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 xiiiPreface xvIntroduction 1Suggested Bibliography 71 Analytical Dynamics of Discrete Systems 131.1 Principle of Virtual Work for a Particle 141.1.1 Nonconstrained Particle 141.1.2 Constrained Particle 151.2 Extension to a System of Particles 171.2.1 Virtual Work Principle for N Particles 171.2.2 The Kinematic Constraints 181.2.3 Concept of Generalized Displacements 201.3 Hamilton’s Principle for Conservative Systems and Lagrange Equations 231.3.1 Structure of Kinetic Energy and Classification of Inertia Forces 271.3.2 Energy Conservation in a System with Scleronomic Constraints 291.3.3 Classification of Generalized Forces 321.4 Lagrange Equations in the General Case 361.5 Lagrange Equations for Impulsive Loading 391.5.1 Impulsive Loading of a Mass Particle 391.5.2 Impulsive Loading for a System of Particles 421.6 Dynamics of Constrained Systems 441.7 Exercises 461.7.1 Solved Exercises 461.7.2 Selected Exercises 53References 542 Undamped Vibrations of n-Degree-of-Freedom Systems 572.1 Linear Vibrations about an Equilibrium Configuration 592.1.1 Vibrations about a Stable Equilibrium Position 592.1.2 Free Vibrations about an Equilibrium Configuration Corresponding to Steady Motion 632.1.3 Vibrations about a Neutrally Stable Equilibrium Position 662.2 Normal Modes of Vibration 672.2.1 Systems with a Stable Equilibrium Configuration 682.2.2 Systems with a Neutrally Stable Equilibrium Position 692.3 Orthogonality of Vibration Eigenmodes 702.3.1 Orthogonality of Elastic Modes with Distinct Frequencies 702.3.2 Degeneracy Theorem and Generalized Orthogonality Relationships 722.3.3 Orthogonality Relationships Including Rigid-body Modes 752.4 Vector and Matrix Spectral Expansions Using Eigenmodes 762.5 Free Vibrations Induced by Nonzero Initial Conditions 772.5.1 Systems with a Stable Equilibrium Position 772.5.2 Systems with Neutrally Stable Equilibrium Position 822.6 Response to Applied Forces: Forced Harmonic Response 832.6.1 Harmonic Response, Impedance and Admittance Matrices 842.6.2 Mode Superposition and Spectral Expansion of the Admittance Matrix 842.6.3 Statically Exact Expansion of the Admittance Matrix 882.6.4 Pseudo-resonance and Resonance 892.6.5 Normal Excitation Modes 902.7 Response to Applied Forces: Response in the Time Domain 912.7.1 Mode Superposition and Normal Equations 912.7.2 Impulse Response and Time Integration of the Normal Equations 922.7.3 Step Response and Time Integration of the Normal Equations 942.7.4 Direct Integration of the Transient Response 952.8 Modal Approximations of Dynamic Responses 952.8.1 Response Truncation and Mode Displacement Method 962.8.2 Mode Acceleration Method 972.8.3 Mode Acceleration and Model Reduction on Selected Coordinates 982.9 Response to Support Motion 1012.9.1 Motion Imposed to a Subset of Degrees of Freedom 1012.9.2 Transformation to Normal Coordinates 1032.9.3 Mechanical Impedance on Supports and Its Statically Exact Expansion 1052.9.4 System Submitted to Global Support Acceleration 1082.9.5 Effective Modal Masses 1092.9.6 Method of Additional Masses 1102.10 Variational Methods for Eigenvalue Characterization 1112.10.1 Rayleigh Quotient 1112.10.2 Principle of Best Approximation to a Given Eigenvalue 1122.10.3 Recurrent Variational Procedure for Eigenvalue Analysis 1132.10.4 Eigensolutions of Constrained Systems: General Comparison Principle or Monotonicity Principle 1142.10.5 Courant’s Minimax Principle to Evaluate Eigenvalues Independently of Each Other 1162.10.6 Rayleigh’s Theorem on Constraints (Eigenvalue Bracketing) 1172.11 Conservative Rotating Systems 1192.11.1 Energy Conservation in the Absence of External Force 1192.11.2 Properties of the Eigensolutions of the Conservative Rotating System 1192.11.3 State-Space Form of Equations of Motion 1212.11.4 Eigenvalue Problem in Symmetrical Form 1232.11.5 Orthogonality Relationships 1262.11.6 Response to Nonzero Initial Conditions 1282.11.7 Response to External Excitation 1302.12 Exercises 1302.12.1 Solved Exercises 1302.12.2 Selected Exercises 143References 1483 Damped Vibrations of n-Degree-of-Freedom Systems 1493.1 Damped Oscillations in Terms of Normal Eigensolutions of the Undamped System 1513.1.1 Normal Equations for a Damped System 1523.1.2 Modal Damping Assumption for Lightly Damped Structures 1533.1.3 Constructing the Damping Matrix through Modal Expansion 1583.2 Forced Harmonic Response 1603.2.1 The Case of Light Viscous Damping 1603.2.2 Hysteretic Damping 1623.2.3 Force Appropriation Testing 1643.2.4 The Characteristic Phase Lag Theory 1703.3 State-Space Formulation of Damped Systems 1743.3.1 Eigenvalue Problem and Solution of the Homogeneous Case 1753.3.2 General Solution for the Nonhomogeneous Case 1783.3.3 Harmonic Response 1793.4 Experimental Methods of Modal Identification 1803.4.1 The Least-Squares Complex Exponential Method 1823.4.2 Discrete Fourier Transform 1873.4.3 The Rational Fraction Polynomial Method 1903.4.4 Estimating the Modes of the Associated Undamped System 1953.4.5 Example: Experimental Modal Analysis of a Bellmouth 1963.5 Exercises 1993.5.1 Solved Exercises 1993.6 Proposed Exercises 207References 2084 Continuous Systems 2114.1 Kinematic Description of the Dynamic Behaviour of Continuous Systems: Hamilton’s Principle 2134.1.1 Definitions 2134.1.2 Strain Evaluation: Green’s Measure 2144.1.3 Stress–Strain Relationships 2194.1.4 Displacement Variational Principle 2214.1.5 Derivation of Equations of Motion 2214.1.6 The Linear Case and Nonlinear Effects 2234.2 Free Vibrations of Linear Continuous Systems and Response to External Excitation 2314.2.1 Eigenvalue Problem 2314.2.2 Orthogonality of Eigensolutions 2334.2.3 Response to External Excitation: Mode Superposition (Homogeneous Spatial Boundary Conditions) 2344.2.4 Response to External Excitation: Mode Superposition (Nonhomogeneous Spatial Boundary Conditions) 2374.2.5 Reciprocity Principle for Harmonic Motion 2414.3 One-Dimensional Continuous Systems 2434.3.1 The Bar in Extension 2444.3.2 Transverse Vibrations of a Taut String 2584.3.3 Transverse Vibration of Beams with No Shear Deflection 2634.3.4 Transverse Vibration of Beams Including Shear Deflection 2774.3.5 Travelling Waves in Beams 2854.4 Bending Vibrations of Thin Plates 2904.4.1 Kinematic Assumptions 2904.4.2 Strain Expressions 2914.4.3 Stress–Strain Relationships 2924.4.4 Definition of Curvatures 2934.4.5 Moment–Curvature Relationships 2934.4.6 Frame Transformation for Bending Moments 2954.4.7 Computation of Strain Energy 2954.4.8 Expression of Hamilton’s Principle 2964.4.9 Plate Equations of Motion Derived from Hamilton’s Principle 2984.4.10 Influence of In-Plane Initial Stresses on Plate Vibration 3034.4.11 Free Vibrations of the Rectangular Plate 3054.4.12 Vibrations of Circular Plates 3084.4.13 An Application of Plate Vibration: The Ultrasonic Wave Motor 3114.5 Wave Propagation in a Homogeneous Elastic Medium 3154.5.1 The Navier Equations in Linear Dynamic Analysis 3164.5.2 Plane Elastic Waves 3184.5.3 Surface Waves 3204.6 Solved Exercises 3274.7 Proposed Exercises 328References 3335 Approximation of Continuous Systems by Displacement Methods 3355.1 The Rayleigh–Ritz Method 3395.1.1 Choice of Approximation Functions 3395.1.2 Discretization of the Displacement Variational Principle 3405.1.3 Computation of Eigensolutions by the Rayleigh–Ritz Method 3425.1.4 Computation of the Response to External Loading by the Rayleigh–Ritz Method 3455.1.5 The Case of Prestressed Structures 3455.2 Applications of the Rayleigh–Ritz Method to Continuous Systems 3465.2.1 The Clamped–Free Uniform Bar 3475.2.2 The Clamped–Free Uniform Beam 3505.2.3 The Uniform Rectangular Plate 3575.3 The Finite Element Method 3625.3.1 The Bar in Extension 3645.3.2 Truss Frames 3715.3.3 Beams in Bending without Shear Deflection 3765.3.4 Three-Dimensional Beam Element without Shear Deflection 3865.3.5 Beams in Bending with Shear Deformation 3925.4 Exercises 3995.4.1 Solved Exercises 3995.4.2 Selected Exercises 406References 4126 Solution Methods for the Eigenvalue Problem 4156.1 General considerations 4196.1.1 Classification of Solution Methods 4206.1.2 Criteria for Selecting the Solution Method 4206.1.3 Accuracy of Eigensolutions and Stopping Criteria 4236.2 Dynamical and Symmetric Iteration Matrices 4256.3 Computing the Determinant: Sturm Sequences 4266.4 Matrix Transformation Methods 4306.4.1 Reduction to a Diagonal Form: Jacobi’s Method 4306.4.2 Reduction to a Tridiagonal Form: Householder’s Method 4346.5 Iteration on Eigenvectors: The Power Algorithm 4366.5.1 Computing the Fundamental Eigensolution 4376.5.2 Determining Higher Modes: Orthogonal Deflation 4416.5.3 Inverse Iteration Form of the Power Method 4436.6 Solution Methods for a Linear Set of Equations 4446.6.1 Nonsingular Linear Systems 4456.6.2 Singular Systems: Nullspace, Solutions and Generalized Inverse 4536.6.3 Singular Matrix and Nullspace 4536.6.4 Solution of Singular Systems 4546.6.5 A Family of Generalized Inverses 4566.6.6 Solution by Generalized Inverses and Finding the Nullspace N 4576.6.7 Taking into Account Linear Constraints 4596.7 Practical Aspects of Inverse Iteration Methods 4606.7.1 Inverse Iteration in Presence of Rigid Body Modes 4606.7.2 Spectral Shifting 4636.8 Subspace Construction Methods 4646.8.1 The Subspace Iteration Method 4646.8.2 The Lanczos Method 4686.9 Dynamic Reduction and Substructuring 4796.9.1 Static Condensation (Guyan–Irons Reduction) 4816.9.2 Craig and Bampton’s Substructuring Method 4846.9.3 McNeal’s Hybrid Synthesis Method 4876.9.4 Rubin’s Substructuring Method 4886.10 Error Bounds to Eigenvalues 4886.10.1 Rayleigh and Schwarz Quotients 4896.10.2 Eigenvalue Bracketing 4916.10.3 Temple–Kato Bounds 4926.11 Sensitivity of Eigensolutions, Model Updating and Dynamic Optimization 4986.11.1 Sensitivity of the Structural Model to Physical Parameters 5016.11.2 Sensitivity of Eigenfrequencies 5026.11.3 Sensitivity of Free Vibration Modes 5026.11.4 Modal Representation of Eigenmode Sensitivity 5046.12 Exercises 5046.12.1 Solved Exercises 5046.12.2 Selected Exercises 505References 5087 Direct Time-Integration Methods 5117.1 Linear Multistep Integration Methods 5137.1.1 Development of Linear Multistep Integration Formulas 5147.1.2 One-Step Methods 5157.1.3 Two-Step Second-Order Methods 5167.1.4 Several-Step Methods 5177.1.5 Numerical Observation of Stability and Accuracy Properties of Simple Time Integration Formulas 5177.1.6 Stability Analysis of Multistep Methods 5187.2 One-Step Formulas for Second-Order Systems: Newmark’s Family 5227.2.1 The Newmark Method 5227.2.2 Consistency of Newmark’s Method 5257.2.3 First-Order Form of Newmark’s Operator – Amplification Matrix 5257.2.4 Matrix Norm and Spectral Radius 5277.2.5 Stability of an Integration Method – Spectral Stability 5287.2.6 Spectral Stability of the Newmark Method 5307.2.7 Oscillatory Behaviour of the Newmark Response 5337.2.8 Measures of Accuracy: Numerical Dissipation and Dispersion 5357.3 Equilibrium Averaging Methods 5397.3.1 Amplification Matrix 5407.3.2 Finite Difference Form of the Time-Marching Formula 5417.3.3 Accuracy Analysis of Equilibrium Averaging Methods 5427.3.4 Stability Domain of Equilibrium Averaging Methods 5437.3.5 Oscillatory Behaviour of the Solution 5447.3.6 Particular Forms of Equilibrium Averaging 5447.4 Energy Conservation 5507.4.1 Application: The Clamped-Free Bar Excited by an end Force 5527.5 Explicit Time Integration Using the Central Difference Algorithm 5567.5.1 Algorithm in Terms of Velocities 5567.5.2 Application Example: The Clamped-Free Bar Excited by an End Load 5597.5.3 Restitution of the Exact Solution by the Central Difference Method 5617.6 The Nonlinear Case 5647.6.1 The Explicit Case 5647.6.2 The Implicit Case 5657.6.3 Time Step Size Control 5717.7 Exercises 573References 575Index 577