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blade geometry and loading characteristics. Con- 

 sequently, researchers are attempting to develop 

 theories and numerical procedures for calculating 

 propeller exciting forces. An analytical approach 

 offers a means to economically evaluate competing 

 propeller-hull design concepts as well as to diagnose 

 at-sea vibration problems and identify corrective 

 measures. 



The present paper concerns recent advances in 

 the theory for propeller induced surface forces. 

 A general three-dimensional boundary intercepting 

 the propeller disturbance field poses a formidable 

 diffraction problem. As a first step, it is 

 necessary to determine both the time-average and 

 unsteady loading on the propeller. All of the 

 components of loading, together with blade thickness, 

 contribute to the propeller induced flow impinging 

 on the hull and the resultant unsteady pressure. 

 Fortunately, as a result of much past work in the 

 analytical prediction of bearing forces , there now 

 exist powerful theoretical methods for calculating 

 unsteady propeller loading in a prescribed nonuniform 

 flow. The analysis rests on a lifting-surface 

 representation of the propeller, explicitly account- 

 ing for number of blades, radial and chordwise 

 distribution of loading, thickness, and skew. While 

 further refinements and improvements, such as the 

 prediction of transient blade surface cavitation, 

 are needed, the calculation of blade loading can 

 now be done with sufficient accuracy to address the 

 surface force analysis. Also, as these improvements 

 in the propeller calculation become available, they 

 can be incorporated into the surface force calcula- 

 tion without fundamental changes. 



Previous analyses of the surface forces are 

 formulated in terms of the diffracted potential 

 flow about the solid boundary in the presence of 

 a given propeller onset flow. To facilitate the 

 analysis, it was necessary to introduce simplified 

 representations of both the propeller and the 

 boundary as outlined by Breslin (1962) and more 

 recently, Vorus (1974) . For example, analytical 

 expressions for the vibratory force produced on a 

 long flat strip and a circular cylinder adjacent 

 a propeller in uniform flow were derived some years 

 ago [Tsakonas et al. (1962) and Breslin (1962)]. 

 These investigations provided useful insights regard- 

 ing the importance of propeller tip clearance and niom- 

 ber of blades. However, such approximate treatments 

 neglect what are now known to be certain essential 

 physics of the propeller-hull interaction. The net 

 force on a long boundary may be deceptively small 

 because of cancellation of large out-of-phase force 

 components developed fore and aft of the propeller. 

 On a hull which terminated in the immediate vicinity 

 of the propeller, such cancellation will not occur. 

 Also, the components of unsteady blade loading at 

 or near blade frequency can produce much larger 

 surface forces than those arising from the steady 

 loading and thickness. Components of blade loading 

 at higher frequencies, while relatively smaller in 

 amplitude, generate field pressures which decay 

 much more slowly, encompassing a large portion of 

 the hull afterbody and resulting in a significant 

 integrated force. For this same reason, an experi- 

 mental determination of the total surface force by 

 measurement of pressures at selected positions on 

 the hull boundary can be disastrously misleading. 

 In view of these circumstances, it is now generally 

 accepted that a satisfactory theory must represent 

 the hull boundary in a reasonably exact fashion. 



accommodate the presence of the free surface, and 

 account for all constituents of propeller loading. 



This paper sets forth a comprehensive theory 

 for propeller-hull interaction and describes proce- 

 dures for calculating the periodic forces acting 

 on the hull surface. The paper is divided into 

 five sections. In the first section, the problem 

 for the diffracted potential flow about the hull 

 is formulated, in which the propeller unsteady 

 disturbance is assumed to be of small amplitude 

 and high frequency. In keeping with the desire for 

 first order results, the high frequency linearized 

 free surface conditon applies. However, the zero 

 normal velocity condition is satisfied exactly at 

 the hull boundary. Formulae for the surface pres- 

 sures and forces may then be expressed in terms of 

 the propeller velocity potential and the unknown 

 diffraction potential. The following section deals 

 with the representation of the propeller. Dipole 

 singularities with strenths related to the blade 

 pressure loading and thickness are distributed over 

 helicoidal surfaces approximating the geometry of 

 the actual blade surfaces. Based on this model, 

 expressions for the field point velocity potential 

 arising from loading and thickness are developed. 

 Examination of these formulae and their asymptotic 

 behavior at large distances reveals important prop- 

 agation characteristics associated with the unsteady 

 blade loading components at and near blade frequency. 



In the subsequent sections, two methods of solu- 

 tion are developed for determining the surface 

 forces. The direct approach consists of distributing 

 time-dependent source singularities over the hull 

 surface with the source strenths determined for a 

 prescribed propeller onset flow using a modified 

 Douglas-Neumann calculation [Hess and Smith (1964) ] . 

 The force on the body is then found by applying 

 the extended Lagally theorem to the hull singulari- 

 ties. In an alternative approach, based on a 

 special application of Green's theorem, the force 

 is obtained by finding the velocity potential at 

 the propeller produced by the hull boundary executing 

 simple oscillatory motion. 



In the final section, a towing tank experiment 

 is described in which blade frequency forces were 

 measured on a body of revolution adjacent to a 

 propeller operating in uniform flow. The simplifi- 

 cations of body shape and propeller loading provided 

 a physical model which could be treated in a reason- 

 ably exact fashion by the theory. Despite these 

 simplifications, certain classical problems were 

 encountered in the design of the experiment including 

 the measurement of a relatively small force, avoid- 

 ance of system resonances in the frequency range of 

 interest, and retrieval of the force signal from 

 background noise. A two-part body design was 

 developed, similar in concept to the separated 

 stern technique mentioned earlier. A heavy after- 

 body attached to the towing strut, behaved as a 

 seismic mass at all but very low frequencies. 

 Forces were measured on a light rigid forebody, 

 supported from the afterbody by a specially designed 

 and dynamically calibrated straingaged flexure 

 assembly. 



Tests were performed with two propellers differing 

 only in blade thickness in order to reveal the 

 separate contributions of loading and thickness. 

 The measured forces (amplitude and phase) were 

 obtained for a range of speeds and advance coeffi- 

 cients and for two positions of the propeller 

 relative to the test body. The results agree 



