Parametric instability of a magnetic pendulum in the presence of a vibrating conducting plate
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ORIGINAL PAPER
Parametric instability of a magnetic pendulum in the presence of a vibrating conducting plate Thomas Boeck Tatiana Becker
· Seyed Loghman Sanjari ·
Received: 2 March 2020 / Accepted: 24 October 2020 © The Author(s) 2020
Abstract A pendulum with an attached permanent magnet swinging in the vicinity of a conductor is a typical experiment for the demonstration of electromagnetic braking and Lenz’ law of induction. When the conductor itself moves, it can transfer energy to the pendulum. An exact analytical model of such an electromagnetic interaction is possible for a flat conducting plate. The eddy currents induced in the plate by a moving magnetic dipole and the resulting force and torque are known analytically in the quasistatic limit, i.e., when the magnetic diffusivity is sufficiently high to ensure an equilibrium of magnetic field advection and diffusion. This allows us to study a simple pendulum with a magnetic dipole moment in the presence of a horizontal plate oscillating in vertical direction. Equilibrium of the pendulum in the vertical position can be realized in three cases considered, i.e., when the magnetic moment is parallel to the rotation axis, or otherwise, its projection onto the plane of motion is either horizontal or vertical. The stability problem T. Becker acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) under Grant No. BE 6553/1-1. T. Boeck (B) · S. L. Sanjari Institute of Thermodynamics and Fluid Mechanics, Technische Universität Ilmenau, P. O. Box 100565, 98684 Ilmenau, Germany e-mail: [email protected] T. Becker Technical Mechanics Group, Technische Universität Ilmenau, P. O. Box 100565, 98684 Ilmenau, Germany e-mail: [email protected]
is described by a differential equation of Mathieu type with a damping term. Instability is only possible when the vibration amplitude and the distance between plate and magnet satisfy certain constraints related to the simultaneous excitation and damping effects of the plate. The nonlinear motion is studied numerically for the case when the magnetic moment and rotation axis are parallel. Chaotic behavior is found when the eigenfrequency is sufficiently small compared to the excitation frequency. The plate oscillation typically has a stabilizing effect on the inverted pendulum. Keywords Magnetic pendulum · Conducting plate · Vibration · Electromagnetic induction · Parametric instability
1 Introduction Time-dependent magnetic fields cause electromagnetic induction, i.e., the generation of eddy currents in a conducting material. Induction is essential for numerous technological applications such as transformers, generators, electric motors and induction heating. A typical experiment of the Lorentz force caused by induction consists of a pendulum with a metal disk that swings next to a magnet [1]. The relative motion between the magnet and the disk leads to the induction of eddy currents that cause a braking of the disk’s motion. However, there are only few cases when the problem of
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