Analysis of synchronization of n metronomes on a cart via describing function method: New results beyond two metronomes

2016 American Control Conference (ACC) Pub Date : 2016-07-06 DOI:10.1109/ACC.2016.7526710

X. Xin, Yoshinori Muraoka, S. Hara

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引用次数: 6

Abstract

This paper aims to analyze the synchronization phenomena in the system consisting of n identical metronomes hanging from an elastically fixed cart by using describing function (DF) method, which deals the driving torque as a discontinuous function of the angle and angular velocity of each metronome. This paper presents a unified DF solution of the system which shows that there exist two types of synchronization with the first harmonic of the angle of each metronome having the same amplitude. One is the out-of-phase synchronization; that is, there exist at least two of all metronomes having a nonzero phase shift. The other is the in-phase synchronization; that is, the phases of all metronomes are the same. Regarding the out-of-phase synchronization, this paper shows that the frequency and amplitude of its first harmonic of the angle of each metronome, are independent of n and the mechanical parameters related to the cart, and are exactly the same as those of the single metronome hanging from a fixed base. This paper shows that the out-of-phase synchronization is stable and presents a new result of the phase shifts of n ?? 1 metronomes with respect to a referenced metronome. The analytical results are validated by numerical simulation for three metronomes.

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用描述函数法分析车上n个节拍器的同步:超越两个节拍器的新结果

本文采用描述函数(DF)方法，将驱动转矩视为每个节拍器的角度和角速度的不连续函数，分析了由悬挂在弹性固定车上的n个相同节拍器组成的系统中的同步现象。本文给出了该系统的统一DF解，该解表明存在两种类型的同步，每个节拍器角度的一阶谐波具有相同的振幅。一是失相同步;也就是说，存在至少两个节拍器具有非零相移。二是同相同步;也就是说，所有节拍器的相位都是相同的。对于失相同步，本文证明了每个节拍器角度的一阶谐波的频率和幅值与n和与小车相关的机械参数无关，与挂在固定底座上的单个节拍器的频率和幅值完全相同。本文证明了该同步是稳定的，并给出了n ??相移的新结果。1个节拍器相对于一个参考节拍器。通过对三个节拍器的数值模拟验证了分析结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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2016 American Control Conference (ACC)

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