台車上擺鐘的同步型式分析

Abstract

Huygens在1665年最早觀察到兩個懸掛在橫梁下擺鐘的同步現象，隨後有許多學者研究不同情形下擺鐘的同步型態，例如滑動台車和旋轉圓盤上的擺鐘。Czolczynski等人於2009年時，從能量觀點分析滑動台車上的 個擺鐘的同步型式。他們將擺鐘依穩態時的相角分類，同相角的擺鐘歸為一組。他們認為 個擺鐘同步時，除了所有擺鐘同相同步和兩兩一組反相同步外，只存在分為3組或5組（各組的擺鐘間同相同步）的同步型式。但是不管是在2018年使用諧和平衡法的簡伯丞，還是在2020年使用多重尺度法的陳芝郁在分析5個擺鐘時皆有發現其它的同步型式。然而，隨著擺鐘數目的持續增加，擺鐘複雜的運作機構使得分析擺鐘的同步型式變得非常的困難。 本論文旨在分析多個擺鐘固定於可水平移動的台車上時，其穩態運動所展現的同步型式。為了簡化分析，本文將結構複雜的無幌擒縱器擺鐘轉化為受衝擊力矩的單擺，並證明其定性行為與擺鐘相符。接著，使用多重尺度法推導單擺系統在小參數擾動下的一階近似解析解，建立振幅與相角之穩態關係式，並與數值積分方法結果交叉驗證。最後利用所得的穩態關係式，分析 個擺鐘可能的同步型式。

In 1665, Huygens was the first to observe the synchronization phenomenon which happened between two pendulum clocks suspended from a common horizontal beam. Since then, many researchers have studied the synchronization of pendulum clocks under various configurations, such as pendulum clocks on carts and rotating disks. In 2009, Czolczynski et al. analyzed the synchronization patterns of multiple pendulums on a cart from an energy-based perspective. They divided the pendulums into groups according to their phase angles in the steady state. Pendulums with identical phase angles are categorized as one cluster. They claimed that, apart from complete in-phase synchronization and pairwise anti-phase synchronization, only synchronization patterns involving three or five clusters are possible. Nonetheless, other types of synchronization for five pendulum clocks were observed by Po-Cheng Chien in 2018 using the harmonic balance method, and by Chih-Yu Chen in 2020 using the method of multiple scales. However, with the increasing number of pendulum clocks, the intricate nature of their mechanisms renders the analysis of their synchronization patterns highly challenging. This thesisaims to analyze the steady-state synchronization behaviors of multiple pendulum clocks on a horizontally movable cart. The complex escapement mechanism of a real pendulum clock is modeled as a single pendulum subjected to impulsive torques. It is demonstrated that this simplified model qualitatively matches the dynamic behavior of a pendulum clock. Using the method of multiple scales, we derive a first-order approximate analytical solution under small perturbations, establishing steady-state relationships between amplitudes and phase angles of the pendulums. These results are cross-validated with numerical integration. Last, we use the derived steady-state relationships to analyze the possible synchronization patterns of multiple pendulums.