论文标题
带有库仑摩擦的移动平台上的耦合元体
Coupled metronomes on a moving platform with Coulomb friction
论文作者
论文摘要
使用理论,实验和模拟的结合,我们在移动平台上重新审视了两个耦合的元体的动力学。我们的实验表明,平台的运动受到库仑型干摩擦力的阻尼,而不是过去通常假设的粘性线性摩擦力。在此结果的提示下,我们开发了一种新的数学模型,该模型以前引入的模型为基础,但在平台上对摩擦的处理方面偏离了它们。我们通过两次计算分析分析模型,并得出决定其长期动态的慢流程。由于平台在某些参数方面的粘板滑移运动,缓慢流的推导是具有挑战性的。慢速流量的模拟揭示了各种长期行为,包括相同的监测器,相位锁定和非相同监测器的相位漂移以及节拍器抑制和死亡的相同相位和反相同步。在这两个状态下,一个或两个监测器以低幅度摆动,以至于它们不再具有逃避机制。我们发现我们的理论,模拟和实验之间有很好的一致性,但强调我们的探索远非详尽。的确,尽管它们的简单性和熟悉程度,但仍有许多关于耦合的MetRonomes动力学的知识。
Using a combination of theory, experiment, and simulation, we revisit the dynamics of two coupled metronomes on a moving platform. Our experiments show that the platform's motion is damped by a dry friction force of Coulomb type, not the viscous linear friction force that has often been assumed in the past. Prompted by this result, we develop a new mathematical model that builds on previously introduced models, but departs from them in its treatment of the friction on the platform. We analyze the model by a two-timescale analysis and derive the slow-flow equations that determine its long-term dynamics. The derivation of the slow flow is challenging, due to the stick-slip motion of the platform in some parameter regimes. Simulations of the slow flow reveal various kinds of long-term behavior including in-phase and antiphase synchronization of identical metronomes, phase locking and phase drift of non-identical metronomes, and metronome suppression and death. In these latter two states, one or both of the metronomes come to swing at such low amplitude that they no longer engage their escapement mechanisms. We find good agreement between our theory, simulations, and experiments, but stress that our exploration is far from exhaustive. Indeed, much still remains to be learned about the dynamics of coupled metronomes, despite their simplicity and familiarity.