Homotopy Analysis Method for Large-Amplitude Free Vibrations of Strongly Nonlinear Generalized Duffing Oscillators

Abstract

In this study, the homotopy analysis method (HAM) is used to solve the generalized Duffing equation. Both the frequencies and periodic solutions of the nonlinear Duffing equation can be explicitly and analytically formulated. Accuracy and validity of the proposed techniques are then verified by comparing the numerical results obtained based on the HAM and numerical integration method. Numerical simulations are extended for even very strong nonlinearities and very good correlations which achieved between the results. Besides, the optimal HAM approach is introduced to accelerate the convergence of solutions

Highly accurate analytical solution for free vibrations of strongly nonlinear Duffing oscillator

“…A a b c ω HAM [8] ω EBM [16] ω FAF [16,29] ω Present ω Exact [ A a b ω HPM [10] ω SAAM [10] ω FAF [16,24] ω EBM [16] ω HP [9] ω Present Runge-Kutta [9] 0.1 and…”

“…Many researchers have applied various approximate methods to analyze different types of nonlinear Duffing oscillator equations. Some of these methods are multiple scales Lindstedt-Poincare method, 7 homotopy analysis method, 8 homotopy Pade technique, 9 stiffness analytical approximation method, 10 homotopy perturbation method, [10][11][12][13][14][15] frequency amplitude formulation, [16][17][18][19][20] energy balance method, 21 straightforward expansion method, 22 global error minimization method, 23 max-min approach, 24 global residue harmonic balance method, [25][26][27][28] variational approach, 29 perturbation method, 30,31 Hamiltonian approach [32][33][34] harmonic balance method, 35 and coupled homotopy-variational approach. 36 The purpose of the current work is to apply a suggested perturbation technique to obtain higher order approximate periodic solutions for strongly nonlinear Duffing oscillators.…”