Mor x+kx 
RESTORING 
COEFFICIENT 
AMPLITUDE 
Fig. 10.1. Restoring coefficient. 
However, for the stochastic case, when the exciting term is stochastic in character, 
we cannot use Egs. 10.3 and 10.2 directly. T. K. Caughey”? formulated this case as 
follows. Nonlinear oscillation can be expressed as 
X+ax+wox+y - 9(x,%,t) = fio), (10.4) 
where 7 is small, g(x,x, 7) includes all nonlinear effects, and f(r) is a random excitation. 
When the equivalent linearized damping coefficient @,, and the linearized restoring 
coefficient w2, are obtained, 
X+ ak + wegx = fi), (10.5) 
and the error e(x, x,t) is expressed by 
e(x, x,t) = (@ —Qeg)X + (Wh — Weg X +1 . B(x, 4,1). (10.6) 
Namely, eq, We, are to be set to minimize the time average 
of 
or 5 1 2) 5 
le“l= — | e(x,x, t)dt. (10.7) 
—T 
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