[en] This work focuses on the stochastic version of the linear Mathieu oscillator with both forced and parametric excitations of small intensity. In this quasi-Hamiltonian oscillator, the concept of energy stored in the oscillator plays a central role and is studied through the first passage time, which is the time required for the system to evolve from a given initial energy to a target energy. This time is a random variable due to the stochastic nature of the loading. The average first passage time has already been studied for this class of oscillator. However, the spread has only been studied under purely parametric excitation. Extending to combinations of both forcing and parametric excitations, this work provides a closed-form solution and a thorough analytical study of the coefficient of variation of the first passage time of the energy in this system. Simple asymptotic solutions are also derived in some particular ranges of parameters corresponding to different regimes.
Gitterman, M., Spring pendulum: parametric excitation vs an external force. Phys A. Stat. Mech. Appl. 389:16 (2010), 3101–3108.
de Sa Caetano, E., Cable vibrations in cable-stayed bridges. I. A. f. B. Engineering, Structural, (eds.) Structural Engineering Document, 2007, IABSE-AIPC-IVBH, 188.
Vanvinckenroye, H., Denoël, V., Monte Carlo simulations of autorotative dynamics of a simple tower crane model. Proceedings of the 14th International Conference on Wind Engineering, Porto Alegre, Brazil, 2015.
Voisin, D., Etudes des effets du vent sur les grues à tour, “Wind effects on tower cranes”. (Ph.D. thesis), 2003, Ecole Polytechique de l'Université de Nantes.
Horton, B.W., Wiercigroch, M., Effects of heave excitation on rotations of a pendulum for wave energy extraction. Kreuzer, E., (eds.) IUTAM Symposium on Fluid-structure Interaction in Ocean Engineering Iutam Bookseries, vol. 8, 2008, Springer Netherlands, Dordrecht, 117–128.
Najdecka, A., Narayanan, S., Wiercigroch, M., Rotary motion of the parametric and planar pendulum under stochastic wave excitation. Int. J. Non Lin. Mech. 71 (2015), 30–38.
Yurchenko, D., Alevras, P., Stochastic dynamics of a parametrically base excited rotating pendulum. Procedia IUTAM 6 (2013), 160–168.
Alevras, P., Yurchenko, D., Stochastic rotational response of a parametric pendulum coupled with an SDOF system. Probabilist. Eng. Mech. 37:0 (2014), 124–131.
Moshchuk, N.K., Ibrahim, R.A., Khasminskii, R.Z., Chow, P.L., Ship capsizing in random sea waves and the mathematical pendulum. IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics, vol. 47, 1995, 299–309.
Troesch, A.W., Falzarano, J.M., Shaw, S.W., Application of global methods for analyzing dynamical systems to ship rolling motion and capsizing. Int. J. Bifurc. Chaos 02:01 (1992), 101–115.
Lee, C.-L., Stell, G., Wang, J., First-passage time distribution and non-Markovian diffusion dynamics of protein folding. J. Chem. Phys. 118:2 (2003), 959–968.
Schuss, Z., Theory and Applications of Stochastic Processes. Applied Mathematical Sciences, vol. 170, 2010, Springer New York, New York, NY.
Vanvinckenroye, H., Denoël, V., Average first-passage time of a quasi-Hamiltonian Mathieu oscillator with parametric and forcing excitations. J. Sound Vib. 406 (2017), 328–345.
T. Primožič Estimating Expected First Passage Times Using Multilevel Monte Carlo Algorithm, MSc in Mathematical and Computational Finance University.
Mallick, K., Marcq, P., On the stochastic pendulum with Ornstein-Uhlenbeck noise. J. Phys. A Math. Gen., 37(17), 2004, 14.
Palleschi, V., Torquati, M.R., Mean first-passage time for random-walk span: comparison between theory and numerical experiment. Phys. Rev. A 40:8 (1989), 4685–4689.
Vanmarcke, E.H., On the distribution of the first-passage time for normal stationary random processes. J. Appl. Mech., 42(1), 1975, 215.
Engelund, S., Rackwitz, R., Lange, C., Approximations of first-passage times for differentiable processes based on higher-order threshold crossings. Probabilist. Eng. Mech. 10:1 (1995), 53–60.
Deng, M., Zhu, W., Feedback minimization of first-passage failure of quasi integrable Hamiltonian systems. Acta Mech. Sin. 23:4 (2007), 437–444.
Zhu, W.Q., Wu, Y.J., First-passage time of duffing oscillator under combined harmonic and white-noise excitations. Nonlinear Dynam. 32:3 (2003), 291–305.
D. Yurchenko, E. Mo, A. Naess, Reliability of strongly nonlinear single degree of freedom dynamic systems by the path integration method, J. Appl. Mech. 75 (6).
Kougioumtzoglou, I.A., Spanos, P.D., Stochastic response analysis of the softening Duffing oscillator and ship capsizing probability determination via a numerical path integral approach. Probabilist. Eng. Mech. 35 (2014), 67–74.
I.A. Kougioumtzoglou, P.D. Spanos, Nonstationary stochastic response determination of nonlinear systems: a Wiener path integral formalism, J. Eng. Mech. 140(9).
Huang, Z., Zhu, W., Suzuki, Y., Stochastic averaging of strongly non-linear oscillators under combined harmonic and white-noise excitations. J. Sound Vib. 238:2 (2000), 233–256.
Spanos, P.D., Kougioumtzoglou, I.A., Galerkin scheme based determination of first-passage probability of nonlinear system response. Struct. Infrastruct. Eng. 10:10 (2014), 1285–1294.
P.D. Spanos, A. Di Matteo, Y. Cheng, A. Pirrotta, J. Li, Galerkin scheme-based determination of survival probability of oscillators with fractional derivative elements, J. Appl. Mech. 83 (12).
Barbato, M., Conte, J.P., Structural reliability applications of nonstationary spectral characteristics. J. Eng. Mech. 137:5 (2011), 371–382.
Canor, T., Caracoglia, L., Denoël, V., Perturbation methods in evolutionary spectral analysis for linear dynamics and equivalent statistical linearization. Probabilist. Eng. Mech. 46 (2016), 1–17.
Thomas, M.U., Some mean first-passage time approximations for the Ornstein-Uhlenbeck process. J. Appl. Probab., 12(3), 1975, 600.
Murthy, K.P.N., Kehr, K.W., Mean first-passage time of random walks on a random lattice. Phys. Rev. A 40:4 (1989), 2082–2087.
B. Dybiec, E. Gudowska-Nowak, P. Hänggi, Lévy-Brownian motion on finite intervals: mean first passage time analysis, Phys. Rev. E - Stat. Nonlinear Soft Matter Phys. 73 (4).
V. Tejedor, O. Bénichou, R. Voituriez, Global mean first-passage times of random walks on complex networks, Phys. Rev. E - Stat. Nonlinear Soft Matter Phys. 80 (6).
Chunbiao, G., Bohou, X., First-passage time of quasi-non-integrable-Hamiltonian system. Acta Mech. Sin. 16:2 (2000), 183–192.
Moshchuk, N., Ibrahim, R., Khasminskii, R., Chow, P., Asymptotic expansion of ship capsizing in random sea waves-I. First-order approximation. Int. J. Non Lin. Mech. 30:5 (1995), 727–740.
Pontryagin, L., Andronov, A., Vitt, A., Appendix: on the statistical treatment of dynamical systems. Moss, F., McClintock, P.V.E., (eds.) Noise in Nonlinear Dynamical Systems Theory of Continuous Fokker-Planck Systems, vol. 1, 1989, Cambridge University Press, 329–348.
Vanvinckenroye, H., Denoël, V., Stochastic rotational stability of tower cranes under gusty winds. 6th International Conference on Structural Engineering, Mechanics and Computation, Cape Town, South Africa, 2016.
