Abstract
We propose a method for finding approximate analytic solutions to autonomous single degree-of-freedom nonlinear oscillator equations. It consists of the harmonic balance with linearization in which Jacobian elliptic functions are used instead of circular trigonometric functions. We show that a simple change of independent variable followed by a careful choice of the form of anharmonic solution enable to obtain highly accurate approximate solutions. In particular our examples show that the proposed method is as easy to use as existing harmonic balance based methods and yet provides substantially greater accuracy.
Highlights
Ordinary differential equations (ODEs in short) are ubiquitous in fundamental science as well as in engineering
We achieve this goal through a simple change of variable based on the properties of the Jacobian elliptic functions, which results to an ODE in which the restoring force is not approximated and which is solved approximately following the method of harmonic balance with linearization (HBwL) or other generalizations of the method of harmonic balance (HB) such as the rational HB
In this paper we have investigated the approximation of periodic solutions to autonomous single degree-offreedom oscillators equations using the Jacobian elliptic function with the objective of improving the method of cubication
Summary
Ordinary differential equations (ODEs in short) are ubiquitous in fundamental science as well as in engineering. In this respect it has been adapted to handling dissipation terms in ODEs [4], and more recently to solving asymmetric oscillator equations [5] Another important class of non-perturbative techniques involves the approximation of the nonlinear restoring force in a given oscillator ODE by some simple forms for which the exact solution can be readily obtained. Our objective is to develop a higher order approximation than the single cnoidale elliptic function which is obtained when a given ODE is first approximated by a Duffing equation We achieve this goal through a simple change of variable based on the properties of the Jacobian elliptic functions, which results to an ODE in which the restoring force is not approximated and which is solved approximately following the method of HBwL or other generalizations of the method of HB such as the rational HB.
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