An accurate relationship between frequency and amplitude to nonlinear oscillations

Abstract

A weighted global error minimization (WGEM) is proposed in this study. The goal is to improve the accuracy of the global error minimization (GEM) based on a weighting function. In addition to the first-order approximation, the fourth-order approximation for the Duffing oscillator is demonstrated by coupling a proper weighting function with the GEM. In order to exhibit the advantage of this modification, the obtained result is compared with both the exact frequency and the outcome of the GEM. The corollary outstandingly reveals that approximations using the WGEM have a lower relative error than those from the GEM in the first-order approximation. Also the modified approach can preserve its accuracy in the fourth-order approximation. The WGEM can be promisingly utilized to other resembling nonlinear problems.

Keywords

• Analytical approximate solution
• nonlinear oscillation
• frequency–amplitude relation
• global error minimization
• weighting function