Ogilvie 



Today we may consider that it has been confirmed, for most practical purposes; 

 some of the evidence will be presented. 



St. Denis and Pierson used an extremely primitive set of equations of mo- 

 tion, and we must now conclude that those equations are quite unacceptable. 

 They were the best available ten years ago, but we can now do much better. 

 The use of second order ordinary differential equations to describe the rigid 

 body motions of a ship is quite artificial. Under appropriate conditions and with 

 proper interpretation, they provide a valid representation, but such equations 

 certainly cannot have constant coefficients in the usual sense. The form of the 

 equations of motion can now be stated with considerable confidence, and this 

 will be done. 



Actually, the discussion of rigid body equations of motion is somewhat of a 

 digression. Basically, having accepted the linear superposition principle, we 

 need only to find a means of determining the transfer function (or frequency re- 

 sponse function) of the ship. This may be done experimentally, in which case 

 the whole subject of equations of motion need not be introduced, or it may be 

 done by the use of hydrodynamic theory, in which case the information provided 

 by the equations of motion comes out automatically. 



Nevertheless there are important reasons for studying the equations of 

 motion per se. On the one hand, the direct experimental procedure treats only 

 input (the exciting waves) and output (the motions). It provides no insight into 

 the particular ship characteristics which cause different ships to respond dif- 

 ferently in a given seaway. On the other hand, the hydrodynamic theory of ship 

 motions is not yet highly enough developed to tell us comprehensively which 

 ship characteristics are most important in seakeeping and why they are so 

 important. 



Perhaps the largest portion of the literature on ship motions during the past 

 decade has been concerned with the calculation of individual elements in the equa- 

 tions of motion. Some of the methods used have been quite sound scientifically, 

 and some of the results have shown quantitative agreement with experiments. 

 For example, the damping (due to wave radiation) in heave or pitch can be ana- 

 lyzed straightforwardly in certain situations, and recently it has been demon- 

 strated how to calculate the added mass or added moment of inertia through 

 knowledge of the damping. 



Although some of these analyses have led to remarkable results, there is 

 also a basic difficulty of principle in using them, and this problem was already 

 clearly pointed out by Peters and Stoker (1954). Since the free surface prob- 

 lems involved must all be linearized before any progress can be made, these 

 authors set out to perform the linearization in a clearly stated, rational way and 

 to investigate the logical consequences of the simplification. They obtained the 

 linear mathematical model from a systematic perturbation analysis, ship beam 

 being the small parameter. The results were disappointing, for they obviously 

 do not correspond to reality: In the lowest order motion solution, there appear 

 undamped resonances in heave and pitch. The physical interpretation of this 

 result is that the wave damping is of higher order (in powers of the small pa- 

 rameter) than the exciting force, restoring force, and inertial reaction force. 



