Period/frequency estimation of a nonlinear oscillator

Abstract

Fast estimation of a nonlinear oscillator is very much needed in engineering, the perturbation method is valid only for weakly nonlinear oscillators. This paper gives some simple formula for this purpose, only simple calculation is needed to have a relatively high accuracy of the estimated frequency.

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…”

“…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…”

Vibration of an axially moving jet in a dry spinning process

“…This paper is a preliminary study for the string-like vibration of a moving jet. A more sophisticated model is needed to exactly study the jet vibration, and the nonlinear system has to be solved by the homotopy perturbation method, [23][24][25][26] and the frequency-amplitude relation becomes much more complex than that discussed in literature. 27 This is a preliminary theoretical analysis on the effect of vibration on the spinning process.…”

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

“…After decades of continuous efforts, some excellent methods appeared in open literature, such as the homotopy perturbation methods, [1][2][3][4][5][6][7][8][9][10][11] the variational iteration method, [12][13][14][15][16][17] the parameter expansion method, 18 the max-min approach, 19 the variational approach, [20][21][22][23] the Hamiltonian approach, 24 the energy balance method, 25 the stability analysis, [26][27][28][29] and the simple formulae for fast prediction of frequency. 30 To solve nonlinear oscillators, Chinese mathematician Prof. Ji-Huan He proposed a simple but effect frequency-amplitude formulation. [31][32][33][34] He's frequency-amplitude formulation was used by many authors with great success.…”