Coupled Ship Motions 



3. EXPRESSION OF THE IMPULSE RESPONSE IN TERMS 

 OF ADDED-MASS AND DAMPING PARAMETERS 



In [5] it was pointed out that F(t), the hydrodynamic force exerted by the 

 body when the body acceleration is S( t) , can be calculated from either the 

 damping or added-mass parameter (for simple-harmonic oscillation) via the 

 Kramers-Kronig relations followed by a Fourier transformation. We will now 

 discuss this point further, including some observations on a later publication [6] 

 which also treats transients and their relations to force parameters. 



Let us recall that according to Eq. (A-5) of [5] we have 



I F(r') e^'^^'dr' = rpp'(a) = rp [p;(a) + ip^(cr)] (3.1) 



where 



F(t) = hydrodynamic heave force exerted by the body on the fluid, per unit 

 step heave velocity of the body at t = 0, F(t) = for t <0; 



p'(<^) = force parameter - p^ +iPd5 



p^((t) = added-mass parameter; 



p^(ct) = damping parameter; 



(T = radian frequency of oscillation; 



T = submerged (or any other) volume of the body for three-dimensional 

 problems, and volume/unit length for two-dimensional problems. 



It follows that 



F(t) = -g J p'(cr)e-i°^*da 



- 00 



'Tp \ r , , -. 



~ ~^J [PmC^) COS o-t + Pd(o-) Sin crtj do- 







= ^Jp;(oo) 7rS(t) + J [Ap;((T) cos crt + p^(ct) sin at] dcr I , (3.2) 



where 



Ap:(cr) = p'(o-) - p'(co) . 



However, it is sufficient to know either p^C^^) or p^C^), due to the Kramers- 

 Kronig relations, and in fact those relations imply the following: 



415 



