Abstract
We study a class of single-degree-of-freedom oscillators whose restoring function is affected by small nonlinearities and excited by stationary Gaussian stochastic processes. We obtain, via the stochastic perturbation technique, approximations of the main statistics of the steady state, which is a random variable, including the first moments, and the correlation and power spectral functions. Additionally, we combine this key information with the principle of maximum entropy to construct approximations of the probability density function of the steady state. We include two numerical examples where the advantages and limitations of the stochastic perturbation method are discussed with regard to certain general properties that must be preserved.
Highlights
We have studied a class of stochastic nonlinear oscillators, whose nonlinear term is defined via a transcendental function
Since the nonlinear term is affected by a small parameter, to conduct our probabilistic analysis, we have approximated the nonlinear term using a Taylor’s polynomial, and we have applied the stochastic perturbation method to obtain the main statistical moments of the stationary solution
Since a key point when applying the perturbation method is the accuracy of the approximations in terms of the size of the perturbative parameter, from the numerical results obtained in the two examples, we have performed a critical analysis checking whether some important general properties of the statistics associated with the stationary solution are cor
Summary
Many vibratory systems are described by nonlinear equations, say N [Y (t)] = Z (t), where the nonlinear operator is defined in terms of a small perturbation , N [·; ]. The goal of this paper is to tackle the stochastic study of oscillators of form (1) in the case that the nonlinear term h is a transcendental function using a polynomial approximation, based on Taylor’s expansions, and to apply the stochastic perturbation method to approximate the main statistical functions of the steady state. Where we will assume that β > 0, the external source, Z (t), is defined via zero-mean Gaussian stationary stochastic process, which corresponds to an important case in the analysis of vibratory systems [8,9]. 3, we perform a numerical analysis through two examples, with criticism about the validity of the results obtained via the stochastic perturbation method.
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