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Equations for the Response Probability Density of Dynamic Systems under Multi-Component Non-Poisson Impulse Process Excitations

Iwankiewicz R.

Machine Dynamics Problems

2008

Vol. 32, No. 4

24-36

The excitation considered in the present paper consists of n statistically independent random trains of impulses, each of whom is driven by a non-Poisson, renewal process with inter-arrival times being the sum of two independent negative-exponential distributed random variables with parameters vv, Vs, µs (S = 1, 2, ..., n). Each of the original impulse processes is recast into a Poisson driven impulse process with the aid of an auxiliary, purely jump stochastic variable. Each auxiliary variable is governed by the stochastic differential equation driven by two independent Poisson processes, with parameters Vs, µs, thus it is tantamount to two Markov states. The Markov chain for the whole problem is constructed by considering the coincidences of the states of the individual jump processes. The necessary so-called jump probability intensity functions are determined for all state variables and all possible jumps. The equations governing the joint probability density-distribution function of the response and of the Markov states of the auxiliary variables are derived from the general integro-differential forward Chapman-Kolmogorov equation. The resulting equations form a set of integro-partial differential equations.