Statistical Dynamics and Reliability Theory for Mechanical Structures

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-graduate of technical universities. It can be also useful to lecturers and mechanical engineers, including designers in different industries.

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The monograph text is based on lectures delivered by author during many years for students of Applied Iechanics Department of Bauman Ioscow State Technical University. The monograph includes also analitical results of scientific research obtained in collaboration with industry. Progress in developing new equipment has called for a better understand­ ing of the physical peculiarities pertaining to the action of designed structures in real conditions. This is necessary for increasing the accuracy of the analysis and making these structures more reliable. It has been found that classical determined perturbations are not principal and that determinism-based methods of classical mechanics prove insufficient for understanding and explaining physical effects that arise at the operation of instruments located on moving objects, the vibration of rocket engines, the motion of a vehicle, and the action of wind and seismic loads. Therefore the necessity arose for devising a new physical model to analyze these dynamic processes and, in particular, for creating a new mathematical apparatus that would allow us to take into account non-deterministic external excitations. The theory of random processes that had been developed well enough as applied to problems of radio engineering and automatic control, where the effect produced by random excitations appeared to be commensurable with that of deterministic excitations and where the ignoring of the random ex­ citations would bring about incorrect results, became such an apparatus.

Contenu

1. Fundamentals of the Probability Theory and the Theory of Random Processes.- 1.1 Brief Information on the Probability Theory.- 1.1.1 Basic Concepts of the Probability Theory.- 1.2 The Distribution Function and the Probability Density of a Random Variable.- 1.3 Numerical Characteristics of Random Quantities and Their Principal Properties.- 1.4 Probability Density Distribution Laws.- 1.5 Determination of the Probability of a Normally Distributed Random Quantity Lying in the Given Range.- 1.7 Complex Random Quantities.- 1.8 Numerical Characteristics of Functions of Random Arguments.- 2. Non-Stationary Random Functions (Processes).- 2.1 Introduction.- 2.2 Probability Characteristics of Non-Stationary Random Functions.- 2.3 Random Function Systems and Their Probability Characteristics.- 2.4 Random Functions Linear Transformations.- 2.5 The Probabilistic Characteristics of the Linear Differential Equations at Non-Stationary Random Disturbances.- 3. Stationary Random Functions (Processes).- 3.1 Probability Characteristics of Stationary Random Functions.- 3.2 The Ergodic Property of a Stationary Random Function.- 3.3 Derivatives and Integrals of Stationary Functions.- 3.3.1 Probability Characteristics of Stationary Random Function Derivatives.- 3.3.2 Probability Characteristics of the Integral of Stationary Random Functions.- 3.4 The Spectral Representation of Stationary Random Processes.- 3.4.1 Spectral Densities of Stationary Function Derivatives.- 3.4.2 Determination of Spectral Density (Examples).- 3.5 Cross-Spectral Densities and their Properties.- 3.6 Determination of the Spectral Densities of the Linear Differential Equations with Constant Coefficients Solutions.- 4. Fundamentals of the Markov Processes Theory.- 4.1 Continuous One-Dimensional Markov Processes.- 4.2 The Fokker-Planck-Kolmogorov Equation.- 4.3 Multidimensional Markov Processes.- 4.4 Determination of the Probability of Attaining a Random Function Possible Values Area Boundaries.- 5. Random Vibrations of Systems with One Degree of Freedom.- 5.1 Free Random Vibrations of Linear Systems.- 5.2 Forced Random Vibrations of Linear Systems.- 5.2.1 Non-Stationary Vibrations.- 5.2.2 Stationary Forced Vibrations.- 5.3 Vibrations Caused by Random Kinematic Excitation.- 5.3.1 Non-Stationary Random Vibrations at Kinematic Excitation.- 5.4 The Problem of Overshoots at Random Vibrations.- 5.5 Nonlinear Random Vibrations.- 5.5.1 The Method of Statistical Linearization.- 5.5.2 The Solution of the Nonlinear Equations with the Use of Markov Processes.- 5.5.3 The Method of Statistical Trials (Monte-Carlo Method).- 6. Random Vibrations of Systems with Finite Number of Degrees of Freedom.- 6.1 Free Random Vibrations of Linear Systems.- 6.2 Vibrations at Random Pulse Loading.- 6.3 Non-Stationary Random Vibrations of Linear Systems.- 6.4 The Method of Principal Coordinates in Non-Stationary Vibrations Analysis.- 6.5 Forced Stationary Random Vibrations of Linear Systems.- 7. Random Vibrations of Strings; Longitudinal and Torsional Vibrations of Straight Rods.- 7.1 Introduction.- 7.2 Equations of Small Vibrations.- 7.3 Solving Equations of Small Vibrations.- 8. Random Vibrations of Rods.- 8.1 Nonlinear Equations of Motion of Three-Dimensional Curvilinear Rods.- 8.2 Equations of the Motion of a Rod in the Attached Coordinate System.- 8.2.1 Equation of Space Motion of a Rod.- 8.2.2 Equation of Plane Motion of a Rod.- 8.2.3 Rods Having Lumped Masses.- 8.3 Equation of Small Vibrations of Rods.- 8.3.1 Equations of Small Vibrations in the Attached Coordinate Frame.- 8.3.2 Equations of Small Vibrations about a Natural State.- 8.4 Determination of Eigenvalues and Eigenvectors.- 8.5 Non-Stationary Random Vibrations of Rods.- 8.6 Stationary Random Vibrations of Rods.- 9. Fundamentals of Reliability Theory.- 9.1 Introduction.- 9.2 Elementary Problems of Reliability Theory.- 9.3 Possible Causes of Failures.- 9.4 Determination of Numerical Values of No-Failure Operation Probability (Reliability).- 9.5 Determination of Reliability at the Linear Dependence of a Stress State on Random Loads.- 9.6 Determination of the Probability of No-Failure Operation at the Nonlinear Dependence of the Random Quantity F on External Loads.- 10. Random Processes at the Action of Random Functions Bounded in Absolute Value.- 10.1 Introduction.- 10.2 Determining the Maximum Values of the Components of the Systems State Vector.- 10.3 Areas of Possible Values of the System State Vector at the Action of Independent Excitations.- 10.4 Projections of the Area of Possible Values of the System State Vector onto Two-Dimensional Planes.- 10.5 Determination of the Maximum Values of Dynamic Reactions.- 10.6 Areas of Possible Values of the System State Vector in the Case of Several Sections of Motion.- 10.7 Areas of Possible Values of the System State Vector at the Action of Dependent Random Excitations.- 10.8 Determination of the Maximum Values of Linear Functionals at Independent Excitations.- 10.9 Maximum Value of a Linear Functional at Dependent Excitations.- 10.10Vibration Protection of Mechanical Systems.- A. Appendices.- A.1 Elementary Generalized Functions.- A.3 Correlation Functions and Spectral Densities Corresponding to Them.- A.4 Hiawatha Designs an Experiment.- References.