Pf ME THEORY 
<i 
EL THEORY 
PERTURBATION 
THEORY 
0 0.1 0.2 0.3 0.4 05 0.6 0.7 0.8 
£&° 
Fig. 12.6. Variation of the mean square of the response with e*. 
Simulation results: * > = 0.05, + = 0.50. 
(From Roberts.°9) 
The equivalent linearization theory is abbreviated as EL theory. In this figure, only the 
perturbation theory appears different from the rest, but here we have to remember that in 
the perturbation method this is the first order approximation and we can improve the ap- 
proximation by increasing the order of the approximation, as indicated in the discussion 
in Section 10.2.2. 
12.6.3 Nonlinear Oscillation in Nonwhite Excitation 
Roberts®! further extended the scope of applicability of this method and solved for 
nonlinear rolling in nonwhite excitation. The equation of motion is now 
Ip + BC) + Kb) = M(t) (12.96) 
or 
¢ + BF(¢) + G@) = x(t) (12.96’) 
where x(t) is a nonwhite excitation, and can have a colored spectrum. 
Again, he adopted the total energy envelope as Eq. 12.86 
62 
v=£+u@) 
where 
¢ 
up) = | Ges 
0 
and considered that this was slowly varying. 
350 
