Vibrations of Shells and Rods PDF
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Vibrations 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
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