Random Vibrations Theory and Practice

by Wirsching, Paul H.; Paez, Thomas L,; Ortiz, Keith

• ISBN13: 9780486450155
• ISBN10: 0486450155
• Format: Paperback
• Copyright: 2006-05-12
• Publisher: Dover Publications

Summary

The most comprehensive text and reference available on the study of random vibrations, this book was designed for graduate students and mechanical, structural, and aerospace engineers. In addition to coverage of background topics in probability, statistics, and random processes, it develops methods for analyzing and controlling random vibrations. 1995 edition.

Author Biography

Paul H. Wirsching is Professor Emeritus of Aerospace and Mechanical Engineering at the University of Arizona. Thomas L. Paez and Keith Ortiz are affiliated with Sandia National Laboratories of Albuquerque, New Mexico.

Table of Contents

Preface	xv
1 Introduction	1	(8)
1.1 History and Motivation,	1	(3)
1.2 Example,	4	(2)
1.3 Scope and Organization,	6	(3)
2 Random Variables	9	(63)
2.1 Essential Probability,	9	(5)
2.1.1 Set Theory,	10	(1)
2.1.2 Axioms of Probability,	11	(1)
2.1.3 Conditional Probability and Independence,	12	(2)
2.2 Single Random Variables,	14	(16)
2.2.1 Discrete Random Variables and Probability Mass Function,	14	(1)
2.2.2 Continuous Random Variables and Probability Density Function,	15	(4)
2.2.3 Cumulative Distribution Functions,	19	(2)
2.2.4 Measures of Central Tendency and Dispersion,	21	(2)
2.2.5 Expected Values,	23	(2)
2.2.6 Moment-Generating Functions,	25	(1)
2.2.7 Gaussian (Normal) Random Variables,	26	(4)
2.3 Jointly Distributed Random Variables,	30	(13)
2.3.1 Joint and Marginal Distributions,	30	(3)
2.3.2 Conditional Distributions and Independence,	33	(3)
2.3.3 Expected Values, Covariance, and Correlation Coefficient,	36	(2)
2.3.4 Linear Dependence,	38	(3)
2.3.5 Joint Normal Distribution,	41	(2)
2.4 Functions of Random Variables,	43	(12)
2.4.1 Change of Variables by Cumulative Distribution Function,	43	(2)
2.4.2 Change of Variables by Probability Density Function,	45	(4)
2.4.3 Multidimensional Change of Variables,	49	(2)
2.4.4 Sums of Random Variables,	51	(4)
2.5 Central Limit Theorem and Distributions Related to Normal Distribution,	55	(13)
2.5.1 De Moivre-Laplace Approximation,	56	(1)
2.5.2 Central Limit Theorem,	57	(1)
2.5.3 Distribution of Sample Mean: Normal,	58	(2)
2.5.4 Distribution of Sample Variance: Chi Squared,	60	(4)
2.5.5 Distributions of Related Ratios of Random Variables: t and F,	64	(1)
2.5.6 Distribution of Extreme Values,	65	(3)
Problems,	68	(4)
3 Random Processes in Time Domain	72	(28)
3.1 Random Process Definitions,	72	(12)
3.1.1 State Spaces and Index Sets,	72	(1)
3.1.2 Ensembles and Ensemble Averages,	73	(7)
3.1.3 Stationary Processes,	80	(2)
3.1.4 Ergodic Processes,	82	(1)
3.1.5 Gaussian Processes,	83	(1)
3.2 Correlation,	84	(7)
3.2.1 Characteristics of Autocorrelation Function,	84	(2)
3.2.2 Cross-Correlation Function and Linear Transformations,	86	(3)
3.2.3 Derivatives of Stationary Processes,	89	(2)
3.3 Some Essential Random Processes,	91	(5)
3.3.1 Harmonic Processes,	91	(3)
3.3.2 Poisson Process,	94	(2)
Problems,	96	(4)
4 Fourier Transforms	100	(23)
4.1 Fourier Transform,	100	(13)
4.1.1 Fourier Series,	101	(3)
4.1.2 Fourier Transforms,	104	(3)
4.1.3 Symmetry,	107	(1)
4.1.4 Basic Theorems,	108	(3)
4.1.5 Correlation, Convolution, and Windowing,	111	(2)
4.2 Dirac Delta Function,	113	(6)
4.2.1 Definition and Properties of Delta Function,	113	(3)
4.2.2 Fourier Transform of Delta Function,	116	(2)
4.2.3 Transforms of Cosine and Sine Functions,	118	(1)
Problems,	119	(4)
5 Random Processes in Frequency Domain	123	(23)
5.1 Spectral Density Function,	123	(16)
5.1.1 Definition,	124	(3)
5.1.2 Relationship with Fourier Transform of X(t),	127	(3)
5.1.3 Practical Issues,	130	(5)
5.1.4 White Noise and Bandpass Filtered Spectra,	135	(4)
5.2 Cross-Spectral Density Function,	139	(7)
5.2.1 Definition,	139	(1)
5.2.2 Coherence Function and Linear Transformations,	140	(1)
5.2.3 Derivatives of Stationary Processes,	141	(2)
Problems,	143	(3)
6 Statistical Properties of Random Processes	146	(28)
6.1 Level Crossings,	146	(16)
6.1.1 Preliminary Remarks,	146	(4)
6.1.2 Derivation of Expected Rate of Level Crossing,	150	(2)
6.1.3 Specializations,	152	(5)
6.1.4 Rice's Narrow-Band Envelope,	157	(5)
6.2 Distributions of Extrema,	162	(9)
6.2.1 Simple Approach to Distribution of Peak,	162	(2)
6.2.2 General Approach to Distribution of Peak,	164	(3)
6.2.3 Approximate Distribution for Height of Rise or Fall,	167	(4)
Problems,	171	(3)
7 Vibration of Single-Degree-of-Freedom Systems	174	(23)
7.1 Free Vibration of Single-Degree-of-Freedom System,	174	(6)
7.1.1 Background,	174	(1)
7.1.2 Harmonic Motion,	174	(3)
7.1.3 Free Vibration of Undamped Single-Degree-of-Freedom System,	177	(1)
7.1.4 Damped Free Vibration of Single-Degree-of-Freedom System,	178	(2)
7.2 Forced Vibration,	180	(10)
7.2.1 Force-Excited System: Harmonic Excitation,	180	(3)
7.2.2 Base-Excited System: Absolute Motion,	183	(1)
7.2.3 Base-Excited System: Relative Motion,	184	(6)
7.3 Background for Response of Single-Degree-of-Freedom System to Random Forces,	190	(7)
7.3.1 Response of Single-Degree-of-Freedom System to Impulsive Forces,	190	(1)
7.3.2 Response of Single-Degree-of-Freedom System to Arbitrary Loading,	191	(1)
7.3.3 Relationship Between h (t) and H(w),	192	(1)
7.3.4 Relationship Between X(w) and F(w),	193	(1)
Problems,	194	(3)
8 Response of Single-Degree-of-Freedom Linear Systems to Random Environments	197	(21)
8.1 Response to Stationary Random Forces,	197	(6)
8.1.1 Preliminary Remarks,	197	(1)
8.1.2 Mean of Response Process,	198	(1)
8.1.3 Autocorrelation of Response Process,	199	(3)
8.1.4 Spectral Density of Response Process,	202	(1)
8.1.5 Distribution of Response Process,	203	(1)
8.2 White-Noise Process as Model for Force,	203	(4)
8.2.1 Definition of Process,	203	(1)
8.2.2 Response of Force-Excited System to White Noise,	204	(2)
8.2.3 Engineering Significance of White-Noise Process,	206	(1)
8.3 Examples of Response of Single-Degree-of-Freedom Systems to Random Forces,	207	(6)
Problems,	213	(5)
9 Random Vibration of Multi-Degree-of-Freedom Systems	218	(36)
9.1 Equations of Motion for Multi-Degree-of-Freedom System,	218	(5)
9.1.1 Introduction,	218	(2)
9.1.2 Equations of Motion,	220	(1)
9.1.3 Transfer Function and Impulse Response Functions for System,	221	(2)
9.2 Direct Model for Determining Response of MDOF System,	223	(11)
9.2.1 Expression for Response,	223	(1)
9.2.2 Mean Value of Response,	224	(1)
9.2.3 Autocorrelation Function,	225	(1)
9.2.4 Spectral Density Function of Response,	226	(1)
9.2.5 Single Response Variable: Special Cases,	227	(7)
9.3 Normal Mode Method,	234	(15)
9.3.1 Preliminary Remarks,	234	(1)
9.3.2 The Eigenvalue Problem: Free-Vibration Solution to Equations of Motion,	234	(1)
9.3.3 Example of Eigenvalue Problem,	235	(2)
9.3.4 Normal Mode Method: Orthogonality Conditions,	237	(4)
9.3.5 Normal Mode Method: Equations of Motion,	241	(1)
9.3.6 Big Payoff for Using Normal Mode Method,	242	(1)
9.3.7 Introduction of Damping into Equations of Motion,	243	(3)
9.3.8 Normal Mode Method: Mode Combination Problem—Some Methods,	246	(2)
9.3.9 Example of Mode Combination Algorithm Results,	248	(1)
9.4 Computer Codes for Analyzing MDOF Structural Systems,	249	(1)
Problems,	250	(4)
10 Design to Avoid Structural Failures Due to Random Vibration	254	(42)
10.1 Three-Sigma Design,	254	(3)
10.1.1 Basic Design Criterion,	254	(2)
10.1.2 More General Statement of Three Sigma Criterion,	256	(1)
10.2 First-Passage Failure,	257	(9)
10.2.1 Introductory Remarks,	257	(2)
10.2.2 Basic Formulation of First-Passage Problem,	259	(1)
10.2.3 First-Passage Failure: Exact Distribution of Largest in Sample of Size n Independent Peaks,	260	(3)
10.2.4 First-Passage Failure: Use of Extreme-Value Distribution to Model Distribution of Largest Peak,	263	(1)
10.2.5 First-Passage Failure: Determination of Design Value Using Concept of Return Period,	264	(2)
10.3 Fatigue: An Introduction,	266	(24)
10.3.1 Physical Process of Fatigue,	266	(1)
10.3.2 Fatigue Strength Models,	267	(4)
10.3.3 Miner's Rule,	271	(3)
10.3.4 Models of Fatigue Damage Under Narrow-Band Random Stress,	274	(9)
10.3.5 Models of Fatigue Damage Under Wide-Band Random Stresses,	283	(5)
10.3.6 Quality of Miner's Rule,	288	(2)
Problems,	290	(6)
11 Introduction to Parameter Estimation	296	(19)
11.1 Estimation and Analysis of Mean,	297	(11)
11.1.1 Maximum Likelihood Estimation,	297	(4)
11.1.2 Bias and Consistency of Mean Estimator,	301	(2)
11.1.3 Sampling Distribution of Mean Estimator and Confidence Intervals on Mean,	303	(4)
11.1.4 Bias in Variance Estimator,	307	(1)
11.2 Other Important Problems in Parameter Estimation,	308	(3)
11.2.1 Sampling Distribution for Variance,	308	(1)
11.2.2 Confidence Intervals for Mean with Unknown Variance,	309	(2)
11.3 Closure,	311	(1)
Problems,	312	(3)
12 Time-Domain Estimation of Random Process Parameters	315	(26)
12.1 Random Process Parameter Estimation via Ensemble Average,	316	(9)
12.1.1 Mean, Variance, and Standard Deviation Function,	316	(2)
12.1.2 Correlation,	318	(2)
12.1.3 Autocorrelation Function,	320	(4)
12.1.4 Cross-Correlation Function,	324	(1)
12.1.5 Covariance and Correlation Coefficient Function,	324	(1)
12.2 Stationary Random Process Parameter Estimation via Temporal Averaging,	325	(5)
12.3 Other Methods for Nonstationary Random Process Analysis,	330	(8)
12.3.1 Direct Analysis of Nonstationary Random Processes,	330	(3)
12.3.2 Indirect Analysis of Nonstationary Random Processes: Method of Shock Response Spectra,	333	(5)
Problems,	338	(3)
13 Discrete Fourier Transform	341	(29)
13.1 Definition of DFT and Fundamental Characteristics,	341	(3)
13.2 Periodicity of DFT; DFT of Real Signal,	344	(6)
13.3 DFT of Continuous-Time Signals: Aliasing and Leakage,	350	(6)
13.4 Data Windows,	356	(5)
13.5 Fast Fourier Transform,	361	(4)
13.6 Generation of Real-Valued, Finite-Duration, Sampled Realizations of Stationary, Normal Random Processes,	365	(3)
Problems,	368	(2)
14 Frequency-Domain Estimation of Random Processes	370	(34)
14.1 Fundamental Concepts in Estimation of Autospectral Density,	370	(11)
14.1.1 Direct Estimation of Autospectral Density,	371	(4)
14.1.2 Maximum Likelihood Estimation,	375	(3)
14.1.3 Bias, Consistency, and Sampling Distribution of Autospectral Density Estimator,	378	(3)
14.2 Practical Aspects of Estimation of Autospectral Density,	381	(5)
14.3 Real-Time Estimation of Spectral Density,	386	(3)
14.4 Estimation of Cross-Spectral Density and Ordinary Coherence,	389	(3)
14.4.1 Cross-Spectral Density Estimation,	389	(1)
14.4.2 Coherence Estimation,	390	(1)
14.4.3 Bias and Consistency of Cross-Spectral Density Estimator,	391	(1)
14.5 Estimation of Frequency Response Function,	392	(9)
14.5.1 Frequency Response in SISO Case,	392	(4)
14.5.2 Sampling Distribution and Variance of Frequency Response Function Estimator,	396	(2)
14.5.3 Frequency Response Function in MISO Case,	398	(3)
Problems,	401	(3)
Appendix A Convergence of Random Processes	404	(12)
A.1 Introductory Comments,	404	(1)
A.2 Modes of Convergence,	405	(2)
A.3 Mean-Square Continuity,	407	(1)
A.4 Mean-Square Differentiability,	407	(2)
A.5 Mean-Square Integrability,	409	(3)
A.6 Ergodicity,	412	(1)
A.7 Central Limit Theorem,	413	(3)
Appendix B Integrals of Transfer Functions	416	(1)
Appendix C Formulas for Approximate Evaluation of Some Integrals Useful in Probability	417	(2)
Appendix D Some Fast Fourier Transform Programs	419	(4)
D.1 C Language,	419	(2)
D.2 Fortran,	421	(2)
Appendix E Tables	423	(10)
E.1 Table of Cumulative Distribution of Standard Normal Random Variable,	423	(3)
E.2 Table of Percentage Points of X2 Distribution,	426	(3)
E.3 Table of Percentage Points of t Distribution,	429	(1)
E.4 Table of Percentage Points of the F Distribution,	430	(3)
References	433	(8)
Index	441