無幌擒縱器擺鐘的同步

Abstract

同步是常見的現象，涵蓋的範圍也相當廣泛，在生物、物理、通訊、醫療、工程等領域皆有重要的應用。Huygens觀察兩個掛在牆上的擺鐘的行為，最早注意到機械系統的同步現象，也引發後續許多的研究。前人的研究大多為利用范德波爾振盪器模擬節拍器或將節拍器內部擺錘所受的驅動力矩假設為一常數以求得擺錘的運動方程式，來探討同步現象。目前尚未發現有學者透過節拍器或擺鐘內部擒縱裝置的幾何形狀推導擺錘的運動方程式。因此本論文嘗試分析以無幌擒縱器為計時機構的擺鐘，由無幌擒縱器的幾何形狀，利用Lagrange方程式推導擺錘的運動方程式，探討同步現象。 本研究發展了無幌擒縱器擺鐘的動態模型，利用無幌擒縱器的幾何形狀、動作特性與Lagrange方程式推導無幌擒縱器擺鐘的運動方程式。首先將無幌擒縱器擺鐘放置於地面上，探討擺錘振幅和自發性週期運動的關係。接著將一個無幌擒縱器擺鐘放置於可自由運動的水平基底上，探討水平基底對無幌擒縱器擺鐘的影響。 在充分了解單一無幌擒縱器擺鐘固定於地面與水平基底上的運動特性後，將多個無幌擒縱器擺鐘放置於可自由水平運動的基底上，利用數值積分，探討其間的交互作用，及可能的同步行為。我們分析初始角度、擺長、基底質量等參數對同步形式的影響，探討不同同步形式的吸引區域，比較不同同步形式的週期。最後，我們利用諧和平衡法，推導不同形式同步解的振幅和頻率。

Synchronization is a common phenomenon and has important applications in many different fields such as biology, physics, communication, medicine, engineering, etc. Huygens was the first one who observed the synchronization of two pendulum clocks hung on the wall. Stimulated by Huygens’ observation, many researchers started to study the synchronization of pendulum clocks. The key part of a pendulum clock is the escapement mechanism, which has been either modeled as a Van der Pol oscillator or as two constant pulses acting on the pendulum in the literature. To the best of our knowledge, no one has derived the governing equations of a pendulum clock taking into the account the geometric shape of the escapement mechanism. In this thesis, we studied the synchronization of pendulum clocks with the deadbeat escapement mechanism. With the help of Lagrange’s equations, we derived the governing equations according to the shapes and dimensions of the parts composing the deadbeat escapement mechanism. We first placed a deadbeat escapement clock on the ground to study the relation between the driving moment and the period of the pendulum. Then we put a deadbeat escapement clock on a plate, which can move freely in the horizontal direction, to investigate the influence of the movement of the plate on the motion of the pendulum clock. After that, we put several pendulum clocks on a plate and use numerical integration to study possible synchronization patterns of the pendulum clocks. We investigated the effects of system parameters, e.g., length of the pendulum, mass of the plate, initial angles, on the synchronization patterns. We examined the domains of attraction of different synchronization patterns. Finally, we employed the method of harmonic balance to determine approximately the amplitude and frequency of different synchronization patterns.