1=1 



Figure 6 - Plot of the Heave Phase Lag for the Damped 

 Circular Cylinder with R/H =0.1 



harmonic oscillators with linear damping; for low frequencies the heave 

 displacement and wave height are in phase, at resonance they are in quad- 

 rature, and at high frequencies they are 180 deg out of phase. 



DISCUSSION AND CONCLUSIONS 



The damped equations of motion as given by Equations [48] to [50] 

 may be solved for an arbitrary body of revolution to obtain the oscillation 

 amplitudes and phases. Except in the vicinity of the resonance frequencies 

 defined by Equations [37] and [38], it should be sufficient to use the sim- 

 pler undamped equations; the resulting oscillations are given by Equations 

 [34] to [36]. Plots of these oscillations are shown in Figures 2 to 6 for 

 a circular cylinder, with the important restriction that the centers of buoy- 

 ancy and gravity coincide. If this restriction is relaxed, a resonance will 

 be introduced into the equations for pitch and surge, but the frequency of 

 this resonance may be kept small by ballasting. The annplitudes at reso- 

 nance are extreme, but the resonance frequency for heave is quite snnall 

 and can be kept out of the practical range of ocean waves by making the 



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