耦合擺鐘的同步形式分析

Abstract

Huygens最早發現擺鐘的同步現象，隨後有許多學者研究多個擺鐘間的同步形式。Czolczynski等人[18]從能量的觀點將 個擺鐘的同步行為分類。他們將 個擺鐘依相位分組，同相位的分為一組，不同相位的視為不同組。特別的，相位差為180度的2個擺鐘，稱為一對反相對。他們認為n個擺鐘同步時，只可能為分為1、3或5組。此外，當n為偶數時，也可能分為n/2對反相對。但是簡伯丞[19]透過數值積分以及諧和平衡法發現可能存在其他種的同步形式。本研究的目的在於釐清這個歧見，並改採多尺度法進行分析。多尺度法是一個較精確的方法，也可以較方便的使用在阻尼系統中。首先利用多尺度法分別推導單一擺鐘固定於地板與固定於水平滑動基底上振幅的解析解，並分析其穩態行為。再發展為n個擺鐘固定於水平滑動基底上的相角關係式並進行討論。最後透過數值積分驗證所得的結果。

Huygens is the first to observe the synchronization of pendulum clocks. Then, several researchers started looking into further insights of the synchronization between several pendulum clocks. Czolczynski et al. [18] categorized the synchronization of n pendulums based on a minimum total energy aspect. They claimed that n synchronized pendulums can be grouped into one, three or five clusters. A cluster may contain only one pendulum. All the pendulums in a cluster are in fully synchronization with one another. There is a fixed phase difference between two clusters. Besides, when n is even, the pendulums can be grouped into n/2 pairs of pendulums synchronized in anti-phase. However, Po-Cheng Chien[19] found out that there may exist other types of synchronization via numerical integration and the method of harmonic balance. This thesis aims to clarify the disagreement. The method of multiple scales, which is more accurate and suitable for a damped system, is employed. The analytical solutions of the steady-state amplitude of a single pendulum fixed on a floor and a horizontal sliding base, are derived respectively. The results are then extended for the case of n pendulums fixed on a horizontal sliding base. Steady state behavior of the n-pendulum and sliding base system is discussed in detail. Finally, numerical integration are applied to verify the derived conclusions.