Random Vibrations
Mechanical vibration systems are sometimes excited by stochastic forces. Such excitations may appear in all kinds of vibration systems, e. g. an automobile driving on a standard road, a centrifuge with erratic charge or a tall building waving in all weath
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FORCED LINEAR VIBRATIONS BY
PETER C. MÜLLER AND
WERNER 0. SCHIERLEN
SPRINGER-VERLAG WIEN GMBH
This work is subject to copyright. AU rights are reserved, whether the whole or part of the material is concemed specifically those of translation, reprinting, re-use of iIlustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks.
©
1977 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien-New York in 1977
ISBN 978-3-211-81487-1 DOI 10.1007/978-3-7091-4356-8
ISBN 978-3-7091-4356-8 (eBook)
PREFACE This textbook contains, with some extensions, our lectures given at the Deparlment of General Mechanics of the International Centre for Mechanical Seiences (CISM) in Udine/Italy during the month ofOctober, 1973. The book is divided into Jour major parts. The first part (Chapter 2, 3) is concerned with the mathematical representation of vibration systems and the corresponding general solution. The second part (Chapter 4) deals with the boundedness and stability of vibration systems. Thus, information on the general behavior of the system is obtained without any specified knowledge of the initial conditions and forcing functions. The third part (Chapter 5, 6) is devoted to deterministic excitation forces. In particular, the harmonic excitation Ieads to the phenomena of resonance, pseudoresonance and absorption. The fourth part (Chapter 7) considers s~ochastic excitation forces. The covariance analysis and the spectral density analysis of random vibrations are presented. Throughout the book examples are inserted for illustration. In conclusion, wr wish to express our gratitude to the International Centre for Mechanical Seiences (CISM) and to Professor Sobrero who invited us to deliver the lecture in Udine. We also acknowledge the support of Professor Magnus from the Institute B of Mechanics at the Technical University Munich. Munich, October 19 73
Peter C. Müller
Werner 0. Schiehlen
5
CHAPTER
1
Introduction
The subject of vibration deals with the oscillatory behavior of physical systems. The interaction of mass and elasticity allows vibration as well as the interaction of induction and capacity. Most vehicles, machines and circuits experience vibration and their design generally requires consideration of their oscillatory behavior. Vibration systems can be characterized as linear or non-linear, as time-invariant or time-variant, as free or forced, as single-degree of freedom or multi-degree of freedom. For linear systems the principle of superposition holds, and the mathematical techniques available for their treatment are well-developed in matrix and control theory. In contrast, for the analysis of nonlinear systems the techniques are only partially developed and they are based mainly on approximation methods.
For linear, time-invariant sys-
tems the concept of modal analysis is available featuring eigenvalues and eigenvectors. In contrary, for the analysis of linear, time-variant systems the fundamental matrix has to be found by numerical integr
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