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Reliability Problems: General Principles and Applications in Mechanics of Solids and Structures pp 1–49Cite as

Random Vibration and First Passage Failure

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Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 317))

Abstract

In the first part of this Chapter. a variety of techniques for predicting nonlinear dynamic system response to random excitation are discussed. These include methods based on modelling the response as a continuous Markov process. leading to diffusion equations, statistical linearization, the method of equivalent nonlinear equations and closure methods. Special attention is paid to the stochastic averaging method, which is a combination of an averaging technique and Markov process modelling. It is shown that the stochastic averaging method is particularly useful for estimating the “first-passage” probability that the system response stays within a safe domain, within a specified period of Lime. Results obtained by this method are presented for oscillators with both linear and nonlinear damping and restoring terms. An alternative technique for solving the first-passage problem, based on the computation of level crossing statistics. is also described: this is especially useful in more general situations, where the stochastic averaging is inapplicable.

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References

  1. Roberts, J. B. and P. D. Spanos: Random Vibration and Statistical Linearization, J. Wiley, Chichester 1990.

    Google Scholar 

  2. Arnold, L.: Stochastic Differential Equations: Theory and Applications, Wiley Interscience, New York 1973.

    Google Scholar 

  3. Roberts, J. B.: Response of nonlinear mechanical systems to random excitation: Markov methods, Shock Vib. Digest, 13 (1981), 17–28.

    Google Scholar 

  4. Atkinson, J. D.: Eigenfunction expansions for randomly excited non-linear systems, J. Sound Vib., 30 (1973), 153–172.

    Article  MATH  Google Scholar 

  5. Caughey, T. K.: Nonlinear theory of random vibrations, Adv. Appl. Mechs., 11 (1971), 209–253.

    Google Scholar 

  6. Caughey, T. K.: On the response of a class of nonlinear oscillators to stochastic excitation. Proc. Coli. Int. Du Centre Nat. de la Rechercher Scient., No 148, Marseille, France (1964), 393–402.

    Google Scholar 

  7. Dimentberg, M. F.: Statistical Dynamics of Nonlinear and Time-Varying systems, Research Studies Press, Taunton, U.K. (1988).

    Google Scholar 

  8. Mayfield, W. W.: A sequence solution to the Fokker-Planck equation. IEEE Trans. Inf. Theory, IT-19(1973), 165–175.

    Article  Google Scholar 

  9. Toland, R. H. and C. Y. Yang: Random walk model for first-passage probability, ASCE J. Engng. Mech. Div., 97 (1971), 791–806.

    Google Scholar 

  10. Roberts, J. B.: First-passage time for oscillators with nonlinear damping, ASME J. App. Mechs, 45 (1978), 175–180.

    Article  MATH  Google Scholar 

  11. Roberts, J. B.: First passage time for randomly excited nonlinear oscillators, J. Sound Vib., 109 (1986), 33–50.

    Article  MATH  Google Scholar 

  12. Langley, R. S.: A finite element method for the statistics of non-linear vibration, J. Sound Vib., 101 (1985), 41–49.

    Article  MATH  Google Scholar 

  13. Bergman, C. A. and Spencer, B. F.: First passage of sliding rigid structures on a frictional foundation, Earthquake Engng. Struct. Dyns. 13 (1985), 281–291.

    Google Scholar 

  14. Wehner, M. F. and W. G. Wolfer: Numerical evaluation of path integral solutions to the Fokker-Planck equations. Phys. Rev., A27 (1983), 2663–2670.

    Article  Google Scholar 

  15. Roberts, J. B. and P. D. Spanos: Stochastic averaging: an approximate method for solving random vibration problems, Int. J. Non-linear Mechs., 21 (1986). 111–134.

    Article  MathSciNet  MATH  Google Scholar 

  16. Caughey, T. K.: On the response of nonlinear oscillators to stochastic excitation, Prob. Engng. Mechs., 1 (1986), 2–4.

    Google Scholar 

  17. Hennig, K. and J. B. Roberts: Averaging methods for randomly excited nonlinear oscillators, in Random Vibration–Status and Recent Developments (Ed. R. H. Lyon) Elsevier, Amsterdam 1986, 143–161.

    Google Scholar 

  18. Stratonovitch, R. L.: Topics in the Theory of Random Noise [2 vols], Gordon and Breach, New York 1964.

    Google Scholar 

  19. Jazwinskii. A.: Stochastic Processes and Filtering Theory, Academic Press, New York (1970).

    Google Scholar 

  20. Khasminskii, R. Z.: A limit theorem for the solutions of differential equations with random right-hand sides, Theory Prob. Apps. 11 (1966), 390–405.

    Google Scholar 

  21. Khasminskii, R. Z.: On the averaging principle for stochastic differential Ito equations, Kibernetika, 9 (1968), 260–279.

    Google Scholar 

  22. Papanicolaou, G. G. and W. Kohler: Asymptotic theory of mixing stochastic ordinary differential equations, Comna Par. App. Maths. 27 (1974), 641–668.

    MathSciNet  MATH  Google Scholar 

  23. Bogaliubov. N. and A. Mitropoisky: Asymptotic Methods in the Theory of Non-linear Oscillations, Gordon and Breach. New York 1961.

    Google Scholar 

  24. Roberts, J. B.: Averaging Methods in Random Vibration, Dept. Struct. Engng.. Technical Univ. Denmark, Report. R 245. 1989.

    Google Scholar 

  25. Roberts, J. B.: First-Passage Probabilities for randomly excited systems–diffusion methods. Prob. Engng. Mechs., 1 (1986), 66–81.

    Article  Google Scholar 

  26. Cramer. H.: On the intersections between the trajectories of a normal stationary stochastic process and a high level. Arkiv fur Matematik, 6 (1966), 337–349.

    Article  MATH  Google Scholar 

  27. Yang, J–N. and M. Shinozuka: First-passage time problem. J. Acoust. Soc. Am., 47 (1970). 393–394.

    Article  Google Scholar 

  28. Franklin, J.N. and E.R. Roderich: Numerical analysis of an elliptic-parabolic partial differential equation, SIAM J. Num. An.. 5 (1968), 680–716.

    Article  Google Scholar 

  29. Dynkin, E.R.: Markov Processes, Springer-Verlag, New York, 1965.

    MATH  Google Scholar 

  30. Bolotin, V.V.: Statistical aspects in the theory of structural stability. Proc. Int. Conf. on Dyn. Stability of Structs., Northwestern Univ.. Ill., Pergamon Press. (1965) 67–81.

    Google Scholar 

  31. Crandall, S.H.. K.L. Chandiramini and R.G. Cook: Some first-passage problems in random vibration, ASME J. App. Mechs., 33 (1966), 532–538.

    Article  Google Scholar 

  32. Bergman, L.A. and B.F. Spencer: Solution of the first passage problem for simple linear and nonlinear oscillators by the finite element method, Dept. Theor. & App. Mechs., Univ. Ill. at Urbana - Champ., Rep. No. 461, 1983.

    Google Scholar 

  33. Gradshteyn, I.S. and I.M. Ryzhik: Tables of Integrals, Series and Products, Academic Press, San Diego, 1980.

    Google Scholar 

  34. Seshadri, V., B.J. West, and K. Lindenberg: Analytic theory of extrema II. Application to nonlinear oscillators, J. Sound Vib., 68 (1980), 553–570.

    Article  Google Scholar 

  35. Roberts, J.B.: First passage Lime for the envelope of a randomly excited linear oscillator, J.Sound Vib., 46 (1976), 1–14.

    Article  Google Scholar 

  36. Roberts, J.B.: First passage time for oscillators with nonlinear restoring forces, J. Sound Vib., 56 (1978). 71–86.

    Article  MATH  Google Scholar 

  37. Spanos, P.D.: Survival probability of nonlinear oscillators subjected to broad-band random distrubances, Int. J. Nonlinear Mechs., 17 (1982), 303–317.

    Article  MathSciNet  MATH  Google Scholar 

  38. Roberts, J.B.: Probability of first passage failure for lightly damped oscillators. Proc. IUTAM Symp. Stochastic Problems in Dynamics (ed: B.L. Clarkson ), Pitman 1976.

    Google Scholar 

  39. Mark, W.D.: On false-alarm probabilities of filtered noise, Proc. IEEE, 54 (1966), 316–317.

    Article  Google Scholar 

  40. Lutes, L.D., Y.T. Chen and S. Tzuang: First passage approximation for simple oscillators, ASCE J. Engng. Mechs. Div., 106 (1980), 1111–1124.

    Google Scholar 

  41. Roberts, J.B.: Firstpassage time for oscillators with nonlinear damping, ASME J. App. Mechs., 45 (1978), 175–180.

    Article  MATH  Google Scholar 

  42. Roberts, J.B.: First passage probability for nonlinear oscillators, ASCE J. Engng. Mechs. Div., 102 (1976), 851–866.

    Google Scholar 

  43. Roberts. J.B.: First-passage time for randomly excfl d nonlinear oscillators, J.Sound Vib., 109 (1986), 33–50.

    Article  MATH  Google Scholar 

  44. Spanos. P.D. and G.P. Solomos: Barrier crossing due to transient excitation. ASCE J. Engng. Mechs., 110 (1984), 20–36.

    Article  Google Scholar 

  45. Roberts. J.B.: An approach to the first-passage problem • in random vibration. J.Sound Vib., 8 (1968), 301–328.

    Article  MATH  Google Scholar 

  46. Roberts. J.B.: Probability of first passage failure for stationary random vibration, AIAA J., 12 (1974), 1636–1643.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Sussex, Sussex, UK

    J. R. Roberts

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  1. J. R. Roberts
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Editors and Affiliations

  1. University of Pavia, Italy

    F. Casciati

  2. University of Sussex, UK

    J. B. Roberts

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© 1991 Springer-Verlag Wien

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Roberts, J.R. (1991). Random Vibration and First Passage Failure. In: Casciati, F., Roberts, J.B. (eds) Reliability Problems: General Principles and Applications in Mechanics of Solids and Structures. International Centre for Mechanical Sciences, vol 317. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2616-5_1

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  • DOI: https://doi.org/10.1007/978-3-7091-2616-5_1

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-82319-4

  • Online ISBN: 978-3-7091-2616-5

  • eBook Packages: Springer Book Archive

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