Haskind^ has found similar terms in the equations of motion although the constants 

 ■p and q are different. This results from the imposition of another surface condition, namely 

 that = on the surface. For the elongated spheroid p and q equal 1/2. 



A recent paper by Korvin-Kroukovsky,^" using the "source" method for computing ship 

 damping, indicated that a ship with fore-and-aft symmetry and a forward velocity can experi- 

 ence no pitching moment due to heaving velocity. As Havelock suggests, the asymmetry in 

 the flow should result in such a coupling moment. The data appearing in Figures 12 and 13 

 confirm this thesis. 



The present experiments offer an opportunity to check the values of the theoretical 

 coupling terms given by Havelock and Haskind. Havelock calculated the coefficient desig- 

 nated as 3' to be about 1/2. Using the mass of a floating spheroid of length and displacement 

 equal to that of our model, Havelock's expression, e = qMU, has been nondimensionalized and 

 plotted in Figure 13. The magnitude of these frequency-independent curves proportional to 

 the forward speed correspond to the minimum point of the experimental curves. However, the 

 experiments indicate a very strong frequency dependence not indicated by the theory. The 

 steep rise at low frequencies appears to coincide with a velocity-wave celerity ratio of 1/4 

 where the character of the waves generated by the oscillating body is known to change mark- 

 edly. 1 1 



Havelock neglects any pitching moment due to heaving acceleration whereas Haskind 

 finds this term to be zero for a symmetrical model under way. However, our experiments in- 

 dicate that such a moment, represented by the term d'z'in the equations of motion. Equation 

 [lb], does exist. In Figure 12 the coefficient d is shown to be sharply peaked at low fre- 

 quencies, the amplitude of the peak increasing with increasing speed as would be expected. 

 However, at higher frequencies, the coefficient is very small and may be unimportant when 

 compared to inertia moments. 



It is not possible to immediately assess the importance of these coupling terms on the 

 motion of a ship in a seaway. Havelock has computed the alteration in the frequencies of 

 free oscillation due to these terms and found them to be only slightly altered even at high 

 speeds of advance. However, this may not be a sufficient criterion to judge the effect on the 

 motions. The answer awaits a detailed study of the solutions of the coupled equations of 

 motion. Such a study could ideally be performed by analogue computer techniques. 



HASKIND-RIMAN EXPERIMENTS 



Haskind and Riman'' have performed experiments which are similar to those described 

 in this report. A symmetrical model with V-sections was supported on a spring whose upper 

 end was constrained to oscillate harmonically in heave. The damping and added mass of the 

 heaving model at zero speed of advance was computed from the relative motions of the top 

 and bottom of the spring. The damping coefficient was found to be independent of oscil- 

 lation amplitude and its frequency dependence is shown in Figure 16. 



22 



