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References 299

Index

AAccelerated fatigue test, 215Accelerogram, 206Active damping, 290Active mass damper (AMD), 281Active strut, 287Active suspension, 280Active truss, 286Active vibration control, 275Angular rate sensor, 243Anisotropic shaft, 238

stability, 241unbalance response, 240

Anisotropic support (rotor), 236Anti-resonance, 32, 251, 276Assembly, 138Assumed modes method, 114, 135Asymptotic method, 88Asynchronous force, 232Autocorrelation, 170Autocovariance, 170

BBackward whirl, 221, 229, 234Bar, 94, 116

finite element, 137Beam

Euler-Bernoulli, 78, 119finite element, 140free-free, 87free vibration, 83prestress, 82, 122simply supported, 85

Beat phenomenon, 10Bending stiffness, 79Beta controller, 293

Bode plots, 7, 34Boundary layer noise, 188Buckling, 72

beam, 96clamped-free beam, 98critical load, 97simply supported beam, 97

CCampbell diagram, 62, 229, 235Cantilever rotor, 245Car on a random road, 191Car suspension

active, 280passive, 271

Cascade analysis, 167Causality, 184Central frequency, 179Central limit theorem, 174Centrifugal pendulum, 76, 271Centrifugal Pendulum Vibration Absorber, 76,

270Co-spectrum, 188Coherence function, 182Collocated control, 276, 278Collocated system, 32Complex coordinates, 220Conical mode, 234Conservation laws, 64Conservation of energy, 50, 66Conservative force, 49Consistent mass matrix, 142Constitutive equation

active strut, 287linear elastic material, 70plane stress, 100

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Constrained system, 34, 149, 153, 277Convection velocity, 189Convergence, 145Convolution integral, 5, 175Coriolis force, 243Correlation

function, 170, 188integral, 175matrix, 185, 187

Covariance matrix, 214Craig-Bampton reduction, 153Critical speed, 220, 230, 231Cross-correlation, 170

role of-, 193Cumulative mean square response, 173

DD’Alembert principle, 49Damping, 23, 24, 121

modal, 23Rayleigh, 24rotating, 221, 224

Davenport spectrum, 190Degree of freedom (d.o.f.), 45, 114Den Hartog, 251Difference equation, 36Discretization, 113Disk, 232Dissipation function, 55Dynamic amplification, 7, 26Dynamic flexibility matrix, 25Dynamic mass, 159, 166Dynamic Vibration Absorber (DVA), 248

EEffective force, 49Effective modal mass, 160Elastic support, 232Envelope (narrow band process), 181Epicycloid, 271Equal peak design (DVA), 251ESP, 243Euler

Bernoulli beam, 78critical buckling load, 97, 123theorem on homogeneous functions, 65

FFast Fourier Transform (FFT), 172Fatigue, 202

random-, 211

Feedthrough, 12, 27Finite elements, 135First-crossing problem, 203Flexural rigidity (plate), 100Forward whirl, 219, 221, 229, 234Fourier transform, 9Fraction of critical damping, 2Fraction of modal strain energy, 290Frahm, 248Frequency Response

Function (FRF), 8FRF estimation, 182

GGaussian process, 174Generalized coordinates, 44, 53, 136Generalized momentum, 66Geometric stiffness matrix, 72, 122

planar beam element, 147Geometric strain energy, 68, 71Gradient height, 190Gradient velocity, 190Gravity loads, 128Green strain tensor, 68Guyan

mass matrix, 154, 165reduction, 147stiffness matrix, 154

Gyroscopic effect, 53, 60, 224Gyroscopic forces, 61

HHalf power bandwidth, 177Hamilton’s principle, 50High-cycle fatigue, 202Holonomic constraint, 45, 64Homogeneous (random field), 187Homogeneous functions, 65

IIgnorable coordinate, 66Impulse response, 3Integral force feedback, 290Interlacing, 33, 276, 291Isolator

by kinematic coupling, 268corner frequency, 266linear-, 260passive-, 261relaxation-, 263six-axis-, 266

302 Index

JJacobi integral, 64Jeffcott rotor, 60, 218Jitter, 265

KKanai-Tajimi spectrum, 179, 199Kinematic constraint, 44Kirchhoff plate, 99Kronecker delta, 21

LLagrange multipliers, 63Lagrange’s equation, 53

with constraints, 63Lagrangian, 51, 52Lagrangian dynamics, 44Laplacian D

Cartesian coordinates, 101polar coordinates, 104

Laval, 220Lead compensator, 277Linear damage theory, 211Linear oscillator

Bode plots, 7dynamic amplification, 7free response, 1impulse response, 3Nyquist plot, 8quality factor, 8random response, 176state space form, 11

Localization matrix, 139Long rotor, 232Lumped mass matrix, 142

MMass matrix

bar, 117bar element, 138beam, 119lumped, 142planar beam element, 141

Master-slave (d.o.f.), 148Maxwell unit, 263Mean square (MS), 170, 171, 192

mass averaged-, 193Memory, 4

Modaldamping, 23decomposition, 23, 91mass, 21, 91participation factor, 159participation matrix, 164truncation, 24

Modal density, 104Modal spread, 268Mode shape, 19Moment-curvature relationship, 100Multi-axis excitation, 162Multiple natural frequencies, 21

NN-storey building, 36, 167

AMD, 281DVA design, 256random response, 195random response with DVA, 258seismic response, 155

Nabla rCartesian coordinates, 101

Narrow band process, 181Natural boundary conditions, 82Natural frequency, 19Nodal

circles, 107, 108diameters, 107, 108lines, 104

Non-conservative force, 51Non-holonomic

constraint, 45, 64Normal modes, 21Nyquist plot, 8, 34

OOperating Basis Earthquake, 208Orthogonal functions, 115Orthogonality, 20, 89

PPainlevé integral, 66Palmgren-Miner criterion, 211Parseval’s theorem, 9Participation factor, 159Peak factor, 202, 206Periodic structures, 108

Index 303

Phase plane, 181Plane truss, 136Plate

circular, 104Kirchhoff, 99rectangular, 102

Pole-zero pattern, 33, 276, 291Power spectral density, see PSDPrestress, 68, 82, 96, 122Principle of stationarity, 125Principle of virtual work, 47Projection matrix, 39PSD

definition, 171estimation, 172input-output (MIMO), 185input-output (SISO), 175matrix, 185, 187one sided-, 173

Pseudo-acceleration spectrum, 206Pseudo-velocity spectrum, 206

QQuality factor, 8, 26, 252Quarter-car model, 271Quasi-static correction, 27, 160

RRainflow, 212Random fatigue, 211Random vibration, 169Rankine’s model, 220Rayleigh

damping, 24distribution, 181Quotient, 21, 91, 124

Rayleigh-Ritz method, 113, 224Reduction

Craig-Bampton, 153Guyan, 147

Relaxation isolator, see IsolatorReliability, 203Residual mode, 27Resonance

frequency, 19linear oscillator, 2rotating force, 109, 110

Response spectrum, 206Rice formulae, 179Rigid body mode, 21, 28Root locus, 264, 277, 285, 291

Rotating force, 108Rotating mode, 107Rotor dynamics, 217Routh-Hurwitz stability, 224

SS-N curve, 211Safe Shut-down

Earthquake, 208Scleronomic constraint, 45Seismic excitation, 156Self-centering, 220Semi-positive definite (matrix), 16Shape function, 114, 135, 141Signal to noise ratio, 184Single axis excitation, 156Sky-hook damper, 280Slave (d.o.f.), 148Spatial coherence, 188Spectral moments, 179SRSS rule, 193, 209Stability, 224Standard deviation, 170State feedback, 281State variables, 12, 271, 280, 284Stationary process, 170Stiffness matrix

bar, 94bar element, 138beam, 120geometric, 122planar beam element, 142

Stodola-Green rotor, 245Strain energy density, 70, 100String, 92Strouhal number, 188Supercritical velocity, 220Support reaction, 159, 165

TTaipei 101, 253Tautochronic problem, 271Threshold crossing, 202Torsional stiffnessTransmissibility, 261, 264Traveling wave, 107, 109, 110Tuned Mass Damper (TMD), 248

UUnbalance response, 219, 231

304 Index

VVariance, 170Vibration isolation, 259Virtual displacement, 46Virtual work, 47von Mises stress, 213

WWhirl, 221, 229White noise

approximation, 177, 210band limited-, 178process, 174

Wind response, 189Wöhler curve, 211

ZZero (transmission-), 34, 250, 268, 276Zero-period acceleration (ZPA), 207

Index 305