Probabilistic Models for Dynamical Systems

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Probabilistic Models for Dynamical Systems

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Description

Now in its second edition, Probabilistic Models for Dynamical Systems expands on the subject of probability theory. Written as an extension to its predecessor, this revised version introduces students to the randomness in variables and time dependent functions, and allows them to solve governing equations.

  • Introduces probabilistic modeling and explores applications in a wide range of engineering fields
  • Identifies and draws on specialized texts and papers published in the literature
  • Develops the theoretical underpinnings and covers approximation methods and numerical methods
  • Presents material relevant to students in various engineering disciplines as well as professionals in the field

This book provides a suitable resource for self-study and can be used as an all-inclusive introduction to probability for engineering. It presents basic concepts, presents history and insight, and highlights applied probability in a practical manner. With updated information, this edition includes new sections, problems, applications, and examples. Biographical summaries spotlight relevant historical figures, providing life sketches, their contributions, relevant quotes, and what makes them noteworthy. A new chapter on control and mechatronics, and over 300 illustrations rounds out the coverage.

ISBN

9781439849897

Publication Date

2013

Publisher

Taylor & Francis (CRC Press)

City

Boca Raton

Disciplines

Mechanical Engineering

Comments

Contents

Preface, xv.

Acknowledgments, xvii.

Authors, xix.

1 Introduction, 1.

1.1 Applications, 1.

1.1.1 Random Vibration of Structures, 1.

1.1.2 Fatigue Life, 3.

1.1.3 Ocean-Wave Forces, 4.

1.1.4 Wind Forces, 6.

1.1.5 Material Properties, 7.

1.1.6 Statistics and Probability, 8.

1.2 Units, 10.

1.3 Organization of the Text, 10.

1.4 Quotes, 11.

1.5 Problems, 11.

2 Events and Probability, 13.

2.1 Sets, 13.

2.1.1 Basic Events, 14.

2.1.2 Operational Rules, 18.

2.2 Probability, 22.

2.2.1 Axioms of Probability, 24.

2.2.2 Extensions from the Axioms, 25.

2.2.3 Conditional Probability, 26.

2.2.4 Statistical Independence, 28.

2.2.5 Total Probability, 29.

2.2.6 Bayes' Theorem, 33.

2.3 Summary, 39.

2.4 Quotes, 39.

2.5 Problems, 40.

3 Random Variable Models, 43.

3.1 Probability Distribution Function, 44.

3.2 Probability Density Function 46.

3.3 Probability Mass Function, 53.

3.4 Mathematical Expectation, 55.

3.4.1 Mean Value, 56.

3.4.2 Variance, 58.

3.5 Useful Continuous Probability Density Functions, 63.

3.5.1 Uniform Density, 65.

3.5.2 Exponential Density, 67.

3.5.3 Normal or Gaussian Density, 77.

3.5.4 Lognormal Density, 80.

3.5.5 Rayleigh Density, 86.

3.6 Discrete Density Functions, 86.

3.6.1 Binomial Density Function, 88.

3.6.2 Poisson Density Function, 89.

3.7 Moment-Generating Function, 91.

3.7.1 Characteristic Function, 94.

3.8 Two Random Variables, 95.

3.8.1 Marginal Densities, 101.

3.8.2 Conditional Density Function, 106.

3.8.3 Total Probability Revisited, 107.

3.8.4 Covariance and Correlation, 114.

3.9 Summary, 114.

3.10 Quotes, 115.

3.11 Problems, 123.

4 Functions of Random Variables, 123.

4.1 Exact Functions of One Variable, 129.

4.2 Functions of Two or More Random Variables, 141.

4.2.1 General Case, 149.

4.3 Approximate Analysis, 149.

4.3.1 Direct Methods, 152.

4.3.2 Mean and Variance of a General Function of X to Order a~, 156.

4.3.3 Mean and Variance of a General Function of n RVs, 163.

4.4 Monte Carlo Methods, 164.

4.4.1 Independent Uniform Random Numbers, 167.

4.4.2 Independent Normal Random Numbers, 168.

4.4.3 A Discretization Procedure, 170.

4.4.4 Generation of Jointly Distributed Random Variables, 171.

4.5 Summary, 171.

4.6 Quotes, 171.

4.7 Problems, 172.

5 Random Processes, 177.

5.1 Basic Random Process Descriptors, 177.

5.2 Ensemble Averaging, 178.

5.3 Stationarity, 183.

5.4 Correlations of Derivatives, 191.

5.5 Fourier Series and Fourier Transforms, 194.

5.5.1 Sifting Theorem, 201.

5.5.2 Time Differentiation, 203.

5.5.3 Convolution Theorem, 203.

5.5.4 Shifting Theorem, 205.

5.5.5 Scaling Theorem, 206.

5.6 Harmonic Processes, 206.

5.7 Power Spectra, 208.

5.7.1 Discussion of the Wiener-Khinchine Theorem, 210.

5.7.2 Power Spectrum Units, 228.

5.8 Narrow- and Broad-Band Processes, 229.

5.8.1 White Noise Processes, 232.

5.9 Interpretations of Correlations and Spectra, 233.

5.10 Spectrum of Derivative, 235.

5.11 Fourier Representation of a Stationary Process, 242.

5.11.1 Borgman's Method of Frequency Discretization, 244.

5.12 Summary, 245.

5.13 Quotes, 246.

5.14 Problems, 251.

6 Single Degree-of-Freedom Vibration, 252.

6.1 Motivating Examples, 252.

6.1.1 Transport of a Satellite, 254.

6.1.2 Rocket, 255.

6.2 Newton's Second Law, 261.

6.3 Free Vibration With No Damping, 263.

6.4 Harmonic Forced Vibration With No Damping, 263.

6.4.1 Resonance, 267.

6.5 Free Vibration with Viscous Damping, 268.

6.6 Forced Harmonic Vibration, 270.

6.6.1 Harmonic Base Excitation, 276.

6.7 Impulse Excitation, 277.

6.8 Arbitrary Loading, 281.

6.9 Frequency Response Function, 284.

6.10 SDOF: The Response to Random Loads, 284.

6.10.1 Mean Value of Response, 285.

6.10.2 Response Correlations, 289.

6.10.3 Response Spectral Density, 298.

6.11 Response to Two Random Loads, 305.

6.12 Summary, 305.

6.13 Quotes, 306.

6.14 Problems, 306.

9.3 Other Failure Laws, 445.

9.3.1 Gamma Failure Law, 447.

9.3.2 Normal Failure Law, 448.

9.3.3 The Wei bull Failure Law, 450.

9.4 Fatigue Life Prediction, 451.

9.4.1 Failure Curves, 454.

9.4.2 Peak Distribution for Stationary Random Process, 456.

9.4.3 Peak Distribution of a Gaussian Process, 458.

9.4.4 Special Cases, 465.

9.5 Summary, 465.

9.6 Quotes, 466.

9.7 Problems, 469.

10 Nonlinear and Stochastic Dynamic Models, 469.

10.0.1 Chapter Overview, 471.

10.0.2 Examples of Nonlinear Vibration, 475.

10.0.3 Approximate Solution of Simple Pendulum, 475.

10.0.4 Exact Solution of Simple Pendulum, 477.

10.0.5 The Duffing and the van der Pol Equations, 477.

10.1 The Phase Plane, 482.

10.1.1 Stability of Equilibria, 486.

10.2 Statistical Equivalent Linearization, 493.

10.2.1 Equivalent Nonlinearization, 495.

10.3 Perturbation Methods, 500.

10.3.1 Lindstedt-Poincare Method, 502.

10.3.2 Forced Oscillations of Quasi-Harmonic Systems, 505.

10.3.3 Jump Phenomena, 507.

10.3.4 Periodic Solutions of Non-Autonomous Systems, 513.

10.3.5 Random Duffing Oscillator, 515.

10.3.6 Subharmonic and Superharmonic Oscillations, 519.

10.4 The Mathieu Equation, 524.

10.5 The van der Pol Equation, 525.

10.5.1 The Unforced van der Pol Equation, 525.

10.5.2 Limit Cycles, 526.

10.5.3 The Forced van der Pol Equation, 530.

10.6 Markov Process-Based Models, 531.

10.6.1 Probability Background, 535.

10.6.2 The Fokker-Planck Equation, 550.

10.7 Summary, 550.

10.8 Quotes, 551.

10.9 Problems, 555.

11 Nonstationary Models, 559.

11.1 Envelope Function Model, 560.

11.1.1 Transient Response, 564.

11.1.2 Mean Square Nonstationary Response, 566.

11.2 Nonstationary Generalizations, 566.

11.2.1 Deterministic Preliminaries, 567.

11.2.2 Discrete Random Model, 568.

11.2.3 Complex-Valued Stochastic Processes, 569.

11.2.4 Continuous Random Model, 570.

11.3 Priestley's Model, 570.

11.3.1 The Stieltjes Integral: An Introduction, 572.

11.3.2 Priestley's Model, 572.

11.4 Oscillator Response, 573.

11.4.1 Stationary Case, 574.

11.4.2 Nonstationary Case, 575.

11.4.3 Undamped. Oscillator, 576.

11.4.4 Underdamped Oscillator, 579.

11.5 Multi Degree-of-Freedom Oscillator Response, 579.

11.5.1 Input Characterization, 580.

11.5.2 Response Characterization, 582.

11.6 Nonstationary and Nonlinear Oscillator, 583.

11.6.1 The Nonstationary and Nonlinear Duffing Equation, 585.

11.7 Summary, 585.

11.8 Quotes, 586.

11.9 Problems.

12 Monte Carlo Methods.

12.1 Introduction.

12.2 Random Number Generation.

12.2.1 Standard Uniform Random Numbers.

12.2.2 Generation of Nonuniform Random Variates.

12.2.3 Composition Method.

12.2.4 Von Neumann's Rejection-Acceptance Method.

12.3 Joint Random Numbers.

12.3.1 Inverse Transform Method.

12.3.2 Linear Transform Method.

12.4 Error Estimates.

12.5 Applications.

12.5.1 Evaluation of Finite-Dimensional Integrals.

12.5.2 Generating a Time History for a Stationary Random Process Defined by a Power Spectral Density.

12.6 Summary.

12.7 Quotes.

12.8 Problems.

13 Fluid-Induced Vibration.

13.1 Ocean Currents and Waves.

13.1.1 Spectral Density.

13.1.2 Ocean Wave Spectral Densities.

13.1.3 Approximation of Spectral Density from Time Series.

13.1.4 Generation of Time Series from a Spectral Density.

13.1.5 Short-Term Statistics.

13.1.6 Long-Term Statistics.

13.1.7 Wave Velocities via Linear Wave Theory.

13.2 Fluid Forces in General.

13.2.1 Wave Force Regime.

13.2.2 Wave Forces on Small Structures - Morison Equation.

13.2.3 Vortex-Induced Vibration.

13.3 Examples.

13.3.1 Static Configuration of a Towing Cable.

13.3.2 Fluid Forces on an Articulated Tower.

13.3.3 Weibull and Gumbel Wave Height Distributions.

13.3.4 Reconstructing a Time Series for a Given Significant Wave Height.

13.4 Available Numerical Codes.

13.5 Summary.

13.6 Quotes.

14 Probabilistic Models in Controls and Mechatronic Systems.

14.1 Concepts of Deterministic Systems.

14.1.1 Introduction to Feedback Control.

14.1.2 Advantages and Disadvantages of Feedback Control.

14.1.3 State-Space Models.

14.1.4 Transfer Function from State-Space Model.

14.1.5 Controllability and Observability.

14.1.6 State Feedback.

14.1.7 State Observer.

14.1.8 State Feedback Control with State Observer.

14.1.9 Multivariable Control.

14.2 Concepts of Stochastic Systems.

14.2.1 Stochastic vs. Random.

14.2.2 Probabilistic Background Concepts.

14.3 Filtering of Random Signals.

14.3.1 Types of Filters.

14.3.2 Ideal vs. Practical Filters.

14.4 White Noise Filters.

14.4.1 White Noise.

14.4.2 Properties of White Noise Filters.

14.5 Stochastic System Models.

14.5.1 MIMO Stochastic System Model.

14.6 The Kalman Filter.

14.6.1 Introduction to Kalman Filter.

14.6.2 Deriving the Kalman Filter Equations.

14.7 Additional Issues.

14.7.1 Extended Kalman Filter.

14.7.2 Optimal Compensators

14.8 Summary.

14.9 Quotes.

Index

Probabilistic Models for Dynamical Systems

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