Harmonic balancing approach to nonlinear oscillations of a punctual charge in the electric field of charged ring

Abstract

The harmonic balance method is used to construct approximate frequency-amplitude relations and periodic solutions to an oscillating charge in the electric field of a ring. By combining linearization of the governing equation with the harmonic balance method, we construct analytical approximations to the oscillation frequencies and periodic solutions for the oscillator. To solve the nonlinear differential equation, firstly we make a change of variable and secondly the differential equation is rewritten in a form that does not contain the squareroot expression. The approximate frequencies obtained are valid for the complete range of oscillation amplitudes and excellent agreement of the approximate frequencies and periodic solutions with the exact ones are demonstrated and discussed.

Nonlinear Oscillations of a Point Charge in the Electric Field of Charged Ring Using a Particular He’s Frequency-Amplitude Formulation

“…In this section we study the precision and validity of the solution method employed in the present work. We compare our results with the exact solution obtained in [4], all numerical calculations have been made with the help of the software MATHE-MATICA. A comparative analysis of the error given by the present method and the error of the best approximation obtained by the Harmonic balancing approach [4], is made.…”

“…(20) it is observed that the relative percentage error tends to 5.6% in the limit A → ∞. The best known approximation was obtained by the Harmonic Balance Method (HBM) in [4], whose approximation is…”

“…For small values of the amplitude A it is possible to tconsider the following approximation for ω ex (A), as has been seen in [4] ω ex (A) ≈ 1 − 9 16…”

“…It is very important in applications to have a version of the frequency (or period) to have a better understanding of the phenomena modeled through differential equations that contain terms with high nonlinearities, and a simple mathematical method is very useful for practical applications. Recently, many different methods have been developed to obtain the approximate solutions of nonlinear systems including the harmonic balance method [1,2,3,4], the energy balance method [5,6], the Hamiltonian approach [7,8], the use of special functions [9,10], the max-min approach [11,12], the variational iteration method [13,14,15,16,17] and homotopy perturbation [18,19,20,21,22,23,24]. An excellent study, in which many of these techniques can be found in detail to solve nonlinear problems of oscillatory type can be seen in [25].…”

Considerations on “Harmonic balancing approach to nonlinear oscillations of a punctual charge in the electric field of charged ring”

“…Beléndez, E. Fernández, J.J. Rodes, R. Fuentes and I. Pascual, Considerations on the 'harmonic balancing approach to nonlinear oscillations of a punctual charge in the electric field of charged ring', Physics Letters A, Vol. 373, Nº 46, 4264-4265 (2009 In a previous paper [1] we obtain approximate periodic solutions to the oscillations of a charge in the electric field of a charged ring by applying the harmonic balance method.…”