Dynamics and Control of Magnetostatic Structures

Abstract

The equations governing the dynamics of magnetostatic structures are formulated using Lagrangian mechanics. A potential energy function of gravitational, strain, and magnetostatic components is defined. The Lagrangian equations of motion are discretized and then linearized about equilibrium points created by the additional magnetostatic energy, leading to a linear system of ordinary differential equations. These equations are characterized by mass, stiffness, damping, gyroscopic, and circulatory effects.Four experiments are conducted. Using the one-degree-of-freedom magnetostatic levitator, the measured static displacement is compared to those predicted by the exact nonlinear solution and the discretized approximate solution.Three experiments are performed with the two-degree-of-freedom, spherical, magnetostatic pendulum: The natural frequencies of the pendulum are predicted and compared with measurements; the pendulum is made to track a desired path using electromagnets to control the motion; and the pendulum's oscillations about new equilibrium points are regulated using electromagnets and velocity feedback to control settling time. In the last experiment, the stability of the controlled system is proven by examining the eigenvalues about the new equilibrium position.