索末菲效应的分岔和对称性讨论
A bifurcation and symmetry discussion of the Sommerfeld effect
摘要：
阿诺德·索末菲效应（Arnold Sommerfeld Effect）是一个有趣的共振捕获和释放系列事件，最初于1902年被证明。一个单一的事件进行了研究，使用一个两个自由度的数学模型的电动机与不平衡安装到横向限制弹簧连接的车。对于所供应的一定功率，通常马达以与刚性基座上的马达一致的速度旋转。然而，在接近推车的固有频率的速度下，它似乎会发生额外的振荡，其中对于单个旋转，它既会加速，然后又会减速。因此，在该效果的标准实验演示中，随着所供应的扭矩力增加或减少，这可能会产生电动机的稳定操作正在失去和获得稳定性的错觉。这不是严格的情况下，相反，小振荡总是存在于系统的解决方案被放大附近的谐振频率。电动机中的不平衡导致单个谐振曲线折回到其自身上，形成两个折叠分叉，这导致滞后以及增加和减小电动机速度之间的不对称。虽然如所概述的基本机制是由于两个稳定和一个不稳定的极限环之间的相互作用，一个更复杂的分叉的情况下观察到的电机中的更高的不平衡。Z2相空间对称性的存在缓和了动力学和分叉图片。
英文摘要：
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.
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