i [x, ^ - X ^] tan 66 + c [x, - x^] tan^6( 

 / D a Da 



2 "■ b 



By applying the "small angle" approximation and neglecting terms of the 

 order of 6o2. Then, 



6V = 4 [x ^ - x,^]6e 

 2 "- a b ■" 



and the force is : 



F = 7Pg[x/ - x^2]6e. 

 The X and y components of the force are: 



F = F cos 66 : -J pg[x ^ - X, ]6e cos 66 

 and 



F = F sin 66 = ^ pg[x ^ - x, ^] 66 sin 66. 

 y 2 ^'- a b 



Again applying the small angle approximation one finds: 



and 



^x ~~ ^'^^\ - \^^^ 



F = 0, 



y 



The hydrostatic spring constants coupling roll to sway and heave are 

 then: 



and 



KH^^ = (A-5) 



•^32=lPgt-a'- \^' ^^-'^ 



To obtain the moment induced by roll motion compute: 



12 2 13 

 Moment of Gained Volume = - (^ x tan 66) {_-=■ x ) ~ ^ x 66, 



z a o a o a 



1 3 

 Moment of Lost Volume = -^ x, 66 



and 



Moment of Original Volume = Wy, 66. 

 In this formula, 



W = weight per unit length 



77 



