Structural Dynamics for Engineers

Contents Biographies xi
Preface xiii
1 Causes and effects of structural vibration 1
1.1 Introduction 1
1.2 Vibration of structures: simple harmonic motion 4
1.3 Nature and dynamic effect of man-made and
environmental forces 6
1.4 Methods of dynamic response analysis 9
1.5 Single- DOF and multi-DOF structures 9
Further reading 12
2 Equivalent one degree-of-freedom systems 15
2.1 Introduction 15
2.2 Modelling structures as 1-DOF systems 15
2.3 Theoretical modelling by equivalent 1-DOF
mass-spring systems 16
2.4 Equivalent 1-DOF mass-spring systems for linearly
elastic line structures 19
2.5 Equivalent 1-DOF mass-spring systems for linearly
elastic continuous beams 39
2.6 First natural frequency of sway structures 47
2.7 Plates 57
2.8 Summary and conclusions 57
References 59
Further reading 59
3 Free vibration of one degree-of-freedom systems 61
3.1 Introduction 61
3.2 Free un-damped rectilinear vibration 61
3.3 Free rectilinear vibration with viscous damping 64
3.4 Evaluation of logarithmic decrement of damping from
the decay function 68
3.5 Free un-damped rotational vibration 70
3.6 Polar movement of inertia of equivalent lumped
mass-spring system of bar element with one free end 72
3.7 Free rotational vibration with viscous damping 76
Further reading 77
4 Forced harmonic vibration of one degree-of-freedom
systems 79
4.1 Introduction 79
4.2 Rectilinear response of 1-DOF system with viscous
damping to harmonic excitation 79
4.3 Response at resonance 83
4.4 Forces transmitted to the foundation by unbalanced
rotating mass in machines and motors 87
4.5 Response to support motion 92
4.6 Rotational response of 1-DOF systems with viscous
damping to harmonic excitation 98
Further reading 101
5 Evaluation of equivalent viscous damping coefficients
by harmonic excitation 103
5.1 Introduction 103
5.2 Evaluation of damping from amplification of static
response at resonance 103
5.3 Vibration at resonance 104
5.4 Evaluation of damping from response functions
obtained by frequency sweeps 106
5.5 Hysteretic damping 112
5.6 The effect and behaviour of air and water at
resonance 114
Further reading 115
6 Response of linear and non-linear one
degree-of-freedom systems to random loading:
time domain analysis 117
6.1 Introduction 117
6.2 Step-by-step integration methods 118
6.3 Dynamic response to turbulent wind 125
6.4 Dynamic response to earthquakes 126
6.5 Dynamic response to impacts caused by falling loads 126
6.6 Response to impulse loading 133
6.7 Incremental equations of motion for multi-DOF
systems 133
References 135
Further reading 135
7 Free vibration of multi-degree-of-freedom systems 137
7.1 Introduction 137
7.2 Eigenvalues and eigenvectors 137
7.3 Determination of free normal mode vibration by
solution of the characteristic equation 138
7.4 Solution of cubic characteristic equations by the
Newton approximation method 141
7.5 Solution of cubic characteristic equations by the
direct method 142
7.6 Two eigenvalue and eigenvector theorems 142
7.7 Iterative optimisation of eigenvectors 146
7.8 The Raleigh quotient 151
7.9 Condensation of the stiffness matrix in lumped mass
analysis 151
7.10 Consistent mass matrices 154
7.11 Orthogonality and normalisation of eigenvectors 156
7.12 Structural instability 159
References 161
Further reading 161
8 Forced harmonic vibration of multi-degree-of-freedom
systems 163
8.1 Introduction 163
8.2 Forced vibration of undamped 2-DOF systems 163
8.3 Forced vibration of damped 2-DOF systems 166
8.4 Forced vibration of multi-DOF systems with orthogonal
damping matrices 169
8.5 Tuned mass dampers 173
References 175
9 Damping matrices for multi-degree-of-freedom systems 177
9.1 Introduction 177
9.2 Incremental equations of motion for multi-DOF
systems 177
9.3 Measurement and evaluation of damping in higher
modes 178
9.4 Damping matrices 179
9.5 Modelling of structural damping by orthogonal
damping matrices 179
Further reading 184
10 The nature and statistical properties of wind 185
10.1 Introduction 185
10.2 The nature of wind 185
10.3 Mean wind speed and variation of mean velocity with
height 187
10.4 Statistical properties of the fluctuating velocity
component of wind 191
10.5 Probability density function and peak factor for
fluctuating component of wind 200
10.6 Cumulative distribution function 201
10.7 Pressure coefficients 201
Further reading 202
11 Dynamic response to turbulent wind:
frequency-domain analysis 203
11.1 Introduction 203
11.2 Aeroelasticity and dynamic response 203
11.3 Dynamic response analysis of aeroelastically stable
structures 204
11.4 Frequency-domain analysis of 1-DOF systems 204
11.5 Relationships between response, drag force and
velocity spectra for 1-DOF systems 205
11.6 Extension of the frequency-domain method to
multi-DOF systems 212
11.7 Summary of expressions used in the
frequency-domain method for multi-DOF systems 215
11.8 Modal force spectra for 2-DOF systems 216
11.9 Modal force spectra for 3-DOF systems 217
11.10 Aerodynamic damping of multi-DOF systems 218
11.11 Simplified wind response analysis of linear multi-DOF
structures in the frequency domain 225
11.12 Concluding remarks on the frequency-domain
method 230
11.13 Vortex shedding of bluff bodies 231
11.14 The phenomenon of lock-in 237
11.15 Random excitation of tapered cylinders by vortices 240
11.16 Suppression of vortex-induced vibration 240
11.17 Dynamic response to the buffeting of wind using
time-integration methods 241
References 243
Further reading 243
12 The nature and properties of earthquakes 245
12.1 Introduction 245
12.2 Types and propagation of seismic waves 245
12.3 Propagation velocity of seismic waves 245
12.4 Recording of earthquakes 248
12.5 Magnitude and intensity of earthquakes 248
12.6 Influence of magnitude and surface geology on
characteristics of earthquakes 249
12.7 Representation of ground motion 252
References 254
Further reading 254
13 Dynamic response to earthquakes: frequency-domain
analysis 255
13.1 Introduction 255
13.2 Construction of response spectra 255
13.3 Tripartite response spectra 256
13.4 Use of response spectra 258
13.5 Response of multi-DOF systems to earthquakes 260
13.6 Deterministic response analysis using response spectra 262
13.7 Dynamic response to earthquakes using
time-domain integration methods 265
13.8 Power spectral density functions for earthquakes 265
13.9 Frequency-domain analysis of single-DOF systems
using power spectra for translational motion 266
13.10 Influence of the dominant frequency of the ground
on the magnitude of structural response 269
13.11 Extension of the frequency-domain method for
translational motion to multi-DOF structures 270
13.12 Response of 1-DOF structures to rocking motion 274
13.13 Frequency-domain analysis of single-DOF systems
using power spectra for rocking motion 275
13.14 Assumed power spectral density function for rocking
motion used in examples 276
13.15 Extension of the frequency-domain method for
rocking motion to multi-DOF structures 279
13.16 Torsional response to seismic motion 282
13.17 Reduction of dynamic response 286
13.18 Soil-structure interaction 288
References 291
Further reading 291
14 Generation of wind and earthquake histories 293
14.1 Introduction 293
14.2 Generation of single wind histories by a Fourier
series 293
14.3 Generation of wind histories by the autoregressive
method 294
14.4 Generation of spatially correlated wind histories 297
14.5 Generation of earthquake histories 299
14.6 Cross-correlation of earthquake histories 303
14.7 Design earthquakes 303
References 305
Index 307