橢圓齒輪生成8字形軌跡運動特性之研究

On the Study of Motion Characteristics of Figure-Eight Trajectory Generated by Elliptical Gears

沈意加(Yi-Jia Shen)

國立虎尾科技大學/工程學院/動力機械工程研究所/碩士(2006年)

摘要

本研究主要是對橢圓齒輪生成之8字形軌跡進行運動特性分析。本文利用向量迴路法、微分幾何以及幾何關係，推導出一對橢圓齒輪各被動桿及耦桿上一點生成8字形軌跡之數學模式，並分析一對橢圓齒輪在不同橢圓離心率時生成8字形軌跡之運動特性。由一對橢圓齒輪生成8字形軌跡之運動特性值，離心率越大時，生成的8字形軌跡，具等速運動區間較大，但8字形軌跡起始與終止之速度及加速度變化大。本研究利用二對橢圓齒輪組成之橢圓齒輪系，建立橢圓齒輪系各被動桿及耦桿上一點生成8字形軌跡之數學模式，並分析橢圓齒輪系在第一對橢圓齒輪與第二對橢圓齒輪具一相位角時，橢圓齒輪系生成8字形軌跡之運動特性值。在不同相位角中，離心率越大，橢圓齒輪系生成之8字形軌跡在起始與終止之速度及加速度也較大。當相位角 ，橢圓齒輪離心率e=0.4時，橢圓齒輪系生成8字形軌跡趨近等速之區間最大。本研究針對一對橢圓齒輪生成8字形軌跡之數學模式進行無因次化。應用變輸入轉速函數觀念，依據設計準則並訂定最佳化之目標函數，設計出最佳輸入轉速函數。由最佳化設計之結果，變輸入轉速函數的應用可改善以等速輸入時，橢圓齒輪耦桿上一點速度及加速度峰值太高之現象，且8字形軌跡上以等速運行之區間也較等速輸入時長。本研究之成果可應用在光纖頭套端面研磨機構上的研磨路徑產生機構。

並列摘要

In this research, the motion characteristics of Figure-eight trajectory generated by elliptical gears are analyzed. By using the vector method, differential geometry, and geometry relationship, the mathematical model of follower link and Figure-eight trajectory which generated by a point on the coupler-link on elliptical gears is established. Moreover, the motion characteristics of Figure-eight trajectory generated by elliptical gears with different eccentricity are analyzed. The results of motion characteristics of Figure-eight trajectory generated by elliptical gears, the eccentricity is greater, Figure-eight trajectory has constant larger velocity part but the range of velocity and acceleration at start and finish are greater. In this research, the elliptical gears train is composed by two pairs elliptical gears is applied, the mathematical model of follower link and Figure-eight trajectory which generated by a point on the couple-link on elliptical gears train is also established. Moreover, the motion characteristics of Figure-eight trajectory generated by elliptical gears train have phase angles are analyzed. In different phase angles, the eccentricity is greater, the velocity and acceleration of Figure-eight trajectory generated by elliptical gears train at start and finish are greater. When the phase angles ???/2 and the eccentricity of elliptical gears e=0.4, the constant velocity part of Figure-eight trajectory generated by elliptical gears train is greater. In this research, normalized mathematical model of Figure-eight trajectory generated by elliptical gears is established. Applying the concept of variable input speed, the rule of design of variable input speed, and the defined optimal objective function, the optimal function of input speed are obtained. By the result of optimal design, variable input speed function can improve the peak value of velocity and acceleration of point on the coupler-link on elliptical gears. The constant velocity part of Figure-eight trajectory is longer than constant input speed. The results of this study can apply to design the path generated mechanism of polishing machine to polish the optical fiber end-face.

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