Mooring and Positioning 'of Vehioles in a Seaway 



U= Uo(s) +U'(s,t) (98) 



V= Vo(s) + V'(s,t) (99) 



T = To(s) +TMs,t) (100) 



«^= <^o(s) + V(s,t) (101) 



e = 6o(s) +€'(s,t) (102) 



where the '-symbol quantities are perturbations about the equilibrium 

 positions (o-subscript terms), and expanding various constituent 

 terms up to first order terms alone, leads to 



sin ^ = sin (4>o+ <{)') ^ sin 4>o + ^^ cos ^q (103) 



cos ^ = cos (<()Q+(f)') e^ cos 4>^ - 4>' s in ^0 (104) 



The hydrodynamic loading terms are expanded in the form 



F = Fq + F^jU' + F^V + F^c^' (105) 



G = Go + GyU' + GvV + G.4)' (106) 



where the partial derivatives of the loading functions with respect 

 to particular velocities are indicated. This can be accomplished 

 when considering the steady state velocity solutions (from Eqs. (93) 

 and (94) when time derivatives equal zero) which, when combined 

 with Eqs. (114), are 



Uo = (107) 



Vo = (108) 



The linear perturbation equations are then given by 



-|xU' = To4>^+T'4»o^-(l+€b)[FuU'+F^V + F^4>'] +F^€' + W^^'sin 4>^ (109) 

 -HlV = T,'+(l+€o)[G^jU» + G^V'+G^<j)'] +G^e'- We<t>' cos <|>o (UO) 



1065 



