289 



^ - ^ I dS H- 



3np 3nj 



V'"P 



V I-I, dS 

 in 



(56) 



where 



fip"*" - (|>p~ is the jump in the propeller potential 

 across the blade and slipstream surfaces. The two 

 terms in (56) can be identified as the contributions 

 from blade loading and thickness, and with further 

 manipulation can be brought into the form of (47) . 



Equation (47) indicates that the velocity corres- 

 ponding to the potential Hj_n is evaluated over the 

 propeller blades and slipstream. The propeller 

 representation by distributions of dipoles directed 

 normal and tangential to the blade pitch surface is 

 the same as previously discussed. In the formula, 

 the velocity induced by the bare hull, VHj^j^, is 

 resolved into components in the directions of the 

 dipoles, multiplied by the dipole strengths, and 

 the products integrated over the blade and slipstream 

 surfaces. The first integral in (47), in time, 

 extracts the nth Fourier harmonic. Both the blade 

 position and the dipole strengths are functions of 

 time. 



In the case of vertical force analyses, an 

 approximation to the improper integral in (47) has 

 been found to yield acceptable results. Let I be 

 defined as 



I = 



inNoj 

 U 



(?-5': 



VHi„ dS- 



(57) 



If the oscillating exponential varies more rapidly 

 than VHj^jj, then the argument of the exponential can 

 be considered as "large" and I can be expanded in 

 an asymptotic series. '^^in should vary relatively 

 slowly aft in the propeller slipstream for vertical 

 oscillation of the bare hull and an asymptotic 

 evaluation should therefore be valid. (Such a 

 treatment may not apply to an athwartship analysis, 

 for example, where a rudder is involved in the bare 

 hull oscillation. ) To proceed with the asymptotic 

 representation, (57) is integrated by parts yielding 



inNu) 



U 



(?-?') 



inMci) 



inNo) 



(C-5') 



inlJu 



V K, 



3 ^ 



V H. dC 

 m 



For the conditions stated, the integral term is 

 higher order. Hence, to one term. 



I ~ 



-r^ np • V H. (5) 

 inllo) i^ in 



(58) 



and (47) reduces to 

 IT /No) 





dt e 



■inNut 



-TT/NCjJ 



dS [p'VTn^ 



inNo) P in 



(59) 



in which the induced flow is evaluated exclusively 

 on the surface of all N propeller blades SpjT. 



6. 



COMPARISON OF THEORY AND EXPERIMENT FOR A BODY 

 OF REVOLUTION 



An experiment was conducted to measure the periodic 

 forces on a body of revolution adjacent to a propel- 

 ler loading provided a configuration which could 

 be treated in a reasonably exact fashion by potential 

 flow theory. As such, the experiment was intended 

 as a fundamental check on the theory and computer- 

 aided numerical procedures. However, it is believed 

 that the experimental technique can be extended in 

 the future to study more general hull geometries 

 and the effects of unsteady propeller loading and 

 transient cavitation. 



In the following sections, the experimental 

 apparatus and procedures are described and the 

 force measurements are compared with the analytical 

 predictions. 



Test Body and Propellers 



The experiments were performed in the DTNSRDC Deep- 

 Water Basin [(22 feet (6.7 m) deep, 51 feet (15.5 m) 

 wide, and 2600 feet (792 m) long)]. Both the body 

 and propeller were supported and towed from Carriage 

 II which has a drive system capable of maintaining 

 speed to within 0.01 knot. 



Forces were measured on the forward half of an 

 ellipsoid of revolution with a length/diameter 

 ratio of 5.65. This "half body" was mounted by a 

 specially designed strain-gaged flexure assembly 

 to the forward end of a massive streamlined after- 

 body, attached to the towing carriage by a single 

 strut. The propeller was driven by the DTNSRDC 

 35-horse-power dynamometer, separately supported 

 from the towing carriage and positioned so that 

 both the propeller shaft and body axes were aligned 

 parallel to the direction of flow as illustrated 

 in Figure 5. 



The half body consisted of a 0.25 inch (0.64 cm) 

 thick fiberglass shell measuring 36.0 inches (0.91 

 ra) in length and 12.75 inches (0.324 m) in maximum 

 diameter. The shell was filled with polyester foam 

 in order to minimize the mass and obtain a high 

 natural frequency, sufficiently above the propeller 

 blade rate frequency range to reduce nonlinear 

 resonance effects. The aluminum, free-flooded 

 afterbody, together with its support strut had a 

 low natural frequency to prevent mechanical vibra- 

 tions from the propeller dynamometer gears and 

 shafts passing through to the body force dynamometer. 

 The towing strut was attached to a large frame , 

 mounted on the propeller dynamometer structure. 

 Slotted pads supporting the frame permitted trans- 

 verse and longitudinal adjustment of the body 

 location and orientation. Vibration isolating 

 mounts were placed in the framework to further in- 

 hibit "pass through" vibrations. 



Vibratory forces were measured for two propellers. 

 DTNSRDC propeller 4118 is a 3-bladed, 12-inch (0.305 

 m) diameter aluminum propeller designed for uniform 

 flow. Propeller 4119 is identical to 4118, except 



