RO,€)=v+] Ayot+ + 34300 
A201 
él Est €2 yi/2 
2 A201 
cP 7 gaan 64300 
+ v2.30, (12.169) 
where e is the so—called band width parameter, as Eq. 12.165. 
As referred to in Section 11.5, item 6, Dalzell*° generated the simulation data of the 
cubic nonlinear response to check the validity of his tri-spectrum analysis!” for the 
Gaussian excitation of the waves, as expressed by the Moskowitz—Pierson type spectrum, 
- 1.25(33)' 
U,x(w) = 50209 exp———_. —— (12.170) 
@W 
where U,,() is a one-sided spectrum, o2 is the spectrum area and variance, and wo is the 
modal frequency. 
Using the same spectrum parameters for nonlinear response as he used before!’ and 
varying the excitation level by changing 0, tog, = 0.125, 0.25, 0.50, 0.75, and 1.0, he 
obtained results for simulated data averaging 10 samples for each case. 
He calculated the probability distribution of maxima and minima and compared his 
results with Eqs. 12.162, 12.163, 12.166, and 12.167. 
Figure 12.17 shows the comparison of probability distribution densities of response 
maxima and minima estimated for the simulations with those of the first and second 
approximations from Eqs. 12.162, 12.163, 12.166, and 12.167. Here z represents the 
normalized maxima and minima and was equal to v—Ajo0, A100 being the normalized 
mean and v the standardized maxima and minima as appeared in Eqs. 12.162 through 
12.167. 
Figures 12.18 and 12.19 compare cumulative distributions of response maxima and 
minima estimated from the simulations with those of the first and second approximations. 
In these two figures, the standard deviation of input 6, (“SIGMA’”) is the same, and the 
band width parameter, e, (“EPSILON”) was different. Here the first approximation is ex- 
pressed as a straight line (the scale of the cumulative probability is so chosen) and is 
marked as the fitted distribution. 
These figures indicate that the final expression for the second approximation gives 
excellent results that check the simulated results very well. 
375 
