A(v) = = | ———. (12.77) 
yy y 
The integration range R is such that U(x) < v. Especially for Duffing type oscillation the 
equation of motion, Eq. 12.69, is expressed by 
eee oo. (12.78) 
On, Wn 
Here ¢ is replaced by nondimensional timet,t = @,t, w, being the undamped natural 
frequency in the linear case (A = 0), and the nondimensionalized equation is 
X' +2 Cx’ +x" +A* x3 =d* +2 CEQ), (12.79) 
where the differentiation is with respect tot and 
t(1 + 4ex2) |? € 
= | ———_|_ ,e* = —. 
1/2 
mcs, jee e pa Zes |S. 
Ox o; k \o, e* Op, 
The joint density distribution for p(x’, x’) and the probability density distribution 
function p(x’) is, from Eq. 12.72, 
foo Na G —4—+—+ (7) 85d (12.80 
p(x, x’) exp aS i % ) 
and 
* 2 12 R* 14 
pe) = | po x')dx' = /2a C exp |- a 2 ae \\ | (12.81) 
—c 
Here C is a coefficient for normalization, 
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