Vol. 9, No. 1, 2014

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ISSN: 1559-3959
Moment Lyapunov exponents and stochastic stability of coupled viscoelastic systems driven by white noise

Jian Deng, Wei-Chau Xie and Mahesh D. Pandey

Vol. 9 (2014), No. 1, 27–50
Abstract

The moment and almost-sure stochastic stability of two-degree-of-freedom coupled viscoelastic systems, under parametric excitation of white noise, are investigated through moment Lyapunov exponents and Lyapunov exponents, respectively. The system of stochastic differential equations of motion is first decoupled by using the method of stochastic averaging for dynamic systems with small damping and weak excitations. Then a new scheme for determining the moment Lyapunov exponents is proposed for a coupled viscoelastic system. The largest Lyapunov exponent is calculated through its relation with moment Lyapunov exponent. The moment and almost-sure stability boundaries and critical excitation are obtained analytically. These analytical results are confirmed by numerical simulation. As an application example, the stochastic stability of flexural-torsional viscoelastic beam is studied. It is found that, under white noise excitation, the parameters of damping β and the viscoelastic intensity γ have stabilizing effects on the moment and almost-sure stability. However, viscosity parameter η plays a destabilizing role. The stability index decreases from positive to negative values with the increase of the amplitude of noise power spectrum, which suggests that the noise destabilize the system. These results are useful in engineering applications.

Keywords
stochastic stability, moment Lyapunov exponents, white noise, viscoelasticity, coupled system
Milestones
Received: 14 April 2013
Revised: 26 December 2013
Accepted: 11 February 2014
Published: 23 March 2014
Authors
Jian Deng
Department of Civil Engineering
Lakehead University
Thunder Bay, ON  P7B 5E1
Canada
Wei-Chau Xie
Department of Civil and Environmental Engineering
University of Waterloo
Waterloo, ON  N2L 3G1
Canada
Mahesh D. Pandey
Department of Civil and Environmental Engineering
University of Waterloo
Waterloo, ON  N2L 3G1
Canada