A dynamic system under parametric excitation in the form of a non-Erlang renewal jump process is considered. The excitation is a random train of nonoverlapping rectangular pulses with equal, deterministic heights. The time intervals between two consecutive jumps up (or down), are the sum of two independent, negative exponential distributed variables; hence, the arrival process may be termed as a generalized Erlang renewal process. The excitation process is governed by the stochastic equation driven by two independent Poisson processes, with different parameters. If the response in a single mode is investigated, the problem is governed in the state space by two stochastic equations, because the stochastic equation for the excitation process is autonomic. However, due to the parametric nature of the excitation, the nonlinear term appears at the right-hand sides of the equations. The equations become linear if the state space is augmented by the products of the original state variables and the excitation variable. Asymptotic mean and mean-square stability as well as asymptotic sample (Lyapunov) stability with probability 1 are investigated. The Lyapunov exponents have been evaluated both by the direct simulation of the stochastic equation governing the natural logarithm of the hyperspherical amplitude process and using the modification of the method wherein the time averaging of the pertinent expressions is replaced by ensemble averaging. It is found that the direct simulation is more suitable and that the asymptotic mean-square stability condition is not overly conservative.
Stochastic Stability of Mechanical Systems Under Renewal Jump Process Parametric Excitation
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the Applied Mechanics Division, June 27, 2003, final revision, September 13, 2004. Editor: R. M. McMeeking. Discussion on the paper should be addressed to the Editor, Prof. Robert M. McMeeking, Journal of Applied Mechanics, Department of Mechanical and Environmental Engineering, University of California-Santa Barbara, Santa Barbara, CA 393106-5070, and will be accepted until four months after final publication in the paper itself in the ASME JOURNAL OF APPLIED MECHANICS.
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Iwankiewicz, R., Nielsen , S. R. K., and Larsen, J. W. (March 15, 2005). "Stochastic Stability of Mechanical Systems Under Renewal Jump Process Parametric Excitation ." ASME. J. Appl. Mech. March 2005; 72(2): 213–221. https://doi.org/10.1115/1.1839591
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