RANDOM VIBRATION AND SPECTRAL ANALYSIS

Transcription

1 RANDOM VIBRATION AND SPECTRAL ANALYSIS

2 SOLID MECHANICS AND ITS APPLICATIONS Volume 33 Series Editor: G.M.L. GLADWELL Solid Mechanics Division, Faculty oj Engineering University oj Waterloo Waterloo, Ontario, Canada N2L 3G1 Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

3 Random Vibration and Spectral Analysis by ANDRE PREUMONT Universite Libre de Bruxelles, Belgium SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

4 Library of Congress Cataloging-in-Publication Data PreUNont, Andre, Random vibration and spectra] analysis I by Andre Preu.ont. p, CN. -- (Solid mechanics and its applications) Includes index. ISBN ISBN (ebook) DOI / RandoN vibration. 2. Stochastic processes. 3. Spectral theory (Mathematics)--Data processing. I. Title. II. Series. OA935.P '.32' dc ISBN Printed on acid-free paper Translation of the French edition "Vibrations Aleatoires et Analyse Spectrale" Premiere edition ISBN CH-1015 Lausanne Tous droits reserves Reproduction interdite All Rights Reserved 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994 Softcover reprint of the hardcover 1st edition 1994 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner,

5 "... Je n'aflirme rien, je me contente de croire qu'ii y a plus de choses possibles qu'on ne pense." Voltaire, Micromegas

6 TABLE OF CONTENTS Preface xv 1 Introduction Overview Organization Notations The Fourier transform Differentiation theorem Translation theorem Parseval's theorem Symmetry, change of scale, duality Harmonic functions Convolution, correlation Convolution integral Correlation integral Example: The leakage References Problems Random Variables Axioms of probability theory Bernoulli's law of large numbers Alternative interpretation Axioms Theorems and definitions Random variable Discrete random variable Continuous random variable Jointly distributed random variables Conditional distribution Functions of Random variables Function of one random variable Function of two random variables The sum of two independent random variables Rayleigh distribution n fundions of n random variables Moments... 27

7 viii Random Vibration and Spectral Analysis Expected value Moments Schwarz inequality Chebyshev's inequality Characterstic function, Cumulants Single random variable Jointly distributed random variables References Problems Random Processes Introduction Specification of a random process Probability density functions Characteristic function Moment functions Cumulant functions Characteristic functional Stationary random process Properties of the correlation functions Differentiation Convergence Continuity Stochastic differentiation Stochastic integrals, Ergodicity Integration Temporal mean Ergodicity theorem Spectral decomposition Fourier transform Power spectral density Examples White noise Ideal low-pass process Process with exponential correlation Construction of a random process with specified power spectral density Cross power spectral density Periodic process References Problems... 55

8 Contents 4 Gaussian Process, Poisson Process 4.1 Gaussian random variable 4.2 The central limit theorem Example Example 2: Binomial distribution. 4.3 Jointly Gaussian random variables Remark Gaussian random vector. 4.5 Gaussian random process 4.6 Poisson process Counting process Uniform Poisson process Non-uniform Poisson process 4.7 Random pulses 4.8 Shot noise. 4.9 References Problems. 5 Random Response of a Single Degree of Freedom Oscilla- tor Response of a linear system Single degree of freedom oscillator Stationary response of a linear system Stationary response of the linear oscillator. White noise approximation Transient response Excitation applied from t = Stationary excitation Spectral moments Definition Computation for the linear oscillator Rice formulae Envelope of a narrow band process Crandall &. Mark's definition Joint distribution of X and X Probability distribution of the envelope References Problems ix

9 x Random Vibration and Spectral Analysis 6 Random Response of Multi Degree of Freedom Systems Some concepts of structural dynamics Equation of motion Input-output relationship Modal decomposition State variable form Structural and hereditary damping Remarks Seismic excitation Equation of motion Effective modal mass Input-Output relationships in the frequency domain Response to a stationary excitation Role of the cross-correlation Response to a stationary seismic excitation Continuous structures Input-Output relationship Structure with normal modes Co-spectrum Example: Boundary layer noise Discretization of the excitation Along-wind response of a tall building Along-wind aerodynamic forces Mean wind Spectrum at a point Davenport spectrum Example Earthquake Response spectrum Cascade analysis Remark on sound pressure level References Problems Input-Output Relationship for Physical Systems Estimation of frequency response functions Coherence function Effect of measurement noise Example Remark References. 141

10 Contents xi 8 Spectral Description of Non-stationary Random Processes Introduction Stationary random process Non-stationary random process Objectives of a spectral description Instantaneous power spectrum Mark's Physical Spectrum Definition and properties Dualit~r, uncertainty principle Relation to the PSD of a stationary process Example: Structural response to a sweep sine Priestley's Evolutionary Spectrum Generalized harmonic analysis Evolutionary spectrum Vector process Input-output relationship State variable form Remarks Applications Structural response to a. sweep sine Transient response of an oscillator Earthquake records Summary References Problems Markov Process Conditional plobability Classification of random processes Smoluchowski equation Process with independent increments Random Walk Wiener process Markov process and state variables Gaussian Markov process Covariance matrix Wide sense Markov process Power spectral density matrix Random walk and diffusion equation Random walk of a free particle Randoln walk of an elastically bound particle One-dimensional Fokker-Planck equation

11 xii Random Vibration and Spectral Analysis Derivation of the Fokker-Planck equation Kolmogorovequation Multi-dimensional Fokker-Planck equation The Brownian motion of an oscillator Replacement of an actual process by a Markov process One-dimensional process Stochastically equivalent systems Multi-dimensional process References Problems Threshold Crossings, Maxima, Envelope and Peak Factor Introduction Threshold crossings Up-crossings of a level b Central frequency Maxima Envelope Crandall & Mark's definition Rice's definition The Hilbert transform Cramer & Leadbetter's definition Discussion Second order joint distribution of the envelope Threshold crossings Clump size First-crossing problem Introduction Independent crossings Independent envelope crossings Approach based on the clump size Vanmarcke's model Extreme point process First-passage problem and Fokker-Planck equation Multidimensional Markov process Fokker-Planck equation of the envelope Kolmogorov equation of the reliability Peak factor Extreme value probability Formulae for the peak factor References Problems

12 Contents 11 Random fatigue 11.1 Introduction Uniaxial loading with zero mean 11.3 Biaxial loading with zero mean 11.4 Finite element formulation Fluctuating stresses Recommended procedure Example References Problems. xiii The Discrete Fourier Transform 12.1 Introduction Consequences of the convolution theorem Periodic continuation Sampling Shannon's theorem, Aliasing Fourier series Orthogonal functions Fourier series Gibbs phenomenon Relation to the Fourier transform Graphical development of the DFT Analytical development of the DFT Definition and properties of the DFT Definition of the DFT and IDFT Properties of the DFT Leakage reduction Power spectrum estimation lOConvolution and correlation via FFT Periodic convolution and correlation Approximation of the continuous convolution 12.1O.3Sectioning Overlap-save Sectioning Overlap-add FFT simulation of Gaussian processes with prescribed PSD 12.12References Problems Bibliography Index

13 Preface I became interested in Random Vibration during the preparation of my PhD dissertation, which was concerned with the seismic response of nuclear reactor cores. I was initiated into this field through the cla.ssical books by Y.K.Lin, S.H.Crandall and a few others. After the completion of my PhD, in 1981, my supervisor M.Gera.din encouraged me to prepare a course in Random Vibration for fourth and fifth year students in Aeronautics, at the University of Liege. There was at the time very little material available in French on that subject. A first draft was produced during 1983 and 1984 and revised in These notes were published by the Presses Poly techniques et Universitaires Romandes (Lausanne, Suisse) in When Kluwer decided to publish an English translation ofthe book in 1992, I had to choose between letting Kluwer translate the French text in-extenso or doing it myself, which would allow me to carry out a sustantial revision of the book. I took the second option and decided to rewrite or delete some of the original text and include new material, based on my personal experience, or reflecting recent technical advances. Chapter 6, devoted to the response of multi degree offreedom structures, has been completely rewritten, and Chapter 11 on random fatigue is entirely new. The computer programs which have been developed in parallel with these chapters have been incorporated in the general purpose finite element software SAMCEF, developed at the University of Liege. All the chapters have been supplemented with a set of problems. I am deeply endebted to Prof. G.M.L.Gladwell from the University of Wa, terloo, who read the manuscript and corrected many mistakes and misuses of the English language. His comments have been invaluable in improving the text. I take this opportunity to thank Prof. Michel Geradin from the University of Liege for his advice, encouragement, and long friendship. I dedicate this book to Prof. Andre Jaumotte, Honorary Rector ofthe University of Brussels. His enthusiastic response to new ideas, and tireless action to promote research, his supportive friendship and his humanism have been a constant stimulus and example. Andre Preumont Bruxelles, November 1993.