Paulling 



IV. SOLUTION OF THE EQUATIONS OF MOTION 



The total system of forces described In the preceeding 

 sections are now introduced into the equations of motion, Eq, (6), 

 Our linearization of the problem has resulted in the subdivision of 

 these forces into two categories: those forces resulting from the 

 wave motion in the presence of the stationary structure, and those 

 resulting from the motion of the structure in a stationary fluid. The 

 former category contains the first term and the first parts of the 

 second and third term on the RHS of Eq, (7). The platform, motion 

 dependent forces are contained in the second parts of terms two and 

 three of the RHS of Eq, (7), plus the hydrostatic and restraint forces. 

 The wave motion depending forces are seen, as a result of the velocity 

 potential assumed to represent the wave motion, Eq, (8), to be sinu- 

 soidal functions of time. If we rearrange the equations of motion into 

 the standard form, placing the motion- dependent terms on the left-hand 

 side and the time dependent forcing terms on the right-hand side the 

 result is a set of six simultaneous second-order differential equations 

 of the form 



6 



2^ [(mjj +ai^)Xj +bijXj +CijXj] = Foi sin(wt +Ci), (25) 



j = l 



where the exciting force amplitude, F^j , is proportional to the wave 

 amplitude, a. The solution of these equations may be expressed in 

 the form 



Xj = Xq. sin (cot + 6j), 



where x^j is proportional to Fqj , therefore to the wave amplitude. 

 The quantity Xpj/a, or the amplitude of response to unit waves 

 varies with wave frequency, since the exciting force FqI is a function 

 of frequency and because the coefficients a, b, c the LHS of (25) 

 may also vary with frequency. The square of this unit response is 

 then the response amplitude operator, which nnay be combined with 

 the wave spectral density function to obtain the platform response to 

 a random seaway. 



V. MODEL EXPERIMENTS 



A number of model experiments have been conducted in the 

 University of California Towing Tank in order to test several parts 

 of the procedures described in the previous sections. The initial 

 objective of the study was to evaluate the tension leg platform, and 

 all experiments deal with this configuration. Initial experiments 

 were made on single cylinder members to test some of the hydro- 

 dynamic force predictions and the linearity of the resultant motions 

 In regular waves. Next, experiments were conducted in regular 



1100 



