دانلود کتاب VIBRATIONS AND STABILITY : advanced theory, analysis, and tools,

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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
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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
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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
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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
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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
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E.1 Friction and Thermal Expansion Coefficients
E.2 Density and Elasticity Constants
References
Index