe train torsional natural frequencies. Some torsional natural frequencies of a system may actually represent thetorsional natural frequency of a subsystem or segment of the complete system model. For example, in motor-gear-compressor trains, the third torsional mode is almost exclusively governed by the stiffness and inertia characteristicsof the motor core such that the first natural frequency of the motor subsystem is almost exactly the same as a higher

Figure 4-21—Torsional Model of a Parallel Shaft Speed Increaser

PD

p

PDp = pinion pitch diameter; PDg = gear pitch diameter; Wf = face width of mating gears.

PD

g

Gear mesh

Extended end

Pinion

Plane of gear contact

Bull gear

1

4 5 6 7

32

Wf/2

Wf/2

The polor mass moments of inertia from removed sections are lumpedat the intersection of gear shaft centerlines and the plane of gear mesh centers.

NOTE 2

NOTE 1



order mode of the overall system. If the mode shapes for the motor segment and the higher order mode of the overallsystem are compared, they will appear nearly identical. Rotating motor exciters can also add an additionalindependent frequency to the torsional system. Accurate prediction of this mode or higher modes is greatlydependent on the number of torsional stations used in modeling the rotor.

It should be noted that, when queried for mass-elastic data, many motor suppliers will provide a simple one-stationmodel consisting of a single inertia and a single stiffness. This is inadequate for most torsional calculations. Instead,the analyst should insist that the motor supplier provide a detail drawing for the motor shaft so that the analyst canconstruct a model having sufficient detail.

For the purpose of torsional modeling, motors and generators can be divided into several groups:

— solid cylindrical shaft construction with laminations placed around the shaft;

— spider construction consisting of webs attached to the base shaft to support the motor core, with laminationssurrounding the spider and core;

— salient pole synchronous rotor construction consisting of central shaft and separate poles bolted or otherwisefixed to the shaft;

— solid synchronous turbo rotor construction where the axial slots are machined into the rotor and copper windingsare inserted into the slots;

— solid rotor technology applied to high-power, high-speed motors.

4.2.3.2 Stiffness and Inertia of Core Region

Electric machinery designed with a solid cylindrical shaft with laminated core construction can be difficult to accuratelymodel because the contribution of the shrunk-on core to the motor’s midspan shaft stiffness is difficult to analyticallypredict. The shear (torsional) stress paths in the area of the motor core are complex and highly dependent on the

Figure 4-22—Torsional Model of a Gear Mesh

KtMESH



exact magnitude of the radial interference fits. Tolerances in these fits may alter the depth of the effective base shaftpenetration into the motor core and may substantially change the torsional stiffness characteristics of the motor. Theeffect of axial contact between the core laminations as a result of axial preload and thermal effects may also requireconsideration in the torsional model. These effects can be represented in the torsional model as a stiffening effect ofthe rotor core.

Electric machinery designed with spider shaft construction requires the following modeling considerations.

a) The inertia of the rotating armature or poles should be distributed along the axial length of the core (minimumnumber of three elements should be used). Both the inertia of the rotating armature and the base shaft should beincorporated into the model.

b) The stiffening effect of the web arms must be added to the base shaft.

c) The stiffness of the armature is normally not considered for low frequency calculations, but can become importantfor higher modes internal to the motor.

Calculation of the torsional stiffness of noncircular cross-sections such as the webbed midspan area of an electricmotor (Figure 4-23) is a relatively complicated problem. Longitudinal webs or spider bars are often placed mid-span ofa motor shaft and are primarily used to support the windings or rotor laminations while allowing sufficient space forcooling air flow. When subject to a torque, the radial webs experience a loading configuration that includes bendingand torsion while the base shaft experiences pure torsion.

Various approaches have been used to account for the torsional stiffening effect of motor core webs for torsionalrotordynamics, including geometric approximate methods and finite element analysis (FEA) techniques. Nestorides[2] presents various methods to account for the increase in torsional stiffness due to radial webs rigidly attached toshafts, including a technique described as the Griffith and Taylor method [6,7]. This method requires dividing up thewebbed cross-section and performing various geometric calculations to arrive at an effective polar second moment ofarea. Nestorides [2] provides a detailed description of this method, including the sectioning techniques and the tabulardata necessary for the calculation.

Equation 4-6 provides an estimate of the torsional stiffness of a webbed shaft section based on the Griffith and Taylormethod. While the Griffith and Taylor method presented by Nestorides [2] requires several steps, the followingequation provides a single equation estimate of the torsional stiffness of a webbed shaft. This interpretation of theGriffith and Taylor method assumes that the rounding off of the outer webs is negligible and that the λ parameterdescribed by Nestorides has a value of unity. Note that this equation also assumes uniform, identical webs that arefully attached to the base shaft along the entire axial length. Additionally, this equation assumes the materialproperties, specifically the shear modulus, G, are identical for both the shaft and web materials. This stiffness equationdoes not account for any welding effects or additional stiffening effects of the laminations.

(4-6)KaGl---

32------d4 1

16------Nb2d2 N

2h2-------- bh

b2

2----–

3

+ +=



where

d is the diameter of base shaft, m (in.);

h is the height of radial web above base shaft, m (in.);

b is the web thickness, m (in.);

l is the axial length of webbed shaft, m (in.);

N is the number of radial webs;

G is the shear modulus of shaft and webs, N/m2 (psi);

Ka is the approximate webbed shaft torsional stiffness, N-m/rad (lbf-in./rad).

The Griffith and Taylor method has shown relatively good agreement with FEA results for various four-and six-webbed motor core configurations [8]. Figure 4-24 provides data from Reference [8] that compares the Griffith andTaylor method against FEA predictions for a six-webbed shaft with varying web thickness, b. Note that Figure 4-24presents a torsional stiffness ratio that represents the torsional stiffness of the webbed shaft, Ka, divided by thetorsional stiffness of the base shaft, Kb. The data show that the stiffening effect from the Griffith and Taylor methodagrees quite well with FEA for the provided webbed geometry.

Note that Equation 4-6 presented above is different than the torsional stiffening equation provided in API 684, 2ndEdition. The equation given in API 684, 2nd Edition has sometimes provided unreliable results [8].

The torsional stiffness of a solid circular cylinder is written as follows:

(4-7)

Figure 4-23—Typical Geometry of a Webbed Motor Rotor and Cross-Section Under the Windings

Web

Base shaft

N = Number of websb

h

d

KtGL---- D4

32---------- =



where

Kt is the torsional stiffness, N-m/rad (lbf-in./rad);

G is the torsional modulus, N/m2 (lbf/in.2);

L is the cylinder axial length, m (in.);

D is the cylinder diameter, m (in.).

The effective diameter of a cylinder that is equal in length and torsional stiffness to the noncircular shaft section canbe calculated as follows:

(4-8)

Figure 4-24—Torsional Stiffness Data for Six-Webbed Shaft with Varying Web Thickness [8]

FEA

Griffith-Taylor

N = 6, D/d = 2.5

d = 8 in. (203.2 mm)

3

2

2.5

1.5

0.5

1

00 0.05 0.1 0.15 0.2

b/d

Tors

ona

stf

fnes

s ra

to, K

a/Kb

0.25 0.3 0.35 0.4

Deff

32LKt

G---------------

0 25

=



where

Deff is the effective diameter, m (in.);

G is the torsional modulus, N/m2 (lbf/in.2);

L is the cylinder axial length, m (in.);

Kt is the torsional stiffness, N-m/rad (lbf-in./rad).

Deff is the diameter of a cylinder that generates the torsional stiffness of the noncircular shaft section. The ratio of theeffective cylinder diameter, Deff, and the base shaft diameter, d, is written as follows:

(4-9)

where

Deff is the effective diameter, m (in.);

d is the diameter of base shaft, m (in.);

Ka is the equivalent stiffness of noncircular cross-section, N-m/rad (lbf-in./rad);

Kb is the stiffness of base shaft, N-m/rad (lbf-in./rad).

The effective diameter of the equivalent cylindrical shaft section is typically about 6 % to 25 % above the base shaftdiameter, d, for some of the more common types of construction of multi-pole synchronous and induction machines.For further discussion see Reference [2].

4.2.3.3 Electromechanical Coupling in the Motor Air-Gap

4.2.3.3.1 General

Electromechanical coupling in the motor air-gap is normally not considered in a torsional model; however, in thosecases where it is considered to be required, techniques have been developed to account for these effects.

Electromagnetic fields in the air-gap of an electric machine create torque between the rotor and stator. In ideal steady-state operation, both the rotational speed and the torque of the rotor are constant. However, in the real system, therotor rotational speed can be constant, but there is always variation of electromagnetic torque due to thecircumferential periodicity of the electromagnetic system. This additional torque component behaves like an excitationtorque and will be considered in Section 4.5. Also, in the real system, the rotor rotational speed is not exactly constantbut oscillates around the uniform motion due to the torsional vibrations of the drive train. The electromagnetic systemreacts to these oscillations and resists the motion. This counteracting electromagnetic torque can be described by thefamiliar torsional stiffness and damping coefficients.

Figure 4-25 shows an example of the electromagnetic stiffness and damping for a 930 kW induction motor in therated operation condition [9]. The supply frequency is 60 Hz and the rotation speed 892 rpm (14.9 Hz). Simulatedresults obtained by finite element analysis. The stiffness and damping are dependent on the oscillation frequency. Inaddition, it is noteworthy that the electromagnetic damping coefficient is negative in the frequency range 49.7 to 59.4Hz. This range is located somewhat below the supply frequency of 60 Hz.

Deff

d--------

Ka

Kb

-----0 25

=



The stiffness and damping coefficients presented in Figure 4-25 are qualitatively representative, at least for inductionand synchronous machines. The electromagnetic parameters determine the absolute values and frequencydependence.

The magnitude of electromagnetic effects on the torsional dynamics is dependent on the mode shape and particularlythe vibration amplitude at the core region of the electric machine. In most cases, the electromechanical coupling isnegligible. However, when the load inertia is large compared to the motor inertia or in those cases where a flywheel ispresent, the effect might be significant. In the previous example, the 930 kW motor is part of a two-inertia system witha large load-to-motor inertia ratio of 1.75 to 1. This leads to large relative amplitude at the rotor core, and further, tosignificant effects of electromechanical coupling. The calculated effect increased the natural frequency of the lowestmode by 4 %.

4.2.3.3.2 Equivalent Circuit Model (ECM)

The electromagnetic torque can be described by a simple parametric model which is usually referred to as anEquivalent Circuit Model (ECM). Such models are used for steady-state and transient analysis. There are severalvariations of this model in use but it is common that the number of parameters is low [10].

4.2.4 Pumps

When formulating a torsional rotor model for a centrifugal pump with an incompressible working fluid, it becomesnecessary to distinguish between dry impeller inertia and wet impeller inertia. This difference is important because aliquid pump exhibits increased rotational inertia due to the fluid within the rotating passages of the pump impeller. Themagnitude of this fluid inertia is noted to be proportional to the specific gravity of the fluid being pumped, the ratio of

Figure 4-25—Torsional Stiffness and Damping Coefficients of a 930 kW Eight-pole Induction Motor with rated Operation Condition [9]

StiffnessDamping

0-100

100

200

300

400

500

0

20 40Frequency [Hz]

Stf

fnes

s [k

Nm

/rad]

Dam

png

[Nm

s/ra

d]

60 80 100



inner to outer diameter of the impeller, the vane angle, the number of vanes, and the frequency of vibration [11,12].This data are sometimes available from the pump manufacturer and can be used to calculate a frequency range forthe corresponding wet and dry torsional natural frequencies.

The primary torsional excitation that is unique to pumps is a vane-passing excitation at impeller vane-passingfrequency [13]. For instance, if the pump impeller contains three vanes, the excitation occurs at 3X. In multi-stagepumps, vane-passing excitations can become quite large if all stages are “timed” in a manner where their vanes allsimultaneously pass a stationary casing structure, such as the cutwater on a volute. Thus, it is prudent to stagger theimpellers in a multi-stage pump.

The liquid pump also provides a finite level of system damping due to viscosity effects of the fluid shearing betweenthe surfaces of the impeller blades and discs. Nordman et al. [12], performed torsional damping measurements andnoted wide variations over speed. Simplified estimations of torsional damping are sometimes used based on impellerpower and speed [11] such that:

(4-10)

where

Ceff is the effective damping, N-m-s/rad (lbf-in.-s/rad);

Trated is the rated torque, N-m (lbf-in.);

is the frequency of evaluation, rad/s;

rated is the rated speed, rad/s.

4.2.5 Hydraulic Variable Speed Drives

4.2.5.1 General

In addition to the well-known electrical variable frequency drives, which are discussed in Section 4.10, variable speedoperation can also be achieved by placing a hydraulic variable speed drive between the motor and the drivenequipment (such as a compressor or pump.) The most basic hydraulic variable speed drive is often called a hydrauliccoupling, and is very similar to an automobile torque converter. A basic hydraulic coupling consists of two wheels—apump and a turbine. The primary wheel (pump) is connected to the driver while the secondary wheel (turbine) isconnected to the load. Torque transfer is accomplished via the hydraulic fluid within the coupling—there is nomechanical connection between the driving and driven shafts. Although the driver speed is constant, the couplingsettings can be adjusted to vary the speed of the load. This adjustment is usually made by varying the fluid fill levelwithin the coupling. Peikert [14] provides more detail on the workings of hydraulic couplings.

The fluid drive influences two factors simultaneously, namely the torsional stiffness through the fluid drive and thespeed relationship between the driver, which operates at a fixed speed, and the variable speed of the fluid coupledrotor. The system natural frequencies become a function of the torsional stiffness of the fluid drive and the referredinertia between the driven equipment and the remainder of the system. There remains some debate as to whether thefluid coupling behaves in a manner similar to a linear or nonlinear elastomeric material. Therefore, the system can beanalyzed in two different ways.

Ceff 2 Trated

rated2

--------------=



4.2.5.2 Modeling Method 1—Low and High Speed Sides Decoupled

The first analysis method evaluates the input speed shaft system, normally consisting of a motor, low-speed coupling,gear, pinion and primary wheel of the fluid drive, as a separate system. The output shaft system, consisting of thesecondary wheel of the fluid drive and the drive output shaft, high-speed coupling, and connected machinery (such asa pump or compressor), is also analyzed as a separate system. Thus, in this method, the hydraulic coupling isassumed to behave in the same manner as in conventional lateral modeling—that is, it separates the train into twodistinct, independent systems.

4.2.5.3 Modeling Method 2—Low and High Speed Sides Coupled Through the Fluid Drive

The second analysis evaluates the entire train in a single model, including the torsional stiffness of the fluid drive. Thestiffness of the fluid drive is a variable that depends upon the transmitted power, output speed of the fluid drive andfrequency of vibration in the torsional system. Depending upon the number of defined operating conditions, there maybe a number of fluid drive stiffness values that have to be evaluated.

These stiffness values can often be obtained from the manufacturer of the fluid drive. Some fluid drive manufacturershave stated that to obtain a proper solution for the torsional frequency analysis, the stiffness to be used must bechosen for the unique conditions of transmitted power, secondary wheel speed and calculated torsional naturalfrequency of the system for a specific mode. The fluid drive is then modeled as a gear having a ratio through the fluiddrive appropriate to the pinion speed and the secondary wheel output speed of the fluid drive.

Initially, this seems like an enormous task as there could be three or four natural frequencies for a typical compressorsystem operating below 300 Hz, and several defined cases of customer operating conditions. Fortunately, the actualanalysis is not as difficult as imagined.

Further, it is interesting to compare the results for a typical compressor system where the two methods were used.The 1st mode of the coupled system became a new mode with a frequency lower than any found in the separatesystem analysis. The 2nd mode of the coupled system appears identical to the 1st mode of the input system used inmethod one. The 3rd mode of the coupled system appears identical to the 1st mode of the output system used inmethod one. Thus, it appears that the coupled system introduces a much lower frequency mode than either of theseparate input or output system models alone. The higher order modes of the coupled system then become individualmodes of the input and output systems that were analyzed.

Although modeling method 2 is more complex than method 1, it is usually not much more accurate. This is becausethe effective torsional stiffness of the hydraulic drive is so low that the new first mode that it introduces to the systemis too low to be excited by any practical excitation mechanisms. Thus, modeling method 1, which is the same asmethod 2 with a zero stiffness assumed for the coupling, is usually accurate enough to be employed. Among thosethat have come to this conclusion include Nestorides [2], ker Wilson [15], and Tuplin [16].

4.2.5.4 Hydrodynamic Variable Speed Planetary Gear Drive (HVSP)

A much more advanced form of fluid drive is the HVSP gearbox, which is shown in Figure 4-26, [17]. This drivewarrants discussion because it is being encountered on a fairly frequent basis in motor-driven compressorapplications and questions abound on how to account for its presence in a torsional vibration analysis.

An example of HVSP’s operation, which is quite complex, is described in detail in [17]. In short, the HVSP contains ahydraulic torque converter and two planetary gears, a stationary one and a revolving one. The input shaft, which isdriven directly by the motor, has its power split into two paths—the primary path (which goes directly to the ring gearof the revolving planetary) and the secondary path (which consists of the torque converter, stationary planetary gear,and carrier of the revolving planetary). The control system that is built into the HVSP controls the output speed (whichis also the driven equipment speed) via control of the hydraulic torque converter. The advantage of using the HVSP isit permits the driven equipment speed to vary significantly even though the motor speed is held constant.



The fact that the revolving planetary gear at the output of the HVSP is a two input, one output device introduces asignificant amount of complexity from a torsional analysis standpoint. That is because the large majority of torsionalanalysis codes assume that the torsional parameters on shafts rotating at different speeds can be converted using theconventional speed squared equations. However, if this were true for the HVSP, that would mean the torsional naturalfrequencies would have to vary with compressor speed. After all, with a fixed speed motor, the speed ratio when thecompressor is running at 10,000 rpm is significantly different than when it is running at 15,000 rpm. Thus, if aconventional torsional analysis code is used to analyze a train containing an HVSP, it would predict that the naturalfrequencies vary with compressor speed. Unfortunately, this is not accurate. Numerous tests and rigorousmathematical derivations have conclusively proven that the natural frequencies are constant, independent ofcompressor speed and speed ratio.

The chief ramification of this is that conventional torsional analysis codes cannot be used to analyze a train containingan HVSP. The presence of the two input, one output rotating planetary gear is the reason for this. While conventionaltorsional codes are perfectly capable of analyzing the large majority of planetary gears in use, that is because inalmost all planetary gears, one of the three primary components (sun gear, carrier, and ring gear) is not allowed torotate, thereby, yielding a fixed speed ratio. In the HVSP’s revolving planetary gear, all three components are rotating,thereby, yielding a variable speed ratio. This variable ratio is what invalidates conventional torsional codes.

Thus, in order to correctly analyze a train containing an HVSP, a special torsional analysis code, which accounts forthe unique characteristics of the HVSP, must be developed. Several organizations have successfully developed suchcodes. A user that does not have access to such a code should contact one of those organizations for assistance.

4.2.6 Screw Compressors

4.2.6.1 General

The principle of compression in a rotary screw compressor requires that the male and female rotor lobes mesh in veryclose proximity to one another but not physically touch each other. Refer to Figure 4-27 for a typical screwcompressor rotor set. As shown in Figure 4-27, the male and female rotors are constructed with multiple helical flutes

Figure 4-26—HVSP Schematic [17]



cut into a cylindrical rotor. The resulting surface is a series of protruding lobes about a base cylinder for the male rotorand valley shaped recesses on the base cylinder surface of the female rotor.

The screw compressor is a positive displacement device in which the male rotor drives the female. Rotary screwcompressors come in two basic designs:

— dry rotary screw compressor, which utilizes a timing gear;

— flooded, which does not use a timing gear.

In the dry rotary screw compressor, no lubricant is used. This style of compressor utilizes a set of timing gearsmounted on the rotor stub ends to position the male rotor relative to the female rotor to prevent rotor-to-rotor contact.These gears provide a speed ratio the same as the lobe ratio between the male and female rotor.

In the flooded screw compressor, a lubricant is intentionally introduced into the inlet of the compressor. The floodedscrew compressor does not use timing gears. The male and female rotors are timed relative to one another byutilizing the lobes as pseudo gear elements with the lubricant within the compressed gas preventing metal-to-metalcontact of the male and female rotors.

In the design most commonly encountered, the male rotor has four lobes and the female rotor has six. Regardless ofwhich drive method is employed, the relative speeds of the two rotors are always inversely proportional to the numberof lobes each contains. Thus, for the most common design, the female rotor rotates at a speed that is 2/3 of the malerotor’s speed. The main torsional excitation in screw compressors occurs at lobe-passing frequency. This is simplythe number of lobes on the male (or female) rotor multiplied by that rotor’s speed.

4.2.6.2 Dry Screw Compressor Rotor Body Modeling

Torsional modeling of a dry screw compressor is fairly straightforward. The connection of the male and female rotorsof a dry screw compressor, which utilizes timing gears, can be torsionally modeled similar to a geared connectionbetween low and high speed shafts. Refer to Figure 4-28 for a typical system model of a dry screw compressor train.The rotor model must account for the torsional stiffness and inertia of the base cylinder plus the influence of the rotor

Figure 4-27—View of Typical Screw Compressor Rotor Pair



lobes. A pseudo diameter is typically used to model the effective torsional stiffness of the lobed center section. Thelobed male and female rotors would each be modeled with individual pseudo diameters. Since a pseudo diametermay not truly represent the correct polar moment of inertia of a rotor, additional lumped inertia may need to bedistributed along the length of the lobed center body. Some computer programs allow this to be accounted for byenabling the user to identify two diameters: one that reflects the torsional stiffness and another which represents thepolar moment of inertia.

4.2.6.3 Wet Screw Compressor Rotor Body Modeling

In the case of wet screw compressors, the male and female rotors are modeled in the same manner as the dry screwcompressor, with some notable additions. The connection of the male and female rotors for a flooded screwcompressor is more unique since no timing gears are used. In most cases, there exists high damping associated witha flooded screw compressor, and this may preclude the need for a torsional analysis. If a flooded screw compressorsystem is analyzed, the coupled screw compressor rotor is modeled as a conventional screw compressor rotor.However, since the lobes of the coupled rotor act as the gears for the driven rotor, the speed referred inertia of thedriven rotor must be distributed along the coupled rotor as discs uniformly distributed along the length of the coupledrotor. Refer to Figure 4-29 for a typical system model of a flooded screw compressor train. The mating stiffness of themesh between male and female lobes is normally ignored since it is both very difficult to model and considered tohave little effect on torsional accuracy.

4.2.7 Integrally Geared Compressors

Integrally geared compressors typically consist of a motor driving a bull gear, which, in turn, drives multiple pinions,each having one or two impellers, and each pinion having a gear ratio relative to the bull gear. Examples of integrallygeared compressors are shown in Figure 4-30 and Figure 4-31. This type of system must be treated as a branchedtorsional system. Several methods are available for analyzing branched torsional systems including simplified systemanalysis, Holzer’s method, matrix-eigenvalue method, Rayleigh-Ritz method, transfer matrix, and other techniques[2,18]. An example of mode shapes for this type of compressor is included in Figure 4-32. The first torsional modeshape has the torsional deflection taking place primarily in the flexible coupling between the motor and bull gear, withthe pinions behaving essentially as rigid body inertias. The second torsional mode shape has the torsional deflectiontaking place primarily between the low-speed pinion mesh and one of the impellers.

Figure 4-28—Typical System Model of a Dry Screw Compressor Train

Compressor Female Rotor

Compressor Male Rotor Coupling Motor7654

1 2 3

8




A steam turbine rotor is normally modeled as a flexible shaft without considering blade flexibility. Turbine rotordesigners typically assume rigid blade-disc systems, whereas blade designers assume a fixed boss. This type ofmodeling is sufficient for most torsional systems especially when considering the types of turbines used in thepetroleum, chemical, and gas industry services where the blade natural frequencies are typically much higher thanthe torsional excitation sources.

However, such assumptions can be inadequate in those cases where the first tangential eigenfrequency of theindependent blade systems is less than or near an excitation frequency such as twice the line frequency of a turbine-generator string, thereby, requiring a coupled blade-disc-rotor vibration analysis [19]. These cases are typicallyrestricted to strings of equipment having turbines with long last stage blades such as large turbine-generator sets aswould be found in a central power station. In such trains, the only component typically requiring the coupled model isthe low pressure turbine. ISO sites some major incidents due to modes of the coupled shaft and blade system thatwere resonant with the grid excitation frequencies that occurred in the 1970s [20]. In such cases, the blades may betreated as individual branch elements. Other techniques are also available [21].

4.2.9 References

[1] Özguven H. N. and Houser D. R., “Mathematical Models used in Gear Dynamics—A Review,” Journal ofSound and Vibration, Volume 121, Number 3, pp 383-411 (1988).

[2] Nestorides, E. J., A Handbook on Torsional Vibration, British Internal Combustion Engine ResearchAssociation (1958).

[3] Gregory, R. W., Harris, S. L., and Munro, R. G., “Dynamic Behaviors of Spur Gears,” Proceedings of theInstitution of Mechanical Engineers, Vol. 178, Part I, pp. 207-226 (1963-64).

[4] Smith, J. D., Gears and Their Vibration: A Basic Approach to Understanding Gear Noise, Marcel Dekker,(1983).

Figure 4-29—Typical System Model of a Flooded Compressor Train

Compressor Female Rotor

Compressor Male Rotor Coupling Motor6543

1 2

7



Figure 4-30—API 672 Integrally Geared Compressor



Figure 4-31—API 672 Integrally Geared Compressor

Figure 4-32—Typical Mode Shapes for an Integrally Geared Compressor

16981.4 cpm

1415.5 cpm

Torsion Mode Shape Plot

Torsion Mode Shape Plot

Axial Location, in.

Axial Location, in.



[5] AGMA Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth.

[6] Griffith, A. A., and Taylor, G. I., Rep. Memor. Adv. Comm. Aero., Lond., nos. 333, 334, and 392 (1917).

[7] Trayer, G. W., and March, H. W., N.A.C.A. Report no. 334 (1944).

[8] Kulhanek, C. D., James S. M., and Hollingsworth, J. R., “Stiffening Effect of Motor Core Webs for TorsionalRotordynamics,” Proceedings of ASME Turbo Expo 2012, Paper GT2012-69967, Copenhagen, Denmark(June 2012).

[9] Holopainen, T. P., Repo, A.-K. and Järvinen, J., “Electromechanical Interaction in Torsional Vibrations of DriveTrain Systems Including an Electrical Machine,” Proceedings of the 8th IFToMM International Conference onRotordynamics, Seoul, Korea, 8 p (September 2010).

[10] Toliyat, H. A. & Kliman, G. B., Handbook of Electric Motors, CRC Press, Boca Raton, Florida (2004).

[11] Imaichi, K., Tsujimoto, Y., and Yoshida, Y., “An Analysis of Unsteady Torque on a Two-Dimensional RadialImpeller,” ASME Journal of Fluids Engineering, Volume 104, pp 228–234 (1982).

[12] Nordman, R., Weiser, P., Frei, A., and Stuerchler, R., “Torsional Vibration in Pump/Driver Shaft Trains: TheRole of External Damping from Pump Impellers,” Proceedings of the Thirteenth International Pump UsersSymposium, Turbomachinery Laboratory, Texas A&M University, College Station, Texas, pp. 61–70 (1996).

[13] Corbo, M. A. and Malanoski, S. B., “Practical Design Against Torsional Vibration,” Proceedings of the Twenty-Fifth Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, TX,pp. 189–222 (September 1996).

[14] Peikert, G. H., “Variable Speed Fluid Couplings Driving Centrifugal Compressors and other CentrifugalMachinery,” Proceedings of the Thirteenth Turbomachinery Symposium, Turbomachinery Laboratory, TexasA&M University, College Station, TX, pp. 59–65 (1984).

[15] Wilson, W. K., Practical Solution of Torsional Vibration Problems, 1, New York, New York: John Wiley & SonsInc. (1956).

[16] Tuplin, W. A., Torsional Vibration—Elementary Theory and Design Calculations, John Wiley & Sons, NewYork, NY (1934).

[17] Wahl, G., “50 MW Power Transmission for variable speed with Variable Planetary Gear,” Voith Turbo,Germany, (2000).

[18] Sankar, S., “On the Torsional Vibration of Branched Systems Using Extended Transfer Matrix Method,” ASMEJournal of Mechanical Design, Volume 101, pp. 546–553 (1979).

[19] Okabe, A., Otawara, Y., Kaneko, R., Matsushita, O., and Namura, K., “An Equivalent Reduced ModelingMethod and Its Application to Shaft-Blade Coupled Torsional Vibration Analysis of a Turbine-Generator Set”,IMechE Journal of Power and Energy, Volume 205, pp. 173–181 (1991).

[20] Mechanical vibration—Torsional vibration of rotating machinery—Part 1: Land-based steam and gas turbinegenerator sets in excess of 50 MW

[21] Al-Bedoor, B. O., “Reduced-order Nonlinear Dynamic Model of Coupled Shaft-Torsional and Blade-BendingVibrations in Rotors,” ASME Journal of Engineering for Gas Turbines and Power, Volume 123, pp. 82–88(2001).



4.3 Reciprocating Machinery

4.3.1 Scope

This section departs from the general format of the entire document pertaining to torsional vibration in that it does notseparate the different aspects of modeling a reciprocating machine and excitation from reciprocating machinery intodifferent sections. The task force was of the opinion that the subject of torsional vibrations of reciprocating machinerywould be better presented as a separate subject complete onto itself. However even though reciprocating machinerycreates torsional excitation and the modeling of reciprocating machinery is unique to itself, there can be othertorsional aspects which are presented in the rest of the document such as transient vibration of synchronous motor-driven reciprocating compressors or electric variable frequency drives (VFD) or torsional excitation associated withmotor fault conditions that must also be considered in addition to the potential nonuniform torque associated withreciprocating machinery.

4.3.2 Modeling of Reciprocating Machinery Modeling of Reciprocating Machinery

4.3.2.1 General

For more complicated geometries such as crankshafts, the following procedure can be used if the mass-elastic modelis not provided by the manufacturer. A crankshaft can be simplified into several main components: the stub shaft thatconnects to a coupling or flywheel, journals where the main bearings are located, webs, and crankpins. A mass-elastic model of the crankshaft is typically created by lumping the rotating and effective reciprocating inertia at eachthrow and calculating the equivalent torsional stiffness between throws. Additional mass stations are created for theflywheel, oil pump, etc., as necessary.

Figure 4-33 shows a basic crankshaft throw. A throw consists of two webs and a crankpin. Depending on the type ofcrankshaft, there may be one or two throws between journals. The crankpin usually drives a connecting rod,crosshead (for compressors) and a piston and piston rod. Engines with power cylinders in a “V” arrangement mayhave two connecting rods at each crankpin, or use an articulated rod design. Integral engine/compressor units canhave two power cylinders articulated off the main connecting rod for the compressor cylinder for a total of threeconnecting rods per throw.

4.3.2.2 Modeling Crankshaft Torsional Stiffness

As is described by Feese and Hill [1], the determination of the effective torsional stiffness between crankshaft throwsis not a simple task. Equations are given in Nestorides [2] and Ker Wilson [3] for calculating the torsional stiffness of acrankshaft. The basic dimensions of the journals, webs, and crankpins are needed, as well as the shear modulus ofthe shaft material. BICERA also developed curves based on test data for various types of crankshafts. These can bemore accurate, but also more complex and are not discussed here.

Carter’s Formula [1]

(4-11)

Ker Wilson’s Formula [3]

(4-12)

KtG

32Lj 0.8Lw+

Dj4 dj

4–------------------------

0.75Lc

Dc4 dc

4–----------------- 1.5R

LwW3-------------+ +

---------------------------------------------------------------------------------=

KtG

32Lj 0.4Dj+

Dj4 dj

4–------------------------

Lc 0.4Dc+

Dc4 dc

4–--------------------------

R 0.2 Dj Dc+ –

LwW3----------------------------------------+ +

--------------------------------------------------------------------------------------------------------------------=



where

dc is the crankpin inside diameter, m (in.);

Dc is the crankpin outside diameter, m (in.);

dj is the journal inside diameter, m (in.);

Dj is the journal outside diameter, m (in.);

G is the shear modulus, N/m2 (lbf-in.2);

Kt is the torsional stiffness, N-m/rad (lbf-in./rad);

Lc is the crankpin length, m (in.);

Lj is the crankshaft journal length, m (in.);

Lw is the web thickness, m (in.);

R is the throw radius, m (in.);

W is the web width, m (in.).

Carter’s formula is applicable to crankshafts with flexible webs and stiff journals and crankpins, while Ker Wilson’sformula is better for stiff webs with flexible journals and crankpins. When conducting a torsional analysis, Ker Wilsonhas suggested using the average of his and Carter’s formulas to determine the stiffness between throws. To calculatethe torsional stiffness of the stub shaft to the centerline of the first throw, the torsional stiffness of the straight shaft

Figure 4-33—Portion of a Typical Crankshaft Throw

1/2Lj

Crankpin

Journal

JournalWeb

R

W

dj

Dj

DC

dC

1/2Lj

LWLW LC



section can be combined in series with twice the torsional stiffness between throws. For coupling hubs or flywheelswith an interference fit, the 1/3 rule should also be applied.

Equation 4-11 and Equation 4-12 were developed before finite element analysis (FEA) was readily available. A finiteelement program can be used to determine the torsional stiffness for a crankshaft section. The simple models shownin Figure 4-34 were developed from the basic dimensions for two different crankshafts and do not include fillet radiiand oil holes. These models are from journal centers, which is the same as the distance between throw centers. Oneend was rigidly fixed and a moment was uniformly applied across the other end. The calculated torsional stiffness isequal to the moment divided by the angle of twist at the free end. It is interesting to note that for both crankshafts, thecalculated torsional stiffness using FEA fell between the values from the Carter and Ker Wilson formulas.

4.3.2.3 Polar Mass Moment of Inertia

The polar mass moment of inertia (commonly referred to as WR2) at each throw depends on the rotating inertia andthe reciprocating mass. The rotating inertia is constant, but the effective inertia of the reciprocating parts actuallyvaries during each crankshaft rotation. This effect is considered to be negligible for most engines except in the case oflarge slow-speed marine applications [4] when calculating the torsional natural frequencies. As is discussed inSection 4.3.3.2, the reciprocating masses also produce dynamic torque that must be included in the applied torque-effort for forced response calculations.

The equivalent inertia, Ieqv, at each throw can be approximated by adding the rotating inertia of the crankshaft section,Irot, to half of the reciprocating mass, Mrecip, times the throw radius, R, squared.

(4-13)

Figure 4-34—Finite Element Models Used to Calculate Torsional Stiffness of Crankshaft Sections

Carter’s Formula = 69.7 x 106 N-m/rad (617 x 106 lbf-in./rad)Ker Wilson’s Formula = 61.46 x 106 N-m/rad (544 x 106 lbf-in./rad)ANSYS Results = 63.72 x 106 N-m/rad (564 x 106 lbf-in./rad)

Lj = 0.1461 m (5.75 in.)Lc = 0.273 m (10.75 in.)Lw = 0.102 m (4 in.)R = 0.216 m (8.5 in.)

Dj = 0.33 m (13 in.)Dc = 0.254 m (10 in.)W = 0.42 m (16.5 in.)G = 79.29 x 106 N/m2

(11.5 x 106 psi in.)

Carter’s Formula = 154.68 x 106 N-m/rad (1369 x 106 bf-in./rad)Ker Wilson’s Formula = 121.35 x 106 N-m/rad (1074 x 106 bf-in./rad)ANSYS Results = 124.96 x 106 N-m/rad (1106 x 106 lbf-in./rad)

Lj = 0.168 m (6.625 in.)Lc = 0.178 m (7 in.)Lw = 0.132 m (5.1875 in.)R = 0.267 m (10.5 in.)

Dj = 0.33 m (13 in.)Dc = 0.305 m (12 in.)W = 0.635 m (25 in.)G = 79.29 x 106 N/m2

(11.5 x 106 psi in.)

Ieqv Irot 0.5 MrecipR2+



where

Ieqv is the equivalent inertia, kg-m2 (lbm-in.2);

Irot is the rotational inertia of crankshaft throw, kg-m2 (lbm-in.2);

Mrecip is the reciprocating mass, kg (lbm);

R is the throw radius, m (in.).

The rotational inertia of the journal and crankpin can be calculated using the equation for a cylinder. Since thecrankpin rotates at the throw radius and not about its center, the parallel axis theorem must also be used. The inertiaof the webs can be estimated with an equation for a rectangular prism. Any rotating counter-weights that may bebolted to a web should also be included. There are solid model 3D CAD packages that can perform this calculation.

Since part of the connecting rod is rotating and part is reciprocating, it is common practice to divide the rod’s totalmass into two portions—a rotating portion and a reciprocating portion. The rotating portion can be calculated usingEquation 4-14. The connecting rod is generally heavier at the crankpin end and lighter at the reciprocating end. If theweight distribution of the connecting rod is unknown, a typical assumption is to use two-thirds of the weight as rotatingand one-third as reciprocating. The rotating mass of the connecting rod is multiplied by the throw radius squared andadded to the crankshaft rotating inertia to obtain the total rotational inertia, Irot. The total reciprocating mass includesthe small end of the connecting rod, cross-head (for compressors), nut, piston, and piston rod. This total mass needsto be used in Equation 4-13 when calculating the throw inertia.

Mrot = [(Lrod – Lcg)/Lrod] x Mrod (4-14)

where

Mrot is the mass of rotating portion of connecting rod, kg (lbm);

Lrod is the total length of connecting rod, m (in.);

Lcg is the distance from crank pin center to rod CG, m (in.);

Mrod is the total mass of connecting rod, kg (lbm).

4.3.3 Torsional Excitations Generated by Reciprocating Machinery

4.3.3.1 General

Reciprocating compressors and engines produce unsteady torque. This torque variation can be much higher than inrotating equipment and flywheels are often used to smooth the torque. The frequencies of the torque excitation shouldbe considered to avoid coincidence with torsional natural frequencies, which could potentially cause problems.

4.3.3.2 Torque Variation Due to Inertial and Gas Forces

From a torsional standpoint, there are two types of forces that cause torque variation at each throw: inertial and gasload. The total force times the distance between the crankshaft centerline and throw centerline is equal to the momentimposed on the crankshaft. At top dead center (TDC) and bottom dead center (BDC), the throw is inline with theconnecting rod and piston so that no moment can be imposed on the crankshaft. At 90 degrees from BDC and TDC,the moment arm is at the maximum length (full crank radius).



The rotating inertia of the crankshaft must be considered in the mass-elastic model, but does not cause any torquevariation. The inertial forces are caused by the reciprocating mass of the connecting rod, cross-head and piston,which are dependent on the crank angular position and cannot be eliminated by balancing. The unbalance forceshave components which vary once per revolution (primary forces), twice per revolution (secondary forces), and threetimes per revolution (tertiary forces). These inertia forces will vary with the speed squared. Per Den Hartog [5],Equation 4-15 can be used to calculate the 1X, 2X, and 3X components of the inertia excitations. It should be notedthat positive torques represent torques being applied to the crankshaft while negative values mean torque is beingtaken from the crankshaft. These signs are important and must be considered when combining inertia excitations withgas load excitations.

(4-15)

where

inertia is the inertia torque acting on crankshaft, N-m (ft-lbf);

ωt is the crank angle (from top dead center), radians;

mrecip is the total mass of reciprocating parts, kg (slugs);

R is the crank radius, m (ft);

L is the connecting rod length, m (ft);

ω is the rotational speed, rad/sec.

Gas load excitations arise because as a reciprocating compressor or engine goes through its performance cycle, thepressure within its cylinders varies periodically, as a function of crank angle. The gas force is equal to the differentialpressure across the piston times the cross-sectional area of the bore. The stroke, or travel of the piston, is equal totwice the crank throw radius. The swept volume for each cylinder is the bore area times the stroke. In order todetermine the gas load excitations, the pressure versus crank angle must be determined for each cylinder over 360degrees for compressor or two-stroke engine and 720 degrees for a four-stroke engine. The torque can then bedetermined versus crank angle by simply multiplying the pressure by the bore area and the crank radius. Anydistortion in the pressure waveform will affect the dynamic torque and torsional response. A third type of curve that isoften seen is called tangential effort (or tangential pressure), which is the torque normalized for unity crank radius andpiston area.

Once the inertia and gas forces have been determined, they must be correctly added together for each cylinder. Thetorque at each throw must then be properly phased for the entire machine. A Fourier analysis can then be performedon these curves to represent the complex wave as a series of sinusoidal curves at various harmonics. At eachharmonic, the amplitude and phase can be calculated, or the values can be presented as coefficients of sine andcosine functions. Compressors and two-stroke engines will have integer harmonics of running speed while four-stroke engines produce both integer and half orders. Depending on the cylinder phasing, certain orders may cancelout while others become dominant when examining the overall torque output from the machine.

Compressor and engine manufacturers will often provide this information in various forms with the performancecalculations. To use their data in a torsional analysis, it is very important to understand the sign convention and if thevalues are only for gas forces or if the effect of reciprocating mass has also been included. Computer programs areused to calculate the torsional excitation for compressors and engines.

inertia12-- mrecip 2 R2 R

2L------ t sin 2t sin– 3R

2L------ 3t sin–=



4.3.3.3 Compressors

Ideal pressure cards are often used for analysis since they can be computed from the compressor and gasproperties. However, ideal cards do not include valve/manifold losses and gas pulsation, which can affect theresulting torque harmonics. Situations where the harmonic content is changed and the torsional excitation by thecompressor could be increased are: valve failure, gas pulsation due to an acoustic resonance, and various loadsteps.

Single-acting (SA) compressors use only one side of the piston while double-acting (DA) cylinders use both the crankand head ends. A compressor valve failure can be analyzed by unloading one end of a cylinder. All load steps mustbe considered in the analysis, such as unloaders, pockets, etc., as these load steps can significantly affect theharmonic content and influence the torsional responses. The maximum horsepower case will not necessarilycorrespond to the maximum torsional excitation at all harmonics. The full range of operating conditions (pressures,flows, gas mole weights, etc.) should also be considered.

The torsional behavior of reciprocating compressors has been found to be highly sensitive to normal tolerancevariations. That is, a compressor that can be shown to be acceptable via analysis of the nominal system could fail ifcertain tolerance conditions were to occur. For that reason, it is recommended that tolerance variations be accountedfor when analyzing a reciprocating compressor. References [6–8] give more detail on how to implement this.

4.3.3.4 Engines

For a two-stroke engine, intake, compression, expansion, and exhaust occur during one revolution of the crankshaft.However, with four-stroke engines, these cycles occur over two revolutions, which causes half-order excitations.Some engines have a choice of firing orders, which can change the strong harmonics. In critical systems, the bestfiring order could be chosen to reduce the torsional response.

Poorly maintained engines will tend to operate at nonideal conditions that can cause high torsional vibration such as:engine misfire, pressure imbalance, ignition problems, and leaks. Misfire is common when the fuel is inconsistent,such as biogas from waste treatment or landfills.

A misfire condition should be analyzed by assuming one cylinder does not fire. Since the response can varysubstantially depending on which cylinder misfires, the worst case should be assumed. In many practical systems,the misfire case is the sternest test of the design. Thus, the misfire case should be evaluated in all torsional analysesinvolving engines.

4.3.4 Steady-state Response Analysis

The steady-state response analysis for reciprocating trains is fairly similar to that for purely rotary trains, which isdescribed in Section 4.4. However, there are several significant differences.

First, in a rotary train, there are common situations where response analysis is not even required. Specifically, if thereare no steady-state interference points in or near the operating speed range, response analysis can usually beomitted. On the other hand, response analysis almost always needs to be performed for a reciprocating train.

Second, in a rotary train, response analysis is only performed at a few specific speeds—the resonant speeds.Conversely, since there are so many active excitations in a reciprocating train, limiting the analysis to a few governingspeeds is often impossible. Thus, most reciprocating trains are analyzed using a “speed sweep” response analysis,as is shown in Figure 4-35.

Third, in a rotary train, there is only one response of interest—that due to the resonant excitation. On the other hand,in a reciprocating train, all excitations need to be included in the response analysis. Additionally, as is shown in Figure4-35, the response obtained by summing the responses due to all the individual excitations, with phasing accountedfor (which is shown by the solid line in the figure), is usually significantly larger than that due to any single excitation



(represented by the dashed lines), even one at resonance. Thus, the overall summed response is the one that shouldbe compared to the acceptability criteria.

Finally, some of the acceptability criteria are different from those normally used for purely rotary trains. In general, theitems that need to be checked for a reciprocating train are as follows:

1) All shafts must have adequate fatigue life. This is similar to rotary systems (and, thus, can be evaluated usingthe procedure given in Section 4.6) except stresses in the crankshaft must also be looked at. This is discussedfurther in Section 4.3.5.

2) Torques in couplings must be below manufacturers’ limits. This is no different from rotary systems.

3) Motors having spider construction must be checked for their ability to handle the imposed cyclic torques. Inreciprocating compressor trains, spider rotors in motors have often been observed to be the weak link. Thus, thespider’s ability to handle the calculated cyclic torques needs to be verified with the motor supplier.

4) Cyclic torques at gear meshes must be below applicable limits (usually specified as a percentage of the torquetransmitted through the mesh).

5) The displacements and accelerations at the free ends of both reciprocating engines and compressors must bebelow applicable limits. There are often auxiliary components (which aren’t included in the torsional model)attached to the free ends of engines or compressors which can break if there is too much motion at the free end.

Figure 4-35—Typical Reciprocating Train Response Plot

4000

1

1Total

2

2

3

3

4

Vbr

ator

y To

rque

Ap

tude

45

5

6

6

7

789101112

X 104

600 800 1000Speed (RPM)

(31.2e6), Element 10

1200 1400 1600 1800



6) The heat dissipation in all elastomeric couplings and viscous dampers must be below the manufacturer’s rating.These components are discussed further in Section 4.3.6 and Section 4.3.7.

7) The fluctuations in motor current must be below applicable limits. In reciprocating compressor trains, if themotor current varies too much, it can cause problems in the electrical system.

Stephens [9] and [10] and Harris and Smalley [11] discuss many of these items in much more detail.

4.3.5 Crankshaft Stress Concentration Factors

The procedure for evaluating shaft stresses in reciprocating equipment is essentially the same as for rotaryequipment, and is given in detail in Section 4.6. The one item that is different when evaluating reciprocatingequipment is the need to also evaluate the stresses in crankshafts. Stress concentration factors (SCFs) are used toaccount for stress risers caused by geometric discontinuities such as shaft steps, keyways, welds, shrink fits, etc. Agood reference for determining theoretical stress concentration factors is Peterson [12]. Note that the SCF is differentfor shear and bending stress, so care should be taken to use the proper value.

For stepped shafts and keyways, the fillet radius and shaft diameter must be known to determine the SCF. The worst-case SCF occurs with a square cut and would be approximately 5.0 for a keyed shaft and 2.0 for a stepped shaft. If akeyway is required, use a fillet radius of at least 2 % of the shaft diameter as described in USAS B17.1, Keys andKeyseats [13]. This will limit the SCF at the base of the keyway to 3.0.

When possible, keyways should be avoided, particularly at coupling hubs, since the shaft diameter in this area is oftenreduced. Instead, an interference fit (heat shrink or hydraulic) should be considered. However, the shrink must besufficient to prevent slippage and galling of the shaft.

Specific SCF details for reciprocating equipment can be found in Nestorides [2], Ker Wilson [3], and Lloyd’s Registry[14]. The typical SCF for a crankshaft ranges from 1.5 to 2.0 in torsion (excluding any keyways). Therefore, use of astress concentration factor of 2.0 is reasonable for crankshafts when the fillet radii are unavailable and no keywaysare present.

When performing a torsional analysis, the fatigue SCF, kF, should be used to calculate the alternating shear stress ina shaft section. The fatigue SCF is usually less than the theoretical SCF, especially for very small radii, and dependson the notch sensitivity, q, which is a function of the material and notch radius. The details for determining q and kF aregiven in Section 4.6.

4.3.6 Viscous Dampers

Viscous dampers (Houdaille type) are often used in reciprocating engines to help limit torsional vibration andcrankshaft stresses [15,16,17]. These dampers are normally intended to protect the engine crankshaft and notnecessarily the driven machinery. To be effective, dampers need to be located at a point with high angular velocity,i.e., near the anti-node of the crankshaft mode.

4.3.6.1 Equivalent Damping and Inertia

A viscous damper consists of a flywheel that rotates inside the housing, which contains a viscous fluid such assilicone oil (Figure 4-36). The housing is bolted directly to the crankshaft and, thereby, rotates with the crankshaft.During normal operation, the housing drives the flywheel through the fluid, such that they rotate together as a rigidbody. However, when torsional vibration occurs, the large inertia of the flywheel prevents it from following the motionof the housing. This causes relative motion between the housing and flywheel and results in a shearing of the fluid.This shearing dissipates energy as heat and acts to damp out the torsional vibration.



The damping characteristics can be adjusted by changing the internal clearances between the housing and flywheel,h1 and h2, and/or the fluid viscosity, . From the Shock and Vibration Handbook [17], the damping constant is

(4-16)

where

is the viscosity, Pa-s (lbf-s/in.2);

b is the width of damper flywheel, m (in.);

c is the damping constant, N-m-s/rad (in.-lbf-s/rad);

r1 is the inside radius of damper flywheel, m (in.);

r2 is the outside radius of damper flywheel, m (in.);

h1, h2 is the damper internal clearances, m (in.).

For a given vibration frequency, ω, the optimum damping coefficient is given by:

copt = Ifly x ω (4-17)

where

copt is the optimum damping coefficient, N-m-s/rad (in.-lbf-s/rad);

Ifly is the flywheel inertia, N-m-s2 (in.-lbf-s2);

ω is the vibration frequency, rad/s.

Figure 4-36—Viscous Damper

Housing h2

r2

r1b

h1

Crankshaft

Viscousfluid

Flywheel

c 2r2

3bh2

-------r2

4 r14–

2h1

--------------+=



Finally, if the damper is designed for optimum damping, then according to Den Hartog [5], the equivalent damperinertia is equal to the housing inertia plus one-half of the flywheel inertia.

(4-18)

where

Id is the damper flywheel inertia, kg-m2 (lbm-in.2);

Ieqv is the equivalent inertia, kg-m2 (lbm-in.2);

Ih is the damper housing inertia, kg-m2 (lbm-in.2).

If the lumped inertia is used for the damper, the system natural frequency calculations will be satisfactory, but will notpredict the internal flywheel resonance, which is usually well-damped and not of concern. The preferred method is tomodel a damper using separate inertias for the internal flywheel and housing. These two inertias are connected byequivalent damping and stiffness properties.

Viscous dampers have a limited service life and require periodic checks and maintenance. The heat build-up insidethe damper should be calculated and compared to the allowable value provided by the manufacturer. In somesituations, it may be beneficial to use two dampers to provide additional damping.

It is not uncommon to find viscous dampers in the field that have failed. As this will effectively eliminate the torsionalstiffness associated with the damper fluid, the torsional natural frequencies may be altered. Accordingly, it may beprudent to consider the impact of a failed damper during the torsional analysis.

4.3.7 Considerations for Components Often Found in Reciprocating Trains

4.3.7.1 Considerations for Low Stiffness or Resilient Couplings

There are several instances where using “soft” couplings could be beneficial in controlling torsional vibration such as:isolation of excitation between components [18], detuning a torsional natural frequency, or adding damping to thesystem. A coupling is termed torsionally “soft” if it has a lower torsional stiffness than the steel shafts that it isconnecting. Several types of soft couplings are: rubber-in-compression, rubber-in-shear, helical-spring, leaf-spring,and magnetic. Although not technically a coupling, a long torque shaft also has a low torsional stiffness.

Several factors must be considered when choosing a soft coupling. For rubber couplings, the trade-off is usuallyincreased maintenance, since the rubber degrades over time due to heat and environmental factors. Variousgeometries (shear element, round and wedge blocks), materials (natural rubber, styrene-butadiene, neoprene, andnitrile), and durometers (50, 60, 70, and 80) are available. Special silicone blocks may be required for hightemperature applications. The proper selection must be made to optimize the torsional stiffness and dampingproperties needed for the system. The actual coupling stiffness can vary from catalog values and this should beconsidered in the torsional analysis.As was described in Section 4.1.6, the torsional stiffness of a rubber-in-compression coupling is nonlinear and varies with transmitted torque and temperature. This nonlinearity can make itdifficult to tune natural frequencies between engine or compressor orders. Special analysis techniques are required tohandle the nonlinearity during transient events such as synchronous motor start-ups [19]. The torsional naturalfrequencies of a system could be different for cold starts and hot restarts.

Running at or near a resonant condition for just a few minutes could elevate the temperature beyond the melting pointof the rubber elements or blocks and damage the coupling. Therefore, the vibratory torque and heat dissipation (fordamped couplings) must be calculated and should also be reviewed by the coupling manufacturer for acceptability.

Ieqv12-- Id Ih+



The heat dissipation is a function of the vibratory torque and frequency and is normally specified in terms of powerloss (Watts).

When applicable, the torsional analysis should consider all reciprocating compressor load steps, possible compressorvalve failure, and engine misfire. These off-design conditions could produce significant torque excitation atfrequencies not considered for normal operation, which could be potentially damaging to the soft coupling. In systemswith an electric motor variable frequency drive (VFD), the torque harmonics should be evaluated and the samplingrate at which the VFD controls the speed should not be coincident with a torsional natural frequency.

4.3.7.2 Considerations for Speed Increaser or Reducers

Special consideration should be given to gearboxes used with reciprocating equipment. Several gearboxmanufacturers recommend limiting the dynamic torque at the gear mesh to 30 % of the rated torque during steady-state operation. To prevent torque reversals at low load conditions, the dynamic torque should not exceed thetransmitted torque.

Transient events such as startups and emergency loaded shutdowns (ESDs) can also cause high dynamic torque atthe gear mesh. Depending on the speed ramp rate during startup, engines and synchronous motors can cause peaktorques at the gear mesh that are much higher than recommended for continuous operation. If the peak torqueexceeds the catalog rating of the gearbox, then the manufacturer should be contacted to discuss potential gear toothdamage from repeated starts.

Care should be taken for gears that are uploaded. The tangential and separating forces acting on the gears should becalculated for various loads and vectorially summed with the weight to compute the resultant. At full load, theuploaded gear will be lifted into the upper portion of the bearings. However, large torque modulation from areciprocating compressor or engine could reduce the tangential force acting in the upward direction to less than theshaft weight. In this case, the gear could be lifted then dropped once per revolution causing high shaft vibration. Thislateral vibration could cause significant dynamic misalignment of the gear teeth and result in tooth overloading andrapid gear wear.

4.3.7.3 Considerations for Connected Auxiliary Equipment and Electric Motor Drivers

Oil pumps driven from the auxiliary end of a reciprocating compressor can fail due to high torsional vibration. Somecompressor manufacturers have published allowable limits versus frequency [9]. In general, overall torsionaloscillation exceeding one degree peak-to-peak should be closely evaluated.

When electric motors are used to power reciprocating compressors, consideration should be given to the nonuniformtorque demands of the reciprocating compressor that may affect the motor current. Torsional oscillation can causecurrent pulsation in the electrical system. Reference [20] makes a recommendation that when the driven load, suchas that of reciprocating type pumps, compressors, etc., requires a variable torque during each revolution, it isrecommended that the combined installation have sufficient inertia in its rotating parts to limit the variations in motorstator current to a value not exceeding 66 % of full-load current.

4.3.8 References

[1] Feese, T., and Hill, C., “Guidelines for Preventing Torsional Vibration Problems in Reciprocating Machinery,”Proceedings of the 2002 Gas Machinery Conference, GMRC, Nashville, Tennessee, (October 2002).

[2] Nestorides, E. J., A Handbook on Torsional Vibration, British Internal Combustion Engine ResearchAssociation, pp. 84–88, (1958).

[3] Wilson, W. K., Practical Solution of Torsional Vibration Problems, 1, New York, New York: John Wiley & SonsInc., (1956).



[4] Pasricha, M. S., and Carnegie, W. D., “Torsional Vibrations in Reciprocating Engines,” Journal of ShipResearch, Vol. 20, No. 1, pp. 32–39, (March 1976).

[5] Den Hartog, J.P., Mechanical Vibrations, New York, New York: Dover Publications, Inc., (1985).

[6] Murray, B. D., Howes, B. C., Chui, J. and Zacharias, V., “Sensitivity of Torsional Analysis to Uncertainty inSystem Mass-Elastic Properties,” Proceedings of the International Pipeline Conference, ASME, pp. 975–982(1996).

[7] Varty, R. R., and Harvey, J. D., “Torsional Vibration Modeling and Analysis Continued,” Proceedings of the2003 Gas Machinery Conference, GMRC (2003).

[8] Corbo, M. A. and Malanoski, S. B., “Torsional and Pulsation Design Audits,” Proceedings of the 2007 GasMachinery Conference, GMRC, Dallas, TX (October 2007).

[9] Stephens, T. J., “Torsional Amplitude Limits for the Auxiliary End of Ariel Reciprocating Compressors,”Proceedings of the 2001 Gas Machinery Conference, GMRC (2001).

[10] Stephens, T. J., “Torsional Case Studies on High Speed Separable Reciprocating Compressors,” Proceedingsof the 2004 Gas Machinery Conference, GMRC, (October 2004).

[11] Harris, R. E. and Smalley A. J., “Proposed Consensus Document for Torsional Vibration Analysis of High-Speed Reciprocating Compressor Installations,” Proceedings of the 2002 Gas Machinery Conference,GMRC, Nashville, TN (October 2002).

[12] Peterson, R. E., Stress Concentration Factors, New York, New York: John Wiley & Sons (1974).

[13] USAS B17.1—1967, “Keys and Keyseats,” American Society of Mechanical Engineers (1973).

[14] Lloyd’s Register of Shipping, “Main & Auxiliary Machinery,” Rules & Regulations for the Classification of Ships,Part 5, London, (2000).

[15] Brenner, Jr., R. C., “A Practical Treatise on Engine Crankshaft Torsional Vibration Control,” Proceedings of theSAE West Coast International Meeting, Society of Automotive Engineers, Portland, Oregon, (1979).

[16] Lily, L. R. C., Diesel Engine Reference Book, London: Butterworth & Co. Publishers Ltd. (1984).

[17] Harris, C. M., Shock & Vibration Handbook, 4th Ed., New York, New York: McGraw-Hill Companies, Inc.,(1996).

[18] Feese, T., “Torsional Vibration Linked to Water Pumping System Failure,” Pumps and System Magazine, pp.44–45, (September 1997).

[19] Feese, T., “Transient Torsional Vibration of a Synchronous Motor Train with a Nonlinear Stiffness Coupling,”Thesis, University of Texas at San Antonio, (December 1996).

[20] NEMA Standard MG 1-1998, Motors and Generators (1998).

4.4 Torsional Analysis

4.4.1 General

The three major types of torsional analyses are as follows:



1) undamped torsional analysis;

2) steady-state torsional response analysis;

3) transient torsional response analysis.

Undamped torsional analysis is the starting point for most torsional studies. In this analysis, the torsional model isanalyzed to determine the train’s undamped torsional natural frequencies and mode shapes. These are then plotted,along with all of the relevant excitation sources in the system, on Campbell diagrams to determine if there are anyinterference points within the train’s operating speed ranges or, if not, what the separation margins are.

If the undamped torsional analysis reveals that the train does not have sufficient separation margins, the interferencepoints that violate the separation margin requirements can sometimes be justified via steady-state response analysis.In such an analysis, a potential resonance condition is evaluated by applying the expected excitation and damping tothe train and calculating the resulting torques and stresses in the shafts and couplings. All components are thenevaluated for infinite life.

Transient torsional response analysis is usually only needed when the train contains an electric motor or generator.The most common transient analysis performed is the start-up analysis of a synchronous motor, an analysis that mustbe performed whenever the train is driven by a synchronous motor that is started across-the-line. Other transientanalyses that are sometimes performed include short circuits of motors or generators, start-ups of induction motors,and generator mal-synchronization. Unlike steady-state analyses, the criterion for acceptability in transient analysesis low-cycle fatigue, or sometimes simple survival.

All of these analyses are described in detail in the succeeding sections. References [1] and [2] provide detaileddescriptions of all three types. Furthermore, References [3] and [4] provide excellent descriptions of the use oftransient analysis to evaluate synchronous motor startups.

4.4.2 Undamped Torsional Natural Frequency Analysis

This type of analysis considers only the inertias and stiffnesses of each component in the equipment train (i.e.damping is ignored). Since in almost all practical turbomachinery, the component damping is small, for all practicalpurposes, the undamped natural frequencies are no different from their damped counterparts. Therefore, theundamped analysis can be used to design the system.

NOTE A damped torsional analysis may be considered in trains with an elastomeric coupling and/or a viscous/rubber damper toexamine the difference between undamped and damped natural frequencies.

Several types of computer programs are available for calculating undamped torsional natural frequencies. Suchsoftware will calculate the undamped torsional modes using the Finite Element Method or the Transfer Matrix (Holzer)Method. Such codes should be properly validated and be sufficiently robust to analyze all systems of interest to theuser. For convenience to the reader, some computational aspects of the Transfer Matrix (Holzer) Method are detailedin 4.4.2.5 as well as its limitations in predicting undamped torsional modes.

The primary results of the undamped torsional natural frequency analysis are the following:

a) undamped train torsional natural frequencies;

b) corresponding train torsional mode shapes.

A conventional presentation of the torsional natural frequencies is made using a Campbell Natural FrequencyInterference Diagram, as shown in Figure 4-37 for a typical motor-gear-compressor train and Figure 4-38 for a typicalsteam turbine/compressor train. Campbell diagrams provide a graphical display of a rotor system’s torsional naturalfrequencies (plotted as horizontal lines) versus the frequencies of potential excitation mechanisms. The reference



operating speed or speed range is also displayed, via vertical lines. The potential excitation frequencies may beconstant, such as electrical line frequency and twice electrical line frequency or they may be variable in nature.Variable frequency excitations may be associated with rotational frequency or multiples of rotating frequency or maybe associated with electrical excitation such as the harmonics of variable frequency electric drives or asynchronousstarting of a synchronous motor. The excitation may be of a steady nature or may be transient in nature such as thetorsional excitation of a train driven by a synchronous motor. Constant frequency excitations appear as horizontallines on a Campbell diagram. Variable frequency excitations appear as sloped lines. The coincidence of any torsionalnatural frequency with any steady state excitation frequency should meet the API separation margin of ±10 %. Insome situations, but not for coupling modes excited by the 1X excitation, this requirement can be overrriden byshowing that the torsional natural frequency is nonresponsive to the excitation in question. Transient torsionalexcitations may be impossible to detune, in which case the system should be evaluated for transient response.

The system rotor mode shape for each natural frequency is a plot of relative angular deflection versus axial distancealong the coupled rotors. These plots are typically normalized to unity based on the location of maximum angulardeflection. Figure 4-39 and Figure 4-40 display several train torsional mode shapes calculated for a motor-gear-compressor train.

Examination of these figures reveals that there are two distinct types of modes in turbomachinery. The first type, asillustrated by the first two modes of Figure 4-39, is referred to as a Coupling Mode. In a Coupling Mode, the largemajority of twisting is occurring in one or more of the system’s couplings, as is seen from the figure. Very little twistingis occurring in the train’s major components—in fact, most of the components are behaving as rigid bodies. Ingeneral, if a train has N couplings, the first N modes will be Coupling Modes. For the typical motor-gear-compressor

Figure 4-37—Campbell Diagram for a Motor-Gear-Compressor System

0

5th TNF (267 Hz)

4th TNF (258.2 Hz) 3rd TNF = 202.8 Hz

2nd TNF = 68.0 Hz

1st TNF = 29.2 Hz

Compressor Speed

1X Rotational

2X Line Frequency (120 Hz)

1X Line Frequency (60 Hz)

Rotational Speed (RPM)

Campbell DiagramTypical Motor Gear Compressor Train

Tors

ona

Nat

ura

Fre

quen

cy (H

z)

Eec

trca

Fre

quen

cy (H

z)

1000 2000 3000 4000 5000 6000 7000 8000 9000

300

250

200

150

100

50

0

300

250

200

150

100

50

0Motor Speed



Figure 4-38—Campbell Diagram for a Steam Turbine Driven Compressor System

1st Stage bladenatural frequency

3rd Train torsional natural frequency

2nd Train torsional natural frequency

1st Turbine lateral critical1st Train torsional natural frequency

1st Compressor lateral critical

90 %

Spe

ed

Mot

or o

pera

tng

spee

d

110

% S

peed



Figure 4-39—Torsional Mode Shapes for a Typical Motor-Gear-Compressor Train

1.0

0.5

0.0

-0.5

-1.0

1.0

0.5

0.0

-0.5

-1.0

1.0

0.5

0.0

0 0

0 0

0 0

-0.5

-1.0

Nor

ma

zed

ang

uar

dsp

acem

ent (

dm

)N

orm

aze

d a

ngu

ar d

spac

emen

t (d

m)

Nor

ma

zed

ang

uar

dsp

acem

ent (

dm

)

Train Axial Rotor Length (in.)

Low

spe

ed c

oup

ngs

Gea

r

Pn

onH

gh s

peed

cou

png

Cen

trfu

ga

Com

pres

sor

Mot

or

40.0 120.0 160.0 240.0 280.080.0

40.0 160.0 200.0 240.0 280.080.0 120.0

200.0

40.0 120.0 200.0 240.0 280.080.0 160.0

Undampedtrain torsionalnatural frequency1st Mode = 1754 CPM

Undampedtrain torsionalnatural frequency2nd Mode = 3474 CPM

Undampedtrain torsionalnatural frequency3rd Mode = 12165 CPM



train, which has two couplings, there will be two Coupling Modes, as is seen in Figure 4-39. Coupling Modes arealmost always of interest since they are usually the most dangerous modes that the train possesses.

The other type of torsional mode is the Component Body Mode. In this type of mode, there is significant twistoccurring in one or more of the component shafts within the train. In general, all modes that are not Coupling Modesare considered to be Component Body Modes. For instance, in Figure 4-39, the third mode is seen to be aComponent Body Mode involving twisting in the motor shaft. Likewise, in Figure 4-40, the fourth and fifth modes arealso Component Body Modes, involving twisting in the gear and compressor shafts, respectively. Mode shapeinformation is important for the proper interpretation of the results. Should a torsional interference exist, a study of thetrain mode shape in question can yield information on nodal point locations and anti-nodal point locations along thedeflected rotor train. This information allows the designer to understand where the system is sensitive to flexibility andwhere it is sensitive to inertia, respectively. The strain energy and kinetic energy, discussed in 4.4.2.1 and 4.4.2.2,also assist the analyst in identifying the component best suited to tune a given mode and which locations are mostimportant when regarding the response of a given mode. This knowledge allows the analyst to determine whichchanges in the system are most appropriate if modification of the system is required.

4.4.2.1 Energy

4.4.2.1.1 General

The concepts of strain and kinetic energy are described in Reference [5].

4.4.2.1.2 Strain Energy

Strain energy is determined on an element-by-element basis for each calculated mode of the system beinganalyzed. Strain Energy is calculated by Equation 4-19.

(4-19)

Figure 4-40—Torsional Mode Shapes for a Typical Motor-Gear-Compressor Train (Continued)

10.0

7.5

5.0

2.5

0

-2.5

-5.0

-7.5

-100.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Time (secs)

Per

Un

t Tor

que

Single Line-to-Line Short Circuit Torque

UKt

2----2=



where

U is the strain energy, N-m (lbf-in.);

Kt is the element torsional spring constant, N-m/rad (in.-lbf/rad);

is the torsional deflection across the element, rad.

The calculated percent strain energy for a given element relative to the total strain energy of the system enables theanalyst to determine the system segments that most significantly affect the natural frequency for a given mode. Theseelement percent strain energies will vary from mode to mode. The following equations will enable one to estimate howa given change in an element will affect the natural frequency of a given mode (Reference [5]). A change can bemade to an element stiffness and/or inertia, however the most common modification is to alter a system naturalfrequency by changing an element stiffness. The general form of the equation is given in Equation 4-20.

(4-20)

where

is the modified system frequency, rad/s;

is the initial system frequency, rad/s;

SE% is the percent strain energy of the element being modified;

KE% is the percent kinetic energy of the element being modified;

Kt is the change in stiffness of the element being modified, N-m/rad (lbf-in./rad);

Kt is the original stiffness of the element being modified, N-m/rad (lbf-in./rad);

Ip is the change in inertia of the element being modified, kg-m2 (lbm-in.2);

Ip is the original inertia of the element being modified.

For the case where only stiffness is modified, Equation 4-20 simplifies to:

(4-21)

Equation 4-21 can be rearranged to solve for the desired stiffness change:

(4-22)

' 100 SE%

Kt

Kt

-------- +

100 KE%Ip

Ip

------- +

-------------------------------------------

12---

=

'

' 100 SE%

Kt

Kt

-------- +

100-------------------------------------------

12---

=

Kt Kt

100'----

2

100–

SE%---------------------------------------=



NOTE The accuracy of Equation 4-22 is subject to the magnitude of the calculated stiffness change. One may find that theactual frequency shift identified in 4-22 does not agree exactly with . This is due to the fact that as the stiffness of the elementchanges so does the strain energy in that element. For stiffness changes of approximately 20 %, the calculated frequency with thepredicted stiffness will be reasonably accurate. For larger changes, a second iteration may be required after the torsionalcalculations are performed with the revised frequency.

If the desired change is a reduction of a system frequency, the section stiffness must be reduced, thus, increasing thetorsional stress in the section. One must consider normal and transient conditions so that overloading of the sectiondoes not occur.

The concept of strain energy is a tool to assist the analyst to understand which segments are most influential inmodifying the natural frequency of a given mode and how the degree of modeling accuracy may influence the resultsof the analysis. If the percent strain energy of a given element is significant compared to the total system strainenergy, then this element will significantly affect the natural frequency of the mode that it is associated with. Accuracyof the model in this location will significantly impact the accuracy of the calculation. Conversely, a low strain energylevel indicates that this element will not significantly impact the accuracy of the results for this mode. One should beaware that a given element may have significant impact on one mode and yet not impact another mode. In the eventof uncertainties regarding the accuracy of modeling, a review of the strain energy at an element in question can assistthe analyst to determine if further refinement of an element’s geometry is warranted.

4.4.2.1.3 Kinetic Energy

Kinetic energy is determined on an element-by-element basis and for each disk represented as a lumped inertia.Equation 4-23 defines the kinetic energy in a system element or disk.

(4-23)

where

V is the kinetic energy, N-m (lbf-in.);

Ip is the polar moment of inertia, kg-m2 (lbm-in.2);

is the angular velocity at the element, rad/s;

gc is the gravitational constant, 1.0 kg-m/N-s2 (386.09 lbm-in./lbf-s2).

The kinetic energy is calculated for each mode of the system being analyzed. When the kinetic energy of an elementor disk is divided by the total system kinetic energy, the percent kinetic energy of the disk can be identified. It is muchmore difficult to tune a torsional frequency by altering the inertia of an element or disk rather than altering the torsionalstiffness of an element. The identification of the location of significant percentage kinetic energy of a disk or elementcan assist to identify locations that are influential in the response of a given torsional mode. If a potential excitationexists at a location of significant kinetic energy and that excitation can occur at the torsional frequency in question,then there is a strong probability that the mode can be excited.

There are several potential sources of torsional excitation from rotating machinery. Fortunately, most of these aresmall in magnitude generating minimal torsional excitation. Tables of potential excitations with associated frequenciesand magnitudes can be found in Table 5 of Reference [1] and Tables 1 and 4 of Reference [2]. Details of theseexcitations are elaborated in the references and the material in 4.5.

'

VIp

2gc

-------- ꞏ 2

=

ꞏ



4.4.2.2 Acceptance Criteria and Tuning a Mass Elastic System

Once the system has been analyzed, the Campbell diagrams need to be checked for the separation margins betweenall interference points (i.e. intersections between a torsional natural frequency and an excitation) and the operatingspeed range. All separation margins need to be 10 % or greater. If any interference points do not meet thisrequirement, the analyst normally has two options for dealing with them. First, a change can be made to the train(normally to one of the couplings) to move the natural frequency in question enough to yield the needed separationmargin. This is often referred to as “tuning” the system. Second, a steady-state response analysis, in accordance with4.4.3, can be performed to demonstrate that the train is capable of withstanding the resonance condition of concern.

Both of the above options are available to the analyst in all situations except one. If the interference point that doesnot meet the separation margin involves the 1X excitation and a Coupling Mode, the steady-state response analysisoption is not available. Instead, the train must be tuned, most likely via a coupling change, to yield the required 10 %separation margin. In a fixed speed train, it is almost always possible to achieve this without having to go to exoticcoupling designs.

In a variable speed system, one should also make every effort to achieve the required separation margin. However,on rare occasions, one may encounter a variable speed system that has such a large speed range (greater than from50 % to 100 % of MCOS) that doing so is virtually impossible—the interference point will lie within the operatingspeed range no matter what is done. In such a case, the end user and OEM should hold discussions to determine theacceptability of the train.

One should use caution when reducing the torsional stiffness of a coupling to ensure that peak stress incurred duringstartup, or other specified off-design conditions, does not exceed the material yield strength.

4.4.2.3 Examples

4.4.2.3.1 Compressor-Gear-Motor

Figure 4-3 presents the general layout of a typical motor-driven compressor train. For this example, a motor runningat 1788 rpm, drives a speed-increasing gear which powers an 8100 rpm compressor. This train is modeled using datasupplied by vendors of the various components (motor, gear, compressor, and couplings) to support a torsionalnatural frequency analysis. The Campbell diagram in Figure 4-37 cross-plots the frequencies of the modes with theshaft running speed. Figure 4-39 and Figure 4-40 present the first five mode shapes. The Campbell diagram indicatesa potential interference, within 10 %, between the fundamental torsional mode and one times motor speed. The plotalso indicates that no interference exists between the one and two times electrical line frequency and any undampedtorsional natural frequency. Examination of the corresponding torsional mode shape indicates that shaft twisting of thethree individual units is minimal; torsional twisting is principally confined to the couplings for this mode. This impliesthat the torsional stiffness of the couplings is significantly lower than the torsional stiffness of the surrounding shafts.Hence, the frequency of this mode is governed by the torsional stiffness of the couplings. Couplings are thetorsionally soft elements in the train, therefore, their torsional stiffness will govern the locations of the fundamental twomodes. In general, machinery trains will have the same number of Coupling Modes as couplings. These modes aretorsionally significant and require de-tuning if they interfere with potential excitation frequencies. Altering the torsionalstiffness of one or both couplings will allow the design engineer to shift the frequency of the potentially problematicmode. Figure 4-41 displays the result of tuning the coupling torsional stiffness where the potentially problematic modehas been tuned away from the operating speed of the unit. Although motor-driven trains typically possess moresources of torsional excitation than a turbine-driven train, they are also often limited to a single operating speed sotorsional detuning is often not difficult to accomplish. Most coupling vendors may be able to adjust the torsionalstiffness of a coupling within a range of at least ±20 %. Note, however, that tuning the coupling stiffness mayadversely impact the service factor of the coupling.

The third mode calculated for this example is typical of motor-controlled modes where calculated angular deflectionsare predominantly found in the low-speed shafting with the largest change in angular deflection occurring through themotor. Note that the node point of the motion is located nearly at motor midspan so that the two ends of the core



vibrate out of phase. This motion is analogous to the out-of-phase free vibration observed in a system composed oftwo masses connected by a single spring. Higher-order modes contain single unit out-of-phase motions similar to themotor controlled-mode. Figure 4-40 displays the fifth mode as a compressor controlled torsional mode.

4.4.2.3.2 Compressor/Turbine

This example considers a steam turbine directly connected to a centrifugal compressor as depicted in Figure 4-6. Atorsional analysis is often not required for this train because the turbine provides a smooth driver with low amplitudetorque pulsations in the frequency range likely to excite a lower torsional natural frequency. Without a major excitationmechanism, torsional natural frequencies will not be significantly amplified. Even in this case, however, aconservative design approach will ensure that there is no interference with the one times operating speed range,particularly with the fundamental (first) torsional natural frequency.

Figure 4-38 and Figure 4-42 present the train Campbell diagram and first three mode shapes for the turbine-compressor train. The Campbell diagram shows no interference between the undamped torsional natural frequenciesand the one times operating speed lines, indicating an acceptable design for this train. Note that the coupling stiffnesscontrolled mode lies well below the operating speed range, while the resonant modes corresponding to the particular

Figure 4-41—Campbell Diagram for a Motor-Gear-Compressor Train After Tuning

25.0

0.0

Tors

ona

Nor

ma

Fre

quen

cy x

102

(CP

M)

Reference Speed x 101 (RPM)0.0 40.0

Fifth mode = 169.39 CPM

Fourth mode = 160.38 CPM

Third mode = 120.42 CPM

1x Compresso

r speed

1x Motor speed

90 %

Spe

ed =

160

3 R

PM

Nor

ma

Spe

ed =

178

1 R

PM

90 %

Spe

ed =

195

9 R

PM

NOTE First mode defined with 1x motor speed by coupling modification.

80.0 120.0 160.0 200.0 240.0

50.0

75.0

100.0

125.0

150.0

175.00

200.0

Second mode = 31.73 CPM

First mode = 12.50 CPM



Figure 4-42—Torsional Mode Shapes for a Typical Steam Turbine-Driven Compressor Train

1.0

0.5

0.0

-0.5

-1.0

Nor

ma

zed

Am

ptu

de (N

DIM

)

Compressorrotor mode

Steam turbinerotor mode

Fundamental mode

3rd mode = 13630 CPM

2nd mode = 12514 CPM

1st mode = 1841 CPM

1.0

0.5

0.0

-0.5

-1.0

Nor

ma

zed

Am

ptu

de (N

DIM

)

1.0

0.5

0.0

0.0

-0.5

-1.0

Nor

ma

zed

Am

ptu

de (N

DIM

)

Cou

png

Com

pres

sor

EndTu

rbne

40.0 120.0 160.0 200.0 240.0 280.080.0

40.0 120.0 160.0 200.0 240.0 280.080.0

40.0 160.0 200.0 240.0 280.080.0 120.0



machines lie above the operating speed range. This is characteristic of most turbine-compressor trains and results inthe typically acceptable torsional characteristics for these trains.

4.4.2.4 Transfer Matrix (Holzer) Method

The Transfer Matrix (Holzer) Method is best described as a method in which the output oscillating torque is calculatedat one end of the train given an input oscillating torque at the other end of the train. The undamped torsional naturalfrequencies of the train may be calculated by noting that the magnitude of the calculated oscillating torque at the freeend of the train becomes zero when the frequency of the oscillating torque matches a train natural frequency. Inmathematical terms, the condition of torsional natural frequency (within a specified torque residual error) is defined asfollows:

(4-24)

where

TN is the torque residual, N-m (in.-lbf);

Ipi is the ith polar moment of inertia, kg-m2 (lbm-in.2);

is the frequency of oscillation, rad/s;

Ai is the ith shaft section coefficient;

gc is the gravitational constant, 1.0 kg-m/N-s2 (386 lbm-in./lbf-s2).

The convergence limit for a typical transfer matrix routine is to within ±0.01 Hertz of the actual analytical value atnatural frequency. Instead of using the magnitude of the torque residual at the end of each iteration as theconvergence dependent variable, most codes search for the torque residual’s crossover points on the frequency axisto within the specified tolerance limit. This method is used because for all modes above the first several (which aretypically controlled by the coupling torsional stiffnesses), the slope of the torque residual curve becomes very steepand may result in excessive computer iteration time if a residual torque magnitude convergence routine is employed.Since the frequencies of interest are generally several hundred CPM or larger, the error in the calculated frequency isless than ± 0.01 % regardless of the magnitude of the torque residual.

4.4.2.5 Limitations of Analysis—Subsystem Natural Frequency

A common occurrence in the torsional response of a rotor train is the presence of a subsystem natural frequency.This condition will yield an additional torsional natural frequency in the train analysis. However, upon closer inspectionof the rotor mode shapes, it can be seen that such a frequency does not represent a true system phenomenon but,rather, is a characteristic frequency of an isolated part of the train. This subsystem natural frequency is essentiallyuncoupled in nature from the remainder of the system and, as such, is not a true train natural frequency. However,this frequency still represents a potentially significant vibration mode of interest in that, given the required excitationinput, an undesirable resonant condition could exist. Additionally, inspection of the residual torque curve indicates thata normal cross-over point exists for a subsystem natural frequency with a finite slope at TN = 0. The following areexamples of such a condition:

a) exciter assemblies;

b) multiple-geared systems, multiple branches;

c) discontinuous systems.

TNIp i

2Ai

gc

----------------

i 1=

N

0= =



4.4.3 Steady-state Forced Response Calculations

In most fixed speed machinery trains, torsional natural frequencies can usually be tuned to yield the requiredseparation margins. However, it may not be possible to tune a system for a variable speed machinery train. This isespecially true if the train contains potential excitation sources that are related to rotational speed, harmonics ofrotational speed, variable electrical frequencies and their harmonics supplied to a motor, excitations that are related tointer-harmonic frequencies in systems with variable frequency drives or “slip” frequencies associated withsynchronous motors.

If after the torsional natural frequencies are determined for a system, there is a defined interference point that doesnot meet the separation margin requirements, and which cannot be sufficiently moved by tuning, a steady-stateresponse analysis must be performed to justify the separation margin violation. Once again, it should be noted thatthis is not an acceptable procedure if the interference point in question involves a Coupling Mode being driven by the1X excitation. For such a case, the system must be tuned to eliminate the violation.

The following evaluation should be performed to determine if a system can be excited and, if so, is the systemsuitably designed to avoid high cycle fatigue failure:

The first step is to evaluate the torsional natural frequency of concern and its associated mode shape along withkinetic energy distribution. If from this observation, it is apparent that the mode cannot be excited by the excitationassociated with the interference point in question, then this condition need not be examined further.

An example of a torsional mode that is unique to one component in the system is shown in Figure 4-43. Figure 4-43 isa mode shape plot of a compressor system similar to that shown in Figure 4-3. Figure 4-43 identifies the 3rd modetorsional natural frequency as 279.2 Hz. It can be seen from the figure that all of the deflection in the system isassociated with the sections which represent the motor. There are no changes that can be made in the system thatwould alter this mode significantly. In order to prove this, a second calculation was performed where only the motorrotor was modeled. The remainder of the system was deleted. Figure 4-44 identifies the 1st mode torsional frequencyof the motor at a frequency of 278.6 Hz. In addition to the frequency, the mode shape of the motor-only model has thesame modal distribution as that of the complete system. In order to excite a 3rd mode torsional natural frequency ofthe system, this motor would have to have an excitation which has a frequency of 278.6 Hz and would require an out-of-phase excitation as shown by the mode shape plot.

Occasionally, a motor-driven compressor system will have a third mode natural frequency that coincides with theoperating speed of the compressor or twice the operating speed of the compressor. Since the system frequency issolely dependent on the motor design, no tuning of components, such as couplings, would affect the frequency of thismode. Additionally, since this mode will only be responsive to excitations at the motor, no excitation located anywherewithin the compressor will be able to excite it. Thus, such an interference can be discarded from consideration.

Similarly, if the excitation is of such a very high frequency, such as the tooth-passing frequency of a gear which maybe over 100 times the rotation speed of a gear or 30 or more times the pinion speed, the higher order torsional modeswill be local to a single component of the system. Additionally, it is nearly impossible to excite modes at frequenciesthis high. It is for these reasons that harmonics above 24 times rotational speed are usually disregarded in theevaluation of a system.

If the review of the mode shapes and potential excitation mechanisms does identify a potential excitation that canexist at a component with high modal displacement or a high percentage of the kinetic energy of a mode, then asteady-state forced response analysis should be performed.

If it has been determined that there is an excitation that can modally excite the system, then the magnitude of thisexcitation must be quantified. Section 4.5 will identify potential torsional excitations from various types of rotatingmachinery. Section 4.3.3 identifies torsional excitations occurring in reciprocating machinery. If information on theexcitation amplitude is available from the component manufacturer (for example, a VFD excitation), then that



amplitude should be used in the analysis. In the more common situation where specific excitation amplitudeinformation is not available, the rules-of-thumb given in Section 4.5 can often be employed.

After the excitation frequency and magnitude are defined, then damping in the system must be quantified. Many ofthese damping mechanisms are frequency and/or torque magnitude dependent. Examples of damping within asystem are hysteretic damping of elastomeric couplings, shaft material hysteresis, “slip damping” at shrink fits, boltedjoints, and couplings, and fluid/disk interaction at pump impellers. An extensive discussion of these damping sourcesand magnitudes can be found in Reference [2].

In general, the actual magnitudes of damping due to the individual sources within a train are usually small and almostimpossible to determine analytically. Accordingly, most analyses are performed with a generic damping ratio that is

Figure 4-43—3rd Torsional Natural Frequency of a Motor-Gear-Compressor System

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1 3 5 9 11 13 15 17 19 21 23 25 27 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85

Re

atve

Dsp

acem

ent

System Station Identification

3rd Torsional Mode Shape Plot of Complete System—Frequency 279.2 HZ

Complete system 3rd Mode



applied to every shaft element within the train. Although there is a wide disparity of opinion on what that ratio shouldbe (as is documented in Reference [2]), the following values are generally felt to be conservative:

— for turbine-generator trains (or other trains employing rigid couplings)—0.5 % of critical damping;

— for ungeared trains employing flexible couplings—1.0 % of critical damping;

— for geared trains having the gear shafts supported on fluid-film journal bearings—1.5 % of critical damping.

In a literature survey reported in Reference [2], the minimum damping value cited by many sources was found to be1.0 %. This sets the baseline of 1.0 % for ungeared systems. Many authors, including Simmons and Smalley [6],have noted that the torsional damping in geared systems is significantly greater than that in ungeared systemsbecause of the beneficial squeeze-film effect that occurs in the fluid-film bearings supporting the gear shafts whentorsional vibration occurs. Although several authors have observed the damping to approximately double, comparedto that of an ungeared system, an increase of 50 % is felt to be more prudent for design purposes.

Finally, Walker [7] has performed extensive experimental studies on the torsional damping in steam turbine-drivengenerators and has made two findings. First, the damping is highly dependent on generator load (tending to increasewith load) and, second, the damping is less than 1.0 % of the critical value. The reason the damping in these trains isbelieved to be less than that in other ungeared trains is the exclusive use of rigid couplings in these trains. It is

Figure 4-44—1st Torsional Natural Frequency of a Motor

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Re

atve

Dsp

acem

ent

System Station Identification

1st Torsional Mode Shape Plot of Motor Only—Frequency 278.6 HZ

Motor 1st Mode



believed that a significant portion of the torsional damping in many trains originates in the flexible couplings. The lackof these couplings in steam turbine-driven generators results in a loss of damping. This is consistent with Reference[8], which recommends that if a motor-driven reciprocating compressor employs a rigid, rather than a flexible,coupling, the assumed damping should be reduced from 1.0 % to 0.5 %.

After the steady-state response analysis is completed, the mean and cyclic stress occurring in every shaft element inthe model should be calculated. The mean stress should be computed from the steady-state torque at the operatingconditions consistent with the resonance being evaluated and the cyclic stress should be determined from thealternating torque magnitudes determined from the response analysis. Appropriate stress concentrations should beincluded in determining the highest stress occurring at the location in question. Finally, the mean and alternatingstress should be superimposed on a Goodman diagram to determine if the design is acceptable. Discussion of thestress calculations can be found in References [1] and [2] and is provided in detail in 4.6. Figure 4-45 is a plot of themagnification factor of all the model stations in a torsional system. The magnification factor is defined as themagnitude of the maximum torsional response divided by the magnitude of the excitation used in the analysis. Themagnification factor can be used to determine the torque in a torsional system after the steady-state forced responseanalysis is completed.

Figure 4-45—A Typical Magnification Factor Plot of a Torsional Steady-State Response Analysis



4.4.4 Transient Torsional Response Analysis

4.4.4.1 General

Excitations that are transient in nature, such as the asynchronous starting of a synchronous motor, startup of aninduction motor, breaker reclosure after a power interruption, or an electrical fault in a motor or generator, all have thepotential to introduce electrical excitation. The electrical excitation is converted to torque in the motor or generator air-gap, which may excite torsional natural frequencies. Some of these events are not possible to avoid such as the startof a synchronous motor. Events such as premature breaker reclosure are preventable with proper time delay relays inthe motor switch gear. Other potential problems are possible to avoid by fault detection relays in the switch gear. As aresult, the only transient analyses normally performed are those of asynchronous startup of a synchronous motor andshort circuit conditions for a motor or generator. Classic references for torsional vibration problems with synchronousmotor start-ups include References [3] and [4], as well as References [9–14].

The nature of transient torsional vibration is somewhat unique in rotating machinery. Some transient torsionalvibration excitations may be caused by one event, such as the breaker reclosure of an electric motor or the shortcircuit of a motor or generator. Other transient torsional vibration may be caused by a rather limited number of events,such as the starting of a synchronous motor.

A failure of a power transmission component may occur in a single transient torsional vibration event. A transienttorsional vibration may go undetected until a certain number of events occur after which a low cycle fatigue failurecould occur. This potential failure can occur years after the initial startup of the machinery. In some cases, there arewarning signs such as the “clatter” of gear teeth revealing complete torque reversals or there may be lateral vibrationin a geared system created by torsional vibration (see 4.9). Usually transient torsional events in directly-on-lineconnected machinery have no outward warning signs.

The design of rotating machinery to avoid low cycle fatigue failure is somewhat unique. Often special couplings ortorque-limiting devices that interrupt torque transmission or fail intentionally are added to the power transmissioncomponents. Such devices are often used on generators where shorts or improper synchronization to existing gridsare seen as potential problems. Often, the stress may be rather high- in the case of synchronous motors, certainstresses may exceed the yield strength of the material in shear. If the potential number of transient torsional eventscan be reliably estimated, then it is acceptable practice to design for a limited number of starts to avoid exceptionallylarge shafts and couplings. As an example, starting of large synchronous motors is done infrequently. Even starting amotor once a week for 20 years would only result in 1040 starts. This philosophy is recognized and accepted in APIstandards.

Figure 4-46 is a typical output of a transient analysis of fault condition, namely a short between two phases of a motor.

4.4.4.2 Transient Analysis of a Synchronous Motor-Driven System

In order to analyze the transient response of an equipment train driven by a synchronous motor, one needs a modelof the equipment train, data to determine the mean and alternating torque developed by the synchronous motor, dataon any torsional damping added to the system, speed/torque requirements of the driven equipment, physicaldimensions of the segments of the equipment train and material properties of the train segments.

The torsional excitation created by the motor is normally modeled in two manners:

1) A plot of mean and alternating torque or a plot of the direct and quadrature axis torque. The direct axis torque isthat torque produced by the motor when the stator poles and the motor rotor poles are directly aligned. Thequadrature axis torque is that produced when the motor rotor poles are located midway between the statorpoles. The mean torque at a given speed is the average of the direct and quadrature torques. The alternatingtorque is one-half of the difference between the direct and quadrature axis torques. These plots are normallybased on an assumed voltage. Figure 4-47 is a typical plot for the torque versus speed characteristics of asynchronous motor. Another method to determine the mean and alternating torque is to create a slip frequency



dependent electrical circuit model of the motor and an electrical circuit of the electrical supply system from thegrid bus to the terminal connections of the motor [14]. A model such as this combined with the load speed vs.torque curve and a torsional model will develop the instantaneous torque which will account for the precisemotor terminal voltage and the position of the motor rotor relative to the stator. Since the motor torque isproportional to the motor terminal voltage squared, this model eliminates the assumption of constant voltagethroughout the motor speed range. A coupling that is introduced into the system to add damping must beaccurately modeled. Many of the couplings introduced to add damping have elastomeric elements, which havenonlinear stiffness and damping. The variable properties of the coupling must be property represented as thetorque varies during the starting sequence including resonance. Section 4.1.6 has addressed this topic. If thetorques are of a magnitude that would create stresses in the power transmission components that would nothave infinite life, then a low cycle fatigue analysis which includes plastic strain, must be performed. Toaccomplish this analysis, the power transmission component dimensions and material properties must beknown. Further details of this analysis are discussed in 4.6.

2) As is the case for a steady-state response analysis, a generic value of damping must also be input to theanalysis code. In general, the values given in 4.4.3 should be used, with one exception. If the system containsgears, the large transient torques that occur when the train passes through resonance with the first mode resultsin the gears undergoing a torque reversal. This simply means that the cyclic torque at the gears is greater thanthe mean torque, causing the teeth to unload and clatter. The clattering of the teeth generates impactivedamping, which acts to damp out torsional vibrations. Thus, it is fairly customary to increase the genericdamping ratio for a geared train from 1.5 % to 2.0 % when analyzing synchronous motor startups. Once thesystem properties and mean and excitation torque are defined, a transient torsional response analysis can beconducted to determine the system response. Figure 4-48 is a plot of the excitation torque of a synchronousmotor during a startup and the response of the system as the compressor train passes through torsionalresonance. Section 4.6 discusses the analysis of the system when the stresses exceed the infinite fatigue life ofthe material.

Figure 4-46—Transient Torsional Motor Fault Analysis Plot

Two-phase (line-to-line) short circuit analysis4 % relative damping

Torque (in.-lbf) = amplitude x 351195.0

Non

-dm

ens

ona

Stre

ss A

mp

tude

(per

Un

t Fu

Loa

d To

rque

)

0.000-10

0

+10

2.00Time (seconds)Decaying torque excitation

at 1 x and 2 x electricalline frequencies



4.4.4.3 Transient Analysis of Motor/Generator Short Circuits

As is discussed in detail in 4.5, when a motor or generator is subjected to a short circuit, large torque pulsations at line(and sometimes twice line) frequency are developed in the air-gap. These excitations die out rapidly and can beexpressed in the form of a decaying exponential equation. The motor or generator supplier can provide such anequation for any type of short circuit (i.e. line-to-line, three-phase, etc.) of interest.

In general, short circuits only cause torsional damage if their excitations generate a resonance with one of the train’storsional modes having significant activity at the motor or generator air-gap (usually the first mode). Since the shortcircuit generates excitations at line and twice line frequency, that means that a short circuit analysis is only needed ifthere are torsional modes in the proximity of line and/or twice line frequency and if those modes have significantactivity at the motor or generator air-gap. Since API requires a separation margin of 10 %, the minimum requirementwould be to perform the short circuit analysis if the separation margin is less than 10 %. However, there are someexperts who advocate a larger margin for this so it may be prudent to expand this value to 15 % or 20 %.

When dealing with a generator or fixed speed motor, determination of the need for this analysis is straightforwardsince line and twice line frequency are constant. However, if the train is driven by a variable speed motor, then theshort circuit excitation frequencies are no longer constant. Instead, the excitations occur at one and two times theelectrical frequency being fed to the motor by the variable frequency drive (VFD). For instance, a train located in theUnited States having a speed range from 50 % to 100 % of MCOS will have an electrical feed frequency that variesfrom 30 Hz to 60 Hz, with the frequency being proportional to the running speed. In this case, the short circuit analysisneeds to be performed if there are any torsional modes in the vicinity of one or two times any electrical frequency thatthe system can operate at. Obviously, the likelihood that a short circuit analysis is going to be needed is greater for avariable speed system than for a fixed speed train. The analysis should be run for the running speed that puts thesystem as close to resonance as possible.

Figure 4-47—Speed Torque Curve for a Synchronous Motor

3.0

2.5

2.0

1.5

1.0

0.5

0.0

-0.5

-1.0

-1.5

Motor torque vs. speed curve 24-Mar-76 13:11:23

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2.0

The upper curve is the Direct Axis Torque.The center curve is the Average Torque.The lower curve is the Quadrature Axis Torque.

Speed in PU



The performance of the short circuit analysis is similar to that for the synchronous motor startup in that the excitationtorque (as defined by the motor/generator vendor’s equation) is applied to the air-gap at time zero and a transientsimulation is run to determine the torque vs. time history for every shaft element in the model. However, in contrast tosynchronous motor startups, there is no concern with fatigue since short circuits are not expected to occur more thana handful of times (if that) in the life of a train. Thus, the acceptability criterion is simply survival. The maximumstresses in all shaft elements must be below their corresponding shear yield strengths and the torques in all couplingsmust be below their max momentary torque ratings.

4.4.5 References

[1] Wachel, J. C. and Szenasi, F. R., “Analysis of Torsional Vibrations in Rotating Machinery,” Proceedings of theTwenty-Second Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, CollegeStation, TX, pp. 127–151 (1993).


[3] Chen, W. J., “Torsional Vibrations of Synchronous Motor Driven Trains Using p-Method,” ASME Journal ofVibration and Acoustics, pp. 152–160 (January 1995).

Figure 4-48—Plot of Synchronous Motor Transient Response Analysis

2.5

0.0

0.0 1.5 3.0 4.5 6.0

Shaft Torque

2

0

ElastomericCoupling DampingFactor

1

0

Motor Speed

2.5

0.0

Shaft Torque

2

0

-2

Motor Torque



[4] Corbo, M. A. and Cook, C. P., “Torsional Vibration Analysis of Synchronous Motor-Driven Turbomachinery,”Proceedings of the Twenty-Ninth Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&MUniversity, College Station, TX, pp. 161–176 (September 2000).

[5] Simmons, H. R., “Vibration Energy: A Quick Approach to Rotor dynamic Optimization,” ASME Paper Number76-PET-60, 1976.

[6] Simmons, H.R. and Smalley, A.J., “Lateral Gear Shaft Dynamics Control Torsional Stresses in Turbine-DrivenCompressor Train,” ASME Journal of Engineering for Gas Turbines and Power, October 1984, pp. 946–951.

[7] Walker, D. N., Torsional Vibration of Turbomachinery, McGraw-Hill, New York, NY (2004).

[8] Southwest Research Institute, “Recommended Practice for Control of Torsional Vibrations for High SpeedSeparable Reciprocating Compressors,” GMRC (May 2002).

[9] Brown, R.N., “A Torsional Vibration Problem As Associated With Synchronous Motor Driven Machines,”ASME Journal of Engineering for Power, Paper No. 59-A-141, pp. 215–220 (July 1960).

[10] Pollard, E.I., “Torsional Response of Systems,” ASME Paper No. 66-WA/Pwr-5 (Dec. 1966).

[11] Godwin, G. L. and Merrill, E. F., “Oscillatory Torques During Synchronous Motor Starting,” IEEE Transactionson Industry and General Applications, Paper No. 70 TP 16-IGA, pp. 258–265 (May/June 1970).

[12] Sohre, J. S., “Transient Torsional Criticals of Synchronous Motor-Drive, High Speed Compressor Units,”ASME Paper No. 66-FE-22 (1965).

[13] Thames, P. B. and Heard, T. C., “Torsional Vibrations in Synchronous Motor-Geared Compressor Drives,”AIEE Paper No. 59-657, pp. 1053–1056 (December 1959).

[14] Mruk, G. K., “Compressor Response to Synchronous Motor Startup,” Proceedings of the SeventhTurbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, TX, pp. 95-101 (1978).

4.5 Torsional Excitation Sources in Rotating Machinery

4.5.1 Continuous Excitations

4.5.1.1 General

There are a number of potential sources of torsional excitation in rotating machinery. Fortunately, most of these aresmall enough in magnitude that they will not have significant impact on the torsional system. Tables of potentialexcitations with associated frequencies and magnitudes can be found in Table 5 of Reference [1] and Table 1 andTable 4 of Reference [2]. Details of these excitations are elaborated in the references and the material which follows.

4.5.1.2 Centrifugal Compressors

Centrifugal compressors are not normally considered significant sources of torsional excitation within a system.References [1] and [2] discuss the magnitudes and frequencies associated with torsional excitations from acentrifugal compressor. The authors recommend that an excitation of 1.0 % of the operating torque be used forsynchronous (1X) frequency excitations and 0.5 % of operating torque be used for nonsynchronous frequencies,which must be defined.

It is well-documented that surge of a centrifugal compressor can lead to lateral vibration of the rotor. Since, in additionto lateral vibration, surge can lead to overheating and excessive loading of bearings, particularly thrust bearings, anti-



surge control systems are usually implemented in centrifugal compressors. It is less clear whether surge can excitetorsional vibrations but, in any case, it is not typically considered in a torsional analysis.

Sub-synchronous vibration associated with stall cells has the potential to develop torque variations within acompressor—however, the magnitudes and frequencies of these torque variations can not be quantified. Stall mayresult in lateral vibration that would alert users to its presence. However, excitation from stall cells is not typicallyconsidered in a torsional analysis.

In accordance with References [1] and [2], the only centrifugal compressor excitations considered in a typicaltorsional analysis are steady-state 1X and 2X excitations arising from generic sources such as unbalance,misalignment, and ellipticity.


The torsional excitations coming from the compressor shafts (i.e. pinion shafts) in an integrally geared compressorare the same as those arising in a centrifugal compressor. In accordance with References [1] and [2], thoseexcitations are 1X (having a magnitude of 1.0 % of operating torque) and 2X (having a magnitude of 0.5 % ofoperating torque). Although the compressor shafts are also excited by the gears at meshing frequency, thosefrequencies are normally too high to generate issues.

There are also reports that aerodynamic excitations arising at the compressor wheels can sometimes be significant.For instance, Stueber et al. [3] report a case in which “high” observed torsional vibrations in a single-stage integrallygeared compressor were attributed to aerodynamic excitations at the compressor wheel. However, when queried, theauthors of Reference [3] were unable to provide an estimate of the magnitude of the aerodynamic excitation, so it isunclear whether the excitation was large enough to be of concern when designing these machines. In general,aerodynamic excitations are not included in a typical torsional analysis of an integrally geared compressor.

4.5.1.4 Steam Turbines

Torsional excitations in steam turbines are similar to those of centrifugal compressors, as identified in Reference [1].That is, the only steam turbine excitations considered in a typical torsional analysis are steady-state 1X and 2Xexcitations arising from the same generic sources that are present in centrifugal compressors. Additionally, steamturbines can also have excitations at nozzle or blade-passing frequency. However, the frequencies associated withthese excitations are normally so high as to be of no practical interest.

4.5.1.5 Gearing

The magnitude of torsional excitations arising in high quality gearing is quite low. References [1] and [2] state it to be1.0 % or less of rated torque. The excitation frequencies occur at one and two times rotational frequency, of both thegear and pinion, and gear mesh frequency. It should be noted that badly worn gearing can cause excitations manytimes the magnitudes defined for high quality gearing. Unfortunately, there is no known method for calculating theexcitation magnitude as a function of the AGMA quality level.

Geared units have some unique features that must be considered when discussing their torsional vibrationcharacteristics. Lateral vibrations and torsional vibrations are normally not related; however, in geared units there is arelationship between these two very different types of machinery vibrations. The compound angle created by the geartooth helix and pressure angles will result in torque variations due to variations in radial displacement, such asdisplacement due to unbalance. Conversely, torque variations will result in radial and axial force variations. Torque istransmitted from one shaft to another in a gear set via radial forces acting between the gear teeth. As a result of thisinteraction, there will be mean tooth forces that are a function of the average torque transmitted by the shafts, andfluctuating tooth forces that are the result of variations in the torque or varying radial vibration. The bearingssupporting the shafts have significant radial loads as a result of these tooth forces. This relationship between torqueand radial load can cause torsional vibrations and radial vibrations to interact as noted above. These interactions aredescribed in more detail in Section 4.9.



The relationship between torque tooth load and radial bearing load means that torsional vibration can impact thedesign of the gear teeth and the shaft support bearings. In the presence of an alternating torque, a significantconsideration is the selection of the direction of rotation of the gear unit to load the various bearings in the mostfavorable way. The weight of the rotating assemblies and the direction of load from the gear teeth can increase orreduce the bearing loads. Large torque fluctuations can cause the radial bearing load to change load vectororientation. The shafts can actually rise and fall in the bearing clearance as a result of radial loads developed byalternating torque.

All gear teeth are subject to cyclic stress loading during normal operation as gear teeth enter and leave the gearmesh. These cyclic stresses result in the eventual failure of the teeth from fatigue. This fatigue failure can bedesigned to occur after so many cycles of operation that the design life is longer than the expected life of themachinery train. However, torsional vibration can cause reversals in shaft torque or large variations with low meantorque. This torsional vibration can cause the teeth to separate. Impact load associated with teeth separating andthen re-contacting can significantly increase the tooth stresses and thus reduce the life of the gear teeth. Even withouttooth separation or impact due to torque reversal, cyclic stresses associated with torsional vibration will reduce thenumber of cycles to tooth failure. A good rule of thumb is that variations in shaft torque, i.e. torsional vibration, shouldnot exceed 30 % of the mean transmitted torque during any mode of operation. Torsional impact loads fromconnected machinery, for instance compressor surge, can excite a torsional transient resonance and causesignificant increase in gear teeth stresses. Transient torsional loads during synchronous motor start-ups are anothercase of tooth loading that can reduce the life of a gear set.

During operation gear teeth are in contact with one another with only a very thin film of oil between the tooth flanks.Therefore, proper gear tooth geometry is essential for smooth transmission of motion from one shaft to the other. Theprofile of the gear flank and proper matching between gear teeth along the face width are important in insuring evenstress distribution on the gear teeth.

Tooth spacing errors are the most significant torsional excitation arising at gear meshes. Such gear tooth spacingerrors normally increase the stress on the gear teeth and most gear rating formulas include a “dynamic factor” toaccount for these loads. The dynamic factor is a function of the gear accuracy and the peripheral speed of the gearteeth. It assumes the connected train is free from significant torsional response. Gear spacing errors are broken intotwo broad categories, tooth-to-tooth and total accumulated. Both have an effect on the gear stress but theaccumulated errors occur closer to shaft speeds and are more likely to effect torsional vibrations. The large mass ofthe gear and pinion shafts limit the amount of forced torsional vibrations transmitted to the shafts and connectedmachinery from gear tooth errors.

For information on tooth accuracy, refer to AGMA 2015-1-A01 and A02, and ISO TR 10064 Parts1–4. Thesedocuments specify accuracy grades and measurement techniques but do not specify accuracy requirements forvarious applications. It is generally recognized that high-speed gears should have an accuracy grade of 5 or better.API 613, 5th Edition specifies an accuracy grade of 4 or better. Note that gears with accuracy grades better than 3 areextremely difficult and expensive to manufacture. When speeds are low, accuracy grades of 6 or less may be used.Gear teeth are manufactured to tight tolerances, and mechanical damage during handling must be avoided. Gearsets operate as pairs of shafts so the accuracy of gear teeth on individual shafts is only as good as the supportstructure, housing, bearings, and foundation, which support the shafts.

API 613 and API 677 are the American Petroleum Institute Standards on gear unit requirements. For more detail ongear terminology and gear rating, refer to the American Gear Manufacture Association standards such as ANSI/AGMA 1010-E95 for terminology on wear and failure of gear teeth, ANSI/AGMA 2001C95 for fundamental ratingmethods for gear teeth, ANSI/AGMA 6010-F97 for rating of enclosed gear units, and ANSI/AGMA 6011-H98 for ratingof enclosed high speed gear units.

4.5.1.6 Constant Speed Induction Motors

Excitations from directly-on-line connected induction motors at operating speed are relatively low. The primaryexcitations occur at line frequency and twice line frequency. Theoretically, there is no excitation at line frequency for a



three-phase motor, and it is difficult to find a mechanism to generate this component. However, there seems to beempirical observations that indicate the existence of line frequency excitations. A conservative approach is to assumethe presence of line frequency excitations until this open question is resolved. The situation is more obvious with twiceline frequency excitations. These excitations are generated, at least, by an unsymmetric voltage supply or by a rotoreccentricity. References [1] and [2] suggest representative magnitudes of 1.0 % of output torque for line frequencyexcitations and 0.5 % for twice line frequency.

The most serious torsional excitations result from premature breaker reclosure or electrical faults. As discussed in4.5.2, transient excitations may also occur during initial energization of the motor. It should be recognized that mostpotential excitations occur at one or two times line frequency. For this reason, it is prudent to avoid system naturalfrequencies, having significant modal amplitude at the motor core, near one or two times electrical line frequency formotor-driven equipment.

4.5.1.7 Electric Motors Driven by Variable Frequency Drives

The unique torsional characteristics and problems associated with trains that are driven by motors controlled byvariable frequency drives (VFDs) are discussed in detail in Section 4.10.

In addition to the excitations that the VFD directly puts out, the presence of the VFD alters the excitations generatedby the motor. As stated in 4.5.1.6, in a fixed speed motor, the primary excitations are at line and twice line frequency,which appear as horizontal lines on a Campbell diagram. However, as pointed out in Reference [2], when driven by aVFD, the motor’s primary excitations become speed-dependent and, thus, appear as positively-sloped lines on aCampbell diagram. Neglecting the effect of slip, the order number (i.e. slope of the line) for the line frequencyexcitations is given by the following:

Nord = 0.5 x Np (4-25)

where

Nord is the order number for line frequency excitations;

Np is the number of poles in motor.

Thus, for a two-pole synchronous motor, line frequency excitations occur at 1X, and for an asynchronous motor at aslightly higher frequency. Obviously, the order number for twice line frequency excitations is simply twice the valueobtained from Equation 4-25.

4.5.1.8 Screw Compressors

Rotary screw compressors are high speed positive displacement compressors. The primary frequency of thispulsation would be the lobe passing frequency of the compressor. The lobe pass frequency can be calculated fromEquation 4-26.

(4-26)

LPFN60----- Nrl=



where

N is the rotor speed, rpm;

LPF is the lobe pass frequency, Hz;

Nrl is the number of rotor lobes.

There is no defined torsional excitation magnitude and no published torsional excitation measurements of screwcompressor torsional excitation. If one were to approximate a magnitude of torsional excitation it would be reasonableto assume that a relationship exists between the dynamic pressure pulsations present in the discharge passage ofthe compressor and the absolute pressure level in the discharge plenum. Historically a magnitude of 10 % of therated torque has been assumed for a variable speed screw compressor. The dynamic torque would be phased tooccur with the rotation of the meshing rotor pairs.

4.5.1.9 Reciprocating Engines and/or Compressors

The torsional excitations associated with reciprocating machinery are described in Section 4.3.3.

4.5.1.10 Electric Generators

As is the case with fixed-speed motors, the primary steady-state excitations arising from generators occur at line andtwice line frequency. Once again, References [1] and [2] suggest representative magnitudes of 1.0 % of output torquefor line frequency excitations and 0.5 % for twice line frequency.

4.5.2 Transient Excitations

4.5.2.1 Asynchronous Starting of Synchronous Motors

4.5.2.1.1 General

Synchronous motors are normally used for higher horsepower applications, where their operation with leading,lagging, or unity power factors is used to balance the power factor in a plant. Synchronous motors can generate largetorsional excitations during start up—from the moment they are initially energized until the time the field is applied andthe motor is synchronized to the line frequency. While this starting cycle is very short (under 30 seconds), every timethe motor is started, the torsional excitation is so strong that the shafting material will often be subjected to stressesabove its endurance limit when resonance with the first torsional mode occurs. After the initial commissioning of theplant, these large motors are usually started only a few times a year. It is possible to design systems driven by asynchronous motor for 1000 or more starts, which may amount to 20 or more years of service. Starting a system oncea week for 20 years would accumulate 1040 starts.

The torsional excitation developed by a synchronous motor will occur at two times the slip frequency when the motoroperates in the asynchronous starting mode until the field is applied and the motor rotor is synchronized to linefrequency. Slip frequency is the difference between the line frequency in the stator and the rotational speed of therotor. Thus, slip frequency varies linearly from line frequency at zero rpm (with the motor energized) to zero when themotor reaches synchronous speed. This means that the excitation frequency is two times line frequency when themotor is initially energized and reduces linearly with speed until the field is applied, at which time the torsionaloscillations disappear. Therefore, the motor has the potential to excite torsional natural frequencies anywherebetween 0 and 100 Hz for 50 Hz electrical systems and 0 to 120 Hz for 60 Hz systems. Since most practical trainshave at least one torsional mode (and often more) in this range, this means that resonance is practically inevitable.The two times the slip frequency torsional excitation excludes synchronous motors that are started by variablefrequency drives (VFDs). The unique torsional characteristics and problems associated with trains that are driven bymotors controlled by VFDs are discussed in detail in Section 4.10.



Synchronous motor drives are often used for high horsepower centrifugal compressor applications. In such systems,the largest transient response normally occurs when the twice slip frequency excitations are in resonance with the 1st

torsional mode. The responses when resonances with higher modes occur are normally insignificant. Typically, the 1st

torsional natural frequency is 30 Hz or less for such a system. If the first mode is at 30 Hz, resonance will occur at25 % slip or 75 % rated speed for a 60 Hz electrical system. Equation 4-27 yields the motor speed at which the twiceslip frequency excitations are in resonance with the first torsional mode:

(4-27)

where

Ncr is the motor rotational speed at which torsional resonance occurs, rpm;

Ns is the synchronized motor rotational speed, rpm;

fn is the first torsional natural frequency, Hz;

fe is the electrical line frequency, Hz.

4.5.2.1.2 Solid Pole Versus Laminated Pole Synchronous Motors

In synchronous motor-driven trains designed in recent years, solid pole motors are often being used in lieu oflaminated pole designs. This trend is driven by the solid pole design providing an advantage from the standpoint ofequipment cost for the motor and associated gearing. Additionally, mechanical stress limitations constrain laminatedpole designs to operating speeds less than or equal to those associated with 6-pole machines, that is 1200 rpm for aconventional 60 Hz electrical system. The more frequent use of solid pole motors has, unfortunately, made theproblem of torsional vibrations with synchronous motor drives more pronounced. This is because the magnitudes ofthe twice slip frequency excitations in a solid pole synchronous motor are significantly larger than those of a laminatedpole design. Additionally, the designer of a laminated pole motor had many design options in the design of theamortisseur windings of the synchronous motor poles but this is not the case for the solid pole design. Also, the olderdesigns that utilized slip rings had the ability to alter the design of the exciter discharge resistor. This is no longer anoption with self-excited solid state exciters. Therefore, some of the older laminated pole designs could bemanufactured so that the peak torque developed for a typical single body geared compressor train would not exceed5 times rated torque. Unfortunately, similar designs with solid pole synchronous motors now easily reach 8–10 timesrated torque. Reference [4] identifies a uniform method for calculating and measuring torque pulsations that occurduring starting of synchronous motors.

4.5.2.1.3 Methods of Reducing Torsional Response in Synchronous Motor-Driven Trains

One of the more common methods of reducing the magnitude of transient torsional vibration in synchronous motor-driven trains is to reduce the torsional stiffness of the coupling between the motor and the component it is driving(often a gear shaft). This acts to de-couple the motor from the rest of the train and, thereby, reduces the peak torquesat resonance. Another method that is often effective is to introduce damping to the train. Elastomeric couplings areoften used since they provide both functions at the same time.

Another method is to increase the rate of acceleration in the region of resonance with the first mode. Since a high rateof acceleration is desirable to both avoid accumulating multiple high torque cycles and to minimize the magnitude ofthe peak torque by accelerating through resonance as quickly as possible, then anything that can be done to increasethe rate of acceleration should be considered. One way to achieve this is to reduce the magnitude of the connectedload on the motor during starting. With a compressor system, this is accomplished by throttling the system tominimum flow, or decreasing the system pressure level. Reducing the connected load allows more of the air gap

Ncr Ns 1fn

2fe

------- –=



torque to be available to accelerate the system inertia to synchronous speed. It is common practice to startcompressors driven by synchronous motors in the unloaded or partial load condition.

Another method that is commonly used is a reduced voltage start. With this, the motor voltage is reduced (normally toaround 75 % to 85 % of full voltage) during starting so that the system passes through resonance with lower voltage,then switches to a higher voltage to accelerate to synchronization. Naturally, one must ensure that the switch is madeafter the train has passed through resonance with the first torsional mode. Such systems may be difficult to design,however, since along with the benefit of reduced pulsating torque, the reduced voltage also carries the penalty ofreduced torque to accelerate the train (aka average torque). Both the average and pulsating torques are proportionalto the square of the voltage. In general, the net effect of using a reduced voltage start is beneficial but the pros andcons of it must be evaluated, on a case-by-case basis, through a low cycle fatigue analysis of the system to see whichstarting option offers more service life.

4.5.2.2 Across-the-Line Starting of Induction Motors

When an induction motor is started across-the-line, it instantaneously generates very large torques at line frequencywhich die out quite quickly. To ensure that these torques do not cause damage to the train, a transient analysis of thestartup condition can be performed, similar to that for the startup of synchronous motors. In general, the excitationtorque is defined by a decaying exponential equation that can be obtained from the motor supplier. This excitation isapplied to the torsional model at the location of the motor core. In general, this analysis is not normally needed unlessthe train has a torsional mode, having significant modal amplitude at the motor core, at or near line frequency. Moredetails on this phenomenon and these analyses are provided in References [5–7].

4.5.2.3 Breaker Reclosure of Electric Motors

Breaker reclosure is a term used to describe the interruption and reapplication of electrical power to an electric motor.The discussion that follows pertains only to induction motors. It is not recommended to attempt reclosure ofsynchronous motors as reclosure transient torques have been reported to be as high as 30 to 40 times rated torque(Reference [8]). Breaker reclosure can occur when the electric power supply is interrupted by faults in the powersystem such as lightning during thunder storms, faults in transformers, or switching power on the electric powersupply grid. These interruptions of electric power supply can, if the conditions are correct, develop high levels ofinstantaneous transient torques. These torques can be as high as 15 to 20 times (References [9–12]) the rated torqueof the motor if the duration of the interruptions and the load characteristics of the driven equipment combine tomaximize the transient torque. Reclosure torques are often reverse in direction. Reverse failures in keys, couplinghubs, etc. can, therefore, be tell-tale symptoms pointing to damage caused by excessive torque from breakerreclosure.

When the power is removed from an induction motor a residual magnetic flux remains within the motor. The period oftime that it takes to reduce the residual motor voltage to 36.8 % of rated voltage is referred to as the motor’s opencircuit time constant (Reference [9]). As a result of this residual flux, the motor will self-generate a terminal voltagethat will rotate with the motor rotor. The generated frequency will be equal to line frequency times the ratio ofinstantaneous motor speed divided by synchronous motor speed. Therefore, if power is reapplied prior to decay ofthe self-generated voltage, the reapplied voltage will have frequency components of both the self-generated voltageand line voltage. As a result of this broad frequency spectrum, there is significant potential of exciting torsional modes,especially modes below line frequency. As the voltage reduces with the motor speed, there is a phase shift betweenthe self-generated voltage and the line voltage. If the power is reapplied to the motor prior to the decay of the residualvoltage, the effective voltage seen by the motor is the vector sum of the residual voltage and the reapplied linevoltage. In the worst case, the residual voltage may not have decayed below rated voltage and could be 180 degreesout-of-phase with the line voltage. If the electrical power is reapplied to the motor under the condition where theresidual voltage is the rated motor voltage and 180 degrees out-of-phase with the line voltage, the effective voltageseen by the motor is 200 % of the rated voltage. Since the motor torque is proportional to the square of the appliedvoltage, the motor is capable of developing 400 % of rated torque.



However, when the motor is deenergized, the driver and driven equipment decelerate. The rate of deceleration is afunction of the level of power absorbed by the driven equipment and the amount of stored energy in the rotatinginertia of the equipment train. If the motor is reenergized at a point in time where the speed of the motor is at alocation where the motor torque is a maximum on its speed-torque curve, then when the motor is reenergized, themotor will develop a torque that is the maximum shown on the speed-torque curve if the effective motor voltage isidentical to the voltage upon which the speed-torque curve was created. However, during a breaker reclosure, theeffective voltage can be much higher than normal starting voltage. If the residual voltage is equal to the line voltageand 180 degrees out-of-phase with the line voltage, and the equipment has slowed to a point where the startingtorque of the motor would normally be 200 % to 250 % of rated torque, then the torque developed by the motor duringreapplication of electrical power could be 8 to 10 times the rated torque. Any torsional resonance in the system couldamplify this magnitude. Torques of this magnitude will create stresses in the power transmission components, such asshafting, couplings, gearing etc., that can exceed these components’ material ultimate strengths and, thereby, causefailure (Reference [9]).

The protection recommended by motor manufacturers is to avoid the reapplication of power until the residual voltageis 25 % to 30 % of the motor’s rated voltage (Reference [13]). NEMA Standard MG1 includes developedrecommendations for breaker reclosure (Reference [14]). As a practical rule, motor vendors recommend the voltageshould not be reapplied until a period of two open circuit time constants has elapsed between the time when thepower is interrupted and the power is reapplied. MG1 also provides guidelines for the phasing of the reappliedvoltage.

It should also be mentioned that any capacitors at the motor terminals used for power factor correction will increasethe value of the open circuit time constant. The motor vendor should be advised of any capacitors in the motor circuit.

4.5.2.4 Short Circuits in Electric Motors and Generators

Torsional natural frequencies of systems with motors and generators can be excited by transient fault conditions dueto short circuits.

Transient torsional excitation can be created by the following short circuit conditions.

a) A three-phase short circuit is a short between all three phases of a three-phase motor or generator that willdevelop a maximum electromagnetic torque at electrical line frequency.

b) A two-phase (or line-to-line) short circuit is a short between any two of the three phases of a three-phase motor orgenerator. This short will develop excitation at electrical line frequency and twice line frequency.

c) A single phase-to-ground short circuit is a short between any one of the three phases of a three-phase motor orgenerator directly to ground. This short will develop excitation at electrical line frequency and twice line frequency

d) A two phase-to-ground short circuit is a short between any two of the three phases of a three-phase motor orgenerator directly to ground. This short will develop excitation at electrical line frequency and twice line frequency.

The three-phase and line-to-line short circuit conditions (which are normally the worst cases) are often analyzed tosize couplings or to meet specification requirements. The other two conditions are rarely analyzed. In any case, theelectrical switch gear must detect such faults rapidly enough so as not to endanger the motor or connectedmachinery. However, as a precaution, it is prudent to avoid system torsional natural frequencies, having significantmodal amplitude at the motor or generator core, at one and two times electrical line frequency.

4.5.2.5 Mal-Synchronization of Generators

Large transient torques are not expected during normal operation of a synchronous generator. However, severalfaults can occur that lead to the creation of a large torque pulsation over a brief time span. These faults can comefrom two distinct sources. The first is a short circuit, which was described in Section 4.5.2.4. The second can be



produced if there are errors in the synchronization of the generator to the power grid. As with short circuits,synchronization errors can be either three phase or single phase.

The existence of fault torques is well known. However, accurate prediction of fault conditions is difficult, especially forextended periods of time following the fault. While the initial fault may be a single incidence, it can be followed by acascading of events each making predictions more inaccurate and unreliable. Kirschbaum [15] developed a series ofequations describing several fault conditions in a synchronous generator. The generator was considered an “ideal”synchronous machine. Other assumptions included a uniform air gap and symmetrical windings. Hizume [16] alsoprovides useful details on this type of analysis.

Examples of the torques produced for two of the fault conditions in a generator (12 MVA, 50 Hz, 8 poles) are plotted inFigure 4-49 and Figure 4-50. Although the three-phase synchronizing torque has roughly the same peak torque levelas the line-to-line short circuit torque, its components are of a single frequency (fundamental) and largelyunidirectional. This applied torque will produce higher shaft stresses and thus more low cycle fatigue damage.Fortunately, these torque levels shown are for a synchronization error of 45 degrees. With current technology, this isvery conservative. Synchronization errors are expected to be less than 10 degrees, given today’s electroniccontrols.

Figure 4-49—Transient Torque Associated with a Generator Line-to-line Short Circuit

-15

-10

-5

0

5

10

15

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Torq

ue (p

.u.)

Time (s)

Single Line-to-line Short Circuit Torque



4.5.3 References


[2] Corbo, M. A. and Malanoski, S. B., “Practical Design against Torsional Vibration,” Proceedings of the Twenty-Fifth Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, TX,pp. 189–223 (1996).

[3] Stueber, R. R., Homji, R., Beckers, C., and Rivadeneira, J. C., “Investigation and Elimination ofAerodynamically Induced Torsional Vibrations on an Integrally-Geared Centrifugal Compressor,” Case StudyPresented at the Thirty-Ninth Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&MUniversity, College Station, TX (October 2010).

[4] IEEE Standard 1255-2000, Guide for the Evaluation of Torque Pulsations During Starting of SynchronousMotors.

[5] Melfi, M. J. and Umans, S. D., “Transients During Line-Starting of Squirrel-Cage Induction Motors,” IEEEPaper PCIC-2010-23 (2010).

[6] Ran, L., Yacamini, R., and Smith, K. S., “Torsional Vibrations in Electrical Induction Motor Drives During Start-Up,” ASME Journal of Vibration and Acoustics, pp. 242–251 (April 1996).

[7] Srinivasan, A. and Weber, P., “Start-Up Simulations for Induction and Synchronous Motor Driven CompressorTrains,” Proceedings of the ASME 2011 International Design Engineering Technical Conference, ASME PaperDETC2011-47672, Washington, DC (August 2011).

[8] Das, J. C., “Effects of Momentary Voltage Dips on the Operation of Induction and Synchronous Motors,” IEEETransactions on Industry Applications, Volume 26, Number 4 (July/August 1990).

Figure 4-50—Transient Torque Associated with Generator Synchronization Error of 45 Degrees

0

2

4

6

8

Torqu

e (p.u

.) Three Phase Synchronization T



[9] Brozek, B., Induction Motor Open Circuit Time Constant, General Dynamics Publication.

[10] Mruk, G. and Halloran, J., “Torques Due to Electrical Reclosures for Induction Motor Driven CentrifugalCompressors,” Proceedings of the Twelfth Annual Meeting of the Vibration Institute, (May 25-27, 1988).

[11] Daugherty, R. H., “Analysis of Transient Electrical Torques and Shaft Torques in Induction Motors as a Resultof Power Supply Disturbances,” IEEE Transactions on Power Apparatus and Systems, pp. 2826–2836 (1982).

[12] Srinivasan, A., “Analytical Investigation of the Effects of Induction Motor Transients on Compressor DriveShafts,” Proceedings of Turbo Expo 2011, ASME Paper GT2011-45033, Vancouver, BC (June 2011).

[13] Daugherty, R. H., “Bus Transfer of AC Induction Motors, A Perspective,” IEEE Paper PCIC-89-07 (1989).

[14] Bottrell, G. W., “Fast Bus-Transfer Techniques for Maintaining Full Plant Protection,” IEEE Paper PCIC-89-08(1989).

[15] Kirschbaum, H. S., “Transient Electrical Torques of Turbine Generators During Short Circuits andSynchronizing,” Electrical Engineering Transactions, Vol 64, pp. 65–70 (February 1945).

[16] Hizume, A., “Transient Torsional Vibration of Steam Turbine and Generator Shafts due to High SpeedReclosing of Electric Power Lines,” ASME Journal of Engineering for Industry, pp. 968–979 (August 1976).

4.6 Fatigue Analysis

4.6.1 High Cycle Fatigue for Continuous Excitation Sources

If the undamped analysis indicates an interference between an undamped torsional natural frequency and a shaftrotative speed or other potential excitation mechanism, occurring within the operating speed range or within thespecified ±10 % margins of the operating speed range, and the train design cannot be altered sufficiently to removethe resonant interference, then a steady-state torsional response and vibratory stress analysis must be performed toensure that rotor shafts and couplings are not overstressed (note: this avenue cannot be pursued if the interferenceinvolves a coupling mode excited by the 1X excitation). Potential areas of vibratory stress concentrations arekeyways, stepped diameters, and couplings. Since there are several elements in couplings that can be overstressed(i.e. flexible elements, bolted flanged connections, spacers, etc.) and since the calculation of stresses in theseelements is rather complex, the structural integrity of couplings must be determined in cooperation with the couplingmanufacturer. In many cases, the coupling supplier will provide the analyst with a max continuous torque rating for thecoupling. In that case, if the sum of the mean torque required to drive the load and the cyclic torque due to thetorsional vibration is below this rating, the coupling will be acceptable.

Results generated from the damped torsional response and vibratory stress analysis may indicate that shafts must bere-sized (or stress concentration factors reduced) to safely accommodate the high levels of vibratory stress resultingfrom operation close to a torsional natural frequency. Figure 4-51 displays a typical plot of calculated oscillatorystresses versus running speed. The two peaks present in this plot indicate excitation of the first and second traintorsional natural frequencies due to the 1 and 2 times operating speed torque pulsations, respectively.

Calculated stresses will be governed by assumptions regarding the level of available torsional damping as well asoverall expected torque excitation levels at given frequencies. These key parameters are normally set with consent ofboth the purchaser and the vendor and are based on experience, measurement, and/or published literature. Once theanalysis has been run, the calculated cyclic stresses should be compared to the shaft material’s endurance limit,derated for surface finish, size, and other effects, to determine the acceptability of the shaft design. A comprehensivereference on torsional fatigue can be found in Reference [1]. Reference [2] discusses the calculation of shaft stress,stress concentrations and material fatigue stress limitation guidelines applicable to high-cycle fatigue. Reference [3]also discusses shaft stress and provides a detailed procedure for calculating the relevant derating factors and theendurance limit.



Although the references should be consulted for a more precise treatment, the general stress analysis procedure isas follows.

— For each shaft of interest, the mean and cyclic torques should be converted to mean and cyclic shear stressvalues using the following equation from strength of materials:

(4-28)

Figure 4-51—Shaft Operating Stress as a Function of Shaft Operating Speed

1/2 x 2nd Torsionalnatural frequency

1 x 1st torsionalnatural frequency

1250 RPM 1587 RPM

Tors

ona

Sha

ft S

tress

, ps

p-p

Operating Speed, RPM

00 500 1000 1500 2000 2500

2000

4000

6000

8000

0

10

20

30

40

50

Torsona

Shaft S

tress, N/m

m2 p-p

T RJ

------------=



where

is the shear stress, N/m2 (psi);

T is the torque, N-m (in.-lbf);

R is the shaft outside radius, m (in.);

J is the shaft polar area moment of inertia, m4 (in.4).

— The cyclic stress should then be increased by the appropriate stress concentration factor for the stress riser inquestion, [5] (in almost all systems, the “weak link” shaft is in a region of stress concentration). For any stressriser, there are two stress concentration factors—the geometric stress concentration factor, kT, and the effectivestress concentration factor, kF. The geometric stress concentration factor, kT, is solely dependent on the geometryof the stress riser and is totally independent of the part’s material and condition. The geometric stressconcentration factor is normally obtained either using a handbook, such as Reference [4], or from finite elementanalysis. It should be noted that since the stress concentration factors of most common stress risers in torsionalsystems, including keyways, are strong functions of the fillet radii, there are significant advantages to using thelargest fillet radii possible.

— The geometric stress concentration factor is determined assuming that the part is made from an ideal materialwhich is isotropic, elastic, and homogeneous. Fortunately, the deviations from these assumptions that occur inreal materials tend to reduce the impact of the stress riser. To account for these real effects, a second stressconcentration factor, kF, which is always less than or equal to kT, is defined as follows:

kF = 1 + q x (kT – 1) (4-29)

where

kF is the effective stress concentration factor;

kT is the geometric stress concentration factor;

q is the material notch sensitivity.

It is seen that the geometric and effective stress concentration factors are related by a parameter, q, whichrepresents how sensitive the material is to notches. The notch sensitivity factor is defined by the above equationand, by definition, is always between zero and one. A material having a notch sensitivity of zero is totallyinsensitive to notches such that its effective stress concentration factor is always 1.0, regardless of notchgeometry. On the other hand, a material having a notch sensitivity of one is extremely sensitive to the presenceof notches and, as a result, its geometric and effective stress concentration factors are equal. References [5] and[6] can both be used to determine the notch sensitivity for a given configuration. Once the notch sensitivity isobtained, it should be combined with the geometric stress concentration factor using the above equation to obtainthe effective stress concentration factor, kF. This is the factor that should be applied to the cyclic stress.

— Since the stress state in each shaft consists of two components, namely a mean and a cyclic stress, these mustbe converted to an equivalent fully-reversing stress (i.e. cyclic stress corresponding to a mean stress of zero).Although there are several methods for making this conversion (Stephens [7] describes several of them), one ofthe simplest and most commonly used is the Goodman diagram. The linear relation given on this diagram can beconverted into the following equation:

Seqv = CYCLIC x SSU / (SSU – MEAN) (4-30)



where

Seqv is the equivalent fully-reversing shear stress, N/m2 (psi);

CYCLIC is the cyclic shear stress—including stress conc. factor, N/m2 (psi);

MEAN is the mean shear stress, N/m2 (psi);

SSU is the material ultimate shear strength, N/m2 (psi).

— The ultimate shear strength for the shaft material can be calculated from the specified ultimate tensile strengthusing the following equation:

SSU = FSH x UTS (4-31)

where

SSU is the material ultimate shear strength, N/m2 (psi);

FSH is the shear factor;

UTS is the ultimate tensile strength, N/m2 (psi).

The shear factor, FSH, is included to reflect the fact that a material’s strength in shear is less than its tensilestrength. This factor is merely the ratio of a material’s shear strength to its tensile strength. There are two majorfailure theories for determining this factor, the maximum shear stress theory which yields a shear factor of 0.50and the distortion energy (aka, von Mises) theory which gives this ratio as 0.577. Both are commonly used intorsional analysis.

— The parameter that the equivalent fully-reversing shear stress must be compared to in order to determinestructural adequacy is the shear endurance limit. The endurance limit must be used since the interference pointsof interest lie either within the unit’s operating speed range or close by. This means that the unit could,theoretically, run at this condition for an indefinite period of time, thereby, accumulating an essentially infinitenumber of fatigue cycles.

Since endurance limit data are often not available, it is common to use the assumption given in Reference [5] thatthe tensile endurance limit for a steel part is simply equal to one-half of its ultimate tensile strength. The generalequation for the shear endurance limit is then:

SSE = 0.5 x UTS x FSH x kA x kB x kC (4-32)



where

SSE is the shear endurance limit, N/m2 (psi);

UTS is the ultimate tensile strength, N/m2 (psi);

FSH is the shear factor;

kA is the surface finish factor;

kB is the size factor;

kC is the reliability factor.

The surface finish factor, kA, accounts for the fact that the susceptibility of a part to fatigue failures can bedrastically reduced by improving its surface finish. This is because the scratches, pits, and machining markswhich are more prevalent in a rough surface add stress concentrations to the ones already present due to partgeometry. Since most published material properties are obtained from tests performed on finely polishedspecimens, the fatigue strengths of most parts will be less than the published values. The surface finish factorrepresents the ratio of the part’s fatigue strength to that of the test specimen, based on surface finishconsiderations.

— The following equation from Reference [5] is one of several methods that can be used for determining thesurface finish factor:

kA = a x UTSb (4-33)

where

kA is the surface finish factor;

UTS is the ultimate tensile strength (MPa = 106 N/m2);

a is the empirical coefficient dependent on manufacturing method (4.50 for machined surfaces);

b is the empirical coefficient dependent on manufacturing method (–0.265 for machined surfaces).

or with English units

UTS is the ultimate tensile strength (ksi);

a is the empirical coefficient dependent on manufacturing method (2.70 for machined surfaces);

b is the empirical coefficient dependent on manufacturing method (–0.265 for machined surfaces).

The size factor, kB, accounts for the empirical observation that when two shafts are manufactured from the samebatch of material and tested at the same level of surface strain, the larger diameter shaft will almost always fail ina lower number of cycles than the other shaft. The primary reason for this is that fatigue failures almost alwaysinitiate at the location of a flaw in the material. Since a shaft containing a larger volume of material is statisticallymore likely to contain such a flaw, the susceptibility of a shaft to fatigue failure increases with shaft diameter.Since most shafts used in practical turbomachines are larger than the 6–8 mm (0.25–0.3 in.) diameter shaftsusually used to generate published strength data, the size effect almost always involves a reduction in strength.



This is accounted for via the size factor, kB, which is defined as the ratio of the strength of the part to the strengthof the test specimen.

— There are many empirical relations for obtaining the size factor in the literature. One, which is provided inReference [5], is as follows:

kB = a x d –0.157 (4-34)

where

kB is the size factor;

a = 1.51;

d is the shaft diameter, mm.

or with English units

a = 0.91;

d is the shaft diameter (in.).

The reliability factor, kC, reflects the statistical fact that if greater reliability is desired, the allowable stress must belowered. Stephens [7] gives reliability factors for several reliability levels. For a reliability level of 99 %, which iscommonly designed for, the reliability factor is 0.814.

— Once all of the relevant strength-reduction factors are known, they should be applied to Equation 4-32 todetermine the shear endurance limit, SSE. This should then be combined with the equivalent fully-reversing shearstress, Seqv, to define a fatigue safety factor, as follows:

SF = SSE / Seqv (4-35)

where

SF is the fatigue safety factor;

SSE is the shear endurance limit, N/m2 (psi);

Seqv is the equivalent fully-reversing shear stress, N/m2 (psi).

In an exhaustive survey reported in Reference [8], it was found that almost all relevant experts in the torsionalvibration field recommend designing for a safety factor of 2.0. Accordingly, it is common practice to require allshaft elements in a train to demonstrate a safety factor of 2.0 or greater.

The above procedure is presented for reference only—it is not intended to be a required procedure for all to employ. Itis recognized that different analysts and organizations have utilized their own procedures for years, and many of themare perfectly satisfactory. However, all procedures should account for mean stress, stress concentration, surfacefinish, and size effects in order to be considered credible.



4.6.2 Low Cycle Fatigue Analysis for Transient Excitation

Whereas the discussion in 4.6.1 relates to the steady-state forced torsional response analysis and, consequently, theinfinite life fatigue strength, in some instances, it becomes necessary to analyze the time-transient characteristics ofthe system torsional response and the finite life fatigue strength (i.e. low cycle fatigue).

A transient torsional analysis (see Section 4.4.4) usually requires a more elaborate computer code. In the past,transient analyses were often performed with a system model that was a reduced version of the model used tocalculate the torsional undamped natural frequencies. The reduced model was used to minimize the computer timerequired to perform the numerical solution of the torsional equations of motion during transient events such as start-up and fault conditions. The frequencies and mode shapes resulting from the reduced model were compared to thefull model’s results prior to proceeding with the transient analysis. Usually the reduction in model scope does notadversely affect the results. With the case of a synchronous motor drive or other motor-driven systems, where thetransient excitation is an electrical excitation originating in the motor, the first mode is usually the only significant modeexcited. However, although reduced models are sometimes still utilized today, the much greater capabilities ofmodern computers often permit the analyst to run transient analyses on the full model.

A transient analysis is often required for the following cases:

a) a synchronous motor that undergoes an asynchronous start-up;

b) a synchronous or induction motor that experiences a short-circuit fault condition (e.q. line-to-line, line-to-ground,etc.) or a synchronizing and/or switching transient (single/multiple reclosure);

c) a variable speed motor drive.

Variable speed motor drives may require a transient torsional analysis because LCI variable frequency drive designsgenerate high-level transient torque pulsations at the beginning of the unit start. This is especially true when the driverinertia is significantly lower than the driven inertia. In these cases, stresses produced during the start will be higher inthe driver shaft as it attempts to accelerate the high inertia of the driven equipment.

Machinery trains with synchronous motor drivers that undergo asynchronous unit starts require a transient torsionalanalysis to determine train response during unit start. This time-transient analysis, for the full period of trainacceleration to normal running speed (synchronizing speed), is sometimes calculated for both full and reducedsynchronous motor terminal voltage, especially if a complete electrical supply system and electrical motor circuitmodel is not available. Figure 4-52 presents a typical plot from a transient torsional analysis.

Since the pulsating component of synchronous motor torque changes linearly in frequency from 2 times electric linefrequency at 0 % speed to 0 Hertz at 100 % speed, and achieves a maximum amplitude at approximately 80 % to95 % speed (Figure 4-53), all system torsional natural frequencies below 120 Hz will be excited for 60-Hertz AC (orbelow 100 Hz for 50-Hertz systems). Typically, amplification of the second and higher modes is minimal and not ofany great concern. The primary concern is the degree of amplification of the motor pulsating torque component upontraversal of the system fundamental (first) torsional mode. Preferably, judicious component selection early in a designproject will allow adequate sizing to account for the transient torques that will be developed by the system. Whentransient vibration characteristics are problematic, special couplings such as elastomeric couplings and other damperor isolator couplings, may be used to reduce the transient torque pulsations. Other strategies for doing so areprovided in Reference [9].Transient analysis of synchronous motor-driven systems is more important in today’senvironment where 4-pole synchronous motors are often applied. The 4-pole design is usually solid pole as opposedto the laminated pole designs used on 6-pole synchronous motors. The laminated pole design offers the motordesigner the opportunity to modify the pole design to minimize oscillating motor torque, an option that is not availablewhen solid pole construction is used. Figure 4-53 and Figure 4-54 provide representative speed-torque curves forlaminated pole and solid pole designs, respectively.



The principal results of the transient torsional analysis are given in terms of the following:

a) maximum vibratory torque response in shafting;

b) maximum alternating shear stress in shafting.

Once these results are obtained, the train must be checked to ensure that it meets the following four criteria.

1) The low cycle fatigue life of all shafts must be high enough that they can withstand the required number of start-ups.

2) The peak torques in all couplings must be below the peak torque ratings for those couplings (obtained from themanufacturer).

3) The peak torques at all interference fits must be below the minimum torque required to initiate relative motion atthat fit.

4) The peak torques at all gear meshes must be deemed acceptable by the gear manufacturer.

Maximum oscillating torque for a synchronous motor can be as high as 1.0 to 1.5 times full load torque. Also, when atrain torsional natural frequency is traversed during startup, the maximum oscillating torque encountered by shaftends can be as high as 5 to 10 times the full load torque. Such levels of torque may mandate the use of largercouplings, larger shafting, and increased torque capacity gearing than would otherwise be required. Care should beexercised in the use of nonkeyed coupling attachments to avoid slippage of coupling(s) during periods of hightransient torque.

Figure 4-52—Transient Torsional Simulation of a Synchronous Motor-Driven Compressor Train

85 % Voltage train acceleration4 % Relative damping

Shaft shear stress (psi) = Amplitude x 1089.0

Non

-dm

ens

ona

stre

ss a

mp

tude

(per

un

t fu

oad

torq

ue)

0.00-10

0

+10

17.50Time (Seconds)

Initiallinesurge

Exitation of2nd torsionalnatural frequency

Exitation of1st torsionalnatural frequency



Although the calculated peak torque response levels on unit starts can be below the design limits for the traincomponents, it is usually necessary to calculate the life of the components relative to low-cycle fatigue life to ensurethe design integrity of the installation. For this analysis, the areas of highest stress, typically the machinery shaft endsand areas of stress concentration, are investigated.

Since stresses occurring during transient events such as synchronous motor startups occur only a finite number oftimes for each start-up, their acceptability is evaluated via a low cycle fatigue analysis, using the S-N curve for theshaft material in question. The basic concept of a fatigue life calculation is that each cycle of a stress level in excessof the material’s endurance limit dissipates a finite amount of the usable life of the shaft. Therefore, by counting thenumber of stress cycles occurring at each torque level, the cumulative damage of each torque level can bedetermined. This method of damage assessment calculates the number of starts each shaft can be exposed to priorto the onset of fatigue failure. Although there are a number of cumulative damage analysis procedures available, thesimplest and most commonly used is the Miner’s summation, which is explained in detail in Reference [3].

Many different fatigue analysis techniques are available for calculating the number of starts that a synchronous motor-driven train can be exposed to. Detailed procedures are given in References [2,3,10,11]. The major differencesusually appear in the S-N curves, fatigue reduction factors, and stress range counting methods. However, because ofthe log-log nature of the S-N curve, these differences can result in vastly different predictions for the allowablenumber of starts for the same torsional response (this is illustrated in detail in Reference [3]). Therefore, thecalculated number of starts will depend on the analytical model employed, the damping level used, safety factors andthe analyst’s experience. Due to these sensitivities in the transient analysis, it is prudent to use conservatism whendefining the required number of starts for a train.

Figure 4-53—Typical Speed-Torque Curve for a Synchronous Motor with Laminated Pole Construction

200 %

150 %

100 %

50 %

Breakaway torque

Pulsating torque

100 % Fullload torque

Available torque

Per

Un

t Tor

que,

%

Per Unit Speed (1.0 = 1800 RPM)0.5 1.0



API Std 541 currently requires 5000 starts. This is normally a reasonable requirement for a motor because it does nottend to add significant cost to the design. However, considering that if a train were started up once a week, everyweek, for 20 years, it would only accumulate 1040 starts, requiring 5000 starts for the remainder of the train is overlyconservative. Equipment of this type would normally only start up a few times a year, rather than once per week.Thus, a more reasonable number of starts (in the range from 1000 to 1500 starts is common) should be specified forthe remainder of the train. (However, the reviewer should be aware of what conservatisms are built into the analysis.)

4.6.3 References

[1] Loewenthal, S. H., “Design of Power Transmitting Shafts,” NASA Reference Publication 1123 (1984).


Figure 4-54—Typical Speed-Torque Curve for a Synchronous Motor with Solid Pole Construction

Average motor air gap torque

Inrush current

Zero peak air gap pulsation torque

Motor power factor

Compressor counter torque

Per unit speed (1.0 = 1800 RPM)

200 %

150 %

100 %

50 %

Per

Un

t Va

ues,

%

0 1.0



[3] Corbo, M. A. and Cook, C. P., “Torsional Vibration Analysis of Synchronous Motor-Driven Turbomachinery,”Proceedings of the Twenty-Ninth Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&MUniversity, College Station, TX, pp. 161–176 (September 2000).

[4] Pilkey, W. D., Peterson’s Stress Concentration Factors, 2nd Edition, John Wiley & Sons, New York, NY(1997).

[5] Shigley, J. E., Budynas, R. G., and Mischke, C. R., Mechanical Engineering Design, 7th Edition, McGraw-Hill,Inc., New York, NY, (2003).

[6] Rolovic, R., Tipton, S. M., and Sorem, J. R., “Multiaxial Stress Concentration in Filleted Shafts,” ASME Journalof Mechanical Design, pp. 300–303 (2001).

[7] Stephens, T. J., “Torsional Analysis and Reciprocating Compressors,” Proceedings of the Thirty-SecondAnnual Meeting of The Vibration Institute, Williamsburg VA, pp. 1–12, (June 2008).

[8] Corbo, M. A. and Malanoski, S. B., “Practical Design against Torsional Vibration,” Proceedings of the Twenty-Fifth Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, TX,pp. 189–223 (1996).

[9] Yeiser, C. W., Huetten, V., Ayoub A., and Rheinboldt, R., “Revamping a Gas Compressor Drive Train From7000 to 8000 HP With a New Synchronous Motor Driver and a Controlled Slip Clutch Mechanism,”Proceedings of the Thirty-Fifth Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&MUniversity, College Station, TX, pp. 33–48 (2006).

[10] Szenasi, F. R. and von Nimitz, W. W., “Transient Analyses of Synchronous Motor Trains,” Proceedings of theSeventh Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station,TX, pp. 111–117 (1978).

[11] Evans, B. F., Smalley, A. J., and Simmons, H. R., “Startup of Synchronous Motor Drive Trains: The Applicationof Transient Torsional Analysis to Cumulative Fatigue Assessment,” ASME Paper 85-DET-122 (1985).

4.7 Contents of a torsional report

4.7.1 General

The purpose of the torsional report is not only to document the results and acceptability (or lack thereof) of the subjectsystem, it must also provide sufficient detail to allow confirmation that the analysis has been performed properly.Additionally, if specified by the purchaser, the report must provide sufficient data on the subject system to allow anindependent third party to duplicate the analysis.

In general, the items that are required in a particular report are dependent on which types of torsional analysis wereperformed. In general, the large majority of analyses performed for turbomachinery trains can be grouped into one ofthe following four categories:

1) undamped analysis (determination of natural frequencies and mode shapes, and generation of Campbelldiagrams);

2) steady-state damped response analysis;

3) transient analysis of start-up or shut-down conditions (such as start-up of a synchronous motor);

4) transient analysis of electrical fault conditions.



Regardless of which analyses are performed, there are certain items that should be included in all torsional reports.These are outlined in the dynamics standard paragraphs and further clarified here.

— Schematic of drive train (clear identification of all rotors, including associated inertias, coupling locations, andspeeds for all shafts).

— Operating speed ranges for all shafts.

— Rotor model:

— model tabulation—to include rotor geometry (shaft element lengths and diameters) and significant externalmasses with polar moments of inertia,

— clear delineation of whether tabulated values are true or equivalent (i.e. speed-converted) values,

— definition of drawings (or other info. sources) used to obtain data used to model the train,

— definition of shaft material properties (density and shear modulus) used in model,

— sketch of rotor model,

— definition of all gear meshes and their speed ratios,

— gear mesh flexibilities,

— when specified—access to drawings to develop independent model,

— delineation between stiffness and mass diameters,

— torsional stiffnesses and assumed geometries (including spacer length and hub bore diameters) for allcouplings (along with assumptions made regarding shaft penetration),

— description of where models for shafts that mate with coupling hubs are assumed to end,

— description of assumptions used to determine motor rotor stiffness in core region,

— summary of the total inertias of each major train component, along with their respective percentages of thetotal train inertia.

— List of computer codes used in the analysis with a brief description of the type of code, e.g. finite element,transfer matrix, etc.

— Summary of all findings and conclusions on the acceptability of the unit.

4.7.2 Reports Containing an Undamped Analysis

An undamped analysis is the simple determination of the train’s undamped natural frequencies and associated modeshapes. These natural frequencies are then plotted, along with all expected excitations, on Campbell diagrams todetermine if the train has any interference points within, or in the vicinity of, its operating speed range. With very fewexceptions, this analysis should be included in the torsional analysis of all trains. The items that should be included inthe report for this type of analysis include the following.

— Summary of calculated natural frequencies (all frequencies from 0 cpm to the natural frequency immediatelyabove N times MCOS, where N is the order number corresponding to the highest order excitation in the system.



The excitation order number might be as high as 24X the output frequency for some VFDs or 12X for areciprocating engine or compressor.).

— Campbell diagram for each shaft system operating at a different speed:

— plot of natural frequencies versus speed,

— operating speed range delineated by vertical lines,

— all excitations included (along with descriptions of source of each excitation),

— all calculated natural frequencies (using rules described above) plotted,

— all interference points within ±10 % of operating speed range clearly labeled,

— all interference points within ±10 % of line and twice line frequency clearly labeled.

— Undamped mode shapes [use equivalent (i.e. all displacements referenced to the speed of one shaft) shapes,rather than true shapes] for all natural frequencies plotted on Campbell diagram.

4.7.3 Reports Containing a Steady-State Damped Response Analysis

If the undamped analysis reveals that the train has any interference points within its operating speed range or within±10 % of its operating speed range, the analyst has two choices for dealing with this, as follows.

— Look into system changes (usually coupling changes) that will eliminate the interference condition (mandatorywhen dealing with coupling-controlled modes excited by the 1X excitation).

— Evaluate the interference points further using steady-state damped response analyses in an attempt to show thatthe train can withstand these points.



In the event that one or more steady-state damped response analyses are performed, the following informationshould be included in the report.

— Definition (speed, excitation, natural frequency, and steady-state torque) of interference point being analyzed.

— Location and amount of excitation torque.

— Tabulation of VFD excitation torques (as percentages of motor rated torque) as function of speed (if applicable).

— Value of modal damping ratio applied to all shaft elements in model for each mode.

— Definition of any local or hysteretic damping coefficients applied to model.

— Tabulation of cyclic torques at all shaft elements.

— Shaft fatigue and yield analysis:

— steady-state and cyclic torques applied to shaft element,

— shaft element diameter,

— definition of any stress risers present,

— nominal steady-state and cyclic shear stresses in shaft element,

— shaft material strength (UTS and YTS),

— equivalent fully-reversing shear stress in shaft element,

— shear factor,

— surface finish factor,

— size factor,

— reliability factor,

— geometric stress concentration factor,

— notch sensitivity,

— effective stress concentration factor,

— modified shear endurance limit (adjusted using factors above),

— fatigue safety factor and equation used to determine,

— acceptability criteria,

— yield strength in shear,

— yield safety factor.



— Coupling torques:

— steady-state and cyclic torques applied to coupling,

— peak torque applied to coupling,

— coupling maximum continuous torque rating.

— Torques at gear meshes:

— steady-state and cyclic torques applied at gear mesh,

— ratio of cyclic to steady-state torque.

— Tabulation of results and safety factors for all interference points analyzed.

4.7.4 Reports Containing a Transient Start-up or Shut-down Analysis

There are certain situations where a transient analysis of the start-up or shut-down condition is required. The mostcommon one is when the train is driven by a synchronous motor. Since the synchronous motor generates largeexcitation torques, at twice slip frequency, during starting, this analysis must always be performed when the train isdriven by a synchronous motor that is started across-the-line. Another situation that sometimes requires this type ofanalysis is when a reciprocating compressor can be shut down in a loaded condition. At one time, constraints oncomputer time almost always required that these analyses be performed using a greatly reduced (i.e. 3- or 5-diskmodel) compared to that used for the undamped analysis. Although modern computer capabilities have essentiallyeliminated this requirement, there are still analysts who prefer to employ reduced models for these analyses.

Since the report items required for these analyses are heavily dependent on the equipment and condition beinganalyzed, it is impossible to generate a list of required items for all possible situations. However, the following list,which is for the most common of these analyses, the synchronous motor start-up, is representative of the number ofitems and detail required.

— Definition of reduced model (if applicable):

— description of methodology employed to reduce model,

— comparison of natural frequencies calculated by reduced and full models.

— Campbell diagram for transient condition.

— Torque vs. speed characteristics:

— motor average torque,

— motor pulsating torque,

— load torque.



— Description of methodology used for handling nonlinear couplings.



— Time step employed in analysis.

— Gear backlash employed in analysis (if applicable).

— Plot of speed vs. time.

— Plot of torque vs. time for key shaft elements.

— (Optional) Blow-up of torque vs. time plots in resonance region.

— Shaft fatigue analysis:

— all info. previously listed for steady-state response analyses,

— methodology for pairing torque peaks (i.e. rain flow method),

— failure theory employed (i.e. stress-life or strain-life),

— -N curve (plot of allowable shear stress vs. number of cycles),

— stress safety factor employed,

— assumptions on how surface finish, size, and stress concentration factors vary as functions of life,

— cumulative damage algorithm employed,

— predicted allowable number of starts,

— table summarizing torques, stresses, and allowable number of starts for each shaft,

— table summarizing cumulative damage evaluation for “weakest link” shaft.

— Shaft yield analysis:

— all info. previously listed for steady-state response analysis.

— Coupling Torques:


— coupling peak torque rating.

— Torques at interference fits:

— peak torque applied to fit,

— torque required to initiate slippage.

4.7.5 Reports Containing a Transient Analysis of Electrical Fault Conditions

There are certain conditions where the transient analysis of an electrical fault condition is performed. By far, the mostcommon electrical faults analyzed are the line-to-line and three-phase short circuit conditions. These are normallyanalyzed when the train’s first torsional mode is near line or twice line frequency (or near one or two times electrical



feed frequency, if a VFD is employed) or when required by the customer specification. If such an analysis isperformed the following items should be included in the report.

— Definition of reduced model (if applicable):

— description of methodology employed to reduce model,

— comparison of natural frequencies calculated by reduced and full models.

— Steady torque of shafting.

— Motor air gap torque equations used to simulate condition being evaluated.



— Description of methodology used for handling nonlinear couplings.

— Time step employed in analysis.

— Gear backlash employed in analysis (if applicable).

— Identification of weak-link locations within each major component.

— Plot of torque vs. time for each major component.

— Max shear stress in each shaft.

— Shear yield strength for each shaft.

— Coupling Torques:


— coupling max momentary torque rating.

— Torques at interference fits:

— peak torque applied to fit,

— torque required to initiate slippage.

4.8 Testing to Determine Torsional Response

4.8.1 Testing for Torsional Natural Frequencies

The exact location of an assembled train’s torsional natural frequencies may be determined by testing either on thetest stand, during string testing, or in the field. Natural frequencies can be determined during start-ups or shutdownsusing various methods. Although torsional testing may be performed on any train, in general, the situations in which itis most likely to be performed are the following.

1) If the train has suffered from torsional issues and/or failures in the field.



2) If the system contains a source of potentially large excitation torques, such as a synchronous motor, variablefrequency drive, or reciprocating engine or compressor.

3) If the torsional analysis indicates that the train may have torsional issues and changes to alleviate those issueswere not implemented in the design phase.

4) If required by the customer specification.

Obviously, the best time for torsional vibration testing arising from a suspected design problem is on the test stand, asthis allows for the manufacturer to potentially design, install, and test a modification. If no string test is scheduled, thenthe fall-back is to test in the field once the train has gone through the initial phases of commissioning.

There are two basic classes of torsional tests, as described in the next two sections—those that measure torque andthose that measure angular displacement. Since both will show peaks at locations of torsional natural frequencies,either may be used for determining natural frequencies. It should be noted that the amount of torque or oscillationmeasured at a specific location is not, by itself, an indicator of the shaft stresses at other locations. To determine thestresses at critical locations, the torsional analysis should be normalized with the test data to evaluate the stressesthroughout the system [1–3].

4.8.2 Torque Measurement

4.8.2.1 Torque Measurement Using Strain Gauges

One of the most common methods for directly measuring torque is to mount strain gauges directly to the shaft. Straingauges are conductive devices whose electrical resistance changes in direct proportion to the strain they aresubjected to. Since a shaft subjected to pure torsion has its maximum and minimum shearing strains on planes thatare oriented at 45 degree angles with respect to the axis of twist, the strain gauges should be oriented in thosedirections. Additionally, as is explained in Reference [4], in order to maximize the gain of the system (i.e. obtain ameasurable signal from displacements that are typically small), most systems consist of four strain gauges arrangedin a “full” Wheatstone bridge circuit. Typically, two of the gauges are placed on the top of the shaft and the other twoon the bottom, in order to cancel out the effects of bending and axial stresses. If done correctly, the output voltage ofthe Wheatstone bridge circuit will be directly proportional to the torque at the gauge location.Once the gauges areattached to the shaft, the system calibration (in millivolts per in.-lbf) should be checked either using shunt calibration(which is explained in Reference [4]) or a mechanical torque test. The importance of doing this cannot be overstated.The authors are familiar with extensive testing programs on critical equipment that were rendered useless because ofthe test engineer’s failure to perform a proper calibration prior to testing.

Since the gauges are attached to the rotating shaft, some means of transmitting the strain gauge signal to a stationaryreceiver must be provided. This is normally accomplished in one of two ways.

1) A slip ring assembly can be mounted on the rotor and a signal cable can be attached to an appropriate dataacquisition system. Slip ring applications are limited by the rotating speed of the rotor. Typical applications arefor machine speeds less than 5000 rpm. They are also prone to wear. Therefore, they are limited life devices,suitable for short-term testing.

2) A more commonly employed technique is the use of wireless telemetry to transmit the output signal to areceiver and the data acquisition system. Such systems can be based on either radio telemetry or rotarytransformer methods. In both cases, the telemetry circuits can be bonded directly to the rotor with fiberglasslaminations. Radio telemetry requires an additional battery powered radio transmitter mounted to the rotor tobroadcast the torsional data. Such an arrangement is only suitable for short duration tests. The rotarytransformer method uses two rotary transformers: one to transmit power to the telemetry circuit by electricalinductive coupling and the other to transfer the torsional data. This approach provides a long-term measurementinstallation [5].



In general, the location where the strain gauges are mounted must meet the following criteria.

1) There must be enough axial room (about 6 in. of straight shaft) for the gauges and the telemetry system.

2) The location must be an “active” area in the mode shape for the torsional mode of interest (most commonly thefirst mode). In other words, there must be a significant amount of twisting at that location. Naturally, the bestplace to put the gauges is where the slope in the mode shape is the highest.

3) The location must be free of stress risers, especially keyways, which would distort the results.

Of course, the optimum location for the mounting of strain gauges, from the standpoint of the mode shape, is often nota practical place to put them. In the large majority of situations, the gauges are mounted either to the spool piece ofthe driver’s coupling or the end of the driver shaft near the interface with the coupling. Since in the majority of practicaltrains, the first mode contains a significant amount of twisting in those locations, the accuracy obtained from suchmounting is usually sufficient.

More detailed tips for the performance of strain gauge testing are provided in Reference [4].

4.8.2.2 Torque Measurement Using Differential Angular Motion

Torque measurement can also be accomplished by measuring the relative angular displacement (using the methodsgiven in Section 4.8.3) between two closely-spaced points on a rotating element. The relative angular displacement isdirectly proportional to the torque transmitted by the shaft section between the two points. For best results, the shaftcross-section between the two measurement locations should be constant. Corcoran and D’Ercole [5] illustrate howthis principle is utilized in several commercial torquemeters.

4.8.3 Angular Displacement Measurement

The second method for measuring torsional vibration is the measurement of the angular displacement at somelocation within the train. In general, these types of measurement are usually easier to make than direct torquemeasurements. With the exception of the torsiograph and the laser, all of these methods are based on frequencymodulation principles. Thus, they all involve attaching a ring with many equally sized, equally spaced optical orelectrical (for example, notches) targets to one of the rotating elements in the train. An optical pick-up or displacementprobe monitoring the target surface then generates many discrete electrical pulses during each shaft revolution. Ifsignificant torsional vibrations occur at the target ring, then the electrical pulses generated by the keyphasor probe willnot be evenly spaced. Signal processing equipment converts the uneven pulse spacing into torsional (angular)displacements.

Although there are various methods available for making these measurements, the one thing they all have in commonis that the measurement location must be at an “active” region of the system (i.e. away from node points) in the modeshape of interest in order to obtain meaningful measurements. Methods for making these measurements include thefollowing.

1) The first commonly used method was the torsiograph, a device which rotates with the shaft (and whoseoperation is described in Reference [6]) and measures angular velocity or displacement. The torsiographoperates on the seismometer principle, with a spring-constrained seismic mass whose relative motioncompared to the stator is converted into an electrical signal by inductive proximity detectors. A limitation to thisdevice is that it must be mounted on the free end of a shaft, which is usually achievable if the system has agearbox, but can present problems in ungeared systems. It also is limited in its ability to measure low frequencyvibrations, due to resonance of its internal spring-mass system, which seldom occurs below 3 Hz. The actualmounting requires precision machining preparation of the shaft end. The axis of rotation of the torsiograph mustbe collinear with the axis of the shaft on which it is mounted. Since torsiographs are no longer manufactured,this methodology is primarily of academic interest.



2) A precision manufactured toothed wheel can be mounted on the coupling spool piece or a free end of the rotorif available. The number of teeth on the wheel will determine the resolution of the measurement. Typically theminimum number of teeth required for an accurate measurement is 60 teeth. The movement of the toothedwheel can be monitored by either two magnetic pickups or two proximity probes mounted 180 degrees apart.Two sensors are necessary to cancel out any radial vibration of the toothed wheel. The output from the probeswill be a series of voltage pulses as the teeth pass by the probe tips. If there is constant angular velocity (i.e. notorsional vibration), the frequency will be the tooth passing frequency. If torsional oscillations begin to occur, thisfrequency will be modulated (refer to Figure 4-55). Suitable instrumentation can be used to detect thismodulation, yielding torsional velocity. This signal can be integrated to yield torsional displacement in degrees(pk-pk) at the frequency of the torsional oscillations. Note that this same technique can be used on the gearteeth of a geared system as long as the gear mesh is not a node for the torsional mode(s) in question.

3) A further development of the toothed wheel makes use of electromagnetic sensing [7]. A stationary gear havinginner teeth is annularly disposed around the rotating gear. The stationary outer gear contains a coil, and themagnetic circuit around the coil changes as the rotating teeth come in and out-of-phase with the stationaryteeth. The use of several teeth in this way compensates for problems associated with any radial movements.This arrangement has been used in the field and applied to various couplings designs as an attachment to thecoupling spacer.

4) A measurement technique that is currently popular is the use of optical encoders. An encoder is a rotary devicethat contains a precision optical disk that produces a number of precisely spaced pulses each revolution. Thepulse pattern is sensed and the signal is demodulated in the same manner as for the toothed wheel method,yielding shaft speed, torsional vibration amplitude, frequency, etc. per Leader and Kelm [2], encoders willgenerally provide better results, with less noise, than using proximity probes on gear teeth, especially whendealing with the extremely small amplitudes (less than 0.1 degree peak-to-peak) typical of turbomachinerytorsional vibration. A drawback of this method is that, like the torsiograph, the encoder typically needs to bemounted to the end of a shaft. However, this can be overcome by using what Leader and Kelm [2] refer to as ameasuring wheel, a precision rubber-coated disk which mounts to the encoder and interfaces with the rotatingshaft. When this technique is used, the encoder can be positioned anywhere on the shaft where there is about5 in. of open axial length [2]. Mounting methods for the measuring wheel are also shown in [2]. In general, it isbest to use two measuring wheel/encoder combinations, located 180 degrees apart, to eliminate the effects ofrelative lateral motion between the shaft and the encoders. Feese and Hill [8] illustrate a similar method forusing encoders when a shaft free end is not available.

5) Optical, infrared or laser transducers can also be used by observing an epoxy bonded tape wrapped around acoupling spool piece. The tape has a photo etched pattern (bar code) that simulates a toothed wheel. The samemeasurement process is used as described with the toothed wheel.

6) A recent development in torsional vibration measurement is the parallel laser rotational vibrometer, which isdescribed in Reference [9]. Single laser versions of this device have been around for several years. However,their functionality and accuracy have not proven dependable. The parallel laser device is a large improvementbecause it eliminates the requirement that a special target be installed on the rotating shaft. Instead, the laserdevice uses the shaft roughness to reflect the laser beam back to the interferometer [9,10]. Although theyexpress a concern about its ability to handle transient measurements during rapid accels or decels, Feese andHill [8] compared the results obtained with a laser vibrometer to those from encoders and found them to beessentially equivalent. The large advantage of using lasers is that they do not require mounting additionalhardware onto the string. A disadvantage is that they do require optical line-of-sight, which may requirealteration or removal of the coupling guard.

The accuracy and resolution of a torsional displacement measurement depends to a large degree on themanufacturing accuracy of the mechanical measurement media. For example, if a proximity probe/gear or laser/tapemedia are selected, the following calculations must be made.



EXAMPLE

Assume a 150 mm (6 in.) diameter gear or tape with 60 gear teeth or 60 tape bars is to be utilized.

— Circumference = 150 mm (6 in.) x 3.1416 = 471.2 mm (18.85 in.).

— Gear or bar spacing = 471.2 mm (18.85 in.) / 60 teeth = 7.853 mm (0.3142 in.)/tooth (this is the spacing fromthe leading edge of a tooth or bar to the leading edge of the next tooth or bar).

— Degree spacing = 18.85 in. / 360 degrees = 1.309 mm (0.0526 in.)/degree.

— For torsional applications, a resolution to 0.1 pk-pk is required. Therefore, the required accuracy tomanufacture the gear teeth or bar spacing is:

— Spacing accuracy = 1.309 mm (0.0526 in.) / degree x 0.10 degree = 0.1309 mm (0.00526 in.).

NOTE Gear teeth should be manufactured as straight cut parallel sided teeth, i.e. no gear profile.

Figure 4-56 and Figure 4-57 are typical installation examples. As is the case with strain gauges, regardless of whichmethod is implemented, it is important that the test equipment be properly calibrated before being put into use. Acommon method of calibration is to use a Hooke’s universal joint calibrator (as shown in Braund [6]) to produce aknown torsional vibration amplitude at 2X shaft speed.

4.8.4 Measurement of Motor Electrical Parameters

There are some instances where the amount of torsional excitation or vibration occurring in a train can be accuratelyestimated from a measurement of motor current and/or voltage. As is described in Reference [11], the most commoncase is the use of measured motor current and voltage to determine the amplitude of the excitation torque being

Figure 4-55—Displacement Measurement Considerations

Vo

tage

Time

t per voltage pulse is not constant;a change in angular velocity

Vo

tage

Time

t per voltage pulse is constant;no change in angular velocity



Figure 4-56—Torsional Measurements Using a Toothed Wheel and Sensors

Figure 4-57—Torsional Vibration Measurements with a Bar Code

Proximity probes

Toothed wheel

Laser transducer

“Bar code” tapeepoxied to spacer

“Bar code” tape



generated in the air gap of a synchronous motor. Section 4.5.2.1 describes the twice slip frequency excitationdeveloped by a synchronous motor when it is starting. Since the magnitude of this excitation is strongly dependent onthe actual motor terminal voltage, the verification of the magnitude of the excitation should be seriously considered ifa decision is made to measure the magnitude of the system torque during resonance. This technique can also beused on a shop floor motor test to verify the prediction of the torsional excitation of a synchronous motor.

If the voltage and current at the motor terminals are accurately measured with precision current and potentialtransformers and these inputs fed into a Hall effect watt transducer, the electrical power to the motor can be easilydetermined. The accuracy of these measurements is obviously dependent upon the accuracy of the measurementinstruments. Due to the high efficiency of electric motors, the actual power at the air gap will be from 96 % to 98 % ofthe measured power at the motor terminals. The trace from the watt transducer will not only identify the magnitude ofthe air gap torque but the frequency and variation in the magnitude of this torque as well. Reference [11] identifies auniform method for calculating and measuring torque pulsations that occur during starting for synchronous motors.

As pointed out by Feese and Hill [8], another component whose excitations can sometimes be determined viameasurement of motor current and voltage is the variable frequency drive (VFD). As stated in several other sectionswithin this document, VFDs can generate variations in motor air-gap torque, often referred to as ripple, that can leadto torsional issues. However, since torque ripple is always accompanied by corresponding motor current ripple,current measurements can sometimes be used to estimate the actual torque ripple the VFD is generating.

There are also some cases, if the torsional vibration is severe, where a measurement of the fluctuation in the motorcurrent can be used to determine the amount of torsional vibration in the train. In a motor-driven system, any torsionalvibration that occurs in the system will be reflected in a variation in the motor current. In many systems, this variationis too small to be of practical use but in some motor-driven reciprocating compressors, the current variation is largeenough to yield meaningful results. Of course, in such a case, the exact relation between current variation andtorsional vibration must be established via calibration.

4.8.5 References

[1] Wachel, J. C., and Szenasi, F. R., “Analysis of Torsional Vibrations in Turbomachinery,” Proceedings of theTwenty-Second Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, CollegeStation, TX, pp. 127–151 (1993).

[2] Leader, M. L. and Kelm, R. D., “Practical Implementation of Torsional Analysis and Field Measurement,”Proceedings of the Vibration Institute National Technical Training Symposium and 28th Annual Meeting,Bloomingdale, IL, pp. 131–168 (June 2004).

[3] Wang, Q., Feese, T., and Pettinato, B. C., “Torsional Natural Frequencies: Measurement vs. Prediction”,Proceedings of the Forty-First Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&MUniversity, College Station, TX, (September 2012).

[4] Corbo, M. A., Cook, C. P., Yeiser, C. W. and Costello, M. J., “Torsional Vibration Analysis and Testing ofSynchronous Motor-Driven Turbomachinery,” Proceedings of the Thirty-First Turbomachinery Symposium,Turbomachinery Laboratory, Texas A&M University, College Station, TX, pp. 153–175 (September 2002).

[5] Corcoran, J. and, D’Ercole, S., “A New Development in Continuous Torque Monitoring Couplings,”Proceedings of the ASME Power Transmission and Gearing Conference, ASME Paper DETC2000/PTG-14455, Baltimore, MD (September 2000).

[6] Braund, D. F., “Torsional Vibration,” Proceedings of the Institution of Mechanical Engineers, pp. 63–72 (1958-59).

[7] Faulkner, A. H., “Torquemeters”, Wiley Encyclopedia of Electrical and Electronics Engineering, J. Webster(ed.), John Wiley and Sons, Inc. (1999).



[8] Feese, T. and Hill, C., “Prevention of Torsional Vibration Problems in Reciprocating Machinery,” Proceedingsof the Thirty-Eighth Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, CollegeStation, TX, pp. 213–238 (September 2009).

[9] Rothberg, S., “Laser Rotational Vibrometers and Their Application in Modal Analysis,” Proceedings of theSociety of Experimental Mechanics, IMAC XXI, Kissimmee, FL (February 2003).

[10] Webb, C. E., and Jones, J. D., Handbook of Laser Technology and Applications: Applications, New York, NewYork, McGraw-Hill Companies (2003).

[11] IEEE Standard 1255-2000, Guide for the Evaluation of Torque Pulsations During Starting of SynchronousMotors (2000).

4.9 Torsional–Lateral Vibration Coupling

4.9.1 General

As is described in the remainder of this document, the conventional approach to rotordynamics is to assume thatlateral and torsional rotordynamic behaviors are completely independent of one another. That is, there is no couplingbetween lateral and torsional modes and the two are evaluated in completely separate analyses. This assumptionprovides acceptable accuracy for all ungeared systems and, also, for the majority of geared systems.

However, in geared systems, there are occasions where torsional motion couples with lateral motion of one of thegear shafts, usually the pinion shaft. There are four practical consequences that have been observed from thiscoupling (1 and 2 being beneficial and 3 and 4 being detrimental), as follows.

1) Torsional issues can be uncovered in the field via lateral monitoring of gear shaft motion. Since there are veryfew drive trains that are currently instrumented to monitor torsional vibration, in an ungeared system, the firstsign of a torsional problem is usually a failure—there are no warning signs. However, if a geared system ismonitored for lateral vibration, which is the norm, torsional issues can often be detected from increased lateralvibration of the gearshafts. Hudson [1], Tripp et al. [2], Leonhard et al. [3], Huetten et al. [4], and Adachi et al. [5]all present case histories where the first sign of a torsional problem was increased radial motion in a gear shaft.

2) If the gearshafts are supported on fluid-film bearings, the lateral motion that accompanies the torsional motiongenerates a squeeze-film effect in the bearings, which acts to damp out the torsional motion. This effect hasbeen observed by many authors, including Simmons and Smalley [6], and results in a significant, often twofold,increase in the system’s torsional damping. For this reason, as is described in Section 4.4, it is customary to uselarger amounts of damping when performing torsional analyses on geared systems.

3) Lateral instabilities involving subsynchronous “torsional modes” can occur [7]. Wachel and Szenasi [8] foundthat if the lateral first mode of a rotor coincides with a torsional mode, stability can be adversely affected. Zhangand Nyqvist [9] observed a gearshaft to go unstable at a frequency far removed from all lateral modes. Theinstability frequency, which was at approximately one-half of running speed, was later found to correspond tothe first torsional mode.

4) The lateral flexibility of the gear shafts and/or their bearings can result in a lowering of the torsional naturalfrequencies, which can result in torsional resonances not predicted by conventional torsional models. As notedby Mayer [10], this effect has been observed to impact the first torsional mode in a number of cement mills.Gunter [11] found that introduction of lateral flexibility into the torsional model caused the torsional naturalfrequencies to drop precipitously (one mode dropped from 28.1 to 17.2 Hz). However, this effect is notuniversally recognized. Schwibinger and Nordmann [12] state that adding lateral degrees of freedom to thetorsional model can move the torsional natural frequencies in either direction, depending on whether the lateraland torsional motions compliment or counteract each other. Additionally, Simmons and Smalley [6] found thatadding lateral degrees of freedom to the torsional model had little impact on any of the torsional modes.



If a manufacturer or end user has reason to believe that either item 3 or 4 from the above list could occur in their drivetrain, the only way to evaluate it is via a coupled torsional-lateral analysis. Although the details of performing such ananalysis are beyond the scope of this document, it basically entails adding lateral degrees of freedom for one or bothgear shafts to the conventional torsional model. Authors who describe the performance of such analyses includeSimmons and Smalley [6], Zhang and Nyqvist [9], Schwibinger and Nordmann [12], and Kita et al. [13].

In general, the decision on whether or not to perform a coupled torsional-lateral analysis in the turbomachinery designphase can be based on the proceeding factors. For the majority of geared systems, the standard practice ofevaluating the lateral and torsional behavior separately is sufficient. When contemplating the necessity of such ananalysis, the following may prove helpful in determining if there is a benefit in performing the coupled torsional-lateralanalysis.

1) If the system does not contain any gears, the analysis is meaningless since there is no torsional-to-lateralcoupling.

2) The likelihood that torsional-to-lateral coupling is going to create problems is related to the lateral flexibility ofthe gear shafts and their supports. The higher the lateral flexibility, the greater the likelihood of problems. Sincegear shafts are normally relatively rigid, this usually comes down to the flexibility of the gear shaft bearings,primarily the one closest to the gear mesh. Problems are most likely to occur when this bearing is unusually soft.

3) Since the stiffnesses of gearshaft bearings tend to increase as the train is loaded, the potential for problems isgreater at light loads.

4) Since the only lateral vibration that can couple with the torsional motion is that which occurs in the plane of thetooth contact force, the only bearing flexibility that is of concern is that in this direction. Any bearing flexibility inthe orthogonal direction has minimal impact.

5) The only torsional modes that are capable of coupling with lateral modes are those that have significant motionat the gear mesh.

6) Likewise, the only lateral modes that can participate are those that have significant motion at the gear mesh (inthe direction of the tooth contact force).

7) Per Wachel and Szenasi [8], if a torsional mode is close to a lateral mode with marginal stability margin, thechances for coupled mode instabilities are greater.

8) Per Zhang and Nyqvist [9], the chance for a coupled mode instability increases when there is a torsional modeclose to one-half of the operating speed of one of the gearshafts.

4.9.2 References

[1] Hudson, J. H., “Lateral Vibration Created by Torsional Coupling of a Centrifugal Compressor System Drivenby a Current Source Drive for a Variable Speed Induction Motor,” Proceedings of the Twenty-FirstTurbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, TX, pp.113–124 (1992).

[2] Tripp, H., Kim, D., and Whitney, R., “A Comprehensive Cause Analysis of a Coupling Failure Induced byTorsional Oscillations in a Variable Speed Motor,” Proceedings of the Twenty-Second TurbomachinerySymposium, Turbomachinery Laboratory, Texas A&M University, College Station, TX, pp. 17–23 (1993).

[3] Leonhard, M. L., Kern, U., and Reischl, K., “Electric Power Supply Exciting Torsional and Lateral Vibrations ofan Integrally Geared Turbocompressor,” Proceedings of the Thirtieth Turbomachinery Symposium,Turbomachinery Laboratory, Texas A&M University, College Station, TX, pp. 49–55 (2001).



[4] Huetten, V., Beer, C., Krause, T., and Demmig, S., “VSDS Motor Inverter Design Concept for CompressorTrains Avoiding Interharmonics in Operating Speed Range,” Proceedings of the First Middle EastTurbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, Doha, Qatar (February2011).

[5] Adachi, A., Tanaka, K., Takahashi, N., and Fukushima, Y., “Torsional-Lateral Coupled Vibration of VSD MotorDriven Centrifugal Compressor System at Inter-Harmonic Frequencies in Voltage Source PWM Inverter,”Proceedings of the Thirteenth Asia Pacific Vibration Conference, University of Canterbury, New Zealand(November 2009).

[6] Simmons, H.R. and Smalley, A.J., “Lateral Gear Shaft Dynamics Control Torsional Stresses in Turbine-DrivenCompressor Train,” ASME Journal of Engineering for Gas Turbines and Power, pp. 946–951 (October 1984).

[7] Lund, J. W., “Critical Speeds, Stability and Response of a Geared Train of Rotors,” ASME Journal ofMechanical Design, pp. 535–539 (July 1978).

[8] Wachel, J. C. and Szenasi, F. R., “Field Verification of Lateral-Torsional Coupling Effects on Rotor Instabilitiesin Centrifugal Compressors,” Proceedings of the Workshop on Rotordynamic Instability Problems in HighPerformance Turbomachinery, Turbomachinery Laboratory, Texas A&M University, College Station, TX (May1980).

[9] Zhang, Y. and Nyqvist, J., “Rotor Instability Due to Coupled Effect of Lateral and Torsional Modes andImproper Bearing Design,” Proceedings of the 1997 ASME Design Engineering Technical Conference,Sacramento, CA, Paper DETC97/VIB-4031 (September 1997).

[10] Mayer, C.B., “Torsional Vibration Problems and Analyses of Cement Industry Drives,” IEEE Transactions onIndustry Applications, pp. 81–89 (January/February 1981).

[11] Gunter, E. J., “Torsional-Lateral Modeling of Coupled Systems by Finite Element Techniques,” Proceedings ofthe Vibration Institute’s Twentieth Annual Meeting (June 1996).

[12] Schwibinger, P. and Nordmann, R., “Influence of Torsional-Lateral Coupling on Stability Behavior of GearedRotor Systems,” Proceedings of the Workshop on Rotordynamic Instability Problems in High-PerformanceTurbomachinery, Turbomachinery Laboratory, Texas A&M University, College Station, TX, pp. 531–553(1986).

[13] Kita, M., Hataya, H., and Tokimasa, Y., “Study of a Rotordynamic Analysis Method that Considers Torsionaland Lateral Coupled Vibrations in Compressor Trains with a Gearbox,” Proceedings of the Thirty-SixthTurbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, TX, pp.31–37 (2007).

4.10 Variable Frequency Drives

4.10.1 Introduction

As is described in a number of References [1,2], employment of a variable frequency drive (VFD) greatly improvesprocess efficiency by allowing the prime mover (i.e. compressor or pump) to run at the optimum speed for theoperating conditions at hand. Unfortunately, this increased efficiency does not always come without a price.References [3–10] are only a sampling of the many documented torsional failures and/or issues that have beendirectly attributable to VFDs in recent years. The causes of a good number of these problems are still not wellunderstood. Accordingly, whenever the train is driven by a VFD, the criticality of performing a thorough torsionalanalysis in the design phase, and the complexity of that analysis, increases substantially.



The fact that there have been a significant number of VFD-related torsional failures recently does not, by any means,suggest that such failures did not occur in the past. Indeed, such failures are reported in [11–13]. The primarydifference between the two groups of failures is that while the mechanisms generating the past failures are relativelywell understood, the causes for many of the more recent failures remain unresolved.

The concern over VFDs is continually increasing as they are being employed more and more in various applications.A number of industries have been increasingly moving from mechanical to electrical drive trains. Industrial plantoperations have been significantly improved using VFDs, since the motor speed can be adjusted to provide maximumefficiency in the system, without harming system performance. In the case of pumps, VFDs eliminate need ofdischarge control valves for throttling. Noise levels in fan systems and piping are also reduced due to optimized flowrates.

Figure 4-58 shows a generic configuration of the overall system that VFDs participate in. The figure shows anislanded power system network, with two turbine–generator sets, each driven by a gas-turbine. These two turbine-generator sets are supplying power to a set of VFD systems, each of which drives a mechanical load. They alsosupply power to other unidentified electrical nonlinear loads. Locations of possible torsional resonances are alsoshown.

VFDs are being increasingly employed in turbomachinery because they permit operation over a significant speedrange. They accomplish this by varying the electrical frequency being fed to the motor (which is constant at linefrequency for fixed speed motors). The resulting motor speed (neglecting the effects of slip) is directly proportional tothe electrical frequency via the following relation:

RPM = 60 x fvfd x 2/Np (4-36)

where

RPM is the motor speed (rpm);

fvfd is the VFD output frequency (Hz);


Thus, for the familiar case of a two-pole motor being fed with line frequency (60 Hz in North America), the motorsynchronous speed is 3600 rpm.

VFDs are seldom designed for steady-state operation in the 0-10 Hz range. In a 60 Hz grid, minimum outputfrequency is usually 15 Hz or higher (25 % of line frequency.) However, when a VFD is employed, a motor can be,and often is, run at speeds that are 5 % to 10 % above the line frequency.

In addition to controlling the electrical frequency being fed to the motor, the VFD also controls the voltage. For mostapplications, the voltage is kept proportional to the frequency, in what is called a constant volts per Hz design (orconstant flux design).

When a VFD is employed in a turbomachinery drive train, the complexity of the torsional analysis increases for threebasic reasons.

1) The larger operating speed range makes it much more difficult to demonstrate the required 10% separationmargin for all interference points, as compared to a fixed speed system.

2) Excitations from the VFD need to be considered in the analysis, as they are a potential cause of torsionalfailures.



3) The variation of the electrical frequency being fed to the motor makes it more likely that a short circuit condition(which generates excitations at one and two times motor electrical frequency) can trigger a resonance with thefirst torsional mode.

All of these issues are explained in detail in the following sections.

4.10.2 VFD Fundamentals

Although the large majority of readers of this technical report are mechanical engineers, it is necessary to understandsome of the electrical engineering basics of VFDs in order to perform a competent torsional analysis when one ispresent. One of the biggest stumbling blocks to doing that is often a communication gap between the electricalengineers designing the VFD and the mechanical engineers performing the torsional analysis. Accordingly, the betterthe analyst understands the electrical workings of the VFD, the greater the likelihood of a good analysis beingperformed.

The primary “building blocks” in VFDs are the power semiconductors. Typical ones are diodes, Silicon-ControlledRectifiers (SCR), Gate Turn-Off Thyristors (GTO), Insulated Gate Bipolar Transistors (IGBT), and Integrated Gate-Commutated Thyristors (IGCT). Their main function is to act as fast-moving switches, constantly moving between the“on” and “off” states. Their switching frequency can be important, especially in voltage source inverters.

The most basic power semiconductor is the diode. Diodes are analogous to mechanical check valves, in that theyonly allow current to flow in one direction. Like check valves, they are passive devices—whether or not they are “on”or “off” is only dependent on the voltage drop across them (i.e. they cannot be actively turned “on” or “off”). All of theother semiconductors employed in VFDs are more complex versions of the diode. Some can be turned “on” but can’tbe turned “off”, and the most complex (IGBTs and IGCTs) can be turned both “on” and “off” at will.

Figure 4-58—Overview Schematic of a VFD System

M

Non-linearloads

M M M

AC

AC

AC

AC AC

AC AC

AC

GTGT GG



There are many references that discuss the electrical workings of a VFD, among them [2,4,14–22]. Figure 4-59shows a simplified representation of a VFD. The main elements and their functions are as follows.

— Power Line (item 1): The power line supplies power to the VFD, with an AC voltage which has essentiallyconstant voltage magnitude and frequency. Power can be supplied through the utility grid or it can be generatedlocally from a local generator system (e.g. in offshore platforms or islanded power plants), as shown in Figure 4-58. The VFD is connected to the power line through a circuit breaker. For safe operation of equipment, this circuitbreaker protects and isolates the VFD from the power line in case of faults. Generally, the circuit breaker isclosed or opened by the VFD controls.

— Transformer (item 2): The transformer converts the power line AC voltage to the input voltage required by theVFD. The transformer input and output windings are usually referred to as primary and secondary, respectively.

— Rectifier (item 3): The rectifier converts the AC voltage supplied from the transformer to a DC voltage or current.When the power flow is always from the grid to the load, the rectifier is built with diode semiconductors, and it issometimes called a diode rectifier or passive front-end, because there is no control required. When the powerflow can also be from the load to the grid, the rectifier is built with controllable devices such as IGBTs and iscalled an active front-end.

— Optional Harmonic Filter (item 4): A harmonic filter is a set of passive components (resistors, inductors andcapacitors) connected between the transformer input and the grid. The values of these components are chosenin such a way that current harmonics below a specified frequency range can be canceled. However, the size ofthese passive components is heavily dependent on the harmonic range to be canceled; the lower thefrequencies, the bigger the harmonic filter.

— DC-Link (item 5): The DC-link decouples the grid and motor sides of the VFD and acts as a temporary energystorage tank. The DC-link is normally either a bank of capacitors (voltage source system) or an inductor (currentsource system). The energy converted by the rectifier is first stored in the DC-link, and then transferred to theinverter (item 6).

— Inverter (item 6): The inverter converts the DC-link signal (either a voltage or a current) to an AC signal havingvariable amplitude and variable frequency. The control unit (item 7) is used to smoothly adjust the voltage appliedto the motor (item 8), depending on the load request. Adjusting the frequency of the output voltage applied to themotor changes the motor speed, per Equation 4-36.

Figure 4-59—Simplified Representation of a VFDs Elements

(1)

Grid M

AC

AC

DC

DC

Control unit

DC

-nk

Opt

ona

harm

onc

fte

r

Opt

ona

harm

onc

fte

r

(2)

(3)

(4)

(5) (6)

(7) (8)

(9)



— Control Unit (item 7): The main function of this block is to adjust the output voltage (amplitude and frequency)being fed to the motor, in order to yield the desired speed and/or torque.

— Optional Harmonic Filter (item 8): An output filter is often necessary to smooth the pulsed output waveform.Especially for voltage source inverter topologies with long cables, an output filter is used to limit the voltage rateof change and transient magnitude. Sometimes the filter is even designed to filter the motor voltage to be almostsinusoidal.

— Electric Motor (item 9): Drive motor of the train receiving the modified AC voltage (amplitude and frequency).

Thus, the primary components of interest in a VFD are the rectifier, DC-Link, and inverter. The rectifier takes theconstant frequency AC signal from the grid and converts it into a DC signal. The DC-Link feeds this signal to theinverter. The inverter takes the DC signal and converts it to a variable frequency AC signal that is fed to the motor.The motor’s speed is directly proportional to that frequency, in accordance with Equation 4-36.

The rectifier has the functionality to convert AC power into DC power. The rectifier for a VSI (voltage source inverter)is typically built up using diodes. The most basic rectifier is the 6-pulse rectifier, which is shown in Figure 4-60.

The 6-pulse rectifier makes six commutations (current transfers from one pair of diodes to another) within one cycle ofthe AC voltage. This results in six distinct voltage pulses in the rectifier’s output for 360 degrees of the inputwaveform, as is shown in the figure. The ripple in the DC voltage and harmonics in the AC line current can bedecreased by using higher pulse numbers.

The “number of pulses” simply refers to the number of commutations during one cycle of the input or outputfrequency. As shown above, the six-pulse configuration includes six switches at the input and six switches at theoutput (in the inverter). So, it only makes sense to speak about 6-pulse, 12-pulse, etc. configurations when referringto an input or output stage consisting of diodes or SCRs. With more sophisticated configurations, such as pulse widthmodulation (which is discussed later), it’s not meaningful to define the “number of pulses.”

A 12-pulse rectifier is based on two 6-pulse circuits, an 18-pulse is based on three, and so on. Usually the 6-pulsecircuits are connected in series but parallel connection is also possible. From the standpoint of the torsional analyst,the larger the number of pulses the lower the torsional excitations coming from the rectifier.

The rectifier for a CSI (current source inverter) is built up by using thyristors (aka SCRs) or IGCTs instead of diodes.However, the basic configuration for a 6-pulse rectifier is still the same as in Figure 4-60 and 6-pulse rectifiers arecombined to obtain 12, 18, 24, etc. pulse rectifiers in the same manner as for VSIs.

The most complex type of rectifier is the active front-end converter, which contains IGBTs and/or IGCTs in addition todiodes. It is referred to as “active” since the switches are actively controlled, similar to those in an inverter for avoltage source inverter (VSI). Because the pulsation of active front end rectifiers is defined by its control software, the

Figure 4-60—Six-pulse Rectifier and the Voltage Waveforms

Uac

DC

AC

+ Udc

Time



number of voltage levels that can be produced at each AC phase terminal is more appropriate than pulses for circuitclassification.

Per Williams et al. [21], the function of the DC-link is to make the DC waveform as smooth as possible so that theinverter is fed a smooth DC waveform. This is achieved through the use of capacitors or inductors. The role of thecapacitor is to smooth out the ripple in the DC voltage waveform by supplying energy whenever the voltage from therectifier drops. Capacitors store energy in the drive and help to make the VFD more immune to voltage sag problems(when the main voltage supply to the VFD drops—usually due to other equipment being switched on or remotefaults).

The inverter is used to convert DC to AC power in a way that can be used to control an electric motor. The inverter isprobably the most advanced block in a VFD system. The most basic inverter is the six-step inverter, which is highlysimilar to the six-pulse rectifier of Figure 4-60. The output of such an inverter also consists of six pulses, as shown inthe figure, so such inverters are also referred to as six-pulse inverters.

The six-step inverter, or more sophisticated variants of it, is still used in modern current source inverters and loadcommutated inverters. However, in voltage source inverters, the inverter usually employs a much more sophisticatedalgorithm, known as pulse width modulation (which is discussed in the next section) to provide signals to the motor. Inpulse width modulation, instead of trying to approximate a sine wave using six pulses per cycle (as is done in six-stepinverters), several hundred voltage pulses are used. It does not take much imagination to realize that the output ofsuch a device would be much closer to an ideal sine wave than that from a six-step inverter.

4.10.3 Major Types of VFD

Although there are more VFD design types than could possibly be enumerated herein, all of them fall into one of thethree following categories:

1) Current Source Inverter (CSI),

2) Load Commutated Inverter (LCI),

3) Voltage Source Inverter (VSI).

In a CSI, and its closely-related cousin, the LCI, the current being fed to the motor is controlled (while the voltage isallowed to vary). Accordingly, the DC-Link in these designs is the energy storage element for current, an inductor.Conversely, in a VSI, the motor voltage is controlled (and the current allowed to vary) and the DC-Link is a capacitor,or bank of capacitors. Thus, although schematic diagrams for VFDs are often indecipherable for mechanicalengineers, the basic type can be discerned by simply looking at the DC-Link. If an inductor, it’s a CSI or LCI and if acapacitor, it’s a VSI.

Per Huetten et al. [7], both basic types have specific advantages and disadvantages and the selection is based onpower and voltage range, complexity, and the reference situation. In general, VSIs are used for the lower powerratings up to 25 MW and the LCI/CSI is the preferred solution for the highest power ratings up to 120 MW. In therange of 15-25 MW, both topologies can be used.

In the gray area between 15 and 25 MW, different considerations can be made on the selection between an LCI andVSI. Decisions are done case-by-case. However the most common factors considered are the harmonics andsubharmonics in the AC supply line current and the need for power factor correction in addition to the torque ripple.

A CSI is a VFD in which the DC-Link is inductive and, as a consequence, the DC current is relatively slow to change.The rectifier controls the DC current magnitude and the inverter selects sequentially the phases of the motor wherethe DC current is fed. The semiconductor switches in a CSI must block either voltage polarity, but are only required toconduct current in one direction. When braking the motor, the recovered power is fed to the AC line supply.



The LCI is a very commonly used CSI topology. The LCI uses thyristors (SCRs) in both the rectifier and inverter. Thethyristors of the rectifier can be turned off (commutated) only by reactive power supplied by the load, that is, the motor(which is where the LCI gets its name). Thus, the motors controlled by LCIs are always synchronous motors, whichhave to be overexcited to operate at a leading power factor. A simple six-pulse LCI is depicted in Figure 4-61 (notethat both the rectifier and inverter are six-pulse designs), although most LCIs tend to employ 12-pulse rectifiers andinverters.

A typical LCI drive supplies power to the motor from about 10 Hz up to a frequency of up to 60 Hz, allowing for fullcontrol of a two-pole motor up to 3600 rpm. Near zero Hz, and zero rpm, an LCI drive has to be supplemented withmeans to start the motor, as the load motor at low rpm cannot develop the necessary back electromotive force (EMF)to commutate the SCRs.

LCIs are used for gas turbine starters or large drive systems with motor power greater than about 20,000 HP to45,000 HP. The simplicity in the design and the scalability contribute heavily to the reliability. In this power range, thelarge reference base of the LCI drives makes the technology to be considered mature enough.

A VSI (Figure 4-62) is a VFD in which the DC-Link is capacitive (consists of multiple capacitors) and, as aconsequence, the DC voltage is relatively stiff. The power semiconductor switches in the inverter (IGBTs or IGCTs)are controlled in such a way that a reasonably sinusoidal voltage is fed to the motor. The semiconductor switches in aVSI must be able to block the DC voltage and be able to conduct current in either direction. For this reason, the IGBTsor IGCTs each have a parallel diode, as is shown in the figure.

VSIs can be found with many different topologies and different output levels. The number of levels is simply equal tothe number of discrete values that the phase voltages being fed to the motor can have. The most common topology isa two-level inverter where the phase voltage can be either the positive or negative value of the DC-link voltage. Thistopology is widely used for all kinds of low voltage VFDs, usually equipped with a 6-pulse diode rectifier and withIGBTs as active switching elements in the inverter.

In the basic two-level inverter, each of the output phases can be fed with two different levels of voltage: DC-plus andDC-minus. A more sophisticated design is the three-level inverter, in which each of the output phases can be fed withthree different levels of voltage: zero, DC-plus, and DC-minus. The next most common design is a five-level inverter,in which each of the output phases can be fed with five different levels of voltage: zero, DC-plus, two times DC-plus,DC-minus, and two times DC-minus.

Multi-level inverter configurations are often found in medium voltage systems. In general, the greater the number oflevels, the more sinusoidal the voltage signal is, and, consequently, the smoother the motor torque is.

Figure 4-61—Six-pulse Load Commutated Inverter

ACgrid

Grid sideconverter(rectifier)

Motor sideconverter(inverter)DC inductor

MSynchronous

motor



The three-level VSI topology is the most common topology for medium voltage VFD systems for high performance.Such VSI systems are built up to 40,000 hp and, depending on the application, this topology is equipped with a multi-pulse rectifier (typically 12 or 24 pulse) or an active front end rectifier. Since the voltage steps on the output arerelatively large, an inverter duty motor that is designed to cope with steep voltage pulse edges is required for the VFDsystem. For retrofits using old motors, an output filter for the inverter is recommended.

For applications where the required shaft power is below 20,000 HP, VSIs are the standard solution due to lowerintegration costs (especially if an induction motor can be used), less problematic AC supply connection, and highcontrol performance.

The large majority of VSIs use pulse width modulation (PWM) to control the voltage being fed to the motor. Per Weberet al. [14], the power conversion logic is responsible for determining the AC voltage waveform that will be applied tothe motor. There are a variety of ways in which this may be accomplished. But if the switching frequency of the IGBTmodules is at least 10 times greater than the desired fundamental frequency to be applied to the motor, then themethod that will give the best reproduction of the desired AC waveform is PWM.

Per Williams et al. [21], the PWM inverter outputs a series of voltage pulses. Each pulse is the same magnitude, andeach pulse width is individually controlled to govern the RMS output voltage. The output voltage waveform and itsrepresentation as a sine wave are dependent on the number of pulses per cycle, known as the carrier frequency.

Per Williams et al. [21], the term modulation index is used to express the output voltage as a percentage of maxoutput voltage. Two figures in [21] show two signals having the same carrier frequency but one having wide centerpulses (modulation index of 95%) and one having narrower center pulses (modulation index of 50 %). By simplyspeeding up or slowing down the rate at which the pulses are switched on and off, it is possible to vary the frequencyof the output waveform. Carrier frequencies are normally in the range of 800 Hz to 20,000 Hz.

The output of a VFD necessarily consists of pulses. This is because the inverter switches have to be either fully on orcompletely off. The modulation scheme uses the duration and spacing of the pulses to maximize the fundamentalwhile minimizing the harmonics. Many schemes, thus, have an output spectrum in which the switching frequencyappears, at a higher frequency than the desired fundamental, and these switching frequency components are offset alittle by the fundamental. These voltage harmonics will drive some current depending on the motor inductance. Thisharmonic current will produce time harmonics in the torque (an issue that will be discussed shortly).

There has been a vast number of pulse patterns developed over the years for PWM schemes. These fall into threecategories:

Figure 4-62—Two-level Voltage Source Inverter Drive with Diode Rectifier

AC

M



1) Sine-Triangle PWM: The oldest and simplest method uses a triangular carrier wave compared to a sinusoidalreference, and generates switching instants at the crossings. The triangular wave’s frequency, known as thecarrier frequency, is set equal to the switching frequency of the IGBT’s while the reference wave has the samefrequency as the electrical frequency being fed to the motor. This method creates a waveform which has afundamental, but has large components at multiples of the switching frequencies plus or minus the fundamental(i.e., inter-harmonics or sidebands). The components of the spectrum are directly predictable from thefrequencies involved. It is a real-time method.

2) Look-Up Method: Another common method is the look-up method. The pulse pattern for each output frequencyand modulation index is stored in memory and read out iteratively. For some VFDs and motors, there can beoptimum pulse patterns for minimizing switching losses and motor losses. But, the pulse pattern repeats itselfidentically as long as the output frequency and modulation index are the same, so there are distinct andpredictable excitations generated.

3) Space Vector Method: A more recent method is the space vector method which came out in the 1980s. In thisscheme, the control keeps track of the stator flux vector (both orientation and amplitude) and then uses somesimple rules to apply voltage to the stator windings in such a way to drive the flux to follow a rotating referencevector. The pulse pattern is not necessarily the same from cycle to cycle so the harmonic spectrum is spread outwith many more components of smaller amplitude. This has the beneficial effect of reducing the perceived motornoise since there is not a dominant frequency.

PWM can be either asynchronous or synchronous. In asynchronous PWM, the switching frequency is held constant,and is not related to the fundamental frequency. In synchronous PWM, the switching frequency is variable, and isalways kept a specific integer multiple of the fundamental frequency.

4.10.4 VFD Torsional Excitations

4.10.4.1 General

If a VFD were “perfect,” it would output current and voltage waveforms to the motor that are perfectly sinusoidal.However, since some distortion always occurs during the AC-to-DC-to-AC conversion process, the output of all VFDswill contain harmonic distortion in both the current and voltage waveforms. Voltage harmonic distortion can result instressing of electrical components such as cables while current harmonic distortion can result in pulsating torques inthe motor air-gap that can excite torsional vibration in the mechanical drive train.

This harmonic distortion is often referred to as “ripple.” This concept can be confusing since there are two distinctkinds of ripple—current ripple and torque ripple. Adding to the confusion is the fact that the two types of ripple havedifferent harmonic numbers, which always differ by plus or minus one. That is, the fifth and seventh harmonic currentripples generate a sixth harmonic torque ripple. Likewise, the eleventh and thirteenth harmonic current ripplesgenerate a twelfth harmonic torque ripple. Readers interested in understanding why this is so should see [2] or [20].

Experience indicates that the electrical engineers at the VFD supplier tend to think in terms of current ripple.Conversely, since the torsional analyst needs to know the torque ripple that is exciting the train, that is the parameterof primary interest to him or her (while current ripple, indeed, generates torque ripple, it is the latter parameter thatneeds to be known in order to perform a torsional analysis). Thus, when dealing with the VFD supplier, the analystmust insist on obtaining data on the torque ripple generated by the VFD. If this is not made clear, there is a stronglikelihood that the data provided by the VFD supplier will be for current ripple, which is not of much use to the analyst.

The design of VFDs has vastly improved since the 1990s. Most of the early application selections were CSI or LCIdrives. These systems were designed to supply an AC current that was a modified sine wave consisting of six stepsper cycle. Due to this stepping, the torque harmonics at six times the output frequency and its multiples exist and canbe quite high. For example if the output frequency of the inverter is 30 Hz, the ripple components will be atfrequencies of 180 Hz, 360 Hz, 540 Hz, etc.



Recent technology improvements in high power IGBTs and IGCTs has enabled the use of Pulse Width Modulated(PWM) drives to power levels of several thousand horsepower. The output phase voltage of a PWM inverter consistsof samples of the intermediate DC voltage. The control system varies the length of the samples (or pulses as they arecommonly called) in order to create an AC current waveform to the motor that is very close to that of a sine wave. Inthe three-level PWM systems that are common in the highest power range, the intermediate DC voltage circuit is splitinto two halves. Thus positive, zero, and negative voltage levels are available for sampling. With three levelsavailable, it is easier to produce sinusoidal phase currents than with two levels.

Though the VSI drive’s torque is very smooth when compared with LCI drives, some excitations at six times theoutput frequency and its multiples exist due to the inequality of the power semiconductor turn on and off delays, whichcauses a brief dead time when the output is not connected to any defined voltage. Further, the measurement errors inthe motor current sensors of the drive may cause small excitations at the output frequency and twice the outputfrequency. Typically VSI air-gap torque components having frequencies less than 50 Hz are smaller than 1.0 % of therated torque of the motor even without speed controller damping.

The main difference between CSI or LCI drives and VSI drives is the behavior of the magnitude of the torsionalexcitations as a function of the load. With CSI and LCI drives, the magnitude almost always depends strongly on theload torque (and, thus, the operating point) but with VSI drives, the load-dependence is usually smaller. Thus, while aVSI drive almost always has lower excitations than an LCI drive at rated torque, it may have higher excitations at noload.

In addition to these integer torque harmonics, as these ripple frequencies are often called, there exist some inter-harmonics. These noninteger harmonics are caused in a VSI drive by the nonidealities in the PWM modulation andthe intermediate DC voltage’s ripple. A similar phenomenon is also present in CSI and LCI drives due to ripple in theDC current.

The intermediate DC voltage is rectified from the three-phase constant frequency (50 or 60 Hz), constant voltage ACsupply. The intermediate voltage is not perfectly smooth but contains some ripple. Typically a 50 Hz fed six-pulserectifier produces DC voltage ripple harmonics at 300 Hz (6x50 Hz) and multiples of it. However, there are usuallycomponents at all multiples of 100 Hz (2x50 Hz) because of the asymmetry of the AC supply’s three-phase system. Ifthe supply frequency is 60 Hz, the components will be at multiples of 120 Hz, the most dominating being at multiplesof 360 Hz. If a 12-pulse rectifier is used, the most dominating frequencies will be multiples of 600 Hz or 720 Hz with 50or 60 Hz supply, respectively.

Per Kaiser et al. [2], the main technique to reduce torque ripple is to decrease the harmonic content of the outputcurrent. That is generally done by increasing the PWM switching frequency, adding additional voltage levels to theoutput voltage, or both. For the cases where this is not possible, application of filters is an alternative option. It iscommon practice to equip VFDs with an LC filter on the output to attenuate some harmonic currents in order toreduce motor torque ripple. This technique is used for LCIs and VSIs. The disadvantage is that the filter causes extraexpense, increased size, and poorer efficiency. For drive systems with high sensitivity to torque ripple, the multi-levelseries cell designs offer an advantage.

4.10.4.2 Torsional Excitations Generated by Current Source Inverters

Current source inverters (CSIs) generate two primary types of torsional excitations, as follows:

1) harmonic excitations,

2) inter-harmonic excitations.

The frequencies of the harmonic excitations are given by the following equation:

fexcite = pi x fvfd x 1, 2, 3…. (4-37)



where

fexcite is the excitation frequency applied to the torsional system, Hz;

fvfd is the VFD output frequency, Hz;

pi is the number of pulses in inverter.

Thus, for the most basic inverter, which has six pulses, we get the familiar result that the excitations are at 6, 12, 18,etc. times the VFD output frequency. It should be noted that many engineers erroneously think that this yieldsexcitations at 6, 12, 18, etc. times running speed on the Campbell diagram. However, that only happens to be truewhen dealing with a two-pole motor. In general, the order numbers on the Campbell diagram corresponding to theseexcitations are given by the following:

Nord = 0.5 x pi x Np x 1, 2, 3…. (4-38)

where

Nord is the order number on the Campbell diagram;

pi is the number of pulses in inverter;


As shown in Hudson [12], the amplitudes of the harmonic excitations are strong functions of VFD output frequency,thereby, making them strongly speed-dependent. Accordingly, when dealing with CSIs, the VFD supplier will oftenprovide the amplitudes of the various harmonic excitations (as percentages of motor rated torque) in tables for variousVFD output frequencies. Sheppard [23] states that the amplitude of the sixth harmonic can be as high as 20 % to30 % of motor rated torque. However, improvements in VFD design in the time since Reference [23] was writtenmake it unlikely that an excitation this large would be observed in a modern CSI.

As has been previously stated, the output voltage of a VSI drive consists of samples taken from the intermediate DCvoltage. Thus, the ripple in the DC voltage is sampled as well. Similarly, the CSI or LCI drive’s inverter samples theDC current with its ripple. This sampling causes intermodulation frequencies to appear in the phase voltages andcurrents and, thus, in the torque.

Per Terens and Grgic [20], in all DC-link type converters, the output section (inverter) is not completely isolated fromthe input section (rectifier). The energy storage capacity of the inductor in the DC-link, is limited, which results in a DCripple. The effect of this is the generation of harmonic currents with noninteger frequency multiples, caused byinterference from the constant line frequency with the speed-dependent VFD output frequency, which are injected intothe supply system.

The frequencies of the inter-harmonic excitations for a CSI are given by the following equation [19, 20, 24]:

fexcite = pr x m x fline ± (pi x n x fvfd) (4-39)

where


fline is the line frequency, Hz;

pr is the number of pulses in rectifier;




pi is the number of pulses in inverter;

m, n is the positive integers.

In general, the number of pulses in the inverter and rectifier are usually equal. Because of the negatively-sloped lineon the Campbell diagram that results (and the problems that it creates), the inter-harmonics involving the minus signare usually the most dangerous. Thus, for the basic CSI which employs a six-pulse rectifier and a six-pulse inverter, ifm and n are taken to be equal, the excitations are at 6, 12, 18, etc. times slip frequency, fline – fvfd. As is noted byReferences [7], [12], and [25], the only inter-harmonics that are normally significant are the 6 and 12 times slipfrequency ones, with the 6 being the predominant one. Like the harmonic excitations, the amplitudes of the inter-harmonic excitations can also be functions of the VFD output frequency [12].

Per Reference [20], these inter-harmonic excitations are typically small in magnitude, provided that the dimensioningof the DC-link storage elements is sufficient. The amplitudes of some of these components can, in worst cases,amount to 10 % of rated torque. Usually, they are about 3.0 % of rated torque.

The relative danger of harmonic and inter-harmonic excitations can be seen by examining typical Campbell diagramsfor each. Figure 4-63 shows the primary harmonic excitations for a six-pulse CSI driving a two-pole induction motor,while Figure 4-64 shows the primary inter-harmonic excitations for the same drive (other excitations of secondaryimportance are also present, but are omitted from the figures for clarity). The operating speed range is from 50 to100 % speed (1800 to 3600 rpm), which is fairly common.

Since the large majority of reported VFD-induced torsional failures have involved the first torsional mode, thelocations of the resonances with the first mode are of primary interest. Examination of Figure 4-63 reveals that theresonances with the first mode generated by the harmonic excitations occur at low speeds, well below the operatingspeed range. These resonances, therefore, are only transient in nature and are not likely to be dangerous.

Figure 4-63—Campbell Diagram Showing VFD Harmonic Excitations

0 1000

100

200

300

400

500

2000Speed, RPM

3000


6X

12XVFD Harmonic Excitations

Freq

uenc

y, H

z

3600



Examination of Figure 4-64 reveals the situation with the inter-harmonic excitations to be drastically different. It isseen that not only are the resonances with the first mode within the operating speed range, they are close to MCOS,where the system’s transmitted torques are highest. This is a very dangerous situation and needs to be considered inthe torsional analysis of all VFD-driven trains. Thus, even though their excitation magnitudes are usually significantlysmaller than those of the harmonic excitations, the inter-harmonic excitations are often the more dangerous of thetwo.

Per Sihler et al. [19], it is important to note that the amplitudes of inter-harmonic pulsating torques are typically low inthe nominal speed range of a compressor train, i.e. less than one percent of the nominal torque. This low level ofelectrical excitation has been confirmed by air-gap torque measurements taken during testing of a 30 MWcompression train [19]. However, because of the potential to excite the first mode at a speed close to MCOS, eveninter-harmonics having amplitudes less than 1.0 % of motor rated torque can be dangerous.

Kaiser et al. [2] studied the level of inter-harmonics produced by various VFD architectures. The magnitudes rangedfrom as high as 8 % for LCIs to less than 1 % for PWM VFDs.

4.10.4.3 Torsional Excitations Generated by Load Commutated Inverters

All of the information presented on harmonic and inter-harmonic excitations in CSIs is equally applicable to LCIs.However, LCIs have one additional characteristic that makes them more likely to create torsional problems thanCSIs—they generate very large transient excitations during the start-up process (similar to synchronous motors).

The LCI cannot start the motor smoothly because it depends on the motor voltage to commutate the inverterthyristors, and there is insufficient voltage to do this below about 10 % speed. In starting mode, the motor current isapplied for a short time, then turned off completely by the rectifier. The thyristors in the inverter recover, and differentones are then gated, after which the current is turned on again. Thus, the torque has intervals of zero during thisprocess, so there are relatively large pulsations. Fortunately, this process, which is sometimes referred to as pulseoperation mode, stops as soon as the motor reaches 10 % speed.

Figure 4-64—Campbell Diagram Showing VFD Inter-Harmonic Excitations

0 1000

100

f1 = 30Hz

200

300

400

500

2000Speed (RPM)

3000


VFD Inter-Harmonic Excitations

3600

6 × (fline-fvfd)

12 × (fline-fvfd)



Per Wolff and Molnar [26], typical excitations for a six-pulse LCI are as follows:

— 6 x VFD output frequency – 20 % of steady-state torque (100 % for 0 % to 10 % speed);



— 24 x VFD output frequency – 1 % of steady-state torque (10 % for 0 % to 10 % speed).

Frei et al. [18] give a Campbell diagram for a system having an LCI-fed synchronous motor drive. For this system,they distinguish two speed ranges—pulse operation (from 0 to 600 rpm) and load commutation (from 600 to 6000rpm). During the initial pulse operation period, pulsating torques at six to twenty-four times the stator feed frequency,and having relatively large amplitudes, occur. In load-controlled operation, this motor, which has a six-phase statorwinding, is subject to pulsating torques having only 12 and 24 times the feed frequency and having moderateamplitudes.

Thus, when dealing with an LCI, there are two different modes to consider: pulse operation mode (also sometimesreferred to as forced commutated mode) and load commutated mode. Since the excitation torques are much larger inpulse operation mode, that mode is often governing. However, since pulse operation mode is always terminated at aspeed below the minimum operating speed, the resonances excited in this mode are transient, not steady-state.Thus, similar to a synchronous motor, whenever there is an LCI in the system, a transient analysis of the start-upcondition must normally be performed to ensure that the mechanical train can handle the transient resonancesoccurring during start-up.

4.10.4.4 Torsional Excitations Generated by Voltage Source Inverters

The excitation magntiudes arising from VSIs are typically smaller than CSIs and LCIs. However, the excitationsarising from VSIs are not as predictable or well-understood as those from CSI-class drives. This has led to a largenumber of torsional failures to be reported in the technical literature involving VSI-style drives. VSI drives should begiven the same level of attention as CSIs and LCIs during a torsional analysis.

Voltage source inverters (VSIs) generate three primary types of torsional excitations, as follows:

1) harmonic excitations,

2) inter-harmonic excitations,

3) broadband excitations.

Per Reference [24], the frequencies of the harmonic excitations are given by the following equation:

fexcite = 6 x fvfd x 1, 2, 3…. (4-40)

where


fvfd is the VFD output frequency, Hz.

Thus, once again, the harmonic excitations are represented by positively-sloped lines on the Campbell diagram andthe sixth harmonic is normally the largest.



The inter-harmonics have two sources. Firstly due to the frequency differences between the line frequency and theVFD output frequency as given by Equation 4-40 and secondly due to the sidebands of the PWM switchingfrequency.

The line frequency related DC voltage ripple and the resulting inter-harmonic excitations are in VSI drives usuallyquite low. However, Huetten et al. [7] have observed significant inter-harmonics between VFD output frequency andline frequency.

Per Song-Manguelle and Nyobe-Yome [24], the frequencies of the inter-harmonic excitations generated by the pulsewidth modulation (PWM) of VSIs are given by the following equation:

fexcite = m x fswitch ± (n x fvfd) (4-41)

where


fswitch is the VFD switching frequency, Hz;


m, n is the specific positive integers.

Thus the VFD output frequency is interacting with VFD switching frequency.

As was the case with CSIs, the inter-harmonics of most interest are usually the ones that are functions of thedifference between the two frequencies. However, in direct contrast to CSIs, where VFD output frequency wasinteracting with line frequency, in a VSI, the interaction is between VFD output frequency and VFD switchingfrequency (or VFD carrier frequency, which is equal to switching frequency). Inter-harmonics are also sometimesreferred to as “sidebands” because when an FFT is performed on the torsional signal, the inter-harmonics appear aspaired frequencies equally straddling harmonics of the carrier/switching frequency.

The integers, m and n obey some rather odd rules. If m is even, then n is simply the multiples of 6, i.e., 6, 12, 18, etc.[24].

If m is odd, then the permissible values of n are defined by the following Equation [24]:

n = 3(2i + 1) (4-42)

where

i is the all nonnegative integers.

Thus, if m is odd, the corresponding values of n are 3, 9, 15, 21, etc.

There are many VSIs that do not strictly obey these equations (for both harmonic and inter-harmonic excitations).Although the VSIs torque is very smooth when compared with LCIs, some excitations at six times the outputfrequency and its multiples exist due to inaccuracy of the power semiconductor turn on and off delays (these are theharmonics given by Equation 4-41). Furthermore, the measurement errors in the drive’s motor current sensors maycause small excitations at the output frequency and twice output frequency.

There have also been occasions where damaging VSI inter-harmonics have been observed between VFD outputfrequency and frequencies other than switching frequency or line frequency. Adachi et al. [10] report on a high lateralvibration problem caused by high oscillating torques in a geared centrifugal compressor train. However, the source of



the air-gap torque pulsation was not the sidebands based on the PWM carrier frequency (4800 Hz). These sidebandswere instead based on the control loop frequency (1024 Hz) or sampling frequency (256 Hz) of two microprocessorsin the VFD panel.

Per Adachi et al. [10], based on the measured lateral vibration and the estimated torsional amplification factor, theexcitation torque being output by the VFD was estimated to be 1.3 % of motor rated torque. This is fairly close to thevalues measured during the VFD-motor combined test at the manufacturer’s shop.

Additionally, counter to intuition, dangerous inter-harmonics are not limited to small values of m and n in Equation 4-42. Tanaka et al. [6] report problems associated with inter-harmonics having n values as high as 75 and 132.References [2] and [10] describe problems instigated by inter-harmonics having n values of 22 and 30, respectively.

Finally, there have been cases where, although the VSI was known to be causing the failure, the mechanism was notunderstood. Feese and Maxfield [8] discuss such a case, where the oscillating torque in the train was observed to beexcessive whenever the VFD was operated at an electrical frequency above the first torsional natural frequency. Theywere not dealing with a resonance condition since, once the electrical frequency exceeded the first torsional mode,the oscillating torques never dropped, regardless of how far above the first mode they went. Although they finallyarrived at an adequate design by making changes to the system, it does not appear that they ever understood whythe excessive torques were occurring.

As was mentioned previously, there are two basic forms of sine-triangle PWM—synchronized and nonsynchronized.Since nonsynchronized PWM involves a constant switching frequency, inter-harmonic excitations involving theswitching frequency are more likely to be significant than with synchronized PWM. Indeed, Tsukakoshi et al. [4]solved a troublesome torsional vibration problem (also reported on in Reference [3]) by changing fromnonsynchronized to synchronized PWM, along with making some changes to the VFD control algorithm.

The third type of excitation that can occur in VSIs is broadband excitation. As stated before, many excitation sourceshave distinct frequencies that are related to the VFD output frequency, line frequency, and/or VFD switchingfrequency. This is due to the cyclic behavior of the system, where the sequence of events repeats itself constantly.However, if the pulse width modulation scheme is based on hysteresis control, then the excitations are no longer atdiscrete frequencies. Hysteresis type control of motor current or torque is a random process and, in theory, there is norepeating pattern. This means that the excitation spectrum does not have any distinct components. Instead, thespectrum is a continuous function, similar to an acoustic noise spectrum. If the noise spectrum has a constant value,it is called white noise.

A white noise spectrum means that there is some excitation at all frequencies but the magnitude is lower than with aspectrum having only a few distinct frequency components. Moreover, due to the random nature, it does not causesimilar amplification of torsional resonances as a distinct harmonic frequency. White noise may temporarily causevibration to increase but then the exciting frequency component disappears or its phase angle changes to a dampingone and the oscillation decreases again. Thus, in most cases, the stresses in the mechanical system do not build upto harmful levels.

If the analyst wishes to account for broadband or white noise excitations in his or her analysis, Kocur and Muench [9]present a method for modeling white noise excitations. Kocur and Muench [9] and Rotondo et al. [27] describe fieldcases where the measured VFD excitation was primarily broadband. In the latter, the drive employed Direct TorqueControl (DTC) modulation. This is an asynchronous, hysteretic control so it has no specific frequency component—rather, the “noise” is spread along a large frequency interval.

Fortunately, air-gap torques of multi-level inverters at broadband spectrum, also referred to as white noise, are usuallyof a very low level that is considered not to be a cause for concern regarding the excitation of torsional naturalfrequencies.



4.10.5 VFD Control System Instabilities

Unfortunately, the excitations generated by the VFD are not the only potentially dangerous items introduced by theVFD. Additionally, as is reported in References [28] and [29], there have been instances where the VFDs controlsystem has interacted with the mechanical train in such a way as to excite the first or second torsional mode in anunstable fashion. This usually occurs if one of the control frequencies coincides with a torsional natural frequency,usually the first mode. This problem is very difficult to prevent beforehand but it can usually be solved by varyingsome of the control parameters in the VFD software.

This section deals with VFDs seen from a system control point of view. Without taking into account any particularbehavior of the semiconductor switches and their effects (as done in previous sections), the linear model of a typicalVFD has the following form:

(4-43)

where

K(s) is the torque transfer function;

Tref is the desired torque, N-m (in.-lbf);

Tact is the actual torque, N-m (in.-lbf);

ω is the frequency at which desired torque is varying, rad/sec;

d is the system time constant, sec;

ζ is the system damping ratio;

s is the complex frequency variable.

From a control systems standpoint, a typical VFD can be viewed as a transfer function, K(s), whose input is the torquereference and whose output is the actual torque. The transfer function represents a typical second-order system,where the parameters are dependent on the different possible control philosophies, motor electrical parameters, andphysical properties. However, by plotting the Bode diagram of the transfer function for some typical cases, it can bestated that most of the currently available VFDs fall in the range shown in Figure 4-65. In this plot, 0 dB means thatK(s) equals 1.0 or that the actual torque is equal to the reference value. Thus, the control bandwidth of the VFD canbe defined as the frequency range over which the Bode plot’s magnitude is “close enough” to 0 dB. Typically, 3 dB isused as the criterion for max deviation.

As can be seen in the figure, the torque control bandwidth can vary from 20 Hz up to 200 Hz. Lower controlbandwidths are typical of LCI-style drives while higher bandwidths can be achieved only using VSI topology andspecific inverter control methods that can guarantee high control performance. The concept of control bandwidth is akey element in order to consider the process and the control loops characterizing it.

K(s)Torque

reference Torque

actual

K s Tact

Tref

-------

d

2----– 2s 2+

s2 2s 2+ +------------------------------------= =



In a typical process control application, it is possible to identify the following main blocks (see Figure 4-66).

The drive power stage and the electric motor characteristics determine the torque control bandwidth, the processcontrol governs the process controlled quantities, while the speed control can be either integrated into the VFDcontrol or done externally among the process control.

Control stability, i.e. the stability of the control of the process, is determined by the following factors:

— understanding of the process dynamics and control requirements;

— understanding of the VFD control capabilities in terms of control performance.

The above considerations result in the following best practices.

— Tune the process control in such a way that it is slower than the speed control.

Figure 4-65—Bode Plot of Torque Transfer Functions for a Selection of Drives

Figure 4-66—Block Diagram of Drive System Control

10090

180Pha

se (d

eg)

Mag

ntu

de (d

B)

270

360

-40

Bode Diagram

Frequency (Hz)

-30

-20

-10

0

10

101 102 103

e.g. Pressure refProcesscontrols

Drivecontrol

Drivepower stage

EM +workingmachine

Process

e.g. PressureSpeed ref SpeedTorque ref Torque



— Have the speed control tuned to be slower than the VFD torque control. This can be critical, especially in largeLNG trains.

— Consider using the VFD in torque controlled mode only, by integrating the speed control directly in the processcontrol. The VFD can still be used for speed monitoring and fast protection.

— Make sure that the speed reference or torque reference given by the process control is free of noise and otherinterfering signals because these will be reproduced in the motor torque and may excite mechanical resonances.

— Make sure that the speed sensor is properly installed. For example, eccentric mounting causes an erroneousshaft rotation frequency component in the measured speed. Further, make sure that lateral vibrations of themotor are not affecting the speed sensor operation. Any erroneous fluctuation in the measured signal maybecome amplified in the speed control and excite torsional resonances. It is naturally possible to attenuate thiskind of interfering signals by filtering but then it may be impossible to use the speed control to damp resonances.

Figure 4-67 will be used to illustrate the concept of electro-mechanical (E/M) interactions—consider a VSI supplyingan induction motor to drive a compressor train. In the electrical system [Va Vb Vc] are the three-phase voltagesapplied on the induction motor, [VDC ia ib ic] are the measured values of the DC-link voltage together with invertercurrents used by the control, [gu gv gw] are the applied switching commands and Tairgap is the air-gap torqueproduced by the induction motor. In the mechanical train, Ωmotor is the motor speed, ΩC the speed of a certain shaftsection and Ref is the external reference for the electric drive which could be either speed or torque.

Regardless of whether the VFD control uses measured signals or estimated ones, the system shown in Figure 4-67behaves as a closed loop control system where the electrical and mechanical parts are interacting between eachother. Oscillations in the motor currents are in general seen as oscillations in the air-gap torque and, depending on themechanical system properties (e.g. mode shapes) and conditions (e.g. load), in the speed of the different mechanicalsections. In the same way, any speed oscillation of the mechanical system and, thus, of the electric motor are seen bythe electrical system as an oscillation of counter EMF. This leads to oscillating currents which have to be handled bythe VFD control; the reaction of the VFD control shall be such that the oscillations are damped or at least not furtherexcited, resulting in control instability. These phenomena, which consider the behavior of the VFD internal controlloops with respect to the mechanical system, lay under the concept of E/M stability.

A good understanding of both the VFD control and the mechanical system dynamics is fundamental to addresspotential issues and interpret the system behavior in a correct way. The outcome of these understandings can beeither addressed by considerations in the design of the mechanical system or by appropriate tuning of the VFDcontrol. Some VFD vendors offer analysis where the VFD system with its control loops and specific design propertiesis combined with the mechanical system. These studies are based on simulations where models provided by therespective manufacturers are combined together; the degree of modeling detail influences the amount of systemdynamics which can be observed.

To summarize the concepts introduced in this section, process stability shall be intended with respect to the followingtwo aspects.

1) Stability of the process control which is designed to achieve certain process dynamics and controlrequirements.

2) Stability of the VFD internal control loops when subjected to electromechanical oscillations.

The fact that the VFD, motor, controller, and mechanical train form a large closed loop system is a good news/badnews situation. The bad news is that, as stated previously, the control system can go unstable and lead to torsionalvibration failures. The good news is that if the controller is designed correctly, it can be designed to damp out torsionalvibrations.



The mechanism by which control loop instabilities can lead to torsional failures is explained by Rotondo et al. [27],who state that a closed loop simulation takes into account the fact that the speed oscillations of the motor are seen bythe electrical system as an oscillation of counter EMF. This leads to oscillating currents, and so to oscillating torques,which are seen by the control system as a disturbance in the torque control loop. In principle, any current or torquecontroller will react to this phenomenon. It must be ensured that the control is damping the oscillations and notexciting them. This can only be done in a closed loop analysis.

Del Puglia et al. [29] used a coupled E/M simulation to explain an observed instability. They describe the instabilitymechanism as follows. The motor voltage control loop sees the torsional oscillations in the back-EMF voltage, whichit rectifies to the DC-link. This results in a pulsation of the energy stored in the DC reactor. The frequencies of the twotorsional modes, thereby, appear in the DC link current. The pulsation of the two torsional frequencies in the DC-linkcurrent will, in turn, appear as pulsation of the same frequency in the motor current and, thus, in the motor torque.Depending on their phase shift vs. the speed oscillations of the motor, those air-gap torque pulsations can furtherexcite the respective torsional resonances.

On the other hand, the VFD controller can be used to actively damp out the torsional vibrations. Cases where this hasbeen successfully employed are reported in References [19] and [30]. By looking at the process from this point ofview, the VFD can be considered as a torque actuator. Typically mechanical oscillations can be detected by the VFDcontrol in the speed signal and in some internal electrical quantities and opportunely used to generate a torque todamp them. The VFD control can be used to actively damp the mechanical oscillations occurring on the shaft,provided that they are within the VFD control bandwidth. Typically the lowest natural torsional resonance frequencyhas to be lower than half of the torque control bandwidth in order to successfully damp the resonance.

Usually the damping can be done by tuning the speed controller properly. Myszynski and Deskur [31] provide rules fortuning the commonly used PID speed controller to damp resonances in a three moment of inertia mechanical system.

When the control is being used to provide torsional damping, the previously expressed desirability of having a slowspeed control is no longer applicable. That is because with a faster speed controller, the system is able to react fasterto load changes. In this way, the oscillations after load or speed changes are quickly attenuated and continuous

Figure 4-67—VSI and the Compressor Drive Train

Ref

Measurementand VSIcontrol

[gu

gv g

w]

[VDC ia ib ic]

[Va Vb Vc]Supply

andconverter

Inductionmotor

Electrical System

Tairgap

Tairgap



excitations coming either from the motor or the driven load are prevented from being amplified by the mechanicalresonances.

Sihler et al. [19] designed a system that they referred to as integrated torsional mode damping (ITMD). This system isbased on a torsional vibration measurement in the mechanical system and an interface to the existing inverter controlof the drive system. The DC-link inductor of the LCI is partially used as an integrated energy storage unit and iscombined with a smart damping controller, which reacts to a torsional vibration by modulating a small amount of thestored energy and sending it to the motor without impacting the normal operation of the system. As a result, the activepower modulation at a torsional natural frequency of the mechanical system has a strong damping effect for torsionalvibrations.

In order to design such a torsional damper or to analytically determine whether control instability could lead totorsional issues, a fully coupled electromechanical simulation, accounting for the properties of the VFD, motor, controlsystem, and drive train, must be performed. Such simulations have been performed by Huetten et al. [1], Tanaka et al.[6], Huetten et al. [7], Rotondo et al. [27], Del Puglia et al. [29], Falomi et al. [32], and Hernes and Gustavsen [33]. Inalmost all of these simulations, the train torsional model was greatly reduced to a model having five disks or less. Thisusually provides acceptable accuracy since the only torsional modes that are significant are those whose frequenciesare within the bandwidth of the drive, which is normally only the first few modes.

Huetten et al. [1] got approximately the same results with the coupled E/M analysis as they did with the “traditional”torsional analysis. Accordingly, they concluded that the uncoupled torsional analysis as specified in API 617 is asufficiently analytical method to predict the torque response in the train elements caused by an inverter operation.Bear in mind that the accuracy of the results is mainly influenced by the accuracy of the expected air-gap torque.

In general, the decision to run an E/M simulation should be based on the complexity of the train and, perhaps, thelevel of “novelties” in the proposed VFD, as compared to similar trains. In general, it is a risk containment action, sothe decision whether to run it or not should be left up to the end user. It should be recommended, not prescribed.

4.10.6 VFDs and Short Circuits

As stated previously, when a fixed speed motor is subjected to a short circuit, it generates torsional excitations at linefrequency and, sometimes, twice line frequency. The situation changes significantly when a VFD is employed, as theelectrical frequency being fed to the motor is now variable, not constant. This means that the chances of a resonancebeing triggered by a short circuit are theoretically greater when a VFD is in the system. For instance, in most motor-driven compressors, the first torsional mode, which is the mode most likely to be excited by a short circuit, liessomewhere between 20 and 30 Hz. This means that in a fixed speed system, short circuit excitations, occurring at 50or 60 Hz, are not likely to trigger a resonance with the first mode. However, in a variable speed system, it is quitecommon for the minimum speed to be 50 % or less of the synchronous speed, meaning that the electrical frequencywill be in the 20 to 30 Hz range when running at a reduced speed. The implication of this, of course, is that there isoften a running speed where the short circuit excitation will be in direct resonance with the first torsional mode. Thus,when a VFD is in the system, short circuits are potentially much more dangerous than in a fixed speed system.

Of course, any analyst who works on VFD-driven systems for any length of time will almost certainly be exposed tothe argument that if a short circuit were to occur, the VFD would sense it, and shut down the system before anydamage could occur. Although this argument is not entirely without merit, there are few instances where it can berelied on to justify not worrying about any short circuit conditions. The truth is the ability of a drive to detect a short andshut everything down before any damage is done is dependent on the type of drive, type of short circuit, and locationof the short circuit.

For instance, the argument does have some merit when considering line-to-line short circuits because if two phasesare shorted together, the motor is still fed with power. Here, in theory, it should make some difference if the motor isbeing fed by a grid or a VFD because a grid will continue to run in the event of a two-phase short circuit and feed themotor until some protection trips. The same is not true for a VFD because if two phases are shorted together, thedrive would immediately detect it and shut off the supply to the motor. However, it is still worthwhile to perform a two-



phase short circuit analysis in order to be conservative. This is also to provide margin for various unforeseen failuremodes of the VFD, which are not easy to predict.

On the other hand, if a VFD is employed, three-phase short circuits are the same as for a fixed-speed motor. A three-phase short circuit means that all three terminals of a motor are shorted together, so it does not make any differenceif the motor is being fed by a grid or a converter.

Thus, the only thing that can be concluded is that the presence of a VFD decreases the likelihood of a short circuitoccurring, as is confirmed by References [2] and [16]. However, as stated previously, the chances of a failureincreases when a VFD is present since there is a much greater chance of resonance with the first mode. Thus, it isrecommended that if a VFD is in the system, the short circuit analysis be performed for the case where the excitationfrequency is at resonance, or as close to resonance as possible, with the first torsional mode.

4.10.7 Practical Methods for Dealing With VFDs

In a perfect world, performing a torsional analysis on a system being driven by a VFD would be fairly straightforward.The VFD supplier would supply the analyst with all the excitation amplitudes and frequencies that the VFD generatesand the analyst would then use them in a conventional torsional analysis, as outlined herein.

But the scenario described above occurs in a very small fraction of real world situations. The realities are as follows.

1) When queried for the harmonic and inter-harmonic excitations generated by their drive, many VFD suppliers willclaim that they are insignificant. However, real world experience suggests that such claims can be overlyoptimistic, for two reasons. First, in many cases, the actual amplitudes in the field are significantly greater thanwhat had been predicted. Second, as is noted by References [19] and [34], especially when the excitations inquestion are inter-harmonics, even excitations less than 1.0 % of motor rated torque (the level below whichmost VFD suppliers consider the excitations to be insignificant) can create problems. Reference [34] states thatexcitations as low as 0.65 % of rated torque can generate issues and some of the failures described in the otherreferences occurred with excitations smaller than that.

2) Other VFD suppliers will refuse to provide the excitations needed for the torsional analysis.

3) Some VFD suppliers will provide test data from a VFD-motor combination that is similar to, but not the same as,the one under consideration. However, as is pointed out by Hudson [12], the excitations are a function of thecombination of the drive and motor. Thus, unless the test is run with the actual drive, motor, and control systemunder consideration, the test data are not usable.

4) When asked for “harmonic data,” many VFD suppliers will provide data on input or output current harmonics.This is of little use to the analyst, who needs torque harmonic data in order to perform the analysis.

5) The VFD supplier will sometimes provide torque harmonic data for only one operating condition, usually therated condition. This can be misleading since the excitation magnitudes can vary greatly as the operating speedis varied, especially if the drive is of the CSI/LCI type.

6) If the VFD supplier provides any excitation data, it will almost always be for the harmonic excitations only. Inter-harmonic excitation data are often difficult to obtain.

Thus analysts will often find themselves “on their own,” especially with respect to the inter-harmonic excitations. Inthis common situation, although the analyst might be tempted to ignore all inter-harmonic excitations, that approachcan be risky. Instead, it is recommended that the analyst perform a “bounding analysis,” as follows.



1) If the drive is a CSI or an LCI, assume the inter-harmonic excitations are given by Equation 4-39 and that theamplitude is 3.0 % of motor rated torque.

2) If the drive is a VSI, the excitation frequencies are too nebulous to even try and guess at what they would be.Instead, the analyst should assume the worst case condition, where the inter-harmonic excitation generates aresonance with the first torsional mode right at MCOS. The excitation amplitude should be taken to be 1.0 % ofmotor rated torque.

Using the above assumptions, the analyst should analyze the train in the usual manner. If the train is capable ofhandling the assumed conditions, then it can be considered capable of handling inter-harmonics. If not, then theassumed situation should be presented to the VFD supplier and assurances sought that the analyzed condition willnot actually occur in the field.

Thus, with that clarified, when there is a VFD in the system, the following courses of action are recommended.

1) Per Kocur and Muench [9], purchase specifications for trains incorporating VFDs should specify upper limits onthe torque modulations from harmonic, inter-harmonic, and white noise produced by the VFD electrical systems.This is something that is done far too infrequently right now.

2) The torsional analysis should account for VFD harmonic and inter-harmonic excitations in the mannerdescribed above. The large majority of the time, the analysis will be done “open loop.” That is, the air-gap torqueexcitations have been determined from an open-loop analysis or test of the drive. This is often the worst casesince the speed control is often capable of decreasing the torque amplitude when the controller is correctlytuned.

3) If the VFD is an LCI, a transient analysis of the start-up condition shall be performed.

4) If there is a condition in which the electrical frequency feeding the motor can be within 10 % of the first torsionalnatural frequency, analyses should be performed for line-to-line and three-phase short circuits.

5) If there is great concern that a control instability could lead to torsional issues, or if it is desired to use the drive’scontrol system to provide electrical damping of torsional vibration, a coupled E/M simulation should beperformed.

6) In all cases, good communication between the VFD supplier, motor supplier, driven component supplier, enduser, and torsional analyst is paramount.

4.10.8 Other Ramifications of Having a VFD in the System

In addition to the issues already mentioned, there are a number of other ramifications of having a VFD in the system,as follows:

1) Not only does the VFD introduce its own excitations to the train, it also alters the steady-state excitationsgenerated by the motor. In a fixed-speed system, those excitations are constant, at line and twice line frequency.However, when a VFD is present, the electrical frequency being fed to the motor is variable and the excitationsbecome positively-sloped lines on the Campbell diagram. Per Corbo and Malanoski [35], the order number forthe line frequency excitations is given by the following:

Nord = 0.5 x Np (4-44)


where

Nord is the order number corresponding to line frequency excitations;


Thus, for a two-pole motor, line frequency excitations occur at 1X, and twice line frequency at 2X.

2) If the motor is a synchronous motor and the VFD is active during starting (often referred to as “soft start”), thetwice slip frequency excitations that are present when a synchronous motor is started across the line areeliminated. The need for a transient analysis of the synchronous motor startup is also eliminated. However, if theVFD generates large excitations of its own during startup, a transient analysis of that may still be required.

3) If the motor is an induction motor and the VFD is active during starting, the large transient excitation torques atline frequency that are generated when an induction motor is started across-the-line are greatly reduced. Thus,a transient analysis of the startup, which is sometimes performed when the motor is started across-the-line, isalmost never needed when soft start is employed.

There are other benefits to using a VFD for soft starting. Per Weber et al. [14], starting a system across-the-line drawslarge inrush currents but a VFD limits the current by providing a soft start capability. The motor is stressed duringstarts and stops due to the internal rotor heating. Proper application of the VFD and motor allows proper start/stopsand low-speed operation without thermal stress. Per Dickau and Perera [5], the inrush current for a VFD is typicallyless than the full load rating of the motor, whereas the inrush current for an across-the-line induction motor start is 5 to6 times the full load rating of the motor. Thus, there are actually times when a VFD is used with a fixed-speedsystem—solely to provide the advantages derived from soft starting.

4.10.9 References

[1] Huetten, V., Zurowski, R. M., and Hilscher, M., “Torsional Interharmonic Interaction Study of 75 MW Direct-Driven VSDS Motor Compressor Trains for LNG Duty,” Proceedings of the Thirty-Seventh TurbomachinerySymposium, Turbomachinery Laboratory, Texas A&M University, pp. 57–66 (2008).

[2] Kaiser, T. F., Osman, R. H., and Dickau, R. O., “Analysis Guide for Variable Frequency Drive OperatedCentrifugal Pumps,” Proceedings of the Twenty-Fourth International Pump Users Symposium,Turbomachinery Laboratory, Texas A&M University, College Station, TX, pp. 81–106 (2008).

[3] Kocur, J. A. and Corcoran, J. P., “VFD Induced Coupling Failure,” Case Study Presented at the Thirty-SeventhTurbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, TX (2008).

[4] Tsukakoshi, M., Al Mamun, M., Hashimura, K., Hosoda, H., Sakaguchi, J., and Ben-Brahim, L., “Novel TorqueRipple Minimization Control for 25 MW Variable Speed Drive System Fed by Multilevel Voltage SourceInverter,” Proceedings of the Thirty-Ninth Turbomachinery Symposium, Turbomachinery Laboratory, TexasA&M University, College Station, TX, pp. 193–200 (October 2010).

[5] Dickau, R. and Perera, L., “Mechanical Vibration Problems With Variable Frequency Drives,” Proceedings ofthe International Pipeline Conference, Calgary, Alberta (October 2000).

[6] Tanaka, K., Nemoto, H., Takahashi, N., Fukushima, Y., Akita, Y., and Tobise, M., “Measurement andSimulation of Forced Torsional Vibration with Inter-Harmonic Frequencies in Variable Speed Drive MotorDriven Compressor,” Proceedings of the Eighth IFToMM International Conference on Rotor Dynamics, Seoul,Korea, pp. 844–851 (September 2010).

[7] Huetten, V., Beer, C., Krause, T., and Demmig, S., “VSDS Motor Inverter Design Concept for CompressorTrains Avoiding Interharmonics in Operating Speed Range,” Proceedings of the First Middle East



Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, Doha, Qatar (February2011).

[8] Feese, T. and Maxfield, R., “Torsional Vibration Problem with Motor/ID Fan System due to PWM VariableFrequency Drive,” Proceedings of the Thirty-Seventh Turbomachinery Symposium, TurbomachinerySymposium, Texas A&M University, pp. 45–56 (2008).

[9] Kocur, J. A. and Muench, M. G., “Impact of Electrical Noise on the Torsional Response of VFD CompressorTrains,” Proceedings of the First Middle East Turbomachinery Symposium, Turbomachinery Laboratory, TexasA&M University, Doha, Qatar (February 2011).

[10] Adachi, A., Tanaka, K., Takahashi, N., and Fukushima, Y., “Torsional-Lateral Coupled Vibration of VSD MotorDriven Centrifugal Compressor System at Inter-Harmonic Frequencies in Voltage Source PWM Inverter,”Proceedings of the Thirteenth Asia Pacific Vibration Conference, University of Canterbury, New Zealand(November 2009).

[11] Hudson, J. H., “Lateral Vibration Created by Torsional Coupling of a Centrifugal Compressor System Drivenby a Current Source Drive for a Variable Speed Induction Motor,” Proceedings of the Twenty-FirstTurbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, TX, pp.113–123 (1992).

[12] Hudson, J. H., “Selection, Design and Field Testing of a 10,500 HP Variable Speed Induction MotorCompressor Drive,” Proceedings of the Twenty-Fifth Turbomachinery Symposium, TurbomachineryLaboratory, Texas A&M University, pp. 25–34 (September 1996).

[13] Tripp, H., Kim, D., and Whitney, R., “A Comprehensive Cause Analysis of a Coupling Failure Induced byTorsional Oscillations in a Variable Speed Motor,” Proceedings of the Twenty-Second TurbomachinerySymposium, Turbomachinery Laboratory, Texas A&M University, College Station, TX, pp. 17–23 (1993).

[14] Weber, W. J., Cuzner, R. M., Ruckstadtler, E., and Smith, J., “Engineering Fundamentals of Multi-MW VariableFrequency Drives—How they Work, Basic Types, and Application Considerations,” Proceedings of the Thirty-First Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, TX,pp. 177–194 (2002).

[15] Baccani, R., Zhang, R., Toma, T., Luretig, A., and Perna, M., “Electric Systems for High Power CompressorTrains in Oil and Gas Applications—System Design, Validation Approach, and Performance,” Proceedings ofthe Thirty-Sixth Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, CollegeStation, TX, pp. 61–68 (2007).

[16] Grgic, A., Heil, W., and Prenner, H., “Large Converter-Fed Adjustable Speed AC Drives for Turbomachines,”Proceedings of the Twenty-First Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&MUniversity, College Station, TX, pp. 103–112 (1992).

[17] Southwest Research Institute, “Recommended Practice for Control of Torsional Vibrations for High SpeedSeparable Reciprocating Compressors,” GMRC, San Antonio, TX (May 2002).

[18] Frei, A., Grgic, A., Heil, W., and Luzi, A., “Design of Pump Shaft Trains Having Variable-Speed ElectricMotors,” Proceedings of the Third International Pump Symposium, Turbomachinery Laboratory, Texas A&MUniversity, College Station, TX, pp. 33–44 (1985).

[19] Sihler, C., Schramm, S., Song-Manguelle, J., Rotondo, P., Del Puglia, S., and Larsen, E., “Torsional ModeDamping for Electrically Driven Gas Compression Trains in Extended Variable Speed Operation,”Proceedings of the Thirty-Eighth Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&MUniversity, College Station, TX, pp. 81–91 (September 2009).



[20] Terens, L. and Grgic, A., “Applying Variable Speed Drives with Static Frequency Converters toTurbomachinery,” Proceedings of the Twenty-Fifth Turbomachinery Symposium, Turbomachinery Laboratory,Texas A&M University, College Station, TX, pp. 35–46 (September 1996).

[21] Williams, S., Baillie, A., and Shipp, D., “Understanding VSD’s with ESP’s—A Practical Checklist,” Proceedingsof the 2003 ESP Workshop, Society of Petroleum Engineers (2003).

[22] Streicher, J. T. and Olive, J. J., “Straight Talk About PWM AC Drive Harmonic Problems and Solutions,”Rockwell Automation, Milwaukee, WI (October 2006).

[23] Sheppard, D.J., “Torsional Vibration Resulting from Adjustable Frequency AC Drives,” IEEE Transactions onIndustry Applications, pp. 812–817 (September/October 1988).

[24] Song-Manguelle, J. and Nyobe-Yome, J. M., “Pulsating Torques in PWM Multi-Megawatt Drives for TorsionalAnalysis of Large Shafts,” IEEE (2008).

[25] Somaini, R., Bidaut, Y., and Baumann, U., “Comparison of Different Variable Speed Compression TrainConfigurations with Respect to Rotordynamic Stability and Torsional Integrity,” Proceedings of the Forty-FirstTurbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, TX(September 2012).

[26] Wolff, F. H. and Molnar, A. J., “Variable-Frequency Drives Multiply Torsional Vibration Problems,” Power, pp.83–85 (June 1985).

[27] Rotondo, P., Andreo, D., Falomi, S., Jorg, P., Lenzi, A., Hattenbach, T., Fioravanti, D., and De Franciscis, S.,“Combined Torsional and Electromechanical Analysis of an LNG Compression Train With Variable SpeedDrive System,” Proceedings of the Thirty-Eighth Turbomachinery Symposium, Turbomachinery Laboratory,Texas A&M University, College Station, TX, pp. 93–101 (September 2009).

[28] Stephens, T. J., “Torsional Design Considerations for Reciprocating Compressors,” Proceedings of the FifthConference of the EFRC, EFRC, Prague, Czech Republic, pp. 1–11 (March 2007).

[29] Del Puglia, S., De Franciscis, S., Van de moortel, S., Joerg, P., Hattenbach, T., Sgro, D., Antonelli, L., andFalomi, S., “Torsional Interaction Optimization in a LNG Train With a Load Commutated Inverter,” Proceedingsof the Eighth IFToMM International Conference on Rotor Dynamics, KIST, Seoul, South Korea, pp. 994–1001(September 2010).

[30] Sihler, C., Schramm, S., Rossi, V., Lenzi, A., and Depau, V., “Electronic Torsional Vibration Elimination forSynchronous Motor Driven Turbomachinery,” Proceedings of ASME Turbo Expo 2011, ASME, Vancouver, BC(June 2011).

[31] Myszynski, R. and Deskur, J., “Damping of Torsional Vibrations in High-Dynamic Industrial Drives,” IEEETransactions on Industrial Electronics, Vol. 57, No. 2, pp. 544–552 (February 2010).

[32] Falomi, S., De Franciscis, S., Fioravanti, D. and Allotta, B., “Electro-mechanical Interaction in the TorsionalDynamics of LNG Compression Trains with LCI Variable Speed Drive,” Proceedings of the Tenth InternationalConference on Vibrations in Rotating Machinery, Institution of Mechanical Engineers, London, UK, pp. 819–828 (September 2012).

[33] Hernes, M. and Gustavsen, B., “Simulation of Shaft Vibrations Due to the Interaction Between Generator-Turbine Train and Power Electronic Converters at the Visund Oil Platform,” Proceedings of PCC-Osaka,IEEE, pp. 1381–1386 (2002).



[34] Song-Manguelle, J., Schroeder, S., Geyer, T., Ekemb, G., and Nyobe-Yome, J. M., “Prediction of MechanicalShaft Failures Due to Pulsating Torques of Variable Frequency Drives,” IEEE Transactions on IndustryApplications, Volume 46, Number 5, pp. 1979–1988 (September/October 2010).



5-1

SECTION 5—BALANCING OF MACHINERY

5.1 Introduction

This chapter is directed toward both manufacturers and users of machinery built to API standards. However, theinformation presented generally applies to all rotating equipment. The purpose is to present the requirements for low-and operating-speed balancing, the reasoning behind these requirements, and the balancing techniques necessaryto achieve these requirements. The information contained in this tutorial is applicable to rotors undergoing routineservice, repair, or re-rating, as well as new rotors.

5.2 Terms and Definitions

5.2.1balancingA procedure for adjusting the radial mass distribution of a rotor to minimize the lateral vibration of the rotor due tounbalanced inertia forces and forces on the bearings, at once-per-revolution frequency (1X).

5.2.2bowA permanent shaft deflection, other than gravitational deflection, such that the geometric shaft centerline is notstraight and will affect the balance state. Other conditions, such as thermal warpage, can impart a semi-permanent orpermanent deflection (bow) to the rotor. This bow may be three-dimensional (corkscrew). Rotor bow can bedetermined by measuring the shaft relative displacement along its length in V blocks. (See 5.2.18 for the definition ofsag, i.e. gravitational deflection. Sag does not affect balance.)

5.2.3calibrationA test during which known values of the measured variable are applied and the resulting output readings are verifiedor justified.

5.2.4calibration weightA weight of known magnitude that is placed on the rotor at a known location in order to measure the resulting changein machine vibration (1X vector) response. In effect, such a procedure calibrates the rotor system (a known input isapplied, and the resultant output is measured) for its sensitivity to unbalance. Calibration weight is sometimes calledtrial weight.

5.2.5central principal axisthe principal axes of inertia that is in the direction of the axis of rotation.

5.2.6eccentricityThe extent to which the center of a perfect circle, which is the best fit to the surface in question, deviates from the truecenterline of the shaft.

5.2.7cylindricity (out-of-roundness)The variation of the outer diameter of a shaft surface when referenced to the true geometric centerline of the shaft.



5.2.8electrical runoutA source of error on the output signal of a proximity probe transducer system resulting from nonuniform electricalconductivity/resistivity/permeability properties of the observed material shaft surface. The nonuniform properties mayresult from scratches, rust, engravings, or stencil marks. Electrical runout is a change in the proximity probe outputsignal that does not result from a probe gap change.

5.2.9heavy spotA term used to describe the position of the unbalance vector at a specified lateral location (in one plane) on a rotor.

5.2.10high spotThe term used to describe the response vector of the rotor shaft due to an unbalance force.

5.2.11unbalance (imbalance)A measure that quantifies how much the rotor mass centerline is displaced from the centerline of rotation (geometriccenterline) resulting from an unequal radial mass distribution on a rotor system. Unbalance is usually given in eitherg-mm or oz-in.

5.2.12influence vectorThe synchronous vibration response vector divided by the calibration weight vector (trial weight vector) at a particularshaft speed. The influence vector represents the transfer function between vibration response and unbalance force.

5.2.13once-per-revolution markA location on the shaft circumference that is monitored by a phase reference transducer. The mark providesindication of a once-per-revolution occurrence (rpm.) The mark can be a keyway, a key, a hole or slot, or a projection.

5.2.14mechanical runoutA source of error on the output signal of a proximity probe transducer system. It is sensed by a probe gap change thatdoes not result from shaft dynamic motion. Common sources include out-of-round shafts, dents, eccentricity, and flatspots.

5.2.15nonsymmetric (asymmetric) rotorA rotor whose cross-section has two different geometric second area moments of inertia. When rotated at any angleabout its geometric center, it may not appear the same unless returned to its original orientation; for example, anelliptical cross-section, a rotor with a keyway, two pole generator or a rotor section with a crack.

5.2.16orbitThe dynamic path of the shaft centerline displacement motion as it vibrates during shaft rotation at a particular radialplane.

5.2.17resonanceThe condition when the frequency of a harmonic (periodic) forcing function coincides with a natural frequency of thestructure (rotor system.) When a rotor accelerates or decelerates through this speed region(s), the observed vibrationcharacteristics may include one or both of the following: (a) a peak in the 1X vibration amplitude and (b) a change inthe phase angle.



5.2.18sagRotor deflection that is due to gravitational loading of a rotor. The sag is also the natural shape of the rotational axis ofa symmetric rotor as it rotates. Sag is not a factor in balancing. (See 5.2.1.3 for the definition of bow. Bow is adeflection of the rotor that does affect balance.)

5.2.19synchronous componentThe portion of a vibration signal that has a frequency equal to the shaft rotational frequency (1X).

5.3 Fundamentals of Low-speed Balancing

5.3.1 General

Achieving a balanced condition for a rotating assembly is a fundamental element in maximizing machinery reliability.If all other rotor forces are negligible, the level of synchronous vibration produced by machinery is related to theunbalance of the assembly, i.e. the lower the unbalance, the lower the synchronous vibration. Lower vibrationsproduce lower forces and stresses on the rotating assemblies, bearings, and support systems, which increasesmachinery reliability.

Unbalance of rotating machinery parts is the most common cause of equipment vibration. The API Subcommittee onMechanical Equipment has, therefore, developed equipment standards for allowable unbalance levels to minimizethe effect of unbalance on overall equipment vibration. Precision dynamic balance of rotating machinery is required toensure that the equipment performs with minimal vibration on both the vendor test stand and in the field.

The low value of unbalance required by the API standards provides the owner with some margin for in-service rotorerosion and fouling, bearing wear, oil system degradation. This conservatism in balancing increases the availability ofthe machinery.

5.3.2 Derivation of Forces Due to Unbalance in a Rigid Rotor

In an ideal rotor, the mass centerline (principal inertial axis) is coincidental with the rotational axis. If installed inbearings with common centerlines, the shaft would rotate without unbalance forces or vibration. In real rotors, the twoaxes are not coincidental. With no bearing support, the rotor will rotate about its mass centerline. When constrainedby the bearings to rotate about the bearing centerline, forces are created as a result of the displacement of the masscenter of the rotor from the rotational axis, which creates the vibration of the rotor.

An analysis of a simple rotor system will illustrate the dominant influences, which produce rotor forces due tounbalance. Consider a rotor, turning with an angular velocity, , that has an unequal distribution of mass about itscenterline. Further assume that the amount of unequal mass distribution can be represented as a single mass locatedat a distance r from the centerline of the shaft. This is an example of a rotor with simple static unbalance. The basicequation for forces due to rotation of a shaft assembly is:

F = U2 (5-1)

where

F is the centripetal force;

U is the unbalance;

is the rotational speed.



The force generated due to a rotor’s unbalance can be shown by the following equation:

For SI units:

(5-2)

For US Customary units:

(5-3)

where

F is the radial force generated due to the residual unbalance of the rotating assembly, N (lbf);

N is the operating speed of the rotating assembly (rpm);

U is the unbalance of the rotating assembly, g-mm (oz-in.).

This relationship illustrates two important characteristics of rotors with unbalance. The force, F, varies linearly with theradius and size of the unbalance but with the square of the shaft speed. This explains why a particular U, for an 1800rpm rotor, produces acceptable unbalance forces and vibration while the identical U, for a 10,800 rpm rotor, isunacceptable. The forces are 36 times as great for the higher speed rotor!

These forces can be significant for relatively small amounts of unbalance. For example, a rotor with a residualunbalance of only 4320 g-mm (6 oz-in.) running at 10,800 rpm will generate a force due to unbalance of over 5525 N(1242 lbf).

Most rotors covered by the API Standards do not have simple static unbalance. However, the fundamental issuesregarding their unbalance follow the basic issues described above.

5.3.3 Units for Expressing Unbalance

The amount of unbalance in a rotating assembly is normally expressed as the product of the unbalance mass (forexample, g and oz) and its distance from the rotating centerline (such as mm and in.). Thus, the units for unbalanceare generally g-mm, oz-in., g-in., and so forth. For example, 10 g-mm of unbalance would equate to a heavy spot ona rotor of 1 g located at a radius of 10 mm from the rotating centerline. See Figure 5-1.

5.3.4 Types of Unbalance

5.3.4.1 General

The various types of rigid rotor unbalance can be understood by describing the relationship between a rotor’s masscenterline (principal inertial axis) and its rotational axis. If these two axes are coincidental, the rotor has zerounbalance and is considered perfectly balanced. The following sections provide a description of the two unbalanceconditions as determined by a typical low-speed balance stand.

F U N2 260------

2

10 6–=

F UN

1000------------

2

1.77=



5.3.4.2 Static Unbalance

Static unbalance is the balance condition where the principal inertial axis is offset a parallel distance, r, from the rotorrotational axis (Figure 5-2). This locates the Center of Gravity (CG) away from the axis of the rotational. Staticunbalance can be corrected in either of two methods. In the first method, a correction weight can be located directly180 opposite the unbalance (CG), and at the appropriate radius from the rotor centerline to equal the unbalance (g-mm). In the second method, the weight is divided and distributed to each end of the rotor. The location of the weightsmust remain in line with the weight location of the first method, i.e. 180 opposite the CG of the rotor.

5.3.4.3 Dynamic (Couple) Unbalance

A “couple,” in static force analysis, is the application of equal forces, applied about the CG of a body, in oppositedirections (Figure 5-3). Rotor couple unbalance is the balance condition where the mass centerline (principal inertialaxis) intersects the rotor rotational axis and this intersection is also the CG. A rotor with couple unbalance will have nostatic unbalance (the CG lies on the rotor rotational axis), but, if rotated, the opposing forces produce vibration in thebearings. Couple unbalance must be addressed by making weight corrections in the same axial plane of theunbalance at each end of the rotor, i.e. 180 apart.

NOTE ISO defines this as a Couple unbalance.

5.3.4.4 Combination of Static and Dynamic Unbalance

When the rotor has a mixture of the two unbalances described above, the central principal axis is not parallel to anddoes not intersect the axis of rotation, as shown in Figure 5-4. This occurs when the static component lies in an axialplane different than the axial plane defined by the couple unbalance forces. This type of unbalance is common.

NOTE ISO defines this as a Dynamic unbalance.

Figure 5-1—Unbalance Expressed as the Product of Weight and Distance

9 oz

12 oz1.5 in.

518.456 g

Unbalance produced is the samein all 3 cases

18 oz-in. = 12,961.4 g-mm

25 mm

2 in.



Figure 5-2—Static Unbalance

Figure 5-3—Dynamic Unbalance

Figure 5-4—Combination of Static and Dynamic Unbalance

S

S

S

Static unbalance

Axes parallel

Principalinertia axis

Shaft axisEnd view

C

C Couple unbalance

Axes intersectPrincipal

inertia axis

Shaft axis

End view

C

CC

C

S

S Static unbalance

C Couple unbalance

Axes do notintersect Principal

inertia axis

Shaft axis

End view

C

S

CC



5.3.5 Causes of Unbalance

There are many reasons for unbalance in a rotor. The most common causes of rotor unbalance are the following.

a) Nonhomogeneous material: On occasion, rotors with cast components such as pump impellers will have blowholes or sand traps which result from the casting process. This condition can also be caused by porosity in therolled or forged material for shafting and impeller disks. These areas are undetectable through visual inspection.Nonetheless, the void created may cause a significant unbalance.

b) Eccentricity: This exists when the geometric centerline of a rotating assembly does not coincide with the axis ofrotation.

c) Lack of rotor symmetry: This is a condition that can result from a casting core shift as in a pump impeller or placingturbine blades with slightly different CG locations 180 apart.

d) Distortion: Although a part may be reasonably well balanced following manufacture, there are many influencesthat may serve to alter its original balance. Common causes of such distortion include stress relief and thermaldistortion. Impellers that have been fabricated by welding can have internal stresses. Any part that has beenshaped by pressing, drawing, bending, extruding, and welding without stress relief, will naturally have high internalstresses. Therefore all such components should be stress relieved prior to balancing. If the rotor or componentparts are not stress relieved prior to balancing, they may undergo stress relief naturally over a period of time, andas a result, the rotor may distort slightly to take a new shape, thus altering the rotor balance.

Distortion that occurs with a change in temperature is called thermal distortion. Although all metals expand whenheated, most rotors, due to minor imperfections and uneven heating, will expand unevenly causing distortion. Thisdistortion is quite common on machines that operate at elevated temperatures (that is, induced draft boiler fansand steam turbines). For this reason, steam turbine shafting is often subjected to a heat stability test. Thermaldistortion can also be induced by uneven cooling such as occurs in a steam turbine rotor after a trip. For thisreason, turning gears are often employed in large turbine driven trains.

e) Stacking errors: Slight variations in mounting or cocking a shrink-fit disk or impeller on its shaft may result inunbalance. This can also result from the stack-up of machining tolerances in a rotating assembly or fromunaccounted for keys or other missing components.

f) Bent rotor shaft: This condition can occur when a rub causes a yield of the shaft material due to thermal effects.This shifts the entire rotor off-center from the axis of rotation. However, rotors that have experienced a hard rub inservice may only have a ‘kinked’ shaft. In this instance, the entire rotor may not be off-center and the unbalancecan be corrected. (See 5.2.3.)

g) Corrosion, erosion, or deposits: Many rotors, particularly those involved in material handling processes, aresubjected to corrosion, erosion, abrasion, deposit buildup, and wear. If the corrosion or wear does not occuruniformly, unbalance will result. In some applications, deposits on rotor/blade surfaces can accumulate and/or beremoved unevenly causing unbalance.

h) Eccentric mounting of balanced components: Items such as couplings and dry gas seal rotors are balanced asindividual items and contain their own residual unbalance. However, they are not added to a rotor, during thebalancing procedure, to determine the effect of their unbalance on the entire assembly. This effect is moresignificant on small rotors as compared to large rotors.

The distribution of these unbalances along the length and circumference of the rotor will be more or less random (seeFigure 5-5). The unbalances presented in Figure 5-5 are identified by the component. For example, the source of theshaft unbalance may be machining errors and/or bow. The impeller/stage unbalance may be created by stackingerrors and/or eccentric mounting and may be either static and/or couple unbalance. As noted in 5.3.4, low-speedbalance machines are able to resolve these unbalances into a combination of static and couple unbalances. This is



made possible since the low-speed balance machine treats the rotor as a rigid body. The extent to which this holdstrue during operation determines the need for flexible rotor balancing (e.g. operating-speed balancing).

In summary, all of the preceding causes of unbalance can exist to some degree in a rotor. However, the vectorsummation of all unbalance at a particular axial location can be considered as a concentration at a point called theheavy spot. Balancing is thus the technique for determining the amount and location of this heavy spot so that anequal amount of weight can be removed at this location or an equal amount of weight added 180 opposite of theheavy spot.

5.3.6 API Standard Balance Specifications

Balancing denotes the attempt to improve the mass distribution of a rotating assembly such that the assembly rotatesin its bearings with minimal unbalance forces and acceptable vibration amplitudes. This goal, however, can beachieved only to a certain degree; even after careful balancing of a given rotor is completed, the rotor will still retain acertain degree of unbalanced mass distribution known as residual unbalance.

The API Standard Paragraphs specifications for residual unbalance were originally adopted from the US Navy,Bureau of Ships Standards.

For SI units, use the following equation:

(g-mm) for Nmc < 25,000 rpm (5-4a)

(g-mm) for Nmc ≥ 25,000 rpm (5-4b)

For US Customary units, use the following equation:

(oz-in.) for Nmc < 25,000 rpm (5-5a)

(oz-in.) for Nmc ≥ 25,000 rpm (5-5b)

Figure 5-5—Unbalance Distribution Resolved into Static and Dynamic Components

U 6350WN----=

UW

3.937--------------=

U4WN

--------=

UW

6250-------------=



where

U is the maximum allowable residual unbalance for each correction plane of the rotor, g-mm (oz-in.). The tolerance for each plane is based on the static weight supported at each end of the rotor;

W is the bearing journal static weight at each end of the rotor, kg (lbf). Note that for relatively uniform rotors, W represents approximately 1/2 the total rotor weight;

Nmc is the maximum continuous speed of the rotor, rpm.NOTE Not the balance speed.

The API Standard Paragraphs for residual unbalance provide an achievable unbalance level that will minimize theeffect of rotating unbalance on overall vibration level. It is important to remember that a rotor’s vibration level andresidual unbalance are two separate measurements used to achieve the same end: a smooth running machine.

5.3.7 Balancing Tolerances

The state of the art in balancing technology is such that it is not uncommon for high-speed turbomachinery rotors tooperate with shaft synchronous vibration levels of 12.5 microns (0.5 mils) or less. Achieving vibration levels this lowrequires sound balance practices and tight manufacturing and assembly tolerances. In establishing balancetolerances, there is always a trade-off between what is practically feasible and what is economical.

There have been a number of organizations creating balancing standards over the years; a few of these are: ISOStandards (International Organization for Standardization) [1], VDI Standards (Society of German Engineers), ANSIStandards (American National Standards Institute), and the Military Standards (MIL-STD-167). All of the standardsshare a common objective of developing a smoother running machine. In the final analysis, however, all balancingstandards specify an allowable eccentricity, or offset weight distribution, from the rotating centerline.

The ISO 1940 Standard [1] is a commonly referenced balancing standard. The ISO Standard provides a series ofrotor classifications as a direct plot of residual unbalance per unit of rotor mass versus service speed (see Figure 5-6).From the ISO specification, turbomachinery rotors are assigned the Grade of 2.5. The ISO Grade 2.5 equates to anAPI allowable unbalance of about 15W/N. This Grade has been found to be too lenient as a requirement for specialpurpose turbomachinery applications. To achieve the same allowable residual unbalance level as the 4W/N APIStandard would require an ISO Grade of 0.67. This comparison can be graphically illustrated as shown in Figure 5-7.

Although the API 4W/N balance tolerance is significantly tighter than that of ISO 1940 Grade G-2.5, this tightertolerance can be achieved with an experienced balancing machine operator and requires little additional time.

The API standards indicate that 4W/N applies to machines operating below 25,000 rpm. Above 25,000 rpm, thebalance tolerance is W/6250. This change is required since 4W/N has no lower bound (it approaches zero as speedincreases). From a practical perspective, there exists a finite lower limit of a balance machine’s ability to determineresidual unbalance. The practical limit of new shop balance machines is approximately 125 mm (5 in.) but only 250mm (10 in.) during normal use. Therefore, the smallest unbalance measurable in a balance machine is:

(5-6)

where

W is the rotor mass, kg (lbm);

S is the balance machine sensitivity, mm (in.);

Ulimit is the unbalance limit, g-mm (oz-in.);

c is the constant, E-3 (16E-6).

Ulimit c S W=



Figure 5-6—Permissible Residual Specific Unbalance Based on Balance Quality Grade G and Service Speed, n (ISO 1940-1:2003) [1]

200.01

0.02

0.05

0.1

0.2

0.5

1

2

5

10

20

50

100

200

500

1000

2000

5000

10,000

20,000

50,000

100,000

50 100 200 500 1000 2000

G 0.16

G 0.4

G 1

G 2.5

G 6.3

G 16

G 40

G 100

G 250

G 630

G 1600

G 4000

500010,000

Per

mss

be

Res

dua

Spe

cfc

Unb

aan

ce, e

per,

g

20,000200,00050,000

Service Speed, n

100,000

NOTE The white area is the generally used area, based on common experience.



Typical values of S range from 5 to 10 in. The API unbalance tolerance (4W/N) and machine unbalance limit (Ulimit)using a sensitivity of 10 in. can be plotted as unbalance versus speed. This is shown in Figure 5-8.

As can be seen, the API 4W/N tolerance cannot be reached at operating speeds of the rotor above 25,000 rpm for abalance machine with a sensitivity of 10 in.

5.3.8 References

[1] ISO 1940-1:2003, “Mechanical vibration—Balance quality requirements for rotors in a constant (rigid) state—Part 1: Specification and verification of balance tolerances,” 2003.

5.4 Low-speed Balancing Machines

5.4.1 General

In choosing a balancing machine for a given rotor, care must be taken to ensure that the balance machine is capableof successfully reaching the required balance tolerance. Several factors are involved in this determination:

— First, factor to consider is that the rotor weight is matched to the balancing machine capability and will be ofsufficient sensitivity to provide good residual unbalance data. In other words, is the balancing machine capable ofproviding the unbalance tolerance required?

— Second, the balancing machine drive system must also be of sufficient power to bring the rotor up to the desiredbalance speed. This is of particular importance with large steam turbine and centrifugal fan rotors.

— Third, there are two basic types of balancing machines, soft-bearing and hard-bearing machines. The term hardor soft refers to the support system used in these machines, not to the type of bearings employed. Figure 5-9illustrates the applicable speed ranges for hard-bearing and soft-bearing balancing machines [1]. Practically, only

Figure 5-7—Comparison of ISO and API Balance Tolerances

Balance gradesISO

1000

100

10

1

0.1

1000

100

10

1

0.1100,00010,000

Service Speed, rpm

Practical balance limit

G 0.16

G 0.4G 0.67 G 1

G 2.5

G 16

G 40

G 100

G 6.3

1W/N

8W/N

16W/N

32W/N

64W/N128W/N

512W/N

256W/N

4W/N

1000100

Max

mum

Res

dua

Unb

aan

ce p

er J

ourn

a W

egh

t, g-

mm

/kg

API



Figure 5-8—Unbalance Versus Speed for API Limits and Balance Machine Limit (Calculated at W = 1 lbm)

Figure 5-9—Applicable Speed Ranges for Hard-bearing and Soft-bearing Balancing Machines [1]

Speed, RPM0 5000 10,000 15,000 20,000 25,000 30,000

Unb

aan

ce, o

z-n.

0.01

0.001

0.0001

API (4W/N)

Balance machineD

spac

emen

t Am

ptu

de

Pha

se A

nge

180

90

0

Soft bearingnatural frequency

Phase angle

Amplitude

Soft bearing speed range

RPM ______

Dsp

acem

ent A

mp

tude

Pha

se A

nge

180

90

0

Hard bearingnatural frequency

Phaseangle

Amplitude

Hardbearingspeedrange

RPM ______Comparing Balancing Speed Range for Hard-bearing

and Soft-bearing Balancing Machines



hard-bearing machines are used in the oil and gas industry for almost all turbomachinery applications. Thebalance procedure should be matched to the machine type being used.

— Fourth, With regard to the bearings, balance machine anti-friction support rollers should not have a diameter thatis within ±5 % of the diameter of the rotor journal and, preferably, not an integer multiple of each other. This is toprevent the roller variations (eccentricity and out-of-roundness) from masking the rotor vibration readings.

— Fifth, the accuracy of the configuration entered into the machine is critical into reaching a successful balance ofthe rotor assembly or component. Balance machines rely on the setup dimensions to accurately transfer theunbalance measurements at the journal/pedestal locations to the planes being used to correct the unbalance.

— Finally, the calibration should be checked to verify the accuracy of the balancing machine. The API residualunbalance check is one method that can be used for this purpose. While not necessary during the low-speedbalancing of components of a larger assembly or assemblies that will be operating-speed balanced, errors madewill make it more difficult to reach the required tolerance during the final balance procedure.

5.4.2 Soft-bearing Balancing Machines

The soft-bearing balancing machine design employs a flexible spring support system on which the rotor assembly orcomponent is mounted, Figure 5-10. The first natural frequency of a soft-bearing support system (including therotating assembly to be balanced) is very low, so actual balancing is done above this system’s natural frequency. Theunbalance in the rotor results in an unrestrained vibratory motion, usually in the horizontal plane, in the supportsystem. This motion is normally measured with velocity transducers mechanically connected to the support system.

Figure 5-10—Typical Soft Bearing Balancing Machine



With soft-bearing balancing machines, different types of rotors of the same weight will produce differentdisplacements as measured by the vibration pickups, depending upon the configuration of the rotors to be balanced.The signals coming from the vibration pickups are dependent not only on the unbalances and on their positions, butalso on the masses and moments of inertia of the rotor. For example, for a given unbalance, if the mass is larger, thedisplacement is smaller (force = mass x acceleration) and so is the signal. The methods employed in a soft-bearingbalancing machine are similar to those found in field balancing.

The residual unbalance can be obtained only after calibrating the measuring devices to the rotor being balancedusing test masses that constitute a known amount of unbalance. Therefore, this type of balancing machine isgenerally used for production applications in which many identical components are successively balanced.

5.4.3 Hard-bearing Balancing Machines

Hard-bearing balancing machines are similar in construction as soft-bearing machines except that the supports aremuch stiffer. This greater stiffness causes the natural frequency of the balancing machine’s rotor bearing system to bewell above the balancing speed. A hard-bearing balancing machine accepts a wider range of rotor weights andconfigurations without requiring re-calibration.

This type of balancing machine measures rotor unbalance using either velocity pickups or strain-gauge transducersand correlates the unbalance to a force. The strain-gauge transducer’s readouts are proportional to the rotorunbalance. Since the readout of hard-bearing machines is unbalance force and not vibration of a spring force system,the readout will be close to the amount of actual unbalance, in a properly calibrated machine. The force a specificunbalance develops at a specific speed is always the same, regardless of the physical attributes of the rotor.Restated, the force is not influenced by the bearing mass, rotor weight, rotor configuration, rotor moment of inertia, orwindage oscillations from the rotating rotor assembly or component is mounted.

5.4.4 Balance Machine Configurations

5.4.4.1 General

In addition to the support stiffness, the balance machine can also be classified as horizontal or vertical referring to theorientation of the piece to be balanced.

5.4.4.2 Horizontal Balance Machine Drives

Horizontal balancing machines typically employ one of three different drive configurations to spin the rotatingassembly. These drive mechanisms include the direct end drive, the wrap-around belt drive, and the tangential beltdrive.

5.4.4.2.1 Direct-end Drive

The direct-end drive includes a drive shaft with either a balanced flexible element coupling or a balanced universaljoint (U-joint) connected to the rotor, Figure 5-11. It is typically used with rotors that have large moments of inertia orhigh windage losses. This drive design is capable of transmitting high torque forces for fast acceleration and safebraking. To attach the drive shaft, a rotor end must be prepared to accept the coupling flange or a U-joint directly ormust be fitted with a light weight adapter hub. With this design, the drive system becomes a part of the rotor and mustbe considered in the balancing accuracy of the system. Before this type of drive can be used to accurately balance arotor, the U-joint assembly, itself, must be balanced. The desired end result is to be able to rotate the U-joint assembly180° in relation to the rotor without any significant variance in the balancing machine readout. To confirm this level ofunbalance, the following procedure can be utilized. After the addition of the first component(s), rotate the driveshaft180° and check the residual unbalance. If the unbalance value changes and exceeds the values allowed in 5.2.6, thedriveshaft is not balanced or the pilot fit of the shaft is incorrect. The error must be compensated before the balancingcan proceed. An acceptable change in unbalance due to the rotation of the drive shaft and unbalance repeatabilityshould be discussed between the parties involved.



A wrap-around belt drive works very well for rotors weighing less than 2250 kg (5000 lbf) and with at least one smoothsurface. For this drive configuration, 2250 kg (5000 lbf) is approximately the maximum rotor weight that will allowadequate torque transmission to bring the rotor up to the required balance speed. A 180° wrap is required to provideadequate torque to rotate the rotor and keep the applied tangential force in the vertical plane of the balancingmachine pedestal, Figure 5-12. A belt drive of this type is considerably more accurate than a direct-coupled end drivein that the drive mechanism does not influence the rotor balance.

Figure 5-11—Low-speed Balance Machine Employing Direct-end Drive Wrap-Around Belt Drive

Figure 5-12—Low-speed Balance Machine with Wrap-Around Belt Drive



5.4.4.2.2 Tangential Belt Drive

A tangential belt drive, either under the rotor or in an over-arm configuration, is frequently used in smaller capacity,high-volume-production oriented balancing machines for rotors weighing less than 450 kg (1000 lbs). This type ofdrive configuration is used to bring the rotor up to the required balancing speed. The drive arm is then moved awayfrom the rotor and the unbalance data are collected.

5.4.5 Balancing Machine Pitfalls

A few other points concerning balancing machines, regardless of drive type, are worth noting. One is to avoid using amachine with anti-friction support bearings having diameters that are equal to the journals of the rotor (and preferablynot an integer multiple of one another.) In this instance, any imperfection that results in nonconcentricity of the outerrace will be interpreted as unbalance by the balancing machine electronics.

In addition, since unbalance force varies as the square of the rotor speed, a balancing speed range that the balancingmachine manufacturer recommends to achieve the desired sensitivity should be chosen and maintained from thestart to the finish of the job. The balance speed chosen is of particular importance on steam and gas turbine rotorswhere the stage buckets must be seated (due to centrifugal force) in their blade-root attachments in order to achieverepeatable balance data.

Balancing machines with a belt drive have also been known to introduce residual magnetism into the rotor fromfriction and slippage between the belt and the rotor or between the belt and the pulley installed on the rotor. Theresidual magnetism levels of the rotor should always be checked after using this type of balancing machine, and therotor degaussed if necessary. The gauss level of the proximity probe area should not exceed 2 gauss as specified byAPI 670.

5.4.6 References

[1] “Fundamentals of Balancing,” Third Edition, April 1990, Schenck Trebel, Deer Park, New York.

5.5 Low-speed Balancing Procedures

5.5.1 General

Every successful balance procedure addresses/follows these key points:

— The balancing machine calibration must be verified prior to starting the balancing procedure. Calibration of thebalance equipment is vital not only to achieve a verifiable balanced condition but also to avoid wasted time andexpense even if a residual unbalance check is to be performed.

— Balancing machines should be capable of providing the location (separation) of the balance planes and thestatic/dynamic unbalance. This information can permit decisions as to how unbalance will be corrected on thepart or assembly to accommodate any existing dynamic behavior.

— For example, a large static correction can be spread across many impellers so as not to heavily grind or focus thecorrection to two impellers. Spreading the correction may improve vibration levels through the 1st critical speed.

— Balance machine bearings should be located at or near the journal surfaces machined to the same runouttolerance as the actual rotor journals. Even a slight variation in the runout will cause a large unbalance correction.

— For example, a 10,000 lbm (4536 kg) rotor operating at 8000 rpm will have a 4W/N tolerance of 5 oz-in. (3600 g-mm) total. That same rotor placed on balancing machine bearings running on surfaces with a 0.002 in. (51 m)



mechanical total indicated runout (TIR) difference than the actual journals, will record an unbalance of 160 oz-in.(115,210 g-mm) due only to the mechanical TIR difference.

— Rotors with temporary bows should be run until the initial readings are stable. Many rotors left sitting in ahorizontal position supported at the journals may take temporary bows that are eliminated following turning.

— Drift in the readings will cause wasted time and effort and may lead to unnecessary grinding or material removal.It is recommended that not only should the readings be stable, but several start and stops be made to confirmthey are repeatable.

— The records of the balancing procedure should include the initial and final values of the unbalance and detaileddescription of the corrections made (amount and location).

— This permits determination of the balance condition of a stored rotor or verification of the balance weights on therotor. Phase reference should also be a permanent reference on the rotor (thrust collar key, scribe mark, etc.).

— Prior to grinding, the balance machine operator should verify the proposed correction by using a temporarycorrection such as clay. This step is recommended to ensure the balance machine indicates the desired level ofresidual unbalance has been achieved.

Balance corrections are made in either of two methods, removal or addition of weight. Both of these methods may beutilized in the process of balancing. Weight may be removed by either grinding or drilling. Care should be exercisedso that the components are not damaged or weakened in the process. If weight is added, it is done by rigidly affixingthe weights to the components. This may be done by the use of bolts, set screws, or pinning. Balance weights mustbe compatible with the component material and suitable for the operating environment. For example, this may requirethe balance weights to be made from material that has corrosion resistance equal to or greater than the item beingbalanced.

Occasionally, rotors to be balanced include previously balanced ancillary components of an assembly, e.g. couplinghub. If these components contain balance weights or corrections, they should not be altered in the process ofbalancing the rotor to avoid affecting the balance condition of the assembly.

5.5.2 Component Balancing


On flexible shaft rotors (those that operate above the first critical speed), it is vital to multi-plane dynamically balanceall of the major components, individually, before assembly. Major components include the shaft, balancing drum, andthe impellers. The components shall be balanced to the same tolerance as is required for the complete assembly. Ifthe rotor is fully assembled without this step, there is a possibility that large unbalances (both static and dynamic) willbe introduced into the assembly. This will require larger corrections in the two or three planes used during theassembly balance. By making corrections at the component level, the need for corrections at the assembly level isreduced as is internal couples that can excite flexural modes during operation.

To minimize assembly balance corrections, each disk, impeller or wheel, part of a rotor assembly, should be balancedindividually. In many cases, the component does not have a shaft for use in the low-speed stand if the single piecebalancing is performed prior to assembly. Therefore, a balancing arbor is required to provide that physical shaft. Thebalancing arbor constrains the balancing piece on a rotating axis which is typically different from the final assembly.The repeatability of the eccentricity between the balancing tool and the final assembly is a major factor in thecomponent balance quality. The balancing engineer reduces the balancing uncertainty by designing a tool whichallows index balancing compensation and highly repeatable positioning. The balancing arbor centering surface shallface the working piece reference surface. This is the linkage to the final shaft and should be concentric to the finalbearings.



The unbalance variation, generated by tooling assembling/disassembling shall be reasonably lower than the residualunbalance upper specification limit. The vendor should quantify the balance repeatability error of the working pieceassociated with the balancing tool arbor. Despite the careful attention given to the balancing of single piece on a well-designed balancing arbor, it is generally not possible to achieve very tight tolerances. A reasonable eccentricityuncertainty between working piece balancing tooling and final shaft is >5 m. The eccentricity uncertainty grows withthe working pieces dimension and weight. For situations where the rotating assembly is operated without userdisassembly, e.g. multistage centrifugal compressors and pumps, the API unbalance 4W/N can be considered atarget for the single piece balancing before assembly rather than a tolerance. The actual upper specification limitshould be set by the vendor in accordance with the balancing system repeatability and reproducibility error chain. Inthese situations, the final balance tolerance will be achieved for the assembly (through sequential step or operating-speed balancing) even if individual components exceeded the tolerance. The residual error is then mainly related tothe balancing machine, bearings and drive set up. For machinery configurations where changes to the assembly bythe user do occur, e.g. pipeline compressors where the overhung impeller is changed according to the need,individual components should be balanced to the API tolerance specification.

5.5.2.2 Mandrel Requirements

Each major rotor component must be individually balanced on a precision ground mandrel. (Note that expandingmandrels are not acceptable for this purpose.) The mandrel shall have a surface finish roughness that does notexceed 0.4 m (16 in.) Ra. The mandrel shall also have no measurable eccentricity when assessed by an indicatorgraduated in 2.5 m (0.0001 in.). The mandrel mass should not exceed 25 % of the mass of the component to bebalanced.

The balance mandrel should be ground between centers to assure concentricity of all diameters throughout its lengthas well as to assure a good smooth surface finish. After grinding, the mandrel must be precision balanced. A trial biasweight may be used to raise the observed residual unbalance readout of the balancing machine. The desired balanceresult is such that no matter at what angular location the bias weight is added, the unbalance readout is always thesame. In this case the residual unbalance of the precision mandrel is as close to zero as possible.

For the single pieces balanced prior to assembly, the balancing mandrel may be embedded into the balancingmachine. The working piece is simply fitted to the centering device designed for high positioning repeatability andaccuracy; the centering device is attached to the balancing machine plate. The balancing machine needs specificcalibration with the centering device attached. The common configuration of such balancing machines is vertical buthorizontal arrangement exists. The great advantages of this configuration are that the uncertainty, repeatability andaccuracy of the balancing tooling (excluding the centering device) is certified by the balancing machine constructorand controlled by standard maintenance and calibration processes. These types of balancing machines, either withvertical or horizontal spindle, orient the working piece in an overhung configuration. This may limit the measuringplane separation relative to the balancing planes. This peculiarity introduces poor dynamic sensitivity which, for thesame size of working piece, is generally less than the traditional arrangement between bearings.

5.5.2.3 Component Preparation and Mounting

After each component is mounted on its mandrel, the axial and radial runouts may be checked to ensure that themounted impeller or hub is not cocked on its mandrel prior to component balancing. If components are not phase-referenced marked, zero phase shall be taken as the component keyway, locking pin, or key block. If the componentsdo not have a reference point such as a key, zero phase shall be permanently identified for use during the assemblybalancing procedure.

Mandrels typically do not have keyways. When a component with a single keyway is to be mandrel-balanced, thekeyway should be filled with an inside-crowned half key or an equivalent compensating correction should be added tothe component so that a proper balance can be achieved. Components with double keyways arranged with 180degree angular offset do not require the keyways to be filled.



5.5.2.4 Special Considerations

Major components, e.g. turbine disks, thrust collars or low flow coefficient impellers (those with an L/D<<1.0), mayreceive a static balance if the component has a balance plane separation that makes dynamic balance impractical.

5.5.3 Low-Speed Balancing Techniques

For rotors operating between the first and second critical speed, there are two principal balancing techniques andeach addresses static and dynamic unbalance.

a) The first technique is a low-speed progressive-stack balance as described in 5.5.4. In this method, no more thantwo major components are added at a time. The static and couple unbalance of the assembly is established ateach step and the appropriate correction is made to the last two added components (two plane balance).

b) The second technique is to incorporate both low and operating-speed balance. In the low-speed balance step, theprogressive stack balance can be skipped in favor of a three-plane balance procedure. In this procedure, it ispreferred to compensate 30 % to 60 % of the static component at the center plane, Schenck [1]. The remainder ofthe correction is divided between the two end planes. Following the low-speed balance, the assembled rotor isplaced in the operating-speed bunker for final balance (see 5.6).

5.5.4 Progressive Component Stack Balancing


Progressive or stack balancing is used to counteract the accumulation of unbalance into a rotor by the introduction ofmajor components to the assembly. The unbalances due to the component mounting arise from two factors:

a) Eccentric mounting of the component on the mandrel will produce unbalance equal to the mass of the componenttimes the eccentricity. If the eccentricity is caused by the mandrel mounting surface and this is corrected on thecomponent (erroneously), then mounting that component on the shaft (if the shaft surface has no eccentricity) willcause an unbalance equal to the correction made at the component level. If the shaft has the same eccentricsurface as the mandrel but the component is installed 180 different with respect to the high spot (then whenmounted on the mandrel), the unbalance created will be twice the correction made at the component level.

b) Components with unequal stiffness in all planes, such as those with single keyways, may deform when shrunkonto the rotor shaft. For such components, considerable deformation and resultant unbalance can occur betweenmandrel balancing using a light shrink fit and stack balancing on the job shaft with a heavy shrink fit.

5.5.4.2 Balance Procedure

Progressive balancing is a procedure that requires several distinct steps of balancing. It is accomplished by adding nomore than two major components for each balance step. Balance corrections are made to the components added ineach step. Axial and radial runouts of the shaft should be checked against internal requirements to avoid excessivebowing or bending of the shaft due to component shrink fits. Large runouts of the shaft may require large correctionsthat may in turn cause unbalance if the bow relaxes during operation.

As each rotor component is stacked into position, an inspection must be made to verify that the components areproperly fitted to the main rotor. Common reasons for an unbalance being introduced include the improper mountingor eccentricity of a component or the shift of a component (impeller) due to the relaxation of an interference fit. Trimbalancing of the complete assembly, if required, is done as necessary to achieve the balance tolerance.



5.5.4.3 Index Balancing

Index balancing is a technique employed during low-speed balancing to identify the unbalance attributable tocomponents of an assembly. The process involves indexing (turning) of one component (or part of an assembly) withrespect to the other during the balance procedure. Vector math is used to identify the unbalance of each of the twocomponents or assemblies. The process is described both graphically and vectorially below.

Index balancing is commonly used in the following situations:

a) Low-speed balancing involving tooling or arbors that will not accompany the component in the assembly. (Ex: Thebalancing of impellers that require an arbor to be placed on the low-speed stand.)

b) The balance of sub-assemblies or components making an assembly that may not remain together during use.(Ex: Pipeline operators that change impellers in an overhung compressor to meet flow demands.)

In the first situation, it is desirable to correct the impeller for unbalance attributable to the impeller only and not due tounbalance from the arbor or its fit. In the second case, an assembly balance where components may be replaced(without rebalancing) does not guarantee that the new assembly will remain in balance. The individual components orsub-assemblies must be corrected independently.

These situations arise since fits and changes in relative angular position between components may create unbalance.As one component is replaced, the newer component may have variations in the fit in different locations and amountswithin the tolerance. Removal and installation of a component may also lead to a different relative angular positionwith respect to the rest of the assembly through error or inability to measure the orientation. If an assembly balance isused in the original configuration, an implied assumption is made that the fits and the angular position will alwaysremain the same.

A simple example is provided to show how fit eccentricities will cause unbalance. Take for example a shaft with anoverhung fit location (see Figure 5-13). The fit location is eccentric to the centerline of the bearings by an amount, e.

The eccentric shaft section has a weight of m. A component of weight, M, is to be installed on the shaft. Assuming thatthe component is balanced with a perfect fit bore, the resulting unbalance produced by the fit eccentricity would be:

U = (m + M) e (5-7)

Balancing these components as an assembly with balance planes located on the component and the right end of theshaft, a correction of U/r would be made to the component at radius, r (assuming a thin disk), Figure 5-14.

Figure 5-13—Fit Eccentricity Related Unbalance

Weight, M

e

Weight, m

Shaft centerline



However, should the component be installed rotated 180, the correction would also be turned and would cause anunbalance of 2U. Additionally, if the component was replaced with an identically perfect component, the correctionwould be lost altogether. The correct balance procedure in this case is to make the correction on the left end of theshaft since it was the shaft's fit that created the unbalance originally. Then subsequent installation and replacement ofthe component would not cause unbalance problems.

The key to the simple example above is identifying which component is the cause of what percentage of theunbalance identified in the low-speed balance machines. This cannot be achieved by balancing the pieces separatelyas is commonly done. Keep in mind that if the shaft was balanced on its own, a correction of only U = m e would bemade at the left end since the component weight of M would not be present.

If the component in the example above also had an unbalance, u, the low-speed balance machine would read a totalunbalance level as the vector sum of u and U at the left end balance plane. As noted, the successful balanceprocedure would then correct the component unbalance, u, on the component and the shaft unbalance, U, on theshaft. Index balancing achieves this separation.

Keeping with the same example, a step-by-step index balance procedure will be performed. A graphical and vectorialdescription of each step will be shown.

The component (impeller) and the shaft are assembled and an arbitrary 0 reference mark is made. The assembly isplaced in the low-speed balance machine (low-speed stand) and readings are taken at the impeller and far end of theshaft (see Figure 5-15).

The readings taken are at plane 1 and at plane 2. (The subscripts denote plane number first and then indexstep second.) These vectors are comprised of the vector sum of the impeller unbalance, and the shaft unbalance,

also both vector quantities. In our simplistic example, the impeller will be treated as a thin disk. In that case, theimpeller unbalance contribution to the plane 2 readings is insignificant. (The impact of this will be highlighted later.)Thus, the reading at plane 1 can be expressed as . The plane 2 reading is solely comprised of shaftunbalance, . and should be recorded at this point.

The next step is removal and indexing of the impeller 180 relative to the shaft. The assembly should be placed in thestand and readings taken (see Figure 5-16).

Since the impeller has been turned 180 relative to the shaft, the unbalance readings taken at plane 1 can now beexpressed as . The unbalance readings at plane 2 remain unchanged per our simplifyingassumption. Vectorially, and are shown in Figure 5-17. At this stage, the vector quantities and arestill unknown.

Figure 5-14—Unbalance Correction to Fit Eccentricity

U/r

Left endbalance plane

r

Right endbalance plane

R11 R21

u1

U1

R11 U1 u1+=U2 R11 R21

R12 U1 u1+=R11 R12 u1 U1



Figure 5-15—Initial Reading of the Index Balancing Method

Figure 5-16—Indexed Component Relative to the Shaft

Plane 1

Plane 2

Reference mark

Plane 1

R11

Plane 2

Unbalance readings

R21

Plane 1

Plane 2

Reference mark

Plane 1

R12

Plane 2

Unbalance readings

R22



Through vector manipulation the following statements are true:

(5-8)

(5-9)

Notice that by either adding or subtracting the two readings taken during the procedure, the unbalance attributable toeither piece is identifiable. This permits correction of the impeller and the shaft by the amounts identified in theequations. Graphically, these vector manipulations are shown in Figure 5-18.

Once these corrections are made to the corresponding parts, the process can be repeated to ensure that balancetolerances have been met. Most low-speed balance machines have software that automates this process andperforms the vector subtraction internally.

The assumption was made early on that the component could be balanced using one plane (its L/D << 1.0). If twoplanes were needed to balance the component or subassembly (the impeller in our example), the balance planesduring the indexing procedure would be set for the locations as required on that part. Since the shaft or remaining partmay not share those balance planes, the shaft (unbalance attributable to that part) would be ignored during theseruns. After completing the index balance of the first component (impeller in our example), the planes would be reset tothose needed to balance the remaining component or subassembly (shaft in our example.) The index balanceprocedure would then be repeated but now the unbalance attributable to the impeller would be ignored.

NOTE That since the impeller was previously balanced, its unbalance levels should be within tolerance.

Finally, literature is available concerning the index balance method. Implementation of the method without automatedsoftware is shown in Wowk [2].

Figure 5-17—Vector Representation of and

u1

u1

U1

R11

R12

R11 R12

R12 R11+ 2U1=

R11 R12– 2u1=



5.5.5 References

[1] “Aspects of Flexible Rotor Balancing,” Third Edition, 1980, Schenck Trebel, Deer Park, New York.

[2] Wowk, V., Machinery Vibration: Balancing, 1994, McGraw-Hill Professional.

5.6 Operating-speed Balance

5.6.1 Introduction

The balancing of flexible rotors (rotors that operate above their first critical speed) at their Operating-speed hasbecome a more acceptable practice in the turbomachinery industry in recent years. This practice is a step changefrom the time when all rotors were low-speed balanced and Operating-speed balance was seen as an expensive,unproven technology. Operating-speed balance requires measuring dynamic forces at the bearings, at Operating-speed and with bearings that closely simulate the actual rotor/bearing dynamics. This technique is especially useful incases where the residual unbalance, remaining from a slow-speed balance, is sensitive to excite bending modesclose to the operating speed.

5.6.2 Applications of Operating-speed Balance

There are conditions where operating-speed balance should be considered as part of the balancing procedure. Theyinclude:

a) rotors, which have exhibited high vibration as they pass through their critical speeds;

b) rotors, which accelerate slowly through their critical speeds (i.e. gas turbines);

c) rotors, which are running on or near a critical speed;

Figure 5-18—Results of Adding and Subtracting Vectors and

u1

u1

U1

U1

R11

R12

R12

R11 R12



d) rotors, which are very sensitive to unbalance;

e) rotors for equipment in extremely critical services;

f) rotors going to inaccessible and offshore locations;

g) very long, flexible rotors;

h) situations where a critical rotor cannot be run in its casing prior to installation;

i) standard procedure for the vendor for their new rotors.

New rotors in most API turbomachinery cannot be considered rigid but can be balanced in a low-speed balancemachine. These rotors would be candidates for receiving an operating-speed balance when conditions b, e, h, and i,as described in 5.2.5 apply. Likewise, existing rotors in service would be likely to receive an Operating-speed balancefor conditions a, c, d, g, and i.

5.6.3 Operating-speed Balance Facility

The two primary elements of an operating-speed balance facility are its bunker and the control system. The facilitytypically includes auxiliary systems such as the lube oil supply (heating, pumping and degassing), vacuum pumps,data collection/analysis and high pressure oil lift system. The rotor is placed in the bunker and balanced under avacuum, Figure 5-19. The bearing pedestals, located in the bunker, are mounted on rails to allow adjustment to fitrotors of different lengths. Each pedestal is equipped with velocity transducers for housing motion and, in someinstallations, proximity probes to measure shaft displacement, Figure 5-20.

The drive system for the bunker is located outside of the vacuum with the drive shaft penetrating the wall, Figure 5-21.A flexible coupling and adapter are used to connect the drive to the rotor. Angular indication is typically shown in thewall at the drive penetration.

The control system operates the vacuum system, regulates speed, sets the lube oil temperature and flow andmonitors health of the bunker. Vibration monitoring accompanies this system (either as an integral part or a separateanalysis package). It is the vibration monitoring package that collects and reduces the data for the balancingprocedure selected by the bunker operator.

5.6.4 Operating-speed Balance Procedure

The contract rotor dynamics analysis should be available prior to attempting an operating-speed balance. Thisanalysis will provide information about the predicted rotor mode shape as it passes through its resonant frequenciesand best location for balance weights to minimize rotor vibration.

There are several methods available to balance a flexible rotor at Operating-speeds. Foiles et al. [1] presents anexcellent survey of the development and rationale behind the methods. One popular method for balancing in vacuumbunkers is the use of the influence coefficients along with a procedure to minimize residual errors of the calculatedsolution, e.g. least squares. The determination of the influence coefficients from a baseline and trial weight runs isdescribed here. The application of the least squares procedure to solve the linear set of equations is described in thereferences of Foiles et al. [1].

NOTE Since the stiffness and mass of the balancing machine bearing pedestals may vary significantly from actual fieldinstallation, the critical speeds as observed in the balancing machine may differ significantly from that observed when the rotor isrun in the field or during the mechanical test.

A typical operating-speed balance procedure is described below.



Figure 5-19—Top Loading High-speed Balance Bunker

Figure 5-20—Close-up of Bearing Pedestal



1) Rotor Assembly—The assembly should include thrust collar with locking nut and any auxiliary equipment suchas power take-off gear, overspeed trip assembly, and toothed wheel for magnetic speed pickup, as applicable.In most cases, the balance facility drive coupling and adapter is adequate to simulate the overhung half weightof the job coupling, insuring the rotor mode shape is simulated at the operating speed. (Note: The pedestalstiffness may lower the second resonant frequency in comparison to field operation.) In some cases, the job-coupling hub with moment simulator may be required, especially for the nondrive end (in the bunker) of drive-through machines.

2) Initial Low-speed Balance—Incremental rotor assembly and balance for rotors is not required as the initialpreparation for operating-speed balance. Instead, a preliminary three-plane, low-speed balance is required tolimit the initial rotor response (see 5.5.3).

3) Bearing, Pedestal, and Instrumentation Selection—The rotor is placed in the balancing machine pedestals in avacuum chamber to reduce the power required to turn the rotor at higher speeds and to reduce heating fromwindage. Pedestals are selected according to rotor weight to be balanced. The smallest pedestal necessary tosupport the rotor weight should be used to maximize pedestal motion to rotor unbalance.

Specially-manufactured bunker or job bearings are necessary to produce the desired modal response. Bunkerbearings are manufactured to fit the pedestal bearing housing without special adapters or machining. In the vastmajority of cases, these are sufficient to achieve an acceptable operating-speed balance. Job bearings may berequested when tests other than balancing will be performed requiring a closer match of the bunker to fielddynamic behavior. When a more accurate match is required, the use of pedestal stiffening (the engagement ofhydraulic rams on each pedestal) should be considered, Figure 5-22. The pedestal stiffening can increase thesupport stiffness and may bring it more in line with the job support stiffness. However, dynamics of the pedestalsshould be examined for their effect on the rotordynamic behavior of the rotor. This may produce undesirable

Figure 5-21—Bunker Drive Shaft (Shown in Red)



resonances and/or response levels that are not representative of the actual application. However, the quality ofunbalance is not affected.

NOTE With pedestal stiffening engaged, pedestal motion is reduced and different balance criteria are required whetherbased on shaft (using proximity probes) or pedestal motion.

4) Low-speed Balance Check in Bunker—The balance of the rotor is checked at low speed. (Bypassing of thisprocedure could result in serious damage to the rotor and/or operating-speed balance machine.) Engineeringjudgment and operator experience are utilized to determine if any corrections need to be made prior to runningthe rotor at its operating speed. Unbalance readings greater than five times the low-speed balance criteriashould be investigated. This is necessary to avoid unwarranted grinding corrections due to possible rotorassembly errors.

5) Relaxation Run—Proper conditioning of the rotor to remove all bows and distortion prior to operating-speedbalance is essential. This relaxation is accomplished by cycling the rotor between zero speed and trip speed (ifvibration levels permit) with the intent to remove temporary bows due to extended horizontal storage ordistortions due to component interference fits.To simulate fits at elevated operating temperatures, relaxationruns above the trip speed to just below overspeed may be considered. (The additional speed above trip speedis to simulate the thermal expansion at operating temperature.) This process may also require the application ofheat to the rotor during the spinning process to better simulate the operating environment of the rotor. The timerequired for stabilization at maximum speed will vary from rotor to rotor. Cycling should continue until bothmagnitude and phase have stabilized.

Figure 5-22—Hydraulic Pedestal Stiffeners Highlighted



6) Baseline Measurement—Once the readings have stabilized a run from start to balance speed is made to definethe baseline shaft readings. The balance speed is maximum continuous speed for motors and motor drivenequipment and trip speed for variable speed drivers and driven equipment.

7) Bunker Drive Assembly Balance Check—To determine the balance of the bunker drive assembly and adapters,the job rotor assembly should be turned 180 relative to the drive components. A run to the balance speed ismade following the indexing. If the drive assembly is perfectly balanced, the magnitude of the readings willremain unchanged but will show a 180 change in phase angle. Since the drive assembly will have some levelof residual unbalance, the acceptable change in baseline readings at any speed is <25 % of the balance criteria.

8) Trial Weight Runs—Trial weights are added to the rotor following the baseline measurement to determine theeffect of unbalance placed at specific points along the rotor (balance planes where corrections will be made) onthe vibration measurements for specified speeds. These measurements are used to determine the influencecoefficients for each balance plane. For example, if placing a 254 g-mm (10 g-in.) trial weight on balance plane#1 produces a change in vibration of 12.7 m (0.5 mils) at pedestal #1, then the influence coefficient associatedwith balance plane #1 relative to pedestal #1 is 0.05 m/g-mm (0.05 mils/g-in.). (This is taken at a single speedand ignoring phase angles for this example.) This step is repeated until trial weights are added to each of thebalance planes.

(5-10)

where

is the influence coefficient ijk;

Vtrial weight is the vibration with trial weight j at probe i and speed k;

Vbaseline is the baseline vibration at probe i and speed k;

TWj is the trial weight added to balance plane j.

For two probe (pedestal) readings (i) taken at three speeds (k) using three balance planes (j), the linear set ofequations is:

(5-11)

vu------

ijk

Vtrial weight Vbaseline– i

TWj

-----------------------------------------------k

=

vu-----

ijk

vu------

111

vu-----

121

vu-----

131

vu------

211

vu-----

221

vu-----

231

vu------

112

vu------

213

vu-----

122

vu-----

223

vu-----

132

vu------

ijk

U1

U2

Uj

VB11

VB21

VB12

VBik

–

11

21

12

ik

=



where

is the influence coefficient for reading i, plane j, and speed k;

Uj is the correction weight for plane j;

VBik is the baseline vibration for reading i and speed k;

is the residual for reading i and speed k;

The above set of linear equations is solved for the correction weights Uj while minimizing the residuals, ik.

9) Addition of Correction Weight—The correction weights calculated above are added to the rotor (typically in atemporary fashion). The rotor is brought up to balance speed and the vibration levels examined. If balancespecifications have been met, the procedure progresses to the next step. If vibration levels are still abovebalance specifications, a new set of solution weights is be calculated using the vibration from this run as the newbaseline for the above equation in step viii. In some instances, new influence coefficients may be required andsteps 7 through 9 are repeated.

The influence coefficient method as described assumes linear behavior of the rotor, i.e. if 25 g-mm produces 12m of vibration then 50 g-mm will produce 24 m. Unfortunately, the actual rotor/bearing system is for smallvibration levels about a steady state position. As the vibration level changes, especially from large levels ofvibration, linear behavior may not be followed. This may invalidate a set of influence coefficients used to start thebalancing process and require a new set to be calculated to further reduce vibration levels in the bunker.

NOTE Field accessible balance holes should not be used in the bunker. These should be left empty to permit maximumflexibility and simplification for field balancing.

10) Following achievement of the specified balance vibration levels, a final run is made with the pedestal stiffeningon (if pedestal vibration was used for balancing or if probes readings are desired for shaft motion). This stepmay be skipped if proximity probes were used to balance the rotor (pedestal stiffening should already be on inthis case).

11) Unless otherwise specified, a rotor that has been operating-speed balanced shall have the unbalance recordedin a low-speed balance machine. No corrections shall be made to the rotor. A permanent mark or part (such asa keyway) shall be used and recorded for the phase reference. This is for future reference to serve as anindicator of the balance state of the rotor. Comparisons to the low-speed measurements can be used after longstorage times or rotor shipment to qualitatively decide on the high-speed balance condition of the rotor using alow-speed stand (which is more readily available).

NOTE The operating-speed balanced rotor will generally not meet the low-speed balance criteria.

5.6.5 Acceptance Criteria

Operating-speed balance criteria are largely based on the experience of the bunker operator and vary depending onthe machine type and measurement employed. A common acceptance criterion employed in bunkers. is pedestalvelocity. Some bunkers also use shaft vibration as measured by proximity probes as an additional acceptanceguideline. Finally, the magnitude of the residual, , is also used by some gas turbine manufacturers for their balanceacceptance.

The velocity for operating-speed balance measured on the pedestals is dependent on the magnitude of the forcetransmitted from the rotor to the bearings and the pedestal stiffness. As the pedestal stiffness increases, the pedestalvelocity will decrease for a given applied force. Therefore, it is important that the acceptance criteria match thepedestal size and rotor weight. To achieve maximum sensitivity, it is recommended that the smallest pedestal size

vu-----

ijk

ik



(based on rotor weight rating) be used. A pedestal criterion of 1.0 mm/s is used by several bunker operators.However, many vary the criterion based on location relative to critical speeds, Operating-speed, rotor weight or toachieve a specified force on the pedestal (in most cases measured as a fraction, X, of the rotor weight or X times ‘W’.

The following example will demonstrate the relationship between the velocity criteria, applied force and pedestalstiffness.

Assume a rotor weight of W = 907 kg (2000 lbm) split evenly on the pedestals. Also assume the rotor is turning at10,000 rpm (167 Hz). If the bearing pedestal velocity is 1.0 mm/s (0.0394 in./s) RMS, then the displacement of thepedestal is given by:

(5-12)

where

V is the velocity of the bearing pedestal, mm/s (in./s);

f is the frequency of vibration, Hz;

Dpk is the peak displacement of the pedestal, m (mils).

For the example above, the pedestal displacement would be 1.35 m (0.053 mils).

To calculate the pedestal force, the pedestal stiffness is needed. Two common pedestal stiffness values are 5.6 x 108

N/m (3.2 x 106 lbf/in.) and 1.33 x 109 N/m (7.6 x 106 lbf/in.). From the relationship:

(5-13)

where

F is the pedestal force, N (lbf);

C = 1 x 10-6 (1 x 10-3);

Dpk is the displacement, m (mils);

Kped is the stiffness, N/m (lbf/in.).

For the pedestal stiffness of 5.6 x 108 N/m (3.2 x 106 lbf/in.), the pedestal force for the above example is:

F = 756 N (170 lbf)

Alternatively, for the pedestal stiffness of 13.3 x 108 N/m (7.6 x 106 lbf/in.), the force is:

F = 1795.5 N (400 lbf)

In this case, the velocity criterion of 1.0 mm/s produces a pedestal force of 0.09W for the softer pedestal and 0.2W forthe stiffer pedestal. However, pedestal stiffness is normally expressed as the static stiffness or for vibrationfrequencies near zero. Pedestal stiffness varies by the vibration frequency by the following relationship:

(5-14)

Dpk1414V

2f-----------------=

F C Dpk Kped=

Kfreq Kstatic gcm2–=



where

Kfreq is the pedestal stiffness at the vibrational frequency, N/m (lbf/in.);

Kstatic is the pedestal static stiffness, N/m (lbf/in.);

m is the pedestal mass, kg (lbm);

is the vibration frequency (for rotor balancing this is equal to rotational speed), rad/s;

gc = 1 (386.04).

Thus, as the balance speed increases, the applied force necessary to generate a given pedestal velocity willdecrease. Given that in the absence of dynamic effects, the force generated by a given unbalance goes up with thespeed squared, a constant pedestal velocity criterion can be viewed as conservative for the higher speeds in the rotoroperating range.

The relationship between balancing machine pedestal stiffness, rotor weight and operating speed for onemanufacturer is shown in Table 5-1.

NOTE Pedestal stiffness is intentionally lower than normal machinery supports in order to have good balancing sensitivity. Themass of the pedestals also affects the rotor support system natural frequencies. These values are approximate and care must beused to remain within limits set by manufacturer of the pedestals. There may be pedestal stiffening available to increase thestiffness values. A particular balance facility may have pedestals that only cover part of the speed range, are not able toaccommodate rotor lengths or outside diameter, or do not have required bearings to fit inside the pedestals for a particularapplication.

5.6.6 Advantages of Operating-speed Balance

Operating-speed balance is the only means of effectively and efficiently balancing flexible rotors. For stiff shaft or rigidbody balancing, the location of the correction relative to the unbalance is not critical. The unbalance of any rigid bodycan be described as static and dynamic and can be completely balanced using only two planes. (Steps are usuallytaken to minimize the amount of correction needed since material removal is limited by component integrity.) Forflexible rotors, the modal deflections dictate the unbalance and correction locations for an effective balance. Theadvantages of operating-speed balance are primarily for reliability. Whether the rotor is going to an inaccessiblelocation making work at the site expensive or a rerate where there is no opportunity to run the rotor prior to installationin the field, operating-speed balance gives that additional confidence of examining the rotor and vibration levels atspeed.

Table 5-1—Relationship Between Machine Pedestal Stiffness, Rotor Weight, and Operating Speed

Pedestal Size

Typical Pedestal Stiffness Range Typical Rotor Weight RangeTypical Speed

RangeUS Customary Units

SI Units US Customary Units

SI Units

lb/in. x 106 N/m lb kg RPM

Small 1.5 to 2.0 250 to 350 100 to 5, 500 45 to 2500 20,000

Medium 2.5 to 4.5 450 to 800 500 to 28,000 225 to 12,700 12,000

Large 5.5 to 10.0 1000 to 1800 1500 to 110,000 680 to 50,000 6000



5.6.7 References

[1] Foiles, W. C., Allaire, P. E. and Gunter, E. J., 1998, “Review: Rotor Balancing,” Shock and Vibration, Vol. 5,Number 5-6, pg. 325–336.

5.7 Keys and Keyways

Keys and keyway clearances are areas that are often overlooked and yet critical to a precision balance job. All keysshould have a top clearance of 0.05 mm to 0.15 mm (0.002 in. to 0.006 in.). Excessive top key clearance will allow thekey to move radially outward during operation, resulting in a change in unbalance. Conversely, insufficient (or zero)top key clearance may prevent the wheel from properly seating on the shaft. A tall key will distort the wheel bore,resulting in a change in unbalance.

During progressive component stack (and any type of) balancing, all empty shaft single keyways must be filled withcrowned half-keys to ensure that unbalance due to unfilled keyways is not compensated for in trim balancing stackedcomponents. The only situation where half keys are not needed is when two identical keys 180 apart are used. Sincethe unbalance created by the keyways will be offset by one another, filling them is not necessary.

5.8 Residual Unbalance Test

The residual unbalance test, perhaps inappropriately named, is actually a test of the low-speed balance machinecalibration. For rotors that are to be low-speed balanced only, this test is critical to ensure that the balancespecifications were actually achieved. For operating-speed balanced rotors, the test has no real meaning since thefinal balance was performed in the balance bunker.

NOTE A residual unbalance test (or calibration) of the operating-speed bunker is not done nor could it be done. Since thebunker relies on influence coefficients (and not a machine stating how much unbalance there is), there is no need to checkcalibration of the method. Of course, calibration of the measurement condition should be ensured.

After completion of the final balancing of the rotating assembly in the low-speed stand, and before removing the rotorfrom the balancing machine, a residual unbalance test should be performed to verify that the residual unbalance ofthe rotor is within the 4W/N tolerance. This test is performed to ensure that the balancing machine readout is correct.Errors can occur due to balancing machine calibration shift and operator mistakes.

This test is accomplished by adding a known trial weight to the rotor in six equally spaced radial positions in eachbalance machine readout plane. The trial weight and radius is selected so that approximately four times the 4W/Nresidual unbalance tolerance (two times 4W/N for hard bearing balancing machines) is produced. The trial weightshould first be positioned at the heavy spot on the rotor (if known) to assist in selecting the proper readout scale onthe balancing machine. The heavy spot location on the rotor is then considered as the zero point on the rotor for thepolar plot.

The rotor is then run up to test speed and the balancing machine amplitude and trial weight location is measured andrecorded on polar graph paper. (Note that the data to be recorded on the polar plot are balancing machine amplitudeversus the angular location of the trial weight, not the balancing machine phase angle.) This test is repeated for alltrial weight positions in each balance plane of the rotor. Each plane’s polar plot of balancing machine amplitudeversus the trial weight location should approximate a true circle that encircles the center of the polar plot. If the plotdoes not approximate a true circle and/or encircle the center of the polar plot, then either the residual unbalance is inerror due to inadequate sensitivity of the balancing machine, the trial weight is smaller than the residual unbalanceindicating that the rotor is not balanced correctly, or a balancing machine fault exists (i.e. a faulty pickup).

API 617 8th Edition Annex 1A contains a detailed description of the procedure and includes several examples of thereadings and residual unbalance calculations. Careful placement of the known unbalance at the correct radius andangle at each interval is essential for this test. The polar point plotted for run-test number one should repeat at the end



of the test indicating that the balancing machine is reading out consistently. A balancing machine that will not read outconsistently for two identical runs cannot be used to determine true residual unbalance.

5.9 Check Balancing

Once a rotor has been balanced in accordance with the above outlined procedures, further balancing should not berequired if handled and stored correctly. Far too often rotors that have been properly balanced to the correcttolerances are placed into storage for long period of time. If stored horizontally without periodic turning, a rotor bowwill be created. Whether this bow is permanent or temporary depends on factors such as shaft dimensions andstorage time. Runout checks prior to their installation should be performed to determine the existence of a bow. Low-speed balance check may also reveal the existence of a bow by showing large static unbalance levels (exceeding the4W/N specification.) During this check balancing procedure it is possible that the rotor will be found to be out oftolerance. Before re-balancing, efforts should be made to relax the bow, typically by running the rotor in the low-speedstand for several hours.

Poor handling during shipment or inadequate shipping containers can also change the balance state of the rotor.Dropping the container or container flexibility can result in a bowed rotor due to large deflections of the rotor. Theselarge deflections can result in slippage of shrink fits on the rotor. The fits can reposition while the rotor is in a bowedshape, thus keeping the rotor in that shape after the drop or excessive distortion of the container is removed.

As discussed, improper storage or handling can create rotor bows that are temporary in nature (can be relaxed withrotor turning), semi-permanent (will relax when brought up to speed and temperature, i.e. the shrink fits relax andgrab the rotor in a straight shape) or permanent (requiring at a minimum a de-staging or removal of all shrink fits fromthe shaft.) If low-speed balancing is performed prior to installation, an unbalance equal to the correction is created forthe first two possibilities. In these instances, the bow will relax under operation and the corrections used to offset thebow will become unbalances. High vibration may be experienced during coastdown following an extended run. Theunbalance from a permanent bow can be addressed but may require extensive corrections. Also, it is very difficult todistinguish between the three possible scenarios, so the possibility of creating an unbalance through balancing priorto installation will remain.

A rotor that has been properly balanced, handled and stored should not need to be re-balanced prior to its installationunless obvious damage or other sound justification is apparent. If the check balance of a rotor prior to its installationreveals an out-of-tolerance residual unbalance condition, then a thorough inspection of the rotor should be performedand additional data (that is, rotor runout maps) should be measured and recorded to ascertain the problem with therotor. If the problem cannot be located and resolved, then to remove all risks associated with temporary bows orrelaxation of fits the rotor should be totally disassembled.

During the check balance, the 4W/N tolerance may be used for rotors that were only low-speed balanced. Correctionsif deemed necessary can be made with standard low-speed procedures.

For rotors that were operating-speed balanced, the readings taken from the check balance need to be comparedagainst those taken immediately after the operating-speed balance. It is likely that an operating-speed balance mayexceed 4W/N. Therefor using a 4W/N target will not be a reliable indicator of the balance state. If the check balancerepeats the setup as used following the operating-speed balance, then if deemed necessary, the operating-speedbalance condition can be approximated by returning the rotor to those readings. (The further out from those readings,the less likely the balance condition will be repeated.) Any determination of the need for corrections or performingthose corrections on a low-speed stand for operating-speed rotors should be performed by an experienced machineryengineer.

5.10 Field Balancing

In some cases, field balancing has been found to be necessary. This is typical of very large steam turbines (over 50MW) and for field-erected equipment that does not lend itself to shop balancing. This, however, should be consideredas a last resort for high-speed rotors.



Field balancing attempts to correct for the combined effect of misalignment, seal rubs, foundation resonance, androtor unbalance. This approach does not address the true cause of the excitation forces on the rotor. Generally, onlyend-plane balance corrections are available, which further limits the effectiveness of this balancing method. With theexception of steam and gas turbines with field accessible balance planes, the coupling is the only readily accessiblelocation. Since the coupling is not designed for balance weight placement, the size of the correction may be limited. Inaddition, correction of high vibration levels through the 1st critical speed on between bearings machines is ineffectivewhen using a coupling or shaft end location. The mode shape (high in the center and low at the shaft ends) does notlend itself to correction in the coupling location.

While field balancing has been done successfully, it has a fairly high failure rate in rotors with multi-plane correctionpoints and may not correct the problem despite the best of efforts. Field balancing should only be attempted byexperienced personnel since the risk of additional harm to the machine exists through the application of inappropriatecorrection weights.


6-1

SECTION 6—STANDARD PARAGRAPHS

6.1 Introduction

The Dynamics Section (6.8) of the Standard Paragraphs V27, 2010 is enclosed in their entirety including the reportingrequirements and magnetic bearing annexes. Standard Paragraphs represent specifications that share commonalitywith several of the mechanical equipment standards. They are intended to provide a starting point for the individualspecification language used by the API task force.

6.2 Paragraphs

SP6.8 Dynamics

SP6.8.1 General

NOTE Refer to API RP 684 API Standard Paragraphs Rotordynamic Tutorial: Lateral Critical Speeds, Unbalance Response,Stability, Train Torsionals, and Rotor Balancing.

SP6.8.1.1 In the design of rotor-bearing systems, consideration shall be given to all potential sources of excitationwhich shall include, but are not limited to, the following:

a) unbalance in the rotor system;

b) fluid destabilizing forces from bearings, seals, and aerodynamics;

c) internal rubs;

d) blade, vane, nozzle and diffuser passing frequencies;

e) gear-tooth meshing and side bands;

f) coupling misalignment;

g) loose rotor-system components;

h) internal friction within the rotor assembly;

i) synchronous excitation from complimentary geared elements;

j) control loop dynamics such as those involving active magnetic bearings and variable frequency drives;

k) electrical line frequency.

NOTE 1 The frequency of a potential source of excitation can be less than, equal to, or greater than the rotational speed of therotor.

NOTE 2 When the frequency of a periodic forcing phenomenon (excitation) applied to a rotor-bearing support system coincideswith a natural frequency of that system, the system will be in a state of resonance. A rotor-bearing support system in resonancecan have the magnitude of its normal vibration amplified. The magnitude of amplification and, in the case of critical speeds, the rateof change of the phase angle with respect to speed, is related to the amount of damping in the system.

SP6.8.1.2 Resonances of structural support systems that are within the vendor’s scope of supply and that affect therotor vibration amplitude shall not occur within the specified operating-speed range or the required separationmargins (SMr) (see SP6.8.2.9). The dynamic characteristics of the structural support shall be considered in theanalysis of the rotor-support system (see SP6.8.2.4.e).


6-2 API TECHNICAL REPORT 684

SP6.8.1.3 If specified, the vendor with unit responsibility shall communicate the existence of any undesirablerunning speeds in the range from zero to trip speed. This shall be illustrated by the use of Campbell Diagram,submitted to the purchaser for review and included in the instruction manual (see Annex __ of the applicable chapter).(Task force to reference Annex or section containing instruction manual.)

NOTE Examples of undesirable speeds are those associated with rotor lateral critical speeds with amplification factors greaterthan or equal to 2.5, train torsionals, and vane and blading modes.

SP6.8.1.4 Lateral analysis requirements specified in SP6.8.2, SP6.8.5, and SP6.8.6 shall be reported perSP6.8.1.4.1 to SP6.8.1.4.3 and Annex XX-I.

SP6.8.1.4.1 The basic rotordynamic report shall be provided.

SP6.8.1.4.2 If specified, the reporting requirements identified as required for independent audit of the results shallbe provided.

SP6.8.1.4.3 If specified, provisions shall be made to provide the purchaser with access to drawings to developindependent models of the rotor, bearings, and seals. These data shall be made available in electronic format.

NOTE This should be requested at time of order, since nondisclosure agreements may be required.

SP6.8.1.5 Torsional analysis requirements specified in SP6.8.7 shall conform to SP6.8.1.5.1 to SP6.8.1.5.3 andAnnex XX-II.

SP6.8.1.5.1 The basic torsional report shall be provided for all covered machines.

SP6.8.1.5.2 If specified, the reporting requirements identified as required for independent audit of the results shallbe provided.

SP6.8.1.5.3 If specified, provisions shall be made to provide the purchaser with access to drawings to developindependent models of the rotors. These data shall be made available in electronic format.

NOTE This should be requested at time of order, since nondisclosure agreements may be required.

SP6.8.2 Lateral Analysis

SP6.8.2.1 Critical speeds and their associated AFs shall be determined by means of a damped unbalanced rotorresponse analysis.

SP6.8.2.2 The location of all critical speeds below the trip speed shall be confirmed on the test stand during themechanical running test (see SP6.8.3.1). The accuracy of the analytical model shall be demonstrated (see SP6.8.3).

SP6.8.2.3 The vendor shall conduct an undamped analysis to identify the undamped critical speeds and determinetheir mode shapes. The analysis shall identify the first four undamped critical speeds and include as a minimum thestiffness range from 0.1X to 10X the expected support stiffness.

SP6.8.2.4 The rotordynamic analysis shall include but shall not be limited to the following.

NOTE The following is a list of items the analyst is to consider. It does not address the details and product of the analysis whichis covered in SP6.8.1.4, SP6.8.2.7, and SP6.8.2.8.

a) Rotor masses and polar and transverse moments of inertia, including coupling halves, and rotor stiffness changesdue to shrunk on components.

b) Material properties as a function of operating temperature variation along the shaft.

●

●

●

●

●



c) Bearing lubricant-film stiffness and damping values including changes due to speed, load, preload, range of oilinlet temperature, maximum to minimum clearances resulting from accumulated assembly tolerances, and theeffect of asymmetrical loading which may be caused by gear forces (including the changes over range ofmaximum to minimum torque), side streams, eccentric clearances, volutes, partial arc admission, etc.

d) For tilt-pad bearings, the pad pivot stiffness.

e) Structure stiffness, mass, and damping characteristics, including effects of excitation frequency over the requiredanalysis range. For machines whose dynamic structural stiffness values are less than or equal to 3.5 times thebearing stiffness values in the range from 0 to 150 % of Nmc, the structure characteristics shall be incorporated asan adequate dynamic system model, calculated frequency dependent structure stiffness and damping values(impedances), or structure stiffness and damping values (impedances) derived from modal or other testing. Thevendor shall state the structure characteristic values used in the analysis and the basis for these values (forexample, modal tests of similar rotor structure systems, or calculated structure stiffness values).

f) Rotational speed, including the various starting-speed detents, operating speed, and load ranges (includingagreed upon test conditions if different from those specified), trip speed, and coast-down conditions.

g) The influence, over the operating range, of the casing shaft end oil seals, if present. Minimum and maximumstiffness will be considered taking into account the tolerance on the component clearance and the oil inlettemperature.

h) The location and orientation of the radial vibration probes which shall be the same in the analysis as in themachine.

i) Squeeze film damper mass, stiffness, and damping values considering the component clearance and centeringtolerance, oil inlet temperature range, and operating eccentricity.

j) For machines equipped with rolling element bearings, the vendor shall state the bearing stiffness and dampingvalues used for the analysis. The basis for these values or the assumptions made in calculating the values shallbe presented.

k) Dry gas seals shall be assumed to have no stiffness or damping.

(Task force to determine the appropriate items to include in the specification.)

SP6.8.2.5 If specified, the vendor with unit responsibility shall provide a train lateral analysis.

SP6.8.2.6 The vendor with unit responsibility shall provide a train lateral analysis for machinery trains with rigidcouplings.

SP6.8.2.7 A separate damped unbalanced response analysis shall be conducted within the speed range of 0 to150 % of Nmc. Unbalance shall analytically be placed at the locations defined in Figure SP-6. For the translatory(symmetric) modes, the unbalance shall be based on the sum of the journal static loads. For conical (asymmetric)modes, these unbalances shall be 180 degrees out of phase and of a magnitude based on the static load on theadjacent bearing. For overhung modes, the unbalances shall be based on the overhung mass. Figure SP-6 showsthe typical mode shapes and indicates the location and definition of Ua for each of the shapes. The magnitude of theunbalances shall be 2 times the value of Ur as calculated by Equation 2a or Equation 2b.

In SI units:

Ur = 6350 W/Nmc for Nmc < 25,000 RPM (2a-1)

Ur = W/ 3.937 for Nmc ≥ 25,000 RPM (2a-2)

●



Figure SP-6—Unbalance Placement

NOTE Rigid (solid) and flexible (dashed) shaft modes shown.

Key

Bearing location and reaction

Unbalance placement

Overhung components (couplings, impellers, etc.)

Ua = 8

W2W1

W2W1

W2W1

W2W1

W1 + W2N

Ua1 = 8 W1N

Ua1 = 8 W1N

Ua2 = 8 W2N

Ua2 = 8 W2N

Ua = 8 W3

W3

W3

W3

N

Ua = 8 W3

1st Mode (W3 > W4)

Coupling Mode

1st Bending

Translatory 1st Rigid

2nd Bending(Unbalance @ Quarterspan)

Conical 2nd Rigid(Unbalance @ Journals)

Overhung Machines

2nd Mode (W3 > W4)

3rd Mode

N

Ua = 8 W4

W4W4

N

Ua1 = 8 W4

W4

N

Ua2 = 8 W3

W3

N

Ua = 8 W1 + W2N



In U.S. Customary units:

Ur = 4 W/Nmc for Nmc < 25,000 RPM (2b-1)

Ur = W/6250 for Nmc ≥ 25,000 RPM (2b-2)

where

Ua = 2 x Ur is input unbalance for the rotordynamic response analysis, g-mm (oz-in.);

Ur is maximum allowable residual unbalance, g-mm (oz-in.);

Nmc is maximum continuous operating speed, rpm;

W is the journal static load in kg (lbm), or for bending modes where the maximum deflection occurs at the shaft ends, the overhung mass (that is the mass of the rotor outboard of the bearing) in kg (lbm) (see Figure SP-6).

NOTE Above 25,000 RPM, the unbalance limit is based on 0.254 m (10 in.) mass displacement, which is in generalagreement with the capabilities of conventional balance machines, and is necessary to invoke for small rotors running at highspeeds.

SP6.8.2.8 Additional analyses shall be made for the verification test. The location of the unbalance shall bedetermined by the vendor. The unbalance shall not be less than 2 times or greater than 8 times the value fromEquations 2a or Equation 2b or as specified in SP6.8.2.8.1. Any test stand parameters which influence the results ofthe analysis shall be included.

SP6.8.2.8.1 For coupling unbalance placement (unbalance based on the coupling half weight), the unbalance shallbe greater or equal to 16 times the value of Equation 2a or Equation 2b.

NOTE For most machines, there will only be one plane readily accessible for the placement of an unbalance; for example, thecoupling flange on a single ended drive machine, or the impeller hub or disk on an integrally geared machine, or expander-compressors. However, some turbomachinery (axial compressors and steam turbines for example) may provide additionalexternally accessible balance planes. For these machines, when there exists the possibility of exciting other critical speeds,multiple runs may be required.

SP6.8.2.9 The damped unbalanced response analysis shall indicate that the machine will meet the followingrequirement:

SMa SMr

where

SMr is required separation margin, %;

SMa is defined in Figure SP-7.

a) If the AF at a particular critical speed is less than 2.5, the response is considered critically damped and noseparation margin is required (SMr = 0).

b) If the AF at a particular critical speed is greater than or equal to 2.5 and that critical speed is below the minimumspeed, the SMr is given by Equation 3.

(3)SMr 17 11

AF 1.5–---------------------–

=



c) If the AF at a particular critical speed is greater than or equal to 2.5 and that critical speed is above the maximumcontinuous speed, the SMr is given by Equation 4.

(4)

SP6.8.2.10 The calculated unbalanced peak-to-peak response at each vibration probe, for each unbalance amountand case as specified in SP6.8.2.7 shall not exceed the mechanical test vibration limit, Avl, of 25.4 m (1.0 mil) orEquation 5, whichever is less, over the range of Nma to Nmc as shown in Figure SP-8.

In SI units:

(5a)


(5b)

where

Avl is mechanical test vibration limit, mm (mil);

Nmc is maximum continuous speed, rpm.

SP6.8.2.11 For each unbalance amount and case as specified in SP6.8.2.7, the calculated major-axis, peak-to-peak response amplitudes at each close clearance location shall be multiplied by a scale factor defined by equation 6.

Scc = Avl /Amax or 6, whichever is less (6)

NOTE To meet the requirements of SP6.8.2.10, the scale factor should be greater than or equal to one.

SP6.8.2.11.1 For each close clearance location, the scaled response shall be less than 75 % of the minimumdesign diametral running clearance over the range from zero to trip speed.

SP6.8.2.11.2 For this evaluation, floating-ring, abradable and compliant seals are not considered to be closeclearance locations. The response amplitude as compared to the running clearance at these locations shall beagreed.

NOTE Running clearances may be different than the assembled clearances with the machine shutdown. Consideration shouldbe given to:

a) centrifugal/thermal growth;

b) bearing lift;

c) rotor sag;

d) nonconcentricity (of stator to the bearings).

SMr 10 17+ 11

AF 1.5–---------------------–

=

Avl 25.4 12,000Nmc

-----------------=

Avl12,000

Nmc

------------------=



Figure SP-7—Rotor Response Plot

Ac1

Ac1

Sa1

Operating-speedrange San

Nc1

Nc1

Nma

Nma

Nmc

Nmc

Ncn

Ncn

N1

N1

N2

N2

AF1

0.707 x Ac1

NOTE The shape of the curve is for illustration only and does not necessarily represent any actual rotor response plot.

is the amplification factor for the first critical speed

Sa1 is the actual separation between Nc1 and the operating speed range;

San is the actual separation between Ncn and the operating speed range;

SMa1 is the actual separation margin of first critical speed, %

= 100 x Sa1 / Nma

SMan is the actual separation margin of nth critical speed, %

= 100 x San / Nmc

= Nc1 / (N2 – N1)

is the final (greater) speed at 0.707 x Ac1;

is the initial (lesser) speed at 0.707 x Ac1;

is the amplitude at Nc1;

is the maximum continuous speed;

is the minimum allowable speed;

is the nth critical speed;

is the rotor first critical speed;

Rotor Speed

Key

Vbr

aton

Am

ptu

de



SP6.8.2.12 If the analysis indicates that if either of the following requirements can not be met:

— the required separation margins;

— the requirements of SP6.8.2.10 and SP6.8.2.11;

and the purchaser and vendor have agreed that all practical design efforts have been exhausted, then acceptableamplitudes, separation margins and amplification factors shall be agreed, subject to the requirements of SP6.8.4.

SP6.8.3 Unbalanced Rotor Response Verification Test

SP6.8.3.1 An unbalanced rotor response test shall be performed as part of the mechanical running test (Note: SeeSection of the applicable chapter), and the results shall be used to verify the analytical model. The actual response ofthe rotor on the test stand to the same arrangement of unbalance and bearing loads as was used in the analysisspecified in SP6.8.2.8 shall be used for determining the validity of the damped unbalanced response analysis. Toaccomplish this, the requirements of SP6.8.3.1.1 through SP6.8.3.1.6 shall be followed.

SP6.8.3.1.1 During the mechanical running test (Note: see Section of the applicable chapter), the amplitudes andphase angle of the shaft vibration from trip to slow roll speed shall be recorded after the four hour run. The recordinginstrumentation resolution shall be at least 0.05 mils (1.25 micron).

NOTE This set of readings is normally taken during a coastdown, with convenient increments of speed such as 50 RPM. Sinceat this point the rotor is balanced, any vibration amplitude and phase detected should be the result of residual unbalance andmechanical and electrical runout.

SP6.8.3.1.2 The unbalance which was used in the analysis performed in SP6.8.2.8 shall be added to the rotor in thelocation used in the analysis.

Figure SP-8—Definition of Speed Range for Probe Scaling

Avl

Vibration limitspeed range

Probe responsewith 2 x Ur unbalance

Amax

Rotor Speed

Vbr

aton

Am

ptu

de

Nma Nmc



SP6.8.3.1.3 The machine shall then be brought up to trip speed after being held at maximum continuous speed forat least 15 minutes and the indicated vibration amplitudes and phase shall be recorded during the coast down usingthe same procedure as SP6.8.3.1.1.

SP6.8.3.1.4 The location of critical speeds below the trip speed shall be established. If a clearly defined responsepeak is not observed during the test, then the critical speeds shall be identified as those in the rotordynamic report.

NOTE Slow roll run out should be vectorially subtracted from the 1X Bode plots to accurately define the location of the criticalspeeds.

SP6.8.3.1.5 The corresponding indicated vibration data taken in accordance with SP6.8.3.1.1 and SP6.8.3.1.4 shallbe vectorially subtracted.

NOTE Check slow roll run out prior to subtraction. This should be nearly identical for both runs.

SP6.8.3.1.6 The results of the mechanical run including the unbalance response verification test shall be comparedwith those from the analytical model specified at SP6.8.2.8.

NOTE It is necessary for probe orientation to be the same for the analysis and the machine for the comparison to be valid.

SP6.8.3.2 Using the unbalance response test results, the vendor shall correct the model if it fails to meet either ofthe following criteria.

a) The actual critical speed determined on test shall not deviate from the corresponding critical speed rangespredicted by analysis by more than ±5 %.

b) The maximum probe response from the results of SP6.8.3.1.5 shall not exceed the predicted ranges.

SP6.8.3.3 The vendor shall determine whether the comparison will be made for absolute or relative motion.

NOTE For absolute motion, bearing housing vibration will need to be vectorially added to relative probe readings. This may berequired for gas turbines, steam turbines, large fans and other machinery that may have soft supports.

SP6.8.3.4 Unless otherwise specified, the verification test of the rotor unbalance shall be performed only on the firstrotor tested, if multiple identical rotors are purchased.

SP6.8.3.5 After correcting the model, the response amplitudes shall be checked against the limits specified inSP6.8.2.10 and SP6.8.2.11.

SP6.8.4 Additional Testing

SP6.8.4.1 Additional testing is required if either of the following conditions exists in the shop verification test data(see SP6.8.3) or the damped unbalanced response analysis (see SP6.8.3.2).

a) Any critical speed fails to meet the SMr requirements (see SP6.8.2.9).

b) The requirements of SP6.8.2.10 and SP6.8.2.11 have not been met.

NOTE When the analysis or test data do not meet the requirements of the standard, additional more stringent testing isrequired. The purpose of this additional testing is to determine on the test stand that the machine will operate successfully.

SP6.8.4.2 Unbalance weights shall be placed as described in SP6.8.2.7; this may require disassembly of themachine. Unbalance magnitudes shall be achieved by adjusting the indicated unbalance that exists in the rotor fromthe initial run to raise the displacement of the rotor at the probe locations to the vibration limit, SP6.8.2.10, at themaximum continuous speed; however, the unbalance used shall be no less than twice nor greater than 8 times the



unbalance limit specified in SP6.8.2.7. The measurements from this test, taken in accordance with SP6.8.3.1.1 andSP6.8.3.1.3, shall meet the following criteria.

a) From zero to trip speed, the shaft deflections shall not exceed 90 % of the minimum design running clearances.

b) Within the operating-speed range, including the SMr, the shaft deflections shall not exceed 55 % of the minimumdesign running clearances or 150 % of the allowable vibration limit at the probes (see SP6.8.2.10).

c) For this evaluation, floating-ring, abradable and compliant seals are not considered to be close clearancelocations. The response amplitude relative to the running clearance at these locations shall be agreed.

SP6.8.4.3 The internal deflection limits specified in SP6.8.4.2 items a through c shall be based on the calculateddisplacement ratios between the probe locations and the areas of concern identified in SP6.8.2.11 based on acorrected model, if required. Acceptance will be based on these calculated displacements or inspection of the seals ifthe machine is opened.

NOTE Internal displacements for these tests are calculated by multiplying these ratios by the peak readings from the probes.

SP6.8.4.4 Damage to any portion of the machine as a result of this testing shall constitute failure of the test. Internalseal rubs that do not cause changes outside the vendor's assembly clearance range do not constitute damage.

SP6.8.5 Level 1 Stability Analysis

SP6.8.5.1 A stability analysis shall be performed on all centrifugal or axial compressors, turbines and/or radial flowrotors that meet the following.

a) Those rotors whose maximum continuous speed is greater than the first undamped critical speed on rigidsupports, FCSR, in accordance with SP6.8.2.3.

b) Those rotors with fixed geometry bearings or oil film ring seals.

The stability analysis shall be calculated at the API defined maximum continuous speed.

NOTE Level I analysis was developed to fulfill two purposes: first, it provides an initial screening to identify rotors that do notrequire a more detailed study. The approach as developed is conservative and not intended as an indication of an unstable rotor.Second, the Level I analysis specifies a standardized procedure applied to all manufacturers similar to that found in SP6.8.2.(Refer to API RP 684 for a detailed explanation.)

SP6.8.5.2 The model used in the Level I analysis shall include the items listed in SP6.8.2.4.

SP6.8.5.3 When tilt pad journal bearings are used, the analysis shall be performed with synchronous tilt padcoefficients.

SP6.8.5.4 For rotors that have quantifiable external radial loading (e.g. integrally geared compressors), the stabilityanalysis shall also include the external loads associated with the operating conditions defined in SP6.8.5.5. For somerotors, the unloaded (or minimal load condition) may represent the worst stability case and shall be considered.

SP6.8.5.5 The anticipated cross-coupling, QA, present in the rotor is defined by the following procedures:




The parameters in Equation 7 shall be determined based on the machine conditions at normal operating point unlessanother operating point is agreed upon.

(7)

Equation 7 is calculated for each impeller of the rotor. QA is equal to the sum of qa for all impellers.


(8)

Equation 8 is calculated for each stage of the rotor. QA is equal to the sum of qa for all stages.

SP6.8.5.6 An analysis shall be performed with a varying amount of cross-coupling introduced at the rotor mid-spanfor between bearing rotors or at the center of gravity of the stage or impeller for single overhung rotors. For doubleoverhung rotors, the cross-coupling shall be placed at each stage or impeller concurrently and should reflect the ratioof the anticipated cross-coupling, (qa, calculated for each impeller or stage).

SP6.8.5.7 The applied cross-coupling shall extend from zero to the minimum of:

a) A level equal to 10 times the anticipated cross-coupling, QA.

b) The amount of the applied cross-coupling required to produce a zero log decrement, Q0. This value can bereached by extrapolation or linear interpolation between two adjacent points on the curve.

SP6.8.5.8 Level I Screening Criteria:


If any of the following criteria apply, a Level II stability analysis shall be performed:

i. Q0 / QA < 2.0.

ii. A < 0.1.

iii. Q0 / QA < 10 and the point defined by CSR and the average density at the normal operating point is located in Region B of Figure SP-9.

Otherwise, the stability is acceptable and no further analyses are required.


If A < 0.1, a Level II stability analysis shall be performed. Otherwise, the stability is acceptable and no further

analyses are required.

SP6.8.6 Level II Stability Analysis

SP6.8.6.1 A Level II analysis, which reflects the actual dynamic forces (both stabilizing and destabilizing) of therotor, shall be performed as required by SP6.8.5.8.

qa

HP BcCDcHcNc

----------------------d

s

----- =

qa

HP BtCDtHtNt

----------------------=



SP6.8.6.2 The Level II analysis shall include the dynamic effects from all sources that contribute to the overallstability of the rotating assembly. These dynamic effects shall replace the anticipated cross-coupling, QA. Thefollowing sources shall be considered:

a) labyrinth seals;

b) damper seals;

c) impeller/blade flow aerodynamic effects;

d) internal friction.

SP6.8.6.2.1 The vendor shall state how the sources are handled in the analysis.

NOTE It is recognized that methods may not be available at present to accurately model the destabilizing effects from allsources listed above.

SP6.8.6.3 The Level II analysis shall be calculated at Nmc.

SP6.8.6.4 The operating conditions defined for the normal operating point shall be extrapolated to Nmc.

Figure SP-9—Level I Screening Criteria

0 20(1.25)

1.0

1.5

2.0

CS

R

Region A

Region B

3.0

2.5

3.5

40(2.5)

60(3.75)

80(5.0)

100(6.25)




NOTE Extrapolated conditions should not fall outside the operating limits (the defined operating map) of the equipment trainsuch as horsepower, discharge pressure, etc.

SP6.8.6.5 The modeling requirements of Level I shall also apply.

SP6.8.6.6 The dynamic coefficients of the labyrinth seals shall be calculated at minimum seal running clearance.

SP6.8.6.7 When calculating the dynamic coefficients of damper seals, the running clearance profile range, which isdetermined by drawing dimensions, manufacturing tolerances and deformations in the seal, seal support and rotor,shall be included.

SP6.8.6.8 The frequency and log decrement of the first forward damped mode shall be calculated progressively forthe following configurations (except for double overhung machines where the first two forward modes shall beconsidered).

a) Rotor and support system only (basic log decrement, b).

b) Each source from SP6.8.6.2 utilized in the analysis.

c) For damper seals, the dependence due to parameters defined in SP6.8.6.7.

d) Complete model including all sources (final log decrement, f).

SP6.8.6.9 Acceptance criteria:

The Level II stability analysis shall indicate that the machine, as calculated in SP6.8.6.1 through SP6.8.6.8, shall havea final log decrement, f, greater than 0.1.

SP6.8.6.10 If after all practical design efforts have been exhausted to achieve the requirements of SP6.8.6.9,acceptable levels of the log decrement, f, shall be agreed upon.

NOTE It should be recognized that other analysis methods and continuously updated acceptance criteria have been usedsuccessfully since the mid-1970s to evaluate rotordynamic stability. The historical data accumulated by machinery manufacturersfor successfully operated machines can conflict with the acceptance criteria of this specification. If such a conflict exists and thevendors can demonstrate that their stability analysis methods and acceptance criteria predict a stable rotor, then the vendor's’criteria should be the guiding principle in the determination of acceptability.

SP6.8.7 Torsional Analysis

SP6.8.7.1 For trains including motors, generators, positive displacement units or gears, the vendor having unitresponsibility shall ensure that a torsional vibration analysis of the complete coupled train is carried out and shall beresponsible for directing any modifications necessary to meet the requirements of SP6.8.7.3 through SP6.8.7.7.

SP6.8.7.2 If specified for direct driven turbine trains, the vendor shall perform a torsional vibration analysis of thecomplete coupled train and shall be responsible for directing any modifications necessary to meet the requirements ofSP6.8.7.3 through SP6.8.7.7.

SP6.8.7.3 For trains covered in SP6.8.7.2, a torsional analysis employing a simplified model (lumped rotor inertiaand stiffness) and 1X excitation is sufficient.

NOTE The intent of the simplified analysis is to calculate the primary (coupling) modes of the system. Primary modes are thoseinfluenced primarily by the coupling torsional stiffness.

●



SP6.8.7.4 Excitation of torsional natural frequencies may come from many sources and should be considered in theanalysis. These sources shall include but are not limited to the following:

a) gear characteristics such as unbalance, pitch line runout, and cumulative pitch error;

b) torsional pulsations due to gear radial vibrations;

c) cyclic process impulses;

d) torsional excitation resulting from electric motors, variable frequency drives and reciprocating and rotary typepositive displacement machines;

e) one and two times electrical line frequency;

f) one and two times operating speed(s).

SP6.8.7.5 Primary (coupling) modes shall be at least 10 % above or 10 % below any 1X excitation frequency(mechanical or electrical) within the specified operating-speed range.

SP6.8.7.6 All other torsional natural frequencies shall preferably be at least 10 % above or 10 % below any possibleexcitation frequency within the specified operating-speed range (from minimum to maximum continuous speed).

SP6.8.7.6.1 Any interference resulting from SP6.8.7.6 shall be shown to have no adverse effect using SP6.8.7.7.

SP6.8.7.7 When torsional resonances are calculated to fall within the margin specified in SP6.8.7.6 (and thepurchaser and the vendor have agreed that all efforts to remove the critical from within the limiting frequency rangehave been exhausted), a steady state stress analysis shall be performed to demonstrate that the resonances have noadverse effect on the complete train.

SP6.8.7.7.1 The analysis shall show that all shaft sections, couplings and gear mesh have infinite life using anagreed upon criteria.

SP6.8.7.8 In addition to the torsional analyses required in SP6.8.7.3 through SP6.8.7.7, the vendor shall perform atransient torsional vibration analysis for synchronous motor driven units, using a time-transient analysis. Therequirements of SP6.8.7.8.1 through SP6.8.7.8.4 shall be followed.

SP6.8.7.8.1 In addition to the parameters used to perform the torsional analysis specified in SP6.8.7.4, the followingshall be included.

a) Motor average torque, as well as pulsating torque (direct and quadrature axis) vs. speed characteristics.

b) Load torque vs. speed characteristics.

c) Electrical system characteristics affecting the motor terminal voltage or the assumptions made concerning theterminal voltage including the method of starting, such as across the line or some method of reduced voltagestarting.

SP6.8.7.8.2 The analysis shall generate the maximum torque as well as a torque vs. time history for each of theshafts in the compressor train.

a) The maximum torques shall be used to evaluate the peak torque capability of coupling components, gearing, andinterference fits of components such as coupling hubs.



b) The torque vs. time history shall be used to develop a cumulative damage fatigue analysis of shafting, keys, andcoupling components.

c) An appropriate cumulative fatigue algorithm shall be used to develop a value for the safe number of starts.

d) The required number of starts shall be agreed.

NOTE The number of starts depends on the analytical model used and the vendor’s experience. Values of 1000–1500 starts arecommon. API Std 541 requires 5000 starts. This is a reasonable assumption for a motor since it does not add significant effort tothe design. The driven equipment, however, would be designed with overkill to meet this requirement. For example, 20-year life,1 start/week = 1040 starts. Equipment of this type normally would start once every few years rather than once per week.

SP6.8.7.8.3 Appropriate fatigue properties and stress concentrations shall be used.

SP6.8.7.8.4 An appropriate cumulative fatigue algorithm shall be used to develop a value for the safe number ofstarts. The safe number of starts shall be as agreed.

SP6.8.7.9 For VFD driven equipment trains, the vendor shall extend the analysis defined in SP6.8.7.3 to SP6.8.7.7as required by SP6.8.7.9.1 to SP6.8.7.9.4.

SP6.8.7.9.1 In addition to the excitations of SP6.8.7.4, the following shall also be considered but is not limited to:

a) integer orders of the drive output frequency;

b) sidebands of the pulse width modulation.

NOTE VFD produced broad band noise floor and feedback generated excitations can cause harmful torsional pulsations.Transient and/or mechanical/electrical coupled analyses can be required to understand the effects of these excitations.

SP6.8.7.9.2 A steady state response analysis shall be performed from zero to maximum continuous speed toquantify the effects of the VFD excitation of SP6.8.7.9.1.

SP6.8.7.9.3 For interferences occurring below the minimum operating speed, an agreed upon criteria shall be usedto establish acceptability of the train.

SP6.8.7.9.4 For interferences occurring within the operating-speed range, the criteria set forth in SP6.8.7.7.1 shallbe used.

SP6.8.7.10 If specified, for motor-driven equipment and trains including an electrical generator, a transient shortcircuit fault analysis shall be performed in accordance with SP6.8.7.10.1 and SP6.8.7.10.2.

SP6.8.7.10.1 The following faults shall be considered but is not limited to:

— Short Circuits:

a) line-to-line,

b) two phase,

c) three phase,

d) line-to-ground,

●



e) line-to-line-to-ground;

— Synchronization (generators):

a) single phase,

b) three phase.

SP6.8.7.10.2 For these fault conditions, generated stresses in the shafting and couplings shall not exceed the lowcycle fatigue limit.

NOTE Due to the timing of the torsional analysis, the vendor and purchaser should hold discussions concerning the impact ofany design changes needed to meet the specification. The analysis for these fault conditions assumes a one time event. Somecomponents as identified by the analysis may need to be replaced following the fault event.

SP6.8.7.11 If specified, alternating torques produced by breaker reclosure shall be shown to have no negativeimpact on the intended operating life of the equipment train.

SP6.8.8 Vibration and Balancing

SP6.8.8.1 Major parts of the rotating element, such as the shaft, balancing drum and impellers, shall be individuallydynamically balanced before assembly, to ISO 1940 Grade G0.67 or better. When a bare shaft with a single keywayis dynamically balanced, the keyway shall be filled with a fully crowned half key, in accordance with ISO 8821.Keyways 180 degrees apart, but not in the same transverse plane, shall also be filled. The initial balance correction tothe bare shaft shall be recorded. The components to be mounted on the shaft (impellers, balance drum, etc.), shallalso be balanced in accordance with the “half-key-convention,” as described in ISO 8821.

SP6.8.8.2 Unless otherwise specified, the rotating element shall be sequentially multiplane dynamically balancedduring assembly. This shall be accomplished after the addition of no more than two major components. Balancingcorrection shall only be applied to the elements added. Minor correction of other components may be required duringthe final trim balancing of the completely assembled element. In the sequential balancing process, any half keys usedin the balancing of the bare shaft (see SP6.8.8.1) shall continue to be used until they are replaced with the final keyand mating element. On rotors with single keyways, the keyway shall be filled with a fully crowned half-key. Theweight of all half-keys used during final balancing of the assembled element shall be recorded on the residualunbalance worksheet (see appropriate annex). The maximum allowable residual unbalance per plane (journal) shallbe calculated as follows:

In SI units:

Umax = 6350 W/Nmc for Nmc < 25,000 RPM (9a-1)

Umax = W/3.937 for Nmc ≥ 25,000 RPM (9a-2)


Umax = 4 W/Nmc for Nmc < 25,000 RPM (9b-1)

Umax = W/6250 for Nmc ≥ 25,000 RPM (9b-2)

●



where

Umax is maximum allowable residual unbalance, g-mm (oz-in.);

Nmc is maximum continuous operating speed, rpm;

W is journal static load in kg (lbm);

NOTE Above 25,000 RPM, the unbalance limit is based on 0.254 m (10 in.) mass displacement, which is in generalagreement with the capabilities of conventional balance machines, and is necessary to invoke for small rotors running at highspeeds.

SP6.8.8.2.1 When the vendors standard assembly procedures require the rotating element to be disassembledafter final balance to allow compressor assembly (i.e. stacked rotors with solid diaphragms and compressor/expanders), the vendor shall, as a minimum, perform the following operations.

a) To ensure the rotor has been assembled concentrically, the vendor shall take axial and/or radial runout readingson the tip of each element (impeller or disc) and at the shaft adjacent to each element when possible. The runouton any element shall not exceed a value agreed upon.

b) The vendor shall balance the rotor to the limits of SP6.8.8.2, Equation 9a or Equation 9b.

c) The vendor shall provide historic unbalance data readings of the change in balance due to disassembly andreassembly. This change in unbalance shall not exceed 4 times the sensitivity of the balance machine. For thispurpose, balance machine sensitivity is 0.254 m (10 in.) maximum.

d) The vendor shall conduct an analysis in accordance with SP6.8.2, to predict the vibration level during testing,using an unbalance equal to that in item b, plus 2 times the average change in balance due to disassembly andreassembly as defined in item c. The results of this analysis shall show that the predicted vibration at designspeed on test shall be no greater than 2 times the requirements of SP6.8.8.8.

NOTE Trim balancing may be done.

SP6.8.8.2.2 If specified, the vendor shall record the balance readings after initial balance for the contract rotor. Therotor shall then be disassembled and reassembled. The rotor shall be check balanced after reassembly to determinethe change in balance due to disassembly and reassembly. This change in balance shall not exceed that defined inSP6.8.8.2.1.c.

SP6.8.8.3 The following options are available concerning operating-speed balancing.

SP6.8.8.3.1 If specified, after low-speed sequential balancing, the rotor shall be operating-speed balanced inaccordance with SP6.8.8.4.

SP6.8.8.3.2 If specified or with purchaser’s approval, completely assembled rotating elements shall be subject tooperating-speed balancing (in accordance with SP6.8.8.4) in lieu of sequential low-speed balancing.

SP6.8.8.4 Operating-speed Balancing Procedure.

SP6.8.8.4.1 The following information shall be provided, prior to high-speed balancing:

a) the contract rotor dynamics analysis;

b) final low-speed balance records;

c) mechanical radial and axial runout checks of the rotor;

●

●

●



d) job and balance stand bearing details.

SP6.8.8.4.2 The rotor shall be supported in bearings of the same type and with similar dynamic characteristics asthose in which it will be supported in service.

NOTE 1 Job bearings can be used when practical.

NOTE 2 Evacuated tilting pad bearing need temporary end seals.

SP6.8.8.4.3 The rotor shall be completely assembled including thrust collars with locking collars and any auxiliaryequipment, such as power take-off gears, overspeed trip assemblies and tachometer rings for governor overspeedswitches. Shaft end seals are not added.

SP6.8.8.4.4 The high-speed drive assembly shall be shown to have an effect less than 25 % of the balancetolerance.

NOTE In most cases, the facility drive coupling and adapter is adequate to simulate the job coupling half moment. In somecases, the job-coupling hub with moment simulator may be required, especially for the outboard ends of drive-through machines.

SP6.8.8.4.5 If specified, two (2) orthogonally mounted radial noncontacting vibration probes shall be mounted nextto the bearings and at mid-shaft.

SP6.8.8.4.6 When noncontacting vibration probes have been specified, structural resonance frequency of theprobes and supports shall be determined after installation of the rotor and probe assemblies in the balance machinewhen nonstandard mounting is used (i.e. cantilevered probe holders).

SP6.8.8.4.7 The smallest available pedestal rated for the rotor weight shall be used without pedestal stiffeningengaged.

NOTE Light rotors used with larger pedestals could require a reduction of the rotor balance criteria.

SP6.8.8.4.8 Prior to operating-speed balance, the complete rotor shall be balance checked at low speed in theoperating-speed facility. If the measured unbalance exceeds five times the maximum allowable residual unbalance forthe rotor, then the cause of the unbalance shall be identified prior to operating-speed balancing.

NOTE The purpose of identifying the unbalance is to increase the likelihood of the rotor successfully traversing its criticalspeed(s) and a successful balance.

SP6.8.8.4.9 Prior to balancing, the rotor residual unbalance shall be stabilized. This shall be accomplished by:

a) record low-speed residual unbalance (amount and phase) before running up in speed;

b) run rotor to a speed equal to trip speed plus 4 % of MCS, hold for 3 minutes;

c) reduce to maximum continuous operating speed and record unbalance readings for each pedestal;

d) reduce speed and record low-speed unbalance again;

e) repeat until readings taken in SP6.8.8.4.10.c and d are consistent.

SP6.8.8.4.10 Field accessible balance holes shall not be used for balance corrections.

SP6.8.8.4.11 Balance weights if used shall be compatible with disk material and suitable for the operatingenvironment.

●



SP6.8.8.4.12 After the rotor is balanced within the tolerances of SP6.8.8.5 repeat the final balance run with thepedestal stiffening engaged.

SP6.8.8.4.13 Upon completion of the balancing, Bode and polar plots for each pedestal velocity and noncontactingprobe (when used) shall be provided for the initial run, stabilized rotor prior to balancing, and final balanced rotor withand without pedestal stiffening. Noncontacting probe data shall be compensated for slow roll mechanical andelectrical runout.

SP6.8.8.5 The acceptance criteria will be agreed upon.

NOTE The criteria are typically based on the operating-speed balance provider’s experience and can be expressed in pedestalvibration, pedestal force or residual unbalance.

SP6.8.8.5.1 When noncontacting vibration probes have been specified in SP6.8.8.4.5, the acceptance criteria forthe readings shall be agreed.

SP6.8.8.6 Unless otherwise specified, a rotor that has been operating-speed balanced shall have the unbalancerecorded in a low-speed balance machine. No corrections shall be made to the rotor. A permanent mark or part (suchas a keyway) shall be used and recorded for the phase reference.

NOTE 1 This is for future reference if a low-speed balance check is performed on the rotor before installation.

NOTE 2 The operating-speed balanced rotor will generally not meet the low-speed balance criteria.

SP6.8.8.7 For a rotor that has been low-speed sequentially balanced (see SP6.8.8.2) and will not be operating-speed balanced, a low-speed residual unbalance check shall be performed in a low-speed balance machine andrecorded in accordance with the residual unbalance worksheet (see appropriate annex for residual unbalance).

SP6.8.8.8 During the mechanical running test of the machine, assembled with the balanced rotor, operating at anyspeed within the specified operating-speed range, the peak-to-peak amplitude of unfiltered vibration in any plane,measured on the shaft adjacent and relative to each radial bearing, shall not exceed the value from SP6.8.2.10equation 5 or 25.4 m (1 mil), whichever is less.

SP6.8.8.8.1 At any speed greater than the maximum continuous speed, up to and including the trip speed of thedriver, the vibration level shall not increase more than 12.7 m (0.5 mil) above the value recorded for each probe atthe maximum continuous speed prior to accelerating to trip.

SP6.8.8.9 Electrical and mechanical runout shall be determined by rotating the rotor through the full 360 degreessupported in V blocks at the journal centers. The combined runout, measured with a noncontacting vibration probe,and the mechanical runout, measured with dial indicators at the centerline of each probe location, shall becontinuously recorded during the rotation. Teflon shall not be used in the V blocks.

NOTE The rotor runout determined above generally may not be reproduced when the rotor is installed in a machine withhydrodynamic bearings. This is due to pad orientation on tilt pad bearings and effect of lubrication in all journal bearings.

SP6.8.8.10 Records of electrical and mechanical runout for the full 360 degrees at each probe location shall beincluded in the mechanical test report (see SP6.8.3.1.1).

SP6.8.8.11 If the vendor can demonstrate that electrical or mechanical runout is present, a maximum the level fromEquation 10 or 6.35 m (0.25 mil), whichever is greater, may be vectorially subtracted from the vibration signalmeasured during the factory test. Where shaft treatment such as metallized aluminum bands have been applied toreduce electrical runout, surface variations (noise) may cause a high frequency noise component which does nothave an applicable vector. The nature of the noise is always additive. In this case, the noise shall be mathematicallysubtracted.



In SI units:

(10)

In U.S. customary units:

(10a)

Symbols

Ac1 is the amplitude at Nc1, m (mil);

Amax is the maximum probe response amplitude (p-p) considering all vibration probes, over the range of Nma to Nmc, for the unbalance amount/case being considered, m (mil);

Avl is the mechanical test vibration limit defined in SP6.8.2.10, m (mil);

AF1 is the amplification factor of the 1st critical speed defined as:

AF1 = Nc1 / (N2 – N1)

Bc = 3;

Bt = 1.5;

C = 9.55 (63);

CSR is the critical speed ratio

= Nmc / FCSR

Dc is the impeller diameter, mm (in.);

Dt is the blade pitch diameter, mm (in);

FCSR is the first undamped critical speed on rigid supports, RPM;

HC is the minimum of diffuser or impeller discharge width per impeller, mm (in.);

Ht is the effective blade height, mm (in.);

HP is the rated power per stage or impeller, Nm/sec (HP);

N is the operating speed, RPM;

Nr is the normal operating speed for calculation of aerodynamic excitation using Equation 7 or 8, RPM;

Nc1 is the rotor first critical speed, RPM;

Ncn is the rotor nth critical speed, RPM;

Rout25.4

4----------- 12,000

Nmc

------------------=

Rout14--- 12,000

Nmc

-----------------=



Nma is the minimum allowable speed, RPM;

Nmc is the maximum continuous speed, RPM;

N1 is the initial (lesser) speed at 0.707 x peak amplitude, RPM;

N2 is the final (greater) speed at 0.707 x peak amplitude, RPM;

QA is the anticipated cross-coupling for the rotor, kN/mm (klbf/in.) defined as:

Q0 is the minimum cross-coupling needed to achieve a log decrement equal to zero for either minimum or maximum component clearance, kN/mm (klbf/in.);

qA is the cross-coupling defined in Equation 7 or Equation 8 for each stage or impeller, kN/mm (klbf/in.);

Rout is the combined mechanical & electrical runout, m (mil);

S is the number of stages or impellers;

Sa1 is the actual separation for 1st critical speed, RPM;

San is the actual separation for nth critical speed, RPM;

Scc is the scale factor for close clearance check;

SM1 is the separation margin for the 1st critical speed, %

= 100 x Sa1 / Nma

SMa is the forced response analysis actual separation margin, %;

SMan is the separation margin for nth critical speed, %;

SMr is the forced response analysis required separation margin, %;

Ua is the input unbalance for the rotordynamic response analysis, g-mm (oz-in.)

= 2 x Ur

Umax is the maximum allowable residual unbalance, g-mm (oz-in.);

Ur is the maximum allowable residual unbalance, g-mm (oz-in.);

W is the journal static load in kg (lbm);

is the logarithmic decrement;

A is the minimum log decrement at the anticipated cross-coupling for either minimum or maximum component clearance;

b is the basic log decrement of the rotor and support system only;

QA qAi

i 1=

S

=



f is the log decrement of the complete rotor support system from the Level II analysis;

ave is the average gas density across the rotor, kg/m3 (lb/ft3);

d is the discharge gas density per stage or impeller, kg/m3 (lbm/ft3);

s is the suction gas density per stage or impeller, kg/m3 (lbm/ft3).

Definitions

Support stiffness and damping is the equivalent oil film to ground complex stiffness characteristics. Pivot stiffnessshould be included in the oil film characteristics.

Structure stiffness and damping refers to the bearing housing to ground equivalent complex stiffness.

Complex stiffness is the notation for the total equivalent stiffness and damping expression, including the cross-coupled terms as required for the hydrodynamic bearing or squeeze damper oil film.

Stability analysis is the determination of the natural frequencies and the corresponding logarithmic decrements (logdecs) of the damped rotor/support system using a complex eigenvalue analysis.

Synchronous tilt pad coefficients are derived from the complex frequency dependent coefficients with thefrequency equal to the rotational speed of the shaft.

Stage refers to an individual turbine or axial compressor blade row.

Hysteresis or internal friction damping causes a phase difference between the stress and strain in any materialunder cyclic loading. This phase difference produces the characteristic hysteric loop on a stress-strain diagram andthus, a potentially destabilizing damping force.

Minimum and maximum clearances for a tilt pad bearing occurs at the maximum and minimum preload conditions,respectively. These can be calculated using the following formulas.

For minimum clearance at maximum preload:

For maximum clearance at minimum preload:

Sub-critical rotors are those with a MCS is less than the first critical speed.

Super-critical rotors are those with a MCS greater than the first critical speed.

Sub-synchronous vibration is any vibration content with a frequency less than the synchronous or 1x operatingspeed.

Preloadmax 1Bearing Radiusmin Shaft Radiusmax–

Pad Boremax Shaft Radiusmax–---------------------------------------------------------------------------------------------–=

Bearing Clearancemin Bearing Radiusmin Shaft Radiusmax–=

Preloadmin 1Bearing Radiusmax Shaft Radiusmin–

Pad Boremin Shaft Radiusmin–----------------------------------------------------------------------------------------------–=

Bearing Clearancemax Bearing Radiusmax Shaft Radiusmin–=



Super-synchronous vibration is any vibration content with a frequency higher than the synchronous or 1x operatingspeed.

Vibration content is the summation of all the various frequency components at any time period required to collectadequate information to process the spectrum content.

Compliant Seal is a seal design that allows rotor or rotor sleeve contact and possible stator element penetrationwithout excessive loss of sealing performance.

Slow Roll is less than 5 % of the normal operating speed or the minimum speed permitted by the speed control.

Annex 1.C Report Requirements for Lateral and Stability Analyses

1. Standard Lateral Analysis and Stability Report

A. Rotor Model

i. Sketch of rotor model.

ii. Clear identification of bearing, shaft end and internal seals, probe, coupling, and disc (impellers, wheels, etc.) locations.

B. Oil Film Bearings and Liquid-film Seals Data (if present)

i. Dynamic coefficients (plot or table) for minimum and maximum stiffness cases vs. speed and power.

ii. In the Level II Stability analysis, the synchronous and/or nonsynchronous coefficients when used by manufacturer.

iii. Identification of coordinate system including direction of rotation.

iv. Bearing type, length, pad arc length, diameter, minimum and maximum clearance, offset, number of pads, load geometry, preload and pivot type and geometry.

v. Bearing load and direction vs. speed and power.

vi. Oil film seal configuration, length, diameter, minimum and maximum clearance, load geometry, and seal geometry.

vii. Oil properties and operating conditions

a. Oil viscosity (two temperature data if a nonstandard ISO Grade)

b. Oil flow rate and/or inlet pressure

c. Inlet operating temperature range and permissive to start temperature

d. Oil specific gravity

e. Seal operating conditions



C. Rolling Element Bearing Data

i. Type and model number

ii. Dynamic coefficients vs. frequency and speed

iii. Bearing loads and preload

D. Bearing Pedestal Data

i. Identify parameters vs. frequency (mass, stiffness and damping)

E. Gas Film Seal Data (does not apply to dry gas seals)

i. Coefficients (when a Level 2 analysis is required) for labyrinth seals, balance piston seal, and/or center bushing seal

ii. Seal type (labyrinth, honeycomb, hole pattern, etc.)

iii. Teeth on rotor, teeth on stator or interlocking

iv. Seal minimum and maximum operating clearance

v. Presence of shunt holes and/or swirl brakes

F. Squeeze Film Dampers

i. Dynamic coefficients (plot or table) for clearance extremes vs. frequency

ii. State static position and whirl eccentricity assumptions or calculation

iii. Identification of coordinate system including direction of whirl

iv. Damper type, length, diameter, minimum and maximum clearance, centering device and end seal type

v. Stiffness values for end seals and centering device (when used)

G. Other Forces Included in the Analysis (Machine Dependent)

i. Motor stator magnetic stiffness

ii. Volute fluid dynamic forces

iii. Partial arc steam loads

iv. Gear mesh loads

NOTE Vendor should state force magnitude and basis of calculation.

H. Analysis Methods

i. List computer codes used in the analysis with a brief description of the type of code, e.g. finite element, CFD, transfer matrix, etc.



I. Undamped Critical Speed Map and Mode Shapes

i. Critical speed vs. support stiffness.

ii. Curves of the support stiffness (i.e. Kxx and Kyy for minimum and maximum stiffness).

iii. Plot, as a minimum, the first 4 critical speeds with the stiffness axis extending to “rigid and soft support” regions.

iv. Show the minimum allowable and maximum continuous speeds.

v. The map shall be displayed as shown in Figure 1.C-1.

Figure 1.C-1—Undamped Critical Speed Map

Nc4r = 56,000 RPM

Nc3r = 47,000 RPM

Nc2r = 28,000 RPM

Nc1r = 7500 RPM

Support Stiffness, b/in.

Key

Support Stiffness (N/mm)

Undamped Critical Speed Map

Und

ampe

d C

rtca

Spe

ed,R

PM

Kxx (min clr.)

104

104

105

103

102

105 106 107 108

104 105 106 107 108

109

Kxx (max clr.)

Kyy (min clr.)

Kyy (max clr.)

NMIN = 9000 RPM

MCOS = 13500 RPM



vi. Undamped mode shapes from the rigid, expected and soft support regions.

vii. For machines that do not have similar support stiffness, the critical speed map shall indicate the

specified reference bearing and its location. For each of the other bearing locations, the bearing

stiffness ratio, relative to the specified reference bearing, shall be defined.

a. The vendor can substitute mode shape plots for the undamped critical speed map and list the

undamped critical speeds and the support stiffness for each of the identified modes.

J. Unbalance Response Predictions

i. Identification of the frequency of each critical speed in the range from 0 to 150 % of Nmc.

ii. Frequency, phase and amplitude (Bode plots) at the vibration probe locations in the range 0 to 150 %

of Nmc resulting from the unbalances specified in SP6.8.2.7 and SP6.8.2.8.

a. If there are no vibration probes near a bearing centerline then the Bode plots shall be shown at the

bearing centerline.

b. Minimum allowable and maximum continuous speed shown.

iii. Tabulation of critical speeds, amplification factor, actual and required separation margin and scale

factor.

iv. Axial location, amount and phase of unbalance weights for each case.

v. Plots of amplitude and phase angle vs. speed at probe locations.

a. For min and max bearing stiffness cases.

b. Pedestal vibration amplitudes for flexible pedestals as defined in SP6.8.2.4.e.

vi. Plots of deflected rotor shape at critical speeds and Nmc for min and max bearing stiffness cases.

vii. A table of the close clearance magnitudes and locations and maximum vibration levels verifying that

SP6.8.2.11.1 has been met.

K. Stability Level 1 Analysis

i. The calculated anticipated cross-coupling, qA, (for each centrifugal impeller or axial stage), total

anticipated cross-coupling, QA, log dec and damped natural frequency at anticipated cross-coupling,

and Q0/QA.

ii. Figure 1.C-2 plot of log dec vs. cross-coupled stiffness for min and max bearing stiffness.

iii. Figure 1.C-3 plot of flexibility ratio vs. average gas density with application point(s) identified on plot.



Figure 1.C-2—Level I Stability Sensitivity Plot

Figure 1.C-3—Stability Experience Plot

Applied Cross-coupled Stiffness, Q KN/mm (Klbf/in.)

Log

Dec

QA Q00

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

04

(22.8)8

(45.7)12

(68.5)16

(91.4)

Minimum

Maximum

A

0 20(1.25)

1.0

1.5

2.0

CS

R

Average gas Density, ave kg/m3 (lbf/ft3)

Region A

Region B

3.0

2.5

3.5

40(2.5)

60(3.75)

80(5.0)

100(6.25)



L. Stability Level II Analysis

i. Description of all assumptions used in the analysis.

ii. Description of all dynamic effects included in the analysis.

iii. Value of log dec and frequency versus component addition for min and max bearing stiffness (defined in SP6.8.6.2-7).

M. Summary Sheet that identifies compliance with API requirements

2. Data Required to Perform Independent Audits of Lateral Analysis and Stability Reports (SP6.8.1.4.2)

A. All of the requirements of 1.C.1 shall be met. This requirement details additional data that must be provided in conjunction with the Standard Report or as an addendum to it.

B. Rotor model

i. Model tabulation to include rotor geometry (including delineation between stiffness and mass diameter) and external masses with weight, polar, and transverse moments of inertia.

ii. The weight, polar and transverse moments of inertia and center of gravity of the impellers, balance piston, shaft end seals, coupling(s), and any other rotating components.

iii. Shaft Material Properties (density and Young’s Modulus with temperature dependence).

iv. Axial pre-loading due to tie bolts.

v. The magnitude and direction of any additional side loads (gears forces, partial arc admission, etc.) over the full operating range.

C. Bearing and liquid-film seal

i. Data to permit independent calculation of bearing coefficients

a. Table 1.C-1 and Figure 1.C-4 and Figure 1.C-5 indicate geometry required for tilt pad bearings.

NOTE Similar dimensions are required for fixed pad bearings when used. API RP 684 can assist in thedetermination of the dimensions needed.

Table 1.C-1—Tilt Pad Bearing Dimensions and Tolerances

Dimension Nominal Tolerance(+) (-)

Journal Diameter (2 x Rj)

Pad Machined Diameter (2 x Rp)

Set Bore (2 x Rb)

Pivot Location ()

Pad Arc Length ()



Figure 1.C-4—Geometry Definitions for Tilt Pad Bearing

Figure 1.C-5—Preloaded Pad

Offset, ø /x

0.5 (Centrally pivoted)

x

X

Y

Ob

Oj

R

W

Loadbetween

pivots

R+c b

ø

Pivot

Op

Oj

cp cb

Journal

Tilting pad

Preload, m = l - cb/cp

Typical m = 0.2 to 0.6(20 % to 60 %)

Rb = R + cb

cb = assembled bearing clearance

cp = pad clearance

Rp = R + cp

cb

Rp

Rb

R



ii. Tilt pad bearing pivot material and pivot type

iii. Seal dimensional data

D. Internal seals (labyrinth, balance piston seal, wear rings, and center bushing seal)

i. Data to permit independent calculation of seal coefficients

a. Dimensional data

b. Inlet swirl ratio

c. Swirl brake type

d. Clearance assumptions

e. Shunt hole location

f. Gas conditions and properties at operating speed

Annex 1.D Report Requirements for Torsional Natural Frequency, Synchronous Torsional Response, and Transient Torsional Response Analyses

1. Standard Torsional Natural Frequency Report (for systems which comply with separation margins)

A. System Torsional Sketch

i. Sketch of torsional system.

ii. Clear identification of individual rotors and their associated inertias, and coupling locations.

B. Shaft Element Data

i. Shaft element length and diameters.

ii. Shaft material properties (material density and shear modulus of elasticity).

C. Lumped Inertia Data

i. Identify inertia magnitude and location.

ii. Identify what each lumped inertia represents.

iii. Identify inertia of each body and the total for the train.

NOTE Inertia and stiffness should be actual values and not referenced to a particular shaft.

D. Coupling Data

i. Stiffness and inertia including tolerances due to manufacturing and assembly.

ii. Description of shaft end model accounting for hub penetration.



E. Analysis Methods

i. List computer codes used in the analysis with a brief description of the type of code, e.g. finite element, Holzer, etc.

F. Torsional Natural Frequencies

i. Table of the torsional natural frequencies up to two times the highest rotor speed including separation margins.

G. Natural Frequency Mode Shapes

i. Plots for all torsional natural frequencies that are less than or equal to two times the highest rotor speed.

H. Campbell Diagram

i. Identify torsional natural frequencies.

ii. Identify operating-speed range(s) with 10 % separation margin for train components.

iii. Identify torsional excitation frequencies.

iv. A typical diagram is shown in Figure 1.D-1.

2. Standard Torsional Natural Frequency Report (For systems which do not comply with separation margin(s))

All of the items in 1 above are required in addition to the following:

A. A statement of the potential torsional excitation mechanism(s), its location, magnitude and frequency. For systems which operate at variable speeds, the excitation mechanism must be evaluated throughout the operating-speed range.

B. Damping levels used in the analysis shall be stated.

C. The peak torques for all couplings and gear mesh(s) shall be identified.

D. The calculated maximum shaft stress for each shaft shall be presented.

E. Shaft stress concentration factors applied shall be listed.

F. Statement of fatigue life acceptance criteria used and conformance.

NOTE For variable speed motor drives refer to item 3.

3. Standard Torsional Natural Frequency Report for Variable Frequency Drives (VFD)

All of the items in 1 above are required (with the addition of identifying the VFD excitation on the Campbell diagram required in 1.H) in addition the following information shall be provided:

A. Train acceleration schedule.



B. The excitation associated with the VFD throughout the entire speed range shall be identified as a percentage of rated driver torque.

C. Damping levels and notch fatigue factors used shall be stated.

D. The peak torques for all couplings and gear mesh(s) shall be identified.

E. The calculated maximum shaft stress for each shaft shall be presented.

Figure 1.D-1—Typical Campbell Diagram

Reference speed x 101, RPM0.0

First Mode = 1754 CPM

1x motor speed

1x compresso

r speed

Second mode = 3474 CPM

Third mode = 12,165 CPM

Fourth mode = 15,491 CPM

Fifth mode = 16,039 CPM

90 %

spe

ed =

160

3 R

PM

Nor

ma

spe

ed =

178

1 R

PM

110

% s

peed

= 1

959

RP

M

0.0

25.0

50.0

75.0

100.0

Tors

ona

Nat

ura

Fre

quen

cy x

102 ,

CP

M

125.0

150.0

175.0

200.0

40.0 80.0 120.0 160.0 200.0 240.0

NOTE Torsional and natural frequency interference with 1x motor speed



F. Shaft stress concentration factors applied shall be listed.

G. Statement of fatigue life acceptance criteria used and conformance.

4. Transient analysis of synchronous motor drives

The standard report associated with a conventional torsional natural frequency analysis shall be provided in accordance with items 1 or 2 above as appropriate. In addition the following shall be provided:

A. The speed torque curve for the motor identifying the mean and alternating torque shall be plotted.

NOTE For a realistic transient analysis the motor speed torque should reflect the expected starting voltage drop andvoltage recovery.

B. A load speed torque curve of the driven equipment identifying the process conditions under which the equipment is required to start shall be included.

C. Damping levels and fatigue factors used shall be stated.

D. The transient torque vs. speed for couplings, gear mesh(s) and selected shaft sections shall be plotted. Peak torques at the couplings and gear mesh shall be identified.

E. The transient stress vs. speed for selected shaft sections with high cycle fatigue (HCF) (endurance limit) and the low cycle fatigue (LCF) limits identified.

F. Results of the damage accumulation calculations as a function of one start.

G. Predicted number of starts to failure for each shaft, coupling(s) and gear mesh.

5. Transient analysis of electric motor/generator short circuit and synchronization

The standard report associated with a conventional torsional natural frequency analysis shall be provided in accordance with items 1 or 2 above as appropriate. In addition the following shall be provided:

A. The torque magnitude and frequency associated with the short circuit fault condition shall be identified.

B. The analysis shall identify the peak torques in all rotors, gear mesh(s) and couplings. The shaft stress at each of the peak torque locations shall be calculated and evaluated using criteria suitable for either high cycle or low cycle fatigue.

C. Damping levels and fatigue factors used shall be stated.

D. The transient torque vs. time for couplings, gear mesh(s) and selected shaft sections shall be plotted. Peak torques at the couplings and gear mesh shall be identified.

E. The transient stress vs. time for selected shaft sections with low cycle fatigue (LCF) identified.

F. A summary shall be included identifying that the shafting, coupling(s) and gear mesh have a finite number of fault cycles.



Annex E—Magnetic Bearings (normative)

E.1 General

NOTE The paragraph numbers used for this Annex closely parallel the paragraph numbers used in API Std 617 Part 1. This ispurely for convenience, and shall not be construed as implying any requirements.

E.1.1 Scope

This standard covers the additional minimum requirements, and modifications to the requirements presented in APIStd 617 Part 1 for machines that have active magnetic bearings.

E.2 Referenced Publications

The editions of the following standards, codes, and specifications that are in effect at the time of publication of thisspecification shall, to the extent specified herein, form a part of this standard. The applicability of changes instandards, codes, and specifications that occur after the inquiry shall be mutually agreed upon by the purchaser andthe vendor.

— EN 55011 Group 1, Class A

— EN 61000-6-2

— ISO 14839, part 1

— ISO 14839, part 3

E.3 Terms, Abbreviated Terms, and Definitions

E.3.1active magnetic bearingMeans to support a rotor, without mechanical contact, using only attractive magnetic forces based upon servofeedback technology which normally consists of sensors, electromagnets, power amplifiers, power supplies, andcontrollers.

E.3.2bearing axisA location and specific direction in which the force acts, or a rotor displacement is measured.

NOTE For horizontal axis machines, the radial bearing axes are typically at ±45 degrees from vertical.

E.3.3actuatorThe portion of the AMB system that applies a force to the rotor by converting a current into a magnetic force.

E.3.4auxiliary bearingA separate bearing system that supports the shaft when the shaft is not levitated by the AMB system, or the AMB isoverloaded. The auxiliary bearing is inactive under normal AMB operation. This bearing system may have a verylimited life, and be considered a consumable machine protection system.

Also known as a “touchdown,” “catcher,” “backup,” and/or “coastdown” bearing.



E.3.5close loop transfer functionRefer to API Std 617 Part 3.

E.3.6sensitivity functionRefer to API Std 617 Part 3.

E.3.7compensatorThe AMB controller, including any input and/or output transformations. The inputs to the compensator are the sensoroutputs. The outputs of the compensator are the current or force commands.

E.3.8AMB control systemDevice which detects and processes the sensor signal and transfers it to the power amplifier in order to regulate themagnetic attractive force to levitate the rotor

E.3.9delevitation Loss of AMB control of rotor position.

E.3.10free-free mapA plot of natural frequencies for the rotating assembly only (no bearings, no seals, etc.) as a function of operatingspeed.

E.3.11landing surface or landing sleeveThe surface on the rotating assembly which is meant to contact the auxiliary bearing surface when the rotor comesinto contact with it.

E.3.12levitationActivating the AMB system to provide currents to the bearing such that the rotor is suspended within the magneticbearing. Can be used to refer to a single axis, or the entire AMB system.

E.3.13local electronicsAny electrical components required by the magnetic bearing system that are not included in the control systemcabinet and are thus located on or near the machine skid.

E.3.14noncollocationRefers to the usual arrangement where the bearing actuator is not located in the same axial location as thecorresponding sensor.

E.3.15plantThe power amplifier(s), actuator(s), rotor/housing dynamics and sensor(s). This term is used refer to a single axis, ormultiple axes jointly.



E.3.16unbalance force rejection controlAny control scheme which is phase and frequency locked to rotor rotation and has the objective of minimizing thesynchronous (and/or integer harmonic) component of the bearing force.

E.3.17transfer functionA mathematical relationship between the input and output of a linear system in terms of frequency.

E.4 General

E.4.1 Dimensions and Units

The dimensional and unit requirements of API Std 617 Part 1 shall apply.

4.2 Statutory Requirements

The statutory requirements of API Std 617 Part 1 shall apply.

4.3 Reserved for Future Use

E.4.4 Basic Design

Active Magnetic Bearing Systems shall be in accordance with 4.4 of API Std 617 Part 1 unless otherwise agreed andwith the additional requirements as follows.

E.4.4.1 Vendor shall supply bearings suitable for the intended operating environment including any liquids that areexpected to enter the bearing system during normal operation and shut down periods.

E.4.4.2 All components shall be suitable for operation, both during shop tuning and testing and under fieldconditions.

E.4.4.3 All leads (power, sensor, speed, and temperature) shall be identified at both the stator end and theconnector end. Identification shall be durable in the intended environment and shall be able to withstand handlingassociated with installation and removal.

E.4.4.4 For expander-compressors, the design shall be such that the magnetic bearing stationary components canbe removed and replaced as a single unit (cartridge design).

E.4.5 Materials

Active Magnetic Bearing Systems shall be in accordance with 4.5 of API Std 617 Part 1 unless otherwise agreed andwith the additional requirements as follows.

E.4.5.1 Electrical insulation of stator windings shall be Class H (180 °C) as a minimum. Overall bearing assemblyshall be rated to Class F (155 °C) as a minimum.

E 4.6 Casings

Active Magnetic Bearing Systems shall be in accordance with 4.6 of API Std 617 Part 1 as applicable, unlessotherwise agreed.

E.4.7 Rotating Elements



Active Magnetic Bearing Systems shall be in accordance with 4.7 of API Std 617 Part 1 as applicable, unlessotherwise agreed, and with the additional requirements as follows.

E.4.7.1 The rotor shaft sensing areas observed by radial and axial shaft displacement sensors shall be as requiredto meet the requirements of the AMB sensor system. This requirement shall replace the relevant portions of API Std617 Part 1 4.7.2, 4.7.3, and 4.7.4.

E.4.7.2 Sufficient area must be provided on the AMB rotor segment to turn the assembled shaft assembly on abalancing machine. The total indicated runout between the surface used for balancing and the surface used by thesensors to determine rotor position shall not exceed 5 microns (0.0002 in.).

E.4.7.3 Rotor landing surfaces shall be either repairable or replaceable, without causing replacement of the entirerotor system.

E.4.8 Dynamics

Active Magnetic Bearing Systems shall be in accordance with 4.8 of API Std 617 Part 1 as applicable, and with theadditional and modified requirements as follows.

E.4.8.1.1 In the design of rotor-bearing and AMB systems, consideration shall also be given to AMB Sensor Runout.

E.4.8.1.2 Analysis requirements specified in E.4.8.2.1 through E.4.8.7 shall be reported per E.4.8.1.2.1 and E.9.

E.4.8.1.2.1 If specified, the reporting requirements identified as required for independent audit of the results shall beprovided.

E.4.8.2 Lateral Analysis

E.4.8.2.1 A Free-Free map shall be generated over the range of 0 to 150 % of Nmc and shall include, as a minimum,all modes below 3 x Nmc.

E.4.8.2.2 The rotordynamics analysis shall also include:

a) any effects of sensor-actuator noncollocation;

b) the complete transfer functions from displacement to force;

c) any negative stiffness effects.

E.4.8.2.3 The rotordynamic analyses shall be conducted with Unbalance Force Rejection Control, when provided,not active.

E.4.8.2.4 The requirements specified in API Std 617 Part 1 4.8.2.10 shall be replaced with the following.

The calculated unbalanced peak-to-peak response at each vibration probe, for each unbalance amount and case asspecified in API Std 617 Part 1 4.8.2.7 shall not exceed the smaller of Equation E.1, or 0.3 times the minimumdiametral close clearance (typically the auxiliary bearing), over the range of Nma to Nmc as shown in API Std 617 Part1, Figure 3.

In SI units:

(E.1a)

●

Avl 3 25.4 12,000Nmc

------------------ =




(E.1b)

E.4.8.2.5 The predicted dynamic forces for each analysis required in API Std 617 Part 1 4.8.2.7, or any combinationthereof leading to the limit specified in E.4.8.2.4, shall be less than an AMB vendor supplied allowable dynamic forcecapacity envelope versus frequency for the given machine. This force capacity envelope shall include an agreedfactor of safety relative to the maximum rated dynamic force, that shall be greater than or equal to 1.5.

E.4.8.2.6 If the analyses indicates that the force limit requirements can not be met, and the purchaser and vendorhave agreed that all practical design and retuning efforts have been exhausted, then acceptable dynamic force levelsshall be agreed, subject to the requirements of API Std 617 Part 1 4.8.4.

E.4.8.3 Closed Loop Transfer Function Model Verification Test

E.4.8.3.1.1 If specified, the unbalance rotor response verification test specified in API Std 617 Part 1 4.8.3 shall bereplaced with the transfer function based procedure described in E.4.8.3.2 through E.4.8.3.11 for the radial bearingaxes.

If the transfer function based approach is specified as replacing API Std 617 Part 1 4.8, the additional unbalancecases specified in API Std 617 Part 1 4.8.2.8 shall not be made.

E.4.8.3.1.2 If specified, both the unbalance rotor response verification test specified in API Std 617 Part 1 4.8.3 andthe transfer function based procedure described in E.4.8.3.2 through E.4.8.3.11 shall be performed for the radialbearing axes.

NOTE The unbalance based procedure specified in API Std 617 Part 1 4.8.3 may require disassembly of the machine for someAMB supported machinery.

E.4.8.3.1.3 The transfer function based procedure described in E.4.8.3.2 through E.4.8.3.11 shall be performed forthe axial axis.

E.4.8.3.2 Analytical closed loop transfer functions shall be calculated on an axis by axis basis as (4 lateral and/or 1axial transfer functions).

E.4.8.3.3 Transfer function measurements shall be performed as part of the mechanical running test and the resultsshall be used to verify the analytical model.

E.4.8.3.4 Closed loop transfer function measurements shall be made as described in E.4.8.3.6. Thesemeasurements shall be made at 0 RPM.

E.4.8.3.5 If specified, a second measurement as described in E.4.8.3.6 shall be made at an additional speed(s) thatshall be agreed.

NOTE Measurements at higher speeds may not be suitable for validation purposes due to the presence of immeasurable forcesfrom unbalance, aerodynamic forces, etc.

E.4.8.3.6 Closed loop transfer functions shall be measured on an axis by axis basis (4 lateral and/or 1 axial transferfunctions). The excitation shall be added to the plant input as shown in Figure E1. The response shall be measured atthe output of the compensator as shown in Figure E1. The closed loop transfer function shall be computed as Cmd/Exc. The required measurements and calculations may be performed externally or internally by the AMB controlsystem using sine excitation.

Avl 3 12,000Nmc

------------------ =

●

●

●



E.4.8.3.7 If specified, the open loop transfer functions shall be measured on an axis by axis basis (4 lateral and/or 1axial transfer functions). The excitation shall be applied to the plant input (Exc) as shown in Figure E1.

E.4.8.3.8 If specified, the lateral transfer function measurements required by E.4.8.3.6 and/or E.4.8.3.7 shall bemade for all 16 lateral transfer functions (each of 4 inputs to each of 4 outputs).

E.4.8.3.9 The results of the transfer function measurements made in E.4.8.3.6 shall be compared with those fromthe analytical model specified in E.4.8.3.2. using the criteria specified in E.4.8.3.10.

E.4.8.3.10 The frequency of radial resonance peaks from the closed loop transfer function up to 1.25 x Nmc, shall notdeviate from the corresponding frequency predicted by the analysis by more than ±5 %, and the measured peakamplitudes must not be greater than 1.0 times, nor less than 0.5 times the predicted amplitudes.

E.4.8.3.11 The frequency of axial resonance peaks from the closed loop transfer function up to 1.25 x Nmc, shall notdeviate from the corresponding frequency predicted by the analysis by more than ±10 %.

E.4.8.3.12 After correcting the model, if required, the response amplitudes shall be checked against the limitsspecified in API Std 617 Part 1 4.8.2.10 and API Std 617 Part 1 4.8.2.11, as well as E.4.8.2.5. The requirements ofE.4.8.5.6 and/or E.4.8.6.4 and E.4.8.7.3 as applicable shall also be checked.

E.4.8.3.13 Unless otherwise specified, the verification test of the rotor unbalance shall be performed only on the firstrotor tested, if multiple identical rotors are purchased.

E.4.8.4 Additional Testing

E.4.8.4.1 Additional testing as described in API Std 617 Part 1 4.8.4 is required if any of the requirements ofE.4.8.3.10 can not be met, and the purchaser and vendor have agreed that all practical design and AMB retuningefforts have been exhausted.

E.4.8.5 Level I Stability Analysis

E.4.8.5.1 A Level I Stability analysis as described in API Std 617 Part 1 4.8.5 shall be performed on all AMBsupported compressors, and with the following additional requirements.

E.4.8.5.2 Level I Screening Criteria

a) All modes up to 2 x Nmc shall be considered.

b) The requirements of API Std 617 Part 1 4.8.5.8 shall apply to all modes below Nmc.

c) For modes above 1.25 Nmc, a Level II analysis shall be performed if A < 0.0.

d) For modes between Nmc and 1.25 Nmc, a Level II analysis shall be performed if A < min allowable given by EquationE.2.

(E2)

E.4.8.5.3 The sensitive function analysis described in ISO 14839-3 shall be performed. The analysis shall beperformed with two times the anticipated cross-coupling, QA as defined in API Std 617 Part 1.

E.4.8.5.4 If the peak values of the sensitivity functions computed in E.4.8.5.3 do not fall within zone A as defined inISO 14839-3, a Level II analysis shall be performed.

●

●

●

min allowable 0.5 0.4Nmode

Nmc

-----------–=



E.4.8.6 Level II Stability Analysis

E.4.8.6.1 The calculations described in API Std 617 Part 1 4.8.6.8 shall be performed for all modes up to 2 x Nmc.

E.4.8.6.2 Acceptance criteria.

The Level II stability analysis shall indicate that the machine, as calculated in E.4.8.6.8.1:

a. Has a final log decrement greater than 0.1 for all modes between 0 and Nmc;

b. Has a final log decrement greater than 0.0 for all modes greater than 1.25 Nmc;

c. Has a final log decrement greater than min allowable given by equation E.3 for any mode between Nmc and 1.25 Nmc.

(E3)

E.4.8.6.3 If after all practical design and retuning efforts have been exhausted to achieve the requirements ofE.4.8.6.2, acceptable levels of the log decrement, f, shall be agreed.

E.4.8.6.4 The sensitive function analysis described in ISO 14839-3 shall be performed as part of the Level IIanalysis. The analysis shall be performed progressively, similar to the procedure described in API Std 617 Part 14.8.6.8.

E.4.8.6.5 The peak values of the sensitivity functions computed in E.4.8.6.4 shall fall within zone A as defined in ISO14839-3.

E.4.8.6.6 If after all practical design and retuning efforts have been exhausted to achieve the requirements ofE.4.8.6.5, acceptable levels of the peak value of the sensitivity function shall be mutually agreed upon

E.4.8.7 Axial Analysis

NOTE Section E.4.8.7 presents additional axial analysis requirements that an AMB system must meet. It does not replace therequirements of API Std 617 Part 1 4.8.7.

E.4.8.7.1 The vendor having train responsibility shall ensure that an axial analysis of the complete coupled train iscarried out and shall be responsible for directing any modifications necessary to meet the requirements.

E.4.8.7.2 A simplified lumped mass model (lumped rotating component masses, stiffnesses and damping) issufficient for this analysis.

E.4.8.7.3 The axial analysis shall consider all major items that affect the axial dynamics including, but not limited to,the following.

a) Rotating components masses, stiffnesses and damping.

b) The axial AMB system.

c) Coupling masses, stiffnesses. and damping.

d) Nonrotating structural stiffness, mass, and damping characteristics as they relate to the axial actuator and sensor,including effects of excitation frequency over the required analysis range. The vendor shall state the structurecharacteristic values used in the analysis and the basis for these values (for example, modal tests of similar rotorstructure systems, or calculated structure stiffness values).

e) Disk flexibility.

min allowable 0.5 0.4Nmode

Nmc

------------–=



f) Seals.

g) Magnetic and aerodynamic centering forces and associated dynamics.

h) Eddy-current limitations on actuator bandwidth

E.4.8.7.4 The damped natural frequencies of all modes with an amplification factor greater than 2.5 shall becalculated.

E.4.8.7.5 The sensitivity function analysis described in ISO 14839-3 shall be performed for the axial system from 0to 1000 Hz.

E.4.8.7.6 The analytical sensitivity functions for rotor dominated shall fall within zone B or better as defined in ISO14839-3. Modes dominated by motion in a flexible coupling may be excluded from this requirement.

E.4.8.8 Vibration and Balancing

Active Magnetic Bearing Systems shall be in accordance with 4.8.8 of API Std 617 Part 1 as applicable, unlessotherwise agreed, and with the additional requirements as follows

E.4.8.8.1 During the mechanical running test of the machine, assembled with the balanced rotor, operating at anyspeed within the specified operating-speed range, the peak-to-peak amplitude of unfiltered vibration in any plane,measured on the shaft adjacent and relative to each radial bearing, shall not exceed the smaller of Equation E.4, or0.3 times the minimum diametral close clearance (typically the auxiliary bearing), over the range of Nma to Nmc asshown in API Std 617 Part 1, Figure 3.

In SI units:

(E4a)


(E4b)

This paragraph replaces the requirements of API Std 617 Part 1 4.8.8.8.

E.4.9 Bearings and Housings

E.4.9.1 General

Radial and axial bearings shall be as specified in the subsequent parts of this specification.

E.4.9.2 Radial Magnetic Bearing System

E.4.9.2.1 The load capacity of the radial bearings shall be designed with sufficient force capability to prevent contactbetween the rotor and any portion of the stator (including the auxiliary bearings) at all speeds from zero to trip atexpected operating conditions.

NOTE This is Expander-Compressor specific, put in Appendix E for Chapter 4.

Avl 3 25.4 12,000Nmc

------------------ =

Avl 3 12,000Nmc

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E.4.9.2.2 Two (2) temperature sensors shall be installed in each radial bearing. One shall be used for overtemperature protection and the other as an installed spare.

E.4.9.3 Axial Magnetic Bearing System

E.4.9.3.1 The expected residual thrust load during normal operation shall be no more than 50 % of the axialmagnetic bearing’s rated load capacity.

NOTE For Expander-Compressors, add:

For expander-compressors with automatic thrust balancing systems, the axial magnetic bearing’s rated load capacity shall be noless than 2 times the largest residual thrust expected using the automatic thrust balancing system.

E.4.9.3.2 If specified, two (2) axial position sensors shall be provided and located such that they can be used toprovide rotor to stator differential expansion information.

E.4.9.3.3 Two (2) temperature sensors shall be installed in each axial bearing. One shall be used for overtemperature protection and the other as an installed spare.

E.4.9.4 Auxiliary Bearing System

E.4.9.4.1 An auxiliary bearing system shall be provided for all machines that use active magnetic bearings.

NOTE Auxiliary bearings are considered to be a consumable machinery protective device.

E.4.9.4.2 The radial auxiliary bearing system shall include a damping mechanism.

E.4.9.4.3 Auxiliary bearing materials and lubricant(s) (if present) shall be compatible with the specified operatingenvironment (both shop testing and in the field) and shall not adversely affect adjacent components.

E.4.9.4.4 The auxiliary bearing system shall be designed to support the rotor, without allowing any contact at closeclearance locations between the rotor and stator, except at the auxiliary bearing, under all of the following conditions:

a) when the AMB system is not energized;

b) during specified transient operating events that exceed the load capacity of the AMB system;

c) during a rotor drop transient following a partial or full AMB failure;

d) during a coastdown from trip speed, under specified operating conditions, with the auxiliary bearing systemproviding the rotor support.

NOTE Rotor displacements with the full force capacity of one or more AMB actuators being applied may need to be considered.

E.4.9.4.5 For the purposes of E.4.9.4.4, floating ring, abradable and compliant seals are not considered to be closeclearance locations. The design requirements at these locations shall be agreed.

E.4.9.4.6 The vendor shall provide analytical predictions confirming that the requirements of E.4.9.4.4 are met.

E.4.9.4.7 The auxiliary bearing system shall be designed to accommodate an agreed number of momentarycontacts due to specified transient operating events that exceed the load capacity of the AMB system withoutrequiring replacement or refurbishment.

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E.4.9.4.8 The auxiliary bearing system shall be designed to accommodate an agreed number of coastdowns fromtrip speed under specified operating conditions without requiring replacement or refurbishment. This number shall notbe less than 2.

E.4.9.4.9 The vendor shall describe the basis for expecting the auxiliary bearing system to meet the designrequirements. This basis may include analytical simulations, as well as field and/or test stand data.

E.4.9.4.10 The vendor shall provide a means for confirming operability of the auxiliary bearing system withoutrequiring machine disassembly.

E.5 Other AMB Subsystems and Components

E.5.1 Monitoring and Control Systems

E.5.1.1 The AMB system shall include a control system.

E.5.1.2 The control system shall consist of an enclosure containing amplifiers, control electronics, and otherequipment necessary for the operation and safety of all magnetic bearings. The control system shall provide alarmand shutdown protective logic for the magnetic bearings, auxiliary bearings, and control cabinet.

E.5.1.3 If specified, an electronic digital communications link(s) shall be provided for connection to purchaser’ssystems. The format(s) and data to be provided shall be agreed.

E.5.1.4 The magnetic bearing control system shall have the capability of moving the rotor both radially and axially inorder to check auxiliary bearing clearances. This check shall be possible with the unit in service but not rotating (i.e.disassembly shall not be required to perform this check).

E.5.1.5 The magnetic bearing control system shall have the capability to record and display the number of largeshaft excursions during machine operation. Consideration shall be given to both operating speed and magnitude ofthe excursion in determining if an excursion is counted.

E.5.1.5.1 The count specified in E.5.1.5 shall be provided as a total since installation, and as a resettable count forthe installed set of auxiliary bearings.

E.5.1.6 The control system shall not emit or be receptive to EMF signals, and shall comply with standards EN 55011Group 1 Class A and EN 61000-6-2.

E.5.1.7 Unless otherwise specified, the control system enclosure shall be designed for bottom entry wiring, andshall be suitable for the area classification and location specified.

E.5.1.8 A means shall be proved for cooling control system components as required.

E.5.1.8.1 If air cooling is used, the control cabinet shall be provided with multiple cooling fans. Failure of a single fanshall not cause over-temperature shutdown to occur.

E.5.1.8.2 If water cooling is used, provision must be made to prevent problems from condensation.

E.5.1.9 Local electronics, if required, shall be provided. Local electronics to be suitable for specified hazardous areaand for specified ambient temperature and humidity range and vibration environment.

E.5.1.10 Vendor shall provide EMF filters on the control cabinet power supply, if required, to avoid contamination ofinput power by magnetic bearing power amplifiers.

E.5.1.11 Vendor’s standard Man Machine Interface, if required, shall be provided. English language shall be used.

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E.5.1.12 A system which will provide power to the AMB system for a minimum required levitation time following lossof normal electric power supply shall be provided. The required levitation time shall be agreed.

E.5.1.13.1 The vendor’s standard uninterruptable power supply/battery backup system shall be provided if acustomer supplied system is not utilized.

E.5.2 Cabling

E.5.2.1 The vendor shall specify cabling requirements for the bearing power and sensor connections. Any electricalor electronic components required to adjust for the installed length shall be included in the vendor’s scope of supply.

NOTE On systems where the cable distance between the machine and the control cabinet are long (100 m to 300 m, or 300 ftto 1000 ft), special consideration should be given to the electrical compensation and type of cable used to insure proper operation.Electrical compensation may also be necessary for shop testing setups.

E.5.2 Rotor Position Sensors

E.5.2.1 Sensors shall be vendor’s standard design with demonstrated operating experience.

E.5.2.2 Sensor components and assembly shall be compatible with the environment to which they are exposed.

E.5.2.3 The radial position sensors shall be located as close to the radial magnetic bearing as possible wherepracticable.

E.6 Inspection, Testing, and Preparation For Shipment

E.6.1 Reserved for future use.

E.6.2 Reserved for future use.

E.6.3 Testing

E.6.3.1 All electronic components shall have a 24-hour burn in prior to shipment.

E.6.3.2 The dry insulation resistance of assembled bearing power coils shall be greater than 50 megohms whentested with a 500 Volt DC megohmmeter.

E.6.3.3 If specified, a “wet” bearing assembly insulation test shall be performed. Procedure and acceptance criteriashall be agreed.

E.6.3.4 The magnetic bearing control system shall be functionally tested prior to shipment. Functional test shallinclude, as a minimum, connection to a simulated load and demonstration of the system’s monitoring functions.

E.6.3.5 If specified, static load capacity tests on all new bearing designs shall be performed. Bearing measured loadcapacity shall be equal to or greater than specified requirement.

E.6.3.6 Static and dynamic tests shall be performed using cables provided by the vendor. In general, these will notbe the same cables as that used in the field. The vendor shall allow for any special tuning adjustments in the design.

NOTE 1 Users may want to consider building up the actual machine skid for the mechanical test to reduce the likelihood of fieldretuning.

NOTE 2 A rotor ring test may also be advisable prior to machine assembly to improve the accuracy of higher rotor modes in themodel.

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●



E.6.3.7 The sensitive function measurement described in ISO 14839-3 (including the axial axis) shall be performedat 0 RPM prior to operating the machine for the mechanical test.

E.6.3.8 The peaks of the radial sensitivity functions measured in E.6.3.7 shall fall within zone A as defined in ISO14839-3.

E.6.3.9 The peaks of the axial sensitivity functions measured in E.6.3.7 shall fall within zone B or better as defined inISO 14839-3.

E.6.3.10 If specified, the sensitive function measurement described in ISO 14839-3 shall be performed at additionalspeed(s). The additional measured sensitivity function peaks shall also fall within the zones specified above.

E.6.3.11 While the equipment is operating at maximum continuous speed, or other speed required by the testagenda, vibration data shall be acquired to determine amplitudes at frequencies other than synchronous. These datashall cover a frequency range from 0.25 to 8 times the maximum continuous speed. If the amplitude of any discrete,nonsynchronous vibration exceeds 50 % of the allowable vibration as defined in 4.8.8.8 of API Std 617 Part 1, thepurchaser and the vendor shall agree on requirements for any additional testing and on the equipment’s acceptability.

This requirement replaces the requirements of API Std 617 Part 2 6.3.1.2.2 and API Std 617 Part 4 6.3.3.3.2.

E.6.3.12 If specified, short-term delevitations at specified operating conditions shall be performed duringmechanical test and/or full load test. The details of this test and the procedures to be followed, including any post-testauxiliary bearing inspection(s), shall be agreed.

NOTE The intent of this paragraph is to provide an option to use short (few second) delevitations at normal operating speed,followed by relevitation and normal AMB system operation, to confirm the basic operational characteristics of the auxiliary bearingsystem. It would be appropriate to stop the machine and perform lift checks and/or inspect the auxiliary bearings between multipletests.

E.6.3.13 If specified, a coastdown from trip speed using the auxiliary bearing system, with the AMB system inactive,shall be performed during mechanical test and/or full load test. The details of this test and the procedures to befollowed, including any post-test auxiliary bearing inspection, shall be agreed.

NOTE If the mechanical test configuration coast-down time is significantly longer than the field coast-down time, this may not bea valid test of the auxiliary bearing system’s performance.

E.6.3.14 Damage to auxiliary bearing system components that did not compromise their ability to prevent contact atother close clearance locations, and did not cause secondary damage, does not constitute failure for the testsspecified by either of E.6.3.12 or E.3.6.13.

NOTE Auxiliary bearings are considered to be a consumable machinery protective device. The AMB supplier should beconsulted for guidance regarding auxiliary bearing reuse, refurbishment or replacement following these tests.

E.7 Reserved for Future Use

E.8 Field, As-installed Analyses

NOTE Due to unmodeled structural dynamics and/or process condition effects, it may be necessary to fine-tune the AMBsystem based on actual machine field experience during commissioning. The intent of this section is to ensure that the finalrotordynamics report reflects the installed tuning.

E.8.1 The final rotordynamics report shall include analyses performed with the AMB control system parametersused in the AMB system at the conclusion of initial commissioning.

E.8.2 The report shall indicate if the rotordynamic analyses with the as-tuned parameters meet the applicablerequirements.

●

●

●



The analyses to be updated with the as-tuned parameters and re-evaluated shall not include those related to theunbalance rotor response verification test described in API Std 617 Part 1 4.8.3 and/or the transfer function basedprocedure described in E.4.8.3.2 through E.4.8.3.11.

E.9 Reporting Requirements For Lateral and Stability Analysis

Active Magnetic Bearing Systems shall be in accordance with Annex C of API Std 617 Part 1 as applicable, and withthe additional requirements as follows:

E.9.1 Standard Lateral Analysis and Stability Report

E.9.1.1 AMB Data

a) General actuator and rotor component dimensions.

b) Auxiliary bearing gap(s) at 0 RPM and over the operating-speed range.

c) Plot of allowable force versus frequency envelope, and identification of factor of safety assumed.

d) Identification of actuator coordinate system (generally ±45 degrees and axial).

e) Plots of AMB system displacement to force transfer functions that were used for the analytical rotordynamicpredictions as amplitude and phase versus frequency (Bode plots). These shall be in physical (sensor-actuator)coordinates. Plots for cross-coupling between the various axes shall be provided if there is significant coupling.

E.9.1.2 Undamped Results

Plot, as a minimum, a free-free map over the range of 0 to 150 % of Nmc and including all modes below 3 x Nmc.

E.9.1.3 Unbalance Response Predictions

Plot predicted bearing forces relative to the allowable force envelope versus speed for each unbalance responsecase.

E.9.1.4 Level I Stability Analysis

a) Plots of predicted ISO 14839 sensitivity functions with 2 x QA applied.

b) The Level I stability plot specified in API Std 617 Part 1 Annex C, Figure C.2, augmented to show all modesrequired to be analyzed.

E.9.1.5 Level II Stability Analysis

a) Values of all log decrements for all modes required to be analyzed.

b) The peak values of the ISO 14839-3 sensitivity functions for each component, and the peak frequencies.

c) Plot(s) of the ISO 14839-3 sensitivity functions versus frequency with all components applied.

E.9.1.6 Axial Analysis

a) List of axial natural frequency and brief description or modeshape plot.

b) Plot of ISO 14839-3 sensitivity function for axial axis.



c) Peak value of sensitivity function plot.

E.9.1.7 Additional Plots

Predicted closed loop transfer functions, with resonance peak frequencies and values annotated, if the optionalclosed loop verification test has been specified.

E.9.1.8 Auxiliary Bearings

Analytic results specified in E.4.9.4.6 confirming no contact at close clearance locations.

E.9.2 Data Required to Perform Independent Audits of Lateral Analysis and Stability Reports

E.9.2.1 AMB Control/Actuator System Data to Permit Independent Analysis

a) Axial locations and angular orientations of sensors and actuators.

b) Coefficients for all required displacement to force transfer functions for lateral and axial analyses over a frequencyrange adequate to perform the required analyses. The force displacement functions include the effects of alldynamic systems such as sensor, compensator, amplifier, sample rate, computational delays, eddy currenteffects, etc. These shall be provided for a coordinate system corresponding to the physical orientations of thesensors and actuators, not a transformed system (such as tilt-translate). If agreed, they may be supplied as firstorder system matrices.

c) Description of any significant variations in specific transfer function coefficients due to gain scheduling or othereffects. A series of overall transfer functions for different speed ranges is also acceptable.

d) Actuator negative magnetic stiffness, and where applied, if required for model.

E.9.2.2 AMB Force Envelope

a) Force versus frequency operating envelope data in tabular format with brief explanation of what specific issueswere considered.

b) Data to permit check on reasonableness of operating force envelope. This might include, for example:

1) actuator nominal inductances;

2) nominal bias currents;

3) nominal power amplifier supply voltage;

4) nominal actuator force/current at 0 RPM for nominal radial load;

5) actuator/rotor materials and pole face area.

E.9.2.3 Auxiliary Bearings

a) Axial locations.

b) Type (i.e. angular contact, bushing), including dimensions, rotor/auxiliary bearing gap, coefficient of friction,geometry and other pertinent information.

c) Force versus deflection curve, including the effects of hard stop if present.

d) Damping provided by the auxiliary bearing and/or compliant mount system.


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Product No. C684101


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