Nonlinear Oscillations of a Magneto Static Spring-Mass

Haiduke Sarafian

doi:10.4236/jemaa.2011.35022

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Journal of Electromagnetic Analysis and ... > Vol.3 No.5, May 2011

Nonlinear Oscillations of a Magneto Static Spring-Mass ()

Haiduke Sarafian

DOI: 10.4236/jemaa.2011.35022 PDF HTML XML 3,879 Downloads 7,559 Views Citations

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

Share and Cite:

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.

Conflicts of Interest

The authors declare no conflicts of interest.

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.