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أخوانى فى الله أحضرت لكم كتاب
Nonlinear Oscillations in Mechanical Engineering - A. Fidlin
ويتناول الموضوعات الأتية :
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 b E XI Contents 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 b b I 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 b XIII 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 F 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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