Boi 
Boe | F(6)dé. (12.33) 
0 
From Eq. 12.31 
p(x,x) = p(x)p(x), (12.34) 
and 
p(x) = of ee (12.35) 
Wo 
p(x) =C ex mae | F(6)dé |, (12.36) 
Wo 
0 
we find the probability distribution density function of the velocity x is a Gaussian. 
More concretely, for a nonlinear spring system, F{X(z)} in Eq. 12.20 is 
FIX,()} = FIX(o} = wf X . ea(X(0)] (12.37) 
and the equation of oscillation Eq. 12.20 is 
K(0) + BX(0) + 3 Xow) + eg{X(}] = Ao. (12.38) 
Here for hard spring type, € > 0, and € is on the order of x(t)/g{x(¢)}, g{x(2)} = 
—g{—x(z)}, and as Lx(z)|>0, x(t)g{x(t)| > 0. Now we define wo as the 
undamped (when # = 0) natural frequency of oscillation. 
The mean square value of this oscillation is 
334 
