Table Of ContentVibrations of Shells and Rods
Springer
Berlin
Heidelberg
New York
Barcelona
HongKong
London
Milan
Paris
Singapore
Tokyo
Khanh Chau Le
Vibrations of Shells
and Rods
With 84 Figures
Dr. habil. Khanh Chau Le
Ruhr U niversitat Bochum
Fakultat fur Bauingenieurwesen
Lehrstuhl fur Allgemeine Mechanik
U niversitatsstra6e 150
44780 Bochum
Germany
ISBN-13: 978-3-642-64179-4 e-ISBN-13: 978-3-642-59911-8
DOI:1O.lO07/978-3-642-59911-8
Library Congress Cataloging-in-Publication Data
Le, Khanh Chau
Vibrations of shells and rods / Khanh Chau Le.
Includes bibliographical references and index.
1. Shells (Engineering)--Vibration. 2. Bars (Engineering)--Vibration.
3. Ealstic plates and shells. 4. Elastic rods and wires. I. Title.
TA660.S5L38 1999 624.1'7762--dc21 99-28999
This work is subject to copyright. All rights are reserved, whether the whole or part of the
material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data
banks. Duplication of this publication or parts thereofis permitted only under the provisions
of the German Copyright Law of September 9, 1965, in its current version, and permission for
use must always be obtained from Springer-Verlag. Violations are liable for prosecution act
under German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1999
Softcover reprint of the hardcover 1s t edition 1999
The use of general descriptive names, registered names, trademarks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from
the relevant protective laws and regulations and therefore free for general use.
Typesetting: camera-ready by author
Cover design: de'blik, Berlin; Production: ProduServ GmbH Verlagsservice, Berlin
SPIN:10680234 62/3020-543210 -Printed on acid -free paper
To my parents
Le Khanh Can and Truong Thi Tan Nhan
Preface
We live in a world of vibrations and waves, without which there would not
be sound, light, radio, television, communication etc. That is why the study
of vibrations and waves is so important in many branches of physics and me
chanics. This book is devoted to the study of small mechanical vibrations of
shells and rods, which are made of elastic or piezoelectric materials. But even
in this very special field there are already many excellent books and mono
graphs written since the monumental work by Rayleigh [47]. The peculiarity
of the present book is that we regard the equations of shells and rods as
two- and one-dimensional approximate equations which can be derived from
the three-dimensional theory by using the variational-asymptotic method.
The latter has been invented especially for those variational problems which
contain small parameters. It turns out that for vibrations of shells and rods
there are many situations in which such small parameters exist. Thus, the
application of the variational-asymptotic method enables one to derive not
only the classical two- and one-dimensional theories of low-frequency vibra
tions of shells and rods, but also the theories of high-frequency (or thickness)
vibrations.
The present book is organized into ten chapters. After the short introduc
tory chapter containing some historical background we provide preparatory
material on tensor analysis, geometry of curves and surfaces, dynamic theo
ries of elasticity and piezoelectricity, and the variational-asymptotic method
in the second chapter. The rest of the book is divided into two nearly
equal parts which treat the theories of low- and high-frequency vibrations,
respectively. Chapters 3-6 present two- and one-dimensional theories of low
frequency vibrations and wave propagation in thin bodies, namely elastic
shells and plates (Chapter 3), elastic rods (Chapter 4), piezoelectric shells
and plates (Chapter 5), and piezoelectric rods (Chapter 6). Chapters 7 and
8 deal with high-frequency vibrations of elastic shells, plates, and rods, and
finally, Chapters 9 and 10 study high-frequency vibrations of piezoelectric
shells, plates, and rods, respectively. To help a reader become more profi
cient, each section ends with problems and exercises, of which some can be
4
solved effectively by using the Mathematica. Difficult problems are marked
with an asterisk. It is not our aim to give complete references on the subject,
which is very large. We cite rather those papers which are directly related
to the methods used in the book.
This book is intended for engineers who deal with vibrations of shells
and rods in their everyday practice but also wish to understand the sub
ject from the mathematical point of view. Some of the results concerning
high-frequency vibrations of shells and rods may be new for them. The
book can serve as a texbook for graduate students who have completed first
year courses in mechanics and mathematics. It may also be interesting for
those mathematicians who seek applications of the variational and asymp
totic methods in elasticity and piezoelectricity. Only a minimum knowledge
in advanced calculus and continuum mechanics is assumed on the part of a
reader.
I would like to express here my deep gratitude to my teacher Prof. V.L.
Berdichevsky (Detroit), who has had a great influence on my development of
the subject. Substantial parts of Chapters 3 and 4 are based on his lectures
and publications. I thank Prof. H. Stumpf (Bochum) for his warm hospital
ity during the writing of the book, and Professors R.J. Knops (Edinburgh),
A.G. Maugin (Paris), W. Pietraszkiewicz (Gdansk), L. Truskinovsky (Min
neapolis), D. Weichert (Aachen) and many other friends and colleagues for
their comments and useful discussions. The competent language assistance
by Mrs. Anne Gale (Springer Verlag) is also gratefully acknowledged.
Last, but not least, thanks are due to my wife and my daughter, without
whose patience and love this book would not have appeared at all.
Bochum, May 1999
K.C. Le
Contents
1 Introduction 9
2 Preliminaries 17
2.1 Tensor analysis .......... 17
2.2 Geometry of curves and surfaces . 24
2.3 Dynamic theory of elasticity . . . 30
2.4 Dynamic theory of piezoelectricity. 37
2.5 Variational-asymptotic method .. 45
I Low-frequency vibrations 57
3 Elastic shells 59
3.1 Two-dimensional equations . 59
3.2 Asymptotic analysis .... 68
3.3 Dispersion of waves in plates . 78
3.4 Frequency spectra of circular plates 89
3.5 Dispersion of waves in cylindrical shells 96
3.6 Frequency spectra of cylindrical shells. 109
3.7 Frequency spectra of spherical shells 118
4 Elastic rods 123
4.1 One-dimensional equations . 123
4.2 Asymptotic analysis 130
4.3 Cross section problems 140
4.4 Dispersion of waves 147
4.5 Frequency spectra . 155
5 Piezoelectric shells 163
5.1 Two-dimensional equations . 163
5.2 Asymptotic analysis .... . 172
6
5.3 Error estimation and comparison ... 183
5.4 Frequency spectra of circular plates . . 193
5.5 Frequency spectra of cylindrical shells . 200
6 Piezoelectric rods 209
6.1 One-dimensional equations 209
6.2 Asymptotic analysis 215
6.3 Cross section problems 224
6.4 Frequency spectra. . 234
6.5 Longitudinal impact 240
II High-frequency vibrations 249
7 Elastic shells 251
7.1 Two-dimensional equations . 251
7.2 Long-wave asymptotic analysis. 257
7.3 Short-wave extrapolation . . . 267
7.4 Dispersion of waves in plates . . 275
7.5 Frequency spectra of plates · 285
7.6 Dispersion of waves in cylindrical shells .296
7.7 Frequency spectra of cylindrical shells . .305
8 Elastic rods 311
8.1 One-dimensional equations 311
8.2 Long-wave asymptotic analysis. 316
8.3 Short-wave extrapolation. 322
8.4 Cross section problems 329
8.5 Dispersion of waves 336
8.6 Frequency spectra . .340
9 Piezoelectric shells 349
9.1 Two-dimensional equations . .349
9.2 Long-wave asymptotic analysis. · 351
9.3 Short-wave extrapolation . . . . .360
9.4 Frequency spectra of circular plates .369
9.5 Frequency spectra of cylindrical shells . .374
10 Piezoelectric rods 381
10.1 One-dimensional equations · 381
10.2 Long-wave asymptotic analysis. .383
10.3 Short-wave extrapolation . . . . .390
7
10.4 Cross section problems . 395
10.5 Frequency spectra . .400
A Material constants 405
A.l Elastic isotropic materials .405
A.2 Piezoelectric crystals . .406
A.3 Piezoceramic materials .408
B List of notations 409
Bibliography 413
Index 419
