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 ...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.展开更多
文摘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.
