Non-linear stochastic dynamics of tension leg platforms

摘要

In this paper, a set of non-linear equations of motion for a single-tendon tension leg platform are developed. The equations of motion consist of partial differential equations representing the transverse and longitudinal response of the tendon. In addition, a mixed formulation partial differential equation describing the surge response of the hull and tendon, coupled with an ordinary differential equation for the pitch response of the rigid hull is presented. Many of the simplifying assumptions used by prior researchers have been eliminated. The hull is modelled as a hollow rigid cylindrical body, and the tendon as a hollow cylindrical beam pinned at its top to the hull and at its bottom to the template connected to the seafloor. The Extended Hamilton's Principle is applied and the Lagrangian is fully developed. Terms include the kinetic energy, bending and membrane strain energies and the potential energy due to gravity and buoyancy. The normalized equations of motion are also detailed. The full derivation with assumptions are presented. The response, analyzed for stochastic wave and current loading, is presented with a planar motion assumption. The tension leg platform will oscillate about its vertical position due to ocean waves. Current will cause a tension leg platform to oscillate about an offset position rather than its vertical position. This offset in the surge direction has a corresponding setdown, the lowering of the hull in the heave direction, which increases the buoyancy forces. This results in a higher tension in the tendons than if the tendon and hull were in a vertical position. Forces on the tendon have been neglected in much of the literature. The responses presented in this work show that the inclusion of forces on the tendon will result in both a greater amplitude and offset position when compared to studies where these forces are neglected. This offset position, which is the surge displacement from the vertical position, is significant in the operation of a tension leg platform. A Monte Carlo simulation was performed on the drag and inertia coefficients in Morison's equation. A uniform random distribution of coefficients was selected from 0.6 to 2.0 for each coefficient. Twenty computer simulations were implemented for each coefficient. The response showed that the offset position and the amplitude are both dependent on the drag coefficient. The surge of the hull shows a maximum offset approximately three times greater for the coefficient that resulted in the maximum displacement than the minimum. The response did not show a significant dependence on the inertia coefficient, however, this is not necessarily true for unsteady current, large hull and tendon diameters, ocean wave frequencies greater than 1 rad/s, and low current velocity.