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Research Papers

Marginal Instability and Intermittency in Stochastic Systems—Part II: Systems With Rapid Random Variations in Parameters

[+] Author and Article Information
M. F. Dimentberg

Department of Mechanical Engineering, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609

A. Hera

Information Technology Division, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609

A. Naess

Centre for Ships and Ocean Structures, and Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

J. Appl. Mech 76(3), 031002 (Mar 05, 2009) (7 pages) doi:10.1115/1.3086593 History: Received January 08, 2007; Revised October 08, 2008; Published March 05, 2009

Dynamic systems with lumped parameters, which experience random temporal variations, are considered. The variations “smear” the boundary between the system’s states, which are dynamically stable and unstable in the classical sense. The system’s response within such a “twilight zone” of marginal instability is found to be of an intermittent nature, with alternating periods of zero (or almost-zero) response and rare short outbreaks. As long as it may be impractical to preclude completely such outbreaks for a designed system, subject to highly uncertain dynamic loads, the corresponding system’s response should be analyzed. Results of such analyses are presented for cases of slow and rapid (broadband) parameter variations in Papers I and II, respectively. The former case has been studied in Paper I (2008, “Marginal Instability and Intermittency in Stochastic Systems—Part I: Systems With Slow Random Variations of Parameters,” ASME J. Appl. Mech., 75(4), pp. 041002) for a linear model of the system using a parabolic approximation for the variations in the vicinity of their peaks (so-called Slepian model) together with Krylov–Bogoliubov averaging for the transient response. This resulted in a solution for the probability density function (PDF) of the response, which was of an intermittent nature indeed due to the specific algorithm of its generation. In the present paper (Paper II), rapid broadband parameter variations are considered, which can be described by the theory of Markov processes. The system is assumed to operate beyond its stochastic instability threshold—although only slightly—and its nonlinear model is used accordingly. The analysis is based on the solution of the Fokker–Planck–Kolmogorov partial differential equation for the relevant stationary PDF of the response. Several such PDFs are analyzed; they are found to have integrable singularities at the origin, indicating an intermittent nature of the response. Asymptotic analysis is performed for the first-passage problem for such response processes with highly singular PDFs, resulting in explicit formulas for an expected time interval between outbreaks in the intermittent response.

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Figures

Grahic Jump Location
Figure 2

(a) Samples of u(t) and v(t) as obtained from numerical simulation for the system with m=1, α=1, β=1, k=1, γ=0.98, and D=1. It illustrates intermittency of type I (u0=0.01 and Δ=1.98). All quantities are nondimensional. (b) Samples of u(t) and v(t) as obtained from numerical simulation for the system with m=1, α=1, β=1, k=1, γ=0.5, and D=20. It illustrates intermittency of type II (u0=0.5, v0=1, and Δ=0.05). All quantities are nondimensional.

Grahic Jump Location
Figure 1

(a) A short sample of response x(t) illustrating the response outbreaks. System parameters are Ω=2, α=0.01, β1=0.02, and δ=0.82. (b) PDF of v, w(v), predicted by theory (Eq. 4) and calculated from simulation data (δ=0.82,β1=0.02). To emphasize the difference between the two plots, only the PDF for v∊[1×10−4,1×102] has been used in the normalization of the results obtained from simulation data. (c) Expected period between outbreaks as shown by Eq. 8, Eq. 8 with correction for w(v), and numerical simulation. The errors between numerical simulation and predicted results using the corrected PDF of v are 5% (δ=0.8), 11% (δ=0.82), 23% (δ=0.85), and 50% (δ=0.90). β1=0.02.

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