s^ = 1 + a-j^-^k-^^ ; s^ = 1 + a22k2 



Then substitution for v and G from Equations [U8a,b] into [i4Ta,b] gives: 

 s^F + a-j^2^i'^ = ^i'"'> B.^i^^'P + s^G = kgO [U9a,b] 



These equations may be solved for F and G in the form: 



DF = s^k^v - a^g^^k^e [50a] 



DG = - a^^k^k^v + s^k^e [50b] 



° = ^1^2 - ^12^21^^2 f50c] 



Direct substitution for F and G from Equations [50a,b] into [U6a,b] then 

 gives as equations of motion, provided D ^ 0: 

 mv + hm'e + (c-, + BS) v - CSe 



+ D"-"- (s2k-]_v - a-j^2^-,_k2e) - BS^G = [51a] 



hmv + 1-6 + LBSv + (c„ + E^S) 9 



+ D"-"- (-a k k^v + s k^e) - LBS^G = [51b] 



21 1 2 1 2 



Similar substitution for F and G in Equations [i+Bajb] gives, 

 provided D ^ 0: 



V = (1 - D"^s-) v + D"^a,-k,e ; 6 = D~^a^,k.v + (l - d"^s )G [52a,b] 



The special case D = requires other methods as will be explained later. 



kg 



