Table Of ContentLocalized Dynamics of
Thin-Walled Shells
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Localized Dynamics of Thin-Walled Shells
Gennadi I. Mikhasev, Petr E. Tovs"k
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Localized Dynamics of
Thin-Walled Shells
Gennadi I. Mikhasev
Belarusian State University
Petr E. Tovstik
Saint Petersburg State University
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Library of Congress Cataloging-in-Publication Data
Names:Mikhasev,G.I.,author.|Tovstik,P.E.,author.
Title:Localizeddynamicsofthin-walledshells/GennadiI.Mikhasevand
PetrE.Tovstik.
Description:Firstedition.|BocRaton:CRCPress,2020.|Series:
Chapman & Hall/CRC monographs and research notes in mathematics |
Includesbibliographicalreferencesandindex.
Identifiers: LCCN 2019060133 | ISBN 9781138069749 (hardback) | ISBN
9781315115467(ebook)
Subjects: LCSH: Shells (Engineering)--Vibration--Mathematical models. |
Thin-walledstructures--Vibration--Mathematicalmodels.|
Localizedwaves.|Wavepackets.|Asymptoticexpansions.
Classification:LCCTA660.S5M4872020|DDC624.1/7762015118--dc23
LCrecordavailableathttps://lccn.loc.gov/2019060133
ISBN:978-1-138-06974-9(hbk)
ISBN:978-1-315-11546-7(ebk)
TypesetinCMR
byNovaTechsetPrivateLimited,Bengaluru&Chennai,India
Contents
Preface xi
Authors xv
1 Introduction 1
1.1 Two-dimensional theories in the dynamics of thin shells: Brief
overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Localized modes in the dynamics of thin-walled structures . 4
1.2.1 Mode localization induced by features in boundary
conditions . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Edge vibrations of thin plates and shells . . . . . . . . 5
1.2.3 Interfacial vibrations in thin plates and shells . . . . . 7
1.2.4 Mode localization induced by inhomogeneity of
geometrical and physical parameters . . . . . . . . . . 7
1.3 Localized parametric vibrations of thin-walled structures . . 8
1.4 Localized waves in thin shells . . . . . . . . . . . . . . . . . . 10
1.5 Asymptotic methods in the analysis of localized dynamics of
thin shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Equations of the two-dimensional theory of shells 17
2.1 Geometric relations . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 ElasticityrelationsandboundaryconditionsfortheKirchhoff–
Love model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Governing equations. Membrane equations . . . . . . . . . . 26
2.5 Qualitative analysis of the frequency spectrum for free
vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Dynamic equations for pre-stressed shells . . . . . . . . . . . 31
2.7 The Timoshenko–Reissner model for a transversely isotropic
shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.8 Timoshenko–Reissner shell inhomogeneous in the thickness
direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
v
vi Contents
2.9 The equivalent single layer model for laminated transversally
isotropic cylindrical shells . . . . . . . . . . . . . . . . . . . 40
2.9.1 Governing equations in terms of displacements . . . . 41
2.9.2 Technical theory equations . . . . . . . . . . . . . . . 43
2.10 On the error of 2D shell theories . . . . . . . . . . . . . . . . 47
3 Localized vibration modes of plates and shells of revolution 51
3.1 On the modes of localized vibrations of shells . . . . . . . . . 52
3.2 Localized vibration modes of a plate with free rectilinear edge 55
3.2.1 The bending vibrations . . . . . . . . . . . . . . . . . 55
3.2.2 The in-plane vibrations . . . . . . . . . . . . . . . . . 57
3.3 Localized vibration modes of a circular plate . . . . . . . . . 58
3.4 Vibrations of a shell of revolution . . . . . . . . . . . . . . . 62
3.5 Non-uniform localized modes of shell vibrations . . . . . . . 65
3.5.1 Effectofinitialmomentlessstressesonthenon-uniform
vibration modes . . . . . . . . . . . . . . . . . . . . . 69
3.6 Oscillating vibration modes localized near the edge . . . . . 71
3.7 Construction algorithm of localized solutions . . . . . . . . . 75
3.8 Vibration modes of a shell of revolution localized near a
parallel lying strictly within the shell . . . . . . . . . . . . . 78
3.9 Vibrationmodesofapre-stressednon-uniformcylindricalshell
localized near a parallel . . . . . . . . . . . . . . . . . . . . . 81
3.9.1 Axisymmetric vibrations . . . . . . . . . . . . . . . . . 82
3.9.2 Non-axisymmetric vibrations . . . . . . . . . . . . . . 84
3.10 Localized modes of axisymmetric vibrations of an infinitely
long non-uniform cylindrical shell resting on an elastic
foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.11 Transverse shear influence on the localized vibration modes of
shells of revolution . . . . . . . . . . . . . . . . . . . . . . . . 91
3.11.1 Effect of boundary conditions on stability of a circular
cylindrical shell . . . . . . . . . . . . . . . . . . . . . . 93
3.11.2 Localized non-uniform vibration modes . . . . . . . . 95
4 Localized vibration modes of cylindrical and conic shells 97
4.1 Equations of localized shells vibrations, and boundary
conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2 Low-frequency vibrations of circular cylindrical shells . . . . 100
4.3 Localized vibrations modes of cylindrical shells . . . . . . . . 102
4.4 Localized vibration modes of a cylindrical panel near a free or
a weakly supported rectilinear edge. . . . . . . . . . . . . . . 110
4.5 Localized vibration modes of shells close to cylindrical ones . 115
4.6 Localized vibration modes of conic shells . . . . . . . . . . . 120
Contents vii
4.7 Low-frequency vibrations of a Timoshenko–Reissner circular
cylindrical shell . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.8 Localized vibration modes of a Timoshenko–Reissner
non-circular cylindrical shell . . . . . . . . . . . . . . . . . . 129
4.9 Localized vibration modes of Timoshenko–Reissner cylindrical
panel with a free or a weakly supported edge . . . . . . . . . 133
5 Localized Parametric Vibrations of Thin Shells 139
5.1 Localized parametric vibrations of cylindrical shells under
periodic axial forces . . . . . . . . . . . . . . . . . . . . . . . 140
5.1.1 Asymptotic approach . . . . . . . . . . . . . . . . . . 141
5.1.2 Reconstruction of the asymptotic expansion . . . . . . 146
5.1.3 Effect of dissipative forces . . . . . . . . . . . . . . . . 150
5.1.4 Parametric instability domains . . . . . . . . . . . . . 150
5.1.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.2 Parametric vibrations of laminated cylindrical shells under
periodic axial forces: effect of shear . . . . . . . . . . . . . . 156
5.2.1 Asymptotic solution . . . . . . . . . . . . . . . . . . . 158
5.2.2 Reconstruction of asymptotic solution . . . . . . . . . 162
5.2.3 Main region of instability . . . . . . . . . . . . . . . . 165
5.3 Parametric vibrations of cylindrical and conical shells under
pulsing pressure . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.3.1 Parametric vibrations of conical shells . . . . . . . . . 166
5.3.2 Parametric vibrations of a cylindrical shell under
pulsing pressure . . . . . . . . . . . . . . . . . . . . . 174
5.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.4 Parametric vibrations of nearly cylindrical shells . . . . . . . 178
5.5 Localized axisymmetric vibrations of long cylindrical shells
resting on elastic foundation . . . . . . . . . . . . . . . . . . 183
6 Wave Packets in Medium-length Cylindrical Shells 189
6.1 Wave packets in a non-circular cylindrical shell . . . . . . . . 190
6.1.1 Setting the problem . . . . . . . . . . . . . . . . . . . 190
6.1.2 Splitting the initial WP . . . . . . . . . . . . . . . . . 192
6.1.3 The initial-boundary-value problem for a fixed WP . . 193
6.1.4 Reduction of the original 2D problem to the sequence
of 1D problems on moving generatrix . . . . . . . . . 194
6.1.5 Solution of the sequence of 1D problems . . . . . . . 197
6.1.6 Integration of the amplitude equation . . . . . . . . . 201
6.1.7 Definition of constants of integration . . . . . . . . . 203
6.2 Analysis of the constructed solutions . . . . . . . . . . . . . 205
6.2.1 Stationary wave packet (localized eigenmode) . . . . 206
6.2.2 Properties of the Hamiltonian system solutions . . . . 207
viii Contents
6.3 The influence of shell geometry on the propagation of wave
packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
6.4 Constructing solutions in the form of running wave packets
based on Maslov’s method . . . . . . . . . . . . . . . . . . . 224
6.4.1 Canonical system of equations and auxiliary initial
conditions . . . . . . . . . . . . . . . . . . . . . . . . . 228
6.4.2 Class of asymptotically equivalent functions . . . . . . 230
6.4.3 Approximatesolutionofthecanonicalsystem(Maslov’s
method) . . . . . . . . . . . . . . . . . . . . . . . . . . 230
6.4.4 The leading approximation . . . . . . . . . . . . . . . 234
6.4.5 Comparison of different solutions . . . . . . . . . . . . 235
7 Effect of External Forces on Wave Packets in Zero Curvature
Shells 237
7.1 Classification of dynamical stress-strain state . . . . . . . . 238
7.2 Effect of external forces on non-stationary localized vibrations
of a medium-length cylindrical shell . . . . . . . . . . . . . . 239
7.2.1 Stationary wave packets . . . . . . . . . . . . . . . . . 242
7.2.1.1 Stationary WPs in the case of non-uniform
hoop stress resultant. . . . . . . . . . . . . . 242
7.2.1.2 Stationary WPs in the case of non-uniform
elastic foundation . . . . . . . . . . . . . . . 243
7.2.2 Effect of stationary non-uniform hoop stress resultants
on dynamics of WPs . . . . . . . . . . . . . . . . . . . 244
7.2.3 Effectofinhomogeneouselasticfoundationondynamics
of WPs . . . . . . . . . . . . . . . . . . . . . . . . . . 245
7.2.4 Effect of dynamic hoop stress resultants on dynamics
of WPs . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.3 Localized families of bending waves in medium-length conical
shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
7.3.1 Eigenmodes of bending vibrations of a beam with
variable characteristics . . . . . . . . . . . . . . . . . . 252
7.3.2 Algorithm for constructing solutions and principal
relations . . . . . . . . . . . . . . . . . . . . . . . . . . 253
7.4 Wave packets in a cylindrical shell pre-stressed by axial forces 255
7.4.1 Algorithm for constructing a solution in the form of
travelling WPs . . . . . . . . . . . . . . . . . . . . . . 256
7.4.2 Resolving equations and relations . . . . . . . . . . . . 259
7.4.3 Solution analysis . . . . . . . . . . . . . . . . . . . . . 260
7.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 262
Contents ix
8 Wave Packets in Long Shells of Revolution Travelling in the
Axial Direction 265
8.1 Governing equations in terms of displacements. Setting the
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
8.2 Classification of integrals of the dynamic equations . . . . . 269
8.2.1 Axisymmetric waves . . . . . . . . . . . . . . . . . . . 270
8.2.1.1 Longitudinal waves . . . . . . . . . . . . . . 270
8.2.1.2 Bending waves . . . . . . . . . . . . . . . . . 271
8.2.2 Non-axisymmetric waves with low variability along a
parallel . . . . . . . . . . . . . . . . . . . . . . . . . . 272
8.2.2.1 Tangential waves . . . . . . . . . . . . . . . . 272
8.2.2.2 Bending waves . . . . . . . . . . . . . . . . . 272
8.2.3 Non-axisymmetric waves with large variability along a
parallel . . . . . . . . . . . . . . . . . . . . . . . . . . 273
8.2.3.1 Tangential waves . . . . . . . . . . . . . . . . 273
8.2.3.2 Bending waves . . . . . . . . . . . . . . . . . 273
8.3 Asymptotic solution as superposition of packets of
axisymmetric flexural and tangential waves . . . . . . . . . . 273
8.3.1 Axisymmetric packets of bending waves . . . . . . . . 274
8.3.2 Axisymmetric packets of longitudinal waves . . . . . . 278
8.3.3 Axisymmetric packets of torsional waves . . . . . . . . 280
8.3.4 Superposition of axisymmetric WPs . . . . . . . . . . 280
8.3.5 Solution Properties. Examples . . . . . . . . . . . . . 282
8.4 Non-axisymmetric WPs in shells of revolution with a small
number of waves in the circumferential direction . . . . . . . 286
8.5 Non-axisymmetric WPs in shells of revolution with a large
number of waves in the circumferential direction . . . . . . . 287
8.5.1 Non-axisymmetric packets of tangential waves. . . . . 288
8.5.2 Non-axisymmetric packets of bending waves . . . . . . 290
8.5.3 Superposition of solutions . . . . . . . . . . . . . . . . 292
8.5.4 The effect of the shell geometry on dynamics of WPs
of tangential waves . . . . . . . . . . . . . . . . . . . 294
8.6 Wave packets in long cylindrical shells with variable
parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
8.6.1 Axisymmetric packets of bending waves . . . . . . . . 297
8.6.1.1 Effect of the material inhomogeneity on the
WP dynamics . . . . . . . . . . . . . . . . . 298
8.6.1.2 Effect of the thickness variation on the WP
dynamics . . . . . . . . . . . . . . . . . . . . 301
8.6.2 Axisymmetric packets of longitudinal waves . . . . . 301
8.6.3 Axisymmetric packets of torsional waves . . . . . . . 304
8.6.4 Non-axisymmetric WPs with a large number of waves
in the circumferential direction . . . . . . . . . . . . . 305
