Kaplan 



these degrees of freedom) is the main characteristic of moorings 

 applied to ships that distinguishes the resulting motions from those 

 of free ships in waves. 



V. SOLUTION OF EQUATIONS 



The equations of motion result from combining all of the 

 constituent terms discussed above, and solutions can be obtained 

 by converting them to a simpler form for sinusoidal waves. Since 

 the exciting forces and moments are sinusoidal functions, the 

 motions will also be sinusoidal with the same frequency. Defining 



— iwt 

 X = xe 



— iwt 

 y = ye , 



— icot 

 z = ze , 



^ui ~ ^e 



iiut 



— ioft 

 Yu, = Ye 



etc, 



the equations of motion are then converted to (complex) algebraic 

 linear equations. In matrix form the equations may be represented 

 by 



U.^31 







^22 



I- 32 



(24) 



for the longitudinal motions, where the coefficient matrix is sym- 



metric , I. e. a,- = a_, , a«_ = a 



by: 



13 



-3! 



23 



^32- 



The matrix elements are defined 



a.. = (- mco + iojN^ + k^) 



^22" 



^23 



a|3 = a^i = m | BG | to 

 (m + r A33 de) 00^ + iojC^N^ + pg J B* di 



^2= ^'^^\'^^i di - icoN.e- pgy B*e de 



= - (V "^y ^^33 ^^ ^0 '^^ ^ '"^^^^ ^^^^ ^^*^^ '^^ 



The lateral equations are represented by 



(25) 

 (26) 



(27) 



(28) 



(29) 



1032 



