Analysis of highly nonlinear oscillation systems using He’s max-min method and comparison with homotopy analysis and energy balance methods

Abstract

Nonlinear functions are crucial points and terms in engineering problems and the solutions of many important physical problems are centered on finding accurate solutions to these functions. In this paper, a new method called max-min method has been presented for deriving accurate/approximate analytical solution to strong nonlinear oscillators. Furthermore, it is shown that a large class of linear or nonlinear differential equations can be solved without the tangible restriction of sensitivity to the degree of the nonlinear term, adding that the method is quite convenient due to reduction in size of calculations. Results obtained by max-min are compared with Homotopy Analysis Method (HAM), energy balance and numerical solution and it is shown that, simply one term is enough to obtain a highly accurate result in contrast to HAM with just one term in series solution. Finally, the phase plane to show the stability of systems is plotted and discussed.

Non-linear vibration of Euler-Bernoulli beams

“…In recent years, some promising approximate analytical solutions have been proposed, such as Frequency Amplitude Formulation [13], Variational Iteration [5,6,14,17], Homotopy-Perturbation [3,4,7,24], Parametrized-Perturbation [18], Max-Min [15,19,29], Differential Transform Method [16], Adomian Decomposition Method [22], Energy Balance [23,30], etc.…”

A local differential transform approach for the cubic nonlinear Duffing oscillator with damping term

“…Various versions of the approximate solution techniques have been used to find solutions of the nonlinear and conservative Duffing equation [6][7][8][9][10][11][12][13]. The homotopy analysis method [6], harmonic balance method [7], the homotopy Padé technique [8], energy balance method [9], coupled homotopy variational approach [10], the Newton harmonic balance method [11], parameter-expanding and max-min approach [12,13], coupling of energy and harmonic balance method [14], Jacobi elliptic functions [15], parameter based perturbation technique [16] have all been used to solve the nonlinear Duffing equation without damping effect. If the Duffing oscillator involves the damping effect, the amplitude of the oscillation decreases with time, then one obtains a nonconservative system.…”

Analytical Solution For Nnonlinear Vibration Of Micro-electromechanical System (mems) By Frequency-amplitude Formulation Method

“…Therefore special techniques should be applied to solve them. Many of these techniques have been performed in recent literatures such as Homotopy Perturbation Method (HPM) [14][15][16][17][18], Homotopy Analysis Method (HAM) [19][20][21], Iteration Perturbation Method (IPM) [22][23][24], Variational Iteration Method (VIM) [25][26][27][28], Differential Transformation Method (DTM) [29][30], Frequency Amplitude Formulation (FAF) [31][32][33], Max-Min Approach (MMA) [34][35][36][37][38].…”