Vassilopoulos and Mandel 



3. Rudders and other control surfaces and mechanisms are assumed 

 "locked" in zero position. 



4. In deriving the equations of motion for pitch and heave the coupling be- 

 tween the latter two and the other degrees of freedom is totally neglected. For 

 seas from directly ahead or astern, this is a reasonably valid assumption. In 

 particular, surge effects are ignored which in turn implies that propeller thrust 

 fluctuations are negligible. 



5. As a consequence of the last statement in 4., the ship speed is assumed 

 to be constant. 



6. Forces and moments due to wind action, tow-lines, etc., are not consid- 

 ered. The external excitation is to be that due to waves only. 



7. Since a linear theory will be developed, the translatory and angular de- 

 partures of the ship from and about an inertial reference are assumed to be 

 very small (first order). 



8. The ship is assumed to be originally on an even keel. 



9. Motion of the ship is assumed to take place in a given, idealized fluid 

 which is unbounded in all directions. 



10. The wave excitation is that due to uniform, infinitely long-crested sinus- 

 oidal waves of small amplitude which come from directly ahead or astern, i.e., 

 the direction of ship motion is taken to be normal to the wave crests. 



Two orthogonal, right-handed systems of coordinates will be employed in 

 the development of the coupled pitching and heaving equations of motion. Con- 

 sistent use of right-handed systems is advantageous because it allows a conven- 

 ient check in the analysis by simply permuting the terms of various expressions. 

 The first system of axes will be fixed in space with its origin located at an arbi- 

 trary point on the still water level. This will be regarded as a Newtonian frame 

 of reference with respect to which the wave configuration and body orientation 

 in space will be referred. The second system of axes, usually referred to as 

 "body" axes, will be fixed in the ship witii a convenient point as origin. In rigid 

 body dynamics, the origin of the 'Tjody" axes is usually chosen to be the center 

 of gravity of the body. However, in the ship problem case it is usually advanta- 

 geous to fix the origin of the "body" axes at the intersection of the midship sec- 

 tion, the longitudinal vertical plane of symmetry and the waterplane through G. 

 This not only simplifies the computation of the hydrostatic and hydrodynamic 

 forces but is also convenient because the midsection plane is fixed in a ship, 

 whereas the position of the center of gravity is variable. 



It is pertinent to note at this point that, from 1954 [8] onwards, Korvin- 

 Kroukovsky assumed for simplicity in all his analyses that the vertical plane of 

 the center of gravity and midship section coincided. Whereas, it is true that the 

 LCG is usually a small fraction of the ship length, this simplification is some- 

 times responsible for wrong interpretations of phase angles, leading to errors 

 up to 10° for certain ships in short wavelengths. 



326 



