Nonlinear Oscillations of a Magneto Static Spring-Mass
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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.
- G. Duffing, “Erzwungene Schwingungen bei Veranderlicher Eigen-frequenz,” F. Vieweg und Sohn, Braunschweig, 1918.
- S. Wolfram, “The Mathematica Book,” 5th Edition, Cambridge University Publications, Cambridge, 2003, and MathematicaTM software V8.0, 2011.
- H. Sarafian, “Static Electric-Spring and Nonlinear Oscillations,” Journal of Electromagnetic Analysis & Applications (JEMAA), Vol. 2, No. 2, February 2010, pp. 75-81.
- J. R. Retiz and F.J. Milford, “Foundations of Electromagnetic Theory,” Addison-Wesley, Hoboken, 1960.
- J. D. Jackson, “Classical Electrodynamics,” 3rd Edition, Wiley, Hoboken, 2005.
- 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.