Lyapunov exponents of parametrically coupled linear two-DOF stochastic systems and related stability problems

The almost-sure asymptotic stability of elastic systems subjected to parametric excitation is studied. The excitation consists of a harmonic function on which a stochastic term is superposed. The effect of the parametric action on the stability of a coupled system of differential equations is studied. By means of stochastic transformations of state norm process, the stability boundaries are determined using the stochastic averaging method and a technique due to Khasminskii. As an application, the problem of coupled flexural–torsional instability of a deep rectangular beam in the presence of fluctuating axial loads and end moments is considered.