Equations 12.46 and 12.47 can be applied to a nonlinear process as long as the 
process is narrow banded. Crandall®* analyzed a hardened spring oscillation system, the 
same as the one analyzed by Caughey* utilizing the Fokker—Planck equation, as was 
shown in Section 12.4. Equation of motion is by Eq. 12.20, 
X(t) + BX(t) + F{X(2)| = fir). 
He obtained the average period t(a), the probability distribution function of the peak val- 
ues pp(a), and the probability distribution function of the envelope p,(a) as functions of 
the amplitude a by introducing the potential energy per unit mass 
x 
V(x) = | F(6)dé. (12.54) 
0 
The solution of the Fokker—Planck equation, Eq. 12.31 (when f(x) is Gaussian) is 
Tea | exe 
p(x,x) =C exp A) ap Vix)? }, (12.55) 
of| 2 
where C is a normalizing constant that makes 
ioe] foo) 
| | Deak) adn = i 
—-Om —O 
The results are, as for the envelope process a(t), 
V(a) = 2 + V(x), (12.56) 
a [2[Via)-Vix)]f2 
Pia. <a)=4 | dx | p(x, x) dx (12.57) 
0 0 
339 
