Summary
We investigate the reliability of a nonlinear pendulum forced by a resonant harmonic excitation and interacting in a random environment. Two types of random perturbations are considered: additive weak noise, and random phase fluctuations of the harmonic resonant forcing. Our goal is to predict the probability of a transition of the response from oscillatory regime to rotatory regime. In the first stage, the noise-free system is analyzed by an averaging method in view of predicting period-1 resonant orbits. By averaging the fast oscillations of the response, these orbits are mapped into equilibrium points in the space of the energy and resonant phase variables. In the second stage, the random fluctuating terms exciting the averaged system are evaluated, leading to a Fokker-Planck-Kolmogorov equation governing the probability density function of the energy and phase variables. This equation is solved asymptotically in the form of a WKB approximationp∼exp(−Q/ε) as the parameter ε characterizing the smallness of the random perturbations tends to zero. The quasipotentialQ is solution of a Hamilton-Jacobi equation, and can be obtained numerically by a method of characteristics. Of critical importance is the evaluation of the minimum difference of quasipotential between the equilibrium point and the boundary across which the transitions occur. We show that this minimum difference determines to logarithmic accuracy the mean first-passage time to the critical boundary and hence the probability of failure of the oscillatory regime. The effects of the two types of random perturbations are analyzed separately.
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Valéry Roy, R. Probabilistic analysis of a nonlinear pendulum. Acta Mechanica 115, 87–101 (1996). https://doi.org/10.1007/BF01187431
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DOI: https://doi.org/10.1007/BF01187431