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This paper concerns the evolutionary transitional probability density functions (PDFs) and the mean upcrossing rates (MCRs) of responses of the non-smoothly damped stochastic oscillator under both modulated random and harmonic excitations. The damping expressed by a non-smooth function of velocity is considered in the stochastic dynamical systems. A procedure is presented by making the time-related weighted residue of the formulated Fokker–Planck–Kolmogorov (FPK) equation vanish in the weak sense. Thus, a group of stochastic ordinary differential equations regarding the undetermined parameters in the assumed solution can be formulated and solved numerically. Three cases are studied under different conditions. The achieved evolutionary transitional PDF and MCR solutions attest to the outstanding effectiveness of the presented procedure even in the tail ends of the PDF solutions. Moreover, the achieved solutions of the stochastic dynamical systems are found to have significant non-Gaussian behaviors due to the interaction of the random and harmonic excitations and the system nonlinearity. Notably, the presented procedure greatly improves the running efficiency when compared with Monte Carlo simulation.
Er et al. (Fri,) studied this question.