Introduction to Finite Element Vibration Analysis

✍ Scribed by Maurice Petyt

• Publisher: Cambridge University Press
• Year: 1990
• Tongue: English
• Leaves: 569
• Category: Library

✦ Synopsis

This book presents an introduction to the mathematical basis of finite element analysis as applied to vibrating systems. Finite element analysis is a technique that is very important in modeling the response of structures to dynamic loads and is widely used in aeronautical, civil and mechanical engineering as well as naval architecture. Commercial computer programs based on this technique already exist. Nevertheless, a knowledge of the mathematical principles involved is necessary before they can be successfully used. Therefore, this book assumes no previous knowledge of finite element techniques by the reader. The author has taught courses on the subject at undergraduate and postgraduate levels. The book has been written in a modular style to make it suitable for use in courses of varying length and level.

✦ Table of Contents

Cover......Page 1
Title......Page 3
Copyright......Page 4
Contents......Page 5
Preface......Page 10
Notation......Page 13
1.1 Dynamic equilibrium......Page 15
1.2 Principle of virtual displacements......Page 18
1.3 Hamilton's principle......Page 19
1.4 Lagrange's equations......Page 24
1.5 Equations of motion for a system with constraints......Page 28
Problems......Page 31
2. Element energy functions......Page 36
2.1 Axial element......Page 37
2.2 Torque element......Page 38
2.3 Beam bending element......Page 41
2.4 Deep beam bending element......Page 43
2.5 Membrane element......Page 44
2.6 Thin plate bending element......Page 47
2.7 Thick plate bending element......Page 49
2.8 Three-dimensional solid......Page 51
2.9 Axisymmetric solid......Page 53
2.10 The dissipation function......Page 55
2.11 Equations of motion and boundary conditions......Page 57
Problems......Page 62
3.1 Rayleigh-Ritz method......Page 67
3.2 Finite element displacement method......Page 77
3.3 Axial vibration of rods......Page 80
3.4 Torsional vibration of shafts......Page 98
3.5 Bending vibration of beams......Page 100
3.6 Vibration of plane frameworks......Page 106
3.7 Vibration of three-dimensional frameworks......Page 115
3.8 Techniques for increasing the accuracy of elements......Page 123
3.9 Shear deformation and rotary inertia effects......Page 128
3.10 Numerical integration......Page 135
3.11 Other considerations for beams......Page 147
Problems......Page 150
4. In-plane vibration of plates......Page 155
4.1 Linear triangular element......Page 157
4.2 Linear rectangular element......Page 163
4.3 Linear quadrilateral element......Page 171
4.4 Area coordinates for triangles......Page 177
4.5 Linear triangle in area coordinates......Page 178
4.6 Increasing the accuracy of elements......Page 180
Problems......Page 186
5.1 Axisymmetric solids......Page 190
5.2 Applied loading......Page 191
5.3 Displacements......Page 194
5.4 Reduced energy expressions......Page 195
5.5 Linear triangular element......Page 196
5.6 Core elements......Page 206
5.7 Arbitrary shaped solids......Page 209
5.8 Rectangular hexahedron......Page 211
5.9 Isoparametric hexahedron......Page 217
5.10 Right pentahedron......Page 222
5.11 Volume coordinates for tetrahedra......Page 226
5.12 Tetrahedron element......Page 229
5.13 Increasing the accuracy of elements......Page 232
Problems......Page 240
6. Flexural vibration of plates......Page 243
6.1 Thin rectangular element (non-conforming)......Page 244
6.2 Thin rectangular element (conforming)......Page 258
6.3 Thick rectangular element......Page 262
6.4 Thin triangular element (non-conforming)......Page 270
6.5 Thin triangular element (conforming)......Page 279
6.5.1 Cartesian coordinates......Page 280
6.5.2 Area coordinates......Page 285
6.6 Thick triangular element......Page 291
6.7 Other plate bending elements......Page 295
Problems......Page 304
7.1 Stiffened plates I......Page 308
7.2 Stiffened plates II......Page 313
7.3 Folded plates I......Page 318
7.4 Folded plates II......Page 321
7.5 Folded plates III......Page 323
Problems......Page 326
8.1 Some preliminaries......Page 329
8.1.1 Orthogonality of eigenvectors......Page 335
8.1.2 Transformation to standard form......Page 336
8.2 Sturm sequences......Page 341
8.3 Orthogonal transformation of a matrix......Page 349
8.4 The Jacobi method......Page 350
8.5.1 Givens' method......Page 354
8.5.2 Householder's method......Page 355
8.6.1 The bisection method......Page 357
8.6.2 Inverse iteration......Page 359
8.7.1 The LR method......Page 364
8.7.2 The QR method......Page 366
8.7.3 The QL method......Page 368
8.8 Reducing the number of degrees of freedom......Page 369
8.8.1 Making use of symmetry......Page 370
8.8.2 Rotationally-periodic structures......Page 373
8.8.3 Elimination of unwanted degrees of freedom......Page 378
8.8.4 Component mode synthesis......Page 383
8.8.4.1 Fixed interface method......Page 384
8.8.4.2 Free interface method......Page 387
8.9.1 Bisection/ inverse iteration......Page 391
8.9.2 Subspace iteration......Page 392
8.9.3 Simultaneous iteration......Page 394
8.9.4 Lanczos' method......Page 395
Problems......Page 398
9.1 Modal analysis......Page 400
9.2.1 Structural damping......Page 401
9.2.2 Viscous damping......Page 402
9.3.1 Modal analysis......Page 405
9.3.2 Direct analysis......Page 416
9.4 Response to periodic excitation......Page 423
9.5.1 Modal analysis......Page 428
9.5.1.1 Central difference method......Page 432
9.5.1.2 The Houbolt method......Page 438
9.5.1.3 The Newmark method......Page 444
9.5.1.4 The Wilson 0 method......Page 450
9.5.2 Direct analysis......Page 453
9.5.2.1 Central difference method......Page 454
9.5.2.2 The Houbolt method......Page 459
9.5.2.3 The Newmark method......Page 460
9.5.2.4 The Wilson 0 method......Page 461
9.5.3 Selecting a time step......Page 462
Problems......Page 463
10.1.1 Representation of the excitation......Page 464
10.1.2 Response of a single degree of freedom system......Page 476
10.1.3 Direct response of a multi-degree of freedom system......Page 480
10.1.4 Modal response of a multi-degree of freedom system......Page 485
10.1.5 Fatigue and failure......Page 486
10.2 Truncation of the modal solution......Page 489
10.2.1 Mode acceleration method......Page 493
10.2.2 Residual flexibility......Page 495
10.3.1 Direct response......Page 496
10.3.2 Modal response......Page 499
10.4.1 Single degree of freedom systems......Page 501
10.4.2 Multi-degree of freedom systems......Page 506
10.5.1 Making use of symmetry......Page 510
10.5.2 Rotationally periodic structures......Page 511
10.5.4 Component mode synthesis......Page 514
11 Computer analysis techniques......Page 516
11.1.1 Pre-processing......Page 518
11.1.2 Solution phase......Page 526
11.1.3 Post-processing......Page 527
11.2 Modelling......Page 528
11.3 Using commercial codes......Page 536
Appendix......Page 540
Answers to problems......Page 545
Bibliography......Page 549
References......Page 551
Index......Page 566