Theory of Vibration Protection

Igor A. Karnovsky • Evgeniy Lebed

Theory of VibrationProtection

Igor A. KarnovskyCoquitlam, BC, Canada

Evgeniy LebedMDA Systems Ltd.Scientific and Engineeringstaff member

Burnaby, BC, Canada

ISBN 978-3-319-28018-9 ISBN 978-3-319-28020-2 (eBook)DOI 10.1007/978-3-319-28020-2

Library of Congress Control Number: 2016938787

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Preface

Decreasing the level of vibration of machines, devices, and equipment is one of the

most important problems of modern engineering. Suppression of harmful vibrations

contributes to the product’s normal functionality, leads to increased product reli-

ability, and reduces the negative impact on the human operator. This is the reason

why suppressing vibrations is a complicated technical issue with far-reaching

implications. The set of methods and means for reducing vibrations is called

vibration protection (VP).

Modern objects for which VP is necessary include engineering structures,

manufacturing equipment, airplanes, ships, and devices on mobile objects, to

name a few. The principal approaches to VP, concepts, and methods remain the

same regardless of the variations in different objects. Modern VP theory encom-

passes a broad scope of ideas, concepts, and methods. The theory of VP is largely

based on the common fundamental laws of vibration theory, theory of structures,

and control system theory and extensively uses the theory of differential equations

and complex analysis.

This book presents a systematic description of vibration protection problems,

which are classified as passive vibration protection, parametric (invariant), and

active vibration protection.

Passive vibration suppression means usage of passive elements only, which do

not have an additional source of energy. The passive vibration protection leads to

three different approaches: vibration isolation, vibration damping, and suppression

of vibration using dynamic absorbers. The passive vibration protection theory uses

the concepts and methods of linear and nonlinear theory of vibration.

One method of vibration protection of mechanical systems is internal vibration

protection: changing the parameters of the system can reduce the level of vibra-

tions. This type of vibrations reduction we will call parametric vibration protection.

The problem is to determine corresponding parameters of the system. Parametric

vibration protection theory is based on the Shchipanov-Luzin invariance principle

and uses the theory of linear differential equations.

v

Active vibration suppression is achieved by the introduction into the system of

additional devices with a source of energy. The problem is to determine additional

exposure as a function of time or function of the current state of the system. Optimal

active vibration protection theory is based on the Pontryagin principle and the Krein

moments method; these methods allow us to take into account the restrictions of the

different types.

This book is targeted for graduate students and engineers working in various

engineering fields. It is assumed that the reader has working knowledge of vibra-

tions theory, complex analysis, and differential equations. Textual material of the

book is compressed, and in many cases the formulas are presented without any

rigorous mathematical proofs. The book has a theoretical orientation, so technical

details of specific VP devices are not discussed.

The book does not present the complete vibration protection theory. The authors

included in the book only well probated models and methods of analysis, which can

be treated as classical. The number of publications devoted to the VP problem is so

large that it is impossible to discuss every interesting work in the restricted volume

of this book. Therefore, we apologize to many authors whose works are not

mentioned here.

The book contains an Introduction, four Parts (17 chapters), and an Appendix.

Introduction contains short information about the source of vibrations. It

describes briefly the types of mechanical exposures and their influence on the

technical objects and on a human. The dynamic models of the vibration protection

objects, as well as principal methods of vibration protection are discussed.

Part I (Chaps. 1–9) considers different approaches to passive vibration protec-

tion. Among them are vibration isolation (Chaps. 1–4), vibration damping (Chap. 5)

and vibration suppression (Chaps. 6 and 7). This part also contains parametric

vibration protection (Chap. 8) and nonlinear vibration protection (Chap. 9).

Part II considers two fundamental methods for optimal control of the dynamic

processes. They are the Pontryagin principle (Chap. 10) and Krein moments method

(Chap. 11). These methods are applied to the problems of active vibration suppres-

sion. Also, this part of the book presents the arbitrary vibration protection system

and its analysis using block diagrams (Chap. 12).

Part III is devoted to the analysis of structures subjected to impact. Chapter 13

presents the analysis of transient vibration of linear dynamic systems using Laplace

transform. Active vibration suppression through forces and kinematic methods as

well as parametric vibration protection is discussed. Chapter 14 describes shock and

spectral theory. Chapter 15 is devoted to vibration protection of mechanical systems

subjected to the force and kinematic random exposures.

Part IV contains two special topics: suppression of vibrations at the source of

their occurrence (Chap. 16) and harmful influence of vibrations on the human

(Chap. 17); Chapter 17 was written together with Т. Моldon (Canada).

The Appendix contains some fundamental data. This includes procedures with

complex numbers and tabulated data for the Laplace transform.

vi Preface

Numbering of equations, (Figures and Tables) has been followed sequentially

throughout the chapter—the first number indicates the chapter; the second number

indicates the number of the figure equation (Figure or Table).

Problems of high complexity are marked with an asterisk*.

Coquitlam, BC, Canada Igor A. Karnovsky

Burnaby, BC, Canada Evgeniy Lebed

October 2015

Preface vii

Acknowledgments

We would like to express our gratitude to everyone who shared with us their

thoughts and ideas that contributed to the development of our book.

The authors are grateful to the numerous friends, colleagues, and co-authors of

their joint publications. The ideas, approaches, and study results, as well as the

concepts of this book, were discussed with them at the earliest stage of work.

One of the authors (I.A.K.) is sincerely grateful to the well-known specialists, his

colleagues, and friends. Among these are Acad. R.Sh. Adamiya (Georgia), prof.

A.E. Bozhko (Ukraine), prof. M.I. Kazakevich (Germany), acad. М.V. Khvingiya

(Georgia), prof. A.O. Rasskazov (Ukraine), prof. V.B. Grinyov (Ukraine), prof.

М.Z. Kolovsky (Russia), prof. S.S. Korablyov (Russia), prof. A.S. Tkachenko

(Ukraine). Although they were not directly involved in the writing of this book,

they were at the very beginning of the research that eventually formed the book.

Their advice, comments, suggestions, and support cannot be overstated.

The authors thank Mark Zhu and Sergey Nartovich for ongoing technical

assistance for computer-related problems.

The authors are grateful to Olga Lebed for her contribution as manager through-

out the period of the work on the book.

The authors will appreciate comments and suggestions to improve the current

edition. All constructive criticism will be accepted with gratitude.

Coquitlam, BC, Canada Igor A. Karnovsky

Burnaby, BC, Canada Evgeniy Lebed

October 2015

ix

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii

Part I Passive Vibration Protection

1 Vibration Isolation of a System with One or More

Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1 Design Diagrams of Vibration Protection Systems . . . . . . . . . . 3

1.2 Linear Viscously Damped System. Harmonic Excitation

and Vibration Protection Criteria . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Simplest Mechanical Model of a Vibration

Protection System . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.2 Force Excitation. Dynamic and Transmissibility

Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 Kinematic Excitation. Overload Vibration Coefficient

and Estimation of Relative Displacement . . . . . . . . . . . 10

1.3 Complex Amplitude Method . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.1 Vector Representation of Harmonic Quantities . . . . . . 15

1.3.2 Single-Axis Vibration Isolator . . . . . . . . . . . . . . . . . . 17

1.3.3 Argand Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3.4 System with Two Degrees of Freedom . . . . . . . . . . . . 20

1.4 Linear Single-Axis Vibration Protection Systems . . . . . . . . . . . 21

1.4.1 Damper with Elastic Suspension. Transmissibility

Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.4.2 Simplification of Vibration Isolators . . . . . . . . . . . . . . 24

1.4.3 Vibration Isolators Which Cannot Be Simplified . . . . . 26

1.4.4 Special Types of Vibration Isolators . . . . . . . . . . . . . . 26

1.5 Vibration Protection System of Quasi-Zero Stiffness . . . . . . . . . 28

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

xi

2 Mechanical Two-Terminal Networks for a System

with Lumped Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.1 Electro-Mechanical Analogies and Dual Circuits . . . . . . . . . . . 37

2.2 Principal Concepts of Mechanical Networks . . . . . . . . . . . . . . . 42

2.2.1 Vector Representation of Harmonic Force . . . . . . . . . . 42

2.2.2 Kinematic Characteristics of Motion . . . . . . . . . . . . . . 42

2.2.3 Impedance and Mobility of Passive Elements . . . . . . . 43

2.3 Construction of Two-Terminal Networks . . . . . . . . . . . . . . . . . 48

2.3.1 Two-Terminal Network for a Simple

Vibration Isolator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3.2 Two-Cascade Vibration Protection System . . . . . . . . . 52

2.3.3 Complex Dynamical System and Its Coplanar

Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.4 Mechanical Network Theorems . . . . . . . . . . . . . . . . . . . . . . . . 55

2.4.1 Combination of Mechanical Elements . . . . . . . . . . . . . 56

2.4.2 Kirchhoff’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.4.3 Reciprocity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.4.4 Superposition Principle . . . . . . . . . . . . . . . . . . . . . . . . 59

2.5 Simplest One-Side m–k–b Vibration Isolator . . . . . . . . . . . . . . 60

2.5.1 Force Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.5.2 Kinematic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.6 Complex One-Sided m–k–b Vibration Isolators . . . . . . . . . . . . . 66

2.6.1 Vibration Isolator with Elastic Suspension . . . . . . . . . . 66

2.6.2 Two-Cascade Vibration Protection System . . . . . . . . . 67

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3 Mechanical Two-Terminal and Multi-Terminal Networks

of Mixed Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.1 Fundamental Characteristics of a Deformable System

with a Vibration Protection Device . . . . . . . . . . . . . . . . . . . . . 75

3.1.1 Input and Transfer Impedance and Mobility . . . . . . . . . 76

3.1.2 Impedance and Mobility Relating

to an Arbitrary Point . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.2 Deformable Support of a Vibration Protection System . . . . . . . 84

3.2.1 Free Vibrations of Systems with a Finite

Number of Degrees of Freedom . . . . . . . . . . . . . . . . . 84

3.2.2 Generalized Model of Support and Its Impedance . . . . 89

3.2.3 Support Models and Effectiveness Coefficient

of Vibration Protection . . . . . . . . . . . . . . . . . . . . . . . . 91

3.3 Optimal Synthesis of the Fundamental Characteristics . . . . . . . 93

3.3.1 Problem Statement of Optimal Synthesis.

Brune’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.3.2 Foster’s Canonical Schemes . . . . . . . . . . . . . . . . . . . . 95

3.3.3 Cauer’s Canonical Schemes . . . . . . . . . . . . . . . . . . . . 100

xii Contents

3.3.4 Support as a Deformable System

with Distributed Mass . . . . . . . . . . . . . . . . . . . . . . . . 104

3.4 Vibration Protection Device as a Mechanical

Four-Terminal Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.4.1 Mechanical Four-Terminal Network for Passive

Elements with Lumped Parameters . . . . . . . . . . . . . . . 111

3.4.2 Connection of an М4ТN with Support

of Impedance Zf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.4.3 Connections of Mechanical Four-Terminal

Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

3.5 Mechanical Multi-Terminal Networks for Passive

Elements with Distributed Parameters . . . . . . . . . . . . . . . . . . . 127

3.5.1 M4TN for Longitudinal Vibration of Rod . . . . . . . . . . 128

3.5.2 Mechanical Eight-Terminal Network for Transversal

Vibration of a Uniform Beam . . . . . . . . . . . . . . . . . . . 130

3.6 Effectiveness of Vibration Protection . . . . . . . . . . . . . . . . . . . . 135

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4 Arbitrary Excitation of Dynamical Systems . . . . . . . . . . . . . . . . . . 141

4.1 Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.1.1 Analysis in the Time Domain . . . . . . . . . . . . . . . . . . . 141

4.1.2 Logarithmic Plot of Frequency Response.

Bode Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.2 Green’s Function and Duhamel’s Integral . . . . . . . . . . . . . . . . . 151

4.2.1 System with Lumped Parameters . . . . . . . . . . . . . . . . 152

4.2.2 System with Distributed Parameters . . . . . . . . . . . . . . 156

4.3 Standardizing Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5 Vibration Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.1 Phenomenological Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.1.1 Models of Material . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.1.2 Complex Modulus of Elasticity . . . . . . . . . . . . . . . . . . 170

5.1.3 Dissipative Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

5.1.4 Dimensionless Parameters of Energy Dissipation . . . . . 172

5.2 Hysteretic Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5.2.1 Hysteresis Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5.2.2 Hysteretic Damping Concept . . . . . . . . . . . . . . . . . . . 178

5.2.3 Forced Vibration of a System with One Degree

of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

5.2.4 Comparison of Viscous and Hysteretic Damping . . . . . 182

5.3 Structural Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

5.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Contents xiii

5.3.2 Energy Dissipation in Systems with Lumped

Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

5.3.3 Energy Dissipation in Systems with Distributed

Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

5.4 Equivalent Viscous Damping . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5.4.1 Absorption Coefficient . . . . . . . . . . . . . . . . . . . . . . . . 189

5.4.2 Equivalent Viscoelastic Model . . . . . . . . . . . . . . . . . . 189

5.5 Vibration of a Beam with Internal Hysteretic Friction . . . . . . . . 191

5.6 Vibration of a Beam with External Damping Coating . . . . . . . . 194

5.6.1 Vibration-Absorbing Layered Structures . . . . . . . . . . . 195

5.6.2 Transverse Vibration of a Two-Layer Beam . . . . . . . . 196

5.7 Aerodynamic Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

5.7.1 The Interaction of a Structure with a Flow . . . . . . . . . . 201

5.7.2 Aerodynamic Reduction of Vibration . . . . . . . . . . . . . 202

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

6 Vibration Suppression of Systems with Lumped Parameters . . . . . 207

6.1 Dynamic Absorber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

6.2 Dynamic Absorbers with Damping . . . . . . . . . . . . . . . . . . . . . 213

6.2.1 Absorber with Viscous Damping . . . . . . . . . . . . . . . . . 214

6.2.2 Viscous Shock Absorber . . . . . . . . . . . . . . . . . . . . . . . 216

6.2.3 Absorber with Coulomb Damping . . . . . . . . . . . . . . . . 217

6.3 Roller Inertia Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

6.4 Absorbers of Torsional Vibration . . . . . . . . . . . . . . . . . . . . . . . 222

6.4.1 Centrifugal Pendulum Vibration Absorber . . . . . . . . . . 222

6.4.2 Pringle’s Vibration Absorber . . . . . . . . . . . . . . . . . . . 226

6.5 Gyroscopic Vibration Absorber . . . . . . . . . . . . . . . . . . . . . . . . 228

6.5.1 Elementary Theory of Gyroscopes . . . . . . . . . . . . . . . 229

6.5.2 Schlick’s Gyroscopic Vibration Absorber . . . . . . . . . . 232

6.6 Impact Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

6.6.1 Pendulum Impact Absorber . . . . . . . . . . . . . . . . . . . . . 235

6.6.2 Floating Impact Absorber . . . . . . . . . . . . . . . . . . . . . . 237

6.6.3 Spring Impact Absorber . . . . . . . . . . . . . . . . . . . . . . . 238

6.7 Autoparametric Vibration Absorber . . . . . . . . . . . . . . . . . . . . . 238

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

7 Vibration Suppression of Structures with Distributed

Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

7.1 Krylov–Duncan Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

7.2 Lumped Vibration Absorber of the Beam . . . . . . . . . . . . . . . . . 250

7.3 Distributed Vibration Absorber . . . . . . . . . . . . . . . . . . . . . . . . 254

7.4 Extension Rod as Absorber . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

xiv Contents

8 Parametric Vibration Protection of Linear Systems . . . . . . . . . . . . 265

8.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

8.2 Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

8.2.1 Shchipanov–Luzin Absolute Invariance . . . . . . . . . . . . 266

8.2.2 Invariance up to ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

8.3 Parametric Vibration Protection of the Spinning Rotor . . . . . . . 271

8.4 Physical Feasibility of the Invariance Conditions . . . . . . . . . . . 275

8.4.1 Uncontrollability of “Perturbation-Coordinate”

Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

8.4.2 Petrov’s Two-Channel Principle . . . . . . . . . . . . . . . . . 277

8.4.3 Dynamic Vibration Absorber . . . . . . . . . . . . . . . . . . . 278

8.5 Parametric Vibration Protection of the Plate

Under a Moving Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

8.5.1 Mathematical Model of a System . . . . . . . . . . . . . . . . 280

8.5.2 Petrov’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

9 Nonlinear Theory of Vibration Protection Systems . . . . . . . . . . . . . 289

9.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

9.1.1 Types of Nonlinearities and Theirs Characteristics . . . . 290

9.1.2 Features of Nonlinear Vibration . . . . . . . . . . . . . . . . . 294

9.2 Harmonic Linearization Method . . . . . . . . . . . . . . . . . . . . . . . 295

9.2.1 Method Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . 295

9.2.2 Coefficients of Harmonic Linearization . . . . . . . . . . . . 300

9.3 Harmonic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

9.3.1 Duffing’s Restoring Force . . . . . . . . . . . . . . . . . . . . . . 303

9.3.2 Nonlinear Restoring Force and Viscous Damping . . . . 307

9.3.3 Linear Restoring Force and Coulomb’s

Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

9.3.4 Internal Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

9.4 Nonlinear Vibration Absorber . . . . . . . . . . . . . . . . . . . . . . . . . 319

9.5 Harmonic Linearization and Mechanical Impedance

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

9.6 Linearization of a System with an Arbitrary Number

of Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

Part II Active Vibration Protection

10 Pontryagin’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

10.1 Active Vibration Protection of Mechanical Systems

as a Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

10.1.1 Mathematical Model of Vibration

Protection Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

Contents xv

10.1.2 Classification of Optimal Vibration

Protection Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 340

10.2 Representation of an Equation of State in Cauchy’s

Matrix Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

10.3 Qualitative Properties of Vibration Protection Systems . . . . . . . 347

10.3.1 Accessibility, Controllability, Normality . . . . . . . . . . . 347

10.3.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

10.4 Pontryagin’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

10.5 Vibration Suppression of a System with Lumped Parameters . . . 357

10.5.1 Vibration Suppression Problems

Without Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 358

10.5.2 Vibration Suppression Problem with Constrained

Exposure. Quadratic Functional, Fixed Time

and Fixed End . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

10.6 Bushaw’s Minimum-Time Problem . . . . . . . . . . . . . . . . . . . . . 369

10.7 Minimum Isochrones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

11 Krein Moments Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

11.1 The Optimal Active Vibration Protection Problem

as the l-moments Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

11.1.1 Formulation of the Problem of Vibration

Suppression as a Moment Problem . . . . . . . . . . . . . . . 386

11.1.2 The l-moments Problem and Numerical

Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

11.2 Time-Optimal Problem for a Linear Oscillator . . . . . . . . . . . . . 393

11.2.1 Constraint of Energy . . . . . . . . . . . . . . . . . . . . . . . . . 393

11.2.2 Control with Magnitude Constraint . . . . . . . . . . . . . . . 395

11.3 Optimal Active Vibration Protection of Continuous

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

11.3.1 Truncated Moments Problem . . . . . . . . . . . . . . . . . . . 398

11.3.2 Vibration Suppression of String. Standardizing

Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

11.3.3 Vibration Suppression of a Beam . . . . . . . . . . . . . . . . 404

11.3.4 Nonlinear Moment Problem . . . . . . . . . . . . . . . . . . . . 413

11.4 Modified Moments Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 415

11.5 Optimal Vibration Suppression of a Plate

as a Mathematical Programming Problem . . . . . . . . . . . . . . . . . 420

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

12 Structural Theory of Vibration Protection Systems . . . . . . . . . . . . 427

12.1 Operator Characteristics of a Dynamical System . . . . . . . . . . . . 428

12.1.1 Types of Operator Characteristics . . . . . . . . . . . . . . . . 428

xvi Contents

12.1.2 Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

12.1.3 Elementary Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

12.1.4 Combination of Blocks. Bode Diagram . . . . . . . . . . . . 441

12.1.5 Block Diagram Transformations . . . . . . . . . . . . . . . . . 448

12.2 Block Diagrams of Vibration Protection Systems . . . . . . . . . . . 450

12.2.1 Representation of b–k and b–m Systems

as Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

12.2.2 Vibration Protection Closed Control System . . . . . . . . 457

12.2.3 Dynamic Vibration Absorber . . . . . . . . . . . . . . . . . . . 463

12.3 Vibration Protection Systems with Additional

Passive Linkages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

12.3.1 Linkage with Negative Stiffness . . . . . . . . . . . . . . . . . 465

12.3.2 Linkage by the Acceleration . . . . . . . . . . . . . . . . . . . . 466

12.4 Vibration Protection Systems with Additional

Active Linkages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

12.4.1 Functional Schemes of Active Vibration

Protection Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

12.4.2 Vibration Protection on the Basis of Excitation.

Invariant System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

12.4.3 Vibration Protection on the Basis of Object State.

Effectiveness Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 471

12.4.4 Block Diagram of Optimal Feedback Vibration

Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

Part III Shock and Transient Vibration

13 Active and Parametric Vibration Protection of TransientVibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

13.1 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

13.2 Heaviside Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

13.3 Active Suppression of Transient Vibration . . . . . . . . . . . . . . . . 501

13.3.1 Step Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

13.3.2 Impulse Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

13.4 Parametric Vibration Suppression . . . . . . . . . . . . . . . . . . . . . . 508

13.4.1 Recurrent Instantaneous Pulses . . . . . . . . . . . . . . . . . . 508

13.4.2 Recurrent Impulses of Finite Duration . . . . . . . . . . . . . 510

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

14 Shock and Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

14.1 Concepts of Shock Excitation . . . . . . . . . . . . . . . . . . . . . . . . . 519

14.1.1 Types of Shock Exposures . . . . . . . . . . . . . . . . . . . . . 519

14.1.2 Different Approaches to the Shock Problem . . . . . . . . 521

Contents xvii

14.1.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

14.1.4 Time and Frequency Domain Concepts . . . . . . . . . . . . 536

14.2 Forced Shock Excitation of Vibration . . . . . . . . . . . . . . . . . . . 537

14.2.1 Heaviside Step Excitation . . . . . . . . . . . . . . . . . . . . . . 538

14.2.2 Step Excitation of Finite Duration . . . . . . . . . . . . . . . . 540

14.2.3 Impulse Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

14.3 Kinematic Shock Excitation of Vibration . . . . . . . . . . . . . . . . . 544

14.3.1 Forms of the Vibration Equation . . . . . . . . . . . . . . . . . 545

14.3.2 Response of a Linear Oscillator to Acceleration

Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

14.4 Spectral Shock Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

14.4.1 Biot’s Dynamic Model of a Structure: Primary

and Residual Shock Spectrum . . . . . . . . . . . . . . . . . . . 549

14.4.2 Response Spectra for the Simplest Vibration

Protection System . . . . . . . . . . . . . . . . . . . . . . . . . . . 551

14.4.3 Spectral Method for Determination of Response . . . . . 552

14.5 Brief Comments on the Various Methods of Analysis . . . . . . . . 554

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

15 Statistical Theory of the Vibration Protection Systems . . . . . . . . . . 561

15.1 Random Processes and Their Characteristics . . . . . . . . . . . . . . 562

15.1.1 Probability Distribution and Probability Density . . . . . 563

15.1.2 Mathematical Expectation and Dispersion . . . . . . . . . . 565

15.1.3 Correlational Function . . . . . . . . . . . . . . . . . . . . . . . . 568

15.2 Stationary Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . 570

15.2.1 Properties of Stationary Random Processes . . . . . . . . . 570

15.2.2 Ergodic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

15.2.3 Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574

15.2.4 Transformations of Random Exposures

by a Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . 577

15.3 Dynamic Random Excitation of a Linear Oscillator . . . . . . . . . 582

15.3.1 Transient Vibration Caused by Impulse Shock . . . . . . . 583

15.3.2 Force Random Excitation . . . . . . . . . . . . . . . . . . . . . . 587

15.4 Kinematic Random Excitation of Linear Oscillator . . . . . . . . . . 591

15.4.1 Harmonic and Polyharmonic Excitations . . . . . . . . . . . 591

15.4.2 Shock Vibration Excitation by a Set

of Damped Harmonics . . . . . . . . . . . . . . . . . . . . . . . . 597

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

xviii Contents

Part IV Special Topics

16 Rotating and Planar Machinery as a Source of Dynamic

Exposures on a Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

16.1 Dynamic Pressure on the Axis of a Rotating Body . . . . . . . . . . 605

16.2 Types of Unbalancing Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . 609

16.2.1 Static Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

16.2.2 Couple Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

16.2.3 Dynamic Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . 610

16.2.4 Quasi-Static Unbalance . . . . . . . . . . . . . . . . . . . . . . . 611

16.3 Shaking Forces of a Slider Crank Mechanism . . . . . . . . . . . . . . 612

16.3.1 Dynamic Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . 614

16.3.2 Elimination of Dynamic Reactions . . . . . . . . . . . . . . . 617

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622

17 Human Operator Under Vibration and Shock . . . . . . . . . . . . . . . . 623

17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

17.1.1 Vibration Exposures and Methods

of Their Transfer on the Person . . . . . . . . . . . . . . . . . . 624

17.1.2 International and National Standards . . . . . . . . . . . . . . 628

17.2 Influence of Vibration Exposure on the Human Subject . . . . . . . 628

17.2.1 Classification of the Adverse Effects

of Vibration on the Person . . . . . . . . . . . . . . . . . . . . . 629

17.2.2 Effect of Vibration on the Human Operator . . . . . . . . . 631

17.3 Vibration Dose Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

17.4 Mechanical Properties and Frequency Characteristics

of the Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

17.4.1 Mechanical Properties of the Human Body . . . . . . . . . 640

17.4.2 Frequency Characteristics of the Human Body . . . . . . . 642

17.5 Models of the Human Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

17.5.1 Basic Dynamic 1D Models . . . . . . . . . . . . . . . . . . . . . 647

17.5.2 Dynamic 2D–3D Models of the Sitting

Human Body at the Collision . . . . . . . . . . . . . . . . . . . 651

17.5.3 Parameters of the Human Body Model . . . . . . . . . . . . 653

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657

Appendix A: Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

Appendix B: Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669

Contents xix

About the Authors

Igor A. Karnovsky, Ph.D., Dr. Sci. is a specialist in structural analysis, theory of

vibration, and optimal control of vibration. He has 40 years of experience in

research, teaching, and consulting in this field and is the author of more than

70 published scientific papers, including two books in Structural Analysis

(published with Springer in 2010–2012) and three handbooks in Structural Dynam-

ics (published with McGraw Hill in 2001–2004). He also holds a number of

vibration control-related patents.

Evgeniy Lebed, Ph.D. is a specialist in applied mathematics and engineering. He

has 10 years of experience in research, teaching, and consulting in this field. The

main sphere of his research interests are qualitative theory of differential equations,

integral transforms, and frequency-domain analysis with application to image and

signal processing. He is the author of 15 published scientific papers and holds a US

patent (2015).

xxi

Introduction

Mechanical Exposure and Vibration Protection Methods

The introduction contains a short summary about sources of vibration and the

objects of vibration protection. Different types of mechanical exposure, influences

on technical objects, and on human operators are briefly described. Dynamical

models of vibration protection objects and fundamental methods of vibration

protection are discussed.

Source of Vibration and Vibration Protection Objects

Amechanical system is the object of study in the theory of vibration protection. The

source of vibration induces mechanical excitations, which in turn are relayed by

connections to vibration protection objects (Fig. 1).

Excitation factors, which are the source of vibrations of the object, can occur for

several different reasons. These reasons are generally grouped into two categories;

internal, which arise due to normal function of the object itself, and external, which

generally do not depend on the functions carried out by the object. Internal

excitation factors can be further classified into two subcategories.

Excitation Factors Arising due to Moving Bodies Examples of moving bodies

include a rotating rotor, reciprocating piston motion, as well as any moving parts

of machinery. Moving parts inside a source usually give rise to dynamic reactions,

which arise in constraints. These connections transmit the dynamic forces on the

different objects, which are associated with the source of vibration, in particular,

objects that are responsible for eliminating or suppressing the vibrations. Hence-

forth, these objects will be referred to as Vibration Protection Objects (VPO).

xxiii

Reducing vibration activity of source vibrations amounts to reducing dynamic

reactions in the constraint. Balancing machinery methods, specifically, static and

dynamic balancing of rotating objects, such as rotors, and their corresponding

automatic balancing, are usually employed to achieve this goal. A detailed classifi-

cation of automated balancing techniques of machinery rotors is presented in [1, 2].

Excitation Factors Caused by Physical and Chemical Processes Originating atthe Source Such processes should include the following: Exhaust processes in

internal combustion and jet engines, processes involving interactions of liquids or

gasses with an engine’s turbine blades, pulsations of liquids and gasses in conduits,

electromagnetic reactions in engines and generators, various technological pro-

cesses (e.g., cutting of metals on powered metal-cutting equipment, processing of

materials in the mining equipment), etc. Changing the settings of the physical and

chemical processes can reduce the vibration activity factors in this group [3, vol. 4].

External factors are not related to an object’s function. These external factors

may include explosions, seismic influences, collisions, temperature fluctuations,

and wind loads.

Let us have a closer look at several examples of vibration protection objects and

influences that act upon them.

1a. An engine with an unbalanced rotor, mounted on a foundation. The vibration

protection problem involves reducing vibrations of the engine’s frame. The

engine’s frame is the object of vibration protection. The source of vibrations

(SoV) is the engine’s rotor. Dynamic excitations are the dynamic reactions of

the rotor’s supports (Fig. 2a, b).

Source ofVibration (SoV)

Vibration ProtectionObject (VPO)

Сonnection betweenSoV and VPO

Fig. 1 Scheme representing an interaction between Source of Vibration (SoV) and Vibration

Protection Object (VPO)

b

Rotor

VPO c

VPO

Rotor-SoV

Connection

Foundation

Rotor-SoV

a

Fig. 2 An unbalanced rotor as a source of vibration and two variation of the vibration protection

problem

xxiv Introduction

1b. For the same system, the goal here is to lower the vibrations of the foundation.

In this case the vibration protection object is the foundation. The source of

vibrations is the same as in the previous case—the unbalanced engine’s rotor.

The dynamic excitations are the dynamic reactions in the system that mounts

the engine to the foundation (Fig. 2c).

2a. Control panel, mounted inside an airplane’s cockpit. The vibration protection

problem is to reduce the vibrations of the control panel. The vibration protec-

tion object is the control panel. The source of vibrations is the aircraft with all

of its parts, which cause the vibrations of the control panel. Dynamic distur-

bances are the kinematic excitations of the points where the control panel is

fixed to the aircraft.

2b. For the same system, we can pose the problem of lowering vibrations of the

airplane’s hull at the location (or locations) where the control panel is mounted.

In this case, the VPO becomes the part of the aircraft to which the control panel

is mounted. The source of vibration in this case arises from multiple, simulta-

neously interacting parts of the aircraft, creating dynamical and acoustic

influences, which act on the VPO.

3. A problem of particular importance is how to properly protect a human

operator of transport equipment from vibrations. This type of problem has

many different types of approaches. In one case, we can choose the seat of the

human operator to be the VPO. In another case we may be interested in

reducing vibrations of an entire cabin; in this case, the cabin becomes the

VPO. Alternatively, we may want to reduce vibrations of the entire transpor-

tation mechanism.

Excitation of the system can be of either force (dynamic) or kinematic nature. Ifvibration of the object is caused by the load (force, torque), which is applied just to

the object, we have a case of force or dynamic excitation. If vibration of the object

is caused by the displacement, velocity, or acceleration of the base, then we have a

case of kinematic excitation. In both cases the vibration of the object depends on the

properties of connection between the object and the foundation. An example of

kinematic excitation is vibration of a pilot of the aircraft caused by the motion of

the seat.

From here on, we refer to general mechanical excitations as force (dynamic) and

kinematic excitations. The simplest case of such excitations is shown in Fig. 3.

m

k

F(t)a

m

k

(t)ξ

bFig. 3 (a) Force (dynamic)

and (b) kinematic excitation

Introduction xxv

Here, m represents the mass of the object, k is the stiffness coefficient of the

connection between foundation and object, and F(t) and ξ(t) refer to force and

kinematic excitations, respectively.

As such, in the case of internal excitation, the kinematic excitation is determined

by the problem formulation. In the case of external excitation, for example,

earthquakes, the kinematic character of excitation is natural.

Mechanical Exposures and Their Influence on TechnicalObjects and Humans

Mechanical exposures are commonly subdivided into three classes: linear overload,

vibrational exposures, and shocks.

Linear Overload

Mechanical effects of kinematic nature that arise during acceleration

(or deceleration) of objects are known as linear overloads. Linear overloads become

particularly prevalent during aircrafts’ takeoffs (or landings) and during an air-

craft’s maneuvers (roll, pitch, and yaw). The two main characteristics of linear

overloads are constant acceleration a0 (Fig. 4) and the maximal rate at which

acceleration grows _a ¼ max da=dt. This characteristic is known as jerk.

In special cases, linear overloads vary linearly in time. Linear overloads are

statically transferred to objects, and this is the primary reason why objects cannot be

protected from independently arising linear overloads. However, if linear overloads

are superimposed onto the vibrational or impact excitation, then the vibration

protection process significantly changes its nature and the characteristics of vibra-

tion protection (VP) devices become more complicated.

Three different types of operating states for VP devices are possible when an

object is fixed to a moving platform, which is able to move with large linear

accelerations in the presence of linear overloads.

Starting State At this stage the VP devices are in a state of stress, and current

overloads provide additional stress on the VP device.

a

t

a0

a

Fig. 4 Graph “linear

overload-time”

xxvi Introduction

Shutting Down the Starting Engines State During this state, the engines that were

initially used to accelerate the mechanism are turned off. The VP device, which was

stressed up to this point, is relaxed and instantaneously releases all of its stored

potential energy. This leads to a shock phenomenon, which could be hazardous to

the VP device.

Deceleration State This state is characterized by the fact that a significant linear

overload is applied to the VP device.

Vibrational Exposure

Force (dynamic) vibration exposures represent force F or torqueM, which act upon

an object. Acceleration (a) of points connected to the source (foundations, aircraft

hull, etc.), their velocities (υ) and displacements (x) represent kinematic vibrational

exposures. All of these exposures are functions of time. These exposures can be of

either stationary (steady-state) or non-stationary (unsteady-state) character.

Stationary Vibration Exposures The simplest exposures of this type have the

form

x tð Þ ¼ x0 sinω0t,

where x(t) is the vibrational force or kinematic exposure, x0 and ω0 represent the

amplitude and frequency of excitation. The period of an oscillation can be deter-

mined from the excitation frequency by T ¼ 2π=ω0.

Harmonic process and corresponding Spectra are shown in Fig. 5ab.

Harmonic force exposures are produced by unbalanced rotors, different types of

vibrators, and piston pumps [4]. Kinematic excitations are produced by vibrations

of the foundation to which the object is mounted [5].

Non-stationary Vibrational Excitation Such effects occur during transient pro-

cesses, originating at the source. For example, dynamic excitations acting upon an

engine’s hull during the rotor’s acceleration can be expressed by

x tð Þ ¼ a ωð Þ sin ω tð Þ tð Þ;where ω(t) represents the rotor’s angular acceleration, as a function of time.

t

xa

Tx0

T

b

x0

x

0ω ω

Fig. 5 Harmonic process

and its corresponding

spectra

Introduction xxvii

Polyharmonic Vibrational Excitation Excitations of this nature are described by

the following expression [3, vol. 1]:

x tð Þ ¼X1k¼1

ak cos kω0tþ bk sin kω0tð Þ:

The set of frequencies kω0 for k ¼ 1, 2, . . . ; of harmonic components, arranged

in ascending order, is called the frequency spectrum of the process. An amplitude

Ak ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2k þ b2k

q, and an initial phase φk, where tanφk ¼ bk=ak, is associated with

each frequency. The set amplitudes, sorted in ascending order of the respective

frequencies, form the amplitude spectra of the process. A typical amplitude spectra

of a polyharmonic excitation is shown in Fig. 6. Such effects usually occur in

machinery containing cyclic mechanisms [3, vols. 1, 4].

Bandwidth of frequenciesωmax � ωmin. The range of frequencies for whichωmax

=ωmin > 10 is referred to as broadband. If the energy spectra is concentrated aroundjust a few frequencies, such excitations are known as narrowband.

Geometric addition of two processes leads to a flat curve called a Lissajous

curve. The appearance of curves depends on correlation between frequencies,

amplitude, and phases of the two processes [3, vol. 1]. A beat is a phenomenon

occurring when two periodic oscillations with slightly different frequencies are

imposed one upon the other. In this case we observe a periodic growth/reduction in

the amplitude of the summed signal. The frequency of the amplitudes change, and

the resulting signal is equal to the difference in frequencies of the two original

signal [6].

The bandwidthωmin � ωmax of a polyharmonic excitation has a profound impact

on vibration protection problems. Depending on this bandwidth, different design

diagrams may be chosen to represent the vibration protection object. The model

should be chosen in such a way that all the eigenfrequencies of the vibrating object

fall into the bandwidth of the excitation spectra [2].

Exposure to high frequency vibrational excitations typically results in acousticvibrational effects. In this case the vibrational excitations are transferred to the

object not only by elements mechanical connections, but also by the surrounding

environment. High acoustic pressure can have a significant impact on high preci-

sion machinery, such as modern day jet engines and supersonic aircraft.

A1A

ω

A2A3

An

1ω 2ω 3ω nω

Fig. 6 Amplitude spectra

of a polyharmonic

excitation

xxviii Introduction

Chaotic Exposure The following expression can be used to characterize chaotic

vibrations:

x tð Þ ¼XNk¼1

ak cosωktþ bk sinωktð Þ:

A polyharmonic process with the ratio of frequencies forming an irrational

number describes a vibrational exposure excited by completely independent

sources.

Random Exposure It often happens that vibrational exposures are not fully deter-

ministic. This is explained by the following. The characteristics of vibrational

exposure can be determined either by calculations, or by in situ measurements. In

both cases, random factors play a significant role, and their influences are impos-

sible to determine beforehand. This is why such vibrational exposures are difficult,

and often impossible, to describe with standard functions. The only way that this

can be achieved is to characterize that process as random, and use the

corresponding characteristics. Some typical examples of random vibrational expo-

sures include pulsations of liquids as they move through pipes, aerodynamic noise

of a jet stream, and a vibrating platform with multiple objects fixed onto itself [7].

Impact Exposure

Impact exposures are classified into dynamic impact excitation (DIE) and kinematic

impact excitation (KIE). DIE implies that a system is under the action of impact

force or torque. KIE implies that a system is influenced by kinematic excitations;

such excitations arise during a rapid change in velocity (i.e., landing of an aircraft).

Both of these excitations are characterized by short temporal durations and signif-

icant maximum values. Oscillations caused by impacts are of unsteady nature.

The graph “force-time,” or “moment-time” for DIE and graph “acceleration-

time” for KIE is called form of impact. On this graph the force (moment, acceler-

ation) varies from zero to the peak value and again back to zero within the duration

of the impact interval. The main properties of an impact’s form include its duration,

amplitude, and spectral characteristics [8].

Influence of Mechanical Exposure on Technical Objects and Humans

Influences of Linear Overloads In their natural form (without any additional

exposures), such exposures lead to static loading of an object. In this case, for

example, linear overloads may lead to false operation of the relay devices.

Influence of Vibrational Excitation The harmful influence of such excitations are

manifested in diverse forms:

Introduction xxix

1. The biggest hazard related to this type of exposure is the appearance of

resonances.

2. Alternating exposures lead to an accumulation of damage in the material. This in

turn leads to an accumulation of fatigue damage and destruction.

3. Vibrational exposures lead to gradual weakening and erosion of fixed joints.

4. In connections with gaps, such exposures cause collisions between contact

surfaces.

5. These exposures result in damage to the structure’s surface layers, and premature

wear on the structure develops.

Particularly hazardous vibrational effects are manifested in the presence of

linear overloads [9].

Influences of shock excitations. Such exposures can lead to brittle fractures.

Resonances may occur during periodic shocks. Fatigue failures can occur in the

case of multiple recurrent shocks [2]. Similar to the case of vibrational exposures,

the addition of linear overloads significantly complicates the function of a vibration

protection system in shock excitations [9].

In the literature one can find numerous examples where different systems failed

to function properly or were even completely destroyed due to vibrational expo-

sures. Such systems range from the simplest to most complicated objects found in

transportation, aviation, civil engineering, structural engineering, etc.

Vibrational influences on a human depend on a number of factors [10]. These

factors include the spectral composition of vibrations, their durations, direction and

location at which they are applied, and finally each individual person’s physical

characteristics. Harmful vibrations are subdivided into two groups:

1. Vibrations influencing a person’s functional state;

2. Vibrations influencing a person’s physiological state

Negative vibrational effects of the first group lead to increased fatigue, increased

time of visual and motor reaction, and disturbance of vestibular reactions and

coordination. Negative vibrational effects of the second group lead to the develop-

ment of nervous diseases, violation of the functions of the cardiovascular system,

violation of the functions of the musculoskeletal system, and degradation of the

muscle tissues and joints.

Vibrational effects on a person’s functional state lead to reduced productivity

and quality, while vibrational effects on a person’s physiological state contribute to

chronic illnesses and even vibrational sickness [10].

Dynamical Models of Vibration Protection Objects

A fundamental characteristic of a dynamical system is the number of degrees of

freedom. The degrees of freedom is the number of independent coordinates that

uniquely determine the position of the system during its oscillation.

xxx Introduction

All structures may be divided into two principal classes according to their

degrees of freedom. They are structures with concentrated and distributed param-

eters (lumped and continuous systems). Members with lumped parameters assume

that the distributed mass of the member itself may be neglected in comparison with

the lumped mass, which is located on the member. The continuous system is

characterized by uniform or non-uniform distribution of mass within its parts.

From a mathematical point of view the difference between the two types of systems

is the following: the systems of the first class are described by ordinary differential

equations, while the systems of the second class are described by partial differential

equations. Examples of the lumped and continuous systems are shown below.

Figure 7a, b shows a massless statically determinate and statically indeterminate

beam with one lumped mass. These structures have one degree of freedom, since

transversal displacement of the lumped mass defines the position of all points of the

beam. A massless beam in Fig. 7c has three degrees of freedom. It can be seen that

introducing additional constraints on the structure increases the stiffness of the

structure, i.e., increases the degrees of static indeterminacy, while introducing

additional masses increases the degrees of freedom.

Figure 7d presents a cantilevered massless beam that is carrying one lumped

mass. However, this case is not a plane bending, but bending combined with torsion

because mass is not applied at the shear center. That is why this structure has two

degrees of freedom, the vertical displacement and angle of rotation in y–z planewith respect to the x-axis. A structure in Fig. 7e presents a massless beam with an

absolutely rigid body. The structure has two degrees of freedom, the lateral dis-

placement y of the body and angle of rotation of the body in y–x plane. Figure 7f

presents a bridge, which contains two absolutely rigid bodies. These bodies are

supported by a pontoon. Corresponding design diagram shows two absolutely rigid

bodies connected by a hinge Cwith elastic support. Therefore, this structure has one

degree of freedom.

Figure 8 presents plane frames and arches. In all cases we assume that no

members of a structure have distributed masses. Since the lumped mass M in

f

C

Pontoon

d x

y

z

a

y1

cy1 y2 y3

e x

y

b

y1

Fig. 7 (a–f) Design diagrams of several different structures

Introduction xxxi

Fig. 8a, b can move in vertical and horizontal directions, these structures have two

degrees of freedom. Figure 8c shows a two-story frame containing absolutely rigid

crossbars (the total mass of each crossbar is M ). This frame may be presented as

shown in Fig. 8d.

Arches with one and three lumped masses are shown in Fig. 8e, f. Taking into

account their vertical and horizontal displacements, the number of degrees of

freedom will be two and six, respectively. For gently sloping arches the horizontal

displacements of the masses may be neglected; in this case the arches should be

considered structures having one and three degrees of freedom in the verticaldirection.

All cases shown in Figs. 7 and 8 present design diagrams for systems with

lumped parameters. Since masses are concentrated, the configuration of a structure

is defined by displacement of each mass as a function of time, i.e., y ¼ y tð Þ, and thebehavior of such structures is described by ordinary differential equations. It is

worth discussing the term “concentrated parameters” for cases 7f (pontoon bridge)

and 8с (two-story frame). In both cases, the mass—in fact, the masses are distrib-

uted along the correspondence members. However, the stiffness of these members

is infinite, and the position of each of these members is defined by only onecoordinate. For the structure in Fig. 7f, such coordinate may be the vertical

displacement of the pontoon or the angle of inclination of the span structure, and

for the two-story frame (Fig. 8с), the horizontal displacements of each crossbar.

The structures with distributed parameters are generally more difficult to ana-

lyze. The simplest structure is a beam with a distributed mass m. In this case a

configuration of the system is determined by displacement of each elementary mass

as a function of time. However, since the masses are distributed, then a displace-

ment of any point is a function of a time t and location x of the point, i.e., y ¼ y x; tð Þ,so the behavior of the structures is described by partial differential equations.

It is possible to have a combination of the members with concentrated and

distributed parameters. Figure 9 shows a frame with a massless strut ВF (m¼ 0),

members AВ and ВС with distributed masses m, and absolutely rigid member СD(EI¼1). The simplest form of vibration is shown by the dotted line.

MEI=∞

MEI=∞

cM

M

d

Mb

e

f

a M

Fig. 8 (a–f) Design diagrams of frames and arches

xxxii Introduction

If in Fig. 7a, we take into account the distributed mass of the beam and the

lumped mass of the body, then the behavior of the system is described by differ-

ential equations—partial derivatives of the beam and ordinary derivatives of

the body.

The diversity of mechanical systems usually makes it necessary to represent

them in conditional forms. To achieve this, we employ three different passive

elements: mass, stiffness and damper. A damper is a mechanism in which energy

is dissipated. Each of the systems in Fig. 7a, b, f may be represented as one degree

of freedom systems, neglecting damping, as shown in Fig. 3.

Let us return to Fig. 7a. The system shown here is described by a second-order

ordinary differential equation. Introduction of two additional masses (Fig. 7c)

increases the number of degrees of freedom by two. This leads to an introduction

of two additional differential equations of second order.

The model of any system with two degrees of freedom (Figs. 7d, f and 8a–e) may

be presented (neglecting damping) as shown in Fig. 10. This model may be applied

for force, as well as kinematic excitations. Stiffness coefficients k1 and k1 depend onthe type of structure and the structure’s boundary conditions. Their derivations are

presented in [11].

The system shown in Fig. 10 is described by two second-order ordinary differ-

ential equations. The order of equations will not change if dampers, parallel to the

elastic elements, are introduced into the system.

Special Case Assume that a damper is attached to an arbitrary point on the system

“massless beam + lumped mass m” (Fig. 11), except directly on the mass.

This system is described by two ordinary differential equations

y1 ¼ �b _y1δ11 � m€y2δ12,y2 ¼ �b _y1δ21 � m€y2δ22:

m2m1k1 k2

Fig. 10 Design diagram of a mechanical system with two degrees of freedom

EI=∞EI, m

EI, m=0

A B C D

F

EI, mFig. 9 Frame with

distributed and lumped

parameters

mEI

y1 y2

b

Fig. 11 Mechanical system

with 1.5 degrees of freedom

Introduction xxxiii

The second equation, for the mass, is second order with respect to y2, while firstequation for the damper is first order with respect to y1. Here δik are unit displace-ments; their calculation is discussed in [12]. The two equations describing this

system can be reduced to one third-order equation, so the total number of degrees of

freedom for this system is 1.5 [13].

An arbitrary vibration protection system can be described by a linear and

nonlinear differential equation. For systems with lumped parameters we have the

ordinary differential equations, while for systems with distributed parameters, we

use partial differential equations. For a linear stiffness element, such as a spring of

zero mass, the applied force and relative displacement of the ends of the element are

proportional. For a linear damping element, which has no mass, the applied force

and relative velocity of the ends of the element are proportional. For a linear system

the superposition principle is valid. Superposition principle means that any factor,

such as reaction or displacement, caused by different loads acting simultaneously,

are equal to the algebraic or geometrical sum of this factor due to each load

separately [14].

Vibration Protection Methods

Three fundamentally different approaches can be used to reduce vibrations in an

object. These approaches are

1. Lowering the source’s vibrational activity;

2. Passive vibration protection;

3. Active vibration protection.

Lowering the Source’s Vibrational Activity The set of methods used to lower

vibrational activity in machines and instrumentation is based on static and dynamic

balancing of rotors and, in general, balancing any moving parts in the machinery

[2, 15].

Passive vibration protection implies the absence of external sources of energy for

devices, which drive the vibration protection process. This type of vibration

protection can be achieved via isolating and damping vibrations, as well as changes

to the structure and parameters of the object. Typically these methods are charac-

terized by vibration isolation, vibration damping, and vibration absorption. Passive

vibration protection systems include the mechanical system itself, as well as

additional masses, elastic elements, devices for dissipating energy, and potentially

other massless elements.

Vibration isolation is a method to reduce oscillations in a mechanical system

(object) where additional devices that weaken connections between the object and

the source of vibrations are introduced into the system [2, 16, 17]. Such devices are

called vibration isolators. If the source of excitation is located inside the object, then

the excitation is force. Otherwise, if the source of excitation is located outside the

xxxiv Introduction

object, then the excitation of the mechanical system is kinematic, and the

corresponding vibration isolation is kinematic. A simplified schematic of a vibra-

tion isolator is shown in Fig. 12а. Weakening of connections between the object

and foundation is achieved by an elastic element.

Vibration damping is a type of method to reduce oscillations in an object that

involves introducing additional devices that facilitate the dissipation of energy [2,

16, 18]. Such devices are called dampers. This method can be interpreted as a way

of altering the object’s structure. A vibration isolator with a damper is shown in

Fig. 12b.

Vibration absorption involves reducing oscillations in a system by introducing

devices called absorbers into the system [2, 16, 19, 20, 21]. Absorbers create an

additional excitation that compensates for the primary excitation and reduces the

object’s vibration by transferring the oscillation energy onto the absorber. An object

m with an elastic element k, damper b, and absorber ma–ka is shown in Fig. 12c. In

all of the cases shown in Fig. 12, oscillations can be caused by dynamic or

kinematic excitations.

In the class of passive vibration protection systems one can identify optimalpassive systems. Here we are talking about the best type of additional device or best

set of system parameters concerning vibration isolation, vibration damping and

vibration absorption. One is free to choose the desired optimality criteria to quantify

the vibration protection process. Some of these criteria may include the minimum

dimensions of the system, the shortest time in which the desired level of oscillations

is achieved, and many others [22, 23].

Changing the Parameters of the Object and Structure of Vibration ProtectionDevices The essence of this method is to tune out the resonant modes. This is

accomplished by changing the frequency of the object’s oscillations without using

additional devices, as well as using additional passive elements, in particular,

employing devices that facilitate energy dissipation. Using these techniques allows

us to eliminate the resonance regime and, as result, to reduce amplitude of

vibration.

Active Vibration Protection refers to an automatic control system in the presence

of additional sources of energy [23–26]. A schematic of a typical active VP system

is shown in Fig. 13. The vibration protection object of the mass m is connected to

support S using block 1 of passive elements. The active part of the VP system

m

k

a

c

b

m

k

am

akb

b

m

k

Fig. 12 Simplest models of

passive vibration protection

Introduction xxxv

contains sensors 2 of state of object, devices 3 for signal conversion, and executive

mechanism 4 (actuator). The system is subjected to force and/or kinematical

excitation.

One major advantage of active vibration protection systems is their ability to

optimally reduce (or eliminate) vibrations while adhering to constraints. For exam-

ple, one can set the goal to suppress vibrations in the shortest possible time while

adhering to the constraint of only consuming a certain amount of energy.

Parametric Vibration Protection This type of vibration protection pertains to

linear dynamic systems subjected to excitations. The types of excitations are not

discussed. This method is based on the Shchipanov-Luzin invariance principle,

which is one of the modern methods of control theory [27, 28]. For a certain set of

parameters, one or more generalized coordinates of the system do not react to the

excitation. In other words, these coordinates are invariant with respect to external

excitation. The Shchipanov-Luzin’s principle provides us with a method to deter-

mine the system parameters which lead to realization of invariance conditions.

Estimating the Effectiveness of Vibration Reduction

The effectiveness of vibration protection can be estimated by the reduced levels of

vibrations of the object or by reduced dynamic loads transmitted upon the object or

foundation. For this purpose the different approaches can be used. Among them,

particularly, are estimation according the kinematical parameters, transmitted

forces, energetic parameters [29].

Assume that a steady-state harmonic process is observed in the system “object-

vibration protection device.” In this case it is convenient to compare the kinemat-

ical parameters at any point a in the presence of a vibration protection device or in

its absence. If the amplitude of vibrational displacement at point a is ya then

k* ¼ yVPDa

ya:

The expression above demonstrates how one can construct a dimensionless coeffi-

cient k* either in terms of the velocity _y: or acceleration y

m(t)x1

x(t)

F(t)

341

2

pasu actu

S

Fig. 13 Functional scheme

for a one-dimensional VP

system: 1—passive

components, 2—sensors,

3—device for signal

conversion, 4—actuator

xxxvi Introduction

k* ¼ _y VPDa

_y a

¼ €yVPDa

€ya

The reduction in vibrations can be characterized by the effectiveness of the vibra-

tion protection coefficient

ke ¼ 1� k*:

As ke increases, the effectiveness of the VP device also increases. In the presence of

a VPD, the resulting vibrations in the system are fully suppressed when ke ¼ 1.

The effectiveness of vibration protection in the case of steady-state forced

vibration subjected toF tð Þ ¼ F0 sinωtmay be evaluated via the dynamic coefficient

(DC), which is the ratio of an amplitude A of sustained period motion to the static

displacement δst of the object, caused by amplitude force F0, i.e., DC ¼ A=δst.Another important indicator of vibration protection effectiveness is the dynamic

response factor, which represents the relation of two forces that are transferred upon

the foundation. These are amplitude of force in the presence of a VP device and the

amplitude of distributing force. A transmissibility coefficient allows us to estimate

the effectiveness of the VP device considering the like parameters (particularly, the

forces) in two different points of a system.

Using these methods, one can construct measures on the effectiveness of a VP

device for kinematic excitation. In this case, the effectiveness coefficients for the

relative and absolute motion should be considered separately. The effectiveness of

vibration protection can be evaluated in the logarithmic scale. The criteria of the

effectiveness of vibration protection on the basis of the energetic parameters take

into account the vibration power, the energy loss, etc. In any case, the effectiveness

criteria of vibration protection is defined as the ratio of two parameters in the

presence of a vibration protection device and its absence.

Frequency Spectrum: Linear, Log, and Decibel Units

In industrial settings, mechanical vibrations are observed in a wide frequency

spectrum. Vibrations with frequencies in the 8–16Hz range are known as low

frequency vibrations, 31.5–63Hz are medium frequency vibrations, and 125–

1000Hz are high frequency vibrations. The entire frequency spectrum is partitioned

into frequency intervals. These intervals are referred to as octaves, and larger

intervals are known as decades.An octave is an interval where the ratio of the upper frequency to the lower

frequency is 2 [30]. If f1 and f2 are the lower and upper frequencies of a band, then

the whole octave (1/1) and its parts are determined as follows:

1=1 octave : f 2 ¼ 2f 1; 1=2 octave : f 2 ¼ffiffiffi2

pf 1 ¼ 1:4142f 1;

1=3 octave : f 2 ¼ffiffiffi23

pf 1 ¼ 1:2599f 1; 1=6 octave : f 2 ¼

ffiffiffi26

pf 1 ¼ 1:1214 f 1:

Introduction xxxvii

The interval in octaves between two frequencies f1 and f2 is the base 2 logarithm of

the frequency ratio:

Octf 1�f 2 ¼ log2 f 2=f 1ð Þ ¼ 3:322 log f 2=f 1ð Þoctave:

Here symbol log represents base 10 logarithm.

For example, if f 1 ¼ 2Hz, f 2 ¼ 32Hz; then interval f 1 � f 2 covered

3:322 logf 2=f 11 ¼ 3:322 log16 ¼ 4octaves:

In industrial settings vibrations are usually observed in 8–10 octaves.

A decade is the interval between two frequencies that have a frequency ratio of

10. The interval in decades between any two frequencies f1 and f2, is the base

10 logarithm of the frequency ratio, i.e.,

Decf 1�f 2 ¼ log f 2=f 1ð Þ:

The frequency characterizing a frequency band [f1, f2] as a whole is usually

represented as a geometric mean of the two frequencies, and is equal to

f gm ¼ffiffiffiffiffiffiffiffif 1f 2

p:

The spectral content of vibrations is evaluated in octaves and one-third of octave

frequency bands. The octaves, three 1/3-octave frequency bands for each octave,

and corresponding geometric mean of the frequencies are presented in Table 1.

Table 1 Boundary values of frequency band, 1/3 frequency bands for each octave, and

corresponding geometric mean frequencies [2]

Boundary values

of frequency band, Hz Geometric mean

frequencies, Hz

Boundary values

of frequency band, Hz Geometric mean

frequencies, HzOctavea 1/3 octaveb Octavea 1/3 octaveb

0.7–1.4 0.7–0.89 0.8 11–22 11.2–14.1 12.5

0.89–1.12 1.0 14.1–17.8 16

1.12–1.4 1.25 17.8–22.4 20

1.4–2.8 1.4–1.78 1.6 22–44 22.4–28.2 25

1.78–2.24 2.0 28.2–35.6 31.5

2.24–2.8 2.5 35.5–44.7 40

2.8–5.6 2.8–3.5 3.15 44–88 44.7–56.2 50

3.5–4.4 4.0 56.2–70.8 63

4.4–5.6 5.0 70.8–89.1 80

5.6–11.2 5.6–7.1 6.3 88–176 89.1–112.2 100

7.1–8.9 8.0 112.2–141.3 125

8.9–11.2 10 141.3–177.8 160af2/f1¼ 2bf 2=f 1 ¼

ffiffiffi23

p ¼ 1:25992

xxxviii Introduction

Existing standards provide data on the maximum allowable vibration levels in

terms of the root-mean-square (rms). Next we present formulas for calculating rms

for several different methods of representing variables.

The rms value of a set of values xi, i ¼ 1, n is the square root of the arithmetic

mean (average) of the squares of the original values, i.e.,

xrms ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

nx21 þ x22 þ � � � þ x2n� �

:

r

The corresponding formula for a continuous function (or waveform) f(t) definedover the interval T1 � t � T2 is

f rms ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

T2 � T1

ðT2

T1

f tð Þ½ �2dt:s

The rms value for a function over all time is

f rms ¼ limT!1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

T

ð T0

f tð Þ½ �2dt:s

The rms value over all time of a periodic function is equal to the RMS of one period

of the function [30]. For example, in the case of f tð Þ ¼ a sinω t, we get

f rms ¼ a=ffiffiffi2

p.

Example The function f tð Þ ¼ a sinω t is considered in interval T. Calculate the

mean square value f2and rms value frms.

The mean square value for a function over all time is

f2 ¼ lim

T!11

T

ð T0

f tð Þ½ �2dt ¼ limT!1

1

T

ð T0

a sinω t½ �2dt

¼ limT!1

a2

T

ð T0

1

21� cos 2ωtð Þdt ¼ a2

2;

so the rms value becomes f rms ¼ a=ffiffiffi2

p.

Three types of units can be used to measure vibrations and graphically represent

the corresponding physical quantities. These units are linear, logarithmic, and

decibel.

Linear units provide a true picture of the vibration components in terms of the

domain. The linear scale allows us to easily extract and evaluate the highest

components in the spectra. At the same time, low frequency component values

could prove to be challenging for analysis. This is because the human eye can

distinguish components in the spectra that are 40–50 times lower than the maximum

component. Any components lower than that are generally indistinguishable.

Introduction xxxix

Therefore, one adapts the linear scale if the spectrum’s components of interest are

all of the same order.

Logarithmic Units If the spectrum contains frequency components of very large

range (several different orders of magnitude), then for their graphical representa-

tion, it is convenient to plot the logarithm of the magnitude on the y-axis, and not

just the magnitude itself. This will allow us to easily interpret and represent on a

graph a signal whose maximum and minimum values differ by more than 5000.

Compared to a linear scale, this will increase the graph’s range by at least a factor of

100. The other advantage of the logarithmic scale is the following: incipient faults

of a complex mechanical system are manifested as spectral components with very

small relative amplitude. The logarithmic scale can allow us to discover this

component and watch its development. Compared to a linear scale, the disadvan-

tage of the logarithmic scale is that one must always remember to take the

exponential of the values when attempting to determine the true amplitudes from

the graph.

Decibel The magnitude of any physical quantity (velocity, pressure, etc.) may be

estimated by comparing it with the standard threshold (or reference level) of this

quantity. The decibel (dB) is a logarithmic unit that is used to express the ratio

between two values of the same physical quantity. The decibel is a dimensionless

parameter determined by the formula:

Lσ ¼ 20lg σ=σ0ð Þ dBð Þ;

where σ is a generalized representation of vibrational acceleration, velocity, dis-

placement, etc., and is measured in the standard corresponding units ISO 1683

(International Organization for Standardization) [31];

σ0 is the reference level corresponding to 0 dB.

Thus, the decibels is a characteristic of oscillations that compares two physical

quantities of the same kind (Table 2).

In this table a, υ, d are current values of the acceleration, velocity and

displacement.

Reference quantity υ0 ¼ 10�9m=s leads to the fact that all indicators of a

vibrational process measured in dB are positive. However, various other

reference quantities are used, in particulary d0 ¼ 8� 10�12 mð Þ; υ0 ¼ 5� 10�8

m=sð Þ, a0 ¼ 3� 10�4 m=s2ð Þ [2].

Table 2 Preferred reference quantities are expressed in SI units (lg¼ log10) [10, 31]

Description Definition (dB) Reference quantity

Vibration acceleration level LA ¼ 20lg a=a0ð Þ a0 ¼ 10�6m=s2

Vibration velocity level LV ¼ 20lg υ=υ0ð Þ υ0 ¼ 10�9m=s

Vibration displacement level LD ¼ 20lg d=d0ð Þ d0 ¼ 10�11m

Vibration force level LF ¼ 20lg F=F0ð Þ F0 ¼ 10�6N

xl Introduction

Decibels and corresponding values of accelerations and velocities are presented

in Table 3.

If the decibel units are used to evaluate vibrational levels, as opposed to linear

units, then much more information about the activity levels of vibration becomes

available. Also, decibels represented on a logarithmic scale are generally more

visually appealing than linear units represented on a logarithmic scale.

Decibels and Their Relation to Amplitude Since the decibel is a relative loga-

rithmic unit of measuring vibration, it allows us to easily perform comparative

measurements. Assume that a measured quantity σ is increased n times. With this,

the level of vibration is increased by xdB,: therefore, Lσ þ x ¼ 20lgnσ

σ0. We can

express this relationship as Lσ þ x ¼ 20lgnþ 20lgσ

σ0, or x ¼ 20lgn: If n ¼ 2, then

x ¼ 6dB: Thus an increase of any kinematic value by 6 dB mean doubling its

amplitude. If n ¼ 10, then x ¼ 20dB:

Now assume that the vibration level is changed by k dB. In this case we have tworelationships:

L1 ¼ 20lgσ1σ0,

L2 ¼ L1 þ k ¼ 20logσ2σ0:

Therefore, k ¼ 20lgσ2σ1. Amplitude ratio

σ2σ1

¼ 10k=20: If k ¼ 3 thenσ2σ1

¼ 1:4125:

These properties allow us to study trends in evolution of vibrations. The relation-

ships between changes in levels of vibrations (in dB) and the corresponding

amplitudes are shown in Table 4.

These data can be presented on a logarithmic scale as shown in Fig. 14

Table 3 Conversion between

decibels, acceleration (m/s2),

and velocity (m/s); Reference

levels defined in ISO 1683

Decibel (dB) Acceleration (m/s2) Velocity (m/s)

�20 10�7 10�10

0 10�6 10�9

20 10�5 10�8

40 10�4 10�7

60 10�3 10�6

80 10�2 10�5

100 10�1 10�4

120 1 10�3

140 10 10�2

160 102 10�1

180 103 1

200 104 10

Introduction xli

Conversion Triangle Let us consider a case of harmonic vibration of frequency

f (in Hz). If we consider the kinematic relationships between displacement (D),velocity (V ) and acceleration (A), then the relationship between their amplitude

values D, V, and A, in standard international units, is A ¼ 2πfð Þ2D, A ¼ 2πf V,V ¼ 2πfD.

Generalized Measurement Units In the case of harmonic vibrations with fre-

quency f (Hz) for an accepted reference quantity, it is easy to establish a relationshipbetween vibration acceleration level LA, velocity LV and displacement LD, mea-

sured in dB. Let the reference quantities be [2]

a0 ¼ 3� 10�4m=s2, υ0 ¼ 5� 10�8m=s, d0 ¼ 8� 10�12m:

Table 4 Changes in vibrations levels (in dB) and the corresponding amplitude ratios

Change in level (dB) Amplitude ratioa Change in level (dB) Amplitude ratioa

0 1 30 31

3 1.4 36 60

6 2 40 100

10 3.1 50 310

12 4 60 1000

18 8 70 3100

20 10 80 10,000

24 16 100 100,000aSome amplitude ratios are rounded

0100

101

102

103

104

105

10 20 30 40 50Change in level (dB)

Am

plitu

de r

atio

60 70 80 90 100

Fig. 14 Changes in vibrations levels (in dB) and the corresponding amplitude ratios

xlii Introduction

We determine an expression for LA in terms of LV. According to the conversional

triangle, we havea ¼ 2π f υ, thereforeLA ¼ 20lga

a0¼ 20lg

2π f υ

3 � 10�4. This expression

contains velocity υ; therefore, the reference quantity for υ0 ¼ 5� 10�8m=s shouldbe introduced in the denominator. After that, the expression for LA becomes

LA ¼ 20lg2π f υ

3� 10�4¼ 20lg

υ

5� 10�8� 2π35� 104

f

!

¼ 20lgυ

5� 10�8

� �þ 20lg

5� 2π

3� 104

� �þ 20lgf :

Finally we get

LA ¼ LV þ 20lgf � 60 dBð Þ:

Relationships between LV and LD, LD and LA may be similarly derived.

Problems

1. Define the following terms: (1) Source of vibration; (2) Vibration protection

object; (3) Two groups of internal factors that cause vibrations; (4) Passive

vibration protection; (5) Active vibration protection; (6) Vibration isolation,

vibration damping, vibration absorption; (7) Force and kinematic excitation;

(8) Decade, octave, decibel; (9) Displacement (velocity, acceleration) level.

2. Explain the idea of parametric vibration protection

3. What are the main elements of the design diagram for passive and active

vibration protection systems?

4. Describe the principal approaches for estimating the effectiveness of vibration

protection.

5. Describe the physical relationships for the principal linear passive elements.

6. Describe the principal parts of the statement of the optimal active control

vibration problem.

7. Establish relationships between vibration velocity level LV, frequency f Hz anddisplacement LD. Give results in dB. Assume the basic levels are

υ0 ¼ 5� 10�8m=s, d0 ¼ 8� 10�12m.

Answer: LV ¼ LD þ 20lgf � 60 dBð Þ:8. Establish relationships between vibration displacement level LD and accelera-

tion LA. Give results in dB. Assume the basic levels are

a0 ¼ 3� 10�4m=s2, d0 ¼ 8� 10�12m.

Answer: LD ¼ LA � 40lgf þ 120 dBð Þ:9. Calculate the number of octaves in a single decade.

Answer: Octf 1�f 2 ¼ log2 f 2=f 1ð Þ ¼ 3:322 lg10 ¼ 3:322 octaves.

Introduction xliii

10. Find the ratio f2/f1 that corresponds to four octaves.

Answer: Octf 1�f 2 ¼ log2 f 2=f 1ð Þ ¼ 3:322 lg f 2=f 1ð Þ ¼ 4 ! lg f 2=f 1ð Þ ¼1:2041 ! f 2=f 1 ¼ 101:2041 ¼ 16:

11. Find the number of decades in the frequency interval 10–160 Hz.

Answer: Decf 1�f 2 ¼ lg f 2=f 1ð Þ ¼ lg16 ¼ 1:204 decades.

12. Compose a conversion table of vibration levels (dB) to the value of the velocity

(m/s); take υ0 ¼ 5� 10�8m=s:

Solution. If Lυ ¼ 90dB, then Lυ ¼ 20lgυ

5� 10�8¼ 90dB ! lg

υ

5� 10�8¼

4:5 ! υ

5� 10�8¼ 104:5 ¼ 31622:7 ! υ ¼ 0:00158 ¼ 0:158� 10�2m=s:

Answer: Conversion table of vibration level Lυ (dB) to the value of the

velocity (m/s); υ0 ¼ 5� 10�8m=s:

dB m/s dB m/s dB m/s dB m/s dB m/s

80 0.050 90 0.158 100 0.50 110 1.58 120 5.0

81 0.056 91 0.177 101 0.56 111 1.77 121 5.6

82 0.063 92 0.199 102 0.63 112 1.99 122 6.3

83 0.071 93 0.223 103 0.71 113 2.23 123 7.1

84 0.079 94 0.251 104 0.79 114 2.51 124 7.9

85 0.089 95 0.281 105 0.89 115 2.81 125 8.9

86 0.099 96 0.316 106 1.00 116 3.16 126 10.0

87 0.112 97 0.354 107 1.12 117 3.54 127 11.2

88 0.026 98 0.397 108 1.26 118 3.97 128 12.6

89 0.141 99 0.446 109 1.41 119 4.46 129 14.1

factor 10�2 10�2 10�2 10�2 10�2

13. At a frequency f ¼ 100Hz, the amplitude of displacements is x ¼ 8mm.

Calculate the vibration acceleration level La (dB). Assume the basic level is

d0 ¼ 8� 10�12 m.

Solution. Lx ¼ 20lgx

d0¼ 20lg

0:008

8�10�12¼ 180dB! La ¼ Lxþ40lgf �120¼

180þ40lg100�120¼ 140dB:Answer: La ¼ 140dB.

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xliv Introduction

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Introduction xlv