Ogilvie 



It is apparent that there can be considerable utility in representing the mo- 

 tion by a set of second order equations, generalized from (3), 



6 



H f jk^'k + ^ikK + ^jk'^k} = fj^t) + gj(t) , j = 1, ..., 6, (4) 



k= 1 



where the fj(t) represent wave-induced excitations and the gj(t) represent all 

 other external forces and constraints. However, it is worth reiterating that 

 these are not really equations of motion in a proper sense. They are valid only 

 if the right hand sides all vary sinusoidally at a single frequency and if the con- 

 stant coefficients on the left have the values appropriate to that frequency. As 

 stated earlier, these equations describe the frequency-domain characteristics 

 of the system, and a non- conventional derivation was presented here to empha- 

 size this point. 



Golovato (1959) gave direct experimental proof that these second order 

 equations cannot be used to describe non- sinusoidal motions. He conducted 

 transient tests with a ship model, giving the model an initial pitch inclination 

 and allowing it then to undergo a transient motion, returning to its equilibrium 

 attitude. He found that the response could not be represented as that of a simple 

 damped spring-mass system, which would have been appropriate to a second 

 order ordinary differential equation with constant coefficients. An even more 

 startling result has recently been produced by Ursell (1954). For a heaving 

 body which is released from a position above its equilibrium height and allowed 

 to come to rest, he found analytically that there are only a finite number of 

 oscillations, after which the body gradually approaches its equilibrium position 

 in a non-oscillatory manner. This cannot be explained in terms of equations such 

 as (4), but it will be shown presently that the true equations are integro-differen- 

 tial equations, and these do allow of such solutions. 



In the full generality of six degrees of freedom, the ordinary differential 

 equations are still not simple to work with. There are 108 "constants" on the 

 left, each being a function of frequency. Also, all of the parameters in general 

 depend on wave heading as well as frequency. 



In order to make the system manageable, several simplifications have been 

 tried by various investigators. The most straightforward is to limit considera- 

 tion to head and following seas. This means that three degrees of freedom can 

 be eliminated,* and the number of coefficients is reduced to 27 — these not being 

 functions of heading. Of course, there is nothing wrong with this simplification, 

 provided one is satisfied with results valid only in head or following waves. 



Frequent attempts have also been made to neglect couplings between modes. 

 This was done, for example, by St. Denis and Pierson (1953), and it has appealed 

 to many investigators since then. However, Gerritsma (1960) has shown this is 

 dangerous. He calculated responses using experimentally obtained frequency- 

 dependent coefficients, both with and without couplings. Some of his results were 



*We are neglecting phenomena such as the unstable rolling which occurs in fol- 

 lowing seas when frequency of encounter equals twice the natural frequency of 

 roll. See Grim (1952), Kerwin (1955), Kinney (1963). 



24 



