Abstract
In this work, the Duffing’s type analytical frequency–amplitude relationship for nonlinear oscillators is derived by using Hés formulation and Jacobi elliptic functions. Comparison of the numerical results obtained from the derived analytical expression using Jacobi elliptic functions with respect to the exact ones is performed by considering weak and strong Duffing’s nonlinear oscillators.
Highlights
It is known that when the rational or irrational functions describe the restoring forces of nonlinear oscillators, the original nonlinear differential equations can be written in an equivalent form using methods that transform the original equations into Duffing-type oscillators and the approximate frequency–amplitude relationship can be determined by the analytical closed-form solutions of the corresponding Duffing-type oscillator
He’s formulation for getting the frequency–amplitude relationship for nonlinear oscillators provides accurate results when the two trial residuals are based on Jacobi elliptic functions
The simplicity of Professor He’s formulation can be expanded to obtain analytical expressions for the frequency of nonlinear oscillators, since most of these can be written as a Duffing-type equation using transformation techniques
Summary
Many dynamic systems that arise in physics and engineering applications are known to be modeled by a homogeneous nonlinear differential equation whose closed-form solution is unknown, numerical or approximate methods have to be used to predict its dynamic response behavior and determine the frequency–amplitude relationship needed to understand its qualitative and quantitative dynamic response.[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] in a few cases, the closed-form solution of some nonlinear oscillators is known. The cubic, the quadratic–cubic, and the cubic–quintic Duffing-type oscillators are those whose analytical closed form solutions are described exactly in terms of Jacobi elliptic functions.[19,20,21,22] It is known that when the rational or irrational functions describe the restoring forces of nonlinear oscillators, the original nonlinear differential equations can be written in an equivalent form using methods that transform the original equations into Duffing-type oscillators and the approximate frequency–amplitude relationship can be determined by the analytical closed-form solutions of the corresponding Duffing-type oscillator. To demonstrate the accuracy achieved by using Jacobi functions to find the approximate frequency–amplitude expression by He’s algorithm, two cases are examined. We provide a brief summary of He’s formulation to derive frequency–amplitude expressions of nonlinear oscillators of the form u€ þfðuÞ 1⁄4 0; uð0Þ 1⁄4 A0; u_ ð0Þ 1⁄4v0. From equation (2), the approximate frequency–amplitude relationship of nonlinear oscillators modeled by an equation of the form y€i þ a0yi þ a1y3i þ a2y5i þ a3y7i þ a22y2i 1⁄4 0; yið0Þ 1⁄4 1; y_i ð0Þ 1⁄4 v (18)
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