Coupled Nonlinear Barge Motions, Part II: Stochastic Models and Stability Analysis

A computationally efficient quasi-two-degree-of-freedom (Q2DOF) stochastic model and a stability analysis of barges in random seas are presented in this paper. Based on the deterministic 2DOF coupled roll-heave model with high-degree polynomial approximation of restoring forces and moments developed in Part I, an attempt is made to further reduce the DOF of the model for efficient stochastic stability analysis by decoupling the heave effects on roll motion, resulting in a one-degree-of-freedom (1DOF) roll-only model. Using the Markov assumption, stochastic differential equations governing the evolution of probability densities of roll-heave and roll responses for the two low-DOF models are derived via the Fokker-Planck formulation. Numerical results of roll responses for the 2DOF and 1DOF models, using direct simulation in the time domain and the path integral solution technique in the probability domain, are compared to determine the effects of neglecting the influence of heave on roll motion and assess the relative computational efforts required. It is observed that the 1DOF model is computationally very efficient and the 2DOF model response predictions are quite accurate. However, the nonlinear roll-heave coupling is found to be significant and needs to be directly taken into account, rendering the 1DOF roll-only model inadequate for practical use. The 2DOF model is impractical for long-duration real-time response computation due to the insurmountable computational effort required. By taking advantage of the observed strong correlation between measured heave and wave elevation in the experimental results, an accurate and efficient Q2DOF model is developed by expressing the heave response in the 2DOF model as a function of wave elevation, thus reducing the effective DOF to unity. This Q2DOF model is essential as it reduces the computational effort by a factor of $10−5$ compared to that of the 2DOF model, thus making practical stochastic analysis possible. A stochastic stability analysis of the barge under operational and survival sea states specified by the U.S. Navy is presented using the Q2DOF model based on first passage time formulation.