Understanding and Prediction of Ship Motions 



may be called "reactive" forces; they are associated with the local disturbance 

 of tJie water, but they are not related to the average rate of transfer of energy. 



If there are two or more modes of motion occurring simultaneously, then, 

 as we have seen, there will be coupling between modes, and we may expect that 

 there will be, say, j -component forces in phase with a.(t) which result from 

 the accelerations and displacements in the k-th mode. This will be demon- 

 strated explicitly in the next chapter. This means that the damping in the case 

 of coupled motions will involve the coefficients /j.*^ and c-^. It is still conven- 

 ient to refer to the coefficients /x*^, h*^, and c-^, for k :|: j, respectively, as 

 added mass, damping, and restoring force coefficients, but it must be remem- 

 bered that all are involved in the damping. 



In the two expressions appearing in (16), the frequency dependence enters 

 only through the integral terms, and it is important to note that these integrals 

 are, respectively, the sine and cosine transforms of the same function, K.^(t). 

 This fact will lead to the establishment in the next chapter of a formula relating 

 added mass and damping coefficients. 



In concluding this chapter, I would comment much as I did at the end of the 

 previous chapter. We can now continue to use the old second order differential 

 equations, as we did years ago. But now we know that we can, when desirable, 

 turn to the true equations of motion, for it is these which give broader meaning 

 physically to the equations which are valid only for sinusoidal motions. We also 

 know that we must allow the "constants" in the differential equations to be func- 

 tions of frequency. And finally we have obtained from this study of the equations 

 of motion some powerful new tools: pulse methods of testing, which are an order 

 of magnitude more efficient than the older methods, and an extremely valuable 

 relation between added mass and damping (to be proved presently). 



PROPERTIES OF TERMS IN THE EQUATIONS 

 OF MOTION 



This chapter will be devoted to some special relationships for the various 

 terms and coefficients in the eqviations of motion. Specifically, the following 

 facts will be proven: 



1. The added mass matrix can be determined from the matrix of damping 

 coefficients, and vice versa. 



2. The exciting forces at zero speed can be deduced from knowledge of the 

 far-field potential for the problem of the ship oscillating in calm water, i.e., the 

 diffraction problem can be avoided. 



3. The diagonal elements of the damping coefficient matrix can be calcu- 

 lated from the same far -field potentials used in (2) above. If the ship has zero 

 speed, all elements of this matrix can in principle be found in this way. 



In other words, if we can find velocity potentials for the six problems cor- 

 responding to the sinusoidal oscillations of a ship in calm water, we can evaluate 



39 



