An analytical method for approximating high-order Galerkin solutions

The high-order Galerkin procedures for computing forced oscillations of nonlinear systems are generally impractical to apply analytically. In this paper, an analytical method for approximating high-order Galerkin solutions is presented, which is applicable to ordinary differential equations with polynomial nonlinearities. The procedure is restricted to oscillations whose predominant Fourier component is the fundamental; hence subharmonic oscillations may be allowed, whereas superharmonic oscillations are excluded. The approach taken is to consider only the first-order effects of the higher harmonics. An example is given which demonstrates that the accuracy of the method can be quite impressive.