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| موضوع: كتاب Engineering Vibration الخميس 21 أكتوبر 2021, 10:04 pm | |
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أخواني في الله أحضرت لكم كتاب Engineering Vibration Fourth Edition DaniEl J. inman University of Michigan
و المحتوى كما يلي :
Contents PREFACE viii 1 INTRODUCTION TO VIBRATION AND THE FREE RESPONSE 1 1.1 Introduction to Free Vibration 2 1.2 Harmonic Motion 13 1.3 Viscous Damping 21 1.4 Modeling and Energy Methods 31 1.5 Stiffness 46 1.6 Measurement 58 1.7 Design Considerations 63 1.8 Stability 68 1.9 Numerical Simulation of the Time Response 72 1.10 Coulomb Friction and the Pendulum 81 Problems 95 MATLAB Engineering Vibration Toolbox 115 Toolbox Problems 116 2 RESPONSE TO HARMONIC EXCITATION 117 2.1 Harmonic Excitation of Undamped Systems 118 2.2 Harmonic Excitation of Damped Systems 130 2.3 Alternative Representations 144 2.4 Base Excitation 151 2.5 Rotating Unbalance 160 2.6 Measurement Devices 166iv Contents 2.7 Other Forms of Damping 170 2.8 Numerical Simulation and Design 180 2.9 Nonlinear Response Properties 188 Problems 197 MATLAB Engineering Vibration Toolbox 214 Toolbox Problems 214 3 GENERAL FORCED RESPONSE 216 3.1 Impulse Response Function 217 3.2 Response to an Arbitrary Input 226 3.3 Response to an Arbitrary Periodic Input 235 3.4 Transform Methods 242 3.5 Response to Random Inputs 247 3.6 Shock Spectrum 255 3.7 Measurement via Transfer Functions 260 3.8 Stability 262 3.9 Numerical Simulation of the Response 267 3.10 Nonlinear Response Properties 279 Problems 287 MATLAB Engineering Vibration Toolbox 301 Toolbox Problems 301 4 MULTIPLE-DEGREE-OF-FREEDOM SYSTEMS 303 4.1 Two-Degree-of-Freedom Model (Undamped) 304 4.2 Eigenvalues and Natural Frequencies 318 4.3 Modal Analysis 332 4.4 More Than Two Degrees of Freedom 340 4.5 Systems with Viscous Damping 356 4.6 Modal Analysis of the Forced Response 362Contents v 4.7 Lagrange’s Equations 369 4.8 Examples 377 4.9 Computational Eigenvalue Problems for Vibration 389 4.10 Numerical Simulation of the Time Response 407 Problems 415 MATLAB Engineering Vibration Toolbox 433 Toolbox Problems 433 5 DESIGN FOR VIBRATION SUPPRESSION 435 5.1 Acceptable Levels of Vibration 436 5.2 Vibration Isolation 442 5.3 Vibration Absorbers 455 5.4 Damping in Vibration Absorption 463 5.5 Optimization 471 5.6 Viscoelastic Damping Treatments 479 5.7 Critical Speeds of Rotating Disks 485 Problems 491 MATLAB Engineering Vibration Toolbox 501 Toolbox Problems 501 6 DISTRIBUTED-PARAMETER SYSTEMS 502 6.1 Vibration of a String or Cable 504 6.2 Modes and Natural Frequencies 508 6.3 Vibration of Rods and Bars 519 6.4 Torsional Vibration 525 6.5 Bending Vibration of a Beam 532 6.6 Vibration of Membranes and Plates 544 6.7 Models of Damping 550 6.8 Modal Analysis of the Forced Response 556vi Contents Problems 566 MATLAB Engineering Vibration Toolbox 572 Toolbox Problems 572 7 VIBRATION TESTING AND EXPERIMENTAL MODAL ANALYSIS 573 7.1 Measurement Hardware 575 7.2 Digital Signal Processing 579 7.3 Random Signal Analysis in Testing 584 7.4 Modal Data Extraction 588 7.5 Modal Parameters by Circle Fitting 591 7.6 Mode Shape Measurement 596 7.7 Vibration Testing for Endurance and Diagnostics 606 7.8 Operational Deflection Shape Measurement 609 Problems 611 MATLAB Engineering Vibration Toolbox 615 Toolbox Problems 616 8 FINITE ELEMENT METHOD 617 8.1 Example: The Bar 619 8.2 Three-Element Bar 625 8.3 Beam Elements 630 8.4 Lumped-Mass Matrices 638 8.5 Trusses 641 8.6 Model Reduction 646 Problems 649 MATLAB Engineering Vibration Toolbox 656 Toolbox Problems 656 APPENDIX A COMPLEX NUMBERS AND FUNCTIONS 657 APPENDIX B LAPLACE TRANSFORMS 663Contents vii APPENDIX C MATRIX BASICS 668 APPENDIX D THE VIBRATION LITERATURE 680 APPENDIX E LIST OF SYMBOLS 682 APPENDIX F CODES AND WEB SITES 687 APPENDIX G ENGINEERING VIBRATION TOOLBOX AND WEB SUPPORT 688 REFERENCES 690 ANSWERS TO SELECTED PROBLEMS 692 INDEX 69 index A Accelerance transfer function, 260–261 Acceleration, simple harmonic motion, 10, 13w Accelerometers, 166f, 167f, 578 Aircraft base excitation example, 157–159 control tab, 113f foot pedal model, 105f jet engine with transverse vibration, 290f landing system, 106f steering-gear mechanism, 104f vibration-induced fatigue, 479 wing distributed-parameter system, 502 engine mount, 107f harmonic excitation example, 169–170 impulse response function, 288f stability example, 71 torsional vibration, 420f vibration examples, 49–52, 373–375 vibration models, 200f, 423f Air damping, 179 Airfoil, 204f Algebraic eigenvalue problem, 400, 402 Aliasing, 581–582 Amplitude, 8 Angular motion, 526f Angular natural frequency, 8 Arbitrary input, general forced response, 226–235 examples, 228–235 problems, 290–292 Arbitrary periodic input, general forced response, 235–242 examples, 237–242 problems, 293–294 Argand plane plots, 591 Assumed mode method, 566 Asymmetric eigenvalue problem, 332 Asymptotic stability, 69, 265, 267 Autocorrelation function, 249 Automobiles base excitation example, 157–159 brake pedal model, 202f drive train vibration analysis, 423f frequency response function, 441f single-degree-of-freedom model, 440f tires and resonance, 117 vibration isolation example, 446–448, 447f vibration response example, 440–441 See also Suspension systems; Vehicles Average value, 20 B banded matrix, 674 bar distributed-parameter systems, 519–525 examples, 520–524 problems, 567–569 finite element method, 619–625 example, 624–625 problems, 649–650 two materials, 568f See also Cantilevered bar baseball bat, 42 base excitation, 151–160 examples, 156–160 harmonic excitation, 151f problems, 205–208 beam bending vibration, 532–544 examples, 537–544 problems, 570 beam elements, finite element method for, 630–638 examples, 634–638 problems, 652–653 beam–mass model, 126 beams Euler–bernoulli, 533–540, 533f shear deformation, 541f single-finite-element model, 631f Timoshenko, 540–544, 541f tip mass, 568f transverse vibration, 533f, 539t bearing housing displacement, 609f beat, 124 beats, two-degree-of-freedom system, 317 bell-shaped curve, 253 bernoulli–Euler beams. See Euler–bernoulli beams bIbO (bounded-input, boundedoutput stability), 263, 265, 267 biharmonic operator, 549 bilinear systems, 83 borel’s theorem, 246 boundary value problem, 506 bounded-input, bounded-output (bIbO) stable, 263, 265, 267 bridge, 651f broadband vibration absorption, 469f buildings ground motion, 207f horizontal vibration example, 348–351, 349f, 351f machine with rotating unbalance, 560f, 571f C Cable vibration, 504–507 example, 507 Camera mount, 222f Cantilevered bar finite element grids, 620f longitudinal vibration, 555f, 620w one-element model, 649f three-element four-node model, 620f Cantilevered beam, 533 applied axial force, 649f driving points, 599f700 Index Cantilevered beam (Continued) measurement points, 602f spring-mass system attached, 652f two-element, three-node mesh, 637f Center of percussion, 39, 40 Characteristic equation, 23, 311, 509 Cholesky decomposition, 318, 389–392 Circle fitting, 591–595 problems, 613–614 Clamped beam, 653f Clamped–clamped bar, 555f, 640f Clamped–free bar. See Cantilevered bar Clamped–free beam. See Cantilevered beam Clamped two-element beam system, 654f Clamped two-step aluminum beam, 653f Coherence function, 587, 588f Coiled spring, stiffness of, 52–53, 53f Complex arithmetic, 466w Complex modes, 404 Complex modulus, 178 Complex stiffness, 178, 479, 480f Computational eigenvalue problems for vibration in MDOF systems, 389–407 Computer-controlled vibration endurance test, 607f Computer disk drive motor, 452f Computer software eigenvalues, 392 numerical simulation, 72, 180 Consistent-mass matrices, 638 Constrained-layer damping, 482 Continuous systems. See lumped- parameter systems Conversation of energy equations, 33 Convolution integral, 227, 228w, 233 Cooling fan, 483f Coulomb damping, 81–82 Coulomb friction free response, 84, 85f, 86f harmonic excitation, 170–173 vibration, 81–88 examples, 85–88 problems, 114–115 Coupling device, 417f Critical damping coefficient, 23 Critically damped motion, 27–31 response, 28f Critical points, 471 Critical speeds, for vibration suppression on rotating disks, 485–491 examples, 488–491 problems, 500–501 Cross-correlation function, 585 Cross-spectral density, 586 Cutoff frequency, 577 D Damped natural frequency, 24 Damped systems eigenvalue problems, 396–399, 402–407 harmonic excitation, 130–144 examples, 134–144 problems, 201–204 single-degree-of-freedom, 23f two-degree-of-freedom, 428f Damping air, 179 Coulomb, 81–82, 170–173 distributed-parameter systems, 550–555 examples, 551–555 problems, 570–571 harmonic excitation, 170–180 examples, 173–180 problems, 210–211 hysteretic, 176 modal, 356–362 models, 180t proportional, 362 vibration absorber with, 463f vibration absorption, 463–470 viscous, 21–31 Damping coefficient, 22, 60–61 amplitude of vibration example, 64–65 Damping ratio, 24, 60–61 Dashpot, 22f Decibel (db), 20 Decoupling equations of motion using modal analysis, 335f Degree of freedom, 4 See also Multiple-degree-offreedom (MDOF) systems; Single-degree-of-freedom systems Design, definition of, 436 Design considerations harmonic excitation example, 185–187 modal approach, 407 range of, 448 robustness, 68, 463 rotor system example, 488–489 vibration, 63–68 examples, 64–68 problems, 111–112 vibration, acceptable levels of, 442 vibration absorbers, 458 vibration suppression, 435–501 Diagnostics, vibration testing for, 606–609 example, 607–608 Digital Fourier analyzer, 579 Digital Fourier transform (DFT), 579 Digital representations of signals, 581f Digital signal processing, 579–584 example, 582 problems, 611 Digital spectral coefficients, 582 Dirac delta function, 219 Discrete systems. See lumped-parameter systems Discretization, 618 Disk drive motor of personal computer, 452f Disk–shaft system, 8w critical speeds for vibration suppression, 485–491 example, 37 harmonic excitation example, 149–150 torsional vibration, 47f Displacement simple harmonic motion, 10, 11w vibration, 437t Displacement transmissibility, 153, 154f, 156, 157f, 442, 443w Distributed-parameter systems, 502–572 bar vibration, 519–525 beam bending vibration, 532–544 cable vibration, 504–507 damping models, 550–555 explanation, 503Index 701 forced response modal analysis, 556–566 membrane vibration, 544–550 modes, 508–518 natural frequencies, 508–518 plate vibration, 544–550 rod vibration, 519–525 string vibration, 504–507 torsional vibration, 525–532 Divergent instability, 69 Divergent response, 69, 69f Diving board, 112f Dot product, 306 Double pendulum, with generalized coordinates, 369f Driving frequency, 118, 124 Driving point, 599 Duhamel integral, 228 Dynamically coupled systems, 375 eigenvalue problems, 389–407 E Effective mass, 35 Eigenfunctions, 510 Eigenvalue problems computational, 389–407, 430–431 damped systems, 402–407 example, 404–407 dynamically coupled systems, 389–392 example, 391–392 two-degree-of-freedom system example, 324–326 using codes, 392–399 Eigenvalues distributed-parameter systems, 510 MDOF systems, 318–332 examples, 320–332 problems, 418–420 Eigenvectors, 320w, 321, 324–326 Elastic damper, 476f Elastic modulus, 46 complex data, 481t measurement, 58 stress–strain curve, 59f temperatures, 481f Electric motor mount, 493f Electronic cabinet with cooling fan, 483f Endurance, vibration testing for, 606–609 Energy methods for modeling vibration, 31–46 examples, 34–46 problems, 104–108 Engineering Vibration Toolbox distributed-parameter systems, 572 eigenvalue problems, 389 finite element method, 656 general forced response, 301–302 harmonic excitation, 214–215 MDOF systems, 433 Runge–Kutta method, 77 vibration, 115–116 vibration testing, 615–616 Ensemble average, 253 Equilibrium points/positions, 81, 83, 87–89, 89f Equivalent viscous damping, 285f Euler–bernoulli beam model, 543 Euler–bernoulli beams, 533–540, 533f Euler method linear and nonlinear equations, 90–91 numerical solution, 74–78 single-degree-of-freedom system, 72 Euler relations, 18, 24, 25w Exciters, 575–576 Expansion method, 346–351 examples, 348–351 Expansion theorem, 347, 566 Expected value, 253 Experimental modal analysis. See Vibration testing F Fast Fourier transform (FFT), 579, 583 FEM. See Finite element method Finite dimensional systems. See lumped-parameter systems Finite element analysis (FEA), 619 Finite element mesh/grid, 618 Finite element method (FEM), 617–656 bar, 619–625 beam elements, 630–638 lumped-mass matrices, 638–641 model reduction, 646–649 three-element bar, 625–630 trusses, 641–646 Finite element model, 619 Finite elements (FE), 618 First mode shape, 314, 348 Flexural vibrations, 532 Floor-mounted compressor, 494f Fluid systems example, 35 natural frequency example, 53–55 Flutter instability, 69, 70f, 267 Forced response. See General forced response Forced response modal analysis distributed-parameter systems, 556–566 examples, 557–566 problems, 571–572 MDOF systems, 362–369 examples, 364–369 problems, 428–429 Force summation method, 32 Force transmissibility, 155, 157f, 442, 443w Forcing frequency, 118 Formula error, 75–76 Fourier coefficients, 236 Fourier representations of signals, 581f Fourier series, 236, 579–580, 580w Fourier transforms, 246, 579 Fragility, 436 Free response, 5, 10 Coulomb friction, 84, 85f, 86f numerical simulation of time response, 72–81 Free vibration, 1 Frequency cutoff, 577 importance of concept, 15 two-degree-of-freedom system, 317 vibration, 437 Frequency response approach to harmonic excitation, 146–148 problems, 205–206 Frequency response curves for mode shapes, 600–601f Frequency response function, 147, 588, 588f Friction coefficients, 82f, 82t702 Index G Gaussian distribution function, 253 General forced response, 216–302 arbitrary input response, 226–235 arbitrary periodic input response, 235–242 impulse response function, 217–226 nonlinear response properties, 279–287 numerical simulation, 267–279 random input response, 247–255 shock spectrum, 255–259 stability, 262–267 transfer functions, 259–262 transform methods, 242–247 Generalized eigenvalue problem, 399 Generalized symmetric eigenvalue problem, 331 Geometric approach to harmonic excitation, 145–146 problems, 205–206 Gibbs phenomenon, 238 Global condition, 89 Global coordinate system, 641 Global mass matrix, 627 Global stiffness matrix, 627 Gravity, spring problems and, 16–17 H Hammer center of percussion, 42 impact, 576–578, 577f impulse, 576 instrumented, 221, 222 Hanning window, 583, 584f Harmonic excitation, 117–215 base excitation, 151–160 damped systems, 130–144 damping, forms of, 170–180 design considerations, 184–187 explanation, 118 frequency response approach, 146–148 geometric approach, 145–146 measurement devices, 166–170 nonlinear response properties, 188–197 numerical simulation, 180–187 rotating unbalance, 160–165 transform approach, 148–151 undamped systems, 118–130 Harmonic motion examples, 16–21 problems, 99–101 representations, 19w vibration, 13–15 Heaviside step functions, 224–225, 257, 244, 272–273 Helical spring spring–mass system natural frequency, 66 stiffness, 48 Helicopter. See Rotorcraft Hertz (Hz), 15 Hooke’s law, 59 Houdaille damper, 469f Humans forearm vibration model, 199f longitudinal vibration, 199f Hysteresis loop, 175, 175f Hysteretic damping, 176 Hysteretic damping constant, 176 I Impact, 221, 223f, 226f, 577f Impact hammer, 576–578 Impulse, definition of, 218 Impulse hammer, 576 Impulse response function general forced response, 216–226 examples, 221–226 problems, 287–302 Inconsistent-mass matrices, 638 Inertia force, 32 Inertia matrix, 308 Infinite-dimensional systems, 503, 515 See also lumped-parameter systems Initial conditions, 10f Inner product, 307 Input frequency, 118 International Organization of Standards (ISO), 436, 437 Inverted pendulum, 112–113, 113f, 266 Isolation problems, 442 K Kelvin–Voigt damping, 554 Kinetic energy, 2 Kronecker delta, 327w L lagrange’s equations, 32 energy method, 43–44 example, 43–45 MDOF systems, 369–377 examples, 371–377 problems, 429–431 lagrange stability, 265 laplace operator, 545 laplace transforms common, 244t convolution type evaluations, 233 Fourier transforms versus, 247 general forced response, 242–247 harmonic excitation, 148–151 laptop computers, 453 lathe MDOF system example, 377–381 moving parts, 378f leaf spring, transverse vibration of, 49, 49f leakage, 583, 584f legs, vibration example, 28–31 levers, vibration model of coupled, 372f linear systems, 7 local coordinate direction, 641 local stability, 90 logarithmic decrement, 60 longitudinal motion, 46 longitudinal vibration, 199f, 519f, 524t, 525t, 555f loss coefficient, 174 loss factor, 174 lumped-mass matrices, 638–641 problems, 654–655 lumped-parameter systems, 503, 524 example, 639–641 M Machinery acceptable vibration levels, 438f health monitoring, 607–608 rotating unbalance examples, 160f, 209f, 492f rubber mount, 206f vibration absorbers, 455 vibration isolation, 476f vibration model, 371f Marginal stability, 68 Mass, frequency of oscillation for measuring, 62–63Index 703 Mass condensation, 648 Mass loading, 576 Mass matrix, 308 Mass moment of inertia, 58 Mass normalized stiffness, 319 Mass ratio versus natural frequency, 461f Mathcad eigenvalues, 393–394, 405–407 general forced response, 269–271, 273, 274, 276, 282, 285–287 harmonic excitation, 181–182, 185 linear and nonlinear equations, 90–93, 195–197 MDOF systems, 408–409, 410–415 Runge–Kutta method, 79 Mathematica eigenvalues, 394–396, 398–399, 406–407 general forced response, 271, 273–274, 276, 278, 283–284, 286–287 harmonic excitation, 187 linear and nonlinear equations, 92–93, 193, 196–197 MDOF systems, 410, 412, 414–415 Runge–Kutta method, 79 MATlAb eigenvalues, 393, 397, 405–407 Engineering Vibration Toolbox, 77, 115–116, 214, 301, 389, 433, 501, 572, 615, 656 general forced response, 270, 272–274, 275, 277, 282, 286 harmonic excitation, 183–184, 186 linear and nonlinear equations, 91–92, 192, 196 MDOF systems, 409, 411, 414 Runge–Kutta method, 77 Matrix inverse, 310 Matrix of mode shapes, 335, 335f Matrix square root, 318 MDOF systems. See Multipledegree-of-freedom (MDOF) systems Measurement hardware, 574f, 575–579, 600f problems, 611 harmonic excitation, 166–170 example, 169–170 problems, 210 transfer functions, 260–262 vibration examples, 59–63 problems, 110–111 Membrane vibration, 544–550, 545f example, 546–550 problems, 570 Method of undetermined coefficients, 120 Mindlin–Timoshenko theory, 550 Min-max problem, 474 Mobility frequency response function, 592f, 592w Modal analysis forced response distributed parameter systems, 556–566 forced response MDOF systems, 362–369 MDOF systems, 332–340 problems, 420–422 See also Vibration testing Modal coordinate system, 334, 336 Modal damping, 356–359, 361w, 404, 550–555 Modal data extraction, 588–591 example, 590–591 problems, 612–613 Modal equations, 334, 336, 551 Modal participation factors, 348 Modal testing. See Vibration testing Modeling, definition of, 31 Modeling methods, vibration, 31–46 examples, 34–46 problems, 104–108 Model reduction, 646–649 example, 648–649 problems, 655 Modes, 355 distributed-parameter systems, 508–518 examples, 512–518 problems, 566–567 Mode shapes clamped–pinned beam, 538f definition, 304 eigenvectors, 326 explanation, 355 first, 314, 348 longitudinal vibration, 525f measurement for vibration testing, 596–606 examples, 599–606 problems, 614–615 nodes, 351 normalizing, 330–331 resonance, 366–367 second, 314 torsional vibration, 532t vibrating string, 515f Mode summation method distributed-parameter systems, 522–524 forced response, 367–369 modal analysis, 346–351 examples, 348–351 modal damping, 358–361, 361w Modulus data, 47t Mounting bracket, 563f Mounts aircraft wing engine, 107f base excitation and, 151–160 electric motor, 493f Multiple-degree-of-freedom (MDOF) systems, 303–434 computational eigenvalue problems, 389–407 eigenvalues, 318–332 examples, 377–389 forced response modal analysis, 362–369 lagrange’s equations, 369–377 modal analysis, 332–340, 346–351 more than two degrees of freedom, 340–346, 341f examples, 343–346 problems, 422–426 natural frequencies, 318–332 numerical simulation, 407–415 two-degree-of-freedom model (undamped), 304–318 viscous damping, 356–362 N Natural frequency aircraft wing, 49–50 angular, 8 damped, 24 distributed-parameter systems, 508–518 examples, 512–518 problems, 567–569 energy method, 42 fluid system, 38 human leg, 29–30 longitudinal vibration, 525t704 Index Natural frequency (Continued) mass ratio versus, 460f MDOF systems, 304, 318–332 examples, 320–332 problems, 418–420 pendulum, 17, 40–42 spring–mass system, 9, 16, 37, 55–56 torsional system, 48 torsional vibration, 532t wheel, 34–35 n-degree-of-freedom system, 341f Neutral plane/surface, 549 Newton’s laws, 9–10, 32 Nodes, in finite element analysis, 618, 619 Nodes of a mode, 351, 515 Nonlinear response properties general forced response, 279–287 examples, 280–287 harmonic excitation, 188–197 examples, 189–197 problems, 213–214 Nonlinear systems, 7 Coulomb friction, 81–88 general forced response problems, 299–301 nonlinear pendulum equations, 89–95 Nonperiodic forces, 217–218 Normalization of eigenvectors, 324–326 Nose cannon, 498f Numerical simulation general forced response, 267–279 examples, 269–279 problems, 298–299 harmonic excitation, 180–187 examples, 181–187 problems, 211–213 MDOF systems, 407–415 examples, 408–415 problems, 431–433 vibration and free response, 72–81 examples, 74–81 problems, 114–115 Numerical solutions concept of, 72–73 Euler method examples, 73–77 sources of error, 75 Nyquist circles, 591 Nyquist frequency, 582 Nyquist plots, 591, 594f O Operational deflection shape (ODS) measurement, 609–611 Optical table with vibration absorber,456f Optimization in vibration suppression examples, 474–479 problems, 498–500 vibration suppression, 471–479 Orthogonality, 236, 322, 325, 521 Orthogonal matrices, 326 Orthonormal vectors, 322, 326, 326w Oscillation decay in, 21 frequency of, for measuring mass and stiffness, 61–63 natural frequency examples, 34–35, 38–39, 48 Oscillatory motion, 10 Overdamped motion, 26–27, 27f Overshoot, 230 P Package, vibration model of dropped, 289f Parts sorting machine, 496f Peak amplitude method, 590f Peak frequency, 143 Peak time, 230 Peak value, 19 Pendulum compound, 39–42, 40f, 41f damped, 202f double, with generalized coordinates, 369f equilibrium positions, 89f examples, 2–4, 35–36, 39–42 inverted, 70–71, 266 nonlinear systems, 89–94 problems, 115 swinging, 8w Performance robustness, 463 Periodic forces, 217 Period T, 15, 124 Personal computer disk drive motor, 452f Phase, 8 Physical constants for common materials, 47t Piezoelectric accelerometers, 167f, 169, 578 Pinned beam, 535 Pitch, 341f Pivot point, 42 Plate vibration, 544–550 Positive definite matrix, 390w Positive semidefinite matrix, 390w Potential energy, 2 Power-line pole with transformer, 296f Power spectral density (PSD), 249–251, 585 Printed circuit board, 495f Probability density function, 253 Proportional damping, 362 Pulse input function, 281f Punch press base excitation example, 159 machine schematic, 386f MDOF system example, 385–389 three-element-bar problem, 651f vibration model, 386f Q Quadratic damping, 179 Quadrature peak picking method, 589f R Radial saw, 461f Radius of deflection, 488f Radius of gyration, 40 Random input, general forced response, 247–255 examples, 252–255 problems, 295 Random signal analysis, 584–587 Random vibration analysis, 585w Rattle space, 448 Rayleigh dissipation function, 107 Receptance matrix, 596–597 Receptance transfer function, 592f Rectilinear system, 46t Reduced-order modeling. See Model reduction Resonance damped systems, 138, 143 distributed systems, 503 explanation, 117–118 importance of concept, 121 MDOF systems, 366–367 modal testing, 575 undamped systems, 125, 125fIndex 705 Response divergent, 69, 69f free, 5, 10, 72–81, 84, 85f, 86f steady-state, 133, 137–138 transient, 133, 137–138 See also Forced response modal analysis; General forced response Response spectrum. See Shock spectrum Rigid-body modes, 352–355, 380 example, 352–355 Robustness, of designs, 68, 463 Rod vibration, 519–525 examples, 520–524 problems, 569–570 Roll, 341f Rolling disk vibration model, 108f Root-mean-square value, 20, 249, 437 Rotating disk critical speeds, 485–491, 485f Rotating unbalance equation, 486w harmonic excitation, 160–165 examples, 163–165 problems, 208–210 model of disk–shaft system, 485f model of machine, 160f, 209f, 492f model of machine in building, 560f, 571f model of motor, 209f Rotational kinetic energy, 34 Rotational system, 46t Rotorcraft and resonance, 117 rotating unbalance example, 164–165 thrust directions, 164f Round-off error, 75 Runge–Kutta method, 75–76 examples, 77–80 general forced response, 272–278 linear and nonlinear equations, 92 Mathematica, 79 MDOF systems, 410–412 single-degree-of-freedom system, 72 S Saddle point, 472, 473f Sample function, 247–255 Sampling theorem, 582 Scalar product, 307 Scanning laser doppler vibrometer (SlDV), 579 Second mode shape, 314 Seismic accelerometer, 166f Self-excited vibrations, 69 Semidefinite systems, 380 Separation of variables, 508 solutions method, 518w Settling time, 230 Shaft and disk. See Disk–shaft system Shaker, 260 Shakers, 575–576 Shannon’s sampling theorem, 582 Shape functions, 621 Shear coefficient, 541 Shear modulus, 541f Ship, fluid system example, 53–54 Shock, 255, 441–442, 442f Shock loading, 217 Shock pulse, 446 Shock spectrum, general forced response, 255–259 examples, 256–259 problems, 295–296 Signal conditioners, 578 Signal processing. See Digital signal processing Signals, representations of, 581f Signum function, 83 Simple harmonic motion, 10, 13w Simple harmonic oscillator, 10 Simple machine part, vibration model of, 371f Simple sine function, 248f Simply supported beam, 535 Sine function, 1 Single-degree-of-freedom curve fit, 588 Single-degree-of-freedom systems, 5, 219w compliance frequency response function, 589f damped, 23f example, 8w external force, 119f response, 219w undamped, 10 Sinusoidal vibration, acceptable limits, 438f Sliding boundary, 535 Sloshing, 39 Software eigenvalues, 392 numerical simulation, 72, 180 Solid damping, 176 Specific damping capacity, 174 Spectral matrix, 328 Spring–mass–damper system deterministic and random excitations, 254w, 255 excitation response, 586w general applied force, 280f magnitude plot, 261f potentially nonlinear elements, 189f square input, 272f total time response, 242f truck suspension system example, 232–233 Spring–mass system, 8w examples, 11, 37–38, 43 gravitational field, 33f harmonic excitation examples, 122–130, 141–142 helical spring natural frequency, 66 kinetic coefficient of friction, 82f natural frequency example, 55–57 nonnegligible mass, 37f problems, 95–98 response of, 9f vibration, 5–13 vibration absorbers, 455 Springs coiled, 52–53, 53f constants, 53t, 56 helical natural frequency, 66 stiffness, 51 leaf, 49–50, 49f manufacture of, 56 static deflection, 6f, 57, 67 stiffness calculation rules, 55f, 56 Stability asymptotic, 69, 265, 266 bIbO, 263, 265, 267 general forced response, 262–267 examples, 266–267 problems, 298 local, 90 marginal, 68 vibration, 68–71, 263w examples, 70–71 problems, 112–113 response, 69f706 Index State matrix, 77, 401 State variables, 77 State vector, 77, 401 Static coupling, 375 Static deflection of spring, 6f, 57, 67 Stationary signals, 248f Steady-state response, 133, 137–138 Steam-pipe system with absorber, 495f Steel, elastic modulus, 59f Step function, 228, 229f Stereo turntable frequency response function, 441f single-degree-of-freedom model, 440f vibration response example, 440–441 Stiffness, 46–57 calculation rules for parallel and series springs, 55f, 56 coiled spring, 49–50, 50f definition, 6 examples, 48–57 frequency of oscillation for measuring, 62–63 helical spring, 48 problems, 108–110 twist, 48 Stiffness matrix, 308 Stinger, 576 Strain gauges, 578 Strain rate damping, 554 Stress–strain curve for elastic modulus, 59f String equation, 506 String vibration, 504–507, 504f Structural damage acceptable vibration levels, 438f vibration measurements, 609 Structural damping, 176 Structural Health Monitoring (SHM), 574 Subway car coupling device, 417f Superposition, 95, 118, 217, 636 Support motion. See base excitation Suspension systems arbitrary input response, 232 base excitation, 159 chassis dynamometer, 289f damped, 108f design of, 42 examples, 11, 48–49, 67–68, 158 harmonic excitation, 150 mass of occupants, 207f model of, 289f multiple-degree-of-freedom systems, 340 speed bump, 291f torsional, 149–150 torsion rod, 100f trifilar, 58f two-degree-of-freedom model, 417f vertical, 67–68 Symmetric eigenvalue problem, 321–322, 320w, 332, 401 Symmetric matrix, 308 Synchronous whirl, 487 T Tangent line method, 73 Telephone lines, 456 Tennis racket, 42 Tensile test, 59 Thin plate theory, 550 3-db down point, 589 Three-element bar, finite element method for, 625–630 examples, 627–630 problems, 652–653 Time history of impulse force, 218f Time response lumped- versus distributedparameter systems, 524 MDOF systems, 407–415 vibration and free response, 72–81 examples, 74–81 Time to peak, 230 Timoshenko beam model, 543 Timoshenko beams, 540–544, 541f Timoshenko shear coefficient, 541 Tires, and resonance, 117 Torsional constant, 527–528 Torsional motion, 46 Torsional system natural frequency, 48–49 two degrees of freedom, 416f Torsional vibration boundary conditions, 528t distributed-parameter systems, 525–532 examples, 528–532 problems, 569–570 shaft, 47f transform approach, 149–150 Transducers, 166, 575, 578 Transfer functions, 149, 260t general forced response, 260–262 problems, 296–297 Transformations, 332 Transform methods general forced response, 242–247 examples, 243–247 problems, 294–295 harmonic excitation, 148–151 problems, 204–205 Transient response, 133, 137 Transmissibility base excitation example, 157–158 displacement, 153, 154f, 155, 157f force, 155, 157f formulas, 443w Transmissibility ratio, 442–455, 446f Transmission lines, 456 Transpose of matrix, 308 Transverse motion, 46 Transverse vibrations, 532 Truck hitting object, 571f loading dirt, 232f pipe stacking, 105f spring–mass–damper system example, 232–233 Trusses, 641–646 problems, 654–655 three-element, 655f Turntable. See Stereo turntable Twist, 48 Two-degree-of-freedom model damped, 428f MDOF systems, 304–318, 305f, 328f examples, 307–318 problems, 415–418 rigid-body translation, 352f vehicle example, 382–385 viscous damping, 361f Two-member framed structure, 642f U Undamped motion, 68–69 Undamped systems harmonic excitation, 118–130 examples, 122–130 problems, 197–201 lagrange stability, 265 single-degree-of-freedom, 10Index 707 two-degree-of-freedom, 304–318 examples, 307–318 Underdamped motion, 24–26 Underdamped solution, 25w Underdamped systems forced response, 140w, 364w response, 26f, 60f vibration example, 30–31 Unrestrained degree of freedom, 352 U-tube manometer, 38f V Valve and rocker arm system, 102f Variance, 249 Vector equation, 307 Vehicles side section, 382f two-degree-of-freedom system example, 382–385 See also Automobiles Velocity, simple harmonic motion, 13, 14w Velocity-squared damping, 179, 193–194, 284–285 Vibration, 1–116 acceptable levels, 436–442 examples, 438–442 problems, 491 consequences of, 436 Coulomb friction, 81–95 description, 1–2 design considerations, 63–68 displacement amplitude, 437t energy methods, 31–46 explanation, 2 frequency ranges, 437t harmonic motion, 13–21 measurement, 58–63 modeling methods, 31–46 nonlinear, 81 nonlinear pendulum equations, 89–94 numerical simulation of time response, 72–81 performance standards, 436 shock versus, 442 spring–mass model, 5–13 stability, 68–71 stiffness, 46–57 viscous damping, 21–31 Vibration absorbers, 455–463 damping, 463f examples, 461–462 problems, 495–496 viscous, 469f Vibration absorption damping, 463–470 problems, 496–498 Vibration dampers, 463 Vibration isolation, 442–455 examples, 446–455 optimization, 476–478 problems, 491–494 transmissibility formulas, 443w Vibration model airplane wing, 50f, 373f coupled levers, 372f punch press, 386f simple machine part, 371f Vibration suppression, 435–501 optimization, 471–479 rotating disk critical speeds, 485–491 vibration absorbers, 455–463 vibration absorption damping, 463–470 vibration isolation, 442–455 Vibration testing, 573–616 circle fitting, 591–595 digital signal processing, 579–584 endurance and diagnostics, 606–609, 607f measurement hardware, 575–579, 575f, 600f modal data extraction, 588–591 mode shape measurement, 596–606 operational deflection shape (ODS) measurement, 609–611 random signal analysis, 584–587 uses of, 574 Virtual displacements, 370 Virtual work, 370 Viscoelastic, definition of, 480 Viscous damping, 21–31 critically damped motion, 27–31 equivalent, 285f examples, 27–31 MDOF systems, 356–362, 376 examples, 357–362, 376–377 problems, 426–428 overdamped motion, 26–27 problems, 101–103 two-degree-of-freedom system, 361f underdamped motion, 24–26 vibration, 21–31 See also Damped systems; Undamped systems; Underdamped systems Viscous vibration absorbers, 469f, 470f W Washing machine, 492f Wave hitting seawall, 292f Wave speed, 506 Whirling, 485, 487 Window function, 583, 584f Wing. See under Aircraft Y Yaw, 341f Young’s modulus. See Elastic modulus Z Zero mode, 380
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