Stability of a Non-Linear Damped Second-Order System with Randomly Fluctuating Restoring Coefficient

In connection with questions concerning possible unstable motion of certain types of marine structures, the non-stationary response of a homogeneous second-order system with randomly fluctuating restoring coefficient and with combined linear and non-linear power-law damping has been studied. New solutions, both in the time-domain and in terms of probability distributions, have been derived for non-stationary response initiated by some disturbance at time zero. The solutions hold asymptotically in the limit of small damping and small random restoring, and have been obtained using stochastic averaging techniques and a Fokker-Planck-Kolmogorov equation.
The solutions have been used to assess the asymptotic behaviour of time-domain realisations of response, of probability distributions of response, and of statistical moments of response as time approaches infinity. It is found that, depending on magnitude and non-linearity of the damping force, response can decrease to zero, can tend to a stationary state of random finite-amplitude response, or can grow unboundedly as time increases.