Flux-gradient symmetry controls harmonic generation and nonlinear electromagnetic damping

The damping of a magnet oscillating in a coil can change qualitatively when the equilibrium position is shifted by only a few centimeters. We show that this behavior is governed by the local symmetry of the magnetic flux gradient: its lowest nonzero spatial variation selects the harmonic content of the induced current, and this harmonic structure determines the functional form of the amplitude decay. A slowly varying amplitude analysis of the coupled mechanical–electrical equations predicts a second-harmonic current with cubic nonlinear damping at the solenoid midpoint, a fundamental-frequency current with predominantly linear damping at off-center positions, and higher-order corrections near the extrema of the flux gradient. Measurements using a neodymium–iron–boron magnet oscillating in a solenoid confirm these spectral signatures and show that the damping half-life varies by a factor of seven across a 3-cm displacement of the equilibrium position. The results identify local field geometry as a practical control parameter for electromagnetic dampers and energy harvesters.