The Analysis of Motions of Semisubmersible Drilling Vessels in Waves
Authors: Burke Ben G. 
Published: December 2007

Abstract
A mathematical model was developed to compute the motions of semisubmersible drilling vessels in waves for a wide variety of semisubmersible configurations. The model was derived from a linear representation of motions, ocean waves, and forces. The semisubmersible is represented as a rigid space frame composed of a number of cylindrical members with arbitrary diameters, lengths and orientations. Forces on the semisubmersible are derived from anchorline properties, and hydrostatic hydrodynamic principles. A solution is obtained for motions in six degrees of freedom for a sinusoidal wave train of arbitrary height, period, direction and water depth. Results from the analysis of three semisubmersibles are compared with results from available model test data to verily the mathematical model.

Introduction
An accurate and complete representation of the response of a drilling vessel to waves is a valuable engineering tool for predicting vessel performance and designing drilling equipment. The performance and designing drilling equipment. The wave response for a floating vessel may be obtained to various degrees of accuracy from model tests or analytical means, as described by Barkley and Korvin-Kroukovsky and as applied by Bain. A review of the works cited shows that the evaluation of the wave response for a particular vessel requires considerable time and effort, either in model construction and testing or in computer programming and calculations. In order to reduce programming and calculations. In order to reduce the amount of time and effort required to evaluate a particular vessel, means were investigated to generalize and automate, on a digital computer, methods for evaluating wave response for vessels of arbitrary configuration. The mathematical model described in this paper is the result of such an investigation for semisubmersible-type drilling vessels. The paper presents a general description of the mathematical model and the basic principles and assumptions from which it was derived. The validity of the model is evaluated by comparing results of the analysis of three semisubmersibles with available model test data.

MATHEMATICAL MODEL
The mathematical model for calculating the motions of a semisubmersible in waves is derived from basic principles and empirical relationships in classical mechanics. All equations are derived for "small amplitude" waves and motions. The nonlinear equations that appear in the problem are replaced by "equivalent" linear equations in order to conform to the linear analysis method used in obtaining a solution. The model is implemented in a computer program that computes vessel response in all six degrees of freedom for a broad range of semisubmersible configurations and wave parameters. The basic elements in the theoretical model are outlined, with a more detailed discussion of the principles and derivations used to obtain the model principles and derivations used to obtain the model presented in the Appendix. presented in the Appendix.

SEMISUBMERSIBLE DESCRIPTION AND EQUATIONS OF MOTION
The semisubmersible is characterized as a space-frame of cylindrical members and is described geometrically by specifying end-coordinates and diameters for all of the members. Specification of the mass, moments of inertia, center of gravity and floating position are required to complete the description. The six equations of motion for the semisubmersible derive from Newton's second law for a rigid body. These differential equations, when written in matrix form, equate the product of the six-component acceleration vector, {x}, and the inertia matrix, I, to a six-component, force-moment vector, {FT}.
SPEJ
P. 311