Motions of Moored Ships in Six Degrees of Freedom 



The first three elements x , x , x in the displacement vector x 

 represent the surge, the sway and the heave of the moored ship, and 

 the next three elements Qj, Q^, and 9 3 represent respectively the roll, 

 the pitch and the yaw of the ship. The force vector f(x) is nonlinear. 

 If J.ts argument x is replaced by -x, f(-x) will in general differ from 

 f(x) in magnitude as well as in sign. Hence f(x) is asymmetric. The 

 quantities (J, q. and y . represent wave frequency, force or moment 

 amplitude and phase angle for the ith element of the vector g(t). 



METHOD OF SOLUTION 



Since the excitation vector is harmonic, we assumed that an 

 approximate steady- state solution for the response of the system (l) 

 may take the following form : 



x. = z. '+ y. (2) 



111 v ' 



- z. + y. cos (ut +<p.) i = 1. . . , 6 (3) 



z. is_a constant introduced to account for the asymmetry of f(x) in (l). 



If f(x) is symmetric, then z. will vanish, y. is a harmonic function 



whose amplitude and phase are V and >p . 



i ' i 



Consider a linear system defined by 



My + Cy + (K + K)y = g(t) (4) 



where K is an unknown matrix. If K is known, this linear system can 

 readily be solved to give : 



Ti =Vw. 2 + w. 2 +6 • (5) 



i = 1 6 



W 



f, = tan - 2 -i±A. (6) 



i w. 



where w.'s are given by 



675 



