A bifurcation and symmetry discussion of the Sommerfeld effect

TL;DR

This paper analyzes the Sommerfeld effect using a mathematical model, revealing bifurcation phenomena, hysteresis, and symmetry effects that explain the resonance capture and release in rotating systems.

Contribution

It provides a detailed bifurcation and symmetry analysis of the Sommerfeld effect, highlighting the role of limit cycles and phase space symmetry in the system's dynamics.

Findings

Resonance curves fold back, creating bifurcations and hysteresis.

Small oscillations are amplified near the natural frequency.

Higher imbalances lead to more complex bifurcation scenarios.

Abstract

The Arnold Sommerfeld effect is an intriguing resonance capture and release series of events originally demonstrated in 1902. A single event is studied using a two degree of freedom mathematical model of a motor with imbalance mounted to laterally restricted spring connected cart. For a certain power supplied, in general the motor rotates at a speed consistent with a motor on a rigid base. However at speeds close to the natural frequency of the cart, it seemingly takes on extra oscillations where for a single rotation it both speeds up and then slows down. Therefore in a standard experimental demonstration of the effect, as the supplied torque force is increased or decreased, this may give the illusion that the stable operation of the motor is losing and gaining stability. This is not strictly the case, instead small oscillations always present in the system solution are amplified near the resonant frequency. The imbalance in the motor causes a single resonance curve to fold back on itself forming two fold bifurcations which leads to hysteresis and an asymmetry between increasing and decreasing the motor speed. Although as outlined the basic mechanism is due the interplay between two stable and one unstable limit cycles, a more complicated bifurcation scenario is observed for higher imbalances in the motor. The presence of a Z2 phase space symmetry tempers the dynamics and bifurcation picture.