MSD OUM SEDO s50 2 (10.41) 
Substituting Eq. 10.41 in Eq. 10.40, gives 
[Xo + 2aXo + wiXo —fit)] + €[X1 + 2aX) + WX) + WHe(Xo)] 
+ €7[X> + 2aX> + wiX2+ wpe(X)] +... 
0. (10.42) 
This equation is independent of €, so each term with powers of € should independently 
be equal to zero, 
Xo + 2aXo + weXo = fit) 
X, + 2aX, + woX, = —we(Xo) (10.43) 
Xp + 2aX2 + wEX2 = — wog(K1) = — w9X 1g" (Xo). 
Usually the first approximation X = Xp + €X, is used. From Eq. 9.16 
oo 
Xo = | h(t) fit —t)dt. Here Xp is the linear response. Accordingly, from the first equation 
—-o 
of Eq. 10.43 
h(t) = —== sin («3 = a?)?r (10.44) 
{@3-2) 
Then from the second equation of Eq. 10.43 
X(t) = (1% | h(t)gXo(t—T)dt , (10.45) 
which is the expression for the first order approximation. 
Here some expectations are 
291 
