Mechanical vibrations : theory and application to structural dynamics

Mechanical Vibrations: Theory and Application to Structural Dynamics, Third Edition is a comprehensively updated new edition of the popular textbook. It presents the theory of vibrations in the context of structural analysis and covers applications in mechanical and aerospace engineering. Although keeping the same overall structure, the content of this new edition has been significantly revised in order to cover new topics, enhance focus on selected important issues, provide sets of exercises and improve the quality of presentation. Without being exhaustive (see the Introduction for a comprehensive list), some key features include: a systematic approach to dynamic reduction and substructuring, based on duality between mechanical and admittance concepts; an introduction to experimental modal analysis and identification methods; an improved, more physical presentation of wave propagation phenomena; a comprehensive presentation of current practice for solving large eigenproblems, focusing on the efficient linear solution of large, sparse and possibly singular systems; a deeply revised description of time integration schemes, providing framework for the rigorous accuracy/stability analysis of now widely used algorithms such as HHT and Generalized; solved exercises and end of chapter homework problems; and, a companion website hosting supplementary material. With revised, coherent and uniform notation, Mechanical Vibrations: Theory and Application to Structural Dynamics, Third Edition is a must-have textbook for graduate students working with vibration in mechanical, aerospace and civil engineering, and is also an excellent reference for researchers and industry practitioners. Cover 1 TItle Page 5 Copyright 6 Contents 7 Foreword 15 Preface 17 Introduction 19 Suggested Bibliography 25 List of main symbols and definitions 27 Chapter 1 Analytical Dynamics of Discrete Systems 31 Definitions 32 1.1 Principle of virtual work for a particle 32 1.1.1 Nonconstrained particle 32 1.1.2 Constrained particle 33 1.2 Extension to a system of particles 35 1.2.1 Virtual work principle for N particles 35 1.2.2 The kinematic constraints 36 1.2.3 Concept of generalized displacements 38 1.3 Hamilton's principle for conservative systems and Lagrange equations 41 1.3.1 Structure of kinetic energy and classification of inertia forces 45 1.3.2 Energy conservation in a system with scleronomic constraints 47 1.3.3 Classification of generalized forces 50 1.4 Lagrange equations in the general case 54 1.5 Lagrange equations for impulsive loading 57 1.5.1 Impulsive loading of a mass particle 57 1.5.2 Impulsive loading for a system of particles 60 1.6 Dynamics of constrained systems 62 1.7 Exercises 64 1.7.1 Solved exercises 64 1.7.2 Selected exercises 71 References 72 Chapter 2 Undamped Vibrations of n-Degree-of-Freedom Systems 75 Definitions 76 2.1 Linear vibrations about an equilibrium configuration 77 2.1.1 Vibrations about a stable equilibrium position 77 2.1.2 Free vibrations about an equilibrium configuration corresponding to steady motion 81 2.1.3 Vibrations about a neutrally stable equilibrium position 84 2.2 Normal modes of vibration 85 2.2.1 Systems with a stable equilibrium configuration 86 2.2.2 Systems with a neutrally stable equilibrium position 87 2.3 Orthogonality of vibration eigenmodes 88 2.3.1 Orthogonality of elastic modes with distinct frequencies 88 2.3.2 Degeneracy theorem and generalized orthogonality relationships 90 2.3.3 Orthogonality relationships including rigid-body modes 93 2.4 Vector and matrix spectral expansions using eigenmodes 94 2.5 Free vibrations induced by nonzero initial conditions 95 2.5.1 Systems with a stable equilibrium position 95 2.5.2 Systems with neutrally stable equilibrium position 100 2.6 Response to applied forces: forced harmonic response 101 2.6.1 Harmonic response, impedance and admittance matrices 102 2.6.2 Mode superposition and spectral expansion of the admittance matrix 102 2.6.3 Statically exact expansion of the admittance matrix 106 2.6.4 Pseudo-resonance and resonance 107 2.6.5 Normal excitation modes 108 2.7 Response to applied forces: response in the time domain 109 2.7.1 Mode superposition and normal equations 109 2.7.2 Impulse response and time integration of the normal equations 110 2.7.3 Step response and time integration of the normal equations 112 2.7.4 Direct integration of the transient response 113 2.8 Modal approximations of dynamic responses 113 2.8.1 Response truncation and mode displacement method 114 2.8.2 Mode acceleration method 115 2.8.3 Mode acceleration and model reduction on selected coordinates 116 2.9 Response to support motion 119 2.9.1 Motion imposed to a subset of degrees of freedom 119 2.9.2 Transformation to normal coordinates 121 2.9.3 Mechanical impedance on supports and its statically exact expansion 123 2.9.4 System submitted to global support acceleration 126 2.9.5 Effective modal masses 127 2.9.6 Method of additional masses 128 2.10 Variational methods for eigenvalue characterization 129 2.10.1 Rayleigh quotient 129 2.10.2 Principle of best approximation to a given eigenvalue 130 2.10.3 Recurrent variational procedure for eigenvalue analysis 131 2.10.4 Eigensolutions of constrained systems: general comparison principle or monotonicity principle 132 2.10.5 Courant's minimax principle to evaluate eigenvalues independently of each other 134 2.10.6 Rayleigh's theorem on constraints (eigenvalue bracketing) 135 2.11 Conservative rotating systems 137 2.11.1 Energy conservation in the absence of external force 137 2.11.2 Properties of the eigensolutions of the conservative rotating system 137 2.11.3 State-space form of equations of motion 139 2.11.4 Eigenvalue problem in symmetrical form 142 2.11.5 Orthogonality relationships 144 2.11.6 Response to nonzero initial conditions 146 2.11.7 Response to external excitation 148 2.12 Exercises 148 2.12.1 Solved exercises 148 2.12.2 Selected exercises 161 References 166 Chapter 3 Damped Vibrations of n-Degree-of-Freedom Systems 167 Definitions 168 3.1 Damped oscillations in terms of normal eigensolutions of the undamped system 169 3.1.1 Normal equations for a damped system 170 3.1.2 Modal damping assumption for lightly damped structures 171 3.1.3 Constructing the damping matrix through modal expansion 176 3.2 Forced harmonic response 178 3.2.1 The case of light viscous damping 178 3.2.2 Hysteretic damping 180 3.2.3 Force appropriation testing 182 3.2.4 The characteristic phase lag theory 188 3.3 State-space formulation of damped systems 192 3.3.1 Eigenvalue problem and solution of the homogeneous case 193 3.3.2 General solution for the nonhomogeneous case 196 3.3.3 Harmonic response 197 3.4 Experimental methods of modal identification 198 3.4.1 The least-squares complex exponential method 200 3.4.2 Discrete Fourier transform 205 3.4.3 The rational fraction polynomial method 208 3.4.4 Estimating the modes of the associated undamped system 213 3.4.5 Example: experimental modal analysis of a bellmouth 214 3.5 Exercises 217 3.5.1 Solved exercises 217 3.6 Proposed exercises 225 References 226 Chapter 4 Continuous Systems 229 Definitions 230 4.1 Kinematic description of the dynamic behaviour of continuous systems: Hamilton's principle 231 4.1.1 Definitions 231 4.1.2 Strain evaluation: Green's measure 232 4.1.3 Stress-strain relationships 237 4.1.4 Displacement variational principle 239 4.1.5 Derivation of equations of motion 239 4.1.6 The linear case and nonlinear effects 241 4.2 Free vibrations of linear continuous systems and response to external excitation 249 4.2.1 Eigenvalue problem 249 4.2.2 Orthogonality of eigensolutions 251 4.2.3 Response to external excitation: mode superposition (homogeneous spatial boundary conditions) 252 4.2.4 Response to external excitation: mode superposition (nonhomogeneous spatial boundary conditions) 255 4.2.5 Reciprocity principle for harmonic motion 259 4.3 One-dimensional continuous systems 261 4.3.1 The bar in extension 262 4.3.2 Transverse vibrations of a taut string 276 4.3.3 Transverse vibration of beams with no shear deflection 281 4.3.4 Transverse vibration of beams including shear deflection 295 4.3.5 Travelling waves in beams 303 4.4 Bending vibrations of thin plates 308 4.4.1 Kinematic assumptions 308 4.4.2 Strain expressions 309 4.4.3 Stress-strain relationships 310 4.4.4 Definition of curvatures 311 4.4.5 Moment-curvature relationships 311 4.4.6 Frame transformation for bending moments 313 4.4.7 Computation of strain energy 313 4.4.8 Expression of Hamilton's principle 314 4.4.9 Plate equations of motion derived from Hamilton's principle 316 4.4.10 Influence of in-plane initial stresses on plate vibration 321 4.4.11 Free vibrations of the rectangular plate 323 4.4.12 Vibrations of circular plates 326 4.4.13 An application of plate vibration: the ultrasonic wave motor 329 4.5 Wave propagation in a homogeneous elastic medium 334 4.5.1 The Navier equations in linear dynamic analysis 334 4.5.2 Plane elastic waves 336 4.5.3 Surface waves 338 4.6 Solved exercises 345 4.7 Proposed exercises 346 References 351 Chapter 5 Approximation of Continuous Systems by Displacement Methods 353 Definitions 355 5.1 The Rayleigh-Ritz method 357 5.1.1 Choice of approximation functions 357 5.1.2 Discretization of the displacement variational principle 358 5.1.3 Computation of eigensolutions by the Rayleigh-Ritz method 360 5.1.4 Computation of the response to external loading by the Rayleigh-Ritz method 363 5.1.5 The case of prestressed structures 363 5.2 Applications of the Rayleigh-Ritz method to continuous systems 364 5.2.1 The clamped鈥揻ree uniform bar 365 5.2.2 The clamped鈥揻ree uniform beam 368 5.2.3 The uniform rectangular plate 375 5.3 The finite element method 381 5.3.1 The bar in extension 382 5.3.2 Truss frames 389 5.3.3 Beams in bending without shear deflection 394 5.3.4 Three-dimensional beam element without shear deflection 404 5.3.5 Beams in bending with shear deformation 410 5.4 Exercises 417 5.4.1 Solved exercises 417 5.4.2 Selected exercises 424 References 430 Chapter 6 Solution Methods for the Eigenvalue Problem 433 Definitions 435 6.1 General considerations 437 6.1.1 Classification of solution methods 438 6.1.2 Criteria for selecting the solution method 438 6.1.3 Accuracy of eigensolutions and stopping criteria 441 6.2 Dynamical and symmetric iteration matrices 443 6.3 Computing the determinant: Sturm sequences 444 6.4 Matrix transformation methods 448 6.4.1 Reduction to a diagonal form: Jacobi's method 448 6.4.2 Reduction to a tridiagonal form: Householder's method 452 6.5 Iteration on eigenvectors: the power algorithm 454 6.5.1 Computing the fundamental eigensolution 455 6.5.2 Determining higher modes: orthogonal deflation 459 6.5.3 Inverse iteration form of the power method 461 6.6 Solution methods for a linear set of equations 462 6.6.1 Nonsingular linear systems 463 6.6.2 Singular systems: nullspace, solutions and generalized inverse 471 6.6.3 Singular matrix and nullspace 471 6.6.4 Solution of singular systems 472 6.6.5 A family of generalized inverses 474 6.6.6 Solution by generalized inverses and finding the nullspace N 475 6.6.7 Taking into account linear constraints 477 6.7 Practical aspects of inverse iteration methods 478 6.7.1 Inverse iteration in presence of rigid body modes 478 6.7.2 Spectral shifting 481 6.8 Subspace construction methods 482 6.8.1 The subspace iteration method 482 6.8.2 The Lanczos method 486 6.9 Dynamic reduction and substructuring 497 6.9.1 Static condensation (Guyan-Irons reduction) 499 6.9.2 Craig and Bampton's substructuring method 502 6.9.3 McNeal's hybrid synthesis method 505 6.9.4 Rubin's substructuring method 506 6.10 Error bounds to eigenvalues 506 6.10.1 Rayleigh and Schwarz quotients 507 6.10.2 Eigenvalue bracketing 509 6.10.3 Temple鈥揔ato bounds 510 6.11 Sensitivity of eigensolutions, model updating and dynamic optimization 516 6.11.1 Sensitivity of the structural model to physical parameters 519 6.11.2 Sensitivity of eigenfrequencies 520 6.11.3 Sensitivity of free vibration modes 520 6.11.4 Modal representation of eigenmode sensitivity 522 6.12 Exercises 522 6.12.1 Solved exercises 522 6.12.2 Selected exercises 523 References 526 Chapter 7 Direct Time-Integration Methods 529 Definitions 531 7.1 Linear multistep integration methods 531 7.1.1 Development of linear multistep integration formulas 532 7.1.2 One-step methods 533 7.1.3 Two-step second-order methods 534 7.1.4 Several-step methods 535 7.1.5 Numerical observation of stability and accuracy properties of simple time integration formulas 535 7.1.6 Stability analysis of multistep methods 536 7.2 One-step formulas for second-order systems: Newmark's family 540 7.2.1 The Newmark method 540 7.2.2 Consistency of Newmark's method 543 7.2.3 First-order form of Newmark's operator-amplification matrix 543 7.2.4 Matrix norm and spectral radius 545 7.2.5 Stability of an integration method-spectral stability 546 7.2.6 Spectral stability of the Newmark method 548 7.2.7 Oscillatory behaviour of the Newmark response 551 7.2.8 Measures of accuracy: numerical dissipation and dispersion 553 7.3 Equilibrium averaging methods 557 7.3.1 Amplification matrix 558 7.3.2 Finite difference form of the time-marching formula 559 7.3.3 Accuracy analysis of equilibrium averaging methods 560 7.3.4 Stability domain of equilibrium averaging methods 561 7.3.5 Oscillatory behaviour of the solution 562 7.3.6 Particular forms of equilibrium averaging 562 7.4 Energy conservation 568 7.4.1 Application: the clamped-free bar excited by an end force 570 7.5 Explicit time integration using the central difference algorithm 574 7.5.1 Algorithm in terms of velocities 574 7.5.2 Application example: the clamped-free bar excited by an end load 577 7.5.3 Restitution of the exact solution by the central difference method 579 7.6 The nonlinear case 582 7.6.1 The explicit case 582 7.6.2 The implicit case 583 7.6.3 Time step size control 589 7.7 Exercises 591 References 593 Author Index 595 Subject Index 599 EULA 617 Mechanical Vibrations: Theory And Application To Structural Dynamics, Third Edition Is A Comprehensively Updated And Reorganized New Edition Of The Popular Textbook. It Presents The Theory Of Vibrations In The Context Of Structural Analysis And Covers Applications In Mechanical And Aerospace Engineering, This New Edition Now Includes The Fundamentals Of Signal Processing And Identification Technique, And Develops The Concepts Of Dynamic Reduction And Substructuring. A More Detailed Discussion Of The Concept Of Eigensolution Sensitivity To Physical Parameters Is Included And The Fundamental Cases Of Wave Propagation In Solids Are Considered. It Also Includes A Chapter On The Finite Element Method For One-dimensional Structures. This New Edition Contains Coherent And Uniform Notation And Now Includes Solved Exercises At The End Of Each Chapter-- Provides More Detailed Information On Eigensolution Sensitivity To Physical Parameters-- 1. Analytical Dynamics Of Discrete Systems -- 2. Undamped Vibrations Of N-degree-of-freedom Systems -- 3. Damped Vibrations Of N-degree-of-freedom Systems -- 4. Continuous Systems -- 5. Approximation Of Continuous Systems By Displacement Methods -- 6. Solution Methods For The Eigenvalue Problem -- 7. Direct Time-integration Methods. Michel Géradin, Daniel J. Rixen. Includes Bibliographical References And Indexes.