Linear versus Non-linear 



In order to clarify this characteristic in the model hier- 

 archy, it is illuminating to introduce a mooring dynamic model. 



Consider, for example, a general deterministic 'linear' 

 model as developed and contributed to by Cummins (1962), 

 Ogilvie (1964) and others which has been found useful in analys- 

 ing moored-ship behavior. The system of equations appears as 



6 



(1) 



7 J(M, . + m, . ) X . + / K, .(t-T)i.(T)dT 



+ C . x.>= X, (t), k = 1, 2, 3 

 kj 3 k' '' ' ' 



M, . is the inertia matrix 



C, . is the matrix of hydrostatic restoring 

 force coefficients 



K . is an impulse response or retardation 

 function 



m, . is a constant inertia (frequency independent) 

 coefficient matrix 



X, (t) is a time-varying exciting force due to winds 



wave, currents and restraining (mooring) forces, 



This system of equations is linear in the sense that the 

 integral involves a superposition or summation. Also, although 

 the above formulation is written in the time domain, there is 

 an equivalent frequency domain description. This can be shown 

 by letting the moored vessel perform simple harmonic motion in 

 response to harmonic excitation. 



17 



