The effects of slowly-varying wave drift forces on the nonlinear dynamics of mooring systems have been studied extensively in the past 30 years. It has been concluded that slowly-varying wave drift may resonate with mooring system natural frequencies. In recent work, we have shown that this resonance phenomenon is only one of several possible nonlinear dynamic interactions between slowly-varying wave drift and mooring systems. We were able to reveal new phenomena based on the design methodology developed at the University of Michigan for autonomous mooring systems and treating slowly-varying wave drift as an external time-varying force in systematic simulations. This methodology involves exhaustive search regarding the nonautonomous excitation, however, and approximations in defining response bifurcations. In this paper, a new approach is developed based on the harmonic balance method, where the response to the slowly-varying wave drift spectrum is modeled by limit cycles of frequency estimated from a limited number of simulations. Thus, it becomes possible to rewrite the nonautonomous system as autonomous and reveal stability properties of the nonautonomous response. Catastrophe sets of the symmetric principal equilibrium, serving as design charts, define regions in the design space where the trajectories of the mooring system are asymptotically stable, limit cycles, or non-periodic. This methodology reveals and proves that mooring systems subjected to slowly-varying wave drift exhibit many nonlinear phenomena, which lead to motions with amplitudes 2–3 orders of magnitude larger than those resulting from linear resonance. A turret mooring system (TMS) is used to demonstrate the harmonic balance methodology developed. The produced catastrophe sets are then compared with numerical results obtained from systematic simulations of the TMS dynamics.
The weight of a chain mooring line in deep water is the main source of mooring line tension. Chain weight also induces a vertical force on the moored vessel. To achieve the desired tension without excessive weight, hybrid mooring lines, such as lighter synthetic fiber ropes with chains, have been proposed. In this paper, the University of Michigan methodology for design of mooring systems is developed to study hybrid line mooring. The effects of hybrid lines on the slow-motion nonlinear dynamics of spread mooring systems (SMS) are revealed. Stability analysis and bifurcation theory are used to determine the changes in SMS dynamics in deep water based on pretension and angle of inclination of the mooring lines for different water depths and synthetic rope materials. Catastrophe sets in two-dimensional parametric design spaces are developed from bifurcation boundaries, which delineate regions of qualitatively different dynamics. Stability analysis defines the morphogeneses occurring as bifurcation boundaries are crossed. The mathematical model of the moored vessel consists of the horizontal plane—surge, sway, and yaw—fifth-order, large drift angle, low-speed maneuvering equations. Mooring lines are modeled quasistatically as nonlinear elastic strings for synthetic ropes and as catenaries for chains, and include nonlinear drag and touchdown. Excitation consists of steady current, wind, and mean wave drift. Numerical applications are limited to steady current, which is adequate for revealing the SMS design depending on the selected parameters. [S0892-7219(00)00804-9]
Various hydrodynamic maneuvering models are available for modeling the slow motion horizontal plane dynamics of mooring and towing systems. In previous work, we compared four representative and widely used maneuvering models and assessed them based on the design methodology for mooring systems developed at the University of Michigan. In this paper, we study the impact of experimental uncertainties in the maneuvering coefficients on mooring system dynamic analysis. Uncertainties in higher order coefficients may even result in sign change as measured by different experimental facilities. This may indicate lack of robustness in maneuvering modeling. In our recent work, maneuvering models were classified in two schools of thought, each having a different set of coefficients subject to uncertainties. The first school is represented by the Abkowitz (A-M) and the Takashina (T-M) models, and the second by the Obokata (O-M) and the Short Wing (SW-M) models. The design methodology developed at the University of Michigan uses time independent global properties of mooring system dynamics to compare the maneuvering models, and assess their sensitivity and robustness. Equilibria, bifurcation sequences and associated morphogeneses, singularities of bifurcations, and secondary equilibrium paths are such global properties. Systematic change of important coefficients in each model shows that, for both schools of thought, sensitivity to first order terms is high while sensitivity to higher order terms is low. Accuracy in measurement of first order terms is high while accuracy in measurement of higher order terms is low. These two tendencies reduce each other’s impact, providing acceptable robustness.
The effect of second-order slowly-varying wave drift (SVWD) forces on the horizontal plane motions of moored floating vessels has been studied for nearly 30 years. Large amplitude oscillations of moored vessels have been observed in the field or predicted numerically. Often, those have been incorrectly attributed to resonance or time-varying excitation from current/wind. In previous work, the authors have shown that resonance is only one of numerous interaction phenomena, and that large amplitude oscillations can be induced by SVWD forces or even time independent excitation. Currently, there is no mathematical theory to study stability and bifurcations of mooring systems subjected to nonautonomous spectral excitation. Thus, in this paper, bifurcation boundaries are approximated by analyzing simulation data from a grid of points in the design space. These boundaries are plotted in the catastrophe sets of the corresponding autonomous system, for which a design methodology has been developed at the University of Michigan since 1985. This approach has revealed a wealth of dynamics phenomena, characterized by static (pitchfork) and dynamic (Hopf) bifurcations. Interaction of SVWD forces with the Hopf bifurcations may result in motions with amplitudes 2–3 orders of magnitude larger than those due to resonance. On the other hand, in other cases the SVWD/Hopf interaction may reduce or even eliminate limit cycles.
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