pee axidca—Ne 
_—— 8 
Roberts calculated numerically, for a Duffing type system, with various €, A*, and 
€* and compared them with the simulated results. Figure 12.5 shows an example of his 
results. He compared p(x) calculated by Eq. 12.81 under the white noise approximation 
with the result obtained by simulated data and found that the agreement is good and quite 
different from the Gaussian distribution for A* = 0. Once p(x) of the displacement x(t) is 
determined, p(T), the probability distribution of tension on the mooring, can be derived as 
8 
(x) 
pe) aes, (12.82) 
aw 
dx 
where the tension is expressed by 
T =w(2). (12.83) 
fe \ » = 0 
4° \ (GAUSSIAN) 
\ 
\ ] 
10% \ 
aerate 
p(X) \ ; 
\ 4 
10° \ 
\ 
: 
107+ 
WHITE NOISE 
10> APPROXIMATION 
4 3 #2 +1 0 1 2 3 4 
Fig. 12.5. Probability density function for displacement response; 
A* = 0.5,€ = 0.05, e* = 1.0, 0 digital simulation estimate. 
(From Roberts.©°) 
346 
