Mechanical Vibrations

Name: Mechanical Vibrations | Theory and Application to Structural Dynamics
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Theory and Application to Structural Dynamics

Michel Geradin Daniel J. Rixen(Autor*in)

Wiley (Verlag)

3. Auflage

Erschienen am 24. Dezember 2014

616 Seiten

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Michel Géradin holds an Engineering Degree inPhysics and a PhD from ULg (University of Liège, Belgium).Successively he has been a research fellow from the BelgianFNRS (1968-1979), Professor of Structural Dynamics atULg (1979-2010) and Unit Head of the European Laboratory forStructural Assessment (ELSA) of the JRC (European Commission Ispra, Italy) (1997-2010). He has also been a VisitingScholar at Stanford University (1973-1974) and Visiting Professorat the University of Colorado (1986-1987).
He developed research activity in finite element methodology,computational methods in structural dynamics and multibodydynamics. He has been a co-author of the finite element softwareSAMCEF and co-founding member of Samtech SA in1986.
He is Doctor Honoris Causa at the Technical University of Lisbon(1996) and École Centrale de Nantes (2007), and anAssociate Member of the Royal Academy of Sciences of Belgium(2000).
He is the co-author of Flexible Multibody Dynamics. A FiniteElement Approach (Wiley, 2000).
Daniel Rixen holds an MSc in Aerospace Vehicle Designfrom the College of Aeronautics in Cranfield (UK) and received hisMechanical Engineering and Doctorate degree from the University ofLiège (Belgium) supported by the Belgium National ResearchFund. After having spent two years as researcher at the Center forAerospace Structures (University of Colorado, Boulder) between 2000and 2012 he chaired the Engineering Dynamic group at the DelftUniversity of Technology (The Netherlands). Since 2012 he heads theInstitute for Applied Mechanics at the Technische UniversitätMünchen (Germany). Next to teaching, his passion comprisesresearch on numerical and simulation methods as well asexperimental techniques, involving structural and multiphysicalapplications in e.g. aerospace, automotive, mechatronics,biodynamics and wind energy. A recurring aspect in hisinvestigation is the interaction between system components such asin domain decomposition for parallel computing or componentsynthesis in dynamic model reduction and in experimentalsubstructuring.

Inhalt

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

Introduction

We owe to Lord Rayleigh the formulation of the principles relative to the theory of vibration such as they are applied and taught nowadays. In his remarkable treatise entitled Theory of Sound and published in 1877 he introduced the fundamental concept of oscillation of a linear system about an equilibrium configuration and showed the existence of vibration eigenmodes and eigenfrequencies for discrete as well as for continuous systems. His work remains valuable in many ways, even though he was concerned with acoustics rather than with structural mechanics.

Because of their constant aim to minimize the weight of flying structures, the pioneers of aeronautics were the first structural designers who needed to get vibration and structural dynamic problems under control. From the twenties onwards, aeronautical engineers had to admit the importance of the mechanics of vibration for predicting the aeroelastic behaviour of aircraft. Since then, the theory of vibration has become a significant subject in aeronautical studies. During the next forty years, they had to limit the scope of their analysis and apply methods that could be handled by the available computational means: the structural models used were either analytical or resulted from a description of the structure in terms of a small number of degrees of freedom by application of transfer or Rayleigh-Ritz techniques.

The appearance and the progressive popularization of computing hardware since 1960 have led to a reconsideration of the entire field of analysis methods for structural dynamics: the traditional methods have been replaced by matrix ones arising from the discretization of variational expressions. In particular, the tremendous advances in the finite element method for setting up structural models gave rise to the development of new computational methods to allow design engineers to cope with always increasing problem sizes.

Today, the elaboration of efficient computational models for the analysis of the dynamic behaviour of structures has become a routine task. To give an example, Figure 1 illustrates the computational prediction of the vibration modes of a stator section of an aircraft engine. The fineness of the finite element model has been adapted in this case for the needs of the associated stress analysis, the latter requiring a level of detail that is not really needed for a modal analysis. The eigenmode represented is a 3-diameter mode exhibiting a global deformation of the structure. What makes the modal analysis of such a structure very difficult is the high level of cyclic symmetry (resulting from the number of stator blades) which is responsible for the appearance of a high number of nearly equal eigenvalues.

Figure 1 Finite element model of a stator section of aircraft engine. Source: Reproduced with permission from Techspace Aero-SAFRAN Group.

Development of computing, acquisition and sensing hardware has led to a similar revolution in the field of experimental techniques for identification of vibrational characteristics of structures. For more than thirty years, experimental modal analysis techniques have been developed which are based either on force appropriation or on arbitrary excitation.

The methods for dynamic analysis, whether they are numerical or experimental, have now taken an important place everywhere in engineering. If they were rapidly accepted in disciplines such as civil engineering, mechanical design, nuclear engineering and automotive production where they are obviously needed, they have now become equally important in the design of any manufactured good, from the micro-electromechanical device to the large wind turbine.

From its origin in the early sixties, the aerospace department of the University of Liège (Belgium) has specialized mainly in structural mechanics in its education programme. This book results from more than twenty years of lecturing on the theory of vibration to the students of this branch. It is also based on experience gathered within the University of Liège's Laboratory for Aerospace Techniques in the development of computational algorithms designed for the dynamic analysis of structures by the finite element method and implemented in the structural analysis code the team of the laboratory has developed since 1965, the SAMCEFT software.1

The content of the book is based on the lecture notes developed over the years by the first author and later formatted and augmented by one of his former students (the second author). This work reflects the teaching and research experience of both authors. In addition to his academic activity at the University of Liège, the first author has also spent several years as head of the European Laboratory for Safety Assessment at the Joint Research Centre in Ispra (Italy). The second author has accumulated until 2012 lecturing and research experience at the Delft Technical University (The Netherlands) and is currently pursuing his career at the Technische Universität München (Germany). The book has been adopted internationally as course reference in several universities.

Due to its very objective, the book has a slightly hybrid character: the concepts of vibration theory are presented mainly with the intention of applying them to dynamic analysis of structures and significant attention is paid to the corresponding methods. Even though the foundations of analytical mechanics are reviewed, a preliminary acquaintance with this subject is necessary. A good knowledge of matrix algebra and theory of complex numbers, calculus, structural mechanics and numerical analysis for linear systems is required. It is also assumed that the reader is familiar with the theory of the single-degree-of-freedom oscillator. However, the presentation of the finite element method is deliberately made simple since its study requires a course of its own. Finally, the very important fields of nonlinear vibration and random vibration have been intentionally omitted in the present text since they are highly specialized subjects.

What is new in this third edition?

Although the overall structure of the book, its organization into individual chapters and the main topics addressed, remain unchanged, this new version is the result of important revision work to achieve major improvements.

Regarding the theoretical content itself, the main changes with respect to the previous edition are the following:

The response of an either discrete or continuous system has been the object of deep rethinking and turned out to be a thread towards the important concepts of dynamic reduction and substructuring. The latter are explained and developed, starting from the observation that the response of a part from the overall system is the result either from an excitation of its support, or from the application of a set of loads at selected points of the structure. Such duality can be exploited in at least two ways. On the one hand, it allows a system description in terms of the classical concepts of mechanical impedance or admittance. On the other hand, it naturally leads to the concept of dynamic substructuring based on an expansion of the response in terms of the spectral content of the impedance and admittance relationships.
Experimental modal analysis is an essential ingredient in structural dynamics since it allows to confirm by experiment the structural properties predicted through numerical modelling. Therefore it was felt necessary to include in this new version of the book the essentials of signal processing and identification techniques that allow us to extract the spectral properties of a linear structure from measured dynamic responses.
In the same spirit, the concept of eigensolution sensitivity to physical parameters has been further detailed since it is the basis for the development of appropriate numerical tools for improving the numerical model of a real dynamic system.
The considerable evolution of the size of the structural systems to be considered for eigenvalue extraction and transient dynamic analysis in the context of large engineering projects had to be reflected and addressed properly. Models reaching the size of several millions of degrees of freedom (such as the one displayed on Figure 1) are now common practice. The efficiency of the eigenvalue solvers (such as the Lanczos method) and implicit time integrators (based on the Newmark family) depends for one part on the tuning of the algorithms themselves, but perhaps even more on the performance of the linear solvers that are used at each solution step. Therefore it was felt necessary to cover in a deeper manner the topic of linear solvers, introducing not only the principle of the algorithms but also their implementation taking into accountthe sparse character of the large sets of equations generated by finite element discretization. Much attention is also brought to the case of singular systems since they frequently occur in the context of structural dynamics.
The link that was made in the previous edition between vibration and wave propagation did not allow the reader to easily grasp the physical nature of the wave propagation phenomena that can occur in a continuous medium. The discussion of the fundamental cases of wave propagation in solids (both in one-dimensional and three-dimensional media) has thus been reviewed and better illustrated in order to improve the didactics of the presentation.
The presentation of the finite element method has still been limited to one-dimensional structures (bars, beams) since the main objective of the book is not to go deeply into finite...

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