Paulling 



of the structure itself. Additional restraint forces are introduced 

 to account for the effects of the mooring or dynamic position system. 

 This total system of external forces is then equated to the mass times 

 acceleration of the body by Newton's second law, yielding a system 

 of coupled differential equations of motion. These equations are then 

 solved to obtain the time -dependent motion of the structure. 



In the present analysis, a linear relationship will be assumed 

 to exist between all forces and the appropriate motion parameters. 

 Two important consequences follow as a result of such an assumption: 



(1) The hydrodynamic forces acting on the structure may be divided 

 Into two independent parts, one depending only on the incident wave 

 motion, and the second depending only on the platform motion. 



(2) A prediction of the platform response to a realistic random sea- 

 way may be obtained by superimposing the responses to the seaway's 

 regular wave components. 



The validity of such a linearization may be tested either 

 empirically or by comparing its results with results of an "exact" 

 analysis. An exact analysis is normally possible only for such a 

 simplified class of geometries that the validity of the comparison 

 for the realistic case is subject to question. We are, therefore, 

 forced to an experimental test. For the present, we have considerable 

 evidence on the usefulness of linear techniques in predicting ship 

 motions as In Gerrltsma [ I960] , and motions of platforms of the 

 present type, Burke [ 1969] , Paulling and Horton [ 1970] . Some 

 further experimental connparisons are given In the present paper. 



II. THE EQUATIONS OF MOTION 



The motion of the platform will be expressed as a small 

 deviation from a mean position, and for this purpose It Is convenient 

 to define two coordinate systems. The first, OXYZ, Is fixed rela- 

 tive to the structure such that O Is located at the structur's center 

 of gravity, Y Is directed vertically upward, and OXZ Is parallel 

 to the mean waterplane. In many cases, we may take advantage of 

 symnnetry to arrange these axes so that one or more of them Is a 

 principal axis of Inertia. Also, In some cases, a designer's co- 

 ordinate system may be used for drafting or other design purposes, 

 which Is parallel to OXYZ but whose origin Is located elsewhere. 

 Quantities defined in this latter system may always be transformed 

 to the OXYZ system by simple coordinate transforms, and It Is 

 assumed that this Is done. 



A second coordinate system, oxyz , is fixed In space such 

 that It occupies the mean position of OXYZ as the platform moves 

 in waves. In general, it is found most convenient to express the 

 Inertial properties and the forces acting on the structure in OXYZ 

 since the geonnetry of the structure is fixed in this system. On the 

 other hand, it is more convenient to express the equations of linear 

 motion in the space system, oxyz, since this Is an Inertial system, 

 and because we ultimately wish to obtain the motion of the platform 



1086 



