m. rades - dynamics of machinery 1

Dynamics of Machinery I

Mircea Rade Universitatea Politehnica Bucureti

2007

Preface

This textbook is based on the first part of the Dynamics of Machinery lecture course given since 1993 to students of the English Stream in the Department of Engineering Sciences (D.E.S.), now F.I.L.S., at the University Politehnica of Bucharest. It grew in time from a postgraduate course taught in Romanian between 1985 and 1990 at the Strength of Materials Chair.

Dynamics of Machinery, as a stand alone subject, was first introduced in the curricula of mechanical engineering at D.E.S. in 1993. To sustain it, we published Dynamics of Machinery in 1995, followed by Dinamica sistemelor rotor-lagre in 1996 and Rotating Machinery in 2003.

As seen from the Table of Contents, this book is application oriented and limited to what can be taught in an one-semester (28 hours) lecture course. It also contains many exercises to support the tutorial, where the students are guided to write simple finite element computer programs in Matlab, and to assist them in solving problems as homework.

The course aims to: (a) increase the knowledge of machinery vibrations; (b) further the understanding of dynamic phenomena in machines; (c) provide the necessary physical basis for the development of engineering solutions to machinery problems; and (d) make the students familiar with machine condition monitoring techniques and fault diagnosis.

As a course taught for non-native speakers, it has been considered useful to reproduce, as language patterns, some sentences from English texts.

Finite element modeling of rotor-bearing systems and hydrodynamic bearings are treated in the second part. Analysis of rolling element bearings, machine condition monitoring and fault diagnosis, balancing of rotors as well as elements of the dynamic analysis of reciprocating machines are presented in the third part. No reference is made to the vibration of discs, impellers and blades.

August 2007 Mircea Rade

Prefa

Lucrarea se bazeaz pe prima parte a cursului de Dinamica mainilor predat din 1993 studenilor Filierei Engleze a Facultii de Inginerie n Limbi Strine (F.I.L.S.) la Universitatea Politehnica Bucureti. Coninutul cursului s-a lrgit n timp, pornind de la un curs postuniversitar organizat ntre 1985 i 1990 n cadrul Catedrei de Rezistena materialelor.

Dinamica mainilor a fost introdus n planul de nvmnt al F.I.L.S. n 1993. Pentru a susine cursul, am publicat Dynamics of Machinery la U. P. B. n 1995, urmat de Dinamica sistemelor rotor-lagre n 1996 i Rotating Machinery n 2005, ultima coninnd materialul ilustrativ utilizat n cadrul cursului.

Dup cum reiese din Tabla de materii, cursul este orientat spre aplicaii inginereti, fiind limitat la ceea ce se poate preda n 28 ore. Materialul prezentat conine multe exerciii rezolvate care susin seminarul, n cadrul cruia studenii sunt ndrumai s scrie programe simple cu elemente finite n Matlab, fiind utile i la rezolvarea temelor de cas.

Cursul are un loc bine definit n planul de nvmnt, urmrind: a) descrierea fenomenelor dinamice specifice mainilor; b) modelarea sistemelor rotor-lagre i analiza acestora cu metoda elementelor finite; c) narmarea studenilor cu baza fizic necesar n rezolvarea problemelor de vibraii ale mainilor; i d) familiarizarea cu metodele de supraveghere a strii mainilor i diagnosticare a defectelor.

Fiind un curs predat unor studeni a cror limb matern nu este limba englez, au fost reproduse unele expresii i fraze din lucrri scrise de vorbitori nativi ai acestei limbi.

n partea a doua se prezint modelarea cu elemente finite a sistemelor rotor-lagre i lagrele hidrodinamice. n partea a treia se trateaz lagrele cu rulmeni, echilibrarea rotorilor, msurarea vibraiilor pentru supravegherea funcionrii mainilor i diagnosticarea defectelor, precum i elemente de dinamica mainilor cu mecanism biel-manivel. Nu se trateaz vibraiile paletelor, discurilor paletate i ale roilor centrifugale.

August 2007 Mircea Rade

Contents

Preface i

Contents iii

1. Rotor-bearing systems 1 1.1 Evolution of rotating machinery 1 1.2 Rotor-bearing dynamics 22 1.3 Rotor precession 24 1.4 Modeling the rotor 26 1.5 Evolution of rotor design philosophy 29 1.6 Historical perspective 32

2. Simple rotors in rigid bearings 39 2.1 Simple rotor models 39

2.2 Symmetric undamped rotors 40

2.2.1 Equations of motion 41

2.2.2 Steady state response 43

2.3 Damped symmetric rotors 46

2.3.1 Effect of viscous external damping 47

2.3.2 Effect of viscous internal damping 54

2.3.3 Combined external and internal damping 62

2.3.4 Gravity loading 65

2.3.5 Effect of shaft bow 66

2.3.6 Rotor precession in rigid bearings 67

2.4 Undamped asymmetric rotors 68 2.4.1 Reference frames 69

2.4.2 Inertia torques on a spinning disc 69

2.4.3 Equations of motion for elastically supported discs 72

2.4.4 Natural frequencies of precession 75

2.4.5 Response to harmonic excitation 81

2.4.6 Campbell diagrams 87

2.4.7 Effect of gyroscopic torque on critical speeds 97

2.4.8 Remarks on the precession of asymmetric rotors 98

MECHANICAL VIBRATIONS iv

3. Simple rotors in flexible bearings 101 3.1 Symmetric rotors in flexible bearings 101

3.1.1 Effect of bearing flexibility 102

3.1.2 Effect of external damping 109

3.1.3 Effect of external and internal damping 117

3.1 4 Effect of bearing damping 119

3.1.5 Combined effect of bearing damping and shaft mass 131 3.2 Symmetric rotors in fluid film bearings 136

3.2.1 Unbalance response 136

3.2.2 Stability of precession motion 142 3.3 Asymmetric rotors in flexible bearings 145 3.3.1 Equations of motion 145

3.3.2 Natural frequencies of precession 148

3.3.3 Unbalance response 152

3.3.4 Effect of bearing damping 156

3.3.5 Mixed modes of precession 158

3.4 Simulation examples 168

4. Rotor dynamic analysis 207 4.1 Undamped critical speeds 207

4.1.1 Effect of support flexibility 207

4.1.2 Critical speed map 209

4.1.3 Influence of stator inertia 217 4.2 Damped critical speeds 219

4.2.1 Linear bearing models 219

4.2.2 Equations of damped motion 220

4.2.3 Eigenvalue problem of damped rotor systems 220

4.2.4 Campbell diagrams 222

4.2.5 Orbits and precession mode shapes 223 4.3 Peak response critical speeds 224 4.4 Stability analysis 227 4.5 Simulation examples 231 4.6 Planar modes of precession 273 Index 283

1. ROTOR-BEARING SYSTEMS

The first part of the Dynamics of Machinery is devoted to rotor-bearing systems, including the effects of seals and bearing supports. The flexibilities of discs and blades are neglected, so that the Rotor Bearing Dynamics does not include the vibration analysis of impellers and bladed-disc assemblies.

1.1. Evolution of rotating machinery

Interest in the vibration of rotating machinery has been due primarily to the fact that more than 80 percent of the problems involve vibration. In the continuing effort to develop more power per kilogram of metal in a machine, designs have approached the physical limits of materials and vibration problems have increased. These, together with the extremely high cost associated with forced outages, for machines with continuous operating regime, have determined the development of research activity and design procedures in two fields of primary practical interest: the Dynamics of Rotor-Bearing Systems and the Vibrations of Bladed Disc Assemblies.

1.1.1 Steam turbines

Of significance for the technical advancement in this field is the development of steam turbines in Europe [1]. From the first single stage impulse turbine built in 1883 by the Swedish engineer Gustaf de Laval (with a speed of 30000 rpm reduced to 3000 rpm by gearing), and the first multistage reaction turbine built in 1884 by Charles Parsons (having a speed of 18000 rpm and an output of 10 HP), to the turbines of today nuclear power stations, the evolution has been spectacular.

Early in 1901 the Brown Boveri Company built a steam turbine of 250 kW at 3000 rpm, coupled directly to an a.c. generator. From 1907 onwards, a double impulse Curtis wheel (invented in 1896) was mounted before the reaction

DYNAMICS OF MACHINERY 2

stage, which was replaced by single-row versions on two to three impulse wheels. In 1914, a turbine of 25 MW at 1000 rpm was the largest single-cylinder steam turbine in the world. The first systematic studies of Rotor Dynamics started in 1916, carried out by professor Aurel Stodola at the Swiss Federal Institute of Tehnology in Zrich.

After 1920, the high price of coal imposed the increase of steam turbine efficiency. Among other means, this was achieved by the reduction in the diameter and the increase in the number of stages, hence by the increase of the shaft length, a major incentive for developing the Dynamics of Rotor-Bearing Systems.

The maximum unit output of a turbine is largely dependent on the available last-stage blade length. The permissible blade length to diameter ratio has an influence on the machine efficiency. Shafts should be as slender as possible, to ensure small rotor diameter and large blade length. Otherwise, increased shaft weight gives rise to an increase in the average specific bearing loading.

Increasing the cross-section of a machine is limited by the mechanical stresses and the size of pieces that can be transported. This is compensated by the increase of the active length, eventually with a tandem arrangement, having a long shaft line, in which the mechanical power is produced in several turbine cylinders.

The first super-pressure three-cylinder (high, intermediate and low pressure) turbine was built by BBC in 1929, and had an output of 36 MW at 3000 rpm. The steam flowed through high pressure and intermediary pressure rotors in opposite directions, to balance the thrust. Rotors, which previously were composed of keyed and shrunk-on wheels on a continuous shaft, started to be welded from solid discs, allowing larger rotor diameters and increased ratings. The increased efficiency of steam turbines lowered the amount of coal required for producing 1 kWh of electrical energy from 0.75 kg during the war to 0.45 kg in 1927. The output of the largest turbines in Europe had reached 50 to 60 MW by the mid twenties, when, for large units, turbines of 1500 rpm were coupled to four-pole generators. A 165 MW two-shaft turboset was built in 1926-1928, with the high-pressure shaft rotating at 1800 rpm, and the low-pressure shaft at 1200 rpm.

In 1948, the largest steam turboset of single-shaft design (Fig. 1.1) had four cylinders, a length of 27 m (without the station service generator), an output of 110 MW and speed of 3000 rpm [2]. In 1950, turbosets of 125 MW were built in Europe and of 230 MW in the U.S.A., then, in 1956 - with ratings of 175 MW, and in 1964 - with ratings of 550 MW and two shafts.

In 1972, the first 1300 MW cross-compound turboset was built at 3600 rpm, provided with two shaft lines for two 722 MVA generators. Figure 1.2 shows a longitudinal section of the high-pressure turbine of a 1300 MW unit at 1800 rpm.

Current designs have generators of 1635 MVA at 1500 rpm, and of 1447 MVA at 3000 rpm. At present time, turbosets of 1700-2000 MW at 1500 or 1800 rpm, and of 1500-1700 MW at 3000 or 3600 rpm are currently built.

1. ROTOR-BEARING SYSTEMS 3


Generally, the shaft line has a length of 8 to m20 in turbosets of 1 to 50 MW, between 25 and m30 in those of 100 to 150 MW, and exceeds m75 in turbosets beyond 1000 MW.

Fig. 1.2 (from [3])

The increase of the rotor length has been accompanied by the increase of the number of stages (or discs on a shaft), and the number of bearings and couplings between shafts in a line. Adding the increase of seal complexity and the problems raised by the non-uniform thermal expansion at start-up, all doubled by strength of materials problems raised by the increase in size, one can easily understand the complexity of the dynamic calculations of the rotors of such machines.

Figure 1.3 shows a typical axial section in an industrial back-pressure turbine of an early design [4]. The steam is expanded in the turbine from the live-steam pressure to the exhaust pressure in two principal parts.

In the first part, the steam is accelerated in the nozzle segments 1, thus gaining kinetic energy, which is utilized in the blades of the impulse wheel 2. The disc of the impulse stage is integral with the shaft. Usually, the nozzles are machined into several segments fixed into the cylinder by a cover ring. The blades of the impulse wheel are milled from chromium steel bars. The roots are fixed into the slot in the impulse wheel with spacers gripping the upset feet of the blades. In some designs, the flat outer ends are welded together in groups, thus forming an interrupted shroud.

The second or reaction part consists of stationary and moving rows of blades 3 fixed with suitably shaped spacers into slots in the casing and rotor.

The glands 4 prevent the steam flowing out of the casing along the shaft. Labyrinth seals allow a very small amount of steam to escape into specially


provided channels. Due to the turbulence of the steam, the pressure drop is sufficiently high to allow the gland to be made relatively short. The labyrinth strips are caulked into grooves in the rotor shaft whereas the corresponding grooves are machined into a separate bushing of the casing. The risk of damaging the rotor by distortion caused by friction in the seals is avoided, as the heat transfer from the tips of the thin labyrinth strips to the shaft is very small.

Fig. 1.3 (from [4])

The balancing piston 5 is positioned between the impulse wheel and the gland at the steam inlet end. The chamber between is interconnected with the exhaust. Generally, the balancing ring is integral with the shaft. In older designs it was shrunk-on but this design can give rise to instability due to rotating dry friction. This arrangement counteracts the axial forces imposed on the rotor by the steam flow.

The bearing 6 at the steam inlet end is a combined thrust and journal bearing, to reduce the rotor length. The thrust part of it acts in both axial directions on the thrust collars 7 to absorb any excess forces of the balancing piston. Usually tilting bronze pads are fitted on flexible steel rings according to the Mitchell principle.

The journal bearing of the combined bearing and that at the opposite end 8 are lined with white metal cast into separate shells. Tilting pad bearings are used in some designs.

The rotor 9 is machined from high-quality steel forging. After the blades are fitted, the rotor is balanced and subjected to a 20 percent overspeed test for a few minutes. A high-alloy chromium steel is used for high pressures and temperatures. Figure 1.4 shows presently used steam turbine rotor designs [5].


Turbines running at high speeds require reduction gearing to drive alternators with 2 or 4 poles, running at 3000 or 1500 rpm (for 50 Hz).

As a rule, the pinion and gear wheel shafts are connected to the driving and driven machines by means of couplings. They must be able to compensate for small errors in alignment and thermal expansion in the machine without affecting the reduction gearing. The coupling hubs are integral with the forged shafts.

Fig. 1.4 (from [5])

The first steam turbine built in Romania in 1953 at Reia, was a 3 MW at 3000 rpm turbine. In 1967, the first two-cylinder 50 MW turbine was built. Twenty years later, the 330 MW four-cylinder condensing turbine was manufactured at I.M.G. Bucureti, under a Rateau-Schneider license. Rotors have a monoblock construction, having the discs in common with the shaft. At present, General Turbo S.A. manufactures 700 MW turbines.

1.1.2 Gas turbines

The development of gas turbines is more recent. From the first gas turbine for airplanes, designed by Whittle in 1937, and the first stationary turbine built by Brown Boveri in 1939, turbines of 80 MW at 3000 rpm and 72 MW at 3600 rpm are found in power plants, while 16 MW turbines are working with blast-furnace gases. The progress is mainly due to blade cooling and limitation of the effects of corrosion and erosion. State-of-the-art gas turbines built by ABB have 265 MW at 3000 rpm and 183 MW at 3600 rpm.

The simplest type of open circuit stationary gas turbine installation comprises a compressor, a combustion chamber, and a gas turbine. In the


arrangement from Fig. 1.5, the compressor and turbine rotors form a single shaft line, while the generator 7 is coupled via a clutch 6. The starter 9 is used to launch the generator when operating as a compensator. The starter 5 is used to launch the turbine while the generator turns. Part of the compressed air is used for the fuel combustion. The remainder (approx. 70%) is used for cooling the shell of the combustion chamber and some components of the turbine, and is mixed with the hot gases.

Fig. 1.5 (from [6])

The volume of the expanded gas in the turbine is much larger than the volume of the compressed air in the compressor, due to the heating in the combustion chamber. The difference between the work produced by the turbine and the work absorbed by compressor and friction losses is the work supplied to the electrical generator. It is a function of the compressor and turbine thermodynamic efficiencies and the turbine inlet temperature.

Fig. 1.6 (from Power, Jan 1980, p.27)

A design with concentric shafts, resembling the aircraft gas turbines, is shown in Fig. 1.6.


Figure 1.7 shows the Rolls-Royce RB.211 turbofan rotors. The three-stage low pressure (LP) turbine drives the single-stage LP fan which has no inlet guide vanes. The single-stage intermediate pressure (IP) turbine drives the seven-stage IP compressor. The single-stage air-cooled high pressure (HP) turbine drives the six-stage HP compressor.

Fig. 1.7 (adapted from [7])

The eight main bearings are located in four rigid panels (not shown). The three thrust ball bearings are grouped in a stiff intermediate casing. Oil squeeze-film damping is provided between each roller bearing and housing to reduce engine vibration. The short HP system needs only two bearings located away from the combustion zone for longer life.

The single-stage LP fan has 33 blades with mid-span clappers and fir-tree roots. The seven-stage IP axial compressor has drum construction. It consists of seven discs electron beam welded into two drums of five and two stages bolted together between stages 5 and 6. The blade retention is by dovetail roots and lockplates. The six-stage HP compressor consists of two electron beam welded drums bolted through the stage 3 disc with blades retained by dovetail roots and lockplates.

The three-shaft concept has two basic advantages: simplicity and rigidity. Each compressor runs at its optimum speed, thus permitting a higher pressure ratio per stage. This results in fewer stages and fewer parts, to attain the pressure ratio, than in the case of alternative designs. The short, large diameter shafts give good vibration characteristics and a very smooth engine. The short carcase and the positioning of the engine mounting points give a very rigid structure. This allows the rotors to run with smaller tip clearances and thus improved efficiency.

Gas turbines manufactured in Romania are: 1) the Viper 632-41, Rolls-Royce license, 8-stage axial compressor and 2-stage turbine at 13,800 rpm; 2) the Alouette III B, Turbomeca license, 422 kW, 33,480 rpm; and 3) the Turmo IV CA, Turbomeca license, 1115 kW.


1.1.3 Axial compressors

Although patents for axial compressors were taken out as long ago as 1884, it is only in the early 1950's that they become the most versatile form for gas-turbine work. In the aircraft field, where high performance is at a premium, the axial compressor is now used exclusively. It is only for some industrial applications that other compressor types offer serious competition.

Fig. 1.8 (from [8])

The axial-flow compressor resembles the axial-flow steam or gas turbine in general appearance. Usually multistage, one observes rows of blades on a single shaft with blade length varying monotonically as the shaft is traversed. The difference is, of course, that the blades are shorter at the outlet end of the compressor, whereas the turbine receives gas or vapour on short blades and exhausts it from long blades.

In Fig. 1.8 the numbers have the following designations: 1 and 13 - bearings, 2 - seals, 3 - prewhirler, 4 - intake duct, 5 - rotor blades, 6 - stator blades, 7 - straightener stator blades, 8 - discharge duct, 9 - diffuser, 10 - coupling, 11 - gas turbine shaft, 12 - drum-type rotor, 14 - stator casing.

In practically all existing axial compressor designs, the rotor is supported by one bearing at the gas inlet end and by a second bearing at the gas delivery end. In aircraft practice, ball and roller bearings are universally used, on account of their


compactness, small lubricating oil requirements, and insensitivity to momentarily cessations of oil flow as may occur during acrobatic flying.

1.1.4 Centrifugal compressors

Although centrifugal compressors are slightly less efficient than axial-flow compressors, they are easier to manufacture and are thus preferred in applications where simplicity, ruggedness, and cheapness are primary requirements. Additionally, a single stage of a centrifugal compressor can produce a pressure ratio of 5 times that of a single stage of an axial-flow compressor. Thus, centrifugal compressors find application in power station plants, petrochemical industry, gas injection and liquefaction, ground-vehicle turbochargers, locomotives, ships, auxiliary power units, etc.

Fig. 1.9 (from [9])

A typical high-pressure compressor design is shown schematically in Fig. 1.9. Apart from shaft, impellers, bearings and coupling, modeled as for other machines, items of major concern in rotor dynamic analyses are the gas labyrinths, the oil ring seals and the aerodynamic cross coupling at impellers. Furthermore, squeeze film dampers are used to stabilize compressors with problems.

Multistage centrifugal compressors have relatively slender shafts. Usually, impellers are mounted on almost half of the rotor length, the other part being necessary for the centre seal, the balance drum, the oil seals, the radial bearings and the thrust bearing. The shaft diameter is kept small to increase the impeller eye. In comparison with the drum rotor of axial compressors, the shaft of centrifugal compressors is more flexible, having relatively low natural frequencies which favour instabilities.


Vibrations of a centrifugal compressor are controlled by: bearings, shaft geometry, gas seals and oil bushings, fluid forces on impellers, and other factors. Squeeze film dampers are used in centrifugal compressors to eliminate instabilities or to alter the speed at which they occur.

In the case of centrifugal compressors, undamped critical speed maps are of little interest. For typical compressor precession modes which are heavily damped, second mode in particular, the damped natural frequency can be as much as 2 to 9 times lower than the expected peak response speed.

Shop testing, carried out after compressor is constructed but before it is commissioned, can reveal problems prior to start-up. Bode plots, obtained during run-up measurements, are used to check that the critical speeds are not within the operating speed range. Separation margins of the critical speeds from the intended operating speed range are defined in API Standard 617; resonances must be 20 percent above the maximum continuous speed and/or 15 percent below the operating speeds [10]. Compliance with present specifications requires calculation of deflections at each seal along the rotor, as a percentage of the total clearance.

Modern multistage compressors are typically designed to operate through and above several critical speeds so as to maximize the work done by a given size machine. For example, a 425 mm diameter impeller for an industrial centrifugal compressor can be designed for a work load well in excess of 2000 HP by running at speeds approaching 9000 rpm. Up to eight stages are used to obtain the required pressure rise. Process compressors and units used for natural gas injection can have discharge pressures of the order of 650 bar and can drive gases with high density. The result of this combination of supercritical speed, high pressure and high work load has been an increasing tendency for such machines to exhibit problems of nonsynchronous rotor whirling. This is why stability analysis is of prime interest.

While many rotating machines operate below the first critical speed (point A in Fig. 1.10), turbomachinery operate above the first critical speed (point B). Until mid seventies any further shift of the resonance - and hence any increase in the maximum number of stages per casing - was precluded by the bearing stability limit. This was then raised by means of stronger bearing designs until operation above the second critical speed became possible (point C).

High pressure compressors operating on fixed lobe bearings could generate a violent shaft whip condition just above twice the first natural frequency. By going to tilting-pad bearings, that threshold speed can be raised to well over two times the first natural frequency. Attempts to raise speed further came up against another stability limit: rotor instability due to gap excitation. Using vortex brakes before labyrinths this boundary has been pushed back and the way is open in principle to still higher speed ratios (point D).


Exhaust-gas turbocharging is used to increase the mean effective pressure (m.e.p.) of diesel engines. It has applications in stationary plants for electricity generation, in ships' auxiliary and propulsion machinery and in railway traction.

Fig. 1.10 (from [11])

One of the oldest applications was in marine engines. In 1923, BBC and the Vulkan shipyard manufactured turbochargers for the 10-cylinder four-stroke engines from the vessels 'Preussen' and 'Hansestadt Danzig'. The engines, which were designed for an uncharged performance of 1700 HP each at 235 rpm provided, when charged, a cruising power of 2400 HP at 275 rpm and a temporary overload of 4025 HP at 320 rpm (for a m.e.p. = 8.4). Turbocharging of two-stroke marine engines began after 1950.

For the relatively short turbocharger rotors, which are almost always equipped with single-stage compressor and turbine wheels, two bearings are sufficient. One of these is a combined radial-axial bearing, the other a pure radial bearing. Two bearing layouts have proved successful on the market: 1) bearings at the shaft ends (external bearings), used predominantly in large machines, and 2) bearings between the compressor and turbine wheel (internal bearings) used mainly for small turbochargers. In both arrangements the axial bearing is located near the compressor wheel, to keep the axial clearance in that region small.


In the variant with external bearings (Fig. 1.11, a), the large distance between the bearings reduces the radial bearing forces and requires smaller clearances at the compressor wheel and turbine wheel. The frictional losses in the bearings are smaller, particularly at part load. The shaft ends can be kept small in diameter and are simple to equip with a lubricating oil pump and centrifuge, thus rendering rolling-contact bearings and self-lubrication possible.

Fig. 1.11

Internal bearings (Fig. 1.11, b) offer advantages in fitting a turbocharger with axial air and gas inlets to the engine. Small turbochargers do not, however, have an axial-flow turbine, but a radial-flow turbine with axial gas outlet. For specific applications internal bearings have advantages, which relate mainly to the wider variety of ways of fitting the turbocharger to the engine.

In automotive applications, a floating bush bearing is used due to size and cost considerations. This type of bearing has a thin bush rotating freely between the journal and the fixed bush, forming two hydrodynamic oil films [12]. This turbocharger shows peculiar behaviour yet to be explained theoretically: 1) it has stable operation at very high shaft speeds, though at lower speeds it can exhibit instability in either a conical mode or an in-phase bending mode; and 2) some designs have a third flexible critical speed, very difficult to balance out; with a high amplification factor, leading to rubbing and bearing distress.


1.1.5 Fans and blowers

Fans can be either radial-flow or axial-flow machines. The ratio discharge pressure vs. suction pressure is defined as the pressure ratio. Fans are designed for pressure ratios lower than or equal to 1.1. Centrifugal fans absorb powers between 0.05 kW and 1 MW, have flow rates up to 3105 m3/h and discharge pressures up to 1000 mm H2O (~104 N/m2). Blowers are single-stage uncooled compressors with pressure ratios between 1.1 and 4, and discharge pressures up to 3.5105 N/m2. Compressors have pressure ratios larger than 4, so they usually require interstage cooling.

Fig. 1.12 (from [13])

Fig. 1.13 (from [13])


The design from Fig. 1.12 is a medium-pressure blower, with labyrinth seals, and overhung design.

The arrangement from Fig. 1.13 is with double suction and single exhaust. The symmetrical rotor has a disc at the middle.

Centrifugal fans used for forced- or induced-draft and primary-air service generally have large diameter rotors, operating from 500 to 900 rpm in pillow-block bearings, supported on structural steel or concrete foundations.

As a rule, the major problem with fans is unbalance caused by 1) uneven buildup or loss of deposited material; and 2) misalignment. Both are characterized by changes in vibration at or near the rotational frequency.

1.1.6 Centrifugal pumps

Centrifugal pumps are used in services involving boiler feed, water injection, reactor charge, etc. Instability problems encountered in the space shuttle hydrogen fuel turbopumps and safety requirements of nuclear main coolant pumps have prompted research interest in annular seals.

It is now recognized that turbulent flow annular seals in multi-stage pumps and in straddle-mounted single-stage pumps have a dramatic effect on the dynamics of the machine. Stiffness and damping properties provided by seals represent the dominant forces exerted on pump shafts, excluding the fluid forces of flow through the impellers, particularly at part-flow operating conditions. For these systems, the hydrodynamics of oil-lubricated journal bearings is dominated by seal properties.

Typical multi-stage centrifugal pumps have more inter-stage fluid annuli than they have journal bearings. The fluid annuli are distributed between the journal bearings where precession amplitudes are highest and can therefore be 'exercised' more as dampers than can be the bearings.

In typical applications, shaft resonant critical speeds are rarely observed at centrifugal pumps because of the high damping capability afforded by seals. Problems encountered with boiler feed pumps have been produced by excessive wear in seals, yielding a decrease in the dynamic forces exerted by the seals.

Centrifugal pumps have comparatively slender shafts and relatively flexible cantilevered bearing housings (Fig. 1.14).

Fine clearance annular seals are used in pumps primarily to prevent leakage between regions of different pressure within the pump. The rotordynamic behaviour of pumps is critically dependent on forces developed by annular seals, between the impeller shroud and the stator, between the impeller back disc and the stator, and between the impeller and diffuser.


Fig. 1.14 (from [14])

1.1.7 Hydraulic turbines

Hydraulic turbines have traditionally been used to convert hydraulic energy into electricity. The first effective radial inward flow reaction turbine was developed around 1850 by Francis, in Lowell, Massachusetts. Around 1880 Pelton invented the split bucket with a central edge for impulse turbines. The modern Pelton turbine with a double elliptic bucket, a notch for the jet and a needle control for the nozzle was first used around 1900.

The axial flow turbine, with adjustable runner blades, was developed by Kaplan in Austria, between 1910-1924. The horizontal bulb turbines have a relatively straighter flow path through the intake and draft tube, with lower friction losses. In the Straflo (straight flow) design, the turbine and generator form an integral unit without a driving shaft.

With hydraulic turbines, despite the low rotating speeds (200-1800 rpm), problems occur owing to the vertical position of most machines, due to transients and cavitation. Rotors are very robust and stiff, problems being raised by bearings and the supporting structure.


Fig. 1.15

The hydro power plant at Grand Coule (U.S.A.) has a 960.000 hp Francis turbine driving a synchronous generator of 718 MVA at 85.7 rpm. The rotor has a diameter in excess of 9 m and a weight exceeding 400 tons, the main shaft having 3.3 m diameter and more than 12 m length.

The worlds largest hydroelectric plant Itaipu, on the Rio Paran, which forms the border between Brazil and Paraguay, near the city of Foz do Iguau, consists of 18 generating sets of 824/737 MVA, driven by Francis turbines, with a total rating of 12,600 MW. Turbines have rotors of 300 tons and 8 m diameter, the main shaft has 150 tons and 2.5 m diameter, while the synchronous generator has 2000 tons and 16 m diameter, running at respectively 90.9 rpm for 50 Hz generators, and 92.3 rpm for 60 Hz generators (Fig. 1.15).

The hydro power plant at Ilha Solteira, Brazil, has sets of 160 MW at 85.8 rpm. The rotor shaft has 6.33 m length, 1.4 m outer diameter and 0.4 m inner diameter. The generator has 495 tons and the Francis turbine has 145 tons. The first critical speed is about 222 rpm.

The hydroelectric power plant at Corbeni-Arge has four Francis turbines with nominal speed 428.6 rpm, gross head 250 m, nominal water flow 20 m3/s and individual rated power 50 MW.

An axial cross-section of a vertical axis Kaplan turbine is presented in Fig. 1.16 where 1 runner with adjustable blades, 2 draft tube, 3 guide vanes, 4 lower guide bearing, 5 stay vanes and ring support, 6 concrete spiral casing, 7 control ring with servo-motor for the stay vanes, 8 thrust bearing, 9 upper guide


bearing, 10 servo-motor for adjustment of runner blades, 11 runner blades control rod inside the turbine shaft, and 12 generator.

Fig. 1.16 (from [15])

The Porile de Fier I hydroelectric power plant has eight Kaplan turbines of 194 MW, head 27 m, nominal water flow 840 m3/s, speed 71.43 rpm, 6 blades and rotor diameter 9.5 m.

The Porile de Fier II hydroelectric power plant has eight double-regulated bulb units type KOT 28-7.45, with the bulb upstream and the turbine overhung downstream. The unit has three guide bearings and a thrust bearing, 16 stator blades and 4 rotor blades, and the following parameters: head 7.45 m, nominal water flow 432 m3/s, rated power 27 MW, rotor diameter 7.5 m.

1.1.8 Turbo-generators

The turbo-alternator was developed by C. E. L. Brown and first marketed by Brown Boveri in 1901. With a cylindrical rotor having embedded windings, it has proven to be the only possible design for high speeds, as when driven direct by a steam turbine. Such alternators are available for ratings between 500 kVA and


20,000 kVA and higher, but are not normally used below 2500 kW, because salient-pole machines with end-shield bearings are more economical. Beyond 2500 kW, an alternator running at 3000 (or 3600) rpm permits a more economical gear to be used than a 1500 (or 1800) rpm alternator for the same turbine [16].

The marked increase in the unit ratings of turbo-generators has not, for the most part, been accompanied by a corresponding increase in the size of machines because of the increase in the specific electric ratings. For example, between 1940-1975, the maximum power of electric generators increased from 100 to 1600 MVA, whereas in 1940 a 3000 rpm turbo-generator weighed 2 kg per kW of output, and its 1975 counterpart weighed only 0.5 kg/kW.

Alternator rotors have been also designed to be progressively longer and more flexible. The forging of a 120 MW rotor had approximately 30 tons and 8 m distance between bearing centres, while a 500 MW rotor had 70 tons and 12 m. Modern rotors have two or three critical speeds below their operating speed of 3000 rpm.

Fig. 1.17 (from [16])

The rotor of small units is a solid cylindrical forging of high-quality steel with slots milled in it to accommodate the field winding. For larger units, several hollow cylinders are fitted over a central draw-bolt threaded at both ends, to which the two shaft extensions are fastened by shrinking. The specially formed winding is a single layer of copper strip insulated with glass-fibre which is pressed and baked into the slots. To secure the end sections, end-bells forged from solid-drawn non-magnetic steel with ventilation holes or slots are used.


Rotors of electrical machines are different from rotors with bladed discs or impellers, being more massive, but occasionally rising problems due to asymmetrical stiffness properties.

Figure 1.17 is a cutaway perspective drawing of a 400 MVA, 3000 rpm generator with water-cooled stator winding and forced hydrogen direct cooling in the rotor. Due to the high flux density and current loadings, generators of over 500 MW employing these cooling methods must have their stator cores mounted in a flexible suspension. This is necessary in order to isolate the foundations from the enormous magnetic vibration forces arising between rotor and stator.

Two-pole generator rotors have axial slots machined to match more closely the principal stiffnesses. They are intended to reduce the parametric vibrations induced by the variation of the cross-section second moment of area about the horizontal axis, during rotation.

The second order (or 'twice per revolution') forced vibration which arises from the dual flexural rigidity is virtually inescapable in a two-pole machine; where the motion is excited by the weight of the rotor. This is a source of considerable difficulty, largely because it can be cured only at the design stage and cannot be 'balanced'. Certain 'trimming' modifications can be made but these present problems of their own. In fact it would be very difficult to design accurately an alternator rotor so as to have axial symmetry in a dynamical sense. The rotor is, in effect, a large rotating electromagnet, having a north pole and a south pole on opposite sides of the rotor and having slots cut in it, in which copper conductors are embedded to provide the magnetic field.

Fig. 1.18 (from [17])

The cross-section of a 120 MW alternator rotor after slotting is shown in Fig. 1.18, a. It is clear from the figure that the flexural rigidity of the shaft is


unlikely to be the same for bending about the horizontal and the vertical neutral axes, even after copper conductors and steel wedges have been placed in the slots.

In attempts to equalize these rigidities, one of two schemes is usually adopted. In the first, the pole faces are slotted as shown in Fig. 1.18, b. In order to maintain the magnetic flux density, the slots in the pole faces are filled with steel bars that are wedged in. The second technique is to build a rotor in the manner of Fig. 1.18, a and then to cut lateral slots across the poles at intervals along the length of the rotor.

Figure 1.19 shows the different cross-sections in a turbo-generator rotor: A-A rectangular slots for field winding and smaller slots in the pole area, and B-B cross-cuts to ensure uniform flexibility with respect to the vertical and horizontal cross-section principal axes.

Fig. 1.19 (from [18])

Alternator rotors are supported in plain bearings. These hydrodynamic bearings present unequal dynamical stiffnesses in the vertical and horizontal directions. Asymmetry of the bearings introduces a split of critical speeds but cannot by itself cause second order vibration.

For small machines, e.g. electrical motors, having relatively low rotational speeds and rolling-ball bearings, the balancing and the dynamic calculation of the rotor does not generally raise problems. On the contrary, large machines, having long and flexible rotors, sliding bearings, seals, pedestals and relatively flexible casings, with high speeds, have determined the continuous development and improvement of dynamic calculations and vibration measurement.


1.2 Rotor-bearing dynamics

Rotor-Bearing Dynamics has got its own status, apart from Mechanical Vibrations and Structural Dynamics, becoming an interdisciplinary research field, as soon as the importance of the effects of bearings and seals on the rotor dynamic response has been recognized.

The scope of Rotor-Bearing Dynamics is the study of the interaction between rotor, stator and the working fluid, for the design, construction and operation of smooth-running machines in which allowable vibration and dynamic stress levels are not overpassed, within the whole operating range.

Smooth machine operation is characterized by small, stable rotor precession orbits, and by the absence of any instability throughout the machine operating range.

In order to understand the dynamic response of a rotating machine it is necessary to have, early in the design stage, information on the following aspects of its behavior:

1. Lateral critical speeds of the rotor-bearing-pedestal-foundation system; effects of the stiffness and damping of bearings, seals, supporting structure and foundation on the location of critical speeds within the machine operating range.

2. Unbalance response: orbits of the rotor precession as a response to different unbalance distributions, throughout the whole operating range of the machine, and vibration amplitudes due to rotor unbalance.

3. Rotor speed at onset of instability: the threshold speed for unstable whirling due to the rotor/bearing and/or working fluid interaction, as well as the consequences of its crossing.

4. Time transient response analysis, to a blade loss, mainly for gas turbine engines operating at supercritical speeds, or when passing through a critical speed.

5. System torsional critical speeds, especially at geared rotors, eventually the transient response of the shaft line to electric disturbances applied to the generator.

Practical measures regarding the balancing and the monitoring of the dynamic state of rotors are added to these:

6. Balancing of rotors: calculation and attachment (removal) of correction masses such that the centrifugal forces on the rotor due to these additional masses and the inherent unbalance forces are in equilibrium.

7. Machinery monitoring: measurement of the parameters characterizing the dynamic state of machines and trending their time evolution, in order to detect any damage, to anticipate serious faults, determining the outage.


The capability of predicting the performances of a rotor-bearing system is dependent firstly on the information about bearing properties, fluid-rotor interaction and the unbalance distribution along the rotor. In this respect, in recent years, important progress has been achieved in determining the dynamic coefficients of bearings and seals, and in the identification of the spatial distribution of unbalance for flexible rotors. The direct result is the development of computer programs helping in modeling most of the dynamic phenomena occurring during the operation of rotating machinery.

Generally, the following dynamic characteristics of rotating machinery are of interest:

a. Rotor lateral critical speeds in the operating range.

b. Unbalance response amplitudes at critical speeds.

c. Threshold speed of instabilities produced by bearings, seals or other fluid-structure interactions.

d. Bearing transmitted forces.

e. The overshoot ratio, of maximum transient response relative to the steady-state response.

f. System torsional critical speeds.

g. Gear dynamic loads.

h. Vibration amplitudes in casing and supporting structure.

The following can be added to this list:

i. Natural frequencies of bladed discs, impellers, wheels.

j. Frequencies and mode shapes of blades and blade buckets.

k. Blade flutter frequencies.

l. Rotating stall and surge thresholds.

m. Noise radiated by rotating machinery.

In the following, only the first three issues are treated. Problems not treated in this book are:

a. Shafts with dissimilar principal moments of inertia;

b. Cracked rotors;

c. Reverse precession due to dry-friction contact between rotor and stator;

d. Partial rubbing conditions;

e. Transient critical-speed transition.


1.3 Rotor precession

The most important sources of machinery vibration are the residual rotor unbalance and rotor instability.

Most rotors have at least two bearings. With horizontal rotors, the rotor weight is distributed between all the bearings. The rotation axis is coincident with the static elastic line under the own weight. If the weight effect is neglected, the rotation axis coincides with the line connecting the bearing centres.

Any rotational asymmetry due to manufacturing, or produced during operation, makes the line connecting the centroids of rotor cross-sections not to coincide with the rotation axis. Hence, as the rotor is brought up in speed, the centrifugal forces due to dissymmetry cause it to deflect. For example, a 50 tons rotor, with its mass centre off-set by 25 m from the axis of rotation, experiences a force of approximately 13 tons force, when rotating at 3000 rpm. The rotating centrifugal forces are transferred to the bearings and their supports, and produce unwanted vibrations.

While the bearings and the casing vibrate, the rotor has a precession motion. For isotropic bearings, at constant speed, the deflected shape of the rotor remains unchanged during the motion, any cross-section traces out a circular whirling orbit. The motion appears as a vibration only when the whirl amplitude is measured in any fixed direction.

Despite the analogy often used in describing vibration and precession, their practical implications are different. The remedy for resonance internal damping is totally inefficient in the case of critical speeds, since the shape of the deflected rotor does not change (or changes very slightly) during the precession motion at constant speed. Moreover, at a critical speed, if the deflections are not limited, a rotor bends rather than damages by fatigue, phenomenon produced by the lateral vibrations. Instead, journal bearings, small clearance liquid seals, or viscous sleeves are the major source of damping in most cases. Without this damping or a similar source, it would be very difficult to pass through a critical speed. That is why bearings and seals play a major role in the dynamics of the rotor systems.

If identical orbits are traced out with successive rotor rotations, the motion is said to be stable precession. If the orbit increases in size with successive rotations, the motion is an unstable whirl. It may subsequently grow until the orbit becomes bounded either by system internal forces, or by some external constraint, e.g. bearing rub, guard ring, shut-down, etc.

Some typical orbits are shown in Fig. 1.20. The circular orbit (Fig. 1.20, a) represents the synchronous whirling of a rotor in isotropic radial supports. The absence of loops within the orbit denotes synchronous whirl.


An elliptical orbit (Fig. 1.20, b) may arise from orthotropic supports, i.e. from dissimilar bearing or pedestal stiffnesses in the horizontal and vertical directions. Inclination of ellipse axes occurs due to cross-coupled stiffnesses and damping properties.

a b c

d e f

Fig. 1.20 (from [19])

If the precession is non-synchronous, i.e. the rotor whirls at a frequency other than the rotational frequency, the orbit will contain a loop as in Fig. 1.20, c, characteristic for the half-frequency whirl due to the instability of motion in hydrodynamic bearings ("oil whirl"). An internal loop indicates that the precession is in the direction of rotation.

Other non-synchronous excitations may occur at several times rotational frequency, giving rise to multi-lobe whirl orbits depicted in Fig. 1.20, d, as in the case of multi-pole electrical generators.

Instabilities such as the half-frequency whirl are frequently bounded. The whirl is initiated by crossing the onset of instability speed, and then it develops in a growing transient, whose radius increases until a new equilibrium orbit is reached (Fig. 1.20, e).

Another type of transient condition is shown in Fig. 1.20, f. The rotor is initially operating in a small stable unbalance whirl condition. The rotor system then receives a transverse shock, and the journal displaces abruptly in a radial


direction within the bearing clearance, but without contacting the bearing surface. Following the impact, the rotor motion is a damped decaying spiral transient, as it returns to its original small unbalance whirl condition.

Many other types of whirl orbits have been observed, such as those associated with system non-linearities and nonsymmetric clearance effects.

1.4 Modeling the rotor

For the mechanical design of rotor, bearings and supporting structure, one has to take into account that they work as a whole, responding together to the dynamic loading, and interacting. The rotor is part of a dynamic system, its behavior being determined by the location and stiffness of bearings, seals, pedestals and foundation, as well as their damping properties. The casing and foundation masses also play an important role.

The rotor is the main part in any piece of rotating machinery. Its function is to generate or transmit power. It consists of a shaft on which such components as turbine wheels, impeller wheels, gears, or the rotor of an electric machine may be mounted. The rotor is never completely rigid and in many applications it is actually quite flexible. However, in practice, rigid rotors are considered to be those running below 1/3 of the first bending critical speed. Elastic rotors operate near or beyond the first bending critical speed, so that the centrifugal forces due to the residual unbalance cause it to deflect.

In most machines, rotors have shafts with axisymmetric cross-section. If, in some parts, the cross-section is not symmetrical, then the bending stiffness with respect to a fixed axis is variable during the rotation giving rise to non-synchronous motions and instabilities (e.g. two-pole generators and cracked rotors). The rotor shaft can be modeled as a Timoshenko-type beam, accounting for the shear and rotational inertia, including also the effect of gyroscopic couples. The discs usually rigid are included by lumped parameters: the mass, and the polar and diametral mass moments of inertia. More advanced calculations consider the disc flexibility. Rotors of individual machines are joined by couplings (locked spline, double-hinged, sliding spline, flex plate).

Bearings are selected as a function of static load and speed, taking into account the dynamic loading, available space, energy losses, simplity of design solution, as well as durability and reliability requirements.

In early studies, bearings were considered as rigid supports (Fig. 1.21, a). Later, their radial stiffness and damping has been taken into account (Fig. 1.21, b). In rolling bearings and air bearings, the damping is usually neglected. The stiffness and damping characteristics of journal bearings are functions of running speed and loading. At rolling bearings, the stiffness is considered independent of speed and


loading. Generally, only the bearing translational radial stiffness is taken into account, the angular stiffness being relatively small (one tenth).

With journal bearings, under steady-state hydrodynamic conditions, the total pressure force equals the static load on the bearing. If the centre of the rotating journal is in motion, as for instance during synchronous precession, additional pressures are set up in the lubricant film, which act as dynamic forces on the journal in addition to the static forces. The dynamic force depends on both the relative displacement and the velocity of the journal centre motion but, in contrast to conventional elastic forces, the dynamic force does not have the same direction as the imposed motion, being phase shifted in space and time.

Fig. 1.21

Resolving the dynamic force into two components along fixed coordinate axes in the bearing, say Oy and Oz, and likewise resolving the journal centre motion into y and z displacements, the dynamic force components may be expressed by:

.

+

=

zy

cccc

zy

kkkk

ff

zzzy

yzyy

zzzy

yzyy

z

y&&

(1.1)

The above equations are exact only for very small amplitudes, but in practice they prove to be valid even for amplitudes as large as a third of the bearing clearance.


The four stiffness coefficients zzzyyzyy k,k,k,k and the four damping coefficients , zzzyyzyy cc,c,c are calculated from lubrication theory by linearizing the non-

linear bearing forces. They are properties of the particular bearing, being functions of the bearing configuration and the lubricant properties. More important, they belong to a given steady-state journal center position, changing with the speed of rotor.

The inequality of cross-coupling stiffnesses zyyz kk is the source of a certain type of self-excited precession known as oil whirl, fractional frequency whirl, or half-frequency whirl. Because of the speed dependence of the eight bearing coefficients, the effective damping is negative at low speeds and may become positive at higher speeds.

Active magnetic bearings are applied in industrial centrifugal compressors, turbo expanders and centrifugal pumps. The principle is an electromagnetic shaft suspension, without physical contact between rotor and stator. Sensors located near the electromagnets observe the rotor position. This rotor position signal feeds into an electronic controller which feeds, in a closed loop, the power amplifiers of the electromagnets.

Short annular seals with gas or fluid are usually considered isotropic. The diagonal terms of their stiffness and damping matrices are equal, while the off-diagonal terms are equal, but with reversed sign. The two force components by which the seals act upon the rotor can be written as

. +

+

=

zy

Mzy

CccC

zy

KkkK

ff

z

y&&&&

&&

(1.2)

The inertial term is negligible at gas seals, where direct stiffnesses can be very small, even negative. For long annular clearance seals, like those used to break down large pressure differences in multi-stage pumps, angular dynamic coefficients are introduced, because the rotor is acted upon by couples, and forces give rise to tilting shaft motions, and moments produce linear displacements.

Radial seals in centrifugal pumps are either balance disks or the radial gap of mechanical seals. Impellers generate motion-dependent forces and moments in the flow fields between impeller tip and casing (volute or diffuser) and in the leakage flow fields developed between impeller shrouds and casing.

Squeeze-film dampers are used in gas turbines as a means of reducing vibrations and transmitted forces due to unbalance. A squeeze film is an annulus of oil supplied between the outer race of a rolling-element bearing (or the bush of a sleeve bearing) and its housing. It can be considered as a parallel element of a vibration isolator, or as a series element in a bearing housing.


Flexible pedestals are considered in the dynamic response of machines especially for blowers and fans, centrifugal pumps and turbosets with flexible casings and cantilevered bearings. The calculation model (Fig. 1.21, c) includes the stiffness and damping of bearing supports and, in some cases, also their equivalent mass.

The foundation, the sole plate and the soil are seldom included in the calculation model (Fig. 1.21, d), their influence on the rotor response being generally smaller. However, in some cases, especially for large fans on concrete pedestals, the elasticity of the subgrade is taken into account.

Generally, for bearings and pedestals, even with frequency independent characteristics, the horizontal stiffness is lower than the vertical stiffness. This anisotropy doubles the number of critical speeds. In some cases, due to the high damping level, the separation of the two criticals in a pair due to orthotropy does not show up in the unbalance response of rotors. It is also possible for some whirl modes, especially the backward ones, to be overcritically damped, thus appearing neither in the natural frequency diagrams nor in those of the unbalance response

1.5 Evolution of rotor design philosophy

Calculation methods and the interpretation of the results of the dynamic analysis of rotors had a spectacular evolution.

Until the late 1950's, calculations were made numerically or using the graphic method developed by Mohr. It was common practice to assume rigid supports and to treat one span at a time in the model. Analysis was limited to the determination of undamped critical speeds and the objective was to avoid having a running speed at a critical speed, in other words, each span was 'tuned' to avoid certain frequencies. The purpose of an undamped analysis was to provide a close, initial estimate of the critical speeds.

Many specifications explicitly require that operating speeds differ from critical speeds by safe margins. In the API Standard 610, the critical speed is required to be at least 20% greater or 15% less than any operating speed [20]. Compliance with such specifications requires that critical speeds be calculated as part of design and selection procedures of rotating machines. In some cases well established procedures are used. In other cases, e.g. machines that handle liquids, like centrifugal pumps, specific calculation procedures are used, the dry running critical speeds being different from the wet running criticals.

It was recognized that large discrepancies existed between calculations and tests, and efforts were made to improve the analyses. The rigid supports were replaced by elastic springs with stiffness equal to that of the oil film in the bearings. Later, the effect of pedestals was added. It was recognized that entire


rotor-bearing systems, rather than single spans, should be analyzed, taking advantage of the advent of high-speed digital computers. After whole systems were studied, it became clear that a change in philosophy was required, involving a switch from tuning to response.

In some cases, the large discrepancies between calculations and tests were diminished introducing the effect of damping and determining the damped critical speeds. Also, calculating the unbalance response, it came out that the speeds at which the radius of synchronous whirling orbits is a maximum referred to as peak response critical speeds are different from both the undamped and damped critical speeds, approaching the latter. It was recognized that not all potentially critical speeds are indeed critical, the large damping in bearing smoothing the unbalance response curves so that the passage through a critical speed may take place without an increase in the response amplitude.

At present, numerical simulations are used in the predictive design stage, and rotor designs are accepted or rejected on the basis of the unbalance response at the journals as a function of running speed. In contrast to lateral vibration, torsional natural frequencies are tuned to avoid coincidence with running speed and known exciting frequencies.

Fig. 1.22

Figure 1.22 shows the response of a rotor journal versus the ratio of natural frequency to running speed. Values calculated for rigid supports are denoted by R1 and R2, while values for bearings treated as elastic springs are indicated by E1, E2 and E3. The continuous line shows the unbalance response calculated considering both the stiffness and the damping in bearings. D1, D2, D3


are the actual critical speeds, measured by the peaks in the unbalance response characteristic. It can be seen that critical speeds based on rigid support calculations can be seriously in error, that criticals calculated assuming elastic supports can be more accurate and that neither calculation can be used to determine a response level because damping has been neglected.

Figure 1.23 shows an analysis of a complete system consisting of high pressure turbine (HPT), intermediate pressure turbine (IPT), low pressure turbine (LPT), generator (G) and exciter (E).

Fig. 1.23

Assuming elastic supports at the bearings, 22 critical speeds were calculated between zero and the running speed. With such a large spectrum of natural frequencies, the desired separation between the operating speed and the critical speeds has limited application, so that there is a need to change the basic philosophy of design and commissioning of a rotating machinery.

In the field of compressors and turbines for industry or power plants, the actual trend to increase the size and the operating speeds has lead to a new generation of machines for which, inevitably, one or two critical speeds are within the range of operating speeds.

As machinery become larger, the elasticity of the lubricant film in bearings and the flexibility of supports play a more important role with respect to


the rotor stiffness, determining a decrease of the critical speeds and their interference with the range of operating speeds. As the bearing dynamic characteristics are not exactly known, critical speeds cannot be determined accurately, so that the traditional criterion aiming at operation at or near to a critical cannot offer the necessary safe limit

In practice it has been found out that one can operate perfectly safe and reliable at well damped critical speeds if the vibration levels do not exceed the allowable levels and if the rotor has not a pronounced sensitivity to mass unbalance. It means that the unbalance response of the rotor can give the most useful information about the soundness of a design solution. Carrying out this calculation for different unbalance distributions, judiciously chosen so as to enhance the deflection at different unbalance critical speeds, it can be established how critical each of the critical is and what measures have to be taken so that the vibration amplitudes remain within normal limits, even in the presence of unbalances that occur during the normal operation (erosion, deposits, component failures, thermal strains, etc.).

1.6 Historical perspective

The first analysis of critical speeds of a uniform elastic shaft has been made in 1869 by Rankine [21], who devised the term critical speed. The phenomenon was incorrectly thought to be an unstable condition, the rotor being unable to run beyond that speed. In this case art preceded science, for in 1895 some commercial centrifuges and steam turbines were already running supercritically. Gustaf de Laval first demonstrated experimentally that a (single stage steam) turbine could operate above the rotors lowest bending resonance speed and supercritical operation could be smoother than subcritical. In many European papers, the rotor model consisting of a central disk, mounted on a massless flexible shaft supported at its ends, is referred to as the Laval rotor. Although the first correct solution for an undamped model has been given by Fppl [22], who was the first to demonstrate analytically that that a rotor could operate supercritically, the confusion persisted until the publication in 1919 of Jeffcotts paper [23] using a model with damping. This simple model is called the Jeffcott rotor in recent papers.

In 1894 Dunkerley [24] published results of his studies on critical speeds of shafts with many discs, and gave his well-known method with its experimental verification. In 1916 Stodola [25] published an analysis of the bearing influence on the flexible shaft whirling. He also introduced the gyroscopic couples on disks.

Hysteretic whirl was first investigated by Newkirk [26] in 1924 during studies about a series of failures of blast-furnace compressors. It was observed that at speeds above the first critical speed, these units would enter into a violent


whirling in which the rotor centerline precessed at a rate equal to the first critical speed. If the unit rotational speed was increased above its initial whirl speed, the whirl amplitude would increase, leading to eventual rotor failure. During the course of the investigation, Kimball in 1924 suggested that internal shaft friction could be responsible for the shaft whirling. In 1925, Kimball and Lovell performed extensive tests on the internal friction of various materials. At the end Newkirk concluded that the internal friction created by shrink fits of the impellers and spacers is a more active cause of the whirl instability than the material hysteresis in the rotating shaft.

In 1925, Newkirk and Taylor [27] observed oil film whirl and resonant whipping. The true upper limit for safe operating speeds has been thus revealed, namely the threshold speed of rotor-bearing instability. This typically occurs at speeds between two and three times the lowest resonant frequency, wherefrom the name of half-frequency and sub-synchronous whirl. The phenomenon was explained only in 1952 by Poritsky [28], who showed that the destabilizing influence comes from the hydrodynamic journal bearing which loses its ability to damp the lowest rotor-bearing bending resonant mode.

In 1933 Smith [29] published a review of the basic rotor dynamics problems, discussing qualitatively the effect of gyroscopic coupling, and simultaneous asymmetries of the bearing and shaft flexibilities. Between 1932-1935, Robertson [30] presented a series of papers on the subjects of bearing whirl, rotor transient whirl, and hysteretic whirl.

In 1946, Prohl [31] published a transfer matrix procedure for determining the critical speeds of a multi-disc single shaft rotor, allowing for the inclusion of gyroscopic effects, but restricted to isotropic elastic supports. Between 1955-1965, Hagg and Sankey [32], Sternlicht [33], Lund [34] and others have developed the theory of hydrodynamic bearings, Yamamoto [35] studied the rolling bearings and Sternlicht [36], Pan and Cheng investigated the rotor instability in gas bearings.

In 1948 Green [37] studied the gyroscopic effect of a rigid disc on the whirling of a flexible overhang rotor, being credited with the initial generalization of Jeffcotts model to account for rigid-body dynamics. In 1957 Downham [38] has experimentally confirmed the existence of backward whirling.

Between 1963-1967, Lund [39] and Glienicke [40] presented values of the linearized stiffness and damping coefficients for a series of hydrodynamic bearings, first presented by Sternlicht in 1959 [33]. Lund [41, 42] expanded the transfer matrix method of Myklestad and Prohl for calculating damped unbalance response and damped natural frequencies of a flexible rotor with asymmetric supports. Ruhl [43] and Nordmann [44] have first used the finite element method for the dynamic analysis of rotor-bearing systems in their doctoral theses, but the first papers using this method were published by Ruhl and Booker [45] in 1972, and Gasch [46] in 1973. Reduction of the finite element model has been used


starting in 1980 by Rouch and Kao [47], and Jcker [48], the latter introducing also the effect of foundation on the rotor response.

The study of the effect of annular fluid seals was initiated by Lomakin [49] in 1958, and then developed by Black [50] and Childs [51]. The effect of gas seals has been studied by Benckert and Wachter [52], and Iwatsubo [53]. The study of instabilities due to unequal gaps between rotor and stator as a result of the rotor eccentricity was initiated by Thomas [54] and Alford [55].

In Romania, the first book with elements of machinery dynamics was published in 1958 by Gh. Buzdugan and L. Hamburger [56]. The lubrication theory has been developed by N. Tipei [57] and V. N. Constantinescu [58-60]. Books on sliding bearings were published by Tipei et al [61] and Constantinescu et al [62]. The first PhD thesis on Rotordynamics was presented in 1971 by M. Rdoi [63], using a computer program developed at INCREST [64], based on Lunds transfer matrix method [65].

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17. Bishop, R. E. D., and Parkinson, A. G., Second Order Vibration of Flexible Shafts, Phil. Trans. Royal Society, Series A, Vol.259, A.1095, pp 1-31, 1965.

18. * * * Caractristiques de construction des alternateurs de grande puissance, Revue ABB, No.1, 11 pag. 1989.

19. Rieger, N. F., and Crofoot, J. F., Vibrations of Rotating Machinery. Part I: Rotor-Bearing Dynamics, The Vibration Institute, Illinois, Nov 1977.

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21. Rankine, W. J. M., On the centrifugal force of rotating shafts, The Engineer, Vol.27, p.249, Apr.1869.

22. Fppl, A., Das Problem der Laval'schen Turbinewelle, Civilingenieur, Vol.41, pp.332-342, 1895.

23. Jeffcott, N., Lateral vibration of loaded shafts in the neighbourhood of a whirling speed The effect of want of balance, Philosophical Magazine, Series 6, Vol.37, pp.304-314, 1919.

24. Dunkerley, S., On the whirling and vibration of shafts, Trans. Roy. Soc. (London), Vol.185, Series A, pp.279-360, 1894.

25. Stodola, A., Neuere Beobachtungen uber die Kritischen Umlaufzahlen von Wellen, Schweizer.Bauzeitung, Vol.68, pp.210-214, 1916.

26. Newkirk, B. L., Shaft whipping, General Electric Review, Vol.27, pp.169-178, 1924.


27. Newkirk, B. L. and Taylor H. D., Oil film whirl An investigation of disturbances on oil films in journal bearings, General Electric Review, Vol.28, 1925.

28. Poritsky, H., Contribution to the theory of oil whip, Trans. ASME, Vol.75, pp.1153-1161, 1953.

29. Smith, D. M., The motion of a rotor carried by a flexible shaft in flexible bearings, Proc. Roy. Soc. London, Series A, Vol.142, pp.92-118, 1933.

30. Robertson, D., The vibration of revolving shafts, Phil. Mag. Series 7, Vol.13, pp.862, 1932; The whirling of shafts, The Engineer, Vol.158, pp.216-217, 228-231, 1934; Transient whirling of a rotor, Phil. Mag., Series 7, Vol.20, pp.793, 1935.

31. Prohl, M. A., A general method for calculating critical speeds of flexible rotors, Trans ASME, Vol.67, J. Appl. Mech., Vol.12, No.3, pp.A142-A148, Sept.1945.

32. Hagg, A. C. and Sankey, G. O., Some dynamic properties of oil-film journal bearings with reference to the unbalance vibration of rotors, Trans. ASME, J. Appl. Mech., Vol.23, pp.302-306, 1956.

33. Sternlicht, B., Elastic and damping properties of cylindrical journal bearings, Trans. ASME, J. Basic Eng., Series D, Vol.81, pp.101-108, 1959.

34. Lund, J. W., The stability of an elastic rotor in journal bearings with flexible damped supports, Trans. ASME, J. Basic Eng., Vol.87, 1965.

35. Yamamoto, T., On the critical speed of a shaft supported in ball bearing, Trans. Soc. Mech. Engrs. (Japan), Vol.20, No.99, pp.750-760, 1954.

36. Sternlicht, B., Gas-lubricated cylindrical journal bearings of the finite length, Trans. ASME, J. Appl. Mech., Paper 61-APM-17, 1961.

37. Green, R. B., Gyroscopic effects on the critical speeds of flexible rotors, Trans. ASME, J. Appl. Mech., Vol.70, pp.369-376, 1948.

38. Downham, E., Theory of shaft whirling. A fundamental approach to shaft whirling, The Engineer, pp.519-522, 552-555, 660-665, 1957.

39. Lund, J. W., Rotor Bearing Dynamics Design Technology. Part III, Design Handbook for Fluid-Film Type Bearings, M.T.I. Report AFSCR 65-TR-45, 1965.

40. Glienicke, J., Feder- und Dmpfungskonstanten von Gleitlagern fr Turbomaschinen und deren Einfluss auf das Schwingungsverhalten eines einfachen Rotors, Dissertation, T. H. Karlsruhe, 1966.

41. Lund, J. W., Stability and damped critical speeds of a flexible rotor in fluid-film bearings, Trans. ASME, J. Engng. Ind., Series B, Vol.96, No.2, pp.509-517, May 1974.


42. Lund, J. W. and Sternlicht, B., Rotor-bearing dynamics with emphasis on attenuation, Trans. ASME, J. Basic Engng., Vol.84, No.4, pp.491, 1962.

43. Ruhl, R. L., Dynamics of distributed parameter turborotor systems: Transfer matrix and finite element techniques, Ph. D. Thesis, Cornell Univ., Ithaca, N. Y., Jan.1970.

44. Nordmann, R., Ein Nherungsverfahren zur Berechnung der Eigenwerte und Eigenformen von Turborotoren mit Gleitlagern, Spalterregung, ausserer und innerer Dmpfung, Dissertation, T. H. Darmstadt, 1974.

45. Ruhl, R. L. and Booker J. F., A finite element model for distributed parameter turborotor systems, Trans. ASME, Series B, J. Eng. Industry, Vol.94, No.1, pp.126-132, Febr.1972.

46. Gasch, R., Unwucht-erzwungene Schwingungen und Stabilitt von Turbinenlufern, Konstruktion, Vol.25, Heft 5, pp.161-168, 1973.

47. Rouch, K. and Kao, J., Dynamic reduction in rotor dynamics by the finite element method, J. Mechanical Design, Vol.102, pp.360-368, 1980.

48. Jcker, M., Vibration analysis of large rotor-bearing-foundation systems using a model condensation for the reduction of unknowns, Proc. Second Int. Conf. "Vibration in Rotating Machinery", Cambridge, U.K., Paper C280, pp.195-202, 1980.

49. Lomakin, A., Calculation of critical number of revolutions and the conditions necessary for dynamic stability of rotors in high-pressure hydraulic machines when taking into account forces originating in sealings, Power and Mechanical Engineering, April 1958 (in Russian).

50. Black, H., Effects of hydraulic forces on annular pressure seals on the vibrations of centrifugal pump rotors, Journal of Mechanical Engineering Science, Vol.11, No.2, pp.206-213, 1969.

51. Childs, D. and Kim C.-H., Analysis and testing of rotordynamic coefficients of turbulent annular seals, J. of Tribology, Vol.107, pp.296-306, 1985.

52. Benckert, H. and Wachter, J., Studies on vibrations stimulated by lateral forces in sealing gaps, AGARD Proc. No.237 Conf. Seal Technology in Gas-Turbine Engines, London, pp.9.1-9.11, 1978.

53. Iwatsubo, T., Evaluation of instability forces of labyrinth seals in turbines or compressors, Rotordynamic Instability Problems in High-Performance Turbomachinery, NASA CP No.2133, pp.139-167, 1980.

54. Thomas, H., Instabile Eigenschwingungen von Turbinenlufern angefacht durch die Spaltstromungen Stopfbuschen und Beschauflungen, Bull. de l'AIM, vol.71, pp.1039-1063, 1958.


55. Alford, J., Protecting turbomachinery from self-excited rotor whirl, Trans. ASME, J. Engng. Power, pp.333-344, 1965.

56. Buzdugan, Gh. and Hamburger, L., Teoria vibraiilor, Editura tehnic, Bucureti, 1958.

57. Tipei, N., Hidro-aerodinamica lubrificaiei, Editura Academiei, Bucureti, 1957.

58. Constantinescu, V. N., Lubrificaia cu gaze, Editura Academiei, Bucureti, 1963.

59. Constantinescu, V. N., Aplicaii industriale ale lagrelor cu aer, Editura Academiei, Bucureti, 1968.

60. Constantinescu, V. N., Teoria lubrificaiei n regim turbulent, Editura Academiei, Bucureti, 1965.

61. Tipei, N., Constantinescu, V. N., Nica, Al., and Bi, O., Lagre cu alunecare, Editura Academiei, Bucureti, 1961.

62. Constantinescu, V. N., Nica, Al., Pascovici, M. D., Ceptureanu, Gh., and Nedelcu, t., Lagre cu alunecare, Editura tehnic, Bucureti, 1980.

63. Rdoi, M., Contribuii la studiul dinamicei i stabilitii rotorilor, cu considerarea influenei reazemelor, Tez de doctorat, Inst. Politehnic Timioara, 1971.

64. Bi, O., Program pentru calculul rspunsului dinamic al unui rotor, INCREST, Bucureti, 1973.

65. Lund, J. W., Rotor-Bearings Dynamic Design Technology Part III: Design Handbook for Fluid-Film Bearings, Mechanical Technology Inc. Report AFAPL-Tr-65-45, 1965.

2. SIMPLE ROTORS IN RIGID BEARINGS

Simple single-disc rotors with massless shafts supported in rigid bearings are considered in this chapter. The effect of damping and gyroscopic couples on the rotor precession is examined in detail.

2.1 Simple rotor models

The simplest flexible rotor consists of a rigid disc, fixed on a flexible shaft of axi-symmetric cross section, supported at the ends in identical bearings. The symmetric rotor, with a massless shaft supported in rigid bearings (Fig. 2.1) is known as the Laval-Jeffcott model [1, 2]. Generally, only the first precession mode is studied for which, because of the symmetry, the disc rotary inertia can be neglected. The model serves to the introduction of the concepts of critical speed and synchronous precession.

Fig. 2.1 Fig. 2.2

The Stodola-Green model [3-5] consists of a flexible shaft with an over-hung disc, not necessarily thin (Fig. 2.2). The model is used to examine the influence of the disc rotary inertia and gyroscopic torques on the rotor precession, the concepts of forward and backward precession, as well as the effect of the unbalance due to the skew mounting of the disc on the shaft.


In the following, the rotor shaft is considered to be rigidly supported. This is possible when the shaft stiffness is much lower than (less than 10% of) the combined stiffness of bearings and pedestals. The model simplification allows the stepwise introduction of the influence of mass unbalance, external and internal damping, and gyroscopic coupling, neglecting the bearing flexibility and damping.

2.2 Symmetric undamped rotors

Consider a rotor which consists of a flexible shaft of circular cross-section, supported at the ends in rigid bearings, and carrying a thin rigid disc in the symmetry plane, at mid distance between bearings (Fig. 2.3, a).

Fig. 2.3

Let the point G be the disc mass centre and point C - the disc geometric centre, where the geometric axis of the shaft intersects the disc plane. The disc has mass m and polar moment of inertia GJ . The bearing line intersects the disc at point O. Denote e=GC the offset of the disc mass centre G with respect to the point C.

2. SIMPLE ROTORS IN RIGID BEARINGS 41

The shaft stiffness coefficient is denoted by k (the ratio of a force applied to the shaft middle and the static deflection produced at the same point). For the symmetric rotor, it is 348 l/EIk = where l is the span between bearings, I is the shaft cross-section second moment of area, and E is the shaft material Young's modulus.

In this section, the shaft mass, the damping forces and the static deflection (of the horizontal shaft) under the disc weight are neglected. It is assumed that the non-rotating shaft is rectilinear and the rotor motion is studied with respect to this static equilibrium position. Later on (Section 2.3.4) the effect of gravity on the horizontal rotor will be studied.

An inertial coordinate system with the origin in O is considered. The Ox axis coincides with the bearing line (axis of the non-rotating shaft). The horizontal axis Oz and the vertical axis Oy are in the disc median plane (Fig. 2.3, b).

The disc motion in its own plane can be described by the variation in time of either the coordinates Cy and Cz of the geometric centre C, or the coordinates

Gy and Gz of the mass centre G.

Under the action of an external torque ( )tM , the disc turns and, at a given time t, the line CG makes an angle (positive anti-clockwise) with the axis Oy.

2.2.1 Equations of motion

The disc equations of motion can be written using d'Alembert's principle. The disc is isolated and subjected to the elastic restoring force due to the shaft flexibility, the external torque, the inertia force and the inertia torque (Fig. 2.3, c) that must be in dynamic equilibrium [6].

The resulting equations of motion are

).t(MezkeykJ

,zkzm,ykym

CCG

CG

CG

=+=+=+

cos -sin

0 0

&&&&&&

(2.1)

The coordinates of the points C and G are related through

.ezz,eyy

CG

CG

sin cos

+=+=

(2.2)

Substituting (2.2) into (2.1) we obtain


).t(M)zy(ekJ

,ememzkzm

,ememykym

CCG

CC

CC

cossin

cos sin

sin cos 2

2

=+=++=+

&&&&&&&

&&&&& (2.3)

Denoting ,imJ GG2 = where Gi is the disc gyration radius with respect to

the spinning axis, the third equation (2.3) can be written

.ie

iz

iy

mk

J)t(M

GG

C

G

C

G cossin

= &&

Because Gie

2. SIMPLE ROTORS IN RIGID BEARINGS 43

Because, in reality, the free motion decays due to the inherent damping and dies out after a short time interval, in the following only the forced steady-state motion will be studied.

2.2.2 Steady state response

Substitution of (2.8) into (2.5) gives the magnitudes of the displacements of point C along the axes Oy and Oz, respectively:

.e/

/emkemzy

n

n

nCC )(1

)( 22

22

2

2

2

==== (2.9)

Due to the disc mass unbalance, the point C moves along a circle of radius

,/

/ezyrn

nCCC )(1

)(2

222

=+= (2.10)

with the angular speed . The above calculations can be written in a more compact form using

complex numbers.

Fig. 2.4

In the plane yOz, Oy is taken as the real axis and Oz as the imaginary axis (Fig. 2.4). The following notations are used

,zyr,zyr

GGG

CCC

i i

+=+=

(2.11)

where .1 i = Multiply

m. rades - dynamics of machinery 1

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