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![كتاب Mechanical Vibration - Analysis, Uncertainties, and Control Empty](https://2img.net/i/fa/empty.gif) | موضوع: كتاب Mechanical Vibration - Analysis, Uncertainties, and Control الأحد 14 أغسطس 2022, 1:43 am | |
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أخواني في الله أحضرت لكم كتاب Mechanical Vibration - Analysis, Uncertainties, and Control FOURTHE DITION Haym Benaroya • Mark Nagurka • Seon Han
![كتاب Mechanical Vibration - Analysis, Uncertainties, and Control M_v_a_11](https://i.servimg.com/u/f61/15/36/74/14/m_v_a_11.jpg) و المحتوى كما يلي :
Contents 1 INTRODUCTION AND BACKGROUND 1 1.1 Challenges and Examples . 2 1.2 Systems and Structures 5 1.3 Basic Concepts of Vibration 5 1.3.1 Modeling for Vibration 6 1.3.2 Idealization and Formulation . 6 1.3.3 Inertia, Stiffness, and Damping 10 1.3.4 Properties of Keyboard Keys . 10 1.3.5 Computational Aspects 11 1.3.6 Is Vibration Good or Bad? 12 1.3.7 Vibration Control . 14 1.4 Types of Vibration . 14 1.4.1 Signal Classification 14 1.4.2 Deterministic Approximations 14 1.4.3 Probability . 15 1.4.4 System Model Uncertainty 15 1.4.5 Random Vibration . 15 1.5 Types of System Models 15 1.5.1 Linear Approximation . 15 1.5.2 Dimensionality . 16 1.5.3 Discrete Models 16 1.5.4 Continuous Models 16 1.5.5 Nonlinear Models . 17 1.6 Basic Dynamics . 17 1.6.1 Statics and Equilibrium 17 1.6.2 The Equations of Motion . 17 1.6.3 Linear Momentum and Impulse . 19 1.6.4 Principles of Work and Energy 19 1.7 Units . 21 1.7.1 Mars Orbiter Loss . 21 1.7.2 U.S. Customary and SI Systems . 21 1.7.3 The Second . 22 1.7.4 Dimensional Analysis . 27 1.8 Concepts Summary . 29 1.9 Quotes 29 2 SINGLE DEGREE-OF-FREEDOM UNDAMPED VIBRATION 31 2.1 Motivating Examples 31 2.1.1 Transport of a Satellite 31 2.1.2 Rocket Propulsion . 32 2.2 Deterministic Modeling . 33 2.2.1 Problem Idealization . 33 2.2.2 Mass, Damping, and Stiffness 34 xvxvi CONTENTS 2.2.3 Deterministic Approximation . 35 2.2.4 Equations of Motion 35 2.2.5 Energy Formulation 43 2.2.6 Representing Harmonic Motion . 46 2.2.7 Solving the Equations of Motion . 50 2.3 Undamped Free Vibration . 50 2.3.1 Alternate Formulation 52 2.3.2 Phase Plane 54 2.4 Harmonically Forced Vibration 55 2.4.1 A Note on Terminology 57 2.4.2 Resonance . 58 2.4.3 Vibration of a Structure in Water 64 2.5 Concepts Summary . 69 2.6 Quotes 69 2.7 Problems 69 3 SINGLE DEGREE-OF-FREEDOM DAMPED VIBRATION 83 3.1 Overview 83 3.2 Introduction to Damping 86 3.3 Damping Models 87 3.3.1 Viscous Damping and Loss Factor 87 3.3.2 Coulomb Damping 89 3.4 Free Vibration with Viscous Damping 91 3.4.1 Critically Damped and Overdamped Systems 92 3.4.2 Some Time Constants . 92 3.4.3 Underdamped Systems 94 3.4.4 Logarithmic Decrement 97 3.4.5 Phase Plane 101 3.5 Free Vibration with Coulomb Damping 101 3.6 Forced Vibration with Viscous Damping . 105 3.7 Forced Harmonic Vibration 106 3.7.1 Response to Harmonic Excitation 106 3.7.2 Harmonic Excitation in Complex Notation . 115 3.7.3 Harmonic Base Excitation 117 3.7.4 Rotating Unbalance 120 3.8 Forced Periodic Vibration . 127 3.8.1 Harmonic/Spectral Analysis . 127 3.8.2 Fourier Series . 127 3.9 Concepts Summary . 129 3.10 Quotes 130 3.11 Problems 130 4 SINGLE DOF VIBRATION: GENERAL LOADING AND ADVANCED TOPICS 137 4.1 Arbitrary Loading: Laplace Transform 137 4.2 Step Loading 142 4.3 Impulsive Excitation 146 4.4 Arbitrary Loading: Convolution Integral . 150 4.5 Introduction to Lagrange’s Equation . 154 4.6 Notions of Randomness 158 4.7 Notions of Control . 159 4.8 The Inverse Problem 159 4.9 A Self-Excited System and Its Stability 160 4.10 Solution Analysis and Design Techniques . 160 4.11 Model of a Bouncing Ball . 166 4.11.1 Time of Contact 167CONTENTS xvii 4.11.2 Stiffness and Damping 167 4.11.3 Natural Frequency & Damping Ratio 168 4.11.4 Approximations 168 4.12 Concepts Summary . 169 4.13 Quotes 169 4.14 Problems 169 5 VARIATIONAL PRINCIPLES AND ANALYTICAL DYNAMICS 175 5.1 Introduction . 175 5.2 Constraints . 176 5.3 Virtual Work 176 5.3.1 Work and Energy . 176 5.3.2 Principle of Virtual Work . 178 5.3.3 D’Alembert’s Principle 182 5.4 Lagrange’s Equation 189 5.4.1 Lagrange’s Equation for Small Oscillations . 193 5.5 Hamilton’s Principle 196 5.6 Lagrange’s Equation with Damping 197 5.7 Jourdain’s Principle 198 5.7.1 Jourdain’s Principle from d’Alembert’s Principle 198 5.8 Concepts Summary . 199 5.9 Quotes 200 5.10 Problems 200 6 MULTI DEGREE-OF-FREEDOM VIBRATION 209 6.1 Motivating Examples 209 6.1.1 Periodic Structures 209 6.1.2 Inverse Problems . 209 6.1.3 Vehicle Vibration Testing . 210 6.1.4 Scope 211 6.2 The Concepts of Stiffness and Flexibility . 211 6.2.1 Influence Coefficients . 211 6.3 Equations of Motion 216 6.3.1 Mass and Stiffness Matrices . 219 6.4 Undamped Vibration 220 6.4.1 Two Degree-of-Freedom Vibration: Direct Method . 220 6.4.2 Harmonically Forced Vibration: Direct Method 226 6.4.3 Undamped Vibration Absorber 229 6.4.4 Beating Oscillations 232 6.5 Free Vibration with Damping: Direct Method 235 6.6 Modal Analysis . 241 6.6.1 Modal Orthogonality . 241 6.6.2 Modal Analysis with Forcing . 244 6.6.3 Modal Analysis with Proportional Damping 248 6.7 Nonproportional Damping . 251 6.7.1 Phase Synchronization 253 6.8 Real and Complex Modes . 256 6.8.1 Modal Analysis vs. Direct Method 257 6.9 Special Cases 258 6.9.1 Unrestrained Systems . 258 6.9.2 Rigid-Body Mode . 258 6.9.3 Repeated Frequencies . 262 6.10 Eigenvalue Geometry 262 6.11 Periodic Structures . 264xviii CONTENTS 6.11.1 Perfect Lattice Models 264 6.11.2 Effects of Imperfection 265 6.12 Inverse Vibration Problem . 265 6.13 Fluid Sloshing in Container 268 6.14 Stability of Motion . 269 6.15 Rayleigh’s Quotient . 271 6.16 Concepts Summary . 273 6.17 Quotes 273 6.18 Problems 274 7 CONTINUOUS MODELS FOR VIBRATION 283 7.1 Discrete to Continuous . 283 7.2 Vibration of Strings 284 7.2.1 Wave Propagation Solution 285 7.2.2 Wave Equation via Hamilton’s Principle 287 7.2.3 Boundary Value Problem . 288 7.2.4 Modal Solution for Fixed-Fixed Boundary Conditions . 289 7.3 Axial Vibration of Beams . 292 7.3.1 Axial Vibration: Newton’s Approach 292 7.3.2 Axial Vibration: Hamilton’s Approach . 293 7.3.3 Simplified Eigenvalue Problem 294 7.3.4 Eigenfunction Expansion Method 295 7.4 Torsional Vibration of Shafts . 299 7.4.1 Torsion of Shaft with Rigid Disk at One End 300 7.5 Transverse Vibration of Beams 301 7.5.1 Timoshenko Beam . 302 7.5.2 Boundary Conditions . 304 7.5.3 Bernoulli-Euler Beam . 304 7.5.4 Orthogonality of the Modes . 308 7.5.5 Nodes and Antinodes . 314 7.6 Other Transverse Beam Vibration Cases . 314 7.6.1 Beam with Axial Forces 314 7.6.2 Beam with Elastic Restraints . 315 7.6.3 Beam on an Elastic Foundation . 316 7.6.4 Beam with a Moving Support 317 7.6.5 Different Boundary Conditions 318 7.6.6 Beam with Traveling Force 320 7.7 Concepts Summary . 320 7.8 Quotes 321 7.9 Problems 321 8 CONTINUOUS MODELS FOR VIBRATION: ADVANCED MODELS 329 8.1 Vibration of Membranes 329 8.1.1 Rectangular Membranes . 329 8.1.2 Circular Membranes 332 8.2 Vibration of Plates . 336 8.2.1 Rectangular Plates 336 8.2.2 Eigenvalue Problem 338 8.3 Approximate Methods . 341 8.3.1 Rayleigh’s Quotient 341 8.3.2 Rayleigh-Ritz Method . 343 8.3.3 Galerkin Method . 348 8.4 Variables Not Separating 350 8.4.1 Nonharmonic, Time-Dependent Boundary Conditions . 350 8.4.2 Pipe Flow with Constant Tension 355CONTENTS xix 8.5 Concepts Summary . 358 8.6 Quotes 358 8.7 Problems 359 9 RANDOM VIBRATION: PROBABILISTIC FORCES 361 9.1 Introduction . 361 9.2 Motivation 363 9.2.1 Random Vibration . 363 9.2.2 Fatigue Life 364 9.2.3 Ocean Wave Forces 365 9.2.4 Wind Forces 366 9.2.5 Material Properties 366 9.2.6 Statistics and Probability . 367 9.3 Random Variables . 368 9.3.1 Probability Distribution 368 9.3.2 Probability Density Function . 369 9.4 Mathematical Expectation . 370 9.4.1 Variance 371 9.5 Useful Probability Densities 372 9.5.1 Uniform Density 372 9.5.2 Exponential Density 373 9.5.3 Normal (Gaussian) Density 373 9.5.4 Lognormal Density 376 9.5.5 Rayleigh Density 377 9.6 Two Random Variables . 377 9.6.1 Covariance and Correlation 377 9.7 Random Processes . 380 9.7.1 Random Process Descriptors . 380 9.7.2 Ensemble Averaging 380 9.7.3 Stationarity 382 9.7.4 Power Spectrum 383 9.7.5 Units 385 9.7.6 Narrow-Band and Broad-Band Processes 386 9.7.7 White-Noise Process 387 9.8 Random Vibration . 388 9.8.1 Formulation and Preliminaries 388 9.8.2 Mean-Value Response . 389 9.8.3 Response Correlations . 389 9.8.4 Response Spectral Density 390 9.9 Stochastic Response of a Linear MDOF System . 393 9.10 Lunar Seismic Structural Analysis 394 9.11 Random Vibration of Continuous Structures . 398 9.12 Monte Carlo Simulation 400 9.12.1 Random Number Generation . 401 9.12.2 Generating Random Variates . 402 9.12.3 Generating Time History for Random Process . 403 9.13 Inverse Vibration with Uncertain Data 404 9.14 Concepts Summary . 407 9.15 Quotes 407 9.16 Problems 407xx CONTENTS 10 VIBRATION CONTROL 413 10.1 Motivation 413 10.2 Approaches to Controlling Vibration . 415 10.2.1 Why Active Control 416 10.3 Feedback Control 418 10.3.1 Disadvantages of Feedback 421 10.4 Performance of Feedback Control Systems 422 10.4.1 Poles and Zeros of a Second-Order System . 425 10.4.2 System Gain 426 10.4.3 Stability of Response . 427 10.5 Control of Response 427 10.5.1 Control Actions 427 10.5.2 Control of Transient Response 428 10.6 Parameter Sensitivity 430 10.7 State Variable Models . 434 10.7.1 Transfer Function from State Equation . 436 10.7.2 Controllability and Observability . 437 10.7.3 State Variable Feedback . 439 10.8 Multivariable Control 440 10.8.1 State and Output Equations . 441 10.8.2 Controllability and Observability . 441 10.8.3 Closed-loop Feedback of MIMO Systems 441 10.9 Stochastic Control . 442 10.10 Concepts Summary . 443 10.11 Quotes 443 10.12 Problems 444 11 NONLINEAR VIBRATION 447 11.1 Introduction . 447 11.2 Physical Examples . 449 11.2.1 Simple Pendulum: Approximate Solution 451 11.2.2 Simple Pendulum: Exact Solution 451 11.2.3 Duffing and van der Pol Equations 452 11.3 The Phase Plane 452 11.3.1 Stability of Equilibria . 454 11.4 Perturbation or Expansion Methods . 459 11.4.1 Lindstedt-Poincaré Method 462 11.4.2 Forced Oscillations of Quasi-Harmonic Systems 464 11.4.3 Jump Phenomenon 466 11.4.4 Periodic Solutions of Nonautonomous Systems . 466 11.4.5 Subharmonic and Superharmonic Oscillations . 468 11.5 Mathieu Equation . 470 11.6 Van der Pol Equation . 474 11.6.1 Unforced van der Pol Equation 475 11.6.2 Limit Cycles 475 11.6.3 Forced van der Pol Equation . 475 11.7 Motion in the Large 478 11.8 Nonlinear Control 480 11.9 Random Duffing Oscillator . 483 11.10 Nonlinear Pendulum: Galerkin Method 484 11.11 Concept Summary . 485 11.12 Quotes 485 11.13 Problems 486CONTENTS xxi A MATHEMATICAL CONCEPTS FOR VIBRATION 489 A.1 Complex Numbers . 489 A.1.1 Complex Number Operations . 489 A.1.2 Absolute Value . 490 A.1.3 Equivalent Representation 490 A.2 Matrices . 490 A.2.1 Matrix Operations . 491 A.2.2 Determinant and Matrix Inverse . 491 A.2.3 Eigenvalues and Eigenvectors . 493 A.2.4 Matrix Derivatives and Integrals . 493 A.3 Taylor Series & Linearization . 494 A.4 Ordinary Differential Equations 494 A.4.1 Solution of Linear Equations . 495 A.4.2 Homogeneous Solution 496 A.4.3 Particular Solution 498 A.5 Laplace Transforms . 499 A.5.1 Borel’s Theorem 500 A.5.2 Partial Fraction Expansion 500 A.5.3 Laplace Transform Table . 501 A.5.4 Initial-Value, Final-Value Theorem . 501 A.6 Fourier Series & Transforms 501 A.6.1 Fourier Series . 501 A.6.2 Fourier Transforms 502 A.7 Partial Differential Equations . 502 B VISCOELASTIC DAMPING 505 B.1 Viscoelastic Materials . 505 B.1.1 Work Done Per Cycle . 506 B.2 Viscoelastic Material Models 506 B.2.1 Maxwell Model 506 B.2.2 Voigt Model 509 B.2.3 Maxwell Standard Linear Model . 511 B.2.4 Stress-Strain Equivalent Model 512 B.2.5 Boltzmann Superposition Model . 512 B.2.6 General Nonviscous Damping 513 B.3 Causality Issues in Damping Models . 517 B.4 Concepts Summary . 518 B.5 Quotes 518 C SOLVING VIBRATION PROBLEMS WITH MATLAB 521 C.1 Introduction . 521 C.2 SDOF Undamped System . 521 C.3 SDOF Damped System . 524 C.4 SDOF Overdamped System 526 C.5 SDOF Undamped System with Harmonic Excitation 528 C.6 SDOF Damped System with Harmonic Excitation . 531 C.7 SDOF Damped System with Base Excitation 533 C.8 SDOF Damped System with Rotating Unbalance 537 C.9 SDOF Damped System with Impulse Input . 540 C.10 SDOF Damped System with Step Input . 541 C.11 SDOF Damped System with Square Pulse Input 544 C.12 SDOF Damped System with Ramp Input 548 C.13 SDOF System with Arbitrary Periodic Input 550 C.14 MDOF Undamped System . 552 C.15 MDOF Damped System 555xxii CONTENTS C.16 General Vibration Solver 558 C.17 Van der Pol Oscillator . 563 C.18 Random Vibration . 565 C.19 Duffing Oscillator with Random Excitation . 567 C.20 Monte Carlo Simulation of a Random System 570 Index 57 Index accelerometer, 139 admittance, 138 aeroelastic forcing, 67 aerodynamic loads, 363 aircraft, 2 aerodynamic loads, 363 amplitude modulation, 235 anelastic, 86 anelasticity, 83 automobile vibration testing, 62, 210 base-excited motion, 227 beam axial vibration, 292 time-dependent boundary conditions, 350 Bernoulli-Euler, 301, 304 Galerkin method, 348 random vibration, 398 Rayleigh’s quotient, 341 Rayleigh-Ritz method, 343 space elevator, 296 Timoshenko, 302 transversely vibrating, 301 beam transversely vibrating flow in pipe, 355 beating, 60, 232 Bernoulli, 310 Bernoulli-Euler random vibration, 398 Bernoulli-Euler beam, 301, 304 axial force, 314 boundary conditions, 305 elastic foundation, 316 elastic restraints, 315 equation of motion, 304 Euler buckling load, 315 flow in pipe, 355 harmonic boundary conditions, 318 moving support, 317 non-separable solution Mindlin-Goodman technique, 350 orthogonality of modes, 308 traveling force, 320 Bessel, 334 Black, 420 Bode, 431 Bode plot, 107 Bode plots, 431 Borel’s theorem, 138, 142 Borgman’s method, 403 bouncing ball, 166 impact, 166 natural frequency and damping, 168 stiffness and damping, 167 time of contact, 167 boundary value problem string, 289 bridges, 61 damping, 84 London Millenium, 113 Buckingham-Pi Theorem, 28 calculus of variations, 175 characteristic equation, 139, 496 one degree-of-freedom, 91 rectangular membrane, 331 two degrees-of-freedom, 221 characteristic matrix, 226, 233 two degrees of freedom, 221 characteristic polynomial, 226 characteristic time, 93 characteristic values, 221 characteristic vectors, 222 closed-loop transfer function, 419 complex numbers, 46, 489 complex conjugate, 489 Euler’s identity, 489 magnitude, 490 reciprocal, 490 computer keyboards, 10 conservation of energy, 21 conservation of linear momentum, 19 conservative force, 20 constraints configuration, 176 holonomic, 176 nonholonomic, 176 rheonomic, 176 scleronomic, 176 velocity, 176 continuous system 573574 INDEX limit of MDOF system, 283 control, 14, 159 active, 416 actuator, 418 block diagram, 418 Bode plots, 431 characteristic time, 425 classical, 417 closed-loop, 418 controllability, 438, 441 feedback, 413 integral, 428 modern, 417 multivariable, 440 negative feedback, 420 nonlinear, 480 observability, 438, 441 open-loop, 418 oscillator, 159 passive, 415 peak overshoot, 423 positive feedback, 419 proportional, 428 proportional plus derivative, 428 proportional plus integral, 428 proportional plus integral plus derivative, 428 settling time, 423 stability, 427 state variable feedback, 439 step response, 422 stochastic, 442 system gain, 426 controllability, 437 convolution integral, 138, 151 Duhamel integral, 150, 249 correlation coefficient, 382 Coulomb, 90 critical damping, 91 critically damped system, 425 d’Alembert, 184 d’Alembert’s principle, 182 for a rigid body, 182 dampers bridge, 84 damping, 10, 86 Boltzmann superposition model, 512 causality, 517 Coulomb, 86, 89 critical, 91, 92, 425 equivalent viscous damping, 89 general linear nonviscous damping, 513 general nonviscous damping, 513 hysteresis loops, 88 loss factor, 87, 506 material, 86 Maxwell model, 506 Maxwell standard linear model, 511 multi degree-of-freedom systems, 250 overdamped, 91 radiation, 86 standard model, 506 stress-strain equivalent model, 512 structural, 86 underdamped, 91, 94 viscoelastic, 86 viscoelastic materials, 506 viscous, 35, 86 Voigt model, 88, 509 Kelvin-Voigt model, 506 damping nonproportional, 251 Den Hartog, 110 Den Hartog force vector diagram, 108 design, 160 spacecraft shipping container, 163 water landing space module, 161 differential equations ordinary, 494 homogeneous solution, 496 non-homogeneous solution, 498 partial, 502 separation of variables, 502 dimensional analysis similitude theory, 27 Dirac, 148 direct method free vibration with damping, 235 harmonic forcing, 226 displacement transmissibility, 118 Duffing equation, 452 forced oscillations, 464 jump phenomenon, 466 perturbation method Lindstedt-Poincaré, 463 random forcing, 483 Duhamel, 153 Duhamel integral, 150 dynamic amplification factor undamped, 55 dynamics, 1 eigenfunction axially vibrating beam orthogonality, 296 string, 290 normalized, 290 eigenvalue problem axially vibrating beam, 294 characteristic roots, 493INDEX 575 circular membrane, 332 geometric interpretation, 262 rectangular membrane, 330 rectangular plate, 338 repeated frequencies, 262 geometric interpretation, 263 string, 289 eigenvalues, 493 two degrees of freedom, 221 eigenvectors, 493 two degrees of freedom, 222 Einstein, 23 elevator cable system, 224 elliptic integral, 451 energy, 176 engineering, 1 ensemble, 380 ensemble averaging, 380 ergodicity, 382 Euler, 98 Euler buckling load, 315 Euler’s formula, 47, 489 expansion theorem, 242 expected value, 370 fatigue life, 364 fault detection, 159 feedback control, 418 performance, 422 Flügge-Lotz, 482 flexibility coefficient, 211 flexibility matrix, 212 fluid-structure interaction, 64 equivalent mechanical models, 268 flow in pipe, 355 flow-oscillator model, 67 fluid sloshing in containers, 268 Mathieu equation, 470 Pierson-Moskowitz spectrum, 383 Pierson-Moskowitz spectrum generation, 403 van der Pol equation, 474 wave forces Morrison equation, 365 force transmissibility, 119 force vector diagram, 108 dominated frequency range, 109 forced vibration direct method undamped, 226 viscous damping, 105 harmonic, 106 harmonic base excitation, 117 harmonic rotating unbalance, 120 periodic not harmonic, 127 forces arbitrary, 150 conservative, 177 impulsive, 146 nonconservative, 191 random, 159 step load, 142, 151 wind, 366 Fourier, 125 Fourier series, 501 Fourier series forcing, 127 square wave, 128 Fourier transforms, 502 Fourier transform pairs, 502 free vibration Coulomb damping, 101 direct method damped, 235 undamped, 220 undamped, 50 viscous damping, 91 free-body diagram, 34, 35 two degree-of-freedom, 216 frequency damped, 94 fundamental, 221 inclusion principle, 267 frequency response function, 116, 389 sensitivity, 432 transfer function, 388 Galerkin, 346 Galerkin method axially vibrating beam, 348 nonlinear pendulum, 484 Galileo, 51 Gauss, 374 generalized coordinate, 189 generalized force, 190 damping, 192 nonconservative, 192 Germain, 340 Gilbreth, 215 Hamilton, 193 Hamilton’s principle, 196 axially vibrating beam, 293 damped case, 218 elastic pendulum, 197 extended principle, 196 physical interpretation, 196 string, 287 Timoshenko beam, 302 two degrees of freedom, 217 harmonic base excitation, 117 displacement transmissibility, 118576 INDEX force transmissibility, 118 harmonic forcing complex notation, 115 damped response, 106 direct method, 226 undamped, 55 harmonic motion, 46 rotating vector representation, 46 Hooke’s law, 10, 35, 211 idealization, 6, 33 impedance, 138 impulse response, 147, 389 inclusion principle, 267 inertia, 10, 18 influence coefficients, 211 flexibility, 211 reciprocity, 213 stiffness, 211 inverse problem deterministic, 266 imperfect system, 404 two degrees-of-freedom, 406 mass and stiffness estimates, 265 single degree-of-freedom system, 159 inverse problems, 209 Jackson, 367 Jeffcott rotor, 123 Jourdain’s principle, 198 principle of virtual power, 198 Kelvin, 513 kinematics, 1 kinetic energy, 43 particle, 20 kinetics, 1 L’Hôpital, 63 Lagrange, 156 Lagrange’s equation, 43, 154, 189, 191 damped case, 198 fluid-conveying pipe, 357 Lagrangian, 191 mass coefficients, 193 simple pendulum, 154 single degree-of-freedom oscillator, 154 small oscillations, 193 stiffness coefficients, 193 two degrees of freedom, 217 Laplace, 143 Laplace transform, 137, 499 Borel’s theorem, 500 impulsive excitation, 147 partial fraction expansion, 500 step loading, 142 table of, 501 Laplacian, 330 Leibnitz, 48 linear momentum, 19 linear superposition, 15, 33, 495 Fourier series solution, 128 lack of for nonlinear problems, 447 linearization, 33 logarithmic decrement, 97 longitudinal vibration forced beam, 295 loss factor, 87, 506 causality, 88 magnification factor, 106 rotating shaft, 124 mass coefficients, 193 mass matrix positive definite, 219 mathematical expectation, 370 mathematical modeling, 33 Mathieu equation, 470 matrix, 490 adjoint, 492 determinant, 491 eigenvalues, 493 eigenvectors, 493 inverse, 492 matrix multiplication, 491 matrix transpose, 491 singular, 492 Maxwell, 507 membrane circular, 332 boundary conditions, 333 eigenvalue problem, 332 modes of vibration, 334 rectangular, 329 boundary conditions, 330 eigenvalue problem, 330 equation of motion, 329 modes of vibration, 331 MEMs, 56 metronome, 38 Miner’s rule, 364 modal analysis, 241 forced, 244 modal analysis vs. direct method, 257 modal coordinates, 242 modal matrix, 242 modal participation factor, 246 proportional damping, 248 random forces, 393 modal participation factor, 246 modal ratios, 222INDEX 577 modes of vibration circular membrane, 334 degenerate, 333 nodes, 223 real vs. complex, 256 rectangular membrane, 331 rigid-body mode, 258 string, 288 two degrees-of-freedom, 221 Monte Carlo simulation, 400 pseudorandom numbers, 401 random number generation, 401 natural frequency, 36 Newton, 40 Newton’s first law of motion, 17 Newton’s second law of motion, 18, 35 axially vibrating beams, 292 rectangular membrane, 329 string, 285 torsional vibration, 37 unrestrained translation, 261 Newton’s third law of motion, 18 Newton-Euler equation, 36 rectangular plate, 337 unrestrained rotation, 260 nodes and antinodes, 314 nondestructive evaluation, 159 nonlinear vibration autonomous system, 448 center, 456 combination harmonics, 469 Duffing equation, 466 jump phenomenon, 466 logarithmic spiral, 456 Mathieu equation, 470 stability, 471 motion in the large, 478 node, 456 non-autonomous system, 448 parametric forcing, 447 perturbation method, 459 Lindstedt-Poincaré, 462 saddle point, 456 stability of equilibria, 454 subharmonic oscillation, 469 trajectories, 455 nonproportional damping, 251 nonseparable partial differential equations nonharmonic boundary conditions, 350 pipe flow, 355 observability, 437 offshore structures, 2, 213 Morrison equation, 365 wave forces, 365 orthogonality of modes with respect to mass matrix, 242 with respect to stiffness matrix, 242 overdamped system, 91, 425 peak overshoot, 423 pendulum base excited, 269 coupled pendula, 232 amplitude modulation, 235 elastic, 197 nonlinear forced, 466 simple, 37, 154, 190 nonlinear, 449 via Lagrange’s equation, 190 periodic structures, 209 periodic systems, 264 imperfect lattice models, 404 imperfect periodicity, 265 perfect lattice models, 264 phase angle damped system, 94, 95 undamped, 51 viscous damping, 107 phase plane, 54 linear oscillator, 101 nonlinear oscillator, 452 phasing between modes, 237, 252 phase synchronization, 253 plate rectangular, 336 assumptions, 336 boundary conditions, 339 eigenvalue problem, 338 equation of motion, 338 flexural rigidity, 338 Newton-Euler equation, 337 Poincaré, 461 poles, 139 potential energy, 20, 43, 177 principal coordinates, 242 principle of virtual work, 178, 179 constraint, 178 virtual displacement, 178 principle of work and energy, 177 probability, 362 correlation coefficient, 377 cumulative distribution function, 368 ensemble averaging, 380 excursion frequency, 362 histogram, 362 mathematical expectation, 370 expected value, 370578 INDEX variance, 371 probability density function, 361, 369 exponential density function, 373 Gaussian or normal, 373 joint density function, 377 lognormal density function, 376 marginal densities, 378 Rayleigh density, 377 standard normal density, 374 uniform density function, 372 probability distribution function joint distribution function, 377 random process, 363 random variable, 363, 368 statistically independent, 378 proportional damping, 248 random process, 380, see probability correlation coefficient, 382 correlation function, 381 autocovariance, 382 cross correlation, 381 stationarity, 382 earthquake spectra, 384 ergodicity, 382 narrow- and broad-band, 386 ocean wave height spectra, 383 spectral density, 383 white-noise process, 387 wind velocity spectra, 384 Wiener-Khintchine formulas, 383 random vibration, 363, 388 beams, 398 Borgman’s method, 403 Duffing equation, 483 fundamental theorem, 390 multi degree-of-freedom systems, 393 response mean, 389 response spectral density, 390 response to white noise, 390 response variance, 390 randomness, 158, 361 aleatory, 361 epistemic, 361 Rayleigh, 342 Rayleigh damping, 249 Rayleigh dissipation function, 198 Rayleigh’s quotient discrete system, 271 three degrees of freedom, 272 two degrees of freedom, 272 transversely vibrating beam, 341 Rayleigh-Ritz method transversely vibrating beam, 343 resonance, 58, 59 electric motors, 63 Reynolds, 67 rigid-body mode, 258 rigid-body motion, 258 rocket ship, 6, 18, 32, 61, 139 booster vibration, 61 rolling disk, 155 rotating shaft, 123 whirling, 123 rotating unbalance, 120 subcritical operation, 123 supercritical operation, 123 satellites, 31 second, 22 secular term, 60 seismic analysis lunar structures, 394 self-excited system, 160 parametrically excited system, 160 stability, 160 sensitivity, 430 control, 433 separation of variables, 502 axially vibrating beam, 294 Bernoulli-Euler beam, 305 circular membrane, 332 rectangular membrane, 330 rectangular plate, 338 string, 288 settling time, 423 shaft frequency dependent boundary condition, 301 natural frequencies, 301 torsional vibration, 36, 299 warping, 299 space elevator, 296 spacecraft vibration due to shipping, 163 spacecraft shipping container design, 163 spectral density, 383 spring inertia, 44 series and parallel, 53 stability, 160, 427 state variables, 434 stationarity, 382 stiffness, 10 stiffness coefficients, 193 stiffness matrix, 212 positive semi-definite, 219 singular, 260 strain energy, 43 string boundary conditions, 289INDEX 579 boundary value problem, 289 equation of motion, 284 Newton’s second law of motion, 285 standing wave, 285, 286 traveling string, 358 wave equation, 285 Hamilton’s principle, 287 wave propagation, 285 synchronicity of motion, 220 system identification, 159, 210 systems, 5 tallest buildings, 7 Taylor, 34 Taylor series, 178, 455, 494 time constants characteristic time, 92, 425 correlation time, 92 relaxation time, 92 Timoshenko, 312 Timoshenko beam, 302 boundary conditions, 304 deflection equation of motion, 304 equations of motion, 303 shear distortion, 302 torsional vibration shaft, 299 transfer function, 138, 147 base-excited system, 417 closed-loop, 419 control, 417, 418 poles and zeros, 422, 425 transverse vibration beam, 301 Timoshenko beam, 302 turbine, 6 underdamped system, 91, 94, 425 units, 21 unrestrained system, 258 constraint matrix, 259 rigid-body mode, 258 rotation, 260 van der Pol, 477 van der Pol equation, 452 flow-oscillator model, 474 forced, 475 frequency entrainment, 476 limit cycle, 475 unforced, 475 variable mass system, 38 variance, 371 variation, 175 variational principles, 175 Vaughan, 291 vibration absorber pipeline, 232 powered hand tools, 104 Stockbridge, 230 undamped case, 229 vibration control, 14, 413 vibration testing, 387 accelerated testing, 388 automobile, 210 bicycles, 391 viscoelastic materials, 505 Boltzmann superposition model, 512 complex modulus, 506 general nonviscous damping, 513 loss modulus, 506 Maxwell model, 506 standard linear model, 506 storage modulus, 505 Voigt model Kelvin-Voigt model, 506 viscous damping factor, 36 viscous damping ratio, 36 Voigt, 510 von Kármán, 65 water landing space module, 161 wave equation, 502 beam, 293 rectangular membrane, 330 Weiner, 442 Wiener-Khintchine formulas, 383 wind-induced oscillations, 230 wind-induced vibration, 250 work, 176 done by conservative force, 20 zeros, 139
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