He set 
ro) 
£ =_vsinto @ =—-/2Visin@ 
a (12.97) 
U(d) = Vcos?@ U(p)? = V2 cos 6 
introducing the phase 6 as shown in Fig. 12.7. After manipulation of the relations, the 
equation of motion, Eq. 12.96’, was inverted into the equation of V and@. When the 
phase process 6 was modified into a process A(t), and the joint process Z = (V, A) was 
made to converge into a Markovian process, the transitional distribution function 
p(Z\Zo; t) was found to be governed by a Fokker—Planck equation of second order. The 
Stationary solution of the Fokker—Planck equation p(Z) was obtained, and from this 
expression V and A were found to be uncoupled, and p(V) was calculated. With the 
relations of V, @ and@, p(V) was modified into p(¢,¢). 
Ww OD? 
U,V vs.0,6 
Fig. 12.7. U, Vvs. 6,9. 
(From Roberts.®") 
For nondimensionalized, nonlinear rolling expressed as 
d+abp+ bolod+¢—-? = x(0), (12.98) 
Roberts calculated the probability density function of the nondimensionalized amplitude 
of rolling A by the ME theory and compared it with the Rayleigh distribution obtained by 
the linear theory as in Fig. 12.8. He®!-®6 also compared his results with the nonlinear sim- 
ulated data obtained by J.F. Dalzell®° to show the validity of his method. Examples are 
shown in Figs. 12.9 through 12.11 for a variety of damping coefficients a and b, where 
Q =a,/Wo, @, is the peak frequency of the excitation, andg,, is standard deviation 
of the input process x(t). Process 3 in the figures is a wide banded excitation for this 
example. He showed many other results of comparison for other types of excitation with 
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