The stationary response and asymptotic stability in probability of an articulated tower under random wave excitation are investigated. The articulated tower is modelled as a SDOF system having stiffness nonlinearity, damping nonlinearity and parametric excitation. Using a stochastic averaging procedure and Fokker-Plank-Kolomogorov equation (FPK), the probability density function of the stationary solution is obtained for random sea state represented by a P-M sea spectrum. The method involves a Van-Der-Pol transformation of the nonlinear equation of motion to convert it to the Ito’s stochastic differential equation with averaged drift and diffusion coefficients. The asymptotic stability in probability of the system is investigated by obtaining the averaged Ito’s equation for the Hamiltonian of the system. The asymptotic stability is examined approximately by investigating the asymptotic behaviour of the diffusion process Y(t) at its two boundaries Y = 0 and ∞. As an illustrative example, an articulated tower in a sea depth of 150 m is considered. The tower consists of hollow cylinder of varying diameter along the height, providing the required buoyancy of the system. Wave forces on the structure are calculated using Morrison’s equation. The stochastic response and the stability conditions are obtained for a sea state represented by P-M spectrum with 16m significant wave height. The results of the study indicate that the probability density of the stationary response obtained by the stochastic averaging procedure is in very good agreement with that obtained from digital simulation. Further, the articulated tower is found to be asymptotically stable under the parametric excitation arising due to hydrodynamic damping.
Lyapunov Function for Nonuniform in Time Global Asymptotic Stability in Probability with Application to Feedback Stabilization
