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أخواني في الله أحضرت لكم كتاب حل كتاب الاهتزازات الميكانيكية Mechanical Vibrations Solution Manual Fifth Edition Singiresu S. Rao University of Miami
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Prefacexi Acknowledgmentsxv List of Symbolsxvi CHAPTER 1 Fundamentals of Vibration1 1.1Preliminary Remarks2 1.2Brief History of the Study of Vibration3 1.2.1Origins of the Study of Vibration 3 1.2.2From Galileo to Rayleigh 6 1.2.3Recent Contributions 9 1.3Importance of the Study of Vibration10 1.4Basic Concepts of Vibration13 1.4.1Vibration 13 1.4.2Elementary Parts of Vibrating Systems 13 1.4.3Number of Degrees of Freedom 14 1.4.4Discrete and Continuous Systems 16 1.5Classification of Vibration16 1.5.1Free and Forced Vibration 17 1.5.2Undamped and Damped Vibration 17 1.5.3Linear and Nonlinear Vibration 17 1.5.4Deterministic and Random Vibration 17 1.6Vibration Analysis Procedure18 1.7Spring Elements22 1.7.1Nonlinear Springs 23 1.7.2Linearization of a Nonlinear Spring 25 1.7.3Spring Constants of Elastic Elements 27 1.7.4Combination of Springs 30 iv 1.7.5Spring Constant Associated with the Restoring Force due to Gravity 39 1.8Mass or Inertia Elements40 1.8.1Combination of Masses 40 1.9Damping Elements45 1.9.1Construction of Viscous Dampers 46 1.9.2Linearization of a Nonlinear Damper 52 1.9.3Combination of Dampers 52 1.10Harmonic Motion54 1.10.1Vectorial Representation of Harmonic Motion 56 1.10.2Complex-Number Representation of Harmonic Motion 57 1.10.3Complex Algebra 58 1.10.4Operations on Harmonic Functions 59 1.10.5Definitions and Terminology 62 1.11Harmonic Analysis64 1.11.1Fourier Series Expansion 64 1.11.2Complex Fourier Series 66 1.11.3Frequency Spectrum 67 1.11.4Time- and Frequency-Domain Representations 68 1.11.5Even and Odd Functions 69 1.11.6Half-Range Expansions 71 1.11.7Numerical Computation of Coefficients 72 1.12Examples Using MATLAB76 1.13Vibration Literature80 Chapter Summary81 References81 Review Questions83 Problems87 Design Projects120 ContentsCONTENTS v CHAPTER 2 Free Vibration of Single-Degree-of-Freedom Systems124 2.1Introduction126 2.2Free Vibration of an Undamped Translational System129 2.2.1Equation of Motion Using Newton s Second Law of Motion 129 2.2.2Equation of Motion Using Other Methods 130 2.2.3Equation of Motion of a Spring-Mass System in Vertical Position 132 2.2.4Solution 133 2.2.5Harmonic Motion 134 2.3Free Vibration of an Undamped Torsional System146 2.3.1Equation of Motion 147 2.3.2Solution 148 2.4Response of First Order Systems and Time Constant151 2.5Rayleigh s Energy Method153 2.6Free Vibration with Viscous Damping158 2.6.1Equation of Motion 158 2.6.2Solution 158 2.6.3Logarithmic Decrement 164 2.6.4Energy Dissipated in Viscous Damping 166 2.6.5Torsional Systems with Viscous Damping 168 2.7Graphical Representation of Characteristic Roots and Corresponding Solutions174 2.7.1Roots of the Characteristic Equation 174 2.7.2Graphical Representation of Roots and Corresponding Solutions 175 2.8Parameter Variations and Root Locus Representations176 2.8.1Interpretations of and in s-plane 176 2.8.2Root Locus and Parameter Variations 179 2.9Free Vibration with Coulomb Damping185 2.9.1Equation of Motion 186 2.9.2Solution 187 2.9.3Torsional Systems with Coulomb Damping 190 v t n , vd, z, 2.10Free Vibration with Hysteretic Damping192 2.11Stability of Systems198 2.12Examples Using MATLAB202 Chapter Summary208 References209 Review Questions209 Problems214 Design Projects256 CHAPTER 3 Harmonically Excited Vibration259 3.1Introduction261 3.2Equation of Motion261 3.3Response of an Undamped System Under Harmonic Force263 3.3.1Total Response 267 3.3.2Beating Phenomenon 267 3.4Response of a Damped System Under Harmonic Force271 3.4.1Total Response 274 3.4.2Quality Factor and Bandwidth 276 3.5Response of a Damped System Under 278 3.6Response of a Damped System Under the Harmonic Motion of the Base281 3.6.1Force Transmitted 283 3.6.2Relative Motion 284 3.7Response of a Damped System Under Rotating Unbalance287 3.8Forced Vibration with Coulomb Damping293 3.9Forced Vibration with Hysteresis Damping298 3.10Forced Motion with Other Types of Damping300 3.11Self-Excitation and Stability Analysis301 3.11.1Dynamic Stability Analysis 301 3.11.2Dynamic Instability Caused by Fluid Flow 305 3.12Transfer-Function Approach313 3.13Solutions Using Laplace Transforms317 3.14Frequency Transfer Functions320 3.14.1Relation Between the General Transfer function T(s) and the Frequency Transfer Function 322 3.14.2Representation of Frequency-Response Characteristics 323 T(iv) F(t) = F0eiVtvi CONTENTS 3.15Examples Using MATLAB326 Chapter Summary332 References332 Review Questions333 Problems336 Design Projects362 CHAPTER 4 Vibration Under General Forcing Conditions363 4.1Introduction364 4.2Response Under a General Periodic Force365 4.2.1First-Order Systems 366 4.2.2Second-Order Systems 372 4.3Response Under a Periodic Force of Irregular Form378 4.4Response Under a Nonperiodic Force380 4.5Convolution Integral381 4.5.1Response to an Impulse 382 4.5.2Response to a General Forcing Condition 385 4.5.3Response to Base Excitation 386 4.6Response Spectrum394 4.6.1Response Spectrum for Base Excitation 396 4.6.2Earthquake Response Spectra 399 4.6.3Design Under a Shock Environment 403 4.7Laplace Transform406 4.7.1Transient and Steady-State Responses 406 4.7.2Response of First-Order Systems 407 4.7.3Response of Second-Order Systems 409 4.7.4Response to Step Force 414 4.7.5Analysis of the Step Response 420 4.7.6Description of Transient Response 421 4.8Numerical Methods428 4.8.1Runge-Kutta Methods 429 4.9Response to Irregular Forcing Conditions Using Numerical Methods431 4.10Examples Using MATLAB436 Chapter Summary440 References440 Review Questions441 Problems444 Design Projects465 CHAPTER 5 Two-Degree-of-Freedom Systems467 5.1Introduction468 5.2Equations of Motion for Forced Vibration472 5.3Free Vibration Analysis of an Undamped System474 5.4Torsional System483 5.5Coordinate Coupling and Principal Coordinates488 5.6Forced-Vibration Analysis494 5.7Semidefinite Systems497 5.8Self-Excitation and Stability Analysis500 5.9Transfer-Function Approach502 5.10Solutions Using Laplace Transform504 5.11Solutions Using Frequency Transfer Functions512 5.12Examples Using MATLAB515 Chapter Summary522 References523 Review Questions523 Problems526 Design Projects552 CHAPTER 6 Multidegree-of-Freedom Systems553 6.1Introduction555 6.2Modeling of Continuous Systems as Multidegreeof-Freedom Systems555 6.3Using Newton s Second Law to Derive Equations of Motion557 6.4Influence Coefficients562 6.4.1Stiffness Influence Coefficients 562 6.4.2Flexibility Influence Coefficients 567 6.4.3Inertia Influence Coefficients 572 6.5Potential and Kinetic Energy Expressions in Matrix Form574 6.6Generalized Coordinates and Generalized Forces576 6.7Using Lagrange s Equations to Derive Equations of Motion577CONTENTS vii 6.8Equations of Motion of Undamped Systems in Matrix Form581 6.9Eigenvalue Problem583 6.10Solution of the Eigenvalue Problem585 6.10.1Solution of the Characteristic (Polynomial) Equation 585 6.10.2Orthogonality of Normal Modes 591 6.10.3Repeated Eigenvalues 594 6.11Expansion Theorem596 6.12Unrestrained Systems596 6.13Free Vibration of Undamped Systems601 6.14Forced Vibration of Undamped Systems Using Modal Analysis603 6.15Forced Vibration of Viscously Damped Systems610 6.16Self-Excitation and Stability Analysis617 6.17Examples Using MATLAB619 Chapter Summary627 References627 Review Questions628 Problems632 Design Project653 CHAPTER 7 Determination of Natural Frequencies and Mode Shapes654 7.1Introduction655 7.2Dunkerley s Formula656 7.3Rayleigh s Method658 7.3.1Properties of Rayleigh s Quotient 659 7.3.2Computation of the Fundamental Natural Frequency 661 7.3.3Fundamental Frequency of Beams and Shafts 663 7.4Holzer s Method666 7.4.1Torsional Systems 666 7.4.2Spring-Mass Systems 669 7.5Matrix Iteration Method670 7.5.1Convergence to the Highest Natural Frequency 672 7.5.2Computation of Intermediate Natural Frequencies 673 7.6Jacobi s Method678 7.7Standard Eigenvalue Problem680 7.7.1Choleski Decomposition 681 7.7.2Other Solution Methods 683 7.8Examples Using MATLAB683 Chapter Summary686 References686 Review Questions688 Problems690 Design Projects698 CHAPTER 8 Continuous Systems699 8.1Introduction700 8.2Transverse Vibration of a String or Cable701 8.2.1Equation of Motion 701 8.2.2Initial and Boundary Conditions 703 8.2.3Free Vibration of a Uniform String 704 8.2.4Free Vibration of a String with Both Ends Fixed 705 8.2.5Traveling-Wave Solution 709 8.3Longitudinal Vibration of a Bar or Rod710 8.3.1Equation of Motion and Solution 710 8.3.2Orthogonality of Normal Functions 713 8.4Torsional Vibration of a Shaft or Rod718 8.5Lateral Vibration of Beams721 8.5.1Equation of Motion 721 8.5.2Initial Conditions 723 8.5.3Free Vibration 723 8.5.4Boundary Conditions 724 8.5.5Orthogonality of Normal Functions 726 8.5.6Forced Vibration 730 8.5.7Effect of Axial Force 732 8.5.8Effects of Rotary Inertia and Shear Deformation 734 8.5.9Other Effects 739 8.6Vibration of Membranes739 8.6.1Equation of Motion 739 8.6.2Initial and Boundary Conditions 741 8.7Rayleigh s Method742 8.8The Rayleigh-Ritz Method745 8.9Examples Using MATLAB748 Chapter Summary751 References751viii CONTENTS Review Questions753 Problems756 Design Project768 CHAPTER 9 Vibration Control769 9.1Introduction770 9.2Vibration Nomograph and Vibration Criteria771 9.3Reduction of Vibration at the Source775 9.4Balancing of Rotating Machines776 9.4.1Single-Plane Balancing 776 9.4.2Two-Plane Balancing 779 9.5Whirling of Rotating Shafts785 9.5.1Equations of Motion 785 9.5.2Critical Speeds 787 9.5.3Response of the System 788 9.5.4Stability Analysis 790 9.6Balancing of Reciprocating Engines792 9.6.1Unbalanced Forces Due to Fluctuations in Gas Pressure 792 9.6.2Unbalanced Forces Due to Inertia of the Moving Parts 793 9.6.3Balancing of Reciprocating Engines 796 9.7Control of Vibration798 9.8Control of Natural Frequencies798 9.9Introduction of Damping799 9.10Vibration Isolation801 9.10.1Vibration Isolation System with Rigid Foundation 804 9.10.2Vibration Isolation System with Base Motion 814 9.10.3Vibration Isolation System with Flexible Foundation 821 9.10.4Vibration Isolation System with Partially Flexible Foundation 822 9.10.5Shock Isolation 824 9.10.6Active Vibration Control 827 9.11Vibration Absorbers832 9.11.1Undamped Dynamic Vibration Absorber 833 9.11.2Damped Dynamic Vibration Absorber 840 9.12Examples Using MATLAB843 Chapter Summary851 References851 Review Questions853 Problems855 Design Project869 CHAPTER 10 Vibration Measurement and Applications870 10.1Introduction871 10.2Transducers873 10.2.1Variable Resistance Transducers 873 10.2.2Piezoelectric Transducers 876 10.2.3Electrodynamic Transducers 877 10.2.4Linear Variable Differential Transformer Transducer 878 10.3Vibration Pickups879 10.3.1Vibrometer 881 10.3.2Accelerometer 882 10.3.3Velometer 886 10.3.4Phase Distortion 888 10.4Frequency-Measuring Instruments890 10.5Vibration Exciters892 10.5.1Mechanical Exciters 892 10.5.2Electrodynamic Shaker 893 10.6Signal Analysis895 10.6.1Spectrum Analyzers 896 10.6.2Bandpass Filter 897 10.6.3Constant-Percent Bandwidth and Constant-Bandwidth Analyzers 898 10.7Dynamic Testing of Machines and Structures900 10.7.1Using Operational Deflection-Shape Measurements 900 10.7.2Using Modal Testing 900 10.8Experimental Modal Analysis900 10.8.1The Basic Idea 900 10.8.2The Necessary Equipment 900 10.8.3Digital Signal Processing 903 10.8.4Analysis of Random Signals 905 10.8.5Determination of Modal Data from Observed Peaks 907 10.8.6Determination of Modal Data from Nyquist Plot 910 10.8.7Measurement of Mode Shapes 912 10.9Machine Condition Monitoring and Diagnosis915CONTENTS ix 10.9.1Vibration Severity Criteria 915 10.9.2Machine Maintenance Techniques 915 10.9.3Machine Condition Monitoring Techniques 916 10.9.4Vibration Monitoring Techniques 918 10.9.5Instrumentation Systems 924 10.9.6Choice of Monitoring Parameter 924 10.10Examples Using MATLAB925 Chapter Summary928 References928 Review Questions930 Problems932 Design Projects938 CHAPTER 11 Numerical Integration Methods in Vibration Analysis939 11.1Introduction940 11.2Finite Difference Method941 11.3Central Difference Method for Single-Degree-ofFreedom Systems942 11.4Runge-Kutta Method for Single-Degree-ofFreedom Systems945 11.5Central Difference Method for Multidegree-ofFreedom Systems947 11.6Finite Difference Method for Continuous Systems951 11.6.1Longitudinal Vibration of Bars 951 11.6.2Transverse Vibration of Beams 955 11.7Runge-Kutta Method for Multidegree-ofFreedom Systems960 11.8Houbolt Method962 11.9Wilson Method965 11.10Newmark Method968 11.11Examples Using MATLAB972 Chapter Summary978 References978 Review Questions979 Problems981 CHAPTER 12 Finite Element Method987 12.1Introduction988 12.2Equations of Motion of an Element989 12.3Mass Matrix, Stiffness Matrix, and Force Vector991 12.3.1Bar Element 991 12.3.2Torsion Element 994 12.3.3Beam Element 995 12.4Transformation of Element Matrices and Vectors998 12.5Equations of Motion of the Complete System of Finite Elements1001 12.6Incorporation of Boundary Conditions1003 12.7Consistent- and Lumped-Mass Matrices1012 12.7.1Lumped-Mass Matrix for a Bar Element 1012 12.7.2Lumped-Mass Matrix for a Beam Element 1012 12.7.3Lumped-Mass Versus Consistent-Mass Matrices 1013 12.8Examples Using MATLAB1015 Chapter Summary1019 References1019 Review Questions1020 Problems1022 Design Projects1034 Chapters 13 and 14 are provided as downloadable files on the Companion Website. CHAPTER 13 Nonlinear Vibration13-1 13.1Introduction13-2 13.2Examples of Nonlinear Vibration Problems13-3 13.2.1Simple Pendulum 13-3 13.2.2Mechanical Chatter, Belt Friction System 13-5 13.2.3Variable Mass System 13-5 13.3Exact Methods13-6 13.4Approximate Analytical Methods13-7 13.4.1Basic Philosophy 13-8 13.4.2Lindstedt s Perturbation Method 13-10 13.4.3Iterative Method 13-13 13.4.4Ritz-Galerkin Method 13-17 13.5Subharmonic and Superharmonic Oscillations13-19 13.5.1Subharmonic Oscillations 13-20 13.5.2Superharmonic Oscillations 13-23 13.6Systems with Time-Dependent Coefficients (Mathieu Equation)13-2413.7Graphical Methods13-29 13.7.1Phase-Plane Representation 13-29 13.7.2Phase Velocity 13-34 13.7.3Method of Constructing Trajectories 13-34 13.7.4Obtaining Time Solution from Phase Plane Trajectories 13-36 13.8Stability of Equilibrium States13-37 13.8.1Stability Analysis 13-37 13.8.2Classification of Singular Points 13-40 13.9Limit Cycles13-41 13.10Chaos13-43 13.10.1Functions with Stable Orbits 13-45 13.10.2Functions with Unstable Orbits 13-45 13.10.3Chaotic Behavior of Duffing s Equation Without the Forcing Term 13-47 13.10.6Chaotic Behavior of Duffing s Equation with the Forcing Term 13-50 13.11Numerical Methods13-52 13.12Examples Using MATLAB13-53 Chapter Summary13-62 References13-62 Review Questions13-64 Problems13-67 Design Projects13-75 CHAPTER 14 Random Vibration14-1 14.1Introduction14-2 14.2Random Variables and Random Processes14-3 14.3Probability Distribution14-4 14.4Mean Value and Standard Deviation14-6 14.5Joint Probability Distribution of Several Random Variables14-7 14.6Correlation Functions of a Random Process14-9 14.7Stationary Random Process14-10 14.8Gaussian Random Process14-14 14.9Fourier Analysis14-16 14.9.1Fourier Series 14-16 14.9.2Fourier Integral 14-19 14.10Power Spectral Density14-23 14.11Wide-Band and Narrow-Band Processes14-25 14.12Response of a Single-Degree-ofFreedom System14-28 14.12.1Impulse-Response Approach 14-28 14.12.2Frequency-Response Approach 14-30 14.12.3Characteristics of the Response Function 14-30 14.13Response Due to Stationary Random Excitations14-31 14.13.1Impulse-Response Approach 14-32 14.13.2Frequency-Response Approach 14-33 14.14Response of a Multidegree-of-Freedom System14-39 14.15Examples Using MATLAB14-46 Chapter Summary14-49 References14-49 Review Questions14-50 Problems14-53 Design Project14-61 APPENDIX A Mathematical Relationships and Material Properties 1036 APPENDIX B Deflection of Beams and Plates 1039 APPENDIX C Matrices 1041 APPENDIX D Laplace Transform 1048 APPENDIX E Units 1056 APPENDIX F Introduction to MATLAB 1059 Answers to Selected Problems 1069 Index 1077 x CONTENTSIndex 1077 A Accelerographs, 399 400 Accelerometer, 882 886 Acoustics , 6 Active vibration control, 827 832 Addition of harmonic motions, 60 61 Adjoint matrix, 1045 Advance, 873 Airfoil, dynamic instability of, 309 310 Amplitude, 62, 584, 806 Analysis, vibration, 18 22 equations, 20 mathematical modeling, 18 motorcycle, mathematical model of, 20 results, interpretation, 20 Aristotle, 4 Aristoxenus, 4 Arrays with special structure, 1061 Attractor, 13-43 13-44 Asymptotically stable system, 198 Autocorrelation function, 14-9, 14-11, 14-32 Axial compressive force, beam subjected to, 733 737 Axial force effect, 732 734 B Band-limited white noise, 14-25 Bandpass filter, 897 898 Bandwidth, 276 278 Bar element, 991 994 Base excitation response spectrum for, 396 399 system response under, 328 330 Basic concepts of vibration, 13 16 Bathtub curve, 915 Beam deflections, 571 Beam element, 995 998 Beams, deflection of, 1039 1040 cantilever beam, 1039 fixed-fixed beam with end displacement, 1039 fixed-fixed beam, 1039 simply supported beam, 1039 1040 Beams, fundamental frequency of, 663 665 Beating phenomenon, 63, 267 271 Belt friction system, 13-5 Bernoulli, Daniel, 7 Bifurcations, 13-46 Bivariate distributions, 14-8 Blast load on building frame, 392 393 Bode diagrams, 323 324 Bonaparte, Napoléon, 7 Boundary conditions, incorporation of, 1003 1012 Boundary curves, 13-28 Building frame response to an earthquake, 402 C Cam-follower mechanism, 44 45, 73 spring mass system for, 126 Cannon analysis, 173 Cantilever beam, 1039 spring constants of, 28 center, 13-30, 13-40 Center of percussion, 150 151 Central difference method for multidegree of freedom systems, 947 951 Centrifugal pump with rotating unbalance, 812 816 rattle space, 812 816 Cepstrum, 922 Chaos, 13-43 13-52 attractor, 13-43 13-44 bifurcations, 13-46 of Duffing s equation, 13-47 13-52 functions with stable orbits, 13-45 functions with unstable orbits, 13-45 13-47 Poincaré section, 13-43 13-44 strange attractors, 13-46 Characteristic (polynomial) equation solution, 585 590 Characteristic roots, graphical representation, 174 176 Chimney, flow-induced vibration of, 311 Choleski decomposition, 681 683 Classification of vibration, 16 18 Clebsch, R. F. A., 8 Coefficients, numerical computation of, 72 76 Coherence function, 907 Column matrix, 1042 Column vector, 1060 Compacting machine, 387 388, 416 418 Complex algebra, 58 Complex damping, 300 Complex Fourier series, 66 67, 14-17 14-18 Complex frequency response, 279, 14-30 Complex numbers, 1062 harmonic motion representation, 57 58 Complex stiffness, 194 Complex vector representation of harmonic motion, 280 Compound pendulum, 148 150, 561 natural frequency of, 148 151 Consistent mass matrices, 1012 1015 Constant bandwidth analyzers, 898 899 Constant damping, 186 Constant percent bandwidth, 898 899 Continuous systems, 16, 699 768, See also Lateral vibration of beams; Longitudinal vibration of bar or rod; Torsional vibration of a shaft or rod dynamic response of plucked string, 707 712 modeling as multidegree of freedom systems, 555 556 transverse vibration of a string or cable, 701 710 Continuous systems, finite difference method for, 951 959 longitudinal vibration of bars, 951 955 pinned-fixed beam, 959 transverse vibration of beams, 955 959 Control, vibration, 769 869 criteria, 771 775 natural frequencies, control of, 798 799 nomograph, 771 775 ranges of vibration, 773 whirling of rotating shafts, 785 791 Conversion of units, 1056 1058 Convolution integral, 365, 381 393, 1052 1055 blast load on building frame, 392 393 compacting machine under linear force, 391 392 rectangular pulse load, 389 390 response of a structure under double impact, 385 response of a structure under impact, 384 response to a general forcing condition, 385 386 response to an impulse, 382 385 response to base excitation, 386 393 step force on a compacting machine, 387 388 time-delayed step force, 388 389 Coordinate coupling, 488 493 Correlation functions of random process, 14-9 14-10 Coulomb damping, 46 forced response of, using MATLAB, 327 328 forced vibration with, 293 297 free vibration response of a system with, 205 free vibration with, 185 192 pulley subjected to, 191 Coulomb, Charles, 7 Please note that references to pages in chapters 13 and 14 appear in the form 13-1, 13-2, etc., and these chapters are provided on the companion Website.1078 INDEX Coupled differential equations, 470 Crane, equivalent k of, 35 Critical damping constant, 159 Critical speeds, 787 Critically damped system, 162 Cycle, 62 DD Alembert, Jean, 7 D Alembert s principle, 130 Damped dynamic vibration absorber, 840 843 Damped equation, 13-14 Damped response using numerical methods, 434 436 Damped single degree of freedom system Bode diagrams of, 324 transfer function, 315 Damped system, 127 forced vibration response of, MATLAB, 623 624 free vibration response of, Laplace transform, 507 511 Damped system response under F(t) = F0ei*t, 278 281 Damped system response under harmonic force, 271 278, See also under Harmonically excited vibration under F(t) = F0ei*t, 278 281 graphical representation, 272 under harmonic motion of base, 281 287 under rotating unbalance, 287 293 total response, 274 276 vectorial representation, 272 Damped system response using Laplace transform, 317 318 Damped vibration, 17 Damping, 799 800 damping matrix, 611 damping ratio, 159, 179 viscoelastic materials use, 799 Damping elements, 45 54 clearance in a bearing, 47 48 combination of dampers, 52 54 Coulomb or dry friction damping, 46 damping constant of journal bearing, 48 50 damping constant of parallel plates, 47 linearization of nonlinear damper, 52 material or solid or hysteretic damping, 46 piston-cylinder dashpot, 50 52 viscous damping, 45 viscous dampers construction, 46 52 De Laval, C. G. P., 9 Decibel, 63 Degree of Freedom, 14 16 Delay time (td), transient response, 425 Design chart of isolation, 809 810 Determinant, 1043 Deterministic vibration, 17 18, 14-2 Diagonal matrix, 1042 Diesel engine, vibration absorber for, 837 838 Differential equations, 313, 1066 1068 Digital signal processing, 903 904 Dirac delta function, 381 Discrete systems, 16 Displacement method, 1003 Displacement transmissibility, 282 283, 806 Dry friction damping, 46 Duffing s equation, 13-13, 13-47 13-50 Duhamel integral, See Convolution integral Dunkerley s formula, 654 658 Dynamic coupling, 490 Dynamic instability caused by fluid flow, 305 313 of an airfoil, 309 310 flow-induced vibration of a chimney, 311 flow-induced vibration reduction, 307 Helical spoilers, 308 Stockbridge damper, 308 Dynamic response of plucked string, 707 712 Dynamic stability analysis, 301 305 Dynamic system, equations of motion of, 613 615 Dynamic testing of machines and structures, 900 Dynamical matrix, 585 E Earthquake response spectra, 399 403 Eccentricity of rotor, probabilistic characteristics of, 14-6 Eigenvalues/Eigenvalue problem, 583 584, 594 596 Eigenvectors, orthonormalization of, 592 594 Equilibrium states, stability, 13-37 13-40 Elastic coupling, 490 Elastic potential energy, 574 576 Electric motor deflection due to rotating unbalance, 290 Electrodynamic Shaker, 893 895 Electrodynamic transducers, 877 878 Element matrices and vectors, transformation, 998 1001 Elementary parts of vibrating systems, 13 14 Energy dissipated in viscous damping, 166 168 Equation of motion, 147 148, 158, 186 187, 261 262 derivation, 577 581 of dynamic system, 613 615 of an element, 989 991 of finite elements, 1001 1003 for forced vibration, 472 473 of three degree of freedom system, 586 of undamped systems in matrix form, 581 582 whirling of rotating shafts, 785 787 Equivalent linearized spring constant, 26 Equivalent mass of a system, 42 44 Equivalent rotational mass, 42 Equivalent translational mass, 42 Ergodic process, 14-13 Euler, Leonard, 7 Euler-Bernoulli theory, 722, 995 Even functions, 69 71 Exciters, vibration, 892 895, 901 due to unbalanced force, 893 electrodynamic shaker, 893 895 mechanical exciters, 892 893 Expansion theorem, 596 Experimental modal analysis, 900 915 basic idea, 900 coherence function, 907 digital signal processing, 903 904 modal data determination from observed peaks, 907 912 mode shapes measurement, 912 915 necessary equipment, 900 903 random signals analysis, 905 907 Explicit integration method, 943 F Fast Fourier Transform (FFT) method, 896, 902, 924 Finishing process, vibratory, 12 Finite difference method, 941 942 for continuous systems, 951 959 Finite element idealization, 10 Finite element method, 987 1035 bar element, 991 994 beam element, 995 998 boundary conditions, incorporation of, 1003 1012 element matrices and vectors, transformation, 998 1001 equations of motion of, 989 991, 1001 1003 Euler-Bernoulli theory, 995 force vector, 991 998 mass matrix, 991 998 stiffness matrix, 991 998 torsion element, 994 995 First order systems, 151 153, 366 372 response of, 407 408 response under periodic force, 367 372 Fixed-free bar, free vibrations of, 714 Fixed-pinned beam, natural frequencies of, 728 731 Flexibility influence coefficients, 567 571 determination, 569 Flexibility matrix of a beam, 571 Flow-induced vibration of a chimney, 311 reduction, 307 Flutter, 305 Focus, 13-40 Force transmissibility, 283 284 Force vector, 991 998 Forced system, steady-state response of, 615 619 Forced vibration, 17, 494 497, 730 731 steady-state response of spring-mass system, 495 497 of viscously damped systems, 610 616INDEX 1079 Forging hammer forced vibration response of, 608 613 response of anvil of, 169 Fourier analysis, 14-16 14-23 complex Fourier series expansion, 14-17 14-18 Fourier integral, 14-19 14-23 of triangular pulse, 14-22 Fourier integral, 14-19 14-23 Fourier series expansion, 64 66, 73 Cam-follower system, 73 complex Fourier series, 66 67 Gibbs phenomenon, 66 graphical representation using MATLAB, 76 78 numerical Fourier analysis, 74 76 periodic function, 65 Fourth-order Runge-Kutta method, 974 975 Frahm tachometer, 9, 890 Francis water turbine, 291 Free vibration, 17 response of two degree of freedom system, 481 482 response using modal analysis, 606 608 Frequency domain representations, 68 69 Frequency-measuring instruments, 890 891 frequency-measuring instruments, 891 multireed-instrument, 890 single-reed instrument, 890 stroboscope, 891 Frequency of damped vibration, 161 Frequency of oscillation, 62 Frequency or characteristic equation, 475 Frequency response approach, 14-30, 14-33 14-39 mean square response, 14-34 power spectral density, 14-33 Frequency spectrum, 67 68 Frequency transfer functions, 320 325 frequency response characteristics representation, 323 325 general transfer function and, 322 323 physical system, 321 solutions using, 512 515 Fullarton tachometer, 890 G Galileo Galilei, 6 9 Galloping, 305 306 Gaussian random process, 14-14 14-16 General forcing conditions, vibration under, 363 466, See also General periodic force, response under; Nonperiodic force, response under; Periodic force; Response spectrum General periodic force, response under, 365 378 first-order systems, 366 372 second-order systems, 367, 372 374 total response under harmonic base excitation, 377 378 General transfer function and frequency transfer function, 322 323 Generalized coordinates, 472, 488, 576 577 Generalized forces, 576 577 Generalized mass matrix, 575 Germain, Sophie, 8 Gibbs phenomenon, 66 Grid points, 941 H Half power points, 276 Half-range expansions, 71 72 Harmonic analysis, 64 76, See also Fourier series expansion even functions, 69 71 frequency domain representations, 68 69 half-range expansions, 71 72 odd functions, 69 71 time domain representations, 68 69 Harmonic base excitation, total response under, 377 378 Harmonic motion, 54 64, 134 146 addition of harmonic motions, 60 complex algebra, 58 complex number representation of, 57 58 impact, free vibration response due to, 141 142 motion of, graphical representation, 136 natural frequency, 142 146 operations on harmonic functions, 59 61 Scotch yoke mechanism, 55 simple harmonic motion, 56 undamped system, phase plane representation, 138 vectorial representation of, 56 57 water tank, harmonic response of, 139 141 Harmonically excited vibration, 259 362 damped system response under F(t) = F0ei*t, 278 281 damped system response under harmonic force, 271 278, See also individual entry equation of motion, 261 262 forced vibration with Coulomb damping, 293 297, See also Coulomb damping hysteresis damping, forced vibration with, 298 310 quadratic damping, 300 quality factor and bandwidth, 276 278 undamped system response under, 263 271 Helical spoilers, 308 Helicopter seat vibration reduction, 774 783 vibration at source, reduction, 775 776 Heterodyne analyzer, 899 History of vibration, 3 10 finite element idealization, 10 from Galileo to Rayleigh, 6 9 origin, 3 5 recent contributions, 9 10 theory of vibration of plates, 7 torsional vibration tests, 8 Hoisting drum, equivalent k of, 34 35 Holzer s method, 666 670 resultant torque versus frequency, 667 spring-mass systems, 669 670 torsional systems, 666 669 Hooke, Robert, 6 Horizontal position, spring-mass system in, 126 Houbolt method, 962 965 for a two degree of freedom system, 964 Hydraulic valve, periodic vibration of, 374 376 Hysteretic damping, 46 forced vibration with, 298 310 free vibration with, 192 198 I Ideal white noise, 14-25 Identity matrix, 1042 Implicit integration methods, 963 Impulse response function, 382 385, 14-28 14-29 Inelastic collision, response to, 411 412 Inertia influence coefficients, 572 573 Influence coefficients, 562 573 flexibility influence coefficients, 567 571 flexibility matrix of a beam, 571 inertia influence coefficients, 572 573 stiffness influence coefficient, 562 567 stiffness matrix of a frame, 566 Introduction to Harmonics, 5 Inverse Laplace transform, 1049 Inverse matrix, 1045 Inverse of the Matrix, 682 Irregular forcing conditions, response to, 431 436 Irregular forcing function, 378 380 Isolation, vibration, 801 802 with base motion, 814 820 damped spring mount, 801 pneumatic rubber mount, 801 system with flexible foundation, 821 822 system with partially flexible foundation, 822 824 types, 802 803 undamped spring mount, 801 with rigid foundation, 804 813, See also Rigid foundation Iteration method, 670 677, 13-13 13-16, See also Matrices: matrix iteration method J Jacobi s method, 678 680 eigenvalue solution using, 679 682, 684 standard eigenvalue problem, 680 683 joint probability distribution of random variables, 14-7 14-9 bivariate distributions, 14-8 multivariate distribution, 14-8 univariate distributions, 14-81080 INDEX Journal bearing, damping constant of, 48 50 Jump phenomenon, 13-16 K Karman vortices, 305 Kinetic energy expressions in matrix form, 574 576 Kirchhoff, G. R., 8 Kronecker delta, 581 LL Hospital s rule, 266 Lagrange, Joseph, 7 Lagrange s equations, 577 581 Laplace transform, 313, 317 320, 365, 406 427, 504 512, 1048 1055 damped system response using, 317 definition, 1048 1049 first-order systems, response of, 407 409 inverse Laplace transform, 1049 partial fractions method, 1050 1052 second order systems, response of, 409 414 shifting theorems, 1050 steady state response using, 319 320 step force, response to, 414 420 transform of derivatives, 1049 1050 transient and steady-state responses, 406 transient response, 421 427, See also individual entry two degree of freedom systems solutions using, 504 512 Laplacian operator, 741 Lateral vibration of beams, 721 736 axial compressive force, beam subjected to, 733 737 boundary conditions, 724 726 equation of motion, 721 fixed-pinned beam, natural frequencies of, 728 731 forced vibration, 730 731 free vibration, 723 724 initial conditions, 723 orthogonality of normal functions, 726 729 simply supported beam, forced vibration, 731 734 Lathe, 469, 488 489 Left half-plane (LHP) yield, 198 Limit cycles, 13-41 13-43 Lindstedt s perturbation method, 13-10 13-12 Linear algebraic equations, solution of, 1065 Linear coordinates, 555 Linear force, compacting machine under, 391 392 Linear springs, 23 25 Linear variable differential transformer (LVDT) transducer, 878 879 Linear vibration, 17 Linearization of nonlinear spring, 25 27 Literature, vibration, 80 81 Local coordinate axis, 998 Logarithmic decrement, 164 166 Longitudinal vibration of bar or rod, 710 718 bar carrying a mass, natural frequencies of, 715 716 bar subjected to initial force, vibrations of, 716 720 boundary conditions, 712 713 equation of motion and solution, 710 712 free vibrations of a fixed-free bar, 714 orthogonality of normal functions, 713 718 Longitudinal vibration of bars, 951 955 Loops, 6 Lumped mass matrices, 1012 1015 Lumped-mass model, 555 M Machine condition monitoring techniques, 916 918 Machine maintenance techniques, 915 916 breakdown maintenance, 915 condition-based maintenance, 916 preventive maintenance, 915 Machine tool support, equivalent spring and damping constants of, 52 53 Machine vibration monitoring techniques, 918 923 Magnification factor, 264, 273 Marine engine propeller system, 485 488 Mass matrix, 991 998 Mass or inertia elements, 40 45 Material damping, 46 Mathematical modeling, 18 Mathieu equation, 13-24 13-29 MATLAB, 326 332, 436 440, 515 522, 619 627, 683 686, 748 751, 843 850, 925 928, 972 978, 1015 1019, 1041 1047, 1059 1068 accelerometer equation plotting, 927 929 arrays and matrices, 1060 arrays with special structure, 1061 autocorrelation function plotting, 14-46 14-48 column vector, 1060 complex numbers, 1062 Coulomb damping, free vibration response of a system with, 205 damped system, forced vibration response of, 623 624 differential equations solution, 1066 1068 eigenvalue problem solution, 515 516, 619, 683 684 finite element analysis of stepped bar, 1015 forced response of a system with Coulomb damping, 327 328 forced vibration response of simply supported beam, plotting, 748 751 Fourier series graphical representation using, 76 78 free vibration response frequency response, plotting, 519 520 functions in, 1062 Gaussian probability distribution function evaluation, 14-48 14-49 general eigenvalue proble, 685 686 impulse response of a structure, 437 438 matrix, 1060 matrix operations, 1061 M-files, 1062 1063 multidegree of freedom system, 619 626 nonlinear differential equation solution, 13-61 nonlinearly damped system solution, 13-57 13-59 nonlinear system under pulse loading solution, 13-59 numerical Fourier analysis using, 79 Nyquist circle plotting, 925 926 pendulum equation solution, 13-53 13-57 plotting of graphs, 1063 1064 program to generate characteristic polynomial, 625 quartic equation roots, 522 railway cars, time response of, 518 519 response under a periodic force, 438 439 response under arbitrary forcing function, 439 roots of a polynomial equation, 622 roots of a quartic equation, 516 roots of transcendental and nonlinear equations, 750 751 row vector, 1062 solution of a single degree of freedom system, 972 solution of multidegree of freedom system, 973 974 special matrices, 1061 spring-mass system, free vibration response of, 203 static deflection, variations of natural frequency and period with, 202 steady-state response of viscously damped system, 330 331 system response under base excitation, 328 330 total response of an undamped system using, 326 327 total response of system under base excitation, 436 437 transmissibility, plotting, 843 undamped system response, 204 variables, 1060 vibration amplitudes of vibration absorber masses, 845 846 Matrices, 1041 1047, 1060, See also individual entries basic operations, 1046 1047, 1061 trace, 1043 transpose of, 1043INDEX 1081 Maximum overshoot (M p ), 423 Mean square response, 14-34 Mean value, 14-6 14-7, 14-32 Measurement and applications, vibration, 870 938 machine condition monitoring and diagnosis, 915 925 measurement scheme, 872 Mechanical chatter, 13-5 Mechanical exciters, 892 893 Method of isoclines, 13-34 13-35 trajectories using, 13-36 Membranes, vibration of, 739 742 equation of motion, 739 741 free vibrations of rectangular membrane, 742 744 initial and boundary conditions, 741 742 membrane under uniform tension, 740 Mersenne, Marin, 6 7 M-files, 1062 1063 Milling cutter, natural frequencies of, 720 Mindlin, R. D., 9 Modal analysis, 596 forced vibration of undamped systems using, 603 610 free vibration response using, 606 608 Modal damping ratio, 612 Modal matrix, 592 Modal testing, 900 915, See also Experimental modal analysis Modal vectors, 475 Mode shapes, 583 determination, 654 698 measurement, 912 915 of three degree of freedom system, 590 Monochord, 4 5 Motor-generator set, absorber for, 838 843 Multidegree of freedom systems, 553 653, 14-39 14-46, See also Influence coefficients; Three degree of freedom system central difference method for, 947 951 continuous systems modeling as, 555 556 equations of motion of undamped systems in matrix form, 581 582 expansion theorem, 596 free vibration of undamped systems, 601 603 generalized coordinates, 576 577 generalized forces, 576 577 Lagrange s equations to derive equations of motion, 577 581 modal analysis, 603 610, See also individual entry natural frequencies of free system, 598 601 Newton s second law to derive equations of motion, 557 562 potential and kinetic energy expressions in matrix form, 574 576 repeated Eigenvalues, 594 596 self-excitation, 617 619 spring-mass-damper system, equations of motion of, 557 560 stability analysis, 617 619 steady-state response of forced system, 615 619 trailer compound pendulum system, equations of motion of, 560 unrestrained systems, 596 599 Multivariate distribution, 14-8 N Narrow-band process, 14-25 14-27 Natural frequencies, 62, 475 determination, 654 698, See also Dunkerley s formula; Holzer s method; Jacobi s method; Rayleigh s method of free system, 598 601 of torsional system, 484, 668 673 Natural mode, two degree of freedom systems, 471 Newmark method, 968 971 Newton, Isaac, 6 Newton s second law, 129 130, 261, 557 562 Nodes, 6, 706, 13-40 13-41 Nomograph, vibration, 771 775 Nondeterministic vibration, 17 18 Nonlinear damper, linearization of, 52 Nonlinear differential equation solution, 13-61 Nonlinear equations, roots of, 1064 Nonlinear springs, 23 25 Nonlinear system under pulse loading solution, 13-59 Nonlinear vibration, 17, 13-1 13-76 approximate analytical methods, 13-7 13-19 equilibrium states, stability, 13-37 13-40 exact methods for, 13-6 13-7 graphical methods, 13-29 13-37 iterative method, 13-13 13-16 Jump phenomenon, 13-16 limit cycles, 13-41 13-43 Lindstedt s perturbation method, 13-10 13-12, 13-25 nonlinear spring characteristics, 13-4 numerical methods, 13-52 13-53 Ritz-Galerkin method, 13-17 13-19 subharmonic oscillations, 13-20 13-22 superharmonic oscillations, 13-23 13-24 time-dependent coefficients, systems with, 13-24 13-29 variable mass system, 13-5 13-6 Nonperiodic force, response under, 365, 380 381, See also Convolution integral; Laplace transform; Numerical methods Normal modes, 470, 591 593 Number-decibel conversion line, 323 Numerical Fourier analysis, 74 76 Numerical integration methods, 939 986 finite difference method, 941 942 single degree of freedom systems, 942 946 Numerical methods, for response under nonperiodic force, 365, 428 431 Nyquist circle plotting, 925 926 Nyquist plot, modal data determination from, 910 912 O Octave band analyzer, 896, 899 Octave, 63 Odd functions, 69 71 Operational deflection shape measurements, 900 Optimally tuned vibration absorber, 842 Orthogonality of normal functions, 591 593, 713 718, 726 729 Orthonormalization of Eigenvectors, 592 594 Overdamped system, 163, 418 420 P Parameter variations, 176 185, See also under Root locus representations Parseval s formula, 14-17, 14-21 Partial fractions method, 1050 1052 Peak time (t p ), 421 Perfectly elastic collision, response to, 412 414 Periodic solutions using Lindstedt s perturbation method, 13-25 Period of beating , 269 Period of oscillation, 62 Periodic force, 365 378, See also General periodic force, response under Periodic vibration of a hydraulic valve, 374 376 Phase angle, 62, 584 Phase distortion, 888 890 Phase plane representation, nonlinear vibration, 13-29 13-34 phase velocity, 13-34 undamped nonlinear system, 13-32 undamped pendulum, 13-31 Phase plane trajectories, time solution from, 13-36 13-37 Phase velocity, 13-34 Philosophiae Naturalis Principia Mathematica, 6 Piezoelectric transducers, 876 877 Pinned-fixed beam, 959 Piston-cylinder dashpot, 50 52 Plane milling cutter, 721 Plano-milling machine structure, 989 Plates, deflection of, 1039 1040 Poincaré section, 13-43 13-44 Poisson, Simeon, 8 Positive definite matrix, 576 Positive definite quadratic forms, 576 Potential energy expressions in matrix form, 574 5761082 INDEX Power spectral density, 14-23 14-25, 14-33 Precision electronic system, vibration control of, 829 830 Precision machine with base motion, design of isolation for, 816 818 Principal coordinates, 472, 488 493 Principal mode, two degree of freedom systems, 470 Principle of conservation of energy, 131 132 Principle of virtual displacements, 130 Probability density curve, 919 Probability distribution, 14-4 14-5 Propeller shaft, 33 Proportional damping, 611 Pseudo spectrum, 397 Pseudo velocity, 397 Pulley subjected to Coulomb damping, 191 Pulley system, 145 146 Pulse load, 389 390 response due to, 390 Pythagoras, 3 4 Q Q factor/Quality factor, 276 278 Quadratic damping, 300 Quefrency domain analysis, 922 R Ramp function, first-order system response due to, 408 409 Random signals analysis, 905 907 Random vibration, 17, 14-1 14-61, See also Stationary random process band-limited white noise, 14-25 correlation functions of, 14-9 14-10 eccentricity of rotor, probabilistic characteristics of, 14-6 Gaussian random process, 14-14 14-16 ideal white noise, 14-25 joint probability distribution, 14-7 14-9 mean value, 14-6 14-7 multidegree of freedom system response, 14-39 14-46 narrow-band process, 14-25 14-27 power spectral density, 14-23 14-25 probability distribution, 14-4 14-5 random processes, 14-3 14-4 random variables, 14-3 14-4 single degree of freedom system response, 14-28 14-31 standard deviation, 14-6 14-7 stationary process, 14-26 stationary random excitations, response due to, 14-31 14-39 wide-band process, 14-25 14-27 Rayleigh, Baron, 8 Rayleigh s method, 153 158, 658 665, 700, 742 745 beams, fundamental frequency of, 663 665 effect of mass, 155, 157 manometer for diesel engine, 153 Rayleigh s quotient, properties of, 659 661 shafts, fundamental frequency of, 663 668 U-tube manometer, 154 Rayleigh-Ritz method, 700, 745 748 Reciprocating engines, balancing, 792 798 reciprocating engines, balancing, 796 798 unbalanced forces due to fluctuations in gas pressure, 792 793 unbalanced forces due to inertia of the moving parts, 793 796 Recoil mechanism, 173 Rectangular pulse load, 389 390 response due to, 390 Recurrence formula, 943 Reference marks, 777 778 Relative motion, 284 287 Repeated Eigenvalues, 594 596 Resonance, 11 Resonant frequencies of vibration absorber, 847 848 Response spectrum, 394 406 for base excitation, 396 399 building frame response to an earthquake, 402 403 design under shock environment, 403 406 earthquake response spectra, 399 403 of sinusoidal pulse, 394 397 water tank subjected to base acceleration, 398 399 Rigid bar connected by springs, equivalent k of, 37 stability of, 201 Rigid foundation, vibration isolation system with, 804 813 design chart of isolation, 809 810 isolator for stereo turntable, 810 813 machine member on, 804 resilient member on, 804 spring support for exhaust fan, 807 808 undamped isolator design, 808 810 vibratory motion of mass, reduction, 806 Rise time (t r ), 422 423 Ritz-Galerkin method, 13-17 13-19 Rod, spring constants of, 27 Root locus representations, 176 185 and parameter variations, 179 185 roots study with variation of c, 181 + in s-plane, 176 179 , in s-plane, 176 179 *d in s-plane, 176 179 * n in s-plane, 176 179 variation of mass, 183 variation of spring constant, 183 Rotary inertia effects, 734 739 Rotating machines, balancing, 776 785 single-plane balancing, 776 779 two-plane balancing, 779 785 Rotating unbalance, 287 293, 830 837, See also under Damped system response under harmonic force Routh-Hurwitz criteria, 502, 790 Row matrix, 1042 Row vector, 1060 Runge-Kutta methods, 429 431 S+ in s-plane, interpretation, 176 179 , in s-plane, interpretation, 176 179 *d in s-plane, interpretation, 176 179 * n in s-plane, interpretation, 176 179 Saddle point, 13-40 13-41 Sample point, 14-3 Sample space, 14-3 Sauveur, Joseph, 6 Scotch yoke mechanism, 55 Second-order systems, 367, 372 374, 409 414 Seismograph, 5 Self-excitation, 301 313, 500 502, 617 619 Semidefinite systems, 497 500, 598 Settling time, transient response, 424 Shafts, fundamental frequency of, 663 668 Shear deformation effects, 734 739 Shearing stress (,), 49 Shock absorber for a motorcycle, 171 172 Shock environment, design under, 403 406 Shock isolation, 824 827 Shock loads, 403 406 Signal analysis, 895 899 Signum function, 187 Simple harmonic motion, 56 Simple pendulum, 14, 39, 13-3 Simply supported beam, 1039 forced vibration, 731 733 natural frequencies of, 737 742 Singing of transmission lines, 305 Single degree of freedom systems, 14, 14-28 14-31 central difference method for, 942 945 characteristics of, 14-30 14-31 free vibration of, 124 258, See also Undamped translational system, free vibration of frequency response approach, 14-30 impulse response approach, 14-28 14-29 Single-plane balancing, 776 779 Singular point, 13-34 Sinusoidal pulse, response spectrum of, 394 397 Solid damping, 46 Space shuttle, vibration testing, 12 Special matrices, 1061 Spectrum analyzers, 896 897 Spring constants of elastic elements, 27 30 Spring elements, 22 39 deformation of spring, 22 equivalent linearized spring constant, 26 linear springs, 23 25 nonlinear springs, 23 25INDEX 1083 spring constant associated with restoring force due to gravity, 39 spring constants of elastic elements, 27 30 Spring-mass-damper system, 262, 557 560 Spring-mass systems, 126 128, 669 670 Springs, combination of, 30 38 equivalent k, 32, 34 35, 37 in parallel, 30 in series, 31 torsional spring constant of a propeller shaft, 33 Spring-supported mass instability on moving belt, 302 Square matrix, 1042 Stability analysis, 301 313, See also Dynamic instability caused by fluid flow two degree of freedom systems, 500 502 dynamic, 301 305 multidegree of freedom systems, 617 619 whirling of rotating shafts, 790 791 Stability of systems, 198 202 asymptotically stable, 198 200 rigid bar, 201 stable, 198 200 unstable, 198 200 Stable focus, 13-40 Stable orbits, functions with, 13-45 Standard deviation, 14-6 14-7 Standard eigenvalue problem, 585 Static deflection, 263 Static equilibrium position, 132 Static unbalance, 776 Stationary random excitations, response due to, 14-31 14-39 Stationary random process, 14-10 14-14 Strange attractors, 13-46 Steady state response, 406 of forced system, 615 619 using Laplace transform, 319 320 Step force, response to, 414 420, See also under Laplace transform Stepped bar, 1015, 1018 Stiffness influence coefficient, 562 567 Stiffness matrix, 566, 991 998 Stockbridge damper, 308 Stodola, Aurel, 9 Stroboscope, 891 Study of vibration, importance, 10 13 Subharmonic oscillations, 13-20 13-22 Superharmonic oscillations, 13-23 13-24 Suspension system, equivalent k of, 32 Symmetric matrix, 682 683, 1043 System response under base excitation, 328 330 T Tapered beam, fundamental frequency of, 744 750 Taylor, Brook, 7 Taylor s series expansion, 25, 310 Temporal averages, 14-14 Thick beam theory, 734 Thin beam theory, 722 Three degree of freedom system, 15 equations of motion of, 586 fundamental frequency of, 661 663 mode shapes of, 590 natural frequencies of, 587 589 natural frequencies of, 673 679 Time constant, 151 153 Time-delayed step force, 388 389 Time-dependent coefficients, systems with, 13-24 13-29 Time domain analysis, 918 Time domain representations, 68 69 Timoshenko beam theory, 734 735 Timoshenko, Stephen, 9 Torsion element, 994 995 Torsional pendulum, 148 Torsional spring constant of a propeller shaft, 33 Torsional system, 483 488, 666 669 with Coulomb damping, 190 192 with discs mounted on a shaft, 483 equations of motion of, 578 579 natural frequencies of, 484 488, 668 673 with viscous damping, 168 174 Torsional vibration of a shaft or rod, 718 721 Torsional vibration, 8, 146 Trace, 1043 Trajectories of simple harmonic oscillator, 13-29 13-30 Trailer compound pendulum system, equations of motion of, 560 Transducers, 873 879, 901 electric resistance strain gage, 873 electrodynamic transducers, 877 878 linear variable differential transformer (LVDT) transducer, 878 879 piezoelectric transducers, 876 877 variable resistance transducers, 873 876 Transfer function approach, 313 316, 425 426, 502 504 Transient response, 261 262, 406, 421 427 Transition curves, 13-28 Transverse vibration of beams, 955 959 Transverse vibration of string or cable, 701 710, See also under Continuous systems Traveling-wave solution, 709 710 Triangular pulse, Fourier transform of, 14-22 Triple pendulum, 576 Tuned vibration absorber, 842 Two degree of freedom systems, 15, 467 552, See also Forced vibration; Laplace transform; Semidefinite systems; Torsional system automobile, frequencies and modes of, 492 495 coordinate coupling and principal coordinates, 488 493 coupled differential equations, 470 equations of motion for forced vibration, 472 473 forced response of, 520 522 free vibration response of, 481 482 Lathe, 469, 488 489 natural mode, 471 normal mode, 470 packaging of an instrument, 471 principal mode, 470 spring-mass-damper system, 472 transfer function approach, 502 504 Two-plane balancing, 779 785, 848 850 U Undamped dynamic vibration absorber, 833 839 effect on the response of machine, 835 for diesel engine, 837 838 for motor-generator set, 838 843 Undamped equation, 13-13, 13-31 Undamped isolator design, 808 810 Undamped system, 127 free vibration analysis, 474 482 free vibration of, 601 606 free vibration response of, 504 507 in matrix form, 581 582 response under harmonic force, 263 271, See also under Harmonically excited vibration total response of, using MATLAB, 326 327 Undamped torsional system, free vibration of, 146 151 Undamped translational system, free vibration of, 129 146 auxiliary or characteristic equation, 134 D Alembert s principle, 130 eigenvalues or characteristic values, 134 mass under virtual displacement, 131 principle of conservation of energy, 130 principle of virtual displacements, 130 using Newton s second law of motion, 129 130 Undamped vibration, 17 Underdamped system, 160, 414 416 Uniform string, free vibration of, 704 705 Unit impulse response of second-order system, 409 Units, 1056 1058 Univariate distributions, 14-8 Unrestrained systems, 499 502, 596 599 Unstable focus, 13-40 Unstable orbits, functions with, 13-45 13-47 Unstable system, 198 V Variable mass system, 13-5 13-6 Variable resistance transducers, 873 8761084 INDEX Vectorial representation of harmonic motion, 56 57 Velometer, 886 887 Vertical position, spring-mass system in, 132 133 Vibrating string, 702 Vibration absorbers, 832 843, 847 848, See also Damped dynamic vibration absorber; Undamped dynamic vibration absorber Vibration pickups, 879 890 Vibration severity of machinery, 773 Vibrometer, 881 882 Viscoelastic materials use, 799 Viscous damping, 45 Cannon analysis, 173 energy dissipated in, 166 168 forced vibration of, 610 616 free vibration with, 158 174 steady-state response of, 330 331 torsional systems with viscous damping, 168 174 torsional systems with, 168 174 W Wallis, John, 6 Whirling of rotating shafts, 785 791 critical speeds, 787 equations of motion, 785 787 shaft carrying an unbalanced rotor, 791 stability analysis, 790 791 system response, 788 790 Wide-band process, 14-25 14-27 Wiener-Khintchine formula, 14-23 Wilson method, 965 968 Wind-induced vibration, 11 Y Young s modulus, 142 143 Z Zero matrix, 1042 Zhang Heng, 5
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