The closed-form stationary probability distribution of the stochastically excited vibro-impact oscillators

Chen, Lincong; Qian, Jiamin; Zhu, Haisheng; Sun, Jian‐Qiao

doi:10.1016/j.jsv.2018.09.061

Cited by 25 publications

(7 citation statements)

References 29 publications

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“…Besides computing the first statistical moments of the response or performing a stability analysis of systems under stochastic vibrations, we must emphasize that the computation of the finite distribution (usually termed "fidis") associated to the stationary solution, and particularly of the stationary PDF, is also a major goal in the realm of vibratory systems with uncertainties. Some interesting contributions in this regard include [25,26]. In [25], the authors first present a complete overview of methods and techniques available to determine the stationary PDF of nonlinear oscillators excited by random functions.…”

Section: Introductionmentioning

confidence: 99%

“…The theoretical analysis is accompanied with several illustrative examples. In the recent contribution [26], authors propose a new method to compute a closed-form solution of stationary PDF of single-degree-of-freedom vibro-impact systems under Gaussian white noise excitation. The density is obtained by solving the Fokker-Planck-Kolmogorov PDE using the iterative method of weighted residue combined with the concepts of the circulatory and potential probability flows.…”

Section: Introductionmentioning

confidence: 99%

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Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques

et al. 2021

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We combine the stochastic perturbation method with the maximum entropy principle to construct approximations of the first probability density function of the steady-state solution of a class of nonlinear oscillators subject to small perturbations in the nonlinear term and driven by a stochastic excitation. The nonlinearity depends both upon position and velocity, and the excitation is given by a stationary Gaussian stochastic process with certain additional properties. Furthermore, we approximate higher-order moments, the variance, and the correlation functions of the solution. The theoretical findings are illustrated via some numerical experiments that confirm that our approximations are reliable.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques

et al. 2021

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“…Among them, the composite cell coordinate system (CCCS) method [13,14] is usually used to obtain the global properties in the deterministic system, containing attractor, basin of attraction, boundary, and so on. e stochastic generalized cell mapping (SGCM) method [15,16] is used to obtain the stochastic response [17][18][19] in the stochastic system. e rest of this paper is as follows.…”

Section: Introductionmentioning

confidence: 99%

Statistical Characteristics of the First Passage Time Analysis for the Gene Regulatory Circuit in Bacillus subtilis by Cell Mapping Method

Wang

Yuan

et al. 2020

Complexity

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In this paper, we will explore the stochastic exit problem for the gene regulatory circuit in B. subtilis affected by colored noise. The stochastic exit problem studies the state transition in B. subtilis (from competent state to vegetative state in this case) through three different quantities: the probability density function of the first passage time, the mean of first passage time, and the reliability function. To satisfy the Markov nature, we convert the colored noise system into the equivalent white noise system. Then, the stochastic generalized cell mapping method can be used to explore the stochastic exit problem. The results indicate that the intensity of noise and system parameters have the effect on the transition from competent to vegetative state in B. subtilis. In addition, the effectiveness of the stochastic generalized cell mapping method is verified by Monte Carlo simulation.

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“…Zhu [33] obtained one solution of stationary response for the single-degree-of-freedom vibro-impact system by the exponential-polynomial closure method. With the help of Zhuravlev-Ivanov transformation and the iterative method of weighted residue, Chen et al [34] discussed the closed-form stationary probability distribution for stochastically excited vibro-impact oscillators. Based on the framework of Galerkin technique, Xie et al [35] studied the transient response of a vibro-impact system under random excitation.…”

Section: Introductionmentioning

confidence: 99%

Steady-State Dynamical Response of a Strongly Nonlinear System with Impact and Coulomb Friction Subjected to Gaussian White Noise Excitation

Yang

Huang

et al. 2020

Shock and Vibration

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The paper is devoted to the steady-state dynamical response analysis of a strongly nonlinear system with impact and Coulomb friction subjected to Gaussian white noise excitation. The Zhuravlev nonsmooth transformation of the state variables combined with the Dirac delta function is utilized to simplify the original system to one without velocity jump. Then, the steady-state probability density functions of the transformed system are derived in terms of the stochastic averaging method of energy envelope. The effectiveness of the presented analytical procedure is verified by those from the Monte Carlo simulation based on the original system. Effects of different restitution coefficients, amplitudes of friction, and noise intensities on the steady-state dynamical responses are investigated in detail. Results show different intensities of Gaussian white noise can affect the peaks value of the probability density functions, whereas the variations of restitution coefficients and amplitudes of friction can induce the occurrence of stochastic P-bifurcation.

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The closed-form stationary probability distribution of the stochastically excited vibro-impact oscillators

Cited by 25 publications

References 29 publications

Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques

Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques

Statistical Characteristics of the First Passage Time Analysis for the Gene Regulatory Circuit in Bacillus subtilis by Cell Mapping Method

Steady-State Dynamical Response of a Strongly Nonlinear System with Impact and Coulomb Friction Subjected to Gaussian White Noise Excitation

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