Secular dynamics of parametrically excited magnets: modeling and experimental validation

Trapping of magnetic micro-particles enables them to be used as end-effectors in various micro-robotic applications. Parametric excitation is an attractive method to trap such particles, and in such a case, the trapped particle obeys the Mathieu equation. In this paper, an approximate solution is first proposed for the Mathieu equation by using Floquet theory. The approximate solution is shown to be a linear combination of harmonics of the frequency of the parametric excitation and the solutions are shown to agree with numerical results to within 1% for different parameters. Next, a simplified second-order linear time-invariant dynamic model is proposed to describe the secular motion of the magnetic particle. Finally, a simple method is proposed to evaluate the effective stiffness experimentally. Subsequently, experiments are performed to validate the proposed model. It is shown that the experimental results of the particle tracking step and ramp input waveforms are in close agreement with that predicted by theory. These results are subsequently employed to extract the effective stiffness and damping coefficient. Both are shown to agree well with the estimates obtained from theory and to depend in the theoretically expected manner on the frequency and amplitude of parametric excitation.