dP 1 
pea) 2 ap le [2 V(a) - veel (x.[2 ,[2{ V(a)- vealf) ae C28) 
0 
Through manipulation, the following were derived: 
pea) =C-t(a)- F(a) exp <4 = pp(avg' T(a) (12.59) 
xe 
and also, 
V 
pp(a) = F(a)-0;? exp 4- ~ ; (12.60) 
x 
where 7% is the expected number of threshold crossings per unit time of the level x = 0 
with positive slope, and can be derived as v4 = Co?; t(a) is the average period as a 
function of amplitude a. 
Crandall showed the general solution for the following two nonlinear cases: 
i) fora hard spring Duffing type system, 
FIK()) = wg {X(e) + x20}, (12.61) 
ii) fora set-up spring system, as in Fig. 12.2, 
F(X) = 0g {X() +e sgn XO}, (12.62) 
340 
