Asymptotic stability of structural systems based on Lyapunov exponents and moment Lyapunov exponents

An asymptotic expansion for the maximal Lyapunov exponent, the exponential growth rate of solutions to a linear stochastic system, and the moment Lyapunov exponent, the asymptotic growth rate of the moments of the response, have been obtained for systems driven by a small intensity real noise process. The systems under consideration are general four-dimensional dynamical systems with two critical modes. Almost-sure and moment stability conditions are obtained provided the natural frequencies of these critical modes are non-commensurable and the infinitesimal generator associated with the noise process has an isolated simple zero eigenvalue. In this paper, the results obtained are applied to a thin rectangular beam under the action of a stochastic follower force and a model of a vehicle traveling over a rough road. The stability regions predicted by the two different criteria are then compared.