Research Papers

Marginal Instability and Intermittency in Stochastic Systems—Part I: Systems With Slow Random Variations of Parameters

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

Mechanical Engineering Department, 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 75(4), 041002 (May 09, 2008) (8 pages) doi:10.1115/1.2910900 History: Received July 05, 2006; Revised November 13, 2007; Published May 09, 2008

Dynamic systems with lumped parameters, which experience random temporal variations, are considered. The variations may “smear” 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 Parts I and II, respectively. In the former case, the “nominal” system—one without variations of parameter(s)—is stable in the classical sense. Its transient response during the “slow” short-term excursions of the parameter(s) into the instability domain is described by a linear model. The analysis is based on Krylov–Bogoliubov averaging over “rapid” time within the response period together with parabolic approximation for the parameter variations in the vicinity of their peaks (so-called Slepian model). Solution to the resulting deterministic transient response problem with random initial condition(s) at the instant of upcrossing the stability boundary yields a relation between peak value(s) of the response(s) and that of the parameter(s); in this way, reliability study for the system is reduced to a probabilistic analysis of the parameter variations. The solutions are obtained for the cases of negative-damping-type instability in a SDOF system and for TDOF systems with potential dynamic instability due to coalescing or merging of natural frequencies; the illustrating examples of applications are rotating shafts with internal damping, two-dimensional galloping of a rigid body in a fluid flow and a row of tubes in a cross flow of fluid. The response is of the intermittent nature due to the way it is generated, with alternating relatively long periods of zero (or almost-zero) response and short outbreaks due to temporary excursions into the instability domain.

Copyright © 2008 by American Society of Mechanical Engineers
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Figure 1

Short-term single-mode dynamic instability: comparison of theory with direct numerical simulation equation 2 for cases Ω=2.0s−1, α=0.16s−1, λ=0.1s−1, and u=2. (a) A short sample of q(t) illustrating “outbreak” in X(t) corresponding to upcrossing level 0.16 (nonzero response within stability domain is due to additional random RHS in the equation, which is needed to provide initial conditions for transients due to instability); (b) theoretical PDF of scaled amplitude A¯p=Ap∕A0 and corresponding histogram as obtained from sample of X(t); (c) histogram of directly measured initial amplitudes—values at the instants of upcrossing—and the corresponding curve of Rayleigh PDF, which provides the best fit of the data; and (d) PDF of “physical”—nonscaled—peak response amplitude as calculated according theoretical solution based on theoretical Rayleigh PDF of initial amplitudes and corresponding histogram of directly measured peak response amplitudes.

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Figure 2

Theoretical PDFs of scaled peak radius of whirl of rotating shaft with temporal random variations of the internal damping for different values of mistuning factor

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Figure 3

Theoretical PDF of scaled peak vertical response of a rigid body in a 2D galloping



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