no 



THEORY OF SEAKEEPING 



k-a^ 





Fig. 2 Transverse sections and inertia coefficients obtained by Lewis (from W'endel, 1950) 



r = s' + ^'7 = '•£■" 



where Q is the angle which the radiu.s vector makes with 

 the water level. The transformed figure is described by 

 F. M. Lewis as 



•r + «y = S + " + z. 



ax 



-id I °^ p-3.-9 



where a\ and as are real coefficients. After separat 

 real and imaginary parts, the co-ordinates x and 

 given by parametric equations 



xjr = (1 + fli) cos Q + 03 cos 3^ 

 yjr = (1 — OiJ sin & — Oa sin Zd 



The half-beam b and the draft d of the resultant 

 are related to the coefficients Oi and Oz by 



h/r 



1 + -; + 



03 



(12) 



ion of 

 ij are 



(13) 

 figure 



(14) 



= 1 - 



tti 



+ 



03 

 ,.4 



The section shapes resulting from this transformation 

 are shown on Fig. 2. Each separate set of curves corre- 

 sponds to a different h/d ratio (labelled on the figure as 

 a/h). The curves are labelled by the values of the added 

 mass coefficient C (defined later). The validity of this 

 transformation is limited to the range of parameter 

 values shown in Table 1. Fig. 3 shows the values of the 



Table 1 



(From Jjaiuhveber and de Maeagno, 1957) 



d/b 

 0.6 

 8 

 10 

 1,4 

 IS 

 2.5 

 50 



* A is sectional area. 



/J = A/2hd* 

 412-0.93 

 353-0 942 

 294-0 957 

 379-0 937 

 425-0.925 

 471-0 914 

 530-0.898 



added mass coefficient C of Lewis forms as a function of 

 the section coefficient /3 and the draft beam ratio. 



F. M. Lewis e.xpressed the added mass in terms of 

 the water mass enclo.sed within a semi-circular contoiu' of 

 radius b as the coefficient 



C, = 



added mass 

 ■Kph-/2 



(15) 



For the ship sections obtained from the foregoing 



transformation the coefficient is e\'aluated as 



C„ 



(1 + a,)- + 303 



(16) 



(1 + oi + a^Y 



The computed values of the coefficient C , are shown in 

 Fig. 3. The coefficient (', is related to A,, the coefficient 

 of accession to inertia bv 



k, 



C.tB' 8.4 



(17) 



where B is the sectional beam, and A is the sectional area 

 A',, = C,, for a semi-circular profile. 



