1.5 
(aa a 
po =] Tia. 0.2 
LINEAR CASE, € = 0 
0.5 RAYLEIGH 
DISTRIBUTION 
0 a GY eo 3.0 
Fig. 12.4. Probability distribution density of the extreme (peak) 
values of oscillator with set—up springs. 
(From Crandall.°4) 
12.6 APPLICATION OF THE FOKKER—-PLANCK EQUATION FOR 
THE ANALYSIS OF SEAKEEPING DATA 
As has already been mentioned, the Fokker—Planck equations have been solved only 
for a restricted number of cases, and this has made the applicability of this equation rather 
difficult in many engineering fields. For example, this method was applied to the nonlin- 
ear system with nonlinear restoring terms and with Gaussian white excitation, as shown 
in the preceding section, but the method has been considered inapplicable to the system 
with nonlinear damping or with colored noise excitation. A few efforts have been made to 
overcome these difficulties, for example by J. B. Roberts.°? His work, *! especially in 
the analysis of nonlinear seakeeping data, will be summarized briefly. 
12.6.1 Nonlinear Analysis of Slow Drift Oscillation of 
Moored Vessels in Random Seas 
Utilizing the known characteristics of the Fokker—Planck equation, Roberts 
analyzed the statistical behavior of the nonlinear slow drift motion of moored vessels. 
Assuming that the waves 7(t) are narrow banded with band width parameter e, and that 
the drifting force D(t) can be regarded as proportional to the square of the wave height, he 
modified the expression of wave height 7(r) and drifting force D(t) to 
n(t) = H(t) cos [wot + o(t)] = hA(t) cos [Wot + o(2)] 
A al {a2 i: on} | cos [wot +a(2)] 
= h[a(t) cos wot — b(t) sin wot], (12.63) 
342 
