Theory of Vibration Protection (eBook)

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 Dynamics (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 a US patent (2015).

Preface 6
Acknowledgments 10
Contents 12
About the Authors 22
Introduction 24
Mechanical Exposure and Vibration Protection Methods 24
Source of Vibration and Vibration Protection Objects 24
Mechanical Exposures and Their Influence on Technical Objects and Humans 27
Linear Overload 27
Vibrational Exposure 28
Impact Exposure 30
Influence of Mechanical Exposure on Technical Objects and Humans 30
Dynamical Models of Vibration Protection Objects 31
Vibration Protection Methods 35
Estimating the Effectiveness of Vibration Reduction 37
Frequency Spectrum: Linear, Log, and Decibel Units 38
Part I: Passive Vibration Protection 47
Chapter 1: Vibration Isolation of a System with One or More Degrees of Freedom 48
1.1 Design Diagrams of Vibration Protection Systems 48
1.2 Linear Viscously Damped System. Harmonic Excitation and Vibration Protection Criteria 50
1.2.1 Simplest Mechanical Model of a Vibration Protection System 51
1.2.2 Force Excitation. Dynamic and Transmissibility Coefficients 51
1.2.3 Kinematic Excitation. Overload Vibration Coefficient and Estimation of Relative Displacement 55
1.3 Complex Amplitude Method 60
1.3.1 Vector Representation of Harmonic Quantities 60
1.3.2 Single-Axis Vibration Isolator 62
1.3.3 Argand Diagram 64
1.3.4 System with Two Degrees of Freedom 65
1.4 Linear Single-Axis Vibration Protection Systems 66
1.4.1 Damper with Elastic Suspension. Transmissibility Coefficient 67
1.4.2 Simplification of Vibration Isolators 69
1.4.3 Vibration Isolators Which Cannot Be Simplified 71
1.4.4 Special Types of Vibration Isolators 71
1.5 Vibration Protection System of Quasi-Zero Stiffness 73
References 80
Chapter 2: Mechanical Two-Terminal Networks for a System with Lumped Parameters 82
2.1 Electro-Mechanical Analogies and Dual Circuits 82
2.2 Principal Concepts of Mechanical Networks 87
2.2.1 Vector Representation of Harmonic Force 87
2.2.2 Kinematic Characteristics of Motion 87
2.2.3 Impedance and Mobility of Passive Elements 88
2.3 Construction of Two-Terminal Networks 93
2.3.1 Two-Terminal Network for a Simple Vibration Isolator 94
2.3.2 Two-Cascade Vibration Protection System 97
2.3.3 Complex Dynamical System and Its Coplanar Network 98
2.4 Mechanical Network Theorems 100
2.4.1 Combination of Mechanical Elements 101
2.4.2 Kirchhoff´s Laws 103
2.4.3 Reciprocity Theorem 104
2.4.4 Superposition Principle 104
2.5 Simplest One-Side m-k-b Vibration Isolator 105
2.5.1 Force Excitation 105
2.5.2 Kinematic Excitation 109
2.6 Complex One-Sided m-k-b Vibration Isolators 111
2.6.1 Vibration Isolator with Elastic Suspension 111
2.6.2 Two-Cascade Vibration Protection System 112
References 118
Chapter 3: Mechanical Two-Terminal and Multi-Terminal Networks of Mixed Systems 120
3.1 Fundamental Characteristics of a Deformable System with a Vibration Protection Device 120
3.1.1 Input and Transfer Impedance and Mobility 121
3.1.2 Impedance and Mobility Relating to an Arbitrary Point 127
3.2 Deformable Support of a Vibration Protection System 129
3.2.1 Free Vibrations of Systems with a Finite Number of Degrees of Freedom 129
3.2.2 Generalized Model of Support and Its Impedance 134
3.2.3 Support Models and Effectiveness Coefficient of Vibration Protection 136
3.3 Optimal Synthesis of the Fundamental Characteristics 138
3.3.1 Problem Statement of Optimal Synthesis. Brune´s Function 139
3.3.2 Foster´s Canonical Schemes 140
3.3.3 Cauer´s Canonical Schemes 145
3.3.4 Support as a Deformable System with Distributed Mass 149
3.4 Vibration Protection Device as a Mechanical Four-Terminal Network 155
3.4.1 Mechanical Four-Terminal Network for Passive Elements with Lumped Parameters 156
3.4.2 Connection of an 4N with Support of Impedance Zf 160
3.4.3 Connections of Mechanical Four-Terminal Networks 161
3.5 Mechanical Multi-Terminal Networks for Passive Elements with Distributed Parameters 172
3.5.1 M4TN for Longitudinal Vibration of Rod 173
3.5.2 Mechanical Eight-Terminal Network for Transversal Vibration of a Uniform Beam 175
3.6 Effectiveness of Vibration Protection 180
References 184
Chapter 4: Arbitrary Excitation of Dynamical Systems 186
4.1 Transfer Function 186
4.1.1 Analysis in the Time Domain 186
4.1.2 Logarithmic Plot of Frequency Response. Bode Diagram 193
4.2 Green´s Function and Duhamel´s Integral 196
4.2.1 System with Lumped Parameters 197
4.2.1.1 Force Excitation 197
4.2.1.2 Kinematic Excitation 200
4.2.2 System with Distributed Parameters 201
4.3 Standardizing Function 204
References 210
Chapter 5: Vibration Damping 211
5.1 Phenomenological Aspects 212
5.1.1 Models of Material 212
5.1.2 Complex Modulus of Elasticity 214
5.1.3 Dissipative Forces 215
5.1.4 Dimensionless Parameters of Energy Dissipation 216
5.2 Hysteretic Damping 220
5.2.1 Hysteresis Loop 220
5.2.2 Hysteretic Damping Concept 222
5.2.3 Forced Vibration of a System with One Degree of Freedom 223
5.2.4 Comparison of Viscous and Hysteretic Damping 226
5.3 Structural Damping 226
5.3.1 General 227
5.3.2 Energy Dissipation in Systems with Lumped Friction 229
5.3.3 Energy Dissipation in Systems with Distributed Friction 230
5.4 Equivalent Viscous Damping 233
5.4.1 Absorption Coefficient 233
5.4.2 Equivalent Viscoelastic Model 233
5.5 Vibration of a Beam with Internal Hysteretic Friction 235
5.6 Vibration of a Beam with External Damping Coating 238
5.6.1 Vibration-Absorbing Layered Structures 239
5.6.2 Transverse Vibration of a Two-Layer Beam 240
5.6.2.1 Bending of a Composite Beam 240
5.6.2.2 Coefficient of Loss of a Composite Beam 242
5.6.2.3 Free and Forced Vibration of a Composite Beam 243
5.7 Aerodynamic Damping 244
5.7.1 The Interaction of a Structure with a Flow 245
5.7.2 Aerodynamic Reduction of Vibration 246
References 248
Chapter 6: Vibration Suppression of Systems with Lumped Parameters 250
6.1 Dynamic Absorber 250
6.2 Dynamic Absorbers with Damping 256
6.2.1 Absorber with Viscous Damping 257
6.2.2 Viscous Shock Absorber 259
6.2.3 Absorber with Coulomb Damping 260
6.3 Roller Inertia Absorbers 262
6.4 Absorbers of Torsional Vibration 265
6.4.1 Centrifugal Pendulum Vibration Absorber 265
6.4.2 Pringle´s Vibration Absorber 269
6.5 Gyroscopic Vibration Absorber 271
6.5.1 Elementary Theory of Gyroscopes 272
6.5.1.1 Free Gyroscope 272
6.5.1.2 Action of a Force Applied to the Axis of a Gyroscope 272
6.5.1.3 Regular Precession of a Heavy Gyroscope 273
6.5.1.4 The Gyroscopic Effect 274
6.5.2 Schlick's Gyroscopic Vibration Absorber 275
6.6 Impact Absorbers 277
6.6.1 Pendulum Impact Absorber 278
6.6.2 Floating Impact Absorber 280
6.6.3 Spring Impact Absorber 281
6.7 Autoparametric Vibration Absorber 281
References 285
Chapter 7: Vibration Suppression of Structures with Distributed Parameters 287
7.1 Krylov-Duncan Method 287
7.2 Lumped Vibration Absorber of the Beam 292
7.3 Distributed Vibration Absorber 296
7.4 Extension Rod as Absorber 299
References 305
Chapter 8: Parametric Vibration Protection of Linear Systems 306
8.1 General 306
8.2 Invariance Principle 307
8.2.1 Shchipanov-Luzin Absolute Invariance 307
8.2.2 Invariance up to epsi 309
8.3 Parametric Vibration Protection of the Spinning Rotor 312
8.4 Physical Feasibility of the Invariance Conditions 316
8.4.1 Uncontrollability of ``Perturbation-Coordinate´´ Channel 316
8.4.2 Petrov´s Two-Channel Principle 318
8.4.3 Dynamic Vibration Absorber 319
8.5 Parametric Vibration Protection of the Plate Under a Moving Load 321
8.5.1 Mathematical Model of a System 321
8.5.2 Petrov´s Principle 325
References 328
Chapter 9: Nonlinear Theory of Vibration Protection Systems 330
9.1 General 330
9.1.1 Types of Nonlinearities and Theirs Characteristics 331
9.1.1.1 Static Nonlinear Characteristics 331
9.1.1.2 Dynamic Nonlinear Characteristics 333
9.1.2 Features of Nonlinear Vibration 335
9.2 Harmonic Linearization Method 336
9.2.1 Method Foundation 336
9.2.2 Coefficients of Harmonic Linearization 341
9.3 Harmonic Excitation 344
9.3.1 Duffing´s Restoring Force 344
9.3.2 Nonlinear Restoring Force and Viscous Damping 348
9.3.3 Linear Restoring Force and Coulomb´s Friction 352
9.3.4 Internal Friction 357
9.4 Nonlinear Vibration Absorber 360
9.5 Harmonic Linearization and Mechanical Impedance Method 363
9.6 Linearization of a System with an Arbitrary Number of Degrees of Freedom 365
References 370
Part II: Active Vibration Protection 372
Chapter 10: Pontryagin´s Principle 373
10.1 Active Vibration Protection of Mechanical Systems as a Control Problem 373
10.1.1 Mathematical Model of Vibration Protection Problem 373
10.1.2 Classification of Optimal Vibration Protection Problems 380
10.2 Representation of an Equation of State in Cauchy´s Matrix Form 381
10.3 Qualitative Properties of Vibration Protection Systems 387
10.3.1 Accessibility, Controllability, Normality 387
10.3.2 Stability 390
10.4 Pontryagin´s Principle 395
10.5 Vibration Suppression of a System with Lumped Parameters 397
10.5.1 Vibration Suppression Problems Without Constraints 398
10.5.1.1 Fixed Terminal Time and Functional of Energy 398
10.5.1.2 Non-Fixed Terminal Time, Combined Functional of Energy and Time 402
10.5.1.3 Fixed Time, Combined Functional of Energy and Coordinates 403
10.5.1.4 General Case: Quadratic Functional and Fixed Time 405
10.5.2 Vibration Suppression Problem with Constrained Exposure. Quadratic Functional, Fixed Time and Fixed End 407
10.6 Bushaw´s Minimum-Time Problem 409
10.7 Minimum Isochrones 417
Problems 420
References 423
Chapter 11: Krein Moments Method 425
11.1 The Optimal Active Vibration Protection Problem as the l-moments Problem 426
11.1.1 Formulation of the Problem of Vibration Suppression as a Moment Problem 426
11.1.2 The l-moments Problem and Numerical Procedures 431
11.2 Time-Optimal Problem for a Linear Oscillator 433
11.2.1 Constraint of Energy 433
11.2.2 Control with Magnitude Constraint 435
11.3 Optimal Active Vibration Protection of Continuous Systems 438
11.3.1 Truncated Moments Problem 438
11.3.2 Vibration Suppression of String. Standardizing Function 438
11.3.3 Vibration Suppression of a Beam 444
11.3.4 Nonlinear Moment Problem 453
11.4 Modified Moments Procedure 455
11.5 Optimal Vibration Suppression of a Plate as a Mathematical Programming Problem 460
Problems 464
References 465
Chapter 12: Structural Theory of Vibration Protection Systems 467
12.1 Operator Characteristics of a Dynamic System 468
12.1.1 Types of Operator Characteristics 468
12.1.2 Transfer Function 472
12.1.3 Elementary Blocks 474
12.1.4 Combination of Blocks. Bode Diagram 481
12.1.5 Block Diagram Transformations 488
12.2 Block Diagrams of Vibration Protection Systems 490
12.2.1 Representation of b-k and b-m Systems as Block Diagram 490
12.2.2 Vibration Protection Closed Control System 497
12.2.3 Dynamic Vibration Absorber 503
12.3 Vibration Protection Systems with Additional Passive Linkages 505
12.3.1 Linkage with Negative Stiffness 505
12.3.2 Linkage by the Acceleration 506
12.4 Vibration Protection Systems with Additional Active Linkages 507
12.4.1 Functional Schemes of Active Vibration Protection Systems 508
12.4.2 Vibration Protection on the Basis of Excitation. Invariant System 509
12.4.3 Vibration Protection on the Basis of Object State. Effectiveness Criteria 511
12.4.4 Block Diagram of Optimal Feedback Vibration Protection 517
References 521
Part III: Shock and Transient Vibration 523
Chapter 13: Active and Parametric Vibration Protection of Transient Vibrations 524
13.1 Laplace Transform 524
13.2 Heaviside Method 530
13.3 Active Suppression of Transient Vibration 540
13.3.1 Step Excitation 540
13.3.2 Impulse Excitation 544
13.4 Parametric Vibration Suppression 547
13.4.1 Recurrent Instantaneous Pulses 547
13.4.2 Recurrent Impulses of Finite Duration 549
References 556
Chapter 14: Shock and Spectral Theory 557
14.1 Concepts of Shock Excitation 557
14.1.1 Types of Shock Exposures 557
14.1.2 Different Approaches to the Shock Problem 559
14.1.3 Fourier Transform 565
14.1.4 Time and Frequency Domain Concepts 574
14.2 Forced Shock Excitation of Vibration 575
14.2.1 Heaviside Step Excitation 576
14.2.2 Step Excitation of Finite Duration 578
14.2.3 Impulse Excitation [15, 17, 25] 581
14.3 Kinematic Shock Excitation of Vibration 582
14.3.1 Forms of the Vibration Equation 583
14.3.2 Response of a Linear Oscillator to Acceleration Impulse 584
14.4 Spectral Shock Theory 586
14.4.1 Biot´s Dynamic Model of a Structure: Primary and Residual Shock Spectrum 587
14.4.2 Response Spectra for the Simplest Vibration Protection System 589
14.4.3 Spectral Method for Determination of Response 590
14.5 Brief Comments on the Various Methods of Analysis 592
References 597
Chapter 15: Statistical Theory of the Vibration Protection Systems 599
15.1 Random Processes and Their Characteristics 600
15.1.1 Probability Distribution and Probability Density 601
15.1.2 Mathematical Expectation and Dispersion 603
15.1.3 Correlational Function 606
15.2 Stationary Random Processes 608
15.2.1 Properties of Stationary Random Processes 608
15.2.2 Ergodic Processes [12] 611
15.2.3 Spectral Density 612
15.2.4 Transformations of Random Exposures by a Linear System 615
15.3 Dynamic Random Excitation of a Linear Oscillator 620
15.3.1 Transient Vibration Caused by Impulse Shock 621
15.3.2 Force Random Excitation 625
15.4 Kinematic Random Excitation of Linear Oscillator 629
15.4.1 Harmonic and Polyharmonic Excitations 629
15.4.2 Shock Vibration Excitation by a Set of Damped Harmonics 635
References 639
Part IV: Special Topics 641
Chapter 16: Rotating and Planar Machinery as a Source of Dynamic Exposures on a Structure 642
16.1 Dynamic Pressure on the Axis of a Rotating Body 642
16.2 Types of Unbalancing Rotor 646
16.2.1 Static Unbalance 646
16.2.2 Couple Unbalance 647
16.2.3 Dynamic Unbalance 647
16.2.4 Quasi-Static Unbalance 648
16.3 Shaking Forces of a Slider Crank Mechanism 649
16.3.1 Dynamic Reactions 651
16.3.2 Elimination of Dynamic Reactions 654
References 659
Chapter 17: Human Operator Under Vibration and Shock 660
17.1 Introduction 660
17.1.1 Vibration Exposures and Methods of Their Transfer on the Person 661
17.1.2 International and National Standards 665
17.2 Influence of Vibration Exposure on the Human Subject 665
17.2.1 Classification of the Adverse Effects of Vibration on the Person 666
17.2.2 Effect of Vibration on the Human Operator 668
17.3 Vibration Dose Value 672
17.4 Mechanical Properties and Frequency Characteristics of the Body 676
17.4.1 Mechanical Properties of the Human Body 677
17.4.2 Frequency Characteristics of the Human Body 679
17.5 Models of the Human Body 682
17.5.1 Basic Dynamic 1D Models 684
17.5.2 Dynamic 2D-3D Models of the Sitting Human Body at the Collision 688
17.5.3 Parameters of the Human Body Model 690
References 694
Appendix A: Complex Numbers 697
A.1 Complex Conjugate Numbers 697
A.2 Algebraic Procedures 698
Appendix B: Laplace Transform 700
References 702
Index 703