Probability Models in Engineering and Science

by Benaroya; Haym

• ISBN13: 9780824723156
• ISBN10: 0824723155
• Format: Hardcover
• Copyright: 2005-06-24
• Publisher: CRC Press

Summary

Certainty exists only in idealized models. Viewed as the quantification of uncertainties, probabilitry and random processes play a significant role in modern engineering, particularly in areas such as structural dynamics. Unlike this book, however, few texts develop applied probability in the practical manner appropriate for engineers. Probability Models in Engineering and Science provides a comprehensive, self-contained introduction to applied probabilistic modeling. The first four chapters present basic concepts in probability and random variables, and while doing so, develop methods for static problems. The remaining chapters address dynamic problems, where time is a critical parameter in the randomness. Highlights of the presentation include numerous examples and illustrations and an engaging, human connection to the subject, achieved through short biographies of some of the key people in the field. End-of-chapter problems help solidify understanding and footnotes to the literature expand the discussions and introduce relevant journals and texts. This book builds the background today's engineers need to deal explicitly with the scatter observed in experimental data and with intricate dynamic behavior. Designed for undergraduate and graduate coursework as well as self-study, the text's coverage of theory, approximation methods, and numerical methods make it equally valuable to practitioners.

Author Biography

Seon Mi Han is a professor with the Department of Mechanical Engineering, Texas Tech University, Lubbock, Texas, USA.

Table of Contents

Introduction	1	(14)
Applications	1	(11)
Random Vibration	2	(1)
Fatigue Life	3	(2)
Ocean-Wave Forces	5	(3)
Wind Forces	8	(1)
Material Properties	8	(3)
Statistics and Probability	11	(1)
Units	12	(1)
Organization of the Text	13	(1)
Problems	13	(2)
Events and Probability	15	(34)
Sets	15	(11)
Basic Events	16	(5)
Operational Rules	21	(5)
Probability	26	(19)
Axioms of Probability	29	(1)
Extensions from the Axioms	30	(1)
Conditional Probability	31	(2)
Statistical Independence	33	(2)
Total Probability	35	(2)
Bayes' Theorem	37	(8)
Concluding Summary	45	(1)
Problems	45	(4)
Random Variable Models	49	(76)
Probability Distribution Function	50	(2)
Probability Density Function	52	(5)
Mathematical Expectation	57	(9)
Variance	60	(6)
Useful Probability Densities	66	(35)
The Uniform Density	66	(3)
The Exponential Density	69	(2)
The Normal (Gaussian) Density	71	(13)
The Lognormal Density	84	(3)
The Rayleigh Density	87	(3)
Probability Density Functions of a Discrete Random Variable	90	(9)
Moment-Generating Functions	99	(2)
Two Random Variables	101	(16)
Covariance and Correlation	110	(7)
Concluding Summary	117	(1)
Problems	118	(7)
Functions of Random Variables	125	(70)
Exact Functions of One Variable	125	(7)
Functions of Two or More RVs	132	(29)
General Case	149	(12)
Approximate Analysis	161	(18)
Direct Methods	161	(4)
Mean and Variance of a General Function of X to Order σ2x	165	(3)
Mean and Variance of a General Function of n RVs	168	(11)
Monte Carlo Methods	179	(9)
Independent Uniform Random Numbers	179	(4)
Independent Normal Random Numbers	183	(2)
A Discretization Procedure	185	(3)
Concluding Summary	188	(1)
Problems	188	(4)
The Standard Normal Table	192	(3)
Random Processes	195	(80)
Basic Random Process Descriptors	195	(1)
Ensemble Averaging	196	(5)
Stationarity	201	(7)
Derivatives of Stationary Processes	208	(2)
Fourier Series and Fourier Transforms	210	(19)
Harmonic Processes	229	(1)
Power Spectra	230	(31)
Narrow- and Broad-Band Processes	254	(3)
White Noise Processes	257	(2)
Spectral Densities of Derivatives of Stationary Random Processes	259	(2)
Fourier Representation of a Random Process	261	(10)
Borgman's Method of Frequency Discretization	267	(4)
Concluding Summary	271	(1)
Problems	271	(4)
Single-Degree-of-Freedom Dynamics	275	(64)
Motivating Examples	277	(1)
Transport of a Satellite	277	(1)
Rocket Ship	277	(1)
Deterministic SDoF Vibration	278	(29)
Free Vibration With No Damping	288	(1)
Harmonic Forced Vibration With No Damping	289	(2)
Free Vibration With Damping	291	(1)
Forced Vibration With Damping	292	(4)
Impulse Excitation	296	(3)
Arbitrary Loading: Convolution	299	(4)
Frequency Response Function	303	(4)
SDoF: The Response to Random Loads	307	(18)
Formulation	307	(1)
Derivation of Equations	308	(1)
Response Correlations	309	(3)
Response Spectral Density	312	(13)
Response to Two Random Loads	325	(8)
Concluding Summary	333	(1)
Problems	333	(6)
Multidegree-of-Freedom Vibration	339	(64)
Deterministic Vibration	339	(12)
Solution by Frequency Response Function	341	(2)
Modal Analysis	343	(6)
Advantages of Modal Analysis	349	(2)
Response to Random Loads	351	(22)
Response due to a Single Random Force	353	(3)
Response to Multiple Random Forces	356	(17)
Periodic Structures	373	(5)
Perfect Lattice Models	375	(3)
Effects of Imperfection	378	(1)
Inverse Vibration	378	(11)
Deterministic Inverse Vibration Problem	381	(3)
Effect of Uncertain Data	384	(5)
Random Eigenvalues	389	(6)
A Two-Degree-of-Freedom Model	393	(2)
Concluding Summary	395	(1)
Problems	395	(8)
Continuous System Vibration	403	(52)
Deterministic Continuous Systems	404	(7)
Strings	404	(2)
Axial Vibration of Beams	406	(2)
Transversely Vibrating Beams	408	(3)
Sturm-Liouville Eigenvalue Problem	411	(9)
Orthogonality	413	(1)
Natural Frequencies and Mode Shapes	414	(6)
Deterministic Vibration	420	(8)
Free Response	420	(2)
Forced Response via Eigenfunction Expansion	422	(6)
Random Vibration of Continuous Systems	428	(11)
Derivation of Response Spectral Density	429	(10)
Beams with Complex Loading	439	(12)
Transverse Vibration of Beam with Axial Force	439	(3)
Transverse Vibration of Beam on Elastic Foundation	442	(4)
Response of a Beam to a Traveling Force	446	(5)
Concluding Summary	451	(1)
Problems	452	(3)
Reliability	455	(60)
Introduction	455	(2)
First Excursion Failure	457	(36)
Exponential Failure Law	462	(3)
Modified Exponential Failure Law	465	(1)
Calculation of Up-Crossing Rate	466	(6)
Narrow-Band Process -- Envelope Function	472	(2)
Rice's Envelope Function for Gaussian Narrow-Band Process X (t)	474	(14)
Other Failure Laws	488	(5)
Fatigue Life Prediction	493	(19)
Failure Curves	495	(2)
Peak Distribution for Stationary Random Process	497	(4)
Peak Distribution of a Gaussian Process	501	(11)
Concluding Summary	512	(1)
Problems	512	(3)
Nonlinear Dynamic Models	515	(76)
Examples of Nonlinear Vibration	516	(3)
Fundamental Nonlinear Equations	519	(2)
Statistical Equivalent Linearization	521	(11)
Equivalent Nonlinearization	530	(2)
Perturbation Methods	532	(23)
Lindstedt-Poincare Method	534	(5)
Forced Oscillations of Quasiharmonic Systems	539	(4)
Jump Phenomenon	543	(1)
Periodic Solutions of Nonautonomous Systems	544	(8)
Random Duffing Oscillator	552	(3)
The van der Pol Equation	555	(6)
Limit Cycles	556	(1)
The Forced van der Pol Equation	557	(4)
Markov Process-Based Models	561	(26)
Probability Background	561	(4)
The Fokker-Planck Equation	565	(22)
Concluding Summary	587	(1)
Problems	587	(4)
Nonstationary Models	591	(40)
Some Applications	592	(6)
Envelope Function Model	598	(9)
Transient Response	599	(6)
MS Nonstationary Response	605	(2)
Nonstationary Generalizations	607	(5)
Discrete Model	608	(2)
Complex-Valued Stochastic Processes	610	(1)
Continuous Model	610	(2)
Priestley's Model	612	(3)
The Stieltjes Integral: An Aside	612	(2)
Priestley's Model	614	(1)
SDoF Oscillator Response	615	(7)
Stationary Case	615	(1)
Nonstationary Case	616	(2)
Undamped Oscillator	618	(1)
Underdamped Oscillator	619	(3)
Multi DoF Oscillator Response	622	(3)
Input Characterization	622	(2)
Response Characterization	624	(1)
Nonstationary and Nonlinear Oscillator	625	(4)
The Nonstationary and Nonlinear Duffing	627	(2)
Concluding Summary	629	(1)
Problems	629	(2)
The Monte Carlo Method	631	(48)
Introduction	631	(4)
Random-Number Generation	635	(24)
Standard Uniform Random Numbers	635	(2)
Generation of Nonuniform Random Variates	637	(11)
Composition Method	648	(3)
Von Neumann's Rejection-Acceptance Method	651	(8)
Joint Random Numbers	659	(5)
Inverse Transform Method	660	(1)
Linear Transform Method	661	(3)
Error Estimates	664	(6)
Applications	670	(6)
Evaluation of Finite-Dimensional Integrals	670	(3)
Generating a Time History for a Stationary Random Process Defined by a Power Spectral Density	673	(3)
Concluding Summary	676	(1)
Problems	676	(3)
Fluid-Induced Vibration	679	(48)
Ocean Currents and Waves	679	(23)
Spectral Density	680	(4)
Ocean Wave Spectral Densities	684	(5)
Approximation of Spectral Density from Time Series	689	(2)
Generation of Time Series from a Spectral Density	691	(1)
Short-Term Statistics	692	(6)
Long-Term Statistics	698	(2)
Wave Velocities via Linear Wave Theory	700	(2)
Fluid Forces in General	702	(11)
Wave Force Regime	703	(2)
Wave Forces on Small Structures -- Morison Equation	705	(6)
Vortex-Induced Vibration	711	(2)
Examples	713	(10)
Static Configuration of a Towing Cable	713	(4)
Fluid Forces on an Articulated Tower	717	(3)
Weibull and Gumbel Wave Height Distributions	720	(2)
Reconstructing Time Series for a Given Significant Wave Height	722	(1)
Available Numerical Codes	723	(4)
Index	727