Secular Dynamics of Parametrically Excited Magnets: Modeling and Experimental Validation

Abstract

Abstract. Trapping of magnetic microparticles enables them to be used as end-effectors in various microrobotic 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. 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 Mathieu stiffness experimentally. 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. 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 those predicted by theory. These results are subsequently employed to extract the Mathieu 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.