Random vibrations of elastic systems

477 Pages·1984·10.27 MB·English

by  V. V. Bolotin (auth.)

Random vibrations of elastic systems
MONOGRAPHS AND TEXTBOOKS ON MECHANICS OF SOLIDS
AND FLUIDS
Editor-in-Chief: G. lE.. Oravas
MECHANICS OF ELASTIC STABILITY
Editor: H.H.E. Leipholz
Also in this series:
H.H.E. Leipholz, Theory of elasticity. 1974.
ISBN 90-286-0193-7
L. Librescu, Elastostatics and kinetics of anisotropic and heterogeneous
shell-type structures. 1975.
ISBN 90-286-0035-3
C.L. Dym, Stability theory and its application to structural mechanics.
1974.
ISBN 90-286-0094-9
K. Huseyin, Nonlinear theory of elastic stability. 1975.
ISBN 90-286-0344-1
H.H.E. Leipholz, Direct variational methods and eigenvalue problems in
engineering. 1977.
ISBN 90-286-0106-6
K. Huseyin, Vibrations and stability of multiple parameter systems. 1978.
ISBN 90-286-0136-8
H.H.E. Leipholz, Stability of elastic systems. 1980.
ISBN 90-286-0050-7
Random vibrations
of elastic systems
by
v.
V. Bolotin
Moscow Energetics Institute
Moscow, USSR
Translated from the Russian edition by
I. Shenkman
English translation edited by
H.H.E. Leipholz
University of Waterloo
Waterloo, Ontario, Canada
1984 SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging in Publication Data
Bolotin. v. V. (Vladimir VasU' evich)
Random vibrations of elastic sy~tems.
(Mechan1cs of elastic stability ; 8)
Translation of: SluchaYnye kolebani~ uprugikh sistem.
BibUography: p.
Includes index.
1. Random vibration. 2. Elasticity. 3. Elastic
ana1ysis (Theory of structures) I. Leipholz. H. H. E.
(Herst H. E.). 1919- • II. Title. III. Series.
QA935.B64213 1984 531'.3823 84-8191
ISBN 978-90-481-8280-0 ISBN 978-94-017-2842-3 (eBook)
DOI 10.1007/978-94-017-2842-3
Copyright
© 1984 by Springer Science+Business Media Dordrecht
Originally published by Martinus Nijhoff Publishers, The Hague. in 1984
Softcover reprint of the hardcover 1st edition 1984
AU rights reserved. No part of this publication may be reproduced, stored in a
retrieval system, or transmitted in any form or by any means, mechanical,
photocopying, recording, or otherwise, without the prior written permis sion of
the publishers, Springer-Science+Business Media, B.V.
Editor's Preface
The subject of random vibrations of elastic systems has gained,
over the past decades, great importance, specifically due to its
relevance to technical problems in hydro- and aero-mechanics.
Such problems involve aircraft, rockets and oil-drilling platforms;
elastic vibrations of structures caused by acoustic radiation of a
jet stream and by seismic disturbances must also be included. Appli
cations of the theory of random vibrations are indeed numerous and
the development of this theory poses a challenge to mathematicians,
mechanicists and engineers. Therefore, a book on random vibrations
by a leading authority such as Dr. V.V. Bolotin must be very welcome
to anybody working in this field. It is not surprising that efforts
were soon made to have the book translated into English.
With pleasure I acknowledge the co-operation of the very
competent translater, I Shenkman; of Mrs. C. Jones, who typeJ the
first draft; and of Th. Brunsting, P. Keskikiikonen and R. Piche,
who read it and suggested where required, corrections and changes.
I express my gratitude to Martinus Nijhoff Publishers BV for entrust
ing me with the task of editing the English translation, and to
F.J. van Drunen, publishers of N. Nijhoff Publishers BV, who so
kindly supported my endeavours. Special acknowledgement is due to
Mrs. L. Strouth, Solid Mechanics Division, University of Waterloo,
for her competent and efficient preparation of the final manuscript.
Last, but not least, thanks to Dr. V.V. Bolotin for his
advice and the updating of the contents of the Russian version of
this book.
H.H. E. Leipholz
University of Waterloo
January 1984
v
Author's Preface
The theory of random vibrations is finding more and more applica
tions in engineering. It is fundamental in the analysis of air
craft subjected to atmospheric turbulence, pressure pulsations in
a turbulent boundary layer, acoustic radiation of a jet stream, etc.;
in the analysis of road vehicles subjected to vibrations when
moving on a rough path, etc. The methods of the theory of random
vibrations are also extensively used in the analysis of tall build
ings and structures subjected to wind pressure, in the analysis of
ships and other (ocean) structures sUbjected to the effect of waves,
and in the analysis of buildings and structures under seismic
effects. Publications on random vibrations and vibration relia
bility in scientific journals have been growing in number. The
theory of random vibrations is one of the rapidly developing branches
of contemporary applied mechanics.
This book considers systematically the problems of the
theory of random vibrations and the methods of solving them, with
particular attention devoted to continuous systems. There are
eight chapters in this book. The first chapter serves as an
introduction by providing a general idea of random loadings of
mechanical systems and of methods for describing these loadings
probabilistically. The second chapter deals with methods to be
used for solving random vibration problems (mostly, those applied
to continuous systems).
The next two chapters are devoted to random vibrations
of linear continuous systems. The third chapter deals with the
mutual coupling of generalized coordinates in elastic and visco-
elastic systems, with the problems of transmission of random vibra
tions through spatially distributed elastic systems, and with vibrations
of elastic-acoustic systems. The fourth chapter describes the
asymptotic method for the analysis of broad-band random vibrations
of linear continuous systems. The specific feature of such vibrations
is that a large number of natural modes are excited simultaneously,
which makes it possible to discover certain new regularities (laws)
and asymptotic estimates for the characteristics of a vibration
field.
The fifth chapter deals with parametrically excited ran
dom vibrations. The methods of investigating the stability of
stochastic systems and methods effective for constructing stochastic
instability regions are described here. Special attention is
devoted to parametric resonances excited by narrow-band stationary
random processes. The sixth chapter treats random vibrations in
nonlinear systems. These chapters describe not only systems with
a finite number of degrees of freedom but also continuous systems.
The last two chapters are devoted to problems associated
with the application of the theory of random vibrations. The
seventh chapter describes the foundations of reliability theory
of mechanical systems as applied to systems subjected to random
vibrations. The methods of this theory are used for the analysis
of optimal random vibration isolation of systems. In the last
chapter, the principles of planning the measurements of random
vibration fields are discussed. The selection of the number of
sensors and their optimal location in vibrating structures, and
also the making of corrections to account for changes in the vibra
tion field due to the introduction of sensors, are considered.
The list of references at the end of the book is not at
all comprehensive, but it gives a general idea of all new schools
of thought and of major developments both in our country and abroad.
Also, literature used in each section of the book is referred to in
footnotes.
This book sums up the author's work on the theory of
random vibrations and its application since 1959. Part of the
material is taken from the author's lectures for senior level
undergraduate and graduate students at the Moscow Energetics Institute.
Some of the problems were developed by the author in co-operation
with his co-workers from the Moscow Energetics Institute, which
is reflected in the references. Parts of the manuscript were
read by V. Chirkov, V. Radin, N. Ginger, V. Volohovsky, V. Moskvin,
V. Chromatov, V. Vasenin, V. Semenov and A. Scherbatov, to whom
the author expresses his sincere thanks.
V.V. Bolotin
June 1978
ix
x
Contents
page
EDITOR'S PREFACE v
AUTHOR'S PREFACE vii
CJ~PTER 1 - RANDOM LOADINGS ACTING ON MECHANICAL SYSTEMS
1.1 Loadings as Random Functions of Til,Ie 1
1.2 Loadings as Random Functions of Spatial
Coordinates and Time 10
1.3 Experimental Data on Certain Stationary
Random Loadings 18
1.4 Experimental Data on Some Non-Stationary
Random Loadings 30
CHAPTER 2 - METHODS IN THE THEORY OF RANDOM VIBRATIONS
2.1 Linear Systems with a Finite Number of
Degrees of Freedom 35
2.2 Spectral Methods in the Theory of
Random Vibrations 46
2.3 Linear Continuous Systems 55
2.4 Methods in the Theory of Markov Processes 66
2.5 Methods of Statistical Simulation 78
CHAPTER 3 - RANDOM VIBRATIONS OF LINEAR CONTINUOUS SYSTEMS
3.1 General Relations for Linear Systems 85
3.2 Random Vibrations in Linear Viscoelastic Systems 96
3.3 Vibrations of a Plate in a Field of
Random Pressures 101
3.4 Random Vibrations of Shells Containing
Compressible Fluid 107
3.5 Approximate ~1ethod of Analysis 118
3.6 Application of the Method of
Spectral Representation 132
CHAPTER 4 - THE ASYMPTOTIC METHOD IN THE THEORY OF
RANDOM VIBRATIONS OF CONTINUOUS SYSTEMS
4.1 Asymptotic Estimates for Natural Frequencies
and Natural Modes 143
4.2 Application of the Asymptotic Method to
Plates and Shells 157
4.3 Theory of the Distribution of Natural
Frequencies of Elastic Systems 168
4.4 Density of Natural Frequencies of Thin
Elastic Shells 181
4.5 Method of Integral Estimates for the Analysis
of Wide-Band Random Vibrations 194
4.6 Application of the Method of Integral Estimates 202
XL