286 



As an order of magnitude formula, one might use 

 (30) for f = " with the correction for the free 

 surface included. This reduces to the complex 

 amplitude 



1.0 



(N) 



N 



Sirp'n^RQ^ 



pAp Jin 

 n 



0.2 



(b+x/s^+z 2+p2+i32) 



, (b+ \Js2+Zg2+p2+b2+4ah) 



s^+z 2+p2+4dh 







s2+z^2+p2 



dp 



(31) 



(which must not be used for hull drafts in way of 

 the propeller, h, which are small, as clearly Zip ^ 

 ->■ as h ->- 0) . In practice, Apjj = afj(p) cos NO + 

 b{j(p) sin N9 , aj^, b^j being the chordwise average in- 

 phase and quadrature blade pressures given by the 

 unsteady lifting surface calculation. 



With the foregoing considerations of the propeller 

 in mind, we now return to the surface force problem 

 for a general three-dimensional hull boundary and 

 prescribed propeller onset flow. In the following 

 section, a procedure is described for determining 

 the diffraction potential and the surface pressures 

 and forces in terms of singularities distributed 

 over the surface of the hull. 



4. A DIRECT APPROACH FOR DETERMINING SURFACE FORCES 



A "frontal attack" on the problem of predicting the 

 vibration forces generated on an arbitrary hull by 

 the induced flow of the propeller, (and its free sur- 

 face image) is to construct the potential of the hull 

 in the presence of these onset flows. This procedure 

 was first applied by Breslin and Eng (1965) to a 

 realistic hull form. At that time, however, only 

 the mean loading and the blade thickness were 

 accounted for in the flow impinging on the hull and 

 the computer time was observed to be excessive. In 

 contrast to these earlier efforts, the propeller 

 flow is now composed of all constituents of loading 

 and the (high frequency) images arising from the 

 presence of the free surface. 



A solution for the potential, (f^, which satisfies 

 equations (3) , (4) , and (7) , is constructed by 

 distributing source singularities, a^ (x)e^"'^"^, 

 over the surface of the hull, such that 



(X) 



4ir 



a„(x'; 



x-x' 



x-x! 



dS(x') 



(32) 



where the region of integration is over the submerged 

 portion of the hull and Xj^ is the distance from an 



FIGURE 4. Approximate moduli of B-F forces on 

 barge-like ship from pressures emanating from 

 mean and B-F loadings on a 5 bladed propeller 

 (in a single screw ship wake) as a function of 

 integration length forward of propeller. 



ASYMPTOTIC VALUE 

 FOR f— ■ <D V 



B-F FORCE COEFFICIENTZ. ARISING 



I I I I I I I I I I I I I I I I 



5 10 15 20 25 30 



INTEGRATION LENGTH, f, FORWARD OF PROPELLER IN RADII 



