Statistical Dynamics and Reliability Theory for Mechanical Structures

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.