Mechanical Vibrations

Michel Géradin holds an Engineering Degree in Physics and a PhD from ULg (University of Liège, Belgium). Successively  he has been a research fellow from the Belgian FNRS (1968–1979),  Professor of Structural Dynamics at ULg (1979–2010) and Unit Head of the European Laboratory for Structural Assessment (ELSA) of the JRC (European Commission  Ispra, Italy) (1997–2010). He has also been a Visiting Scholar at Stanford University (1973-1974) and Visiting Professor at the University of Colorado (1986-1987).  He developed research activity in finite element methodology, computational methods in structural dynamics and multibody dynamics. He has been a co-author of the finite element software SAMCEF and co-founding member  of Samtech SA in 1986.  He is Doctor Honoris Causa at the Technical University of Lisbon (1996) and École Centrale de Nantes (2007), and an Associate  Member of the Royal Academy of Sciences of Belgium (2000). He is the co-author of Flexible Multibody Dynamics. A  Finite Element Approach (Wiley, 2000).  Daniel Rixen holds an MSc in Aerospace Vehicle Design from the College of Aeronautics in Cranfield (UK) and received his Mechanical Engineering and Doctorate degree from the University of Liège (Belgium) supported by the Belgium National Research Fund. After having spent two years as researcher at the Center for Aerospace Structures (University of Colorado, Boulder) between 2000 and 2012 he chaired the Engineering Dynamic group at the Delft University of Technology (The Netherlands). Since 2012 he heads the Institute for Applied Mechanics at the Technische Universität München (Germany). Next to teaching, his passion comprises research on numerical and simulation methods as well as experimental techniques, involving structural and multiphysical applications in e.g. aerospace, automotive, mechatronics, biodynamics and wind energy. A recurring aspect in his investigation is the interaction between system components such as in domain decomposition for parallel computing or component synthesis in dynamic model reduction and in experimental substructuring.

Foreword xiii

Preface xv

Introduction 1

Suggested Bibliography 7

1 Analytical Dynamics of Discrete Systems 13

1.1 Principle of Virtual Work for a Particle 14

1.1.1 Nonconstrained Particle 14

1.1.2 Constrained Particle 15

1.2 Extension to a System of Particles 17

1.2.1 Virtual Work Principle for N Particles 17

1.2.2 The Kinematic Constraints 18

1.2.3 Concept of Generalized Displacements 20

1.3 Hamilton’s Principle for Conservative Systems and Lagrange Equations 23

1.3.1 Structure of Kinetic Energy and Classification of Inertia Forces 27

1.3.2 Energy Conservation in a System with Scleronomic Constraints 29

1.3.3 Classification of Generalized Forces 32

1.4 Lagrange Equations in the General Case 36

1.5 Lagrange Equations for Impulsive Loading 39

1.5.1 Impulsive Loading of a Mass Particle 39

1.5.2 Impulsive Loading for a System of Particles 42

1.6 Dynamics of Constrained Systems 44

1.7 Exercises 46

1.7.1 Solved Exercises 46

1.7.2 Selected Exercises 53

References 54

2 Undamped Vibrations of n-Degree-of-Freedom Systems 57

2.1 Linear Vibrations about an Equilibrium Configuration 59

2.1.1 Vibrations about a Stable Equilibrium Position 59

2.1.2 Free Vibrations about an Equilibrium Configuration Corresponding to Steady Motion 63

2.1.3 Vibrations about a Neutrally Stable Equilibrium Position 66

2.2 Normal Modes of Vibration 67

2.2.1 Systems with a Stable Equilibrium Configuration 68

2.2.2 Systems with a Neutrally Stable Equilibrium Position 69

2.3 Orthogonality of Vibration Eigenmodes 70

2.3.1 Orthogonality of Elastic Modes with Distinct Frequencies 70

2.3.2 Degeneracy Theorem and Generalized Orthogonality Relationships 72

2.3.3 Orthogonality Relationships Including Rigid-body Modes 75

2.4 Vector and Matrix Spectral Expansions Using Eigenmodes 76

2.5 Free Vibrations Induced by Nonzero Initial Conditions 77

2.5.1 Systems with a Stable Equilibrium Position 77

2.5.2 Systems with Neutrally Stable Equilibrium Position 82

2.6 Response to Applied Forces: Forced Harmonic Response 83

2.6.1 Harmonic Response, Impedance and Admittance Matrices 84

2.6.2 Mode Superposition and Spectral Expansion of the Admittance Matrix 84

2.6.3 Statically Exact Expansion of the Admittance Matrix 88

2.6.4 Pseudo-resonance and Resonance 89

2.6.5 Normal Excitation Modes 90

2.7 Response to Applied Forces: Response in the Time Domain 91

2.7.1 Mode Superposition and Normal Equations 91

2.7.2 Impulse Response and Time Integration of the Normal Equations 92

2.7.3 Step Response and Time Integration of the Normal Equations 94

2.7.4 Direct Integration of the Transient Response 95

2.8 Modal Approximations of Dynamic Responses 95

2.8.1 Response Truncation and Mode Displacement Method 96

2.8.2 Mode Acceleration Method 97

2.8.3 Mode Acceleration and Model Reduction on Selected Coordinates 98

2.9 Response to Support Motion 101

2.9.1 Motion Imposed to a Subset of Degrees of Freedom 101

2.9.2 Transformation to Normal Coordinates 103

2.9.3 Mechanical Impedance on Supports and Its Statically Exact Expansion 105

2.9.4 System Submitted to Global Support Acceleration 108

2.9.5 Effective Modal Masses 109

2.9.6 Method of Additional Masses 110

2.10 Variational Methods for Eigenvalue Characterization 111

2.10.1 Rayleigh Quotient 111

2.10.2 Principle of Best Approximation to a Given Eigenvalue 112

2.10.3 Recurrent Variational Procedure for Eigenvalue Analysis 113

2.10.4 Eigensolutions of Constrained Systems: General Comparison Principle or Monotonicity Principle 114

2.10.5 Courant’s Minimax Principle to Evaluate Eigenvalues Independently of Each Other 116

2.10.6 Rayleigh’s Theorem on Constraints (Eigenvalue Bracketing) 117

2.11 Conservative Rotating Systems 119

2.11.1 Energy Conservation in the Absence of External Force 119

2.11.2 Properties of the Eigensolutions of the Conservative Rotating System 119

2.11.3 State-Space Form of Equations of Motion 121

2.11.4 Eigenvalue Problem in Symmetrical Form 123

2.11.5 Orthogonality Relationships 126

2.11.6 Response to Nonzero Initial Conditions 128

2.11.7 Response to External Excitation 130

2.12 Exercises 130

2.12.1 Solved Exercises 130

2.12.2 Selected Exercises 143

References 148

3 Damped Vibrations of n-Degree-of-Freedom Systems 149

3.1 Damped Oscillations in Terms of Normal Eigensolutions of the Undamped System 151

3.1.1 Normal Equations for a Damped System 152

3.1.2 Modal Damping Assumption for Lightly Damped Structures 153

3.1.3 Constructing the Damping Matrix through Modal Expansion 158

3.2 Forced Harmonic Response 160

3.2.1 The Case of Light Viscous Damping 160

3.2.2 Hysteretic Damping 162

3.2.3 Force Appropriation Testing 164

3.2.4 The Characteristic Phase Lag Theory 170

3.3 State-Space Formulation of Damped Systems 174

3.3.1 Eigenvalue Problem and Solution of the Homogeneous Case 175

3.3.2 General Solution for the Nonhomogeneous Case 178

3.3.3 Harmonic Response 179

3.4 Experimental Methods of Modal Identification 180

3.4.1 The Least-Squares Complex Exponential Method 182

3.4.2 Discrete Fourier Transform 187

3.4.3 The Rational Fraction Polynomial Method 190

3.4.4 Estimating the Modes of the Associated Undamped System 195

3.4.5 Example: Experimental Modal Analysis of a Bellmouth 196

3.5 Exercises 199

3.5.1 Solved Exercises 199

3.6 Proposed Exercises 207

References 208

4 Continuous Systems 211

4.1 Kinematic Description of the Dynamic Behaviour of Continuous Systems: Hamilton’s Principle 213

4.1.1 Definitions 213

4.1.2 Strain Evaluation: Green’s Measure 214

4.1.3 Stress–Strain Relationships 219

4.1.4 Displacement Variational Principle 221

4.1.5 Derivation of Equations of Motion 221

4.1.6 The Linear Case and Nonlinear Effects 223

4.2 Free Vibrations of Linear Continuous Systems and Response to External Excitation 231

4.2.1 Eigenvalue Problem 231

4.2.2 Orthogonality of Eigensolutions 233

4.2.3 Response to External Excitation: Mode Superposition (Homogeneous Spatial Boundary Conditions) 234

4.2.4 Response to External Excitation: Mode Superposition (Nonhomogeneous Spatial Boundary Conditions) 237

4.2.5 Reciprocity Principle for Harmonic Motion 241

4.3 One-Dimensional Continuous Systems 243

4.3.1 The Bar in Extension 244

4.3.2 Transverse Vibrations of a Taut String 258

4.3.3 Transverse Vibration of Beams with No Shear Deflection 263

4.3.4 Transverse Vibration of Beams Including Shear Deflection 277

4.3.5 Travelling Waves in Beams 285

4.4 Bending Vibrations of Thin Plates 290

4.4.1 Kinematic Assumptions 290

4.4.2 Strain Expressions 291

4.4.3 Stress–Strain Relationships 292

4.4.4 Definition of Curvatures 293

4.4.5 Moment–Curvature Relationships 293

4.4.6 Frame Transformation for Bending Moments 295

4.4.7 Computation of Strain Energy 295

4.4.8 Expression of Hamilton’s Principle 296

4.4.9 Plate Equations of Motion Derived from Hamilton’s Principle 298

4.4.10 Influence of In-Plane Initial Stresses on Plate Vibration 303

4.4.11 Free Vibrations of the Rectangular Plate 305

4.4.12 Vibrations of Circular Plates 308

4.4.13 An Application of Plate Vibration: The Ultrasonic Wave Motor 311

4.5 Wave Propagation in a Homogeneous Elastic Medium 315

4.5.1 The Navier Equations in Linear Dynamic Analysis 316

4.5.2 Plane Elastic Waves 318

4.5.3 Surface Waves 320

4.6 Solved Exercises 327

4.7 Proposed Exercises 328

References 333

5 Approximation of Continuous Systems by Displacement Methods 335

5.1 The Rayleigh–Ritz Method 339

5.1.1 Choice of Approximation Functions 339

5.1.2 Discretization of the Displacement Variational Principle 340

5.1.3 Computation of Eigensolutions by the Rayleigh–Ritz Method 342

5.1.4 Computation of the Response to External Loading by the Rayleigh–Ritz Method 345

5.1.5 The Case of Prestressed Structures 345

5.2 Applications of the Rayleigh–Ritz Method to Continuous Systems 346

5.2.1 The Clamped–Free Uniform Bar 347

5.2.2 The Clamped–Free Uniform Beam 350

5.2.3 The Uniform Rectangular Plate 357

5.3 The Finite Element Method 362

5.3.1 The Bar in Extension 364

5.3.2 Truss Frames 371

5.3.3 Beams in Bending without Shear Deflection 376

5.3.4 Three-Dimensional Beam Element without Shear Deflection 386

5.3.5 Beams in Bending with Shear Deformation 392

5.4 Exercises 399

5.4.1 Solved Exercises 399

5.4.2 Selected Exercises 406

References 412

6 Solution Methods for the Eigenvalue Problem 415

6.1 General considerations 419

6.1.1 Classification of Solution Methods 420

6.1.2 Criteria for Selecting the Solution Method 420

6.1.3 Accuracy of Eigensolutions and Stopping Criteria 423

6.2 Dynamical and Symmetric Iteration Matrices 425

6.3 Computing the Determinant: Sturm Sequences 426

6.4 Matrix Transformation Methods 430

6.4.1 Reduction to a Diagonal Form: Jacobi’s Method 430

6.4.2 Reduction to a Tridiagonal Form: Householder’s Method 434

6.5 Iteration on Eigenvectors: The Power Algorithm 436

6.5.1 Computing the Fundamental Eigensolution 437

6.5.2 Determining Higher Modes: Orthogonal Deflation 441

6.5.3 Inverse Iteration Form of the Power Method 443

6.6 Solution Methods for a Linear Set of Equations 444

6.6.1 Nonsingular Linear Systems 445

6.6.2 Singular Systems: Nullspace, Solutions and Generalized Inverse 453

6.6.3 Singular Matrix and Nullspace 453

6.6.4 Solution of Singular Systems 454

6.6.5 A Family of Generalized Inverses 456

6.6.6 Solution by Generalized Inverses and Finding the Nullspace N 457

6.6.7 Taking into Account Linear Constraints 459

6.7 Practical Aspects of Inverse Iteration Methods 460

6.7.1 Inverse Iteration in Presence of Rigid Body Modes 460

6.7.2 Spectral Shifting 463

6.8 Subspace Construction Methods 464

6.8.1 The Subspace Iteration Method 464

6.8.2 The Lanczos Method 468

6.9 Dynamic Reduction and Substructuring 479

6.9.1 Static Condensation (Guyan–Irons Reduction) 481

6.9.2 Craig and Bampton’s Substructuring Method 484

6.9.3 McNeal’s Hybrid Synthesis Method 487

6.9.4 Rubin’s Substructuring Method 488

6.10 Error Bounds to Eigenvalues 488

6.10.1 Rayleigh and Schwarz Quotients 489

6.10.2 Eigenvalue Bracketing 491

6.10.3 Temple–Kato Bounds 492

6.11 Sensitivity of Eigensolutions, Model Updating and Dynamic Optimization 498

6.11.1 Sensitivity of the Structural Model to Physical Parameters 501

6.11.2 Sensitivity of Eigenfrequencies 502

6.11.3 Sensitivity of Free Vibration Modes 502

6.11.4 Modal Representation of Eigenmode Sensitivity 504

6.12 Exercises 504

6.12.1 Solved Exercises 504

6.12.2 Selected Exercises 505

References 508

7 Direct Time-Integration Methods 511

7.1 Linear Multistep Integration Methods 513

7.1.1 Development of Linear Multistep Integration Formulas 514

7.1.2 One-Step Methods 515

7.1.3 Two-Step Second-Order Methods 516

7.1.4 Several-Step Methods 517

7.1.5 Numerical Observation of Stability and Accuracy Properties of Simple Time Integration Formulas 517

7.1.6 Stability Analysis of Multistep Methods 518

7.2 One-Step Formulas for Second-Order Systems: Newmark’s Family 522

7.2.1 The Newmark Method 522

7.2.2 Consistency of Newmark’s Method 525

7.2.3 First-Order Form of Newmark’s Operator – Amplification Matrix 525

7.2.4 Matrix Norm and Spectral Radius 527

7.2.5 Stability of an Integration Method – Spectral Stability 528

7.2.6 Spectral Stability of the Newmark Method 530

7.2.7 Oscillatory Behaviour of the Newmark Response 533

7.2.8 Measures of Accuracy: Numerical Dissipation and Dispersion 535

7.3 Equilibrium Averaging Methods 539

7.3.1 Amplification Matrix 540

7.3.2 Finite Difference Form of the Time-Marching Formula 541

7.3.3 Accuracy Analysis of Equilibrium Averaging Methods 542

7.3.4 Stability Domain of Equilibrium Averaging Methods 543

7.3.5 Oscillatory Behaviour of the Solution 544

7.3.6 Particular Forms of Equilibrium Averaging 544

7.4 Energy Conservation 550

7.4.1 Application: The Clamped-Free Bar Excited by an end Force 552

7.5 Explicit Time Integration Using the Central Difference Algorithm 556

7.5.1 Algorithm in Terms of Velocities 556

7.5.2 Application Example: The Clamped-Free Bar Excited by an End Load 559

7.5.3 Restitution of the Exact Solution by the Central Difference Method 561

7.6 The Nonlinear Case 564

7.6.1 The Explicit Case 564

7.6.2 The Implicit Case 565

7.6.3 Time Step Size Control 571

7.7 Exercises 573

References 575

Index 577