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[资源] Vibration Analysis and Structural Dynamics for Civil Engineers

contents
Part I
Essentials  1
1 Introduction  3
1.1  Definitions, aims and general concepts 3
1.2  Basic features of a vibrating system, and further concepts 4
Tutorial questions 7
2 Single degree-of-freedom systems  9
2.1  Basic equation of motion 9
2.2  Free vibration response 9
2.3  Equivalent spring stiffnesses for various
structural and mechanical systems 19
2.4  Response to harmonic excitation 24
Tutorial questions 30
3 Systems with more than one degree of freedom  33
3.1  Introductory remark 33
3.2  Equations of motion 33
3.3  Techniques for assembling the stiffness matrix 35
vi  Contents
3.4  The flexibility formulation of the equations of
motion and assembly of the flexibility matrix 40
3.5  Determination of natural frequencies and mode shapes 44
3.6  The flexibility formulation of the eigenvalue problem 46
3.7  Worked examples 47
3.8  The modal matrix 56
3.9  Orthogonality of eigenvectors 57
3.10  Generalized mass and stiffness matrices 59
3.11  Worked examples 61
3.12  Modal analysis 64
3.13  Worked example 69
Tutorial questions 73
4 Continuous systems  75
4.1  Introduction 75
4.2  Transverse vibration of strings 76
4.2.1  General formulation 76
4.2.2  Free vibration 77
4.3  Axial vibration of rods 81
4.4  Flexural vibration of beams 86
4.4.1  General formulation 86
4.4.2  Boundary conditions 88
4.4.3  Free vibration 88
4.5  Orthogonality of natural modes of vibration 92
4.6  Dynamic response by the method of modal analysis 97
5 Finite-element vibration analysis  103
5.1  The finite-element formulation 103
5.1.1  Introduction 103
5.1.2  Displacements 103
5.1.3  Strains 104
5.1.4  Stresses 105
5.1.5  Nodal actions and element stiffness matrix 106
5.1.6  Dynamic formulation and element mass matrix 107
5.2  Stiffness and consistent mass matrices
for some common finite elements 109
5.2.1  Truss element 109
5.2.2  Beam element 111
5.2.3  Rectangular plane–stress element 116
Contents  vii
5.3  Assembly of the system equations of motion 122
5.3.1  Procedure in brief 122
5.3.2  Illustration of assembly 122
5.3.3  Accounting for support conditions 124
5.3.4  Numerical example 125
References 127
Part II
Group-theoretic formulations  129
6 Basic concepts of symmetry groups and representation
theory  131
6.1  Symmetry groups 131
6.2  Group tables and classes 132
6.3  Representations of symmetry groups 135
6.4  Character tables 135
6.5  Group algebra 137
6.6  Idempotents 137
6.7  Applications 139
References 140
7 Rectilinear models  141
7.1  Introduction 141
7.2  A shaft–disc torsional system 141
7.2.1  Symmetry-adapted functions 142
7.2.2  Symmetry-adapted stiffness matrices 143
7.2.3  Eigenvalues 146
7.3  A spring–mass extensional system 147
7.3.1  Symmetry-adapted functions 148
7.3.2  Symmetry-adapted stiffness matrices 149
7.3.3  Eigenvalues 151
7.4  Conclusions 152
8 Plane structural grids  153
8.1  Introduction 153
8.2  Rectangular configurations 154
8.2.1  Symmetry-adapted functions 154
8.2.2  Symmetry-adapted flexibility coefficients 156
8.2.3  Eigenvalues 158
viii  Contents
8.3  Square configurations 160
8.3.1  Symmetry-adapted functions 161
8.3.2  Symmetry-adapted flexibility coefficients 162
8.3.3  Eigenvalues 165
8.4  Conclusion 167
9 High-tension cable nets  169
9.1  Basic assumptions and geometric formulation 169
9.2  Outline of computational scheme 171
9.3  Illustrative examples 172
9.4  Symmetry-adapted functions 174
9.4.1  Rectangular 24-node configuration 174
9.4.2  Square 16-node configuration 177
9.5  Symmetry-adapted flexibility matrices 179
9.5.1  Basis-vector plots and subspace properties 179
9.5.2  Equilibrium considerations and
symmetry-adapted flexibility coefficients 184
9.5.3  Derivation of the static equilibrium matrices 185
9.6  Subspace mass matrices 192
9.7  Eigenvalues, eigenvectors and mode shapes 193
9.8  Summary and concluding remarks 194
References 196
10 Finite-difference formulations for plates  197
10.1  General finite-difference formulation for plate vibration 197
10.2  Group-theoretic implementation 199
10.3  Application to rectangular and square plates 201
10.3.1  Basis vectors for the rectangular plate 203
10.3.2  Basis vectors for the square plate 205
10.3.3  Nodal sets 207
10.4  Finite-difference equations for generator
nodes of the basis vectors 210
10.4.1  Generator nodes 210
10.4.2  Basis finite-difference equations
for the rectangular plate 212
10.4.3  Basis finite-difference equations for
the square plate 213
Contents  ix
10.5  Symmetry-adapted finite-difference
equations and system eigenvalues 214
10.5.1  Rectangular plate 214
10.5.2  Square plate 218
10.5.3  Remarks on the structure of symmetryadapted eigenvalue matrices 221
10.6  Concluding remarks 222
References 223
11 Finite-element formulations for symmetric elements  225
11.1  Group-theoretic formulation for finite elements 225
11.2  Coordinate system, node numbering
and positive directions 226
11.3  Symmetry-adapted nodal freedoms 229
11.4  Displacement field decomposition 233
11.5  Subspace shape functions 237
11.6  Subspace element matrices 238
11.7  Final element matrices 241
11.8  Concluding remarks 246
References 247

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