onze SIMULATION 
e = 
e b= 03 
° b=10 
0.20 
0.05 
0 0.01 0.02 0.03 0.04 
Ow 
Fig. 12.12. Variation of standard deviation of roll og with standard deviation of wave 
input o,,: @ = 0.01; Q = 0.90; b = 0.1, 0.3, and 1.0; Process 3. 
(From Roberts.°") 
12.7 PROBABILITY DENSITY FUNCTIONS OF AMPLITUDES, EXTREME 
VALUES IN RELATION WITH THE FUNCTIONAL POLYNOMIALS 
12.7.1 Narrow-Banded Case 
As was mentioned in Section 11.1, when a nonlinear response z(t) was expanded by 
the Voltera expansions (or functional expansions) 
Cc foo) 
z(t) = y. i 5 | Pn, 2) 2 - key) XE — 1X G— 2) x 
n=] —-0 
—S——— 
n 
5 USE) GA G1 oo oo Uy (222) 
the terms for n>2 can be considered as the modifying terms of the Taylor expansion of 
this process around its linear term for n=1. If we take until n=2, and using small €, some- 
times the response z(t) is expressed as 
z(t) = | gi(t) x(t—T) at +€ | | 22(T1, T2)xX(t —T1)xX(t—T2) dtydTz. (12.100) 
—c _ 
The derivatives in terms of time is 
co 
2(t) = | g(t) xX(t—T) att 2 | 
-o —_ 
82(T1,T2)X(t—T1)x(t—T2) dtydt2, (12.101) 
_—— 8 
8 
354 
