Motions of Moored Ships in Six Degrees of Freedom 



However, it can be shown that not all k^j's are independent. In order 

 to avoid this difficulty, we choose in this case, 



k.. = if i ^ j (13) 



Then, k can be uniquely determined as follows : 

 ii 



i r 2Tr 



k.. = — — I L cos(0+f.)d0 (14) 



TTY. J i 



Six more equations are furnished to determine z^ by averaging the 

 nonlinear equations over one cycle which leads to 



•2TT _ 

 2TTKz + / f(x)d0 = i = 1.....6 (15) 



o 



J 



Thus the solution of the nonlinear system is reduced to the solution of 

 Equations (5) - (7), (14) and (15). They are nonlinear algebraic 

 equations but can be solved numerically by the following iteration 

 approach. First, set z^ = and assume a set of values k^. Then 

 Equation (7) can be solved by simple matrix inversion and y^ and 

 \D^ are determined from Equations (5) and (6). Now a new set of 

 values of kii and z^ are calculated from (14) and (15). This proce- 

 dure can be repeated until required accuracy is reached. 



SUMMARY AND DISCUSSION 



An approach to the determination of an approximate solut- 

 ion for the steady-state response of moored ships in six degrees of 

 freedom has been formulated. This approach can be applied to moor- 

 ed structures in open sea as well as moored ships in harbors. In this 

 paper, the degrees of freedom of the system is specified as six. 

 However, this approach is still valid for degrees of freedom other 

 than six. 



The accuracy of an approximate analysis is difficult to pre- 

 dict in general. A different version of this approach where f(x) is 

 symmetric has been employed to problems which possess known exact 

 solution l_6j , it shows that the accuracy of this approach is well 

 within the limits of practical engineering usefulness. 



677 



