2. Linearize equations as above, transform to frequency domain for 

 quasi-linear solution where equations are frequency dependent. 



{Af} = Re ({F} e^'^^) 



{Ail = Re ({Q} e^'^'') (4) 



(-a)^[M] + io) [C] + [K]){Q} = {F} 



3. Direct numerical integration of nonlinear time domain equations. 



[M] {q} = {f} - {g} (5) 



where [M] is the position dependent virtual mass matrix. 



{f} represents the time and position dependent external 

 forces (dragloads, point loads, etc.) 



{g} represents the internal loads from the elements, 



reflecting the material and geometric nonlinearities and 

 material damping. 



The finite element method allows direct calculation of any of these terms 

 for the mooring lines given the material properties (EA, mass, etc.), the 

 nodal positions and the unstretched element lengths. The effects of lumped 

 bodies; such as, buoys, platforms and ships, can be readily included if they 

 are described in functional or tabular form appropriate to the solution form. 

 Special rigid link multi-point constraints are used to tie the bodies into the 

 mooring model. 



Some Demonstration Solutions 



Single Degree of Freedom System With Geometric Nonlinearity 

 Figure 1 shows a single degree of freedom system composed of two linear 

 springs attached to a single mass point. It also shows the nonlinearity 



31 



