Hiroshi Tanaka

Ph.D., P.Eng.
Professor, Civil Engineering

CVG 5154 / 83.538 - Random Vibration

Course Contents:

Chapter 1     Introduction

• Random Vibration of Structures
• Concept of Spectral Analyses

Chapter 2     Mathematical Preparation

2.1. Theory of Random Variables
• Probability Functions
• Statistical Quantities
• Joint Probability Functions
• Binomial, Gaussian, Weibull and Other Distributions

2.2. Harmonic Analyses
• Fourier Series and Fourier Integral
• Power Spectral Density

2.3. Theory of Random Processes
• Correlation Functions
• Wiener-Khinchine Relation
• Cross-correlation, Cross-spectrum and Coherence

Chapter 3     Elements of Structural Dynamics

3.1. Introduction
• Dynamic Problems in Civil Engineering
• Effects of Dynamic Behaviour of Structures
• Damping Characteristics

3.2. Linear SDOF Systems
• Frequency Response Function
• Impulse Response Function
• Frequency Domain Analysis and Time Domain Analysis
• Duhamel Integral and Numerical Integration Techniques

3.3. Dynamics of Distributed Mass Systems
• Dynamic Analysis of Elastic Beams
• Modal Analysis

3.4. Dynamics of MDOF Systems
• Analysis of TDOF Systems
• Modal Analysis of MDOF Systems

Chapter 4     Stochastic Response of Dynamic Systems

4.1. Response to A Single Random Loading

4.2. Response to Two or More Random Loading

4.3. Response to Distributed Loading

4.4. Wind Excitation of Structures

4.5. Earthquake Excitation of Structures

Chapter 5     Analyses of Random Data

5.1. Discrete Fourier Approximation
• Time History and Time Series
• Discrete Fourier Transform
• Fourier Spectra and Nyquist Frequency

5.2. Fast Fourier Transform Technique

5.3. Power Spectra and Auto-correlations

5.4. Smoothing Techniques and Data Preparation

5.5. Wavelet Analysis

Chapter 6     Extreme Value Analyses

6.1 Probability Distribution of Extreme Values
• Estimation from the Parent Population
• Fisher-Tippet Approach
• Prediction of Wind Induced Structural Response

6.2 Probability of Threshold Crossings
• Rice's Theory
• Probability Distribution of Peaks
• Probability Distribution of Envelopes
• Yield Failure and Fatigue Failure of Structures

Chapter 7     Simulation of Random Processes

7.1 Monte Carlo Method

7.2 Stationary Gaussian Random Processes

7.3 Non-stationary Gaussian Random Processes

7.4 Random Pulse Train Processes

7.5 Multi-Variate Random Processes

Chapter 8     Nonlinear Vibration

8.1 Introduction

8.2 Numerical Integration Techniques

8.3 Nonlinear Random Vibration
• Nonlinear Vibration
• Fokker-Planck Equation
• Perturbation Method
• Equivalent Linearization Method

Chapter 9     Special Topics

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References:

1. A.H.S. Ang & W.H. Tang, Probability Concepts in Engineering Planning and Design (in 2 volumes), John Wiley & Sons, 1975/1984.
2. J.R. Benjamin & C.A. Cornell, Probability, Statistics and Decision for Civil Engineers, McGraw-Hill, 1970.
3. J.S. Bendat & A.G. Piersol, Random Data: Analysis & Measurement Procedures, Wiley-Interscience, 1971/1986.
4. R.W. Clough & J. Penzien, Dynamics of Structures, McGraw-Hill, 1975/1993.
5. J.W. Smith, Vibration of Structures - Applications in Civil Engineering Design, Chapman & Hall, 1988.
6. J.D. Robson, An Introduction to Random Vibration in Mechanical Systems, Academic Press, 1963. Elsevier, 1963/1970.
7. S.H. Crandall (ed.), Random Vibration (in 2 volumes), MIT Press, 1958/1963.
8. C.Y. Yang, Random Vibration of Structures, John Wiley & Sons, 1986.
9. D.E. Newland, An Introduction to Random Vibrations, Spectral & Wavelet Analysis, Longman, 1975/1984/1993.
10. Y.K. Lin, Probabilistic Theory of Structural Dynamics, McGraw-Hill, 1967/1995.
11. P.H. Wirsching, T.L. Paez & K. Ortiz, Random Vibrations: Theory and Practice, John Wiley & Sons, 1995.
12. N.C. Nigam, Introduction to Random Vibrations, MIT Press, 1983.
13. S.O. Rice, Mathematical Analysis of Random Noise, Bell System Technical Journal, vol. 23 & 24, 1944-45; Reprinted in Selected Papers on Noise and Stochastic Processes (N.Wax, ed.), Dover, 1954.
14. R.B. Blackman & J.W. Tukey, The Measurement of Power Spectra from the point of View of Communications Engineering, Dover, 1958.
15. E.J. Gumbel, Statistics of Extremes, Columbia University Press, 1958.
16. E.O. Brigham, The Fast Fourier Transform, Prentice-Hall, 1974.
17. H.D.Madsen, S. Krenk & N.C. Lind, Methods of Structural Safety, Prentice-Hall, 1986.
18. J.Éaèko, M.Bílý & J. Bukoveczky, Random Process: Measurement, Analysis and Simulation, Elsevier, 1988.
19. T.T. Soong & M. Grigoriu, Random Vibration of Mechanical and Structural Systems, Prentice Hall, 1993.

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