Short Communication
Determination of periodic solution for a u1/3 force by He's modified Lindstedt–Poincaré method

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Abstract

This paper applies He's modified Lindstedt–Poincaré method to determine the periodic solutions of oscillators in a u1/3 force. The result obtained and comparison with analytical solution provides confirmation for the validity of He's Modified Lindstedt–Poincaré method.

Introduction

In nonlinear analysis, perturbation methods are well-established tools to study diverse aspects of nonlinear problems. Surveys of the early literature with numerous references, and useful bibliographies, have been given by Nayfeh [1], Mickens [2], Jordan and Smith [3] and Hagedorn [4]. However, the use of perturbation theory in many important practical problems is invalid, or it simply breaks down for parameters beyond a certain specified range. Therefore, new analytical techniques should be developed to overcome these shortcomings. Such a new technique should work over a large range of parameters and yield accurate analytical approximate solutions beyond the coverage and ability of the classical perturbation methods. For example, some extensions of the Lindstedt–Poincaré perturbation method to strongly nonlinear systems, so-called He's Modified Lindstedt–Poincaré method, have been proposed; see the comprehensive book by He [5] and the references therein. In He's Modified Lindstedt–Poincaré method, a constant, rather than the nonlinear frequency, is expanded in powers of the expanding parameter to avoid the occurrence of secular terms in the perturbation series solution. The results show that the obtained approximate solutions are uniformly valid on the whole solution domain and they are suitable not only for weakly nonlinear systems, but also for strongly nonlinear systems.

There also exists a wide range of literature dealing with the approximate determination of periodic solutions for nonlinear problems by using a mixture of methodologies [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].

The purpose of this paper is the determination of the periodic solutions to nonlinear oscillator equations for which the elastic restoring forces are non-polynomial functions of the displacement by applying He's Modified Lindstedt–Poincaré method. This class of equations represents a new class of nonlinear oscillating systems which were first studied in detail by Mickens [18].

Section snippets

He's modified Lindstedt–Poincaré method

Currently, we will study the properties of the periodic solutions to certain nonlinear oscillators by applying He's modified Lindstedt–Poincaré method for which the elastic restoring forces are non-polynomial functions of the displacement. In particular, this term is chosen to bef(x)=-x1/3.As it is well known that the Lindstedt–Poincaré method [1], [2], [3], [4] gives uniformly valid asymptotic expansions for the periodic solutions of weakly nonlinear oscillators, in general, the technique is

Conclusion

In summary, we conclude that the method illustrated here is very accurate for entire solution domain. It is extremely simple and easy to use. We think that the method have great potential which still needs further development.

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  • Cited by (0)

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