Considering the platform and the sea to be a single dynamic system, We can describe the interaction between a platform and the sea with Lagrange's energy function. And then, using Hamilton's method, we can abo govern the nonlinear coupling eguations of the platform rolling and pitching in the regular sea. Finally, the steady-state responses of a platform transporting in the regular sea can be solved by the multiple scales that is the advanced method of the nonlinear perturbation with multidegree of freedom.
It is very interesting to study the safety of a platform that is moving or being towed by the tugboat in the regular sea. In the past, one always discussed the motion of a platform (with linear and single-degree of freedom system) after its motion equations (with nonlinear coupling and multidegree of freedom) had be simplily transformed (Wilson, 1984). It is an approach method that cannot reveal the essential feature of the motion responses of a platform, that may be internal responce in the multidegree of freedom or external resonance forced by the waves. The method of multiple scales is used in this paper (Nayfeh, 1979, 1981), that is the best method to solve the nonlinear coupling eguations of the platform with multidegree of freedom.
We can employ two Cartesian Coordinate systems: one fixed in an intertial space and the other fixed in the platform with its origin at the mass center, as shown in Fig. 1. The two coordinate systems coincide initially. Considering the platform and the sea to be a single dynamic system, we suppose that the motion of the fluid is entirely due to the motion of the platform and neglect the effects of viscosity.
