Understanding and Prediction of Ship Motions 



already reproduced here in Fig. 5. The effects of couplings between pitch and 

 heave modes are clearly not negligible. Couplings of pitch and heave with surge 

 may be negligible. 



Much effort has been devoted to calculating some of the coefficients in these 

 equations, and in fact the following chapters will be concerned with this problem. 

 We defer consideration of such analyses for the moment, until we have discussed 

 the nature of the true equations of motion in the time domain. 



Equations in the Time Domain 



We would like to find equations of motion which are valid whatever the na- 

 ture of the seaway; we want to avoid the difficulty encountered with Eq. (4), viz., 

 that the forcing functions had to depend sinusoidally on time. From the nature 

 of Eq. (4), in particular from the frequency dependence of the coefficients. Tick 

 (1959) suggested that the true equations would involve convolution integrals. In 

 fact, this had already been demonstrated many years earlier by Haskind (1946). 

 Unfortunately, there were some errors in Haskind's work, but this basic conclu- 

 sion was correct. 



If we are to obtain the actual equations of motion, we must start by formu- 

 lating the complete mathematical problem involving the dynamics of the ship (as 

 a rigid body), the description of the sea, and the hydrodynamics of the ship -water 

 interactions. This general problem will be treated to some extent in later chap- 

 ters, and other authors at this meeting will devote their papers to it. For pres- 

 ent purposes, we shall look simply at the form of the equations, and for this we 

 follow closely the work of Cummins (1962). The net result will be a set of equa- 

 tions analogous to (4) in that there will be several undetermined parameters and 

 functions. These must be determined either from experiments or from separate 

 hydrodynamic analyses. 



Cummins makes one major assumption: linearity of the system. This 

 means much more than the linearity of Eq. (4). In that case, linearity implied 

 that if the ship were subjected to a sum of two excitations, both sinusoidal at the 

 same frequency , the total response would be the sum of the separate responses. 

 Now the assumption is extended to cover excitations of any nature. In particular, 

 if a ship is given an impulse of some kind, it will have a certain response lasting 

 much longer than the duration of the impulse. If the ship experiences a succes- 

 sion of impulses, its response at any time is assumed to be the sum of its re- 

 sponses to the individual impulses, each response being calculated with an appro- 

 priate time lag from the instant of the corresponding impulse. These impulses 

 can be considered as occurring closer and closer together, \intil finally one in- 

 tegrates the responses, rather than summing them. 



This is an approach to water wave problems which was very popular in the 

 days of Kelvin, but which is generally out of style today. However, modern un- 

 derstanding of analogous problems in control theory makes this approach more 

 useful than ever. In a sense, we find that the existence of the free surface 

 causes the physical system to have a "memory": What happens at one instant of 

 time affects the system for all later times. This, of course, is very obvious; 



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