Simple equations for heave and pitch are 



d*z dz 



O + m 33 ) h N33 — + pgA w z = P gr m A w E zz e i{at -^\ (1) 



dP dt 



dty # 



(Jy + m 55 ) — + N 55 — + P gDM L G+ = pg&JDMLGEs&w-'J. (2) 



dP dt 



"Relative motion" in heave is considered by the following expression: 



d*z d* d 



m h w 3 3 — (2 - r) + N Z3 — - r) + P gA w z = pgr^^^e^ 1 '^ (3) 



tft* dP dt 



where 



d 2 r dr 



— = - w 2 r m e iut ; — = ico r m e iu «. 

 ^2 ^ 



Known hydrostatic coupling terms were neglected for simplicity, although more recent 

 research in hydrodynamics has led to the conclusion, that this involves an appreciable 

 loss of accuracy [4]. It was realized comparatively early that nonlinearities of the 

 restoring moment in roll must be considered, although the application of this finding 

 to forced motions was slow. Special phenomena later required more attention to 

 coupling terms and finally rheo-linear effects were treated. 



The computation of damping forces by hydrodynamic means was initiated by 

 Schuler [12] and Holstein [13], and put on safe ground by Havelock [14]. 



Although quite a bit had been achieved in this way there was an uneasy feeling 

 about the applicability of the formal apparatus to concrete design problems, stemming 

 from the distrust in the Froude-Krylov hypothesis underlying the computation of the 

 exciting forces. 



It turned out that the wholly submerged body was a mechanical model easier 

 to handle than the surface ships. The classical theory of solids moving in a liquid 

 developed mainly by Kirchhoff and Kelvin was rediscovered for our purposes after 

 it had been considered (by our profession) as a kind of mathematical exercise only. 

 Progress was essentially due to the application of the concept of hydrodynamic singu- 

 larities. Lagally's theorem and its generalization by Cummins [15] were important 

 steps in this direction. Reference is made, for example, to the work by Haskind [16] 

 who asserted, not quite justly, that he had found the solution of the hydrodynamic 

 problem neglected by Krylov. 



In the present writer's opinion the procedure of building up equations of motion 

 by intuitive synthesis will continue, notwithstanding its obvious shortcoming, i.e., the 

 need to add new terms in the pertinent equations when dealing with special applications, 

 such as ship stabilization [17]. Nonetheless, there exists an encouraging analogy with 

 flight mechanics, the theory of which has reached a comparatively very satisfactory state 

 by much the same path. It should not be overlooked that the directional stability of 

 ships has been successfully attacked only after methods borrowed from aerodynamics 

 have been applied. 



The use of hydrodynamic singularities is one method of attacking the boundary 

 value problem involved. Since, however, one ordinarily uses a first approximation only 

 when substituting effects due to singularities for effects caused by bodies, it makes sense 



65 



