134 



THEORY OF SEAKEEPING 



Fig. 28 Variation of correction coefficient for damping in 

 heave in short vt^aves, kr(/\) with ratio of draft /wave length 

 with section coefficient, fJ, as parameter (from Haskind, 1946) 



motor-driven scotch yoke and a spring. The theory of 

 the test and the method of computing added masses and 

 damping forces were described clearly by Haskind and 

 Riman (194(j). The measured dependence of these on 

 the circular frec[uency u are shown in Fig. 25. Here m" 

 denotes the added mass and b the coefficient of damping 

 force, corresponding to b in etiuation (1). Comparison 

 of the calculated and experimental values is given in 

 Figs. 26 and 27. Here the ordinate m"g/A is the coeffi- 

 cient of accession to inertia, where A is the displacement. 

 The ordinate in Fig. 27 is the ratio of the damping b at a 

 gi\'en frecjuency co to the asymptotic \'alue S for \'erv long 

 waves; i.e., small values of w, which Haskind defined 

 theoretically by the equation 



- 9 



(53) 



where Au, is the area of the waterplane. The experiments 

 were made at zero forward velocity. 



Superficially, the curves in Figs. 26 and 27 show agree- 

 ment. The general trend of \'ariation over a wide range 

 of frequencies oj is well represented. This wide range, 

 however, is of fittle practical interest. For a ship of the 

 Mariner type in head seas the parameter ui-L/'lirg will 

 range from 0.6 to 2.5, with the value of 1.8 approximately 

 corresponding to the synchronous condition. At this 

 value of abscissa the calculated damping coefficient ap- 



pears to be some 40 per cent abo\-e the experimentally 

 measured one. The error in the added mass appears to 

 be approximately 80 per cent. 



6.4 Computations of Haskind. While the formula- 

 tions of Haskind's (lU-16) theory are applicable to any 

 form of oscillatory ship motion, the actual application 

 was limited to coupled pitching and heaving. Haskind 

 applied his method to a ship with lines defined by equa- 

 tion (50). The Z(z)-function is defined in this case as 



Z{z) = 1 



— z 



(54) 



where ^ is the depth l)elow LWL antl d is the draft. The 

 index n is related to the section coefficient P by 



(55) 



n + 1 



The forms of ship sections corresponding to various 

 values of /3 are shown in the insert in Fig. 28. (In par- 

 ticular, n = 1, /3 = 0.5 represents a straight-sided V-sec- 

 tion; n = 0.5, |8 = 0.33 represents a concave-sided V.) 



Attention should be called to the fact that use of equa- 

 tion (50) gives ship lines with the same \'alue of the index 

 n, or section coefficient /?, throughout the ship's length. 

 The insert in Fig. 28 should not be confused, therefore, 

 with a conventional body plan. 



Haskind (1946) expressed damping in a pure heaving 

 motion at zero speed in the form 



"'"-'^ly-^ 



(66) 



where b is the damping due to very long waves given 

 by equation (53). K2'^{d/\) is the correction coefficient 

 for the wa^-e length as compared with the draft d of a 

 ship; values of it are gi\'en in Fig. 28. The L/\ ratios 

 most important in ship operation in head seas are from 

 0.75 to 1.5. It is suggested that the reader concentrate 

 his attention on the corresponding range of d/\ of 0.05 

 to 0.08. The rapid increase of damping with decrease 

 of section coefficient shown in Fig. 28 is notewortlty. 



The correction coefficient k,{L/\) for wave length as 

 compared with ship length depends on the form of the 

 waterlines. Haskind made computations for water lines 

 defined by the eciuations 



X{.x) = 1 for < .1- < yl 



X{x) 



1 



1 



1 - 



for yl ^ X ^ I, (57) 



where / is the half length L/2, and y is the proportion of 

 parallel section. The coefficients y and m define the full- 

 ness coefficient a of the waterline 



m 



1 



(58) 



1 + m 1 + 7'" 



The values of the coefficients kj(L/\) for the case 

 7 = are repi'oduced in Fig. 29. Again it should be 

 emphasized that the range of L/'\ \-alues is much wider 

 than is needed, since the practical range of L/\ is within 

 the limits 0.5 to 2.0. 



