Abstract
The Equivalent Linearization Method (ELM) with a weighted averaging is applied to analyze five undamped oscillator systems with nonlinearities. The results obtained via this method are compared with the ones achieved by Parameterized Perturbation Method (PPM), Min–Max Approach (MMA), Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Energy Balance Method (EBM), Harmonic Balance Method (HBM), 4th-Order Runge-Kutta Method, and the exact ones. The obtained results demonstrate that this method is very convenient for solving nonlinear equations and also can be successfully applied to a lot of practical engineering and physical problems.
Highlights
Nonlinear oscillation problem is very important in the physical science, mechanical structures, and other kinds of mathematical sciences
Most of real systems are modeled by nonlinear differential equations which are important issues in mechanical structures, mathematical physics, and engineering
Many researchers have been working on various analytical methods for solving nonlinear oscillation systems in the last decades, such as Homotopy Perturbation Method (HPM) [1,2,3,4,5], Max–Min Approach (MMA) [5,6,7,8,9,10], Variational Iteration Method (VIM) [2, 5], Energy Balance Method (EBM) [5, 11,12,13,14,15,16], Amplitude-Frequency Formulation (AFF) [5,6,7, 15, 17, 18], Improved Amplitude-Frequency Formulation (IAFF) [5], Parameter Expansion Method (PEM) [5, 7, 18,19,20], Homotopy Analysis Method (HAM) [5, 21], Modified Homotopy Perturbation Method (MHPM) [5, 6], Modified LindstedtPoincare Method [22], Harmonic Balance Method [23, 24], and combined Newton’s Method with the Harmonic Balance Method [25]
Summary
Nonlinear oscillation problem is very important in the physical science, mechanical structures, and other kinds of mathematical sciences. Anh proposed a new way for determining averaging values: instead of using conventional averaging process the author introduced weighted coefficient functions h(t) [31]; by this manner, the averaging value is calculated in a new way called the weighted averaging value. This proposed method has been applied effectively to analyze some strongly nonlinear oscillations such as the nonlinear Duffing oscillator with third, fifth, and seventh powers of the amplitude, the strongly nonlinear oscillators in forms (1 + εu2)ü + u = 0 and ü + u3/(1 + u2) = 0, and the cubic Duffing with discontinuity [33]. Accuracy and validity of results are presented by comparing the results with the ones obtained by the other well-known techniques and the exact and numerical ones
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