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Random perturbations in nature described by non-Gaussian excitation models are far more widely applied and development prospects than that of Gaussian excitation models in practice. However, the stochastic dynamics research of non-Gaussian excitation is still not very mature. In this work, radial-basis-function-neural-network (RBFNN) is applied for calculating the stationary response of single-degree-of-freedom (SDOF) nonlinear system excited by Poisson white noise. Specifically, the trial probability-density-function (PDF) solution of reduced generalized-Fokker–Plank–Kolmogorov (GFPK) equation is constructed by a suitable number of Gaussian basis functions (GBFs) with a fixed set of means and standard deviations. Subsequently, an approximate squared error of the GFPK equation in a finite domain is considered. Together with the normalization condition, the approximate squared error can be minimized by establishing a Lagrangian function, and then the optimal weight coefficients associated with the approximate PDF solution are solved from a system of linear algebraic equations. For demonstrating the effectiveness of the proposed procedure, two specific examples are presented. The corresponding reduced GFPK equation is truncated with higher order for the strong non-Gaussian case. The precision of the analytical solution is verified against the Monte Carlo simulation (MCS) data. In addition, all the results indicate that RBFNN shows fairly high efficiency under the premise of ensuring high precision in the whole computational procedure.

Poisson white noise is a more accurate representation than ideal Gaussian white noise for simulating the noise background during early bearing failures. In the context of strong Poisson white noise interference, the failure mode of rolling bearings usually remains undetermined. This paper proposes a new method for diagnosing unknown bearing faults in bearings, diagnosis is proposed, which effectively identifies unknown faults across various components. The theoretical characteristics of Poisson white noise are initially analyzed and discussed. Then, to extract potential fault characteristic information, a false characteristic frequency of bearing is constructed in the response spectrum, and the response results at this false fault characteristic are obtained. Finally, coherent resonance (CR) is introduced, and false frequencies caused by false faults are eliminated by defining a quality factor. To verify the effectiveness of the method, the experimental and simulation results were compared with the decomposition results of the SVMD algorithm. The SNR of the experimental signals for outer and inner ring faults under variable speed conditions increased to 8.62dB and 11.74dB, respectively. Results indicate show that this method not only successfully identifies fault features, but also exhibits a strong noise reduction effect.

We review some recent developments in white noise analysis and quantum probability. We pay a special attention to spaces of test and generalized functionals of some Lévy white noises, as well as to the structure of quantum white noise on these spaces.