By using the following relations, as provided by Ogilvie 

 (1964), the frequency domain relations are obtained. 



K, .(t) = — I b, .(o)) cos (jjtdto, 



^J IT I ^J 



^; 



™k1 " aj^ (co') + — J ^ki^^-^ ^^^ to'tdt, 



where 



a, . = added mass coefficient (frequency dependent) 

 (jo ' = one particular but artibrary value of co 



b, . = damping coefficient (frequency dependent). 



Thus, the third category in model hierarchy (Figure 1) 

 which distinguishes between frequency domain and time domain 

 model descriptions is shown to be a case in which one is a 

 subset of the other. 



At this point, we no longer talk about the system of equations 

 in a general way but instead we focus our attention on various 

 terms of the equations and examine each in the light of the 

 linear-non-linear description. We have already examined the 

 impulse response term. 



Consider the various restoration sub-models. The restoration 

 forces consist essentially of two components. There are those 

 that are due to bouyancy effects in roll, pitch and heave and 

 those that are due to the mooring lines. For small motion ampli- 

 tudes, these restoration forces are often approximated by linear 

 functions. But the roll moment is well-known to be non-linear with 

 respect to roll displacement , and the components due to the mooring 

 lines are often non-linear with respect to geometry and/or material 

 properties or both. For example, the mooring restoration forces 

 in the case of a ship at a conventional pier where a combination 

 of mooring lines and fenders result is a highly non-linear restor- 

 ation function. 



18 



