Statistical Dynamics and Reliability Theory for Mechanical Structures

by Svetlitsky, Valery A.; Reshetov, N. L.

• ISBN13: 9783540442974
• ISBN10: 3540442979
• eBook ISBN(s): 9783540458265
• Format: Hardcover
• Copyright: 2003-02-01
• Publisher: Springer Nature
• More Book Details
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Summary

The theory of random processes is an integral part of the analysis and synthesis of complex engineering systems. This textbook systematically presents the fundamentals of statistical dynamics and reliability theory. The theory of Markovian processes used during the analysis of random dynamic processes in mechanical systems is described in detail. Examples are machines, instruments and structures loaded with perturbations. The reliability and lifetime of those objects depend on how properly these perturbations are taken into account. Random vibrations with finite and infinite numbers of degrees of freedom are analyzed as well as the theory and numerical methods of non-stationary processes under the conditions of statistical indeterminacy. This textbook is addressed to students and post-graduates of technical universities. It can also be useful to lecturers and mechanical engineers, including designers in different industries.

Table of Contents

Introduction	1	(10)
Fundamentals of the Probability Theory and the Theory of Random Processes	11	(34)
Brief Information on the Probability Theory	11	(3)
Basic Concepts of the Probability Theory	11	(3)
The Distribution Function and the Probability Density of a Random Variable	14	(4)
Numerical Characteristics of Random Quantities and Their Principal Properties	18	(4)
Probability Density Distribution Laws	22	(8)
Determination of the Probability of a Normally Distributed Random Quantity Lying in the Given Range	30	(10)
Complex Random Quantities	40	(2)
Numerical Characteristics of Functions of Random Arguments	42	(3)
Non-Stationary Random Functions (Processes)	45	(28)
Introduction	45	(3)
Probability Characteristics of Non-Stationary Random Functions	48	(7)
Random Function Systems and Their Probability Characteristics	55	(4)
Random Functions Linear Transformations	59	(6)
The Probabilistic Characteristics of the Linear Differential Equations at Non-Stationary Random Disturbances	65	(8)
Stationary Random Functions (Processes)	73	(28)
Probability Characteristics of Stationary Random Functions	73	(4)
The Ergodic Property of a Stationary Random Function	77	(4)
Derivatives and Integrals of Stationary Functions	81	(3)
Probability Characteristics of Stationary Random Function Derivatives	81	(2)
Probability Characteristics of the Integral of Stationary Random Functions	83	(1)
The Spectral Representation of Stationary Random Processes	84	(8)
Spectral Densities of Stationary Function Derivatives	88	(1)
Determination of Spectral Density (Examples)	88	(4)
Cross-Spectral Densities and their Properties	92	(3)
Determination of the Spectral Densities of the Linear Differential Equations with Constant Coefficients Solutions	95	(6)
Fundamentals of the Markov Processes Theory	101	(30)
Continuous One-Dimensional Markov Processes	101	(2)
The Fokker-Planck-Kolmogorov Equation	103	(16)
Multidimensional Markov Processes	119	(4)
Determination of the Probability of Attaining a Random Function Possible Values Area Boundaries	123	(8)
Random Vibrations of Systems with One Degree of Freedom	131	(66)
Free Random Vibrations of Linear Systems	131	(5)
Forced Random Vibrations of Linear Systems	136	(22)
Non-Stationary Vibrations	136	(16)
Stationary Forced Vibrations	152	(6)
Vibrations Caused by Random Kinematic Excitation	158	(13)
Non-Stationary Random Vibrations at Kinematic Excitation	159	(12)
The Problem of Overshoots at Random Vibrations	171	(11)
Nonlinear Random Vibrations	182	(15)
The Method of Statistical Linearization	183	(6)
The Solution of the Nonlinear Equations with the Use of Markov Processes	189	(3)
The Method of Statistical Trials (Monte-Carlo Method)	192	(5)
Random Vibrations of Systems with Finite Number of Degrees of Freedom	197	(62)
Free Random Vibrations of Linear Systems	197	(5)
Vibrations at Random Pulse Loading	202	(15)
Non-Stationary Random Vibrations of Linear Systems	217	(5)
The Method of Principal Coordinates in Non-Stationary Vibrations Analysis	222	(18)
Forced Stationary Random Vibrations of Linear Systems	240	(19)
Random Vibrations of Strings; Longitudinal and Torsional Vibrations of Straight Rods	259	(20)
Introduction	259	(4)
Equations of Small Vibrations	263	(4)
Solving Equations of Small Vibrations	267	(12)
Random Vibrations of Rods	279	(34)
Nonlinear Equations of Motion of Three-Dimensional Curvilinear Rods	281	(5)
Equations of the Motion of a Rod in the Attached Coordinate System	286	(4)
Equation of Space Motion of a Rod	286	(2)
Equation of Plane Motion of a Rod	288	(1)
Rods Having Lumped Masses	289	(1)
Equation of Small Vibrations of Rods	290	(7)
Equations of Small Vibrations in the Attached Coordinate Frame	293	(1)
Equations of Small Vibrations about a Natural State	294	(3)
Determination of Eigenvalues and Eigenvectors	297	(2)
Non-Stationary Random Vibrations of Rods	299	(4)
Stationary Random Vibrations of Rods	303	(10)
Fundamentals of Reliability Theory	313	(36)
Introduction	313	(9)
Elementary Problems of Reliability Theory	322	(2)
Possible Causes of Failures	324	(1)
Determination of Numerical Values of No-Failure Operation Probability (Reliability)	325	(12)
Determination of Reliability at the Linear Dependence of a Stress State on Random Loads	337	(2)
Determination of the Probability of No-Failure Operation at the Nonlinear Dependence of the Random Quantity F on External Loads	339	(10)
Random Processes at the Action of Random Functions Bounded in Absolute Value	349	(82)
Introduction	349	(10)
Determining the Maximum Values of the Components of the Systems State Vector	359	(2)
Areas of Possible Values of the System State Vector at the Action of Independent Excitations	361	(11)
Projections of the Area of Possible Values of the System State Vector onto Two-Dimensional Planes	372	(2)
Determination of the Maximum Values of Dynamic Reactions	374	(6)
Areas of Possible Values of the System State Vector in the Case of Several Sections of Motion	380	(4)
Areas of Possible Values of the System State Vector at the Action of Dependent Random Excitations	384	(19)
Determination of the Maximum Values of Linear Functionals at Independent Excitations	403	(9)
Maximum Value of a Linear Functional at Dependent Excitations	412	(6)
Vibration Protection of Mechanical Systems	418	(13)
A. Appendices	431	(12)
A.1 Elementary Generalized Functions	431	(3)
A.2 Values of Integrals Jn	434	(2)
A.3 Correlation Functions and Spectral Densities Corresponding to Them	436	(2)
A.4 Hiawatha Designs an Experiment	438	(5)
References	443	(2)
Index	445