E[Z(t1)] =0, ElZ(t1)Z(t2)] = R2t2-t1), 
E[Z(t1)Z(t2)Z(t3)] = 0, 
E[Z(t1)Z(t2)Z(t3)Z(t4)] = R2(t1 — t2)R2(t3 — t4) 
+ R(t; — t3)R2(t2 — t4) + Ro(t) — t4)R2(t2 — 13). 
(10.49) 
From these relations and the assumption that the external force is white in the important 
range of frequency in which the response shows a peak, 
1 
Rt) = 5 Wod() 
(0(t) is Dirac’s function.) Using these relations in the second equation of Eq. 10.46 gives 
E[X?] = 0% -—3e04. (10.50) 
It is interesting that the same result was derived by the equivalent linearization method, 
Eq. 10.17. 
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