Vibration of Axially Loaded Structures

Vibration of Axially Loaded Structures
اسم المؤلف
LAWRENCE N. VIRGIN
التاريخ
1 نوفمبر 2018
المشاهدات
التقييم
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Vibration of Axially Loaded Structures
LAWRENCE N. VIRGIN
Duke University
Contents
Foreword page xiii
Preface xv
1 Context: The Point of Departure . 1
2 Elements of Classical Mechanics . 8
2.1 Introduction 8
2.2 Newton’s Second Law 8
2.3 Energy and Work 10
2.4 Virtual Work and D’Alembert’s Principle 11
2.5 Hamilton’s Principle and Lagrange’s Equations 13
2.5.1 Constraints 15
2.5.2 Conservation Laws 15
2.6 Nonconservative Forces and Energy Dissipation 17
2.6.1 Damping 18
2.6.2 Time-Dependent Forces 19
2.7 Strain Energy 20
references 21
3 Dynamics in the Vicinity of Equilibrium 22
3.1 The Linear Oscillator 22
3.2 Oscillator with a Slow Sweep of Frequency 28
3.3 Dynamics and Stability 30
3.3.1 Stability Concepts 30
3.4 Bifurcations 32
3.4.1 The Saddle-Node Bifurcation 33
3.4.2 Bifurcations from a Trivial Equilibrium 34
3.4.3 Initial Imperfections 35
3.4.4 Bifurcations of Maps 37
3.5 A Simple Demonstration Model 37
3.6 Experiments 41
references 43
viiviii Contents
4 Higher-Order Systems 45
4.1 Introduction 45
4.2 Multiple-Degree-of-Freedom Systems 45
4.2.1 The Algebraic Eigenvalue Problem 46
4.2.2 Normal Modes 47
4.2.3 Equilibrium, Linearization, and Stability 48
4.2.4 Routh–Hurwitz Criterion 54
4.2.5 Lyapunov Functions 55
4.2.6 Rayleigh’s Quotient 56
4.3 Distributed Systems 57
4.3.1 The Differential Eigenvalue Problem 59
4.3.2 Solution Methods 60
4.3.3 Context Revisited 64
references 64
5 Discrete-Link Models 66
5.1 Introduction 66
5.2 An Inverted Pendulum 66
5.2.1 Static Behavior 67
5.2.2 Geometric Imperfections 69
5.2.3 Dynamic Behavior 70
5.2.4 A Note on Inertia 74
5.3 A Discrete-Strut Model 75
5.4 An Asymmetric Model 80
5.5 A Three-Bar Model 82
5.6 A Snap-Through Model 84
5.7 Augusti’s Model 88
5.8 Multiple Loads 91
5.9 Load-Dependent Supports 93
5.10 Path Following and Continuation 94
references 95
6 Strings, Cables, and Membranes 97
6.1 Introduction 97
6.2 The Stretched String 97
6.2.1 The Wave Equation 97
6.2.2 Traveling-Wave Solution 100
6.2.3 Energy Considerations and Rayleigh’s Principle 101
6.3 A Suspended Cable 102
6.3.1 The Hanging Chain 106
6.4 A Rectangular Membrane 108
references 109Contents ix
7 Continuous Struts . 111
7.1 Introduction 111
7.2 Basic Formulation 111
7.2.1 The Response 113
7.2.2 The Temporal Solution 114
7.2.3 The Spatial Solution 116
7.3 Rayleigh’s Quotient 120
7.4 Rayleigh–Ritz Analysis 120
7.5 A Galerkin Approach 124
7.6 Higher Modes 126
7.7 Rotating Beams 131
7.8 A Strut with a Tangential Load 134
7.9 Self-Weight 136
7.9.1 A Hanging Beam 138
7.9.2 Experiments 139
7.10 Thermal Loading 142
7.11 Other Effects 143
references 143
8 Other Column-Type Structures . 147
8.1 A Beam on an Elastic Foundation 147
8.2 Elastically Restrained Supports 149
8.3 Beams with Variable Cross Section 150
8.4 Modal Coupling 154
8.5 Flexural–Torsional Buckling and Vibration 157
8.6 Type of Loading 161
8.7 A Continuous Arch 162
references 164
9 Frames . 166
9.1 A Beam with General Boundary Conditions 166
9.2 The Stiffness Method 168
9.3 A Self-Strained Frame Example 172
9.4 Modal Analysis 174
9.5 Large-Deflection Analysis 177
9.6 A Tubular Structure 178
references 181
10 Plates . 183
10.1 Introduction 183
10.1.1 Brief Review of the Classical Theory 183
10.1.2 Strain Energy 187
10.1.3 Boundary and Initial Conditions 188x Contents
10.1.4 The Simplest Case 190
10.1.5 Initial Imperfections 193
10.2 The Ritz and Finite-Element Approaches 193
10.3 A Fully Clamped Plate 196
10.4 Moderately Large Deflections 198
10.5 Postbuckling 199
10.6 Mode Jumping 205
10.6.1 Introduction 205
10.6.2 The Analytic Approach 205
10.6.3 Finite-Element Transient Results 209
10.7 Cylindrical Shells 209
references 212
11 Nondestructive Testing . 216
11.1 Introduction 216
11.1.1 The Southwell Plot 217
11.1.2 Examples 219
11.2 Some Background 222
11.3 Snap-Through Revisited 225
11.4 Range of Prediction 228
11.5 A Box Column 230
11.6 Plates and Shells 231
references 234
12 Highly Deformed Structures 237
12.1 Introduction to the Elastica 237
12.2 The Governing Equations 239
12.3 Case Study A: Self-Weight Loading Revisited 240
12.3.1 Numerical Results 241
12.3.2 Experiments 242
12.4 Case Study B: A Heavy Beam 243
12.4.1 Numerical Results 244
12.4.2 Experiments 245
12.5 Case Study C: A Pinched Loop 248
12.6 Case Study D: A Beam Loaded by a Cable 251
12.7 The Softening Loop Revisited 256
references 259
13 Suddenly Applied Loads 261
13.1 Load Classification 261
13.2 Back to Link Models 262
13.3 Dynamic Buckling of a Plate 267
13.4 A Type of Escaping Motion 268
13.5 Impulsive Loading 272Contents xi
13.5.1 Equilibrium Behavior 273
13.5.2 Behavior under Sudden Loading 274
13.6 Snap-Through of a Curved Panel 275
references 279
14 Harmonic Loading: Parametric Excitation 282
14.1 An Oscillating End Load 282
14.2 The Variational Equation 283
14.3 Mathieu’s Equation 286
14.4 Pulsating Axial Loads on Shells 288
14.4.1 A Curved Panel 289
14.4.2 A Cylindrical Shell 289
references 292
15 Harmonic Loading: Transverse Excitation 294
15.1 Introduction: Resonance Effects 294
15.1.1 A Single-Mode Approximation 295
15.1.2 Beyond Buckling 296
15.2 The Poincare Section ´ 297
15.3 Continuous Systems 299
15.4 An Application to Vibration Isolation 304
15.4.1 Postbuckling of a Strut Revisited 305
15.4.2 Experimental Verification 306
15.4.3 The Forced Response 307
15.5 Forced Excitation of the Thermally Buckled Plate 308
references 310
16 Nonlinear Vibration . 312
PART I: FREE VIBRATION 312
16.1 Introduction 312
16.2 Abstract Models 313
16.3 A Mass Between Stretched Springs 315
16.4 Nonlinear Vibration of Strings 319
16.5 Nonlinear Vibration of Beams 320
16.6 Nonlinear Vibration of a Plate 322
16.7 Nonlinear Vibration in Cylindrical Shells 324
PART II: FORCED VIBRATION 325
16.8 Nonlinear Forced Vibration of Strings 325
16.9 Nonlinear Forced Vibration of Beams 326
16.10 Persistent Snap-Through Behavior in a Plate 330
16.11 A Panel in Supersonic Flow 334
16.12 Chaotic Behavior 337
references 344
Index 34
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