The almost-sure asymptotic stability of a class of coupled multi-degrees-of-freedom systems subjected to parametric excitation by an ergodic stochastic process of small intensity is studied. Explicit asymptotic expressions for the largest Lyapunov exponent for various values of the system parameters are obtained by using a combination of the method of stochastic averaging and a well-known procedure due to Khas’minskii, from which the asymptotic stability boundaries are determined. As an application, the example of the flexural-torsional instability of a thin elastic beam acted upon by a stochastically fluctuating load at the central cross-section of the beam is investigated.

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