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.
Author(s): Maurice Petyt
Publisher: CUP
Year: 1990
Language: English
Pages: 574
Contents......Page 5
Preface......Page 11
Notation......Page 14
1.1 Dynamic equilibrium......Page 17
1.2 Principle of virtual displacements......Page 20
1.3 Hamilton's principle......Page 21
1.4 Lagrange's equations......Page 26
1.5 Equations of motion for a system with constraints......Page 30
Problems......Page 33
2 Element energy functions......Page 38
2.1 Axial element......Page 39
2.2 Torque element......Page 40
2.3 Beam bending element......Page 43
2.4 Deep beam bending element......Page 45
2.5 Membrane element......Page 46
2.6 Thin plate bending element......Page 49
2.7 Thick plate bending element......Page 51
2.8 Three-dimensional solid......Page 53
2.9 Axisymmetric solid......Page 55
2.10 The dissipation function......Page 57
2.11 Equations of motion and boundary conditions......Page 59
Problems......Page 64
3.1 Rayleigh-Ritz method......Page 69
3.2 Finite element displacement method......Page 79
3.3 Axial vibration of rods......Page 82
3.4 Torsional vibration of shafts......Page 100
3.5 Bending vibration of beams......Page 102
3.6 Vibration of plane frameworks......Page 108
3.7 Vibration of three-dimensional frameworks......Page 117
3.8 Techniques for increasing the accuracy of elements......Page 125
3.9 Shear deformation and rotary inertia effects......Page 130
3.10 Numerical integration......Page 137
3.11 Other considerations for beams......Page 149
Problems......Page 152
4 In-plane vibration of plates......Page 157
4.1 Linear triangular element......Page 159
4.2 Linear rectangular element......Page 165
4.3 Linear quadrilateral element......Page 173
4.4 Area coordinates for triangles......Page 179
4.5 Linear triangle in area coordinates......Page 180
4.6 Increasing the accuracy of elements......Page 182
Problems......Page 188
5.1 Axisymmetric solids......Page 192
5.2 Applied loading......Page 193
5.3 Displacements......Page 196
5.4 Reduced energy expressions......Page 197
5.5 Linear triangular element......Page 198
5.6 Core elements......Page 208
5.7 Arbitrary shaped solids......Page 211
5.8 Rectangular hexahedron......Page 213
5.9 Isoparametric hexahedron......Page 219
5.10 Right pentahedron......Page 224
5.11 Volume coordinates for tetrahedra......Page 228
5.12 Tetrahedron element......Page 231
5.13 Increasing the accuracy of elements......Page 234
Problems......Page 242
6 Flexural vibration of plates......Page 245
6.1 Thin rectangular element (non-conforming)......Page 246
6.2 Thin rectangular element (conforming)......Page 260
6.3 Thick rectangular element......Page 264
6.4 Thin triangular element (non-conforming)......Page 272
6.5 Thin triangular element (conforming)......Page 281
6.5.1 Cartesian coordinates......Page 282
6.5.2 Area coordinates......Page 287
6.6 Thick triangular element......Page 293
6.7 Other plate bending elements......Page 297
Problems......Page 306
7.1 Stiffened plates I......Page 310
7.2 Stiffened plates II......Page 315
7.3 Folded plates I......Page 320
7.4 Folded plates II......Page 323
7.5 Folded plates III......Page 325
Problems......Page 328
8.1 Some preliminaries......Page 331
8.1.1 Orthogonality of eigenvectors......Page 337
8.1.2 Transformation to standard form......Page 338
8.2 Sturm sequences......Page 343
8.3 Orthogonal transformation of a matrix......Page 351
8.4 The Jacobi method......Page 352
8.5.1 Givens' method......Page 356
8.5.2 Householder's method......Page 357
8.6.1 The bisection method......Page 359
8.6.2 Inverse iteration......Page 361
8.7.1 The LR method......Page 366
8.7.2 The QR method......Page 368
8.7.3 The QL method......Page 370
8.8 Reducing the number of degrees of freedom......Page 371
8.8.1 Making use of symmetry......Page 372
8.8.2 Rotationally-periodic structures......Page 375
8.8.3 Elimination of unwanted degrees of freedom......Page 380
8.8.4 Component mode synthesis......Page 385
8.8.4.1 Fixed interface method......Page 386
8.8.4.2 Free interface method......Page 389
8.9.1 Bisection/ inverse iteration......Page 393
8.9.2 Subspace iteration......Page 394
8.9.3 Simultaneous iteration......Page 396
8.9.4 Lanczos' method......Page 397
Problems......Page 400
9.1 Modal analysis......Page 402
9.2.1 Structural damping......Page 403
9.2.2 Viscous damping......Page 404
9.3.1 Modal analysis......Page 407
9.3.2 Direct analysis......Page 418
9.4 Response to periodic excitation......Page 425
9.5.1 Modal analysis......Page 430
9.5.1.1 Central difference method......Page 434
9.5.1.2 The Houbolt method......Page 440
9.5.1.3 The Newmark method......Page 446
9.5.1.4 The Wilson 0 method......Page 452
9.5.2 Direct analysis......Page 455
9.5.2.1 Central difference method......Page 456
9.5.2.2 The Houbolt method......Page 461
9.5.2.3 The Newmark method......Page 462
9.5.2.4 The Wilson 0 method......Page 463
9.5.3 Selecting a time step......Page 464
Problems......Page 465
10.1.1 Representation of the excitation......Page 466
10.1.2 Response of a single degree of freedom system......Page 478
10.1.3 Direct response of a multi-degree of freedom system......Page 482
10.1.4 Modal response of a multi-degree of freedom system......Page 487
10.1.5 Fatigue and failure......Page 488
10.2 Truncation of the modal solution......Page 491
10.2.1 Mode acceleration method......Page 495
10.2.2 Residual flexibility......Page 497
10.3.1 Direct response......Page 498
10.3.2 Modal response......Page 501
10.4.1 Single degree of freedom systems......Page 503
10.4.2 Multi-degree of freedom systems......Page 508
10.5.1 Making use of symmetry......Page 512
10.5.2 Rotationally periodic structures......Page 513
10.5.4 Component mode synthesis......Page 516
11 Computer analysis techniques......Page 518
11.1.1 Pre-processing......Page 520
11.1.2 Solution phase......Page 528
11.1.3 Post-processing......Page 529
11.2 Modelling......Page 530
11.3 Using commercial codes......Page 538
Appendix......Page 543
Answers to problems......Page 549
Bibliography......Page 553
References......Page 555
Index......Page 571
