كتاب Nonlinear Oscillations in Mechanical Engineering

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Nonlinear Oscillations in Mechanical Engineering - A. Fidlin

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1 Introduction 1
11 Usual Sources of Nonlinearity in Mechanical Engineering 1
111 Geometrical Nonlinearities 1
112 Physical Nonlinearities 2
113 Structural or Designed Nonlinearities 3
114 Constraints 4
115 Nonlinearity of Friction 6
12 The Basic Ideas of the Perturbation Analysis 8
121 Variation of Free Constants and Systems in the Standard Form 8
122 Standard Averaging as an Almost Identical Transformation 10
123 Method of Multiple Scales 13
124 Direct Separation of Motions 15
125 Relationship etween These Methods 16
13 Examples of Elementary Nonlinear Problems Solved by Standard
Averaging 17
131 Instability and Self Excited Oscillations in the Van Der Pol’s
Equation 17
132 The Main Resonance in a System with a Small Cubic Nonlinearity 19
133 Secondary Resonances in the System with Cubic Nonlinearity and
Strong Excitation 21
14 Axiomatic Theory of Collisions 24
141 Impulsive Motion of the Point Mass 24
142 Impulsive Motion of a System of Point Masses 25
143 Impulsive Motion of a Rigid Body 27
144 Collinear Collision of Two Point Masses 29
145 Direct Collisions in Mechanical Systems with Ideal Constraints 31
146 Concluding Remarks 33
2 Oscillations in Systems with Dry Friction 35
21 Self xcited Oscillations of the Mass-on-Moving-Belt 38
211 The Problem Description; Equations of Motion 38
212 Types of Motion 40
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213 Pure Slip Oscillations 42
214 Stick-slip Oscillations 44
215 Discussion of the Results 52
216 Concluding Remarks 54
22 Friction Induced Flutter 55
221 Mathematical Basics of Flutter in a System with Two Degrees of
Freedom 55
222 Wobbling of an Elastically Supported Friction Disc 56
223 On the Unstable Behavior of an Asymmetrically Supported Disc
(Brake Squeal) 61
224 Conclusions 64
23 Vibration Induced Displacement Averaging in Discontinuous Systems 64
231 A Simple Example of the Vibration Induced Displacement 65
232 Mathematical Basics for the First Order Averaging of the Constant
Order Discontinuous Regimes 67
233 The Elementary Example of the Vibration Induced Displacement
The First Order Approximation 69
234 Discussion of the Results 70
235 Conclusions 71
24 Vibration Induced Displacement of a Resonant Friction Slider 72
241 Problem Description 72
242 Equations of Motion 73
243 Illustration to System’s Behavior 76
244 Transformation to the Principal Form Amplitude of the Resonator 77
245 Motion of the Slider: Preparing for Averaging 81
246 Performance in Dependence of Parameters; Comparison etween
Analytic Prediction and Numerical Simulations 85
247 Conclusions 87
3 Systems with Almost Elastic Collisions 89
31 The Basic Ideas of Discontinuous Averaging Unfolding
Transformations 91
311 The Basic Idea of the Unfolding Transformation for the Mass
Limited at One Side 91
312 The Unfolding Transformation and Averaging in Case of Slightly
Inelastic Collisions 92
313 Comparison etween Analytic and Numeric Predictions for the
Oscillator Limited from One Side 96
314 Unfolding Transformation and Averaging for the Free Mass in a
Clearance 97
32 The “Mass-on-Moving-Belt” Limited at One Side: First Order
Approximation100
33 Second Order Approximation in Systems with Almost Elastic
Collisions103
331 General Mathematical Approach103
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Contents
332 The Second Order Approximation for the Amplitude of the Mass
on Moving Belt Limited from One Side 106
333 Discussion of the Results and Comparisons with Numeric
Experiments 108
34 The “Mass on Moving Belt” in a clearance 109
341 The Governing Equations and the Unfolding Transformation 110
342 Analyzing the Unperturbed System and Introducing Energy as the
Slow Variable 111
343 Discussion of the Results 114
35 Resonance of the Impact Oscillator Limited at One Side under External
Excitation 116
351 Equations of Motion and the Unfolding Transformation 116
352 Resonances in the Almost Linear System 118
353 Averaging in the Vicinities of the Almost Linear Resonances 119
354 Stability of the Stationary Solutions 121
355 Discussion of the Results, Comparison Between Analytic and
Numeric Predictions 122
36 Nonlinear Resonance of the Externally Excited Oscillator in a
Clearance 124
361 Equations of Motion and the Unfolding Transformation 125
362 Analyzing the Unperturbed System and Introducing Slow and Fast
Variables 126
363 Resonances in the Significantly Nonlinear System 128
364 Averaging in the Vicinity of the Main Nonlinear Resonance 129
365 Equations Governing the Slow Motions; Discussion of the Results 131
366 Comparison etween Analytic and Numeric predictions 133
37 Conclusions 134
4 Systems with Strong Dissipation Due to High Damping or Inelastic
Collisions 137
41 Averaging in Systems with Strong Linear Damping 138
411 The Basic Idea 138
412 Averaging in Systems with Strong Damping with Respect to one
or Several Variables 141
42 Linear Resonance in a Strongly Damped System with Two Degrees of
Freedom 143
421 Equations of Motion 143
422 Perturbation Analysis Transformation to the Form suitable for
Averaging 145
423 Equations of the First Order Approximation Discussion of the
Approach 148
424 Comparison with the Numeric Experiment Discussion of the
Results 150
43 Averaging in Systems with Inelastic Collisions: Basic Ideas and General
Approach 152
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XIV Contents
431 Basic Types of Motion in Systems with Inelastic Collisions:
Elementary Examples 152
432 On the Practical Importance of Regimes with Long Contact 158
433 Regimes with Long Contacts as an Example of the Variable Order
Discontinuous Systems 160
434 Variable Order Discontinuous Systems in the Standard Form 163
44 Basic Regime with Long Contacts for the Mass in a Resonantly Excited
Frame 165
441 Equations of Motion 165
442 Perturbation Analysis Transformation to the Form Suitable for
Averaging166
443 Equations of the First Order Approximation Discussion of the
Results168
45 The Basic Regime with Long Contacts for the Mass over the Resonantly
Excited Base 171
451 Equations of Motion 171
452 The Master and the Slave Variables; the Unperturbed Solution173
453 Equations of the First Order Approximation Discussion of the
Results175
46 Conclusions178
5 Short Notes on the Significantly Nonlinear Resonance181
51 The Basic Example of the Nonlinear Resonance184
511 Elementary Analysis and Natural Scale for the Resonance
Domain184
512 The Basic Regimes of the Equivalent Pendulum187
513 Stability of the Stationary Resonance189
514 Resonant Motions: Averaging with Respect to the Oscillations of
the Equivalent Pendulum 192
52 Nonlinear Resonant Crusher with Almost Elastic Collisions197
521 Problem Description Equations of Motion197
522 The Unfolding Transformation The Main Resonance 200
523 Averaging with Respect to the Fast Rotating Phase Stationary
Regimes201
53 Nonlinear Resonant Crusher with Inelastic Collisions203
531 Problem Description Equations of Motion203
532 The Regularizing Transformation The Main Resonance 206
533 Averaging with Respect to the Fast Rotating Phase Stationary
Regimes208
54 Conclusions210
6 High Frequency Excitation: Basic Ideas and Elementary Effects 213
61 Classification of Systems with HF Excitation Weakly Excited Systems 214
611 Classification of Systems with HF Excitation214
612 Systems with Weak HF Excitation215
613 The Weakly Excited Pendulum216
Contents XV
62 A Strongly Excited Pendulum with the Oscillating Suspension Point
Stiffening, Softening and Biasing 218
621 A Pendulum with the Vertically Vibrating Suspension Point:
Equations Governing the Slow Motions 218
622 Discussion of the Results for the Vertically Excited Pendulum 219
623 The Pendulum With the Horizontally Vibrating Suspension Point:
Equations of Slow Motions and System’s Behavior 221
624 The Pendulum Excited both Vertically and Horizontally 223
63 Shifted Resonances of the Pendulum The Overlapped Slow Excitation
and the Slowly Modulated HF Excitation 226
631 Two Types of Bi-harmonic Excitation 226
632 Obtaining Equations Governing the Slow Motions of the Pendulum 227
633 The Effect of the Overlapped slow Excitation 228
634 The Effect of the Slowly Modulated HF Excitation 231
635 Using the Slowly Modulated HF Excitation in Order to Quench the
Slow Excitation 232
64 The First Generalization and the Exceptional Role of the Terms
Depending on the Velocity in Systems with HF Excitation 234
641 The Basic Equation of the Vibrational Mechanics 234
642 A Remark on the Exceptional Role of the Terms Depending on the
Velocities 238
65 Smoothening of Dry Friction in Presence of HF Excitation Quenching
of the Friction Induced Oscillations 239
651 Smoothening of Dry Friction: A Simple Example 239
652 Slow Translation of a Particle on the Elliptically Vibrating Plane 243
653 Quenching of the Self Excited Oscillations Caused by the Negative
Friction Gradient 246
66 On the Misbehavior of the “Optimally” Controlled Pendulum under the
Influence of the HF Excitation 249
661 Description of the Problem, Equations Governing the Mechanical
Subsystem 249
662 The Optimal State-Feedback Control 251
663 System’s Behavior in Presence of the Strong HF Excitation: Numeric
Results 252
664 Transformation of the System to the Form Suitable for Averaging 253
665 The First Order Approximation; the Stationary Pendulum’s Tilt 256
666 The Second Order Approximation; the Stationary Position of the
Cart 257
667 Discussion of the Results 260
668 A Robust Control with Averaging Observer 262
7 Systems with High- requency Excitation: Advanced Analysis and
Generalizations 265
71 Systems with Strong Excitation General Analysis 266
72 Two Mathematical Examples of Systems with Strong Excitation 272
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XVI Contents
721 A System with One Degree of Freedom and Strong HF Excitation
Depending on the Velocity272
722 A System with Two Degrees of Freedom and a Skew Symmetric
HF Excitation Depending on the Velocities 275
73 The Lowest Natural Frequencies of an Elastic Rod with Periodic
Structure 277
74 Response of a One Degree of Freedom Nonlinear System to a Strong
HFExternal and Parametric Excitation Due to Oscillating Inertia 280
741 The Governing Equations and Their Transformation to the Basic
Mathematical Form280
742 Obtaining the Equations Governing Slow Motion281
743 Discussion of the Results282
75 Systems with Very strong Excitation in the Special Case of Fast
Oscillating Inertial Coefficients 284
76 Response of a One Degree of Freedom Nonlinear System to Very Strong
HF External and Strong Parametric Excitation due to Oscillating Inertia 286
761 Obtaining the Equations Governing the Slow Motion 287
762 Discussion of the Results288
763 Large Solutions 289
77 Dynamics of a Two Link Pendulum with a Fast Rotating Second Link 290
771 Equations of Motion and Their Transformation to the Basic Form
for Systems with Very Strong Excitation 291
772 Obtaining Equations Governing the Slow Motion292
773 Discussion of the Results295
774 A Short Remark on the Practical Importance of the Considered
Solutions297
78 Conclusions298
Appendixes299
Appendix I: The first Bogoliubov’s Theorem for Standard Averaging 299
Appendix II: On the Attractive Properties of the Asymptotically Stable
Equilibrium of the Averaged System 303
Appendix III: Averaging of Systems with Short Strong Perturbations307
Appendix IV: Averaging of Systems with Small Discontinuities of the Right
Hand Sides317
Appendix V: Averaging of Systems with Small Discontinuities of the
Unknown Function321
Appendix VI: Averaging of Variable Order Discontinuous Systems328
Appendix VII: Hierarchic Averaging in Systems with a Semi Slow Rotating
Phase 336
Appendix VIII: Averaging in Systems with Strong High Frequency
Excitation340
References345
Index

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