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 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.

带有一个在导体附近摆动的永久磁铁的摆锤是一个典型的实验，用于演示电磁制动和伦兹感应定律。当导体本身移动时，它可以将能量传递到摆锤上。这种电磁相互作用的精确分析模型对于平板导电板是可能的。在准静态极限中，即当磁扩散率足够高以确保磁场对流和扩散的平衡时，通过移动的磁偶极子在板中感应出的涡流以及所产生的力和转矩在解析上是已知的。这使我们能够在存在沿垂直方向振动的水平平板的情况下研究具有磁偶极矩的简单摆。在三种情况下，即在磁矩平行于旋转轴时，或者在其他情况下，其在运动平面上的投影可以是水平的或垂直的，都可以在三种情况下实现摆在垂直位置的平衡。稳定性问题由带有阻尼项的Mathieu型微分方程描述。仅当振动幅度和板与磁体之间的距离满足与板的同时激发和阻尼效应相关的某些约束时，不稳定才可能发生。对于磁矩和旋转轴平行的情况，对非线性运动进行了数值研究。当本征频率与激励频率相比足够小时，就会发现混沌行为。板振动通常对倒立摆具有稳定作用。