Mechanical Vibrations in Spacecraft Design
Bearbeitet von J. Jaap Wijker
1. Auflage 2003. Buch. xvi, 436 S. Hardcover ISBN 978 3 540 40530 6
Format (B x L): 15,5 x 23,5 cm Gewicht: 1800 g
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1 Introduction ... 1
1.1 Why Another Book about Mechanical Vibrations? ... 1
1.2 A Short Overview of Theory ... 5
1.2.1 Single Degree of Freedom (sdof) Systems...5
1.2.2 Damped Vibrations...6
1.2.3 Multi-Degrees of Freedom (mdof) Dynamic Systems ...7
1.2.4 Modal Analysis ...8
1.2.5 Modal Effective Mass ...9
1.2.6 Response Analysis...9
1.2.7 Transient Response Analysis...10
1.2.8 Random Vibrations...11
1.2.9 Shock-Response Spectrum...12
1.2.10 Acoustic Loads, Structural Responses ...13
1.2.11 Statistical Energy Analysis...13
1.2.12 Inertia-Relief ...14
1.2.13 Mode Acceleration Method...14
1.2.14 Residual Vectors...14
1.2.15 Dynamic Model Reduction ...15
1.2.16 Component Model Synthesis ...16
1.2.17 Load Transformation Matrices ...17
1.3 Problems... 17
1.3.1 Problem 1 ...17
2 Single Degree of Freedom System... 19
2.1 Introduction ... 19
2.2 Undamped Sdof System... 20
2.2.1 Solution of an Sdof System with Initial Conditions...23
x Table of Contents
2.2.2 Solution of an Sdof System with Applied Forces ...24
2.3 Damped Vibration and the Damping Ratio... 27
2.3.1 Solution of the Sdof System in the Time Domain...29
2.3.2 Solution of the Damped Sdof System with Applied Forces...32
2.3.3 Solution in the Frequency Domain...34
2.3.4 State Space Representation of the Sdof System...42
2.4 Problems... 45
2.4.1 Problem 1 ...45
2.4.2 Problem 2 ...46
2.4.3 Problem 3 ...46
2.4.4 Problem 4 ...46
2.4.5 Problem 5 ...47
2.4.6 Problem 6 ...47
2.4.7 Problem 7 ...47
2.4.8 Problem 8 ...48
3 Damping Models ... 49
3.1 Introduction ... 49
3.2 Damped Vibration ... 50
3.2.1 Linear Damping...50
3.2.2 Viscous Damping ...51
3.2.3 Structural Damping ...51
3.2.4 Loss Factor ...52
3.3 Amplification Factor ... 53
3.3.1 Modal Viscous Damping...53
3.3.2 Modal Structural Damping...54
3.3.3 Discussion of Modal Damping...55
3.4 Method of Determining Damping from Measurements... 56
3.4.1 The Half-Power Point Method ...56
3.5 Problems... 57
3.5.1 Problem 1 ...57
3.5.2 Problem 2 ...58
4 Multi-Degrees of Freedom Linear Dynamic Systems ... 61
4.1 Introduction ... 61
4.2 Derivation of the Equations of Motion ... 62
4.2.1 Undamped Equations of Motion with Newton’s Law...62
4.2.2 Undamped Equations of Motion using Energies...64
4.2.3 Undamped Equations of Motion using Lagrange’s Equations...65
4.2.4 Damped Equations of Motion using Lagrange’s Equations ...67
4.3 Finite Element Method... 72
4.4 Problems... 72
4.4.1 Problem 1 ...72
4.4.2 Problem 2 ...73
5 Modal Analysis ... 75
5.1 Introduction ... 75
5.2 Undamped Linear Dynamic Systems... 75
5.2.1 Natural Frequencies and Mode Shapes ...76
5.2.2 Orthogonality Relations of Modes ...79
5.2.3 Rigid-Body Modes ...82
5.2.4 Left Eigenvectors ...86
5.3 Damped Linear Dynamic Systems... 88
5.3.1 The State Vector ...88
5.3.2 Eigenvalue Problem ...90
5.3.3 Eigenvectors ...91
5.4 Problems... 94
5.4.1 Problem 1 ...94
5.4.2 Problem 2 ...95
6 Natural Frequencies, an Approximation ... 97
6.1 Introduction ... 97
6.2 Static Displacement Method ... 97
6.3 Rayleigh’s Quotient ... 100
6.4 Dunkerley’s Method ... 103
6.5 Problems... 109
6.5.1 Problem 1 ...109
6.5.2 Problem 2 ...109
6.5.3 Problem 3 ...109
6.5.4 Problem 4 ...110
6.5.5 Problem 5 ...111
7 Modal Effective Mass ... 113
7.1 Introduction ... 113
7.2 Enforced Acceleration... 113
7.3 Modal Effective Masses of an Mdof System ... 116
7.4 Problems... 124
xii Table of Contents
7.4.1 Problem 1 ...124
7.4.2 Problem 2 ...125
8 Response Analysis ... 127
8.1 Introduction ... 127
8.2 Forces and Enforced Acceleration ... 127
8.2.1 Relative Motions ...128
8.2.2 Absolute Motions ...140
8.2.3 Large-Mass Approach ...142
8.3 Problems... 148
8.3.1 Problem 1 ...148
8.3.2 Problem 2 ...149
8.3.3 Problem 3 ...150
9 Transient-Response Analysis ... 151
9.1 Introduction ... 151
9.2 Numerical Time Integration ... 153
9.2.1 Discrete Solution Convolution Integral...153
9.2.2 Explicit Time-Integration Method...155
9.2.3 Implicit Time-Integration Methods ...155
9.2.4 Stability ...155
9.3 Explicit Time-Integration... 156
9.3.1 Central Difference Method...156
9.3.2 Runge–Kutta Formulae for First-Order Differential Equations ...158
9.3.3 Runge–Kutta–Nyström Method for S-O Differential Equations ...161
9.4 Implicit Time Integration ... 161
9.4.1 Houbolt Method ...162
9.4.2 Wilson–theta Method ...164
9.4.3 Newmark–beta Method ...166
9.4.4 The Hughes, Hilber and Taylor (HHT) alpha–Method ...168
9.4.5 The Wood, Bossak and Zienkiewicz (WBZ) alpha–Method ...169
9.4.6 The Generalised–alpha Algorithm ...170
9.5 Piecewise Linear Method... 171
9.6 Problems... 172
9.6.1 Problem 1 ...172
9.6.2 Problem 2 ...174
9.6.3 Problem 3 ...174
10 Shock-Response Spectrum... 175
10.1 Introduction ... 175
10.2 ... Enforced Acceleration176 10.3 Numerical Calculation of the SRS, the Piecewise Exact Method... 178
10.4 Response Analysis in Combination with Shock-Response Spectra... 183
10.5 Matching Shock Spectra with Synthesised Time Histories ... 192
10.6 Problems... 201
10.6.1 Problem 1 ...201
10.6.2 Problem 2 ...202
11 Random Vibration of Linear Dynamic Systems ... 203
11.1 Introduction ... 203
11.2 Random Process ... 203
11.3 Power-Spectral Density... 209
11.4 Deterministic Linear Dynamic System ... 214
11.4.1 Force-Loaded Sdof System ...216
11.4.2 Enforced Acceleration ...218
11.4.3 Multi-Inputs and Single Output (MISO) ...225
11.5 Deterministic Mdof Linear Dynamic System ... 226
11.5.1 Random Forces...226
11.5.2 Random Base Excitation ...229
11.5.3 Random Stresses and Forces ...231
11.6 Analysis of Narrow-Band Processes ... 236
11.6.1 Crossings ...236
11.6.2 Fatigue Damage due to Random Excitation...240
11.7 Some Practical Aspects ... 243
11.8 Problems... 246
11.8.1 Problem 1 ...246
11.8.2 Problem 2 ...246
11.8.3 Problem 3 ...247
11.8.4 Problem 4 ...247
11.8.5 Problem 5 ...248
12 Low-Frequency Acoustic Loads, Structural Responses... 249
12.1 Introduction ... 249
12.2 Acoustic Loads... 249
12.3 Equations of Motion... 251
12.4 Problems... 262
xiv Table of Contents
12.4.1 Problem 1 ...262
12.4.2 Problem 2 ...262
13 Statistical Energy Analysis... 265
13.1 Introduction ... 265
13.2 Some Basics about Averaged Quantities... 266
13.3 Two Coupled Oscillators... 272
13.4 Multimode Subsystems ... 279
13.5 SEA Parameters ... 285
13.5.1 Subsystem Modal Densities ...285
13.5.2 Source Power Input ...290
13.5.3 Subsystem Energies...291
13.5.4 Damping Loss Factor ...296
13.5.5 Coupling Loss Factor ...297
13.6 Stresses and Strains ... 300
13.7 Problems... 301
13.7.1 Problem 1 ...301
13.7.2 Problem 2 ...301
13.7.3 Problem 3 ...302
13.7.4 Problem 4 ...302
13.7.5 Problem 5 ...302
13.7.6 Problem 6 ...303
13.7.7 Problem 7 ...303
13.7.8 Problem 8 ...304
14 Free-free Dynamic Systems, Inertia Relief ... 305
14.1 Introduction ... 305
14.2 Relative Motion... 305
14.3 Relative Forces... 306
14.4 Flexibility Matrix ... 309
14.5 Problems... 312
14.5.1 Problem 1 ...312
15 Mode Acceleration Method... 315
15.1 Introduction ... 315
15.2 Decomposition of Flexibility and Mass Matrix ... 315
15.2.1 Decomposition of the Flexibility Matrix...315
15.2.2 Decomposition of the Mass Matrix...317
15.2.3 Convergence Properties of Reconstructed Matrices ...318
15.3 Mode Acceleration Method ... 320
15.4 Problems... 327
15.4.1 Problem 1 ...327
15.4.2 Problem 2 ...327
15.4.3 Problem 3 ...328
15.4.4 Problem 4 ...328
15.4.5 Problem 5 ...329
16 Residual Vectors... 333
16.1 Introduction ... 333
16.2 Residual Vectors... 333
16.2.1 Dickens Method ...333
16.2.2 Rose Method ...336
16.3 Problems... 342
16.3.1 Problem 1 ...342
17 Dynamic Model Reduction Methods ... 345
17.1 Introduction ... 345
17.2 Static Condensation Method ... 346
17.2.1 Improved Calculation of Eliminated Dofs ...352
17.3 Dynamic Reduction... 353
17.4 Improved Reduced System (IRS) ... 354
17.5 Craig–Bampton Reduced Models... 357
17.6 Generalised Dynamic Reduction ... 360
17.7 System Equivalent Reduction Expansion Process (SEREP) ... 364
17.8 Ritz Vectors ... 367
17.9 Conclusion ... 369
18 Component Mode Synthesis ... 371
18.1 Introduction ... 371
18.2 The Unified CMS Method ... 372
18.2.1 Modal Truncation ...373
18.2.2 General Synthesis of Two Components ...374
18.2.3 General Example ...376
18.3 Special CMS Methods ... 381
18.3.1 Craig–Bampton Fixed-Interface Method ...381
18.3.2 Free-Interface Method...386
xvi Table of Contents
18.3.3 General-Purpose CMS Method ...393
18.4 Problems... 398
18.4.1 Problem 1 ...398
18.4.2 Problem 2 ...399
19 Load Transformation Matrices ... 401
19.1 Introduction ... 401
19.2 Reduced Model with Boundary Conditions... 402
19.3 Reduced Free-Free Dynamic Model ... 406
19.4 Continuous Dynamic Systems ... 411
19.5 Problems... 414
19.5.1 Problem 1 ...414
19.5.2 Problem 2 ...416
19.5.3 Problem 3 ...416
19.5.4 Problem 4 ...417
References ... 419
Author Index ... 429
Subject Index ... 433
