342 



THEORY OF SEAKEEPING 





1 1 



i 2 3 4 



Forward Speed, V, Ft/Sec. CAgainst the Waves) 



Fig. 15 Comparison of computed and experimentally measured 



exciting force and moment amplitudes for ETT Model 1445 in 



waves 5 ft by 1.5 in. 



perimental data described in Appendix 2 of the 1955 

 paper. A very good agreement between calculated and 

 experimental values was obtained except at zero and very 

 low forward speeds, Fig. 15. It has been obser\-ed in re- 

 cent years that at such speeds there is often interference 

 from the waves reflected by the walls of the towing tank, 

 while at greater speeds this interference is no longer en- 

 countered and the experimental data become more re- 

 lialile. 



Forces Due to Body Motions 



The pressure in tlie fluid due to the bod.v's own motion 

 is, from equations (11) and (13) 



d06,„ . r2 2pvrr 

 Vbm = P -^ = - pz^ - cos a — cos a 



= — ffi'T COS a 



pvr COS a 



(28) 



at the bodj' surface where R = r. The distribution of 

 vertical forces is obtained by substituting equation (28) in 

 (10) 



'dF\ -K „. 



— ) — — p -rh) — pTvrrv 

 \dx Jim - 



(29) 



The apparent vertical velocity ;' of the center of the 

 circular section, given by equation (8), consists of three 

 parts 



V = z + ^0 - ve 



where the first is the heaving velocity, the second is the 



vertical velocity due to the angular velocity of pitching, 

 and the third is the vertical velocity due to the instan- 

 taneous angle of trim 6 at the control planes. Since ^ is 

 a function of time and f = — V 



V = z + ^d - 2]' 



(30) 



After substituting equations (8), (21) and (30) in equa- 

 tion (29), the following set of six terms is obtained 



<iF 



ll-T Jhm 



+ 2 



P 7, '- 



IT ., 

 P - 1- 



(i) 



(ii) 



(31) 



Equation (31) replaces equation (39) of the 1955 paper. 

 Terms (i) and (ii) are identical with (3) and (6) of equa- 

 tions (39) and term (iii) is the sum of the earlier (1) and 

 (5) with the sign of the latter corrected. Terms (iv), (v), 

 and (vi) are twice terms (4), (7), and (2), respectively, 

 of the earlier equation (39), which were incorrect because 

 substitution oi R = r had been made before differentia- 

 tion with respect to time. 



The factor (pTrr-/2) in the first three terms of equation 

 (31) is evidently the virtual mass of an element of body 

 length, and by introducing kt and ki on the basis of the 

 reasoning outlined in connection with exciting forces, it 

 can be expressed as (pSk-zkA). The factor (p7rr tan 13) in 

 the last three terms is the derivative with respect to ^ of 

 (pxr^/2) and so it can be expressed as d(pSk2ki)/d^. The 

 total force due to the body's own motion is then 



F,„ = -pf SkMz + ke - ■2Ve)d^ 



+ Vpj '^-^ a + ^9- id - Vd)di (32) 



and the moment is 



M,m = -pf SkMi + ^0 - '2Ve)^, dt 



+ 1 



. r dSkr. 



'''' a + te - ve)d^ (33) 



where the integration is carried over the length of the 

 hull. 



Since Ft„ and jl/i,„ are functions of z and 6 and their 

 derivatives, the terms of equations (32) and (33) may be 

 transposed to the left-hand side of the coupled equations 

 of motion. Their contributions to the coefficients of these 

 equations are designated by the subscript 1. Thus 



