Hilbert Transform–Based Stochastic Averaging Technique for Determining the Survival Probability of Nonlinear Oscillators
Publication: Journal of Engineering Mechanics
Volume 145, Issue 10
Abstract
A Hilbert transform based definition of the response amplitude of randomly excited nonlinear oscillators has been proposed recently to address certain limitations associated with the standard stochastic averaging solution treatment. In comparison to standard stochastic averaging, the requirement of a priori determination of an equivalent natural frequency is bypassed, yielding flexibility in the ensuing analysis and potentially higher accuracy. In this paper, relying on the Hilbert transform based stochastic averaging, a semianalytical technique is developed for determining the time-dependent survival probability and first-passage time probability density function of stochastically excited nonlinear oscillators, even endowed with fractional derivative terms. To this aim, a Galerkin scheme based on the orthogonality of the confluent hypergeometric functions is utilized to solve approximately the backward Kolmogorov partial differential equation governing the survival probability of the oscillator response. Further, two distinct approximations for the equivalent instantaneous natural frequency are introduced, while their accuracy with respect to oscillator first-passage time statistics is also assessed. The hardening Duffing and the bilinear stiffness nonlinear oscillators, both with and without fractional derivative elements, are considered in the numerical examples for ascertaining the reliability of the technique. For comparison, the analytical results are examined in relation to pertinent Monte Carlo simulation data.
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Acknowledgments
The authors kindly acknowledge the support by the Brazilian Federal Agency for Coordination of Improvement of Higher Education Personnel (CAPES) (Award No. BEX/13406-13-2).
References
Au, S., and Y. Wang. 2014. Engineering risk assessment with subset simulation. Chichester, UK: Wiley.
Barbato, M., and J. P. Conte. 2011. “Structural reliability applications of nonstationary spectral characteristics.” J. Eng. Mech. 137 (5): 371–382. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000238.
Beck, A., and R. Melchers. 2004. “On the ensemble crossing rate approach to time variant reliability analysis of uncertain structures.” Probab. Eng. Mech. 19 (1): 9–19. https://doi.org/10.1016/j.probengmech.2003.11.018.
Belizzi, S., and R. Bouc. 1999. “Analysis of multi-degree of freedom strongly non-linear mechanical systems with random input part I: Non-linear modes and stochastic averaging.” Probab. Eng. Mech. 14 (3): 229–244. https://doi.org/10.1016/S0266-8920(98)00007-1.
Bergman, L. A., and J. C. Heinrich. 1981. “On the moments of time to first passage of the linear oscillator.” Earthquake Eng. Struct. Dyn. 9 (3): 197–204. https://doi.org/10.1002/eqe.4290090302.
Di Matteo, A., I. Kougioumtzoglou, A. Pirrotta, P. Spanos, and M. Di Paola. 2014. “Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral.” Probab. Eng. Mech. 38 (Oct): 127–135. https://doi.org/10.1016/j.probengmech.2014.07.001.
Di Paola, M., G. Failla, A. Pirrotta, A. Sofi, and M. Zingales. 2013. “The mechanically based non-local elasticity: An overview of main results and future challenges.” Philos. Trans. R. Soc. London A: Math. Phys. Eng. Sci. 371 (1993): 20120433. https://doi.org/10.1098/rsta.2012.0433.
Di Paola, M., A. Pirrotta, and A. Valenza. 2011. “Visco-elastic behavior through fractional calculus: An easier method for best fitting experimental results.” Mech. Mater. 43 (12): 799–806. https://doi.org/10.1016/j.mechmat.2011.08.016.
Dostal, L., E. Kreuzer, and N. S. Namachchivaya. 2012. “Non-standard stochastic averaging of large-amplitude ship rolling in random seas.” Proc. R. Soc. A: Math. Phys. Eng. Sci. 468 (2148): 4146–4173. https://doi.org/10.1098/rspa.2012.0258.
Feldman, M. 2011. “Hilbert transform in vibration analysis.” Mech. Syst. Signal Process. 25 (3): 735–802. https://doi.org/10.1016/j.ymssp.2010.07.018.
Freidlin, M. I., and A. D. Wentzell. 1998. Random perturbations of dynamical systems. 2nd ed. Dordrecht, Netherlands: Springer.
Freidlin, M. I., and A. D. Wentzell. 2004. “Diffusion processes on an open book and the averaging principle.” Stochastic Processes Appl. 113 (1): 101–126. https://doi.org/10.1016/j.spa.2004.03.009.
Grigoriu, M. 2002. Stochastic calculus: Applications in science and engineering. Boston: Birkhauser.
Iourtchenko, D., E. Mo, and A. Naess. 2008. “Reliability of strongly nonlinear single degree of freedom dynamic systems by the path integration method.” J. Appl. Mech. 75 (6): 061016. https://doi.org/10.1115/1.2967896.
Khasminskii, R. Z. 1966. “A limit theorem for the solutions of differential equations with random right-hand sides.” Theory Probab. Appl. 11 (3): 390–406.
Khasminskii, R. Z. 1968. “On the principles of averaging for stochastic differential equations.” Kybernetica 4: 260–279.
Koh, C., and J. Kelly. 1990. “Application of fractional derivatives to seismic analysis of base-isolated models.” Earthquake Eng. Struct. Dyn. 19 (2): 229–241. https://doi.org/10.1002/eqe.4290190207.
Kougioumtzoglou, I., and P. Spanos. 2009. “An approximate approach for nonlinear system response determination under evolutionary stochastic excitation.” Curr. Sci. 97 (8): 1203–1211.
Kougioumtzoglou, I., and P. Spanos. 2013. “Response and first-passage statistics of nonlinear oscillators via a numerical path integral approach.” J. Eng. Mech. 139 (9): 1207–1217. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000564.
Kougioumtzoglou, I., and P. Spanos. 2014. “Stochastic response analysis of the softening Duffing oscillator and ship capsizing probability determination via a numerical path integral approach.” Probab. Eng. Mech. 35 (Jan): 67–74. https://doi.org/10.1016/j.probengmech.2013.06.001.
Kougioumtzoglou, I., Y. Zhang, and M. Beer. 2016. “Softening Duffing oscillator reliability assessment subject to evolutionary stochastic excitation.” ASCE-ASME J. Risk Uncertainty Eng. Syst. Part A: Civ. Eng. 2 (2): C4015001. https://doi.org/10.1061/AJRUA6.0000828.
Krenk, S., H. Madsen, and P. Madsen. 1983. “Stationary and transient response envelopes.” J. Eng. Mech. 109 (1): 263–277. https://doi.org/10.1061/(ASCE)0733-9399(1983)109:1(263).
Langley, R. 1986. “On various definitions of the envelope of a random process.” J. Sound Vib. 105 (3): 503–512. https://doi.org/10.1016/0022-460X(86)90175-6.
Li, J., and J. Chen. 2009. Stochastic dynamics of structures. Chichester, UK: Wiley.
Lin, Y. 1967. Probabilistic theory of structural dynamics. New York: McGraw-Hill.
Lutes, L., and S. Sarkani. 2004. Random vibrations: Analysis of structural and mechanical systems. Amsterdam, Netherlands: Elsevier.
Namachchivaya, N. S. 1991. “Co-dimension two bifurcations in the presence of noise.” J. Appl. Mech. 58 (1): 259–265. https://doi.org/10.1115/1.2897161.
Namachchivaya, N. S. 2016. “Random dynamical systems: Addressing uncertainty, nonlinearity and predictability.” Meccanica 51 (12): 2975–2995. https://doi.org/10.1007/s11012-016-0570-4.
Oldham, K., and J. Spanier. 2006. “The fractional calculus: Theory and applications of differentiation and integration to arbitrary order.” In Dover books on mathematics. Mineola, NY: Dover Publications.
Red-Horse, J., and P. Spanos. 1992. “A generalization to stochastic averaging in random vibration.” Int. J. Non Linear Mech. 27 (1): 85–101. https://doi.org/10.1016/0020-7462(92)90025-3.
Roberts, J., and P. Spanos. 1986. “Stochastic averaging: An approximate method of solving random vibration problems.” Int. J. Non Linear Mech. 21 (2): 111–134. https://doi.org/10.1016/0020-7462(86)90025-9.
Spanos, P. 1982a. “Numerics for common first-passage problem.” J. Eng. Mech. Div. 108 (5): 864–882.
Spanos, P. 1982b. “Survival probability of non-linear oscillators subjected to broad-band random disturbances.” Int. J. Non Linear Mech. 17 (5): 303–317. https://doi.org/10.1016/0020-7462(82)90001-4.
Spanos, P., A. Di Matteo, Y. Cheng, A. Pirrota, and J. Li. 2016. “Galerkin scheme-based determination of survival probability of oscillators with fractional derivative elements.” J. Appl. Mech. 83 (12): 121003. https://doi.org/10.1115/1.4034460.
Spanos, P., and I. Kougioumtzoglou. 2014a. “Galerkin scheme based determination of first-passage probability of nonlinear system response.” Struct. Infrastruct. Eng. 10 (10): 1285–1294. https://doi.org/10.1080/15732479.2013.791328.
Spanos, P., and I. Kougioumtzoglou. 2014b. “Survival probability determination of nonlinear oscillators subject to evolutionary stochastic excitation.” ASME J. Appl. Mech. 81 (5): 051016. https://doi.org/10.1115/1.4026182.
Spanos, P., I. Kougioumtzoglou, K. dos Santos, and A. Beck. 2018. “Stochastic averaging of nonlinear oscillators: Hilbert transform perspective.” J. Eng. Mech. 144 (2): 04017173. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001410.
Stratonovich, R. L. 1963. Vol. 1 of Topics in the theory of random noise. Philadelphia: Gordon Breach.
Stratonovich, R. L. 1967. Vol. 2 of Topics in the theory of random noise. Philadelphia: Gordon Breach.
Tarasov, V. 2017. “Fractional mechanics of elastic solids: Continuum aspects.” J. Eng. Mech. 143 (5): D4016001. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001074.
Vanmarcke, E. 1975. “On the distribution of the first-passage time for normal stationary random processes.” J. Appl. Mech. 42 (1): 215–220. https://doi.org/10.1115/1.3423521.
Vanvinckenroye, H., and V. Denoël. 2017. “Average first-passage time of a quasi-Hamiltonian Mathieu oscillator with parametric and forcing excitations.” J. Sound Vib. 406: 328–345. https://doi.org/10.1016/j.jsv.2017.06.012.
Vanvinckenroye, H., and V. Denoël. 2018. “Second-order moment of the first passage time of a quasi-Hamiltonian oscillator with stochastic parametric and forcing excitations.” J. Sound Vib. 427: 178–187. https://doi.org/10.1016/j.jsv.2018.03.001.
Vanvinckenroye, H., I. A. Kougioumtzoglou, and V. Denoël. 2019. “Reliability function determination of nonlinear oscillators under evolutionary stochastic excitation via a Galerkin projection technique.” Nonlinear Dyn. 95 (1): 293–308. https://doi.org/10.1007/s11071-018-4564-8.
Zhu, W. 1996. “Recent developments and applications of the stochastic averaging method in random vibration.” Supplement, J. Appl. Mech. 49 (S10): S72–S80. https://doi.org/10.1115/1.3101980.
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Published In
Journal of Engineering Mechanics
Volume 145 • Issue 10 • October 2019
Copyright
©2019 American Society of Civil Engineers.
History
Received: Oct 18, 2018
Accepted: Feb 11, 2019
Published online: Jul 31, 2019
Published in print: Oct 1, 2019
Discussion open until: Dec 31, 2019
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