Nonlinear Oscillations of a Magneto Static Spring-Mass

Haiduke Sarafian

doi:10.4236/jemaa.2011.35022

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JEMAA> Vol.3 No.5, May 2011

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Nonlinear Oscillations of a Magneto Static Spring-Mass

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DOI: 10.4236/jemaa.2011.35022 3,641 Downloads 7,295 Views Citations

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Haiduke Sarafian

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ABSTRACT

The Duffing equation describes the oscillations of a damped nonlinear oscillator [1]. Its non-linearity is confined to a one coordinate-dependent cubic term. Its applications describing a mechanical system is limited e.g. oscillations of a theoretical weightless-spring. We propose generalizing the mathematical features of the Duffing equation by including in addition to the cubic term unlimited number of odd powers of coordinate-dependent terms. The proposed generalization describes a true mass-less magneto static-spring capable of performing highly non-linear oscillations. The equation describing the motion is a super non-linear ODE. Utilizing Mathematica [2] we solve the equation numerically displaying its time series. We investigate the impact of the proposed generalization on a handful of kinematic quantities. For a comprehensive understanding utilizing Mathematica animation we bring to life the non-linear oscillations.

KEYWORDS

Extended Duffing Equation, Magneto Static Spring, Damped Nonlinear Oscillations, Mathematica

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

H. Sarafian, "Nonlinear Oscillations of a Magneto Static Spring-Mass," Journal of Electromagnetic Analysis and Applications, Vol. 3 No. 5, 2011, pp. 133-139. doi: 10.4236/jemaa.2011.35022.

References

[1]	G. Duffing, “Erzwungene Schwingungen bei Veranderlicher Eigen-frequenz,” F. Vieweg und Sohn, Braunschweig, 1918.

[2]	S. Wolfram, “The Mathematica Book,” 5th Edition, Cambridge University Publications, Cambridge, 2003, and MathematicaTM software V8.0, 2011.

[3]	H. Sarafian, “Static Electric-Spring and Nonlinear Oscillations,” Journal of Electromagnetic Analysis & Applications (JEMAA), Vol. 2, No. 2, February 2010, pp. 75-81.

[4]	J. R. Retiz and F.J. Milford, “Foundations of Electromagnetic Theory,” Addison-Wesley, Hoboken, 1960.

[5]	J. D. Jackson, “Classical Electrodynamics,” 3rd Edition, Wiley, Hoboken, 2005.

[6]	H. Sarafian, “Dynamic Dipole-Dipole Magnetic Interaction and Damped Nonlinear Oscillations,” Journal of Electromagnetic Analysis & Applications (JEMAA), Vol. 1, No. 4, 2009, pp. 195-204.