He’s frequency–amplitude formulation with average residuals for nonlinear oscillators

Abstract

In He's frequency-amplitude formulation, the relationship between frequency and amplitude of a nonlinear oscillator can be obtained through residuals of two trial solutions. Although a high accurate solution can be obtained, this method has some space to be further improved. Here, we show that the calculation of the residuals can be further simplified without loss of accuracy. Duffing oscillator with high nonlinearity is used as an example to show the solution process and accuracy.

The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators

“…He's frequency formulation was initially inspired by an ancient Chinese algorithm 36 ; now this formulation has gained much attention and has been under development, with various modifications appearing in open literature. [37][38][39][40][41][42][43][44][45] We define a residual function R(t) as…”

“…The coefficient of u in equation (41) is larger than zero, so equation (41) has a periodic solution. When u ( 1, equation (40) becomes u 00 þ u 3 ¼ 0; uð0Þ ¼ A; u 0 ð0Þ ¼ 0…”

“…30,31 The most important property of a nonlinear oscillator is the relationship between the frequency and its amplitude, the simplest method to estimate the frequency-amplitude relationship might be He's frequency formulation [32][33][34] and the max-min approach, 35,36 which are still under development and many modifications were proposed to improve the accuracy. [37][38][39][40][41][42][43][44][45] Consider a nonlinear oscillator in the form u 00 þ fðuÞ ¼ 0; uð0Þ ¼ A; u 0 ð0Þ ¼ 0…”

Approximate frequency–amplitude relationship for a singular oscillator

“…When necessary, different frequency-amplitude relationship results, such as equations (26) and (29) can be utilized to get an average one. Furthermore, combined with equation (19) or (21) we get the acceptable period solution.…”

Frequency and solution of an oscillator with a damping

“…with a velocity-dependent damping. This oscillator can be solved by various methods, [10][11][12][13] among which the variational iteration method accompanied by Laplace transform 14 and the modified He's amplitude-frequency formulation 15 have been caught much attention due to their simple solution process and high accurate solutions.…”