PDF solution of nonlinear oscillators subject to multiplicative Poisson pulse excitation on displacement

Abstract

A solution procedure for the stationary probability density function (PDF) of the responses of nonlinear oscillators subjected to Poisson white noises is formulated with exponential-polynomial closure (EPC) method. The effectiveness of the solution procedure is investigated with nonlinear oscillators subjected to both external and multiplicative Poisson white noises at different levels of system nonlinearity, excitation intensity, and impulse arrival rates. Numerical results show that the PDFs obtained with the EPC procedure are in good agreement with those from Monte Carlo simulation.

Nonzero mean PDF solution of nonlinear oscillators under external Gaussian white noise

“…The zero mean response of different nonlinear oscillators has been widely investigated with the EPC method [17,18]. In this paper, The PDF of the nonzero mean response is examined using different types of oscillators under external Gaussian white noise.…”

“…The PDF solution of velocity has zero mean and the PDF distribution is symmetric about its zero mean. According to (18), the PDF solution of velocity is Gaussian PDF. The symmetry and zero-mean property of the PDF of velocity are not influenced by the addition of the constant m.…”

“…In particular, the nonzero mean PDF solutions as well as the tail behavior of the PDF solutions are scarcely mentioned in existing references. In this paper, the nonzero mean PDF solution is tentatively obtained with the exponential-polynomial closure (EPC) method by solving the well-known Fokker-Planck-Kolmogorov (FPK) equation [17,18]. In order to evaluate the effectiveness of the proposed method, different types of oscillators is further investigated in the case of nonzero mean response.…”

Probabilistic response of dynamical systems based on the global attractor with the compatible cell mapping method

“…However, exact analytical solutions for the PDF of the response are only available for very restricted classes of nonlinear systems [6][7][8][9][10][11]. Various approximate and numerical procedures have been developed to solve PDF solutions for the FPK equation or the KF equation [12][13][14][15][16][17][18][19][20][21][22][23][24][25], such as stochastic averaging method, path integral method, cell mapping method and exponential-polynomial closure method. Among these methods, the cell mapping method has been demonstrated to be a very efficient tool due to its ability of global analysis of the strongly nonlinear systems.…”

Direct evaluation of jumps for nonlinear systems under external and multiplicative impulses

“…A Poissonian white noise process is in fact represented by a train of Dirac's deltas of random amplitude occurring at random time instants, and it is frequently used to model a wide range of phenomena in stochastic dynamics (Tung, 1967, Roberts, 1966, Lin, 1963, Merchant, 1964, Liepmann, 1952). Due to this characteristic, many papers have been then devoted to the study of systems under multiplicative Poisson white noise input (Caddemi and Di Paola, 1995, Di Matteo et al., 2014, Di Paola and Pirrotta, 2004, Er et al., 2009, Ibrahim, 1985, Iwankiewicz, 2003, Papadimitriou et al., 1999, Pirrotta, 2005, 2007, Proppe, 2002, Sun et al., 2013, Zheng and Zhu, 2010), which thus require the solution of differential equations with parametric Dirac's delta forces input.…”