A simplified He’s frequency–amplitude formulation for nonlinear oscillators

Abstract

Derived from an ancient Chinese algorithm, He’s frequency–amplitude formulation is an effective approach to finding an approximate solution of a nonlinear oscillator. In this article, based on He’s formulation, a simplified formulation is proposed. Some nonlinear oscillators are adopted as examples to demonstrate the solving process using this simplified formulation. Through the demonstration, it can be seen that the solving process is simplified.

A simple modified harmonic balance method for strongly nonlinear oscillator with cubic non-linearity and harmonic restoring force

“…Homotopy perturbation method was applied by others 13,14 appropriately. Also, many researchers were applied the harmonic balance method, 15 the energy balance method, 16 He's frequency formulation method, 17,18 the variational iteration method, [19][20][21][22] the homotopy analysis method, 23,24 VIM-Pade technique 25,26 etc. successfully to solve strongly nonlinear problems.…”

An Efficient Analytical Approach for the Periodicity of Nano/Microelectromechanical Systems’ Oscillators

“…In the past decades, several techniques have been proposed to get the approximate analytic solution of N/ MEMS problems such as the homotopy perturbation method (HPM), higher-order HPM [8], Taylor series [9], energy balance technique [10], spreading residual harmonic balance method [11], higher-order Hamiltonian method [12], Adomian decomposition method (ADM) [13], Li-He modified HPM [14], modified ADM [15], variational approach [16], Galerkin decomposition method [17], and so on. It is also noted that, besides these methods, there are various analytical techniques for getting the approximate solution to the nonlinear equations, for example, the He-Laplace method [18], global residual harmonic balance method [19], integral transform-based methods [20][21][22], max-min approach [23], frequency-amplitude formulation method [24], Hamiltonian approach [25], and others [26][27][28][29]. Moreover, there have been several review articles that have appeared on the analytical methods for oscillatory problems during the past decade [30][31][32].…”