VIBRATIONS AND STABILITY : advanced theory, analysis, and tools, 3rd edition

This book ties together classical and modern topics of advanced vibration analysis in an interesting and lucid manner. It is intended for advanced students, university teachers, and researchers in mechanical/structural engineering dynamics, and professional engineers aiming at strengthening their theoretical foundation. Assuming background knowledge in elementary vibrations, it provides a set of useful tools for understanding and analyzing the more complex dynamical phenomena that can be met in engineering and scientific practice. Summarizing first basic linear vibration theory, it progresses over various types of nonlinearity and nonlinear interaction to bifurcation analysis, chaotic vibrations, and special high-frequency effects. It trains the student to analyze simple models, recognize nonlinear phenomena, and work with advanced tools such as perturbation analysis and bifurcation analysis. Focusing attention on a limited number of simple, generic models, the book covers the qualitative behavior of a wide variety of mechanical/structural systems; the key examples dealt with can be fabricated as simple physical models for classroom demonstrations. Explaining theory in terms of relevant examples from real systems, this book is user-friendly and meets the increasing interest in, and significance of, nonlinear dynamics in mechanical/structural engineering. This 3rd edition adds many new topics, sections, figures, exercise problems, literature references, and other updates; it is backed up by a separately available booklet with worked out solutions to exercise problems. Preface Contents About the Author Notations 1 Vibration Basics 1.1 Introduction 1.2 Single Degree of Freedom Systems 1.2.1 Undamped Free Vibrations 1.2.2 Damped Free Vibrations 1.2.3 Harmonic Forcing 1.2.4 Arbitrary Forcing 1.3 Multiple Degree of Freedom Systems 1.3.1 Equations of Motion 1.3.2 Undamped Free Vibrations 1.3.3 Orthogonality of Modes 1.3.4 Damped Free Vibrations 1.3.5 Harmonically Forced Vibrations, No Damping 1.3.6 Harmonically Forced Vibrations, Damping Included 1.3.7 General Periodic Forcing 1.3.8 Arbitrary Forcing, Transients 1.4 Continuous Systems 1.4.1 Equations of Motion 1.4.2 Undamped Free Vibrations 1.4.3 Orthogonality of Modes 1.4.4 Normal Coordinates 1.4.5 Forced Vibrations, No Damping 1.4.6 Forced Vibrations, Damping Included 1.4.7 Complex-Valued Eigenvalues and Mode Shapes 1.4.8 Rayleigh’s Method 1.4.9 Ritz Method 1.5 Energy Methods for Setting up Equations of Motion 1.5.1 Lagrange’s Equations 1.5.2 Hamilton’s Principle 1.5.3 From PDEs to ODEs: Mode Shape Expansion 1.5.4 Using Lagrange’s Equations with Continuous Systems 1.6 Nondimensionalized Equations of Motion 1.7 Damping: Types, Measures, Parameter Relations 1.7.1 Damping in Equations of Motion 1.7.2 Damping Models 1.7.3 Damping Measures and Their Relations 1.7.4 Damping Influence on Free Vibration Decay 1.7.5 Damping Influence on Resonance Buildup 1.7.6 Estimating Mass/Stiffness-Proportional Damping Constants 1.7.7 Mass/Stiffness Damping Proportionality Constants for Beams 1.8 The Stiffness and Flexibility Methods for Deriving Equations of Motion 1.8.1 Common Basis 1.8.2 The Flexibility Method 1.8.3 The Stiffness Method 1.8.4 Maxwell’s Reciprocity Theorem 1.9 Classification of Forces and Systems 1.9.1 Force Classification 1.9.2 System Classification 1.9.3 Stability Assessment 1.10 Problems 2 Eigenvalue Problems of Vibrations and Stability 2.1 Introduction 2.2 The Algebraic EVP 2.2.1 Mathematical Form 2.2.2 Properties of Eigenvalues and Eigenvectors 2.2.3 Methods of Solution 2.3 The Differential EVP 2.3.1 Mathematical Form 2.4 Stability-Related EVPs 2.4.1 The Clamped-Hinged Euler Column 2.4.2 The Paradox of Follower-Loading 2.4.3 Buckling by Gravity 2.5 Vibration-Related EVPs 2.5.1 Axial Vibrations of Straight Rods 2.5.2 Flexural Vibrations of Beams 2.6 Concepts of Differential EVPs 2.6.1 Multiplicity 2.6.2 Boundary Conditions: Essential or Natural/Suppressible 2.6.3 Function Classes: Eigen-, Test-, and Admissible Functions 2.6.4 Adjointness 2.6.5 Definiteness 2.6.6 Orthogonality 2.6.7 Three Classes of EVPs 2.6.8 The Rayleigh Quotient 2.7 Properties of Eigenvalues and Eigenfunctions 2.7.1 Real-Valueness of Eigenvalues 2.7.2 Sign of the Eigenvalues 2.7.3 Orthogonality of Eigenfunctions 2.7.4 Minimum Properties of the Eigenvalues 2.7.5 The Comparison Theorem 2.7.6 The Inclusion Theorem for One-Term EVPs 2.8 Methods of Solution 2.8.1 Closed-Form Solutions 2.8.2 The Method of Eigenfunction Iteration 2.8.3 The Rayleigh–Ritz Method 2.8.4 The Finite Difference Method 2.8.5 Collocation 2.8.6 Composite EVPs: Dunkerley’s and Southwell’s Formulas 2.8.7 The Rayleigh Quotient Estimate and Its Accuracy 2.8.8 Other Methods 2.9 Problems 3 Nonlinear Vibrations: Classical Local Theory 3.1 Introduction 3.2 Sources of Nonlinearity 3.2.1 Geometrical Nonlinearities 3.2.2 Material Nonlinearities 3.2.3 Nonlinear Body Forces 3.2.4 Physical Configuration Nonlinearities 3.3 Main Example: Pendulum with an Oscillating Support 3.3.1 Equation of Motion 3.4 Qualitative Analysis of the Unforced Response 3.4.1 Recasting the Equations into First-Order Form 3.4.2 The Phase Plane 3.4.3 Singular Points 3.4.4 Stability of Singular Points 3.4.5 On the Behavior of Orbits Near Singular Points 3.5 Quantitative Analysis 3.5.1 Approximate Methods 3.5.2 On the “Small Parameter” in Perturbation Analysis 3.5.3 The Straightforward Expansion 3.5.4 The Method of Multiple Scales 3.5.5 The Method of Averaging 3.5.6 The Method of Harmonic Balance 3.6 The Forced Response – Multiple Scales Analysis 3.6.1 Posing the Problem 3.6.2 Perturbation Equations 3.6.3 The Non-resonant Case 3.6.4 The Near-Resonant Case 3.6.5 Stability of Stationary Solutions 3.6.6 Discussing Stationary Responses 3.7 Externally Excited Duffing Systems 3.7.1 Two Physical Examples 3.7.2 Primary Resonance, Weak Excitations 3.7.3 Non-resonant Hard Excitations 3.7.4 Obtaining Forced Responses by Averaging 3.7.5 Multiple Scales Analysis with Strong Nonlinearity 3.8 Two More Classical Nonlinear Oscillators 3.8.1 The Van Der Pol Oscillator 3.8.2 The Rayleigh Oscillator 3.9 Vibro-Impact Analysis Using Discontinuous Transformations 3.9.1 The Unfolding Discontinuous Transformation: Basic Idea 3.9.2 Averaging for Vibro-Impact Systems: General Procedure 3.9.3 Example 1: Damped Harmonic Oscillator Impacting a Stop 3.9.4 Example 2: Mass in a Clearance 3.9.5 Example 3: Self-excited Friction Oscillator with a One-Sided Stop 3.9.6 Example 4: Self-excited Friction Oscillator in a Clearance 3.9.7 Second-Order Analysis 3.10 Summing Up 3.11 Problems 4 Nonlinear Multiple-DOF Systems: Local Analysis 4.1 Introduction 4.2 The Autoparametric Vibration Absorber 4.2.1 The System 4.2.2 First-Order Approximate Response 4.2.3 Frequency and Force Responses 4.2.4 Concluding Remarks on the Vibration Absorber 4.3 Nonlinear Mode-Coupling of Non-shallow Arches 4.3.1 The Model 4.3.2 Linear Response and Stability 4.3.3 Nonlinear Response and Stability 4.4 Other Systems Possessing Internal Resonance 4.5 The Follower-Loaded Double Pendulum 4.5.1 The Model 4.5.2 The Zero Solution and Its Stability 4.5.3 Periodic Solutions 4.5.4 Non-periodic and Non-zero Static Solutions 4.5.5 Summing Up 4.6 Pendulum with a Sliding Disk 4.6.1 Introduction 4.6.2 The System 4.6.3 Equations of Motion 4.6.4 Inspecting the Equations of Motion 4.6.5 Seeking Quasi-statical Equilibriums by Averaging 4.7 String with a Sliding Point Mass 4.7.1 Model System and Equations of Motion 4.7.2 Illustration of System Behavior 4.7.3 Response to Near-Resonant Base Excitation 4.7.4 Response to Slow Frequency-Sweeps 4.7.5 Response to Near-Resonant Axial Excitation 4.7.6 Non-trivial Effects of Rotary Inertia 4.7.7 Summing Up 4.8 Vibration-Induced Fluid Flow in Pipes 4.9 Problems 5 Bifurcation Analysis 5.1 Introduction 5.2 Systems, Bifurcations, and Bifurcation Conditions 5.2.1 Systems 5.2.2 Bifurcations 5.2.3 Bifurcation Conditions: Structural Instability 5.3 Codimension One Bifurcations of Equilibriums 5.3.1 The Pitchfork Bifurcation 5.3.2 The Saddle-node Bifurcation 5.3.3 The Transcritical Bifurcation 5.3.4 The Hopf Bifurcation 5.4 Codimension One Bifurcations for N-dimensional Systems 5.4.1 Saddle-Node Conditions 5.4.2 Transcritical and Pitchfork Conditions 5.4.3 Hopf Conditions 5.5 Center Manifold Reduction 5.5.1 The Center Manifold Theorem 5.5.2 Implications of the Theorem 5.5.3 Computing the Center Manifold Reduction 5.5.4 An Example 5.5.5 Summing Up on Center Manifold Reduction 5.6 Normal Form Reduction 5.7 Bifurcating Periodic Solutions 5.8 Grouping Bifurcations According to their Effect 5.9 On the Stability of Bifurcations to Perturbations 5.9.1 Stability of a Saddle-node Bifurcation 5.9.2 Stability of a Supercritical Pitchfork Bifurcation 5.10 Summing Up on Different Notions of Stability 5.11 Graphing Bifurcations: Numerical Continuation Techniques 5.11.1 Sequential Continuation 5.11.2 Pseudo-arclength Continuation 5.11.3 Locating Bifurcation Points 5.12 Bifurcation Analysis and Continuation in Lab Experiments 5.13 Nonlinear Normal Modes 5.14 Examples of Bifurcating Systems 5.14.1 Midplane Stretching (Duffing’s Equation) 5.14.2 Pendulum with a Moving Support (Parametric Excitation) 5.14.3 The Autoparametric Vibration Absorber 5.14.4 The Partially Follower-loaded Double Pendulum 5.15 Problems 6 Chaotic Vibrations 6.1 Introduction 6.2 First Example of a Chaotic System 6.3 Tools for Detecting Chaotic Vibrations 6.3.1 Phase Planes 6.3.2 Frequency Spectra 6.3.3 Poincaré Maps 6.3.4 Lyapunov Exponents 6.3.5 Horizons of Predictability 6.3.6 Attractor Dimension 6.3.7 Basins of Attraction 6.3.8 Summary on Detection Tools 6.4 Universal Routes to Chaos 6.4.1 The Period-Doubling Route 6.4.2 The Quasiperiodic Route 6.4.3 The Transient Route 6.4.4 The Intermittency Route 6.4.5 Summary on the Routes to Chaos 6.5 Tools for Predicting the Onset of Chaos 6.5.1 Criteria Related to the Universal Routes of Chaos 6.5.2 Searching for Homoclinic Tangles and Smale Horseshoes 6.5.3 The Melnikov Criterion 6.5.4 Criteria Based on Local Perturbation Analysis 6.5.5 Criteria for Conservative Chaos 6.6 Mechanical Systems and Chaos 6.6.1 The Lorenz System (D = 3) 6.6.2 Duffing-Type Systems (D = 3) 6.6.3 Pendulum-Type Systems (D = 3) 6.6.4 Piecewise Linear Systems (D ≥ 3) 6.6.5 Coupled Autonomous Systems (D ≥ 4) 6.6.6 Autoparametric Systems (D ≥ 5) 6.6.7 High-Order Systems (D greaterthan 5) 6.6.8 Other Systems 6.7 Elastostatical Chaos 6.8 Spatial and Spatiotemporal Chaos 6.9 Controlling Chaos 6.10 Closing Comments 6.11 Problems 7 Special Effects of High-Frequency Excitation 7.1 Introduction 7.2 The Method of Direct Separation of Motions (MDSM) 7.2.1 Outline of the MDSM 7.2.2 The Concept of Vibrational Force 7.2.3 The MDSM Compared to Classic Perturbation Approaches 7.3 Simple Examples 7.3.1 Pendulum on a Vibrating Support (Stiffening and Biasing) 7.3.2 Mass on a Vibrating Plane (Smoothening and Biasing) 7.3.3 Brumberg’s Pipe (Smoothening and Biasing) 7.4 A Slight but Useful Generalization 7.5 A Fairly General Class of Discrete Systems 7.5.1 The System 7.5.2 Example Functions 7.5.3 The Averaged System Governing the ‘Slow’ Motions 7.5.4 Interpretation of Averaged Forcing Terms 7.5.5 The Effects 7.5.6 Stiffening 7.5.7 Biasing 7.5.8 Smoothening 7.5.9 Effects of Multiple HF Excitation Frequencies 7.5.10 Effects of Strong Damping 7.6 A General Class of Linear Continuous Systems 7.6.1 The Generalized No-Resonance Prediction (GNRP) 7.6.2 The Generalized Analytical Resonance Prediction (GARP) 7.6.3 Example 1: Clamped String with HF Base Excitation 7.6.4 Example 2: Square Membrane with In-Plane HF Excitation 7.7 Specific Systems and Results – Some Examples 7.7.1 Using HF Excitation to Quench Friction-Induced Vibrations 7.7.2 Displacement Due to HF Excitation and Asymmetric Friction 7.7.3 Chelomei’s Pendulum – Resolving a Paradox 7.7.4 Stiffening of a Flexible String 7.8 Summing Up 7.9 Problems Appendix A: Performing Numerical Simulation Outline placeholder A.1. Solving Differential Equations A.2. Computing Chaos-Related Quantities A.3. Interfacing with the ODE-Solver A.4. Locating Software on the Internet Appendix B: Major Exercises Outline placeholder B.1 Tension Control of Rotating Shafts B.1.1 Mathematical Model B.1.2 Eigenvalue Problem, Natural Frequencies and Mode Shapes B.1.3 Discretizations, Choice of Control Law B.1.4 Local Bifurcation Analysis for a Balanced Shaft B.1.5 Quantitative Analysis of the Controlled System B.1.6 Using a Dither Signal for Open-Loop Control B.1.7 Numerical Analysis of the Controlled System B.1.8 Conclusions B.2 Vibrations of a Spring-Tensioned Beam B.2.1 Mathematical Model B.2.2 Eigenvalue Problem, Natural Frequencies and Mode Shapes B.2.3 Discrete Models B.2.4 Local Bifurcation Analysis for the Unloaded System B.2.5 Quantitative Analysis of the Loaded System B.2.6 Numerical Analysis B.2.7 Conclusions B.3 Dynamics of a Microbeam B.3.1 System Description B.3.2 Mathematical Model B.3.3 Eigenvalue Problem, Natural Frequencies and Mode Shapes B.3.4 Discrete Models, Mode Shape Expansion B.3.5 Local Bifurcation Analysis for the Statically Loaded System B.3.6 Quantitative Analysis of the Loaded System B.3.7 Numerical Analysis B.3.8 Conclusions Appendix C: Mathematical Formulas Outline placeholder C.1 Formulas Typically Used in Perturbation Analysis C.1.1 Complex Numbers C.1.2 Powers of Two-Term Sums C.1.3 Averaging Integrals C.1.4 Dirac’s Delta Function C.1.5 Fourier Series of a Periodic Function C.2 Formulas Used in Stability Analysis C.2.1 Sylvester’s Criterion C.2.2 the Routh-Hurwitz Criterion C.2.3 Mathieu’s Equation: Stability of the Zero-Solution Appendix D: Natural Frequencies and Mode Shapes for Structural Elements Outline placeholder D.1 Strings D.2 Rods D.2.1 Longitudinal Vibrations D.2.2 Torsional Vibrations D.3 Beams D.3.1 Bernoulli-Euler Beam Theory D.3.2 Timoshenko Beam Theory D.4 Rings D.4.1 In-plane Bending D.4.2 Out-of-Plane Bending D.4.3 Extension D.5 Membranes D.5.1 Rectangular Membranes D.5.2 Circular Membranes D.6 Plates D.6.1 Rectangular Plates D.6.2 Circular Plates D.7 Other Structures Appendix E: Properties of Engineering Materials Outline placeholder E.1 Friction and Thermal Expansion Coefficients E.2 Density and Elasticity Constants References Index 'Vibrations and Stability' is aimed at third to fifth-year undergraduates and post­ graduates in mechanical or structural engineering. The book covers a range of subjects relevant for a one-or two-semester course in advanced vibrations and stability. Also, it can be used for self-study, e. g. , by students on master or PhD projects, researchers, and professional engineers. The focus is on nonlinear phe­ nomena and tools, covering the themes of local perturbation analysis (Chaps. 3 and 4), bifurcation analysis (Chap. 5), global analysis I chaos theory (Chap. 6), and special high-frequency effects (Chap. 7). The ground for nonlinear analysis is laid with a brief summary of elementary linear vibration theory (Chap. 1), and a treatment of differential eigenvalue problems in some depth (Chap. 2). Also, there are exercise problems and extensive bibliographic references to serve the needs of both students and more experienced users; major exercises for course-work; and appendices on numerical simulation, standard mathematical formulas, vibration properties of basic structural elements, and properties of engineering materials. This Second Edition is a revised and expanded version of the first edition (pub­ lished by McGraw-Hill in 1997), reflecting the experience gathered during its now six years in service as a classroom or self-study text for students and researchers. The second edition contains a major new chapter (7), three new appendices, many new exercise problems, more than 120 new and updated bibliographic references, and hundreds of minor updates, corrections, and clarifications. An ideal text for students that ties together classical and modern topics of advanced vibration analysis in an interesting and lucid manner. It trains the student to analyze simple models, recognize nonlinear phenomena and work with advanced tools such as perturbation analysis and bifurcation analysis.