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Index

σ method, 116n degrees-of-freedom system, 108

aberration, 152absolutely

integrable function, 36, 115invariable in time, 131

acceleration, 9, 111, 172Earth gravity, 175, 301, 359transitional, 172

accuracy, 49, 65, 119, 133, 303, 323, 355accurate

eigenvalues, 62estimation, 72series, 55solution, 13, 63

acousticexcitation, 8vibrations, 112

actionimpulse, 3in Hamilton sense, 20load, 9, 123, 151, 361

acute angle conditions, 5adhesive-plastic deformation, 15adjacent

maximums, 116peaks, 112

adjoined material, 19aeroelastic modes, 9aerospace, 235Airy functions, 241algebraic

equation, 1, 11, 55, 59, 63, 73, 78, 79, 97,107, 173, 184, 214, 243, 271, 273

linear, 56, 285nonlinear, 83

function, 136system, 12, 158, 187

algorithmcomputational, 294equilibrium interaction, 7finite differences, 355numerical, 41Runge-Kutta, 361

alloy, 167aluminum, 319, 322, 355amplification, 100, 102amplitude, 9

-frequency characteristics, 107, 121-modulated wave, 8buckling, 124characteristics, 325, 351, 355estimation, 304excitation, 214, 217, 240, 258, 260, 283,

307, 361Fourier series, 112harmonic

frequency, 120isolated, 118

initial irregularity, 154load, 276maximal, 116pre-buckled beam, 6series, 110vibrations, 236, 326

free, 121small, 286

analysisbifurcation, 7, 137chaotic vibrations, 294conservative system, 75deformation, 123dynamics, 27

391

392 Index

equilibrium, 11post-buckled, 46

error, 65frequency period, 107harmonical, 112perturbation, 126perturbed motion, 54rectangular shell, 2spectral, 95stability, 42, 138, 235

balance point, 130statics, 15structural, 88Theodore von Karman equation, 2

analyticalcomposite object, 54function, 12, 57, 58, 63, 131

differentiable, 113investigation, 8method, 305representation, 61solution, 100

Andronov“self-excited vibration”, 130

Andronov-Hopf bifurcation, 216, 219, 231,255, 261, 264, 270, 276, 277, 279, 294,310

angleacute conditions, 5deflection, 18flow, 9loading, 255pitch, 7rotation, 28sector, 209

angularfrequency, 109plate, 8

anisotropicsolid body, 4

anisotropic material, 15anisotropy, 237anti-symmetric modes, 8antiphase, 271, 274, 279aperiodic

attractor, 230load, 125

applicationFourier transformation, 249method

Bubnov-Galerkin, 185, 294discrete, 59Galerkin, 57, 64, 73MBG, 1–3, 5, 6, 13

non-ideal load, 47stability criterion, 45Taylor series, 65

approachboundary problem, 2Bubnov, 62

-Galerkin, 183Euler, 226Galerkin, 9isoparametric, 74linear, 154Melnikov-type, 6numerical-perturbation, 9static, 324statical, 42stroboscope mechanism, 116variational, 45

approximatefunction, 56, 75, 79, 240method, 4, 54, 77model, 139shell, 95solution, 1, 5, 13, 55, 57–59, 63, 65, 67

convergence, 13approximation, 1, 61

N-termed, 67O(h2), 323, 357, 360, 370accuracy, 119Bubnov-Galerkin method, 287finite differences, 207first mode, 6function, 55harmonic series, 113high order, 41, 76higher order, 95, 96, 120, 249, 281initial, 83numerical, 108parabolic, 116

arbitrarycurve, 28displacement, 52dynamical system, 130function, 37functional, 1loading parameter, 43parameter, 55position, 7superposition, 113supersonic flow, 9

arc, 210slice, 206, 212

areaattraction, 129chaos, 286

Index 393

chaotic, 230, 279harmonic vibrations, 290

artificial excitation, 246asymmetric, 372

imperfection, 153problem, 155vibrations, 127

asymmetry, 369asymptote, 47asymptotic trajectory, 231asymptotically stable, 128, 130asynchronic chaotic vibration, 279attraction, 129, 229

basin, 6boundary, 130

attractor, 135, 211, 220, 226, 309“axiom A”, 229aperiodic, 230chaotic, 6merging crises, 7strange, 8, 228

auto-parametric, 9autonomous

system, 129autonomous system, 8, 130auxiliary convergent operator, 5average

curvature, 151sense, 59, 287

averagedshell, 264solution, 241

averagingprocedure, 8technique, 6

axialcompression, 153curve, 197deflection, 152displacement, 52force, 51load, 10, 136, 151strain, 236stress, 297, 320

axiallyaccelerating beam, 7compressed

rod, 10shell, 154

symmetricshallow shell, 3vibrations, 127

axiom A attractor, 229axis

deflection, 166longitudinal, 6, 51perpendicular, 16real, 38rod, 47rotation, 206symmetry, 84, 91

axisymmetricmode, 236vibrations, 9

baffle plate, 8balance, 48, 124, 128, 130–132, 227

point, 124, 125, 127trajectory, 138

ball, 48-type

clamping, 206resistance contour, 216support, 28, 239, 243

support, 323, 325, 349band

loading, 3base

arbitrary, 5elastic, 48rectangular, 9vertical excitation, 7

basicsystem, 5, 125theorem of MBG, 2

beam, 6, 48-impacting mass, 370-mass structure, 7Bernoulli, 193Bernoulli-type, 357buckled, 6

undamped, 7constrainded, 297deflection, 302, 303driven, 300Euler-Bernoulli, 358length, 366light, 232material, 305, 313motion, 353pinned, 358pre-buckled, 6quadrant, 362separation, 322slip, 321structural member, 9thickness, 321Timoshenko, 320

394 Index

vibrations, 173bearing, 7Belousov-Zhabotinskiy reaction, 232bending, 143, 251

moment, 268stiffness, 300, 316strain, 236

Bernardconvection, 232instability, 229

Bernoullibeam, 193, 357hypothesis, 320

bifurcation, 43, 50, 84analysis, 137, 235Andronov-Hopf, 219, 255, 261, 264, 276,

279, 294, 310branch, 125chaos interlacing, 216collapse, 332five period doubling, 219fold-type, 136Hopf, 8, 9, 227, 228, 266, 269, 330, 371lack, 264period doubling, 307period tripling, 306period-doubling, 251pitchfork, 9point, 47, 50, 139, 154, 236series, 355set, 7static, 123, 135symmetric, 138trace, 355triple, 325tripling period, 313zone, 212, 215, 217, 249

bifurcationalbehaviour, 6determination, 45stability, 41, 49

loss, 53bisectrix, 210, 212“black box”, 129blue sky catastrophe, 7body

contacting, 175deformable, 42, 47impacting, 191lumped, 171, 189, 191, 369solid, 4, 49

border, 52, 110boundary, 9, 28, 89, 301

attraction, 130

conditions, 4, 11, 13, 28, 46, 52, 56, 63, 65,71, 73, 79, 96, 106, 126, 141, 152, 165,168, 179, 183, 184, 190, 206, 216, 238,243, 254, 272, 282, 284, 286, 291, 311,322, 357, 370

coordinates, 114fractal basin, 6geometrical, 52integral, 22, 75loading, 254merging crises, 7point, 153problem, 2, 76, 120, 144, 160region, 57space, 225stability, 125transition, 37

boundedattractor, 211by

closed curve, 16contour, 205

frequency, 109region, 20, 33

bridge, 10Bryan energetical criterion, 54Bubnov-Galerkin method, 1, 3, 10, 11, 13, 41,

55, 56, 74, 75, 79, 95, 96, 106, 141, 144,160, 165, 168, 183, 185, 190, 235, 240,241, 248, 253, 270, 273, 281, 286, 291,294

buckledbeam, 6equilibrium state, 50non-perfect shell, 124rod, 51

buckled beam, 7buckling, 123, 125, 128, 140, 143, 154, 156,

168dependence, 152dynamic, 124dynamics, 8load, 124, 144, 146, 155, 158, 160

depreciation, 153static, 166

shell, 188snap-through, 9stresses, 297, 320

buckling load, 124by-product, 4

canal, 231cantilever beam, 7, 297, 319, 320cascade, 7, 219, 226, 354

Index 395

period-doubling, 9catastrophe, 135, 136

blue sky, 7elliptic, 137higher order, 139

“catch up”, 123Cauchy

-Dirichlet problem, 2problem, 187, 208, 243

centerof

beam deflection, 303curvature, 16shell, 144

centraladdition, 101, 103deflection, 191element, 87, 100

square, 41hole, 208layer, 17, 168maximum, 116point, 124, 287shell, 210, 256surface, 16, 18, 23, 98, 272, 285

chain mapping, 271chaos, 9, 123, 132, 133, 135, 139, 205, 212,

219, 221, 223, 232, 235, 251, 259, 262,268, 271, 276, 278, 279, 297, 307, 313,320, 330, 354, 357

“real”, 283area, 286bifurcation interlacing, 216control, 274development, 135horseshoe, 8onset, 6route, 8

intermittency, 7slight degree, 326subduction, 230surface, 270time-spatial, 226, 279transformation, 230

chaoticarea, 230attractor, 309behaviour, 9dynamics, 6, 7, 9, 274, 355, 367extras, 231flow, 226impacting, 358mapping, 233motion, 132, 135, 228, 258, 275, 329

state, 353subset isolation, 279threshold, 7vibrations, 214, 217, 223, 231, 236, 251,

278, 287, 291, 294, 297, 305, 307zone, 212, 249, 255

Chebyshev polynomials, 65Chetayev theorem, 128circle

section, 235circle section, 28, 151circuit, 11circular

coordinates, 161, 274, 279cylindrical shell, 10, 237intersection, 240, 272

clamped-free boundary, 357beam, 6, 307edge, 305, 366end, 311plate, 8shell, 3, 212, 216, 218

clamping, 180, 210, 239, 301-clamping, 360-for, 349conditions, 322edge, 282hinged, 302moveable, 216pressure, 299sliding, 208symmetric, 362

“clamping-free edge”, 360classical

Galerkin method, 62shells theory, 16Vorovich’s monograph, 13

close-up, 51body contact, 175

closedcurve, 16form, 54shell, 9, 235, 237, 245, 254, 272trajectory, 130

coefficientamplification, 101, 102damping, 25, 206, 282, 301, 321, 325, 357,

359, 361, 370deformation, 299density, 36, 99, 148determination, 57dissipation, 359dynamical, 188

396 Index

Fourier, 114, 115periodic, 241Poisson, 17, 163, 240, 272proportionality, 299, 301rise variation, 181stiffness, 15, 83, 84, 90, 92, 99, 105, 145

flexural, 34variable, 54

coexistence of equilibria, 43coherent methods, 65coincidence, 366collapse

of bifurcation, 327period tripling, 313

collapse of bifurcation, 355collapse of synchronized motion, 307collateral tails, 110collocation method, 11, 55, 65combination

linear, 2, 79, 362combined thermal effect, 9compaction of thickness, 19compatibility of

conditions, 208deformations, 26, 28, 55, 241

completely stable, 43complex

dynamical approach, 42dynamics, 355half-plane, 227modal model, 298structure, 89value, 36vibrations, 216, 294

complexity, 42temporal, 8

componentharmonic, 108, 118

composite, 41, 156beam, 320function, 78

composite beam, 297composite laminated plate, 9composite object, 54composite shell, 163, 165composition catastrophe, 136compressed rod, 10compressibility condition, 226compressible liquid, 60compression, 17, 153, 230compressional damping, 297compressional vibrations, 319computation

elastic strain energy, 10

harmonics, 120manifold, 7

computationalalgorithm, 2, 6, 28, 85, 294, 300, 301, 320,

355, 372method, 3, 58, 283results, 349time, 15, 63

computations, 3, 73, 117economics, 15stability, 42

concentrated mass, 9condition

accurate series, 55boundary, 28, 52, 58, 63, 73, 79, 106, 152,

165, 168, 183, 184, 190, 216, 238, 239,243, 272, 282, 284, 286, 291, 311, 322,357, 360

geometrical, 52uniform, 64

compressibility, 226convergence, 62Dirichlet, 113Dirichlet kernel, 115existence, 47fixing, 49initial, 65, 144, 178, 179, 233, 324intersection, 174irregularity, 233minimization, 68multiplicity, 119necessary, 13, 48orthogonality, 56solution

exact, 65stability, 286

stationarity, 46sufficient, 7, 139symmetry, 208

conditional branch, 43conditioning, 71conditioning equation, 27conditions, 20, 69, 263

boundary, 4, 11, 13, 28, 46Dirichlet, 113equilibrium, 42load, 9MBG, 5sufficient, 7

conicalshell, 358surface, 203

console beam, 326constant

Index 397

acceleration, 176Feigenbaum, 268Nyquist, 118rigidity, 272stiffness, 281velocity, 193, 197, 200, 357

constraints, 52, 83beam, 297ideal, 42one-sided, 171, 190, 203two-sided, 176

contactarea, 188between beams, 301continuous, 173load, 187loss, 191one-sided, 299pressure, 299, 308, 310, 314, 316shell-mass, 194stress, 322surface, 48, 185, 191vibrations, 298zone, 193, 194, 301, 320, 355

continual system, 207continuity conditions, 46continuous

contact, 173, 194function, 72, 178

differentiable, 38functional, 37inner product, 59model, 34parameter, 110partially, 113potential operator, 5signal, 117solution, 4spectrum, 354system, 41, 271, 288, 298, 324, 357total energy, 46

continuum, 46contour

bounded, 205curve, 180

contractual rarefaction, 19contradistinction, 101control, 153, 220

beam, 297chaos, 271, 274, 279electromechanical, 8parameter, 6, 91, 95, 100, 106, 136, 166,

205, 211, 219, 230, 255, 264, 305, 357point, 65, 258

controllable synchronization, 279controlled horseshoe chaos, 8controller, 298, 320convection, 227convective flow, 63convergence, 37, 59, 69, 72, 83, 95, 114, 118,

215, 246, 250approximate solution, 13Bubnov-Galerkin method, 248, 273condition, 62energetical, 71finite differences, 291Fourier series, 113, 248geometrical, 230method

finite differences, 293MBG, 2, 4, 5Newton, 191Runge-Kutta, 286

rate, 63convergent

curve, 246energy, 70Feyer summation, 113Fourier series, 69, 115linear space, 37operator, 5, 6

convex surface, 48convexity, 161convolution theorem, 117coordinates, 17, 30, 40, 42, 45, 74, 80, 103,

109, 120, 155, 160, 164, 172, 181, 185,226, 227, 260, 272, 281, 321, 358

boundary, 114circled, 264circular, 161, 279generalized, 137loading, 254polar, 205spatial, 1, 28, 107, 210, 240, 283, 294, 324,

357Coriolis acceleration, 172correction of solution, 83cosine, 109Couette flow, 227coupled

beams, 298moduli, 302non-linear equations, 9oscillators, 172thermoelasticity problems, 6

crack, 9, 297, 319crisis, 7, 230criterion

398 Index

energetical, 41, 49, 51, 53, 55exact, 61stability, 45, 124, 125variational, 47Volmir, 144, 168, 255

criticalbifurcation point, 50load, 3, 42, 43, 47, 56, 76, 85, 89, 91, 92,

125, 136, 188, 245, 246, 254, 263, 274,293

longitudinal, 279parameter, 42, 189pressure, 237state, 45, 55, 168surface size, 104time, 372

critical loading, 188cross

-typeheterogeneity, 147non-homogeneity, 91

addition, 41, 90section, 189, 235, 237, 251, 266

cryogenic fuel tank, 236cubical cell, 232curvature, 16, 152, 240, 246, 268, 272, 282,

299average, 151non-dimensional, 140radius, 28, 181, 206

curve, 11, 42, 43, 85, 87, 92, 100, 119, 124,143, 144, 153, 154, 156, 166, 168, 179,189, 192, 194, 196, 200, 209, 222, 246,254, 313, 321, 366

“deflection-loading”, 89axial, 197characteristic, 47closed, 16control, 278equilibrium, 44, 45family, 137load-relief, 325logistic, 266monotonic, 154parameterized, 137response, 236separating, 130space-located, 28

curved shallow, 9curvilinear circle, 243cutout, 30, 35cycle, 229, 231

limit, 9unirotational, 287

cyclic-fold bifurcation, 9load problem, 3

cylindricalpanel, 178shell, 3, 9, 28, 128, 151, 152, 154, 172, 235,

237, 240, 243, 245, 254, 255, 264, 273intersection, 278

d’Alembert principle, 174, 300damage, 9damped, 127

beam, 7, 300, 319, 320sandwich, 298vibrations, 305, 312, 317

damping, 100, 127, 133, 135, 139, 143, 168,194, 246, 272, 320, 357, 362, 366, 370

amplitude, 120coefficient, 25, 206, 240, 282, 325, 361compresional, 297linear, 126

decomposition, 114, 251function, 69, 113geometrical, 84

defect, 298definition, 36, 37, 49, 59

axial force, 52extra displacement, 53global stability, 260inner product, 12limit, 112non-homogeneity, 1stability loss, 54static deformation, 18weighting function, 74

deflectedrod, 51shell, 278surface, 48

deflection, 18, 47, 53, 78, 84, 85, 88, 96, 123,124, 151, 153, 156, 158, 161, 166, 168,175, 181, 188, 189, 193, 200, 258, 266,271, 278, 282, 321, 324, 357, 358, 361,372

-loading dependence, 78-time relation, 255amplitude, 153asymmetry, 369averaged, 261beam, 301, 303

hinged, 305central, 191critical, 169description, 299

Index 399

distribution, 322dynamical, 172, 197equilibrium, 95finite, 6function, 240initial, 152isoclines, 209, 214maximum, 274minimum, 223over-critical, 274, 279pre-critical, 254shell, 2, 161, 194, 211small, 313state, 83surface, 197, 308, 325time history, 107, 306, 309, 312, 315transversal, 173

“deflection-loading”, 78“deflection-loading” curve, 89deformable

body, 47, 171surface, 174, 175system, 42

deformation, 16, 17, 20, 85, 123, 126, 299,301, 358

adhesive-plastic, 15axial, 298compatibility, 26, 55, 241diagram, 313elastic, 48equation, 23, 28field, 3flexural, 153function, 22intensity, 303, 304, 321, 325, 349linear, 47local, 175, 203modulus, 321non-

axial, 237linear, 237

passive, 136piezoelectric actuator, 320shell, 28surface, 17, 18, 174

degree of heterogeneity, 144delay in

control, 8Duffing eqaution, 7

delta type functions, 117density, 19, 28, 272, 321, 354, 359

coefficient, 99, 145constant, 281fluid, 225

parameter, 161periodic points, 233shell, 36weight, 301

dependencycontinuous solution, 4dynamic coefficient, 103functional, 28harmonic frequency, 119load-time, 188nonlinear, 101Young modulus, 1

derivative, 72generalized, 3, 38high order, 39, 113normal, 38partial, 240

describing function, 115destabilization, 139diagonal

direction, 61matrix elements, 74

diagrambifurcation, 7, 8chaotic zone, 6

diameter, 176, 208diffusion, 61dimension

contact, 189fractal, 8Lyapunov, 10reduction, 74spatial, 8

dimensionalparameter, 179

dimensionlessboundary conditions, 370parameters, 359

Dirac delta, 60Dirichlet

conditions, 113kernel, 114, 115theorem, 56

Dirichlet theorem, 46discontinuity, 216

first order, 217, 223point, 113, 114

discontinuous load, 237discrete, 59

phase space, 129signal, 117solution, 6spectrum, 354system, 136, 139, 288

400 Index

value, 65, 254discretization method, 358dispersion, 134displacement component method, 320dissipation

coefficient, 359force, 139of energy, 134, 136, 225

dissipativebeam, 319, 355force, 306system, 272, 361vibrations, 135, 372

distortion, 236distributed

load, 168, 191mass, 319

distribution ofamplitudes, 135energy, 146error, 66load, 254, 274pressure, 156temperature, 4

disturbance, 43, 225dispersion, 134

Donnell theorem, 10double

asymptotic trajectory, 231integral, 30, 33precision, 108

doublingbifurcation, 332five period, 219inifinite series, 230period, 214, 230point, 269torus, 7

“dovetail”, 137driven

beam, 300, 357, 361shell, 254

Duffing equation, 7dynamic

amplification, 102, 103buckling, 8, 123, 124, 144, 160

area, 125critical loading, 3stability, 135, 191, 237

dynamicalanalysis, 27approach, 42behavior, 9, 15, 172, 308criterion, 45, 140, 168

Volmir, 255deflection, 197four-dimensional, 8hypothesis, 28load, 357method, 54problem, 1, 3, 5, 6, 159reaction, 173system, 128

autonomic, 130infinite DOF, 120

dynamically stable, 125dynamics

beam, 301chaotic, 6, 7, 9, 274complex, 355equation, 106flying objects, 11nonlinear, 297of mapping, 230shell, 15, 288

rectangular, 190stability, 123

dynamism coefficient, 101, 104

edge, 9, 19, 30clamping, 282free, 180, 283load, 357loaded, 33simply supported, 28

effectivenumerical method, 6velocity estimation, 5

eigen-elements system, 5functions, 66, 127values, 45, 62vectors, 45, 99

elastic-plastic

beam, 355deformations, 299material, 302

-plastic material, 322-plastic problem, 313base, 48beam, 317, 320, 349bearing, 7foundation, 9limit, 157material, 15problem, 325shell, 190

Index 401

stability, 41, 51, 136stop, 358strain energy, 10surface, 173system, 54, 152

deformations, 49stability loss, 125

wave, 27elasticity, 20

modulus, 16, 19, 321theory, 15, 38, 49volume, 299

electromagnetic field, 171element

additional, 19, 84, 102central, 41, 88, 100cross-type, 91finite

linear, 63method, 6, 11, 183

non-homogeneous, 92rigid, 85rotational, 25shell, 15, 27, 30, 56, 83

separated, 32strengthening, 100surface, 90

elementarycatastrophe, 139potential energy, 164works, 20

elementary bifurcations, 136elliptic

-type partial equation, 2catastrophe, 137

energeticbarrier, 154space, 5

energeticalcriterion, 46, 49, 51, 53, 55finite norm, 73method, 11, 74norm, 5, 70

energeticallynormalized, 273orthonormalized, 69, 283

energy, 69“pumping”, 125cascade, 354convergence, 70criterion, 41dissipation, 126, 134, 136, 225, 319distribution, 146kinetic, 20, 25

linearization, 55operator, 73potential, 10, 50, 136, 139, 164

total, 48, 51source, 127system, 46, 54transfer, 9

Englis-Bolotin method, 173enhances, 135equation, 10, 194

algebraic, 73, 79, 214, 243beam, 297, 301deformations, 28differential, 47

partial, 1, 60dimensionless, 359dynamics, 106equilibrium, 300forces, 164fourth order, 241fundamental, 29, 181heat transfer, 4Holmes-Duffing, 6hyperbolic-type, 2initial, 96integral, 57

-differential, 8linear, 49, 285

differential, 67linearized, 45, 155matrix, 13motion, 20, 151Navier-Stokes, 2, 225Newton, 129non-

coupled, 75dimensional, 165, 179, 238

numerical integration, 120operational, 70partial differential, 9, 210PDE, 240, 271phase-modulation, 7quasi-linear, 5shell, 163solution, 71stability, 54, 152variational, 25, 26

equilibrium, 45, 46, 48, 51, 241closely located, 55conditions, 42deflection, 95equation, 56, 300interaction, 7

402 Index

point, 49, 54, 75, 134, 139, 153, 155, 156,246

rectilinear form, 53state, 11, 43, 44, 50, 125, 130, 137, 191,

317, 320equivalent

deformation, 18Galerkin method, 64

equivocal function, 78error, 2, 12, 57, 61, 63, 141, 305

distribution, 66estimation, 6, 67, 119, 237extremes, 65, 71finite differences method, 303function, 56inner product, 59MR, 5weighting method, 64, 66

estimatedcritical loading, 47, 76equilibrium, 46solution, 63total energy, 49

estimation, 175coefficients, 13eigenvalues, 45, 99error, 2, 67, 72, 119, 237frequency

spectrum, 108, 120vibrations, 98

impulse influence, 3Lyapunov exponents, 8MBG, 6numerical, 123solution, 65stability, 76strength, 41

Euler-Bernoulli

beam, 320, 357hypothesis, 298, 299, 319

-type coefficient, 188equations, 1load, 140method, 47, 54, 226numbers, 112rod, 138

evolutional problem, 2, 3, 5exact

criterion, 61model, 15solution, 5, 56, 63–65, 69–71, 76, 108, 119,

183, 210excitation, 211, 236, 255

amplitude, 214, 217, 258, 307artificial, 246force, 260frequency, 219, 251, 264, 269, 325, 357,

361, 369harmonic, 208, 305initial, 107large, 44mechanical, 9mode, 9parametric, 8sign-changeable moment, 223stiff, 132transverse, 320vertical base, 7

excitedbeam, 297, 358

cantilever, 320cantilever system, 7shell, 205

existence ofsolution, 271chaos, 9chaotic attractor, 6, 7solution, 3, 4, 56

experimentaldata, 128load estimation, 188stand, 124

explosion of density point, 36external

edge, 30excitation, 293, 297, 320

harmonic, 9force, 6, 20, 25, 47, 48, 51, 56, 75, 126, 212,

225frequency, 308load, 50, 127, 206, 237, 240, 245, 261, 266,

271, 282, 313, 370harmonic, 254

pressure, 235reaction, 173tank, 236

extortion, 134extrapolation, 84, 158extremal

frequency, 119point, 60, 89resistance, 41

factorization, 74Faedo-Galerkin method, 2feedback, 298feedback controller, 320

Index 403

Feigenbaumconstant, 268scenario, 218, 225, 262, 268transition, 332zone, 212

Feyer summation, 113fibre, 16field

deformation, 3gravitational, 176potential, 228thermal, 9

fifth order approximation, 3filtrate action, 116filtrate property, 31, 39finite

-time stability, 123, 125bending stiffness, 316deflection, 6destabilization, 139differences algorithm, 355differences method, 3, 63, 76, 141, 143, 183,

207, 210, 281, 283, 288, 294, 323, 360error, 303

dimensional problem, 370elements method, 73, 357energetical norm, 73series approximation, 120

first-order

characteristics, 107discontinuity, 217Lagrange equation, 175ODE, 243

mode, 6, 8type elasticity, 17

fixing conditions, 49flatter, 130, 135, 139flexible

beam, 372construction, 171Euler-Bernoulli beam, 357plate, 18rib, 28, 180, 239, 282shell, 89, 220, 237, 281, 286, 294

flexuraldeformation, 153mode, 9stiffness, 15, 19, 34, 35, 83vibrations, 8, 237

flow, 60, 124, 226coefficient, 299convective, 63Couette, 227

deformation, 303, 321, 350rate, 60supersonic, 9

fluctuations, 48, 84, 133, 135fluid, 171, 271, 358

mechanics, 57, 60flutter panel, 9focusing action, 115, 116fold-type

bifurcation, 136catastrophe, 138stability, 138

forcecompressing, 47conservative, 46dissipation, 139dissipative, 306excitation, 45, 160, 215, 240, 260, 272, 287,

352external, 20, 25, 48, 126, 225frequency, 283inertial, 42, 123, 187initial, 52internal, 56iterational, 171longitudinal, 360moment, 29moving, 173, 176, 202normal, 8periodic, 357periodic with impacts, 7potential, 75self-interaction, 195statically determinable, 52supercritical, 358vector, 126weighting, 172

forcedHolmes-Duffing equation, 6orthogonality, 56response, 236shell, 281

formdiscrete, 59matrix, 74non-

dimensional, 15, 177linear, 2

normal, 191operational, 288Timoshenko, 53variational inequality, 4

Fouriercoefficients, 114, 115

404 Index

series, 10, 73, 112, 113, 156, 248transform, 36, 69, 109, 117, 120, 249, 273

fractal, 229basin, 6orbit, 7structure, 268

Frechet differential, 5free

boundary, 302contour, 216edge, 180, 283, 366, 372end, 305from stresses, 125rotation, 6support, 239, 273, 322, 349vibrations, 9, 106, 107, 120, 127, 222, 319

flexural, 237frequency, 212

frequencyangular, 109beam mode, 6bifurcation, 355characteristics, 362definition, 110excitation, 208, 215, 240, 272, 274, 283,

313, 325, 357, 361external, 308fundamental, 304harmonics, 107hidden, 112input function, 277natural, 9, 96, 249, 297, 320Nyquist, 120peak, 118period, 107perturbation, 140radiation, 112region, 118spectrum, 120, 264, 281, 287, 291, 305, 370synchronization, 221, 232vibrations, 55, 98, 100, 206

free, 305linear, 276

friction, 298full

convergence, 287synchronization, 221, 279, 307

functionAiry, 241amplitude, 110analytical, 12, 58, 113, 131approximate, 75autocorrelation, 8boundary conditions, 106

composite, 78contact, 299continuous, 178deflection, 173, 240deformation, 22delta type, 117equivocal, 78filtering property, 31focusing action, 115generalized, 15, 36–38Heaviside, 31impulse, 40in antiphase, 274inverse, 35minimization, 71nonlinear, 130orthonormal, 65periodic, 117, 133positively defined, 47potential, 136stepping, 34strain, 282stresses, 27, 206test, 61, 62, 64, 65, 67, 68, 72unity, 33variation, 23, 24, 152velocity, 322weighting, 59, 60, 62, 63

functional, 37, 38, 50, 70, 129dependency, 28minimum, 68operator, 55singularity, 1

fundamentalamplitude, 305energetical criterion, 49equation, 29, 181, 297, 320excitation period, 310frequency, 304, 325

gain, 50Galerkin

-Pietrov method, 5method, 6, 9, 41, 55, 57, 62, 63, 65–67, 69,

73, 74, 84coherent, 65equivalent, 64modification, 63traditional, 64, 73

projection, 7, 57solution, 63, 71

gas-fluid interaction, 3Gauss method, 107, 141, 187general

Index 405

function, 113solution, 136stability loss, 125

generalization, 4, 36, 37, 75generalized

coordinates, 137derivative, 3functions, 15, 36–38method

Galerkin, 63Melnikov, 7

geometricconditions, 29nonlinearity, 3, 9, 47, 235, 237, 298parameters, 148, 160

geometricalboundary, 52, 55convergence, 230decomposition, 84imperfection, 152

globalcritical load, 261equilibrium, 135function, 62shell behavior, 93stability

loss, 270stability loss, 41, 260

Green formula, 60

Henon model, 230half-

positive function, 47trajectory, 131wave, 254, 260, 264

Hamilton principle, 20, 25, 297, 320Hamiltonian system, 7hard stability loss, 85hardening, 300, 303, 324harmonic

analysis, 113convergence, 95distortion, 236excitation, 9, 208force, 274form, 304frequency, 107, 119, 120input function, 276load, 205, 254, 256, 283oscillator, 133resonance, 229vibrations, 8, 107, 127, 212, 214, 217, 219,

226, 259, 270, 277, 279, 286, 290, 293,298, 317

harmonical analysis, 112harmonically

excited, 205, 216, 281shell, 211

harmonics, 99, 100, 112, 121heat

conductions, 60exchanger tubes, 8field, 171transfer, 4, 11

Heaviside function, 31, 34Hessenberg form, 99heteroclinic orbit, 7heterogeneity, 144, 147, 163heterogeneous shell, 144, 146, 148hidden frequencies, 109high

-amplitude vibrations, 139accuracy, 112frequency, 358

higher-approximation method, 168approximation, 2, 106, 120, 160, 183, 190,

249, 281modes, 101

Hilbert space, 73hinged

beam, 313hinged beam, 304hinged clamping, 302hinged support, 180hole, 208Holmes-Duffing equation, 6holonomic constraints, 174homoclinic orbit, 7homogeneous

conditions, 243equation, 246initial state, 54plate, 8, 95, 105, 107, 121shell, 28, 35, 84, 87, 89, 90, 92, 100, 141,

143, 145, 149, 163, 165, 166, 168solution, 240, 242

Hook law, 15, 17, 305Hopf bifurcation, 8, 9, 139, 219, 226, 228, 266,

269, 307, 312, 327, 328, 330, 355, 371horseshoe, 7, 229Housholder transform, 99hybrid

control, 298, 320form, 165, 178, 240

hydrodynamics, 226, 229, 271, 355hyperbolic

-type equation, 2

406 Index

catastrophe, 137, 139equation, 4, 6

hyperbolic equation, 2hypersonic vehicle, 236hypothesis, 16

Bernoulli, 320dynamical, 28Euler-Bernoulli, 298Kirchhoff-Love, 179linearity, 95straight normals, 299

hysteresis, 325

ideal interpolating function, 118identification, 249, 275, 297, 320, 357, 372identity function, 55impact, 171, 174, 176, 190, 311, 317, 325, 358,

369chaotic, 358Faedo’s, 2force, 7, 123, 319two-sided model, 7wave, 171, 357

impacting tubes, 8impactor, 357imperfection, 47, 63, 124, 138, 140, 151–153,

155, 246method, 54

implicit Newmark method, 9impulse

action, 3function, 38–40load, 127singular, 117

in large stability loss, 188in-span attached mass, 319inclination, 110inclusion, 43

sub-differential, 4inertia, 25, 123, 164, 193

beam, 306inertial force, 42, 139, 171, 187infinite

dimensional problem, 360dimensional system, 7, 57, 120, 210, 281,

357Fourier series, 115series, 112, 117

infinitely small damping, 139infinity, 38, 54, 114inflection point, 124, 137initial

approximation, 84, 158, 246axial force, 53

boundaryconditions, 58, 206, 291problem, 2, 281

buckling amplitude, 124conditions, 13, 28, 62, 65, 96, 130, 144, 159,

168, 178, 179, 191, 233, 240, 243, 254,286, 302, 304, 322, 324, 360

configuration, 42deflection, 152displacement, 107equation, 77equilibrium, 46, 49, 50general solution, 136imperfection, 140, 154irregularities, 152parameters, 7state, 44, 52, 54, 76system, 45time, 109, 317velocity, 176

inner product, 57, 59, 61, 67input

data, 110, 116, 118function, 279

longitudinal, 274periodic, 271

signal, 117, 129instability, 8, 45, 133, 135, 136, 138

aerodynamic, 139Bernard, Taylor, 229internal, 9region, 42snap-through, 10solution, 63static, 41

integrable mapping, 230, 233integral, 26, 28, 32, 53, 55, 67, 97, 183, 323

-differential equation, 8boundary, 22, 24, 75Bubnov-Galerkin method, 77, 80, 185, 242,

284computation, 178double, 30equation, 173

partial, 57estimation, 304Fourier, 113manifold, 130matrix, 133multiple frequencies, 325representation, 244

integration, 20, 33, 61, 108, 286by parts, 22, 25, 28, 38, 75numerical, 67, 237

Index 407

procedure, 120Runge-Kutta method, 9series, 113space, 301step, 141, 178, 191with weights, 115

intensity of deformation, 299interference, 111, 112interior, 31interlacing trajectories, 268intermittency, 7, 231, 233internal

crisis, 230force, 52, 56, 75instability, 9resonance, 7, 8, 236self-balancing load, 320

interpolating function, 118interpolation, 111, 208

parabolic, 116intersection, 174

circural, 272Poincare, 287

inversefunction, 35matrix, 99, 291rule, 67

irregularity, 111, 154, 328isolated

chaotic subset, 279harmonic, 118matrix, 133maximum, 110peak, 112, 120point, 113, 294

isoparametric approach, 74isotropic, 16

material, 15, 163, 166, 299plate, 9solid body, 4

iteration, 83, 107iterative method, 246

“jump”, 210jump down phenomenon, 7, 322

Kantor criterion, 140kernel, 114, 115kinematic

loading, 43nonlinearity, 298viscosity, 225

kinematical model, 29, 178, 266, 271, 300kinetic energy, 20, 25, 164

Kirchhoff-Love model, 3, 16, 17, 29, 163, 178,180, 266, 271

Lagrange-Dirichlet theorem, 46Lagrange

equation, 175multipliers, 112, 174theorem, 54, 75

Lamerey diagram, 231laminar

flow, 226layer, 61phase, 231

Landau-Hopf scenario, 218, 225, 227, 262large sense, 44lateral surface, 30law

Hook, 15, 17, 305linear temperature, 4mass conservation, 60non-linear control, 8nonlinearity, 124Runge, 141, 143, 291

least squares method, 55, 61, 64Lebesque-class, 3light

beam, 232limit

cycle, 9, 130, 131, 139, 231definition, 112point, 137, 154stability, 128

limiting mechanism, 134linear, 95

approach, 154beam, 7, 358combination, 174, 184, 217, 219, 362, 367,

371damping, 126deformation, 47differential equation, 11, 194equation, 49, 83, 152, 273, 285

algebraic, 214, 243differential, 67

finite element, 63model, 171operator, 129

differential, 73PDE, 241quadratic regulator, 298, 320self-coupled problem, 1solution, 58strain, 300, 303, 324strengthness, 322, 355

408 Index

system, 42, 68vibrations, 121, 191, 206, 212, 276viscoelastic material, 8

linear mode, 8linearity, 124linearity hypothesis, 95linearization, 47, 55, 237linearized

equation, 45, 47, 56, 153, 155, 227Poincare transformation, 231

linearlyelastic, 316, 325, 349, 355independent, 13, 68, 71, 74, 240, 284, 362

liquid, 60, 61, 225little sense, 43load

-deflection relation, 154-relief, 325-time dependence, 140, 188acceleration, 198asymmetric, 151axial, 136buckling, 124, 140, 152, 155, 158, 161conservative, 136constraints, 42contact, 187critical, 3, 42, 43, 47, 76, 89, 136, 210, 245,

261, 263, 279, 293cyclic, 3external, 240, 272, 282, 370finite-time, 127harmonic, 254, 281impact, 319impulse, 127in antiphase, 275monotonic increasing, 85non-

homogeneous, 246symmetric, 266

normal, 177, 188one-parameter, 138outer, 126parameter, 206post-critical, 209pre-critical, 143pressure, 321self-balancing, 320sign changeable, 361sign changing, 218sinusoidal, 317spatial, 8static, 125, 237

buckling, 166time-dependent, 237

transversal, 78, 84, 123, 184, 243, 278, 285,300, 304, 322, 358

uniform, 245“load-buckling”, 166“load-deflection” diagram, 254loading, 54, 80, 96

angle, 246, 254, 270critical, 45, 90, 91, 93, 188degree, 278external, 6, 50harmonic, 205monotonic, 42sinusoidal, 270static, 88, 245symmetric, 155zone, 254

“loading-deflection characteristics”, 84

loading-deflection characteristics, 91local

deformation, 203extremes, 104, 160, 219, 261potential energy, 136stability, 85, 91, 260

loss, 41, 149stability loss, 270thickness change, 19weighting function, 62

locallyintegrable, 37, 38stable, 44

locking phase phenomenon, 309locus

equilibrium, 42, 45, 46, 48single, 43

longitudinalcoordinate, 238force, 360impact, 369input function, 274, 276, 279load, 277movement, 52rod axis, 6, 51vibrations, 320wave, 260

loop, 78, 268, 325Lorenz model, 230, 231, 253, 287loss, 3loss of

contact, 191, 305stability, 44, 51, 55, 85, 124, 125, 137, 140,

149, 152, 153, 156, 163, 168, 188, 189,218, 254, 257, 270, 274

loss of

Index 409

synchronization, 232low scale motion, 353lumped

body, 7, 171, 173, 176, 177, 179, 182,189–191, 194, 196, 200, 369

Lyapunovexponents, 8, 10, 212, 221, 233, 249, 274,

287sense, 125, 127stability, 130, 140, 188

magneto-elastic beam, 7Mandelbrodt fractal, 229manifold, 7, 130, 229, 230, 248, 274, 320map, 249, 307

Poincare, 8, 274, 287, 291mapping, 120, 226, 229, 231, 233

chain, 271Markov chains, 129mass conservation law, 60material

adjoined, 19beam, 298, 320cutouts, 28deformation, 47density, 359elastic

non-linear, 355isotropic, 163layer, 167non-linearity, 302shell, 15, 16, 36viscoelastic, 8

mathematical modeling, 123Mathieu-Hill equation, 127matrix

equation, 13form, 58, 74integral, 133inverse, 291operator, 126quadratic, 243reversing, 285stable, 134toroidal, 134tridiagonal, 99

maximalenergy, 46, 48

maximumamplitude, 110, 116deflection, 210, 255, 274, 278error, 71isolated, 110load, 87, 158

local, 219Lyapunov exponent, 8

MBG, 1–6, 10, 11, 13measurable function, 3medium, 25, 100, 143, 171, 240

composite, 41dissipative, 298, 320

Melnikov method, 6membrane, 205mesh, 74, 141, 360, 370

rectangular, 303, 323method, 160

σ , 116“set-up”, 209approximate, 4, 55, 171Bubnov-Galerkin, 1, 10, 13, 56, 77, 96, 157,

168, 190, 253, 273, 283collocation, 11, 65computational, 58, 283controlling, 279convergence, 118discovering hidden frequencies, 112discrete, 59discretization, 358displacement component, 297, 320effectiveness, 63energetical, 51, 74Englis-Bolotin, 173Euler, 47finite differences, 3, 63, 207, 210, 294, 323,

360finite elements, 6, 73finte elements, 357Galerkin, 5, 7, 9, 62, 64, 66, 67, 73Gauss, 107harmonic analysis, 113high order approximations, 41identification, 320least squares, 61Lyapunov exponent, 8MBG, 4, 10, 12momentum-impulse, 61Newton, 76numerical, 5, 78, 208partially reversible, 55Rayleigh-Ritz, 56, 67, 69, 71reduction, 31relaxation, 324Ritz, 1, 140

-Timoshenko, 183Runge-Kutta, 9, 141, 187, 208, 235, 243smoothing, 116spectral, 63, 74spectroscope, 112

410 Index

variated elasticity parameters, 302variated parameters, 304variation, 321variational, 2, 67weighting errors, 58

mid-plane, 8minimal

error, 65loading, 56potential energy, 48square representation, 61

minimization, 65function, 71principle, 10

minimumenergy, 48, 139functional, 68infinite, 113variation, 47

modalmodel, 298, 320non-linear equation, 9portrait, 266, 268, 362technique, 236

“modal portrait”, 268model

approximate, 139continuous, 34Euler-Bernoulli, 300geometrically linear, 171Kirchhoff-Love, 3, 17, 29, 163, 178, 180,

266Lorenz, 231, 253mathematical, 128, 129, 180, 271modal, 298momentous-less, 237non-linear, 297, 320non-perfect shell, 124parameters, 100shell, 15, 152spatially discrete, 6Timoshenko, 3, 319two-sided impact, 7

modesantisymmetric, 8approximation, 6axisymmetric, 236flexural, 9orthogonal, 357vibrations, 100–102, 105

modificationMBG method, 11Rayleigh-Ritz method, 56

modulated signal, 117

modulation, 232modulus, 35

elasticity, 16, 19, 321, 323inverse, 36shear, 302Young, 1, 166, 359

molecule motion, 228moment

bending, 268load, 153time, 108

momentous-less model, 237momentum-impulse method, 61motion

m-periodic, 133“basic”, 125beam, 300, 321chaotic, 6, 135, 258, 275equation, 9, 15, 20, 26, 28, 30, 319initial, 230laminar, 226low scale, 353non-disturbed, 128perturbed, 54regular, 274, 278shell, 124, 177stability, 42, 126stationary, 130synchronized, 221, 307time history, 107trajectory, 125transition, 287transverse, 7

moving load, 195“moving load”, 171MR convergence, 5multi-

DOF system, 358frequency vibrations, 274layered, 319periodic, 134

multibody system, 324multifrequent vibrations, 134multiple

shooting, 358multiple-layered, 15, 166, 297multiplicity condition, 119multipliers, 26, 63, 117, 301

naturalfrequency, 9, 235, 249, 255, 297, 319vibrations, 95, 99

Navier-Stokes equations, 2, 225neighborhood

Index 411

equilibrium, 50, 54point, 47spectral, 10

Newhouse scenario, 225, 228Newton

-Raphson method, 83, 237equation, 129method, 76, 83, 158, 191

node, 67, 74, 360equation, 142, 323interpolation, 208

noise, 225, 328component, 353

non-autonomous

system, 7-autonomous system, 129-axiall deformation, 237-axially symmetric shell, 205-clamped beam end, 317-closed trajectory, 131-compressed rib, 180-dimensional

amplitude, 153curvature, 140equation, 179, 238form, 15, 121, 151, 177, 182, 281parameter, 29, 188, 206, 282, 302, 321system, 272time, 109velocity, 226

-disturbed motion, 128-harmonical vibrations, 121-homogeneity, 15, 20, 29, 41, 76, 87–90, 99,

101, 237perforation-type, 92

-homogeneouselement, 98load, 246material, 299plate, 104shell, 6, 15, 19, 84, 87, 90–92, 96state, 54surface, 102

-impacting motion, 358-linear

-elastic, 313beam, 297, 319deformation, 237dynamics, 262elastic bearing, 7equation, 76, 184operator, 301physically, 313

response, 236strongly, 84vibrations, 9, 95, 124

-linear diffusion, 61-linearity, 236, 298

structural, 301-linearly elastic beam, 349-loading state, 49-perfect shell, 124-potential operator, 2-rectangular window, 118-resistant element, 132-self-coupled operator, 2-static branch, 125-stationary

constraints, 174problem, 210, 241, 248

-stretched rib, 180-symmetric

difference, 304load, 151, 237, 266matrix, 99

-synchronized vibrations, 221-trivial solution, 127-uniform

beam, 319-uniform pressure, 237-uniformity, 237

load, 158parameter, 160

non-stationary

heat problem, 11perturbations, 128problem, 2, 57, 61

uniformbeam, 297loading, 271outer pressure, 151shell, 161

non-coupledequations

differential, 75non-homogeneity

definition, 1surface

total size, 104system, 3

non-homogeneousshell

square, 100non-linear

beamsandwich, 355

412 Index

conditionsboundary, 6

elasticproblem, 325

equationalgebraic, 83

vibrationslongitudinal, 320

nonlinear, 5beam, 307behavior, 236dependency, 101dynamical problem, 3dynamics, 205, 281equation, 45, 78, 152, 156, 158

operational, 2function, 130ODE, 243operator, 129, 238, 240, 272, 282, 360oscillator, 373PDE, 246problem

MBG, 4structure, 42system, 135vibrations, 95, 106, 107, 248

nonlinearity, 130, 139, 313“soft”, 90geometrical, 47material, 321strong, 83

nonstatic state, 279nonunique solution, 246norm, 65

energetic, 5energetical, 70finite energetical, 73uniform, 2

normalforce, 8form, 7, 107, 160, 187, 191harmonic excitation, 10line, 28load, 177, 188perpendicular deformation, 20stresses, 297, 320to shell, 238

normalization, 74, 85normalized

energetically, 273function, 113surface, 87

numerical-perturbation approach, 9

agreement, 188algorithm, 2, 3, 5, 8, 41, 76, 78, 83, 119,

120, 123, 208, 271analysis, 84data, 118integration, 67, 237solution, 108, 210

Nyquistconstant, 118frequency, 120

odd-order cycle, 230ODE, 243

Cauchy problem, 246one

-dimensionaldynamics, 54mapping, 231

-dimensional torus, 229-frequency vibrations, 219, 286-sided

constraints, 171, 190, 203contact, 299spring, 358

-to-one internal resonance, 8-to-two internal resonance, 7DOF, 130

onset of chaos, 6operational equation, 2, 70operator, 179, 321

convergent, 5, 6differential, 141eigenfunctions, 127energy, 73functional, 55linear differential, 73matrix, 126non-self-coupled, 2nonlinear, 238, 240, 272, 282, 301, 360positively defined, 68symmetrical, 70

optical vibrations, 112optimization, 8orbit

chaotic, 8periodic, 84, 227quasi-periodic, 7singular, 130two-dimensional, 269

orbital stability, 130ordinary

differential equation, 1, 7, 9, 11ordinary differential equation, 107orthogonal

Index 413

beam mode, 357collocation, 60, 67equation, 59, 75functions, 55, 69, 74

orthogonality, 55, 67, 127orthogonalization, 56, 62, 74orthonormal functions, 65, 283oscillator, 135, 172

harmonic, 133impact, 8

outer pressure, 151, 159, 240over-critical

deflection, 274, 279equilibrium, 76state, 231

panel, 9, 178, 193, 195parabola, 158parabolic

approximation, 116equation, 2, 4, 5extrapolation, 84

parallelepiped cutout, 30parametric

analysis, 298, 320resonance, 7, 125, 127

partialderivative, 226, 240equation, 2, 7, 9, 11, 57, 59

partiallycontinuous, 113reversible method, 55

partition element, 281passive deformation, 136PDE, 271, 324PDE to ODE transition, 210peak, 110, 112, 116, 120

degree, 118separated, 119

pending onset of chaos, 7perfect

construction, 125filter, 118shell compression, 155

perfect construction, 123perforated plate, 8perforation, 92perforation-type non-homogeneity, 92period

2π , 133-doubling

bifurcation, 9, 231doubling, 7, 214, 219, 230, 251, 307, 309,

313, 325, 332, 369

excitation, 256, 287frequency, 107sampling, 118trebling, 251tripling, 313tripling bifurcation, 306, 307trippling phenomenon, 317

periodiccoefficient, 241cycle, 230dynamics, 9, 355excitation

transverse, 321Fourier transform, 117function, 117, 127, 133, 271, 274, 279motion, 96, 130, 132orbit, 84, 219, 227

homoclinic, 7variation, 306vibrations, 139, 287window, 219

periodicallydriven beam, 300excited plate, 8supported beam, 297

periodicity, 113perpendicular

cutouts, 28deformation, 20ribs, 147shell, 16, 30

perturbation, 47, 128, 140, 229perturbed

manifold, 7motion, 54

phase-modulation, 7-space system, 129dislocation, 133, 134laminar, 231locking, 309plane, 131, 132, 188, 220portrait, 130, 139, 219, 256, 266, 269, 274,

287, 291, 325, 328, 350, 355trajectory, 129, 130

phenomenablue sky catastrophe, 7

physicalcontinuum, 46explanation, 326interpretation, 6nonlinearity, 3, 47, 313surface, 74

physically non-linear, 299

414 Index

piecewisefunction, 178linear polynomials, 73

piezoelectric-active control, 298, 320actuator, 298, 320sensor, 298, 320

pitch angle, 7pitchfork bifurcation, 9plane, 175

-space manifold, 229bifurcational, 6control parameter, 248modal, 268perpendicular, 28

plasticflow coefficient, 299strain, 155

plasticity, 302plate, 95, 200, 271, 307, 319

analysis, 4baffle, 8chaotic

behavior, 9deflection, 123deformation, 56dynamics, 8, 120homogeneous, 105

square, 107perforation, 92rectangular, 41, 178soft, 100thin flexible, 18three-layered, 297vibrations, 102

Poincare map, 8, 219, 231, 256, 274, 287, 291,310, 313, 325, 335, 352, 355, 362

pointbalance, 125, 127, 128, 131bifurcation, 152, 155, 236, 254center, 107characteristic, 20contact, 172contour, 179critical, 227equilibrium, 75, 139explosion, 36extremal, 89isolated, 294mass, 176material, 299saddle-node, 231shell, 16, 83, 84spectrum, 5

stability, 264stationary, 1

Poisson coefficient, 17, 163, 206, 240, 272,282, 321

polar coordinates, 205polynomial, 65, 73

-type solution, 73second order, 84

Pomeau-Manneville scenario, 218, 225, 262population growth, 230positive Frechet differential, 5positively defined operator, 68, 70post-

buckledrod, 75state, 45system, 46, 50

buckling, 47, 76critical

load, 144state, 217, 248, 256, 257, 261

post-critical load, 209potential, 54

energy, 10, 20, 48, 49, 51, 139, 164field, 228force, 42, 75function, 136operator, 5

powerfrequency, 274plant, 172series, 297series expansion, 320spectrum, 8, 212, 214, 220, 249, 251, 256,

268, 291, 295, 305, 308, 310, 325, 326,332, 350, 355, 364

pre-buckled beam, 6compressed shell, 10critical

deflection, 254load, 143state, 217, 237, 260stresses, 125

precision, 108pressure

clamping, 299contact, 187, 301, 305, 308, 310, 314, 316,

328distribution, 156external, 235in contact, 298load, 321non-uniform, 151

Index 415

outer, 127, 155, 159sensor, 205zone, 251, 255

primary parametric excitation, 8principle

conservation, 48d’Alembert, 174, 300energetical, 51Hamilton, 20, 25, 297Lagrange, 75minimization, 10possible displacements, 52Runge, 120, 248variational, 67

prismatic rod, 4problem

asymmetric, 155boundary, 76, 178buckling, 123Cauchy, 187, 208, 243dynamic, 6dynamical, 159Euler, 47linear MR, 5non-stationary heat, 11simplification, 30stability

limit, 128motion, 127shell, 141

staticinstability, 41

statics, 59, 124, 274symmetry, 65three-dimensional, 16variational, 56, 74

procedureapproximation, 3averaging, 8Bubnov-Galerkin, 1, 77, 97, 183, 242, 284error estimation, 67Feyer, 113orthogonalization, 56Runge-Kutta, 107, 120

processcomputational, 85convergence, 95, 119deformation, 325deformation analysis, 123dynamical analysis, 27iterative, 83linearization, 47perforation, 92

projection method, 1, 7, 11, 57

proof, 46, 71, 226solution existence, 3

pure orthogonalization method, 75

quadraticfunctional construction, 2linear regulator, 298matrix, 243regulator, 320

quadrature, 36, 59, 67quasi-

harmonic vibrations, 350linear equation, 5periodic

dynamics, 355motion, 132, 229, 258orbit, 7turbulence, 227vibrations, 261, 264, 325, 350

periodicity, 123, 132, 232quasi-periodicity, 133

radial slice, 206, 208, 212radiation, 112random

direction, 161factor, 156frequency response, 236parameter, 283trajectory, 139vibrations, 132

randomness, 134rate of

convergence, 5, 63flow, 60MBG, 6

Rayleigh-Bernard convection, 227method, 69

-Galerkin, 11-Ritz, 55, 56, 67, 68, 71, 76-Ritz-Timoshenko, 11

solution, 71reaction

Belousov-Zhabotinskiy, 232dynamical, 175force, 173, 192

reactor, 172real chaos, 283realization of

ideal system, 47MBG, 6

rectangularbase, 9

416 Index

contact area, 189mesh, 303, 323plate, 8, 41, 178shell, 2, 16, 95, 100, 140, 281, 286, 291, 294

spherical, 84stability, 123

surface, 185rectilinear equilibrium form, 51, 53recurrent summation, 111reduction

dimension, 74method, 11, 31three-dimensional problem, 15

regularflow, 226motion, 274, 278vibrations, 231, 297wave, 325

regularity condition, 4, 233regularization, 330regulator, 298, 320relaxation method, 324reliability, 209, 210, 247, 270, 297relief, 313resistance, 41, 131

contour, 210, 216, 219time, 221

resonance, 90, 92, 93, 127, 134, 139, 147auto-parametric, 9high harmonic, 229internal, 236parametric, 7, 125subharmonic, 8

resonantfrequency, 236mode, 8

Reynolds number, 226rib, 15, 239

flexible, 180incompressible, 282not compressed, 28shell modelisation, 91width, 91, 101, 105

rigidbody, 171, 191

impact, 190element, 85shell, 89stability, 274

rigidity, 272Ritz

-Timoshenko method, 77, 183method, 2, 140solution, 71

rocket, 172, 235boosters, 236

rod, 11, 136, 140axially compressed, 10axis, 47

longitudinal, 51border, 52calculation, 56Euler, 138extra extension, 52post-buckled, 75prismatic, 4stability, 51supercritical force, 358thin, 51

rotaryinertia, 297, 320shell, 127, 155

rotation, 17, 181axis, 206free, 6point, 28

roughapproximation, 84solution, 54, 58

roughness, 203route to chaos, 7, 8, 10, 355RTN, 228Ruelle

-Takens-Feigenbaum scenario, 269-Newhouse scenario, 212, 218scenario, 269

-Takens-Newhouse scenario, 262

scenario, 228Ruelle-Takens scenario, 269Runge

-Kutta method, 9, 107, 120, 141, 143, 160,187, 191, 208, 235, 243, 273, 281, 291,357, 361

law, 141, 143, 291principle, 120, 248rule, 208, 361

saddle-node point, 231sampling, 117, 118Sanders-Koiter theorem, 10sandwich

beam, 319, 320sandwich beam, 297, 298scenario

Feigenbaum, 218, 262harmonic-chaotic motion, 225

Index 417

Landau-Hopf, 218, 225, 227, 228, 262Pomeau-Manneville, 231, 262Ruelle-Takens, 269

-Newhouse, 212, 218, 262to chaos, 317, 330, 355

scheme, 32Cauchy-Dirichlet, 2mode superposition, 7non-homogeneity, 92numerical integration, 3, 67, 83

second-order

approximation, 116derivative, 73determinant, 297, 320equation, 2, 7, 141, 184system, 131

typeelasticity, 17intermittency, 232system, 44

sector-type shell, 205, 212, 214, 218, 220vibrations, 216

self-balancing load, 320coupled

linear problem, 1operator, 5

dynamics, 225excited vibrations, 130, 132, 135interaction force, 195transformation, 92

separation of variations, 23series

approximated solutions, 5Fourier, 10, 69, 73, 113, 115, 156minimizing, 70power, 8, 320Taylor, 65trigonometric, 83, 157

“set-up” method, 209set-up method, 246, 248, 305shallow elastic shell, 9Sharkovskiy

diagram, 307theorem, 307, 317

shearmodulus, 299, 302, 321transverse, 297, 320

shell, 20, 41-mass

contact, 194interaction, 191, 197system, 188, 190

-typeconstruction, 180structure, 235, 236

“stiff”, 93, 101averaged, 264behavior, 91bending, 251buckling, 126curvature, 282cylindrical, 28, 235, 272, 273deflection, 161, 259

isocline, 211deformation, 28dynamics, 120, 292

irregular, 9edge, 153element, 30, 83flexible, 240, 294free vibrations, 127function, 57geometry, 140harmonically excited, 211height, 140heterogeneous, 144, 146homogeneous, 35, 85, 90, 100, 165imperfection, 153isotropic material, 17jump, 209kinetic energy, 25layer, 163load, 158loading, 88model, 15motion, 27, 124, 177non-homogeneous, 29, 89, 93, 96, 104parameter, 90perfect, 155pipe, 153point, 16, 79, 84radius, 212resistance, 206response, 261rise, 182section, 18sector-type, 205, 210, 216shallow, 3snap, 140spherical, 179, 287stability, 54, 75, 77, 123, 128, 152, 191, 237state

critical, 55initial, 76

strengthened, 156surface, 30, 78, 84, 86, 98, 243, 285

418 Index

thickness, 124, 158thin, 15, 140to cutout transition, 34uniformly loaded, 221vertex, 208vibrations, 248

shuttle Challenger, 236simple-support, 8, 9, 323, 360simple-support beam, 319Simpson rule, 304sine, 72, 109, 117singular

impulse, 117phase trajectory, 130point, 157trajectory, 131, 132

singularities, 120singularity, 1, 131, 156

stable, 137sinusoidal load, 254, 270, 361skippings, 44slender beam, 6, 7, 319sliding clamping, 208sloping parameter, 206, 209, 210, 216, 217Smale

-Birkhoff homoclinic theorem, 7horseshoe, 7, 229

small sense, 90smart layer, 320smooth

function, 115reduction, 160surface, 48, 173thin shell, 156

smoothness, 4snap-through, 129snap-through buckling, 9Sobolev space, 4soft

characteristics, 121excitation, 133Hopf bifurcation, 269loading, 43regime, 129RTN scenario, 229shell, 84, 88, 101, 145

“soft”, 89“soft” shell, 91, 93solid body, 4, 47, 49solution, 4, 12

accuracy, 56, 63approximate, 5, 57, 58, 67behaviour, 61convergence, 241

correction, 83differential equation, 1, 10estimation, 65exact, 56, 62, 70, 119, 183, 210existence, 3Galerkin method, 69homogeneous, 240initially general, 136mass dynamics, 198MR application, 2periodic, 130, 358problem, 109region, 74rough, 54scheme, 7stable, 286statics, 209uniqueness, 271

solvability, 3, 4, 55space, 1, 37, 42, 303, 323

-located curve, 28-plane manifold, 229boundary, 225contact, 188five-dimensional, 139Hilbert, 73of integration, 301point mass, 36shuttle Challenger, 236Sobolev, 4three-dimensional, 238

spatialbehaviour, 61coordinate, 324coordinates, 1, 28, 107, 160, 210, 240, 283,

294, 357dimension, 8discrete model, 6problem, 4system, 271thermal load, 8variable, 207vibrations, 256

spectralanalysis, 95, 120frequency, 274method, 63, 74neighborhood, 10

spectroscope method, 112spectrum

frequency, 370spectrum power series, 8spherical shell, 100, 188, 216, 220, 281, 286SPM scenario, 231

Index 419

“spots”, 277spring, 42, 357springing support, 203square

addition, 84element, 41, 88, 102matrix, 99mesh, 141plate, 8

homogeneous, 107, 121representation, 61shell, 91

SRTN, 228stability, 43, 75, 84, 90, 124, 125, 130, 138,

139, 155analysis, 235bifurcational, 49, 270conditions, 361criterion, 45dynamic, 320dynamical, 8, 123, 191elastic, 41, 51equation, 54limit, 128loss, 42, 53, 85, 133, 149, 153, 156, 163,

168, 188, 189, 226, 254, 257, 260bifurcational, 51

Lyapunov, 140motion, 127rigid, 274shell, 3, 152, 237solution, 143, 273, 286statical, 30, 47, 77, 165stiff, 218, 264structural, 136transient, 9trivial, 7

stabilization, 103, 220solution, 7

stable limit cycle, 227state

-space approximation, 1chaos, 287critical, 45, 55, 168deflection, 83equilibrium, 11, 43, 44, 50, 54, 125, 137,

154, 191, 300, 320initial, 52, 76non-loaded, 49nonstatic, 279post-critical, 248, 256pre-critical, 217, 237stability, 47stress, 46, 127, 157

unstable, 231static, 18

approach, 324balance, 132behaviour, 15bifurcation, 123, 139buckling load, 166equilibrium, 154instability, 41load, 10, 125, 245

critical, 248motion, 279point of balance, 124problem, 6, 235

nonlinear, 4stability, 30, 135stresses, 193

staticalapproach, 42boundary condition, 55criterion, 45loading, 87method, 48, 54stability, 47, 76, 77, 85

statics, 107, 124analysis, 15problem, 4

stationarity, 46stationary

energy, 46motion, 130points, 1problem, 2, 5, 59, 61, 274

steady-statechaos, 7response, 358stability, 9time history, 372vibrations, 107

steering vibrations, 274stepping function, 34stiff

characteristics, 121shell, 98, 145stability loss, 257, 261, 264, 270

stiff shell, 91, 93stiffness, 15, 100

bending, 300, 316characteristics, 89coefficient, 104, 145constant, 281cutout, 35element, 84, 92, 102flexural, 19, 34, 35, 83

420 Index

parameters, 30shell, 91

stochasticdisturbance, 132system, 129

stochasticity, 132straight normals hypothesis, 299strain, 123, 155

bending, axial, 236continuity conditions, 46elastic, 10flexural, 153function, 271, 282hardening, 303, 324linear, 300

strange attactor, 228strange attractor, 8, 135, 309strength

limit, 157strengthness, 322stress, 42, 152

-deflection, 108-deformation, 49-strain, 95, 157axial, 297, 320concentration, 124contact, 322function, 27, 206, 271initial, 53intensity, 303, 321measure, 193state, 127, 128to deformation transition, 22virtual, 46

stretched rib, 28stroboscope, 116stroke, 231strong

bending state, 237convergence theorem, 5deflection, 156filtrate action, 116focusing action, 115nonlinearity, 83

stronglycondensed series, 116nonlinear, 84

subcritical Hopf bifurcation, 9, 231subduction of chaos, 230subharmonic

motion, 8resonance, 10vibrations, 294

sudden

buckling, 124configuration transition, 191contact lack, 298jump, 188, 189prolongation, 294snap-through, 129, 155tripple bifurcation, 325

sufficient conditions, 7, 49, 139super-harmonic response, 236superposition, 7, 113, 129supersonic flow, 9support

ball-type, 28, 243contour, 216

free, 239, 273, 322edge, 283

springing, 203supported

periodically, 297rectangular plate, 8simply, 9

suppressing vibrations, 8surface, 32

balance, 139central, 16, 18, 23, 98contact, 48, 185, 189deflection, 197, 305, 310deformable, 175deformation, 358expansion, 101flow, 60interaction, 195irregularities, 175mean, 28middle, 124neutral, 172non-homogeneity, 102of chaos, 270parameter, 90physical, 74rotation, 17shell, 78, 84, 86, 243, 285smooth, 173transverse, 9unity, 178

suspension bridge, 135symmetric

bifurcation, 138clamping, 362deflections, 152, 210deformation, 237loading, 155matrix, 99operator, 69

Index 421

shell, 3, 19, 127, 205symmetrical

convergence, 120matrix, 99operator, 70

symmetryaxis, 84, 91condition, 208problem, 65

synchronic motion, 279synchronization, 123, 133–135, 220, 229, 279,

297, 306, 307, 313, 328-chaos, 233of frequencies, 223, 232

synchronization-intermittency-chaos, 233synchronized vibrations, 134, 307synchronous action, 205system, 193

n degrees-of-freedom, 95, 108“beam-impacting mass”, 370algebraic, 12, 158autonomic, 8basic, 5bifurcation, 129boundary, 128chaotic dynamics, 9characteristic point, 20conservative, 42, 45, 49, 75continuous, 41, 271, 288, 298coupled oscillators, 172deformable, 30, 42deformations, 49deterministic, 305, 358discrete, 136, 139dissipative, 272, 361disturbance, 43eigenelements, 5elastic, 152energy, 54

entire, 46potential, 51

equations, 56, 57, 59, 64, 285equilibrium, 44functions, 62Hamiltonian, 7hidden frequencies, 109lumped, 203mechanical, 95, 205multi-DOF, 358multibody, 324nonlinear, 129, 135, 243PDE, 240, 266perfect, 138post-buckled, 46, 50

resistance, 131shell-mass, 190stability, 10, 140, 145stable, 126structural mechanics, 7vibrating, 121with non-homogeneity, 3

system, complete, 72systems, 139

T index, 178“tail”, 110Takens scenario, 225, 228tangential plane, 180target-oriented excitation, 220Taylor

instability, 229series, 65

techniquecomputational, 6, 42numerical, 2perturbation, 7

technological defect, 298temperature

filed, 3gradient, 232linear law, 4

temporal solution trajectory, 7test

functions, 13, 58, 61, 62, 65, 67, 68, 71, 72,273, 283

solution, 64, 74theorem

Chetayev, 128convolution, 117decomposition, 113Dirichlet, 46, 56fundamental, 3Lagrange, 54

-Dirichlet, 46MBG, 2RTN, 228Sanders-Koiter, 10Sharkovskiy, 317Smale-Birkhoff, 7solution uniqueness, 4strong convergence, 5

theorycatastrophe, 136elasticity, 15, 20, 38, 49fractal, 8generalized functions, 36heat conduction, 60nonlinear, 8, 17

422 Index

numerical methods, 5stability, 54thermal-elasticity, 4

thermal-elasticity, 4field, 9load, 8

thermoelasticity-type equation, 6thick plate, 9thickness

beam, 301modification, 30plate, 4rarefaction, 19shell, 16, 19, 91, 124

thin-walled

construction, 4layer, 61shell, 156, 188structure, 41, 172

plate, 4, 8, 18rod, 51shell, 9, 15, 140, 151, 153, 235

third order approximation, 2Thom’s list, 139three

-dimensionalequation, 226problem, 16shell, 272solution, 66space, 238, 281, 321

-impacts per motion, 358-layer wafer, 156-layered

beam, 297package, 305, 307

-mode interactions, 7time, 168, 301, 321

-dependentmoment, 222process, 42

-dependent loading, 358-independent

load, 237-spatial chaos, 226, 279finite, 128function, 26history, 214, 220, 249, 293

time history, 192Timoshenko

beam, 297equation, 176

method, 55model, 3, 319shell, 271, 358

titanium alloy, 167topological

similarity, 138structure, 131, 132

topology, 135, 229tori, 7toroidal matrix, 133, 134torque, 205, 221, 223, 360torsion moment, 268torus, 134, 229

doubling, 7total

energy, 49, 53potential, 50, 51

potential energy, 48trajectory, 174, 220, 229

balance, 138double asymptotic, 231mass, 193motion, 20phase, 130random, 139temporal solution, 7tripling, 268

transferenergy, 9heat, 4, 11

transformFourier, 36, 69, 109, 117, 120Housholder, 99

transientmapping, 120sampling period, 118stability, 9

transitionboundary, 37chaotic-harmonic, 276configuration, 191cutout-shell, 34force, 123harmonic-chaotic, 287into chaos, 262, 297, 315, 332process, 128rod, 51stresses-deformations, 22system, 1to equilibrium, 43

transitory process, 129, 154transversal

deflection, 255deformation coefficient, 299

Index 423

displacement, 51homoclinic orbits, 7impact, 190load, 84, 123, 184, 235, 240, 243, 300, 304,

322, 357pressure, 237

transversallydriven beam, 300excited, 205vibrating mass, 173

transversedisplacement, 236load, 136, 144, 148, 168, 272, 276, 278, 283,

285loading, 358motion, 7surface, 9vibrations, 6, 297, 320

travelling, 8trebling, 251triangle, 181tridiagonal matrix, 99trigonometric series, 83trilayered shell, 166, 168triple bifurcation, 313, 325, 330, 355trivial stability, 7tube heat exchanger, 8turbulence, 132, 134, 225, 226, 231turbulent

flow, 226layer, 61solution, 226

twisting, 51two

-dimensionalorbit, 269problem, 16, 57system, 130, 139, 205

-form flatter, 139-frequency vibrations, 258, 261, 294-layered

package, 313-modes interaction, 9-period motion, 133-point boundary problem, 358-sided

constaints, 182constraints, 176impact model, 7

-to-one resonance, 9DOF system, 174post-buckled states, 45

two-dimensionalproblem

abstractive problem, 11two-layered

packagesynchronization, 317

umbilici catastrophe, 139unbounded motion, 8uncoupled

beams, 297oscillator, 373

undamped buckled beam, 7uniform

approximation, 55boundary condition, 64, 286boundary conditions, 127compression, 155convergence, 59, 251load, 221, 245norm, 2pressure, 156, 237shell, 161stretching, 8

unique function, 299uniqueness of solution, 4, 271, 358unirotational cycle, 287unity

function, 31, 33, 39surface, 178

unloading, 44, 155unstable equilibrium, 43

variablecoefficient, 54load, 272parameter, 30pitch angle, 7shell parameters, 15step, 120

variation, 3, 21, 50, 55, 75, 152, 212, 301coefficient, 181contact pressure, 312contact surface, 48damping, 362deflection, 189, 197energy, 51finite, 113function, 23, 24liquid molecules, 61method, 194, 321resonant frequency, 236

variational, 2approach, 45, 62criterion, 47differentiation, 28

424 Index

equation, 15, 25, 26, 55estimation, 51function, 22inequality, 4method, 11, 67, 77, 183problem, 56, 74

variations separation, 23vector of displacement, 126velocity

constant, 200estimation, 5flow, 8function, 322liquid molecules, 61motion, 194non-dimensional, 226projection, 198surface point, 29

vertex, 208vertical

base excitation, 7displacement, 175load, 9movement, 48velocity component, 190, 192

virtual work, 174viscoelastic

-passive control, 298plate, 8

viscoelastic-passive control, 320Volmir criterion, 144, 168, 255volume, 60

elasticity, 299, 303elasticity modulus, 321

wafer, 156

waffled rib, 15wall, 291“wash-out” phase portrait, 329wave, 152, 154, 167, 255, 260, 278

amplitude-travelling, 8damped, 305elastic, 27impact, 171propagation, 297, 320

weakconvergence, 5depreciation of amplitude, 111turbulence, 134

weaklydamped, 7non-linear vibrations, 95

weight, 110, 115body, 176density, 301mass, 187

weightingerrors, 64–66

method, 58force, 172functions, 13, 59, 60, 62, 63, 73

weights, 61, 67weld conditions, 299WEM, 58wind-type load, 156, 160“wrinkle”, 138“wrinkle-type”, 137“wrinkle-type” catastrophe, 138, 139

Young modulus, 1, 163, 168, 299, 301, 302,359