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First-passage failure of MDOF nonlinear oscillator. (English) Zbl 1237.70126

Summary: First-passage failure of multiple-degree-of-freedom nonlinear oscillators with lightly nonlinear dampings and strongly nonlinear stiffness subject to additive and/or parametric Gaussian white noise excitations is studied. First, by using the stochastic averaging method based on the generalized harmonic functions, the averaged Itô stochastic differential equation for the amplitudes of the nonlinear oscillators can be derived. Then the associated backward Kolmogorov equation of the conditional reliability function is established, and the conditional reliability is approximately expressed as a series expansion in terms of Kummer functions with time-dependent coefficients. By using the Galerkin method, the time dependent coefficients of the associated conditional reliability function can be solved by a set of differential equations. Finally, the proposed procedure is applied to Duffing-Van der Pol systems under external and/or parametric excitations of Gaussian white noises. The results are also verified by those obtained from Monte Carlo simulation of the original system. The effects of system parameters on first-passage failure are discussed briefly.

MSC:

70L05 Random vibrations in mechanics of particles and systems
74H50 Random vibrations in dynamical problems in solid mechanics
45R05 Random integral equations
33C20 Generalized hypergeometric series, \({}_pF_q\)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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[1] Anagnos T, Kiremidjian A S. A review of earthquake occurrence models for seismic hazard analysis. Prob Eng Mech, 1988, 3: 3–11 · doi:10.1016/0266-8920(88)90002-1
[2] Price W G, Bishop R E D. Probabilistic Theory of Ship Dynamics. London: Chapman and Hall, 1974 · Zbl 0294.70027
[3] Simiu E, Scanlan R H. Wind Effects on Structures: An Introduction to Wind Engineering. New York: John Wiley & Sons, 1986
[4] Dodds C J, Robson J D. The description of road surface roughness. J Sound Vib, 1973, 31: 175–183 · Zbl 0267.70011 · doi:10.1016/S0022-460X(73)80373-6
[5] Kamesh K M, Robson J D. The application of isotropy in road surface modeling. J Sound Vib, 1978, 57: 80–100
[6] Health A N. Application of the isotropic road roughness assumption. J Sound Vib, 1987, 115: 131–144 · doi:10.1016/0022-460X(87)90495-0
[7] Bharucha-Reid A T. Elements of the Theory of Markov Processes and Their Applications. New York: McGraw-Hill, 1960 · Zbl 0095.32803
[8] Cox D R, Miller H D. The Theory of Stochastic Processes. London: Chapman and Hall, 1965 · Zbl 0149.12902
[9] Zhu W Q. Nonlinear stochastic dynamics and control in Hamiltonian formulation. Appl Mech Rev ASME, 2006, 59: 230–248 · doi:10.1115/1.2193137
[10] Zhu W Q, Huang Z L, Deng M L. First-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems. Int J Non-Linear Mech, 2003, 38: 1133–1148 · Zbl 1348.70065 · doi:10.1016/S0020-7462(02)00058-6
[11] Gan C B, Zhu W Q. First-passage failure of quasi-no-integrable-Hamiltonian systems. Int J Non-Linear Mech, 2001, 36: 209–220 · Zbl 1342.70050 · doi:10.1016/S0020-7462(00)00006-8
[12] Labou M. Evaluation of failure probabilities of mechanical systems under seismic action by the Monte-Carlo simulation method. Strength Mater, 2008, 40: 635–647 · doi:10.1007/s11223-008-9082-3
[13] Chen L C, Zhu W Q. First passage failure of dynamical power systems under random perturbations. Sci China Tech Sci, 2010, 53: 2495–2500 · Zbl 1205.78033 · doi:10.1007/s11431-010-4070-9
[14] Helstrom C W, Isley C T. Two notes on a Markoff envelope process. IRE Trans Inform Theory, 1959, 5: 139–140 · doi:10.1109/TIT.1959.1057511
[15] Rosenblueth E, Bustamante J I. Duration of structural response to earthquakes. J Eng Mech Div ASCE, 1962, 88: 75–105
[16] Gray A H. First-passage time in a random vibrational system. J Appl Mech, 1966, 33: 187–191 · Zbl 0147.16501 · doi:10.1115/1.3624977
[17] Abramovitz M, Stegun I A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1966
[18] Spanos P T D. Survival probability of nonlinear oscillators subjected to broad-band random disturbances. Int J Non-Linear Mech, 1982, 17: 303–317 · Zbl 0498.73095 · doi:10.1016/0020-7462(82)90001-4
[19] Xu Z, Cheung Y K. Averaging method using generalized harmonic functions for strongly nonlinear oscillators. J Sound Vib, 1994, 174: 563–576 · Zbl 0945.70534 · doi:10.1006/jsvi.1994.1294
[20] Stratonovich R L. Topics in the Theory of Random Noise. New York: Gordon and Breach, 1963 · Zbl 0119.14502
[21] Khasminskii R Z. A limit theorem for solutions of differential equations with random right-hand sides. Theory Probab Appl, 1966, 11: 390–406 · doi:10.1137/1111038
[22] Spanos P T D. Numerics for common first-passage problem. J Eng Mech Div ASCE, 1982, 108: 864–882
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