P s = -2pgl Ze la)t (36) 



T_ = 



_P8V [(7^f) " (5 o-^]^ la,t (37) 



The linearized forces and moment about the center of gravity due to the ith 

 mooring line are given by ignoring the inertia of the mooring line and the 

 viscous forces on the line. The total mooring forces and moment on the body 

 are given as the summation of the effects from all mooring lines as 



n 



I F x (i) = (-k xx X - k xz Z + k xQ e) e iut (38) 



i=l 

 n 

 I F ,(i> = (" k ,v X " k„Z + k fl 9) e iwt (39) 



zx zz zl 



1=1 



I M Q (i) = (k Qx X + k Qz Z - k eQ 9) e ia)t (40) 



1=1 



where k xx , k^, etc. are linear spring constants obtained by considering 

 elemental statical mechanics among the displacement of the mooring line 

 cylinder, the elongation, and the change of angle of the mooring line. Sub- 

 stitution of equations (23), (24), (25), and (29) to (40) into equations (26), 

 (27), and (28) yields a system of linear equations with respect to the same 

 number of unknown quantities. This system of equations may be solved simul- 

 taneously for all the unknowns by numerical techniques. 



Although only simple cases are demonstrated, researchers indicate that 

 the numerical method is particularly useful for investigating the response of, 

 and the wave transformation by floating objects having asymmetrical cross- 

 sectional shapes, even in a water region with irregular bottom boundary 

 configurations. In addition, a great advantage of this method is that the 

 more general, yet rather complicated, problems such as multiple floating 

 objects can be easily analyzed with only a little modification of the flow and 

 boundary conditions. 



To verify this numerical technique, Yamamoto, Yoshida, and Ijima (1980) 

 performed large two-dimensional model tests at flume dimensions of 12 feet 

 wide, 15 feet deep, and 342 feet long. The experimental results are compared 

 with the numerically calculated results for the transmission coefficient, C f , 

 for a circular cylinder (Fig. 17) and a rectangular cylinder (Fig. 18). 

 Excellent agreement between the theory and experiments for the circular 

 cylinder was found for all the transmission coefficients and body motions. 

 The agreements for the rectangular cylinder were also excellent except at 

 resonant frequencies of the floating body. Theoretically, the cross-spring 

 moored rectangular cylinder transmitted no waves associated with the resonance 

 in heave motion at W/L =0.21. At the resonant frequency, the measured wave 

 transmission dropped to 0.40 only, not zero. The reason for this was that the 

 viscous damping due to the flow separation at the sharp corners appeared to 

 dampen the heaving motion of the rectangular cylinder. These comparisons 

 indicate that the theory represents the real flow problem as long as the flow 

 separation is small at the resonant frequencies of the body motion. 



47 



