Ogilvie 



a^(t) does not exist, provided only that we replace 3{a,,} by 3{a,,}/ia;. There 

 will also be a singvilarity at co - o, and so the modified (15) is not valid for oj-o. 

 It is also shown in Appendix B that o.^C^) will generally be zero vinless some of 

 the Cj^'s are zero. This is, of course, quite reasonable, since the c-^'s are 

 restoring force coefficients (even though they are not hydrostatic restoring force 

 coefficients). 



We have now shown that the two types of equations of motion are equivalent 

 for both sinusoidal and transient motions, provided only that the system is sta- 

 ble. In the third situation of interest to us, viz., a ship moving in a stationary 

 random sea, the usual arguments of generalized harmonic analysis can be used 

 to show that the two descriptions are again equivalent. Actually, in order to 

 carry out the conventional spectral analysis, we need only to be certain that the 

 assumption of linearity is valid, and evidence was presented in Chapter II to 

 show that it is indeed valid in at least certain modes of motion. The value here 

 of having equations of motion is in the capability which they provide for predict- 

 ing the effects of various parameters on the spectral properties of the ship. 

 They also enable us to develop test procedures, which have been called "pulse 

 techniques," which are an order of magnitude more efficient than regular wave 

 tests in determining the frequency domain characteristics of a ship hull. See 

 Cummins and Smith (1964). 



From (14) or (15), it is natural to define the following quantities: 

 * 1 f 



MjkC"^^ = added mass coefficient = ^J.■^ - — K.^(t) sin ojt dt ; 



(16) 



CO 



h*-^(ui) - damping coefficient = b-^ + I K-^(t) cos o^: dt . 



Of course, following Cummins, we previously defined m-,^ as an added mass. 

 This situation merely shows how arbitrary the definition of this quantity is. The 

 added mass defined in (16) depends on frequency and speed, and so in one sense 

 it is not so natural as the previous definition. However, for the special case of 

 sinusoidal oscillations, it is as reasonable a definition as the other. 



In the time -domain equations, it was not possible to identify any one quan- 

 tity as a "damping coefficient"; some or all of the damping was included in the 

 forces represented by the convolution integral of (13'). In the case of sinusoidal 

 motions, we can readily pick out certain quantities which we identify as "damp- 

 ing coefficients," but we must be careful in interpreting the label thus applied. 

 If there is sinusoidal motion in just one mode, say the k-th mode, then the aver- 

 age rate at which the ship performs work on the water depends only on the com- 

 ponent of force which is in phase with the velocity in that mode; in other words, 

 the dissipation of energy depends only* on bj^^, which is properly a damping 

 coefficient. The force components in phase with acceleration or displacement 



*This is true only in the reference frame in which the water is streaming past 

 the ship. Otherwise ship resistance is involved with the work done. 



38 



