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Coupled Nonlinear Barge Motions, Part I: Deterministic Models Development, Identification and Calibration

INTRODUCTION This paper focuses on the development of optimal deterministic, nonlinearly coupled barge motion models, identification of their system parameters and calibration of their prediction capability using experimental results. The ultimate objective is to develop accurate yet sufficiently low degree-offreedom stochastic models suitable for efficient probabilistic stability and reliability analyses of US Naval barges for preliminary design and operation guideline development (see Part II). First a three-degree-of-freedom (3DOF) fully coupled Roll-Heave-Sway model, which features realistic and practical high-degree polynomial approximations of rigid body motion relations, hydrostatic and hydrodynamic force-moment specifically suitable for barges, is examined. The hydrostatic force-moment relationship includes effects of the barge’s sharp edge and combined roll-heave states, and the hydrodynamic terms are in a “Morison” type quadratic form. System parameters of the 3DOF model are identified using physical model test results from several regular wave cases. The predictive capability of the model is then calibrated using results from a random wave test case. Recognizing the negligible sway influence on coupled roll and heave motions and overall barge stability, and in an attempt to reduce anticipated stochastic computational efforts in stability analysis, a 2DOF Roll-Heave model is derived by uncoupling sway from the roll-heave governing equations of motion. Time domain simulations are conducted using the (3DOF) Roll-Heave-Sway and the (2DOF) Roll-Heave models for regular and random wave cases to validate the model assumptions and to assess their (numerical) prediction capabilities. In the design of ship-to-shore transport cargo barges, it is essential to determine barge stability for a range of operational and survival sea conditions. In general, the barges will operate in a variety of directional seastates. However, the most unstable scenario is if the barges broach and become broadside to the waves in the so called “beam seas” and may incur large amplitude three-degrees of freedom (3DOF) -roll, heave and sway motions with the possibility of capsizing [1]. Many researchers further reduced the DOF of the systems to that of roll only by taking advantage of the dominant roll behavior [27]. Parameters for the coefficients of nonlinear roll motions were determined [8] and the roll motions characteristics of full scale ships were examined [9]. A stochastic approach to the analysis of noisy periodic roll motions was proposed [10]. This paper presents a deterministic 3DOF Roll-HeaveSway model [7, 11], and a corresponding two-degree-offreedom (2DOF) Roll-Heave model [12-13], to predict barge motion responses. These low DOF models, with high-degree polynomial approximations of force and moment relationships, capable of capturing the important nonlinear characteristics of the coupled nonlinear responses for large roll angle motions, will be used in the development of efficient stochastic models for preliminary design and response predictions under operational and survival conditions (see Part II). In research conducted earlier at Oregon State University, a one-degree-of-freedom (1DOF) system [10] was developed to model pure roll motion of a barge in random beam seas. Nonlinearities of the model include the righting moment and fluid-structure viscous effects. Hydrodynamic and structural damping effects were approximated by a linear term plus a Copyright © 2004 by ASME 1 “Morison” type quadratic term [14]. The righting moment included nonlinear stiffness terms to provide a more accurate restoring moment at larger roll angles. This 1DOF model was compared with measured barge motion data and was found capable of reasonable predictions in terms of statistical moments, spectral densities, and histograms. ( ) ( ) M I dt d F mv dt d = = ω ; (1) An inertial coordinate system is placed at the location of the prescribed body-fixed "roll center" of the barge under static equilibrium. Note the inertial coordinate system coincides with the body-fixed (moving) coordinate system initially. Static roll righting moments and heave buoyant restoring forces are calculated as a function of the position and rotation of the barge about the roll center. Equilibrium of forces and moments are considered about the roll center (the position of which is time dependent with respect to the inertia coordinates) with heave and sway directions respect to the inertial coordinates. In this study, we focus our discussion on a three-degreesof-freedom (3DOF) deterministic model including the nonlinear coupling effects of roll, heave and sway motions [1, 11]. This 3DOF model is expected to improve the predictive capability at large roll angles over the 1DOF system because the heave and sway coupling effects with the roll through hydrostatics and rigid body kinematics are included, and significantly higher degree polynomial approximations are employed. The equations of motion of the rigid barge including hydrostatics are first derived. Waves are then applied and terms modeling the hydrodynamic properties are added. Relative motion effects of the barge with respect to the free surface are included. The effects due to hydrostatics are represented with sufficiently high degree polynomials in the model. Various degree polynomials were examined to identify an optimum fit. Because the edges of the barge are sharp, fairly high degree polynomials are required. The coupling effects of sway on roll-heave response prediction are examined using the Roll-Heave-Sway model and a corresponding (2DOF) rollheave model with similar parameters. The body-fixed coordinates are defined such that X = Surge, Y = Sway, Z = Heave, φ = Roll, Θ = Pitch, and ψ = Yaw (Fig. 1). For the (body-fixed) coordinate system origin, the roll center is at the center of gravity of barge and the coordinate system axes are aligned with the principal axes of inertia. One of the main objectives in this study is to extend the equations of motion for a SDOF system in roll to a multi-DOF system. For a symmetric barge in beam seas, the dominant response will be in sway, heave and roll. The surge, pitch and yaw motions become negligible [11-13]. Equation (1) now becomes,     − − =       − + =     − − =