Vibration of Axially-Loaded Structures

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Foreword......Page 15
Acknowledgments......Page 17
1 Context: The Point of Departure......Page 19
2.2 Newton’s Second Law......Page 26
2.3 Energy and Work......Page 28
2.4 Virtual Work and D’Alembert’s Principle......Page 29
2.5 Hamilton’s Principle and Lagrange’s Equations......Page 31
2.5.2 Conservation Laws......Page 33
2.6 Nonconservative Forces and Energy Dissipation......Page 35
2.6.1 Damping......Page 36
2.6.2 Time-Dependent Forces......Page 37
2.7 Strain Energy......Page 38
References......Page 39
3.1 The Linear Oscillator......Page 40
3.2 Oscillator with a Slow Sweep of Frequency......Page 46
3.3.1 Stability Concepts......Page 48
3.4 Bifurcations......Page 50
3.4.1 The Saddle-Node Bifurcation......Page 51
3.4.2 Bifurcations from a Trivial Equilibrium......Page 52
3.4.3 Initial Imperfections......Page 53
3.5 A Simple Demonstration Model......Page 55
3.6 Experiments......Page 59
References......Page 61
4.2 Multiple-Degree-of-Freedom Systems......Page 63
4.2.1 The Algebraic Eigenvalue Problem......Page 64
4.2.2 Normal Modes......Page 65
4.2.3 Equilibrium, Linearization, and Stability......Page 66
4.2.4 Routh–Hurwitz Criterion......Page 72
4.2.5 Lyapunov Functions......Page 73
4.2.6 Rayleigh’s Quotient......Page 74
4.3 Distributed Systems......Page 75
4.3.1 The Differential Eigenvalue Problem......Page 77
Modal Analysis and Truncation......Page 78
Weighted Residuals–Galerkin......Page 80
The Finite-Element Method......Page 81
References......Page 82
5.2 An Inverted Pendulum......Page 84
5.2.1 Static Behavior......Page 85
5.2.2 Geometric Imperfections......Page 87
5.2.3 Dynamic Behavior......Page 88
5.2.4 A Note on Inertia......Page 92
5.3 A Discrete-Strut Model......Page 93
5.4 An Asymmetric Model......Page 98
5.5 A Three-Bar Model......Page 100
5.6 A Snap-Through Model......Page 102
5.7 Augusti’s Model......Page 106
5.8 Multiple Loads......Page 109
5.9 Load-Dependent Supports......Page 111
5.10 Path Following and Continuation......Page 112
References......Page 113
6.2.1 The Wave Equation......Page 115
6.2.2 Traveling-Wave Solution......Page 118
6.2.3 Energy Considerations and Rayleigh’s Principle......Page 119
6.3 A Suspended Cable......Page 120
6.3.1 The Hanging Chain......Page 124
6.4 A Rectangular Membrane......Page 126
References......Page 127
7.2 Basic Formulation......Page 129
7.2.1 The Response......Page 131
7.2.2 The Temporal Solution......Page 132
7.2.3 The Spatial Solution......Page 134
7.4 Rayleigh–Ritz Analysis......Page 138
7.5 A Galerkin Approach......Page 142
7.6 Higher Modes......Page 144
7.7 Rotating Beams......Page 149
7.8 A Strut with a Tangential Load......Page 152
7.9 Self-Weight......Page 154
7.9.1 A Hanging Beam......Page 156
7.9.2 Experiments......Page 157
7.10 Thermal Loading......Page 160
7.11 Other Effects......Page 161
References......Page 162
8.1 A Beam on an Elastic Foundation......Page 165
8.2 Elastically Restrained Supports......Page 167
8.3 Beams with Variable Cross Section......Page 168
8.4 Modal Coupling......Page 172
8.5 Flexural–Torsional Buckling and Vibration......Page 175
8.6 Type of Loading......Page 179
8.7 A Continuous Arch......Page 180
References......Page 182
9.1 A Beam with General Boundary Conditions......Page 184
9.2 The Stiffness Method......Page 186
9.3 A Self-Strained Frame Example......Page 190
9.4 Modal Analysis......Page 192
9.5 Large-Deflection Analysis......Page 195
9.6 A Tubular Structure......Page 196
References......Page 199
10.1.1 Brief Review of the Classical Theory......Page 201
10.1.2 Strain Energy......Page 205
10.1.3 Boundary and Initial Conditions......Page 206
10.1.4 The Simplest Case......Page 208
10.2 The Ritz and Finite-Element Approaches......Page 211
10.3 A Fully Clamped Plate......Page 214
10.4 Moderately Large Deflections......Page 216
10.5 Postbuckling......Page 217
Experimental Description......Page 221
10.6.2 The Analytic Approach......Page 223
Equilibrium Paths......Page 224
Free Vibration......Page 225
10.7 Cylindrical Shells......Page 227
References......Page 230
11.1 Introduction......Page 234
11.1.1 The Southwell Plot......Page 235
11.1.2 Examples......Page 237
11.2 Some Background......Page 240
11.3 Snap-Through Revisited......Page 243
11.4 Range of Prediction......Page 246
11.5 A Box Column......Page 248
11.6 Plates and Shells......Page 249
References......Page 252
12.1 Introduction to the Elastica......Page 255
12.2 The Governing Equations......Page 257
12.3 Case Study A: Self-Weight Loading Revisited......Page 258
12.3.1 Numerical Results......Page 259
12.3.2 Experiments......Page 260
12.4 Case Study B: A Heavy Beam......Page 261
12.4.1 Numerical Results......Page 262
12.4.2 Experiments......Page 263
12.5 Case Study C: A Pinched Loop......Page 266
12.6 Case Study D: A Beam Loaded by a Cable......Page 269
12.7 The Softening Loop Revisited......Page 274
References......Page 277
13.1 Load Classification......Page 279
13.2 Back to Link Models......Page 280
13.3 Dynamic Buckling of a Plate......Page 285
13.4 A Type of Escaping Motion......Page 286
13.5 Impulsive Loading......Page 290
13.5.1 Equilibrium Behavior......Page 291
13.5.2 Behavior under Sudden Loading......Page 292
13.6 Snap-Through of a Curved Panel......Page 293
References......Page 297
14.1 An Oscillating End Load......Page 300
14.2 The Variational Equation......Page 301
14.3 Mathieu’s Equation......Page 304
14.4 Pulsating Axial Loads on Shells......Page 306
14.4.2 A Cylindrical Shell......Page 307
References......Page 310
15.1 Introduction: Resonance Effects......Page 312
15.1.1 A Single-Mode Approximation......Page 313
15.1.2 Beyond Buckling......Page 314
15.2 The Poincar Section......Page 315
15.3 Continuous Systems......Page 317
15.4 An Application to Vibration Isolation......Page 322
15.4.1 Postbuckling of a Strut Revisited......Page 323
15.4.2 Experimental Verification......Page 324
15.4.3 The Forced Response......Page 325
15.5 Forced Excitation of the Thermally Buckled Plate......Page 326
References......Page 328
16.1 Introduction......Page 330
16.2 Abstract Models......Page 331
16.3 A Mass Between Stretched Springs......Page 333
16.4 Nonlinear Vibration of Strings......Page 337
16.5 Nonlinear Vibration of Beams......Page 338
16.6 Nonlinear Vibration of a Plate......Page 340
16.7 Nonlinear Vibration in Cylindrical Shells......Page 342
16.8 Nonlinear Forced Vibration of Strings......Page 343
16.9 Nonlinear Forced Vibration of Beams......Page 344
16.10 Persistent Snap-Through Behavior in a Plate......Page 348
16.11 A Panel in Supersonic Flow......Page 352
16.12 Chaotic Behavior......Page 355
Epilogue......Page 361
References......Page 362
Index......Page 365