1 1 
e*= ewfoe, x,y)=— —=( 4+y? (12.92) 
090, Polx,y) ee? 7 | y’) 
x(t) x(t) if 3 
OSS a i aaa 0o= —lo-w6 ’ 
00 W000 4 
Go is the standard deviation of response when € = 0,€ * = €/wo is the nondimensional 
nonlinearity parameter, and po(x,y) is the result whene* = 0 (C = 1 when €* =0). 
From this result, the statistical moments of the response can be calculated. For 
example, Roberts obtained 
2 
(eo) 1 
o = 26|| G8 e? (1—erf 6)| -6| (12.93) 
i) 
u 1 
7 (3e*)!/2 . 
He calculated the equivalent linear damping, as discussed in Section 10.1, using the 
general expression Eq. 12.91 and calculated 
o* (J1+12e*-1) 
— = —————-_ (12.94) 
(oF) 6e€* 
for n = 2 in Eq. 12.91, which is a closer approximation of o* than Eg. 10.17. He also 
calculated the corresponding perturbation solution as 
2 
Ae oeiste (12.95) 
0 
This is slightly different from Eq. 10.50, which was shown for smalle. All three results 
are compared in Fig. 12.6. Robert calls his method, which introduced the energy envelope 
V(t) as Eq. 12.86, the Markov envelope theory, and indicated in Fig. 12.6 as ME theory. 
349 
