Igor A. Karnovsky · Evgeniy Lebed Theory of Vibration Protection
Jul 07, 2016
Igor A. Karnovsky Evgeniy Lebed
Theory of Vibration Protection
Theory of Vibration Protection
Igor A. Karnovsky Evgeniy Lebed
Theory of VibrationProtection
Igor A. KarnovskyCoquitlam, BC, Canada
Evgeniy LebedMDA Systems Ltd.Scientic 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 products 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 classied 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 elds. 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 specic 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. Itdescribes briey the types of mechanical exposures and their inuence 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. 19) considers different approaches to passive vibration protec-tion. Among them are vibration isolation (Chaps. 14), 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 dynamicprocesses. 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 13presents 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 oftheir occurrence (Chap. 16) and harmful inuence of vibrations on the human
(Chap. 17); Chapter 17 was written together with .ldon (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 chapterthe rst number indicates the chapter; the second number
indicates the number of the gure 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..V. Khvingiya(Georgia), prof. A.O. Rasskazov (Ukraine), prof. V.B. Grinyov (Ukraine), prof.
.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
Coefcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Kinematic Excitation. Overload Vibration Coefcient
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
Coefcient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.2 Simplication of Vibration Isolators . . . . . . . . . . . . . . 24
1.4.3 Vibration Isolators Which Cannot Be Simplied . . . . . 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 Kirchhoffs Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.4.3 Reciprocity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.4.4 Superposition Principle . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5 Simplest One-Side mkb Vibration Isolator . . . . . . . . . . . . . . 602.5.1 Force Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.5.2 Kinematic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.6 Complex One-Sided mkb Vibration Isolators . . . . . . . . . . . . . 662.6.1 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.1 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 Coefcient
of Vibration Protection . . . . . . . . . . . . . . . . . . . . . . . . 91
3.3 Optimal Synthesis of the Fundamental Characteristics . . . . . . . 93
3.3.1 Problem Statement of Optimal Synthesis.
Brunes Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.3.2 Fosters Canonical Schemes . . . . . . . . . . . . . . . . . . . . 95
3.3.3 Cauers 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 an4N with Supportof 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 Greens Function and Duhamels 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 Coefcient . . . . . . . . . . . . . . . . . . . . . . . . 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 Pringles Vibration Absorber . . . . . . . . . . . . . . . . . . . 226
6.5 Gyroscopic Vibration Absorber . . . . . . . . . . . . . . . . . . . . . . . . 228
6.5.1 Elementary Theory of Gyroscopes . . . . . . . . . . . . . . . 229
6.5.2 Schlicks 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 KrylovDuncan 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 ShchipanovLuzin Absolute Invariance . . . . . . . . . . . . 266
8.2.2 Invariance up to . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2688.3 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 Petrovs 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 Petrovs 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 Coefcients of Harmonic Linearization . . . . . . . . . . . . 300
9.3 Harmonic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
9.3.1 Dufngs Restoring Force . . . . . . . . . . . . . . . . . . . . . . 303
9.3.2 Nonlinear Restoring Force and Viscous Damping . . . . 307
9.3.3 Linear Restoring Force and Coulombs
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 Pontryagins Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33310.1 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 Classication of Optimal Vibration
Protection Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 340
10.2 Representation of an Equation of State in Cauchys
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 Pontryagins 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 Bushaws 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38611.1.1 Formulation of the Problem of Vibration
Suppression as a Moment Problem . . . . . . . . . . . . . . . 386
11.1.2 The l-moments Problem and NumericalProcedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Modied 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 bk and bm 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51914.1 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 Biots 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 . . . . . . . . . . 56115.1 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 Inuence of Vibration Exposure on the Human Subject . . . . . . . 628
17.2.1 Classication 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 2D3D 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 eld and is the author of more than
70 published scientic papers, including two books in Structural Analysis
(published with Springer in 20102012) and three handbooks in Structural Dynam-
ics (published with McGraw Hill in 20012004). 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 eld. 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 scientic 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, inuences
on technical objects, and on human operators are briey 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 classied into two subcategories.
Excitation Factors Arising due to Moving Bodies Examples of moving bodiesinclude 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, specically, static and
dynamic balancing of rotating objects, such as rotors, and their corresponding
automatic balancing, are usually employed to achieve this goal. A detailed classi-
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 ininternal combustion and jet engines, processes involving interactions of liquids or
gasses with an engines 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 objects function. These external factors
may include explosions, seismic inuences, collisions, temperature uctuations,
and wind loads.
Let us have a closer look at several examples of vibration protection objects and
inuences that act upon them.
1a. An engine with an unbalanced rotor, mounted on a foundation. The vibration
protection problem involves reducing vibrations of the engines frame. The
engines frame is the object of vibration protection. The source of vibrations(SoV) is the engines rotor. Dynamic excitations are the dynamic reactions of
the rotors supports (Fig. 2a, b).
Source ofVibration (SoV)
Vibration ProtectionObject (VPO)
onnection betweenSoV and VPO
Fig. 1 Scheme representing an interaction between Source of Vibration (SoV) and VibrationProtection 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 protectionproblem
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 casethe unbalanced engines 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 airplanes 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
xed to the aircraft.
2b. For the same system, we can pose the problem of lowering vibrations of the
airplanes 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
inuences, 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)
bFig. 3 (a) Force (dynamic)and (b) kinematic excitation
Introduction xxv
Here, m represents the mass of the object, k is the stiffness coefcient of theconnection between foundation and object, and F(t) and (t) refer to force andkinematic 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 Inuence 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-
crafts maneuvers (roll, pitch, and yaw). The two main characteristics of linear
overloads are constant acceleration a0 (Fig. 4) and the maximal rate at whichacceleration grows _a 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 signicantly 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 xed 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 currentoverloads provide additional stress on the VP device.
a
t
a0a
Fig. 4 Graph linearoverload-time
xxvi Introduction
Shutting Down the Starting Engines State During this state, the engines that wereinitially 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 signicant linearoverload is applied to the VP device.
Vibrational Exposure
Force (dynamic) vibration exposures represent force F or torqueM, which act uponan object. Acceleration (a) of points connected to the source (foundations, aircrafthull, etc.), their velocities () and displacements (x) represent kinematic vibrationalexposures. 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 t x0 sin0t,where x(t) is the vibrational force or kinematic exposure, x0 and 0 represent theamplitude and frequency of excitation. The period of an oscillation can be deter-
mined from the excitation frequency by T 2=0.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
engines hull during the rotors acceleration can be expressed by
x t a sin t t ;where (t) represents the rotors angular acceleration, as a function of time.
t
xa
Tx0
T
b
x0
x
0
Fig. 5 Harmonic processand its corresponding
spectra
Introduction xxvii
Polyharmonic Vibrational Excitation Excitations of this nature are described by
the following expression [3, vol. 1]:
x t X1k1
ak cos k0t bk sin k0t :
The set of frequencies k0 for k 1, 2, . . . ; of harmonic components, arrangedin ascending order, is called the frequency spectrum of the process. An amplitude
Ak a2k b2k
q, and an initial phase k, where tank 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 frequenciesmax min. The range of frequencies for whichmax=min > 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 at 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 bandwidthmin max of a polyharmonic excitation has a profound impacton 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 theobject not only by elements mechanical connections, but also by the surrounding
environment. High acoustic pressure can have a signicant impact on high preci-
sion machinery, such as modern day jet engines and supersonic aircraft.
A1A
A2A3
An
1 2 3 n
Fig. 6 Amplitude spectraof a polyharmonic
excitation
xxviii Introduction
Chaotic Exposure The following expression can be used to characterize chaoticvibrations:
x t XNk1
ak coskt bk sinkt :
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 signicant role, and their inuences are impos-
sible to determine beforehand. This is why such vibrational exposures are difcult,
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 thecorresponding 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 xed onto itself [7].
Impact Exposure
Impact exposures are classied 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 inuenced 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 impacts form include its duration,
amplitude, and spectral characteristics [8].
Inuence of Mechanical Exposure on Technical Objects and Humans
Inuences of Linear Overloads In their natural form (without any additionalexposures), 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.
Inuence of Vibrational Excitation The harmful inuence of such excitations aremanifested 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 xed joints.
4. In connections with gaps, such exposures cause collisions between contact
surfaces.
5. These exposures result in damage to the structures surface layers, and premature
wear on the structure develops.
Particularly hazardous vibrational effects are manifested in the presence of
linear overloads [9].
Inuences 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 signicantly complicates the function of a vibration
protection system in shock excitations [9].
In the literature one can nd 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 inuences 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 nally each individual persons physical
characteristics. Harmful vibrations are subdivided into two groups:
1. Vibrations inuencing a persons functional state;
2. Vibrations inuencing a persons physiological state
Negative vibrational effects of the rst 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 persons functional state lead to reduced productivity
and quality, while vibrational effects on a persons 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 rst 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 denes 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 yz planewith respect to the x-axis. A structure in Fig. 7e presents a massless beam with anabsolutely 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 yx plane. Figure 7fpresents 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 onedegree 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 (af) 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 asshown 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 horizontaldisplacements 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 conguration of a structure
is dened by displacement of each mass as a function of time, i.e., y y t , and thebehavior of such structures is described by ordinary differential equations. It isworth discussing the term concentrated parameters for cases 7f (pontoon bridge)
and 8 (two-story frame). In both cases, the massin fact, the masses are distrib-uted along the correspondence members. However, the stiffness of these members
is innite, and the position of each of these members is dened 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. 8), the horizontal displacements of each crossbar.The structures with distributed parameters are generally more difcult to ana-
lyze. The simplest structure is a beam with a distributed mass m. In this case aconguration 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 y x; t ,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 F (m 0),members A and with distributed masses m, and absolutely rigid member D(EI1). The simplest form of vibration is shown by the dotted line.
MEI=
MEI=
cM
M
d
Mb
e
f
a M
Fig. 8 (af) 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 equationspartial 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 8ae) may
be presented (neglecting damping) as shown in Fig. 10. This model may be applied
for force, as well as kinematic excitations. Stiffness coefcients k1 and k1 depend onthe type of structure and the structures 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 systemmassless beam + lumped mass m (Fig. 11), except directly on the mass.
This system is described by two ordinary differential equations
y1 b _y111 my212,y2 b _y121 my222:
m2m1k1 k2
Fig. 10 Design diagram of a mechanical system with two degrees of freedom
EI=EI, m
EI, m=0A B
C D
F
EI, mFig. 9 Frame withdistributed and lumped
parameters
mEI
y1 y2
b
Fig. 11 Mechanical systemwith 1.5 degrees of freedom
Introduction xxxiii
The second equation, for the mass, is second order with respect to y2, while rstequation for the damper is rst order with respect to y1. Here ik 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 sources vibrational activity;
2. Passive vibration protection;
3. Active vibration protection.
Lowering the Sources 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 simplied schematic of a vibra-
tion isolator is shown in Fig. 12. Weakening of connections between the objectand 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 objects 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
objects vibration by transferring the oscillation energy onto the absorber. An object
m with an elastic element k, damper b, and absorber maka is shown in Fig. 12c. Inall 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 objects 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 [2326]. A schematic of a typical active VP system
is shown in Fig. 13. The vibration protection object of the mass m is connected tosupport 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 ofpassive 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 externalexcitation. The Shchipanov-Luzins 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 inits absence. If the amplitude of vibrational displacement at point a is ya then
k* yVPDa
ya:
The expression above demonstrates how one can construct a dimensionless coef-
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 schemefor a one-dimensional VP
system: 1passive
components, 2sensors,
3device for signal
conversion, 4actuator
xxxvi Introduction
k* _yVPDa
_y a y
VPDa
ya
The reduction in vibrations can be characterized by the effectiveness of the vibra-
tion protection coefcient
ke 1 k*:
As ke increases, the effectiveness of the VP device also increases. In the presence ofa VPD, the resulting vibrations in the system are fully suppressed when ke 1.
The effectiveness of vibration protection in the case of steady-state forced
vibration subjected toF t F0 sintmay be evaluated via the dynamic coefcient(DC), which is the ratio of an amplitude A of sustained period motion to the staticdisplacement st of the object, caused by amplitude force F0, i.e., DC A=st.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 coefcient 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 coefcients 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 dened 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 816Hz range are known as low
frequency vibrations, 31.563Hz 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 largerintervals 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, thenthe whole octave (1/1) and its parts are determined as follows:
1=1 octave : f 2 2f 1; 1=2 octave : f 2 2
pf 1 1:4142f 1;
1=3 octave : f 2 23
pf 1 1:2599f 1; 1=6 octave : f 2
26
pf 1 1:1214 f 1:
Introduction xxxvii
The interval in octaves between two frequencies f1 and f2 is the base 2 logarithm ofthe frequency ratio:
Octf 1f 2 log2 f 2=f 1 3:322 log f 2=f 1 octave:
Here symbol log represents base 10 logarithm.
For example, if f 1 2Hz, f 2 32Hz; then interval f 1 f 2 covered
3:322 logf 2=f 11 3:322 log16 4octaves:
In industrial settings vibrations are usually observed in 810 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 base10 logarithm of the frequency ratio, i.e.,
Decf 1f 2 log f 2=f 1 :
The frequency characterizing a frequency band [f1, f2] as a whole is usuallyrepresented as a geometric mean of the two frequencies, and is equal to
f gm f 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, andcorresponding 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
0.71.4 0.70.89 0.8 1122 11.214.1 12.5
0.891.12 1.0 14.117.8 16
1.121.4 1.25 17.822.4 20
1.42.8 1.41.78 1.6 2244 22.428.2 25
1.782.24 2.0 28.235.6 31.5
2.242.8 2.5 35.544.7 40
2.85.6 2.83.5 3.15 4488 44.756.2 50
3.54.4 4.0 56.270.8 63
4.45.6 5.0 70.889.1 80
5.611.2 5.67.1 6.3 88176 89.1112.2 100
7.18.9 8.0 112.2141.3 125
8.911.2 10 141.3177.8 160af2/f1 2bf 2=f 1
23
p 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 rmsfor several different methods of representing variables.
The rms value of a set of values xi, i 1, n is the square root of the arithmeticmean (average) of the squares of the original values, i.e.,
xrms 1
nx21 x22 x2n
:
rThe corresponding formula for a continuous function (or waveform) f(t) denedover the interval T1 t T2 is
f rms
1
T2 T1
T2T1
f t 2dt:s
The rms value for a function over all time is
f rms limT!1
1
T
T0
f t 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 t a sin t, we getf rms a=
2
p.
Example The function f t a sin t is considered in interval T. Calculate themean square value f
2and rms value frms.
The mean square value for a function over all time is
f2 lim
T!11
T
T0
f t 2dt limT!1
1
T
T0
a sin t 2dt
limT!1
a2
T
T0
1
21 cos 2t dt a
2
2;
so the rms value becomes f rms a=2
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 4050 times lower than the maximum
component. Any components lower than that are generally indistinguishable.
Introduction xxxix
Therefore, one adapts the linear scale if the spectrums 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 graphs 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:
L 20lg =0 dB ;
where 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];
0 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, , d are current values of the acceleration, velocity anddisplacement.
Reference quantity 0 109m=s leads to the fact that all indicators of avibrational process measured in dB are positive. However, various other
reference quantities are used, in particulary d0 8 1012 m ; 0 5 108m=s , a0 3 104 m=s2 [2].
Table 2 Preferred reference quantities are expressed in SI units (lg log10) [10, 31]Description Denition (dB) Reference quantity
Vibration acceleration level LA 20lg a=a0 a0 106m=s2Vibration velocity level LV 20lg =0 0 109m=sVibration displacement level LD 20lg d=d0 d0 1011mVibration force level LF 20lg F=F0 F0 106N
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 is increased n times. With this,
the level of vibration is increased by xdB,: therefore, L x 20lg n0. We can
express this relationship as L x 20lgn 20lg 0, or x 20lgn: If n 2, then
x 6dB: Thus an increase of any kinematic value by 6 dB mean doubling itsamplitude. If n 10, then x 20dB:
Now assume that the vibration level is changed by k dB. In this case we have tworelationships:
L1 20lg10,
L2 L1 k 20log20:
Therefore, k 20lg21. Amplitude ratio
21 10k=20: If k 3 then 2
1 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 betweendecibels, acceleration (m/s2),
and velocity (m/s); Reference
levels dened in ISO 1683
Decibel (dB) Acceleration (m/s2) Velocity (m/s)
20 107 10100 106 109
20 105 108
40 104 107
60 103 106
80 102 105
100 101 104
120 1 103
140 10 102
160 102 101
180 103 1
200 104 10
Introduction xli
Conversion Triangle Let us consider a case of harmonic vibration of frequencyf (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 2f 2D, A 2f V,V 2fD.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 3 104m=s2, 0 5 108m=s, d0 8 1012m:
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 2 f , thereforeLA 20lg aa0 20lg 2 f
3 104. This expressioncontains velocity ; therefore, the reference quantity for 0 5 108m=s shouldbe introduced in the denominator. After that, the expression for LA becomes
LA 20lg 2 f 3 104 20lg
5 108 2
35 104 f
!
20lg 5 108
20lg 5 23 104
20lgf :
Finally we get
LA LV 20lgf 60 dB :
Relationships between LV and LD, LD and LA may be similarly derived.
Problems
1. Dene 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