HYDRODYNAMIC FORCES 



125 



models in conjunction with measurements of the rudder 

 forces. Similar data were reported more recently by 

 Lewis and Eskigian (1954/55) on additional models in 

 which the skeg areas were varied. The foundations of 

 the test procedure also are more completely co\'ered in 

 this latter work. The difficulty in using such data for 

 practical design purposes lies in the fact that the results 

 cannot be generalized readily, while the number and 

 types of models tested are not sufhciently large to cover 

 all practical reciuirements. Such empirical data, ob- 

 tained without an underlying hydrodynamic theory, are 

 particularly difficult to interpret because of the compli- 

 cated relation.ship between the hull's contributions and 

 that of the appendages, such as skegs, rudders, and pro- 

 pellers. 



In the case of lateral motion the principle of compari- 

 son with ellipsoids is identical with that described 

 earlier in connection with motions in the plane of sym- 

 metry; i.e., surging, heaving, and pitching. Davidson 

 and Schiff (194G) used the coefhcients of accession to 

 inertia as given by Lamb for ellipsoids, but the effects of 

 appendages appear to have been neglected. 



An attempt to evaluate the yawing moment caused by 

 obliciue waves was made by Davidson (1948) as a part of 

 an investigation of ship broaching in a following sea. 

 Davidson used an intermediate step between study of the 

 body as a whole and the strip theory. Lateral forces re- 

 sulting from changes in displacement and distribution of 

 the lateral (with respect to ship) components of wave 

 orbital velocities were obtained on a strip basis. The 

 final effect of these on a ship was evaluated by making 

 use of the coefficients obtained on the rotating arm as a 

 part of the previous work of Davidson and Schiff. 



An important contribution to the study of lateral forces 

 by means of the strip theory was made recentl.y by Grim 

 (1953a, 1956). LTsing mathematical definitions of ship 

 sections previously developed by F. AL Lewis (1929), 

 Grim found the solution for inertial and damping forces 

 in motions in lateral directions and in rolling. Land- 

 weber (1957) and Landweber and de Macagno (1957) also 

 presented derivations of added masses and added mo- 

 ments of inertia in lateral motions. The results were 

 given in a simple form, suitable for calculation of forces 

 and moments. These appear to be the only sources of 

 information to date on the basis of which the strip theory 

 can be applied to lateral motion. As yet no experimental 

 verification has been made. 



In the work just cited the effects of the free surface 

 were taken into account for two asymptotic cases of very 

 low and very high freciuencies. It has not yet been pos- 

 sible to evaluate the hydrodynamic forces in lateral mo- 

 tion and in rolling for the intermediate range of fre- 

 quencies. Grim (1956) considered that the deri\'ation 

 valid for the low frequency is directly applicable to ship 

 motions among waves. Wendel (1950) and Landweber, 

 on the other hand, gave added-mass data for high fre- 

 quencies, and their data may be considered as directly 

 applicable to the study of ship vibrations. 



Further research, directed to evaluation of added 



masses and damping in lateral and rolling motions at 

 all frequencies, is needed. 



Although Grim's work goes a long way toward supply- 

 ing the desired information for normal sections, it still 

 is necessary to obtain information on the effect of ap- 

 pendages. It should be noted that hydrodynamic solu- 

 tions have been obtained successfully in two extreme 

 cases: (a) A ship in heaving and pitching, where the 

 hull can be represented by sfiurces and sinks or doublets; 

 and (b) an airplane wing described by a distriliution of 

 vortices. No satisfactory approach has been de\'eloped 

 for a mixed condition, such as a ship with appendages in 

 lateral motion. Fedyaevsky and Sobolev (1957) treated 

 a ship in lateral motion by means of a low aspect-ratio 

 wing theory. 



The added masses of typical afterbody ship sections 

 with large deadwood or skegs can be estimated by replac- 

 ing a true section with a polygonal one and applying a 

 Schwartz-Christoffel transformation. Following Grim 

 (1956), the results of such an analysis may be assumed to 

 be valid for low frecjuencies. Alternately, the added 

 masses of ship sections formed of curved lines in conjunc- 

 tion with deadwood, skegs, or propeller bossings can be 

 obtained by electrical analogy (Section 3.11). I'nfor- 

 tunately, no method of evaluating the damping proper- 

 ties of such sections appears to be available. Theoretical 

 and experimental research in this field is needed. 



Well-separated appendages, such as spade-type rud- 

 ders, or short wing-like skegs (with leading edges) often 

 used to support rudders on multiscrew naval ships, can 

 be treated as separate hydrofoils, with an empirical fac- 

 tor for increase of effective span due to abutment against 

 the hull. Also, an assessment must be made of local 

 water velocities. A certain amount of information on 

 such appendages will be found in Mandel (1953) and 

 Becker and Brock (1958). 



5 Forces and Moments in Rolling 



The general introduction given in Section 1.1 with re- 

 gard to pitching and heaving applies to rolling as well. 

 In the simple e(|uation of motion (1), the heaving dis- 

 placement z is merely replaced by the angle of roll, desig- 

 nated by (f), so that it becomes 



A(f + B(j) + C(t) = L cos o)t 



(■25) 



The forces and moments are caused by changes in 

 water pressure, in tiu'u caused by the angle of roll 0, ve- 

 locity of roll (/), and acceleration in roll (j>. The resultant 

 moments are correspondingly classified as restoring, 

 damping, and inertial ones, the latter usually expressed 

 in terms of the added or hydrodynamic moment of iner- 

 tia coefficient fcu. The amplitude of the rolling moment 

 is here designated by L. As in the case of pitching, all 

 forces and moments involved in rolling also can be con- 

 sidered as the sum of forces and moments caused by ship 

 rolling in smooth water and those exerted by waves on a 

 restrained ship. Direct data on the forces and moments 

 in rolling are extremely scarce. Most of the practical 



