DISCUSSION 



Equations of motion have no-w been obtained, based on the assumption 

 that wave annplitude, body motions, and spar radii are small quantities. If 

 damping is neglected, there are two resonance frequencies (as given by- 

 Equations [I7a] and [I7b]) at which infinite amplitudes of motion will oc- 

 cur. Of course, there does exist damiping which prevents the responses 

 from actually being infinite. That part of the damping due to generation of 

 outgoing waves is small (of second order in terms of spar radii) and so 

 is of importance only near resonance. Generally there is also a viscous 

 damping which nnay be of importance throughout the interesting range of 

 frequencies, or possibly only near resonance, or perhaps not at all. This 

 damping is associated only with axial velocities of the spars, but its ef- 

 fects appear in each of the modes of motion for which there is a resonance. 

 The relative importance of viscous danaping can be determined in individ- 

 ual cases only by actually carrying out solutions of the equations of motion. 



If it is desired to minimize the motions of the structure over a range 

 of practical wave lengths, the nnost effective procedure is to attempt to re- 

 nnove the natural frequencies fronn the desired range. If this is not possi- 

 ble, the natural frequencies should be chosen as frequencies for which the 

 incident wave annplitudes are snnallest. In practice, this will generally be 

 equivalent to reducing the natural frequencies as low as possible. 



From Equation [ 17a] we see that the heave natural frequency is 

 proportional to the radius of the spars at the undisturbed waterline. (S(0) 

 is the cross- sectional area at the equilibrium waterline.) Assuming that 

 the total mass, Mq + NM, is approximately fixed, we have no other param- 

 eter to adjust; thus we would mininnize S(0) as far as possible. 



In the case of the other resonance frequency, we see from Equation 

 [I7b] that, apparently, there are several parameters available for adjust- 

 ment. We note that if the center of buoyancy and center of gravity coin- 

 cide, then 



PSj-MqZq- m(z)zd: 



•^ T 







29 



