Coupled Nonlinear Barge Motions: Part Ii: Deterministic Models 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 highdegree 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 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 US Navy is presented using the Q2DOF model based on first passage time formulation. INTRODUCTION The stability of ship-to-shore cargo barges under various sea conditions is important to design engineers, especially those of the US Navy. As discussed in Part I, while a barge in general experiences multidirectional sea conditions in the ocean, one of the most critical scenarios leading to capsizing is beam sea. A significant number of researchers have examined the roll stability of ships in beam seas from a stochastic perspective [1-7]. Robert [1, 2] analyzed the roll motion of a ship using the Fokker-Planck (FP) formulation to obtain the probability distribution of the response. Robert et al [3] proposed an averaging approximation to reduce the order of the FP equations from two to one to reduce the computational effort. Dahle et al [4] developed a simple probabilistic model and computed the probability of capsizing under specified sea states. Lin and Yim [5] modeled the roll motion of a ship by the FP equation and studied the effects of noise on Copyright © 2004 by ASME 1 deterministic regular wave loads. They showed, similar to the deterministic cases demonstrated by Falzarano et al [6] and Nayfeh and Sanchez [7], the ship motion to be governed by two diverse dynamical regions – homoclinic and heteroclinic, where the heteroclinic region relates to capsizing. They also examined chaotic response behavior with noise via probability density functions. Kwon et al [8] analyzed the roll motion of a ship subjected to an equivalent white noise ocean wave model. Their study focused on the mean upcrossing times for a vessel with nonlinear righting moment and damping. Cai et al [9] analyzed the nonlinear roll response of a ship to stationary Gaussian random waves with non-white broadband spectra. The total roll energy was approximated as a Markov process, using a modified version of quasi-conservative averaging. They treated the capsizing of the ship as a first passage problem. In this paper we begin the study the barge motions under beam sea by first deriving corresponding stochastic models of the deterministic coupled Roll-Heave (2DOF) model developed in Part I and developing a pure Roll (1DOF) in a following section. The path integral solution is employed to numerically obtain the evolutions of barge response probability densities as a solution to the corresponding FP equation of these models. Importance of coupling effects of heave on roll motion is examined by comparing numerical results obtained from the 2DOF and 1DOF models in both time and probability domains. A quasi-2DOF (Q2DOF) model is then developed to take advantage of the observed heave and wave elevation relationship in modeling the roll-heave coupling effects while keeping the number of governing equations to unity. Stability analysis of the barge in terms of reliability against capsizing under various sea states is performed using a first passage time formulation and the quasi-2DOF model. Governing Equations for Roll-Heave and Roll Models 2DOF Roll-Heave Model -We start with the deterministic 2DOF model governing the dynamics of fluid-structure interaction behavior of a barge in beam sea derived in Part I. Recall that the model retains the nonlinear coupling effects between roll and heave but removes the tertiary sway effect from equilibrium consideration. The hydrostatic terms are represented efficiently and accurately in the form of highdegree (13 in roll and 12 in heave) polynomials to represent the characteristics of restoring force and moment. Hydrodynamic terms are in a “Morison” type quadratic form.