Breslin, Savitsky, and Tsakonas 



The solutions of this pair of equations are written (again in Korv^n- 

 Kroukovsky's notation) in the compact complex variable form: 



PS 



QR 



(3) 



where 



/M„P - F„R\ ,^, 



^ = l-ps - qf; " 



P(&j) = -2w^ + ibo) + c 



Q(co) = -d • w^ + Leco + g 



R(co) - -D • 0)2 + iEoi + G 



S(o)) - -Aaj2 + iBo; + c . 



(4) 



(5) 



Inspection of (3) and (4) shows that the response in heave and pitch are both 

 linear combinations of the forces and moments and response or transfer func- 

 tions of the body. Consider only the response in heave (pitch follows in com- 

 plete analogy) which may be written 



Z(t) = Zf + z„ 



where 



and 



° \ PS - QR 



-M, 



° \PS- QR 





-M $ ( oj) e 



(6) 

 (7) 



(8) 



The complex functions 0)^ and o^^ are called frequency response or transfer 

 functions in heave per unit applied force and moment and are evaluated in terms 

 of an amplitude and phase angle in the form: 



Of = Af(co) e-^'("); 



Ajo,) e-^^(") 



where 



Af(a>) 



SCo;) 



T(c^) 



^J^) 



Q(«) 



T(oj) 



(9) 



(10) 



and, for brevity T( w) = PS - QR . 



As is well known, these unit response functions depend only on the body co- 

 efficients themselves and not on the forcing functions. In what follows, it will be 

 necessary to consider the forcing functions characteristics in some detail. 



464 



