288 



(2) = 



«S ''V 



) dS + 



ju2 + (a)p)2 Tn^ • VH-^ 



3<() 



dS 



(42) 



r 



Vp 



inNu 



(e-5') 



dS' e 



V H, 



(47) 



in which again the first term exhibits the same 

 form as for a submerged body and the second term 

 accounts for the intersection with the free surface. 

 If it is assumed that 3(t> /3z = on z = (rigid 

 wall free surface condition for the steady flow 

 about the hull, i.e., low Froude number approximation), 

 then from (42) 



F, (2) = -p a<; T^ (<fp + *p ) dS (43) 



All of the variables in (47) pertain to the propeller 

 except Kin- ^in ^^ ^^^ amplitude of the fluid 

 velocity potential due to the bare hull travelling 

 backwards with speed U across the water surface 

 and oscillating with unit amplitude in the ith 

 direction and at the frequency nNw. Since the 

 details of the derivation of this formula may be 

 found in the cited literature we will only outline 

 major steps as follows. 



The second term in (45) can be rewritten using 

 the following vector identity 



and the total vertical force, F (D + F., (2) 



'^n ^n ' 

 becomes 



(V^ 



(>„) (Hi • n) = (V<f)j^ • a^) (V3 • n) + 



Vx[(t)n(a^ X Vs)] • n - (}>„ Vx (a^^ x Vg) 



(48) 



= -P 



inNuz a_ + Oo t— 

 " ^ 3z 



dS 



(44) 



As noted earlier, the first term under the integral 

 will dominate because of the large multiplying 

 factor nNo). This will be confirmed in the calculated 

 example to be presented subsequently. First, how- 

 ever, we outline an alternative approach for 

 determining the vibratory hull force which avoids 

 the need to solve for the diffraction potential. 



Only the last term contributes to (45) , because V^ 

 • n = (steady flow hull boundary condition) and, 

 by Stokes' theorem 



Vx [(<{, (aj_ X V )] 



n dS = 



fa X Vg)d£ 



(49) 



where the line integral is taken along the hull 

 waterline on which <i>^ = 0. Consequently, Eq. (45) 

 becomes 



[inNtocti - Vx(ai x V^) ] - n dS 



(50) 



5. AN ALTERNATIVE METHOD FOR DETER.MINING THE 

 VIBRATORY HULL FORCES 



Vorus (1971, 1974, 1976) has developed an alternative 

 procedure for determining the vibratory hull surface 

 forces which eliminates the need to solve for the 

 hull diffraction potential in the presence of the 

 propeller onset flow. The ith oscillatory force 

 or moment, Fj^_,, exerted by the pressure on the 

 hull may be written from (12) and (13) as 



F. = p 

 in 



(inHoj(t>„ + Vg • V(j)j,) n • Uj, dS 



where the a^ are defined as 



(45) 



and, upon introducing the function H^^ which satis- 

 fies 



V^H. =0 in fluid domain 

 in 



H. =0 z = 0, outside S 



(51) 

 (52) 



V H, = n • [inNu) a. - Vx(aiXV )] on S(53) 

 in X J- -= 



V H. -> as X ■+ ", z < 

 in ' 



equation (50) is given by 



F. = p 

 in 



V Ki„ dS 



(54) 



(55) 



ai = 1 

 C2 = J 

 "3 = ^- 



aij = y k - z j 

 05 = z i - X k 



«6 = ^' j - y i 



(46) 



Vorus has shown that the solution for F^^, with no 

 additional approximation, is given by the formula 



•n/No) 



NU) ,. -inNut 



in IT 



dt e 



dS(C,p,6 + a) 



-Tl/NU) 



This form can be identified as one of the terms in 

 Green's theorem applied to the functions 41^ and 

 Hin in the fluid domain bounded by the hull surface 

 S, the free surface z = 0, and the surfaces of the 

 propeller blades Sp , and slipstream, Spj^, which 

 yields 



N 



HO) 



dt e 



-inNiot 



-Ti/Nto 



