The frequency-amplitude formulation with ω4 for fast insight into a nonlinear oscillator

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…”

The homotopy perturbation method for a nonlinear oscillator with a damping

“…Equation (1) can describe the string vibration or the membrane vibration under damping. There are many analytical methods for solving effectively nonlinear oscillators, for example, the variational iteration method, [1][2][3][4] the variational approach, [5][6][7][8] the homotopy perturbation method, [9][10][11][12][13][14][15][16][17][18] He's frequency formulation [19][20][21][22][23] and others, but most methods are limited to conservative oscillators, and it is difficult to solve nonlinear oscillators with damping terms. This paper shows that the homotopy perturbation method is also valid for solving equation 1analytically.…”

He’s frequency–amplitude formulation for nonlinear oscillator with damping

“…It was hardly possible for scientists all over the world to solve nonlinear oscillators accurately in early stage of nonlinear science. Now the condition is changed, some effective methods have been developed for the analyses of nonlinear oscillators, such as the variational approach, [2][3][4][5][6] the homotopy perturbation methods, [7][8][9][10][11][12][13] the variational iteration method, [14][15][16][17] He's frequency-amplitude formulation, [18][19][20][21] the parameter expansion method, 22 the Hamiltonian approach, 23 the stability analysis, 24 the max-min approach, 25 the energy balance method, 26 and so on. Although most of the methods are suitable for conservative oscillators, there is some literature focusing on damping cases.…”