Euler-Savary方程式在平面直接接觸機構的應用

Abstract

平面機構的運動學理論中，Euler-Savary方程式（ESE）作為一個經典又簡潔的公式，被廣泛使用在定位平面連桿機構中耦桿點移動路徑的曲率中心，進而合成特定的連桿機構，卻鮮少有人將其應用在直接接觸機構。在齒輪機構中，應用ESE可以更加瞭解齒形的形成過程及齒輪嚙合的原理；在凸輪機構中，將凸輪輪廓與從動件的接觸點視為在平面上運動的點，ESE也可以透過凸輪輪廓與從動件間的相對運動定位未知凸輪輪廓的曲率中心。過去ESE在盤形凸輪輪廓曲率上的應用之所以窒礙難行，是由於凸輪與從動件相對運動的反曲點圓難以被找到。 本文透過將平面機構中的相對運動轉變為瞬心線之間的相對滾動，定位桿件之間相對運動的反曲點圓，最後再將ESE應用在決定直接接觸機構輪廓的曲率中心。此方法不僅能夠快速地求得曲率半徑，也能同時求得機構輪廓，是個有效瞭解直接接觸機構相對運動且簡化輪廓曲率計算及合成的方法。

In the theory of kinematics of planar mechanism, the Euler-Savary equation is a classical and concise formula extensively utilized to locate the center of curvature of coupler point path of planar linkages, moreover, in the synthesis of specific linkages. Nonetheless, limited research has been done on the application of the ESE to direct contact mechanisms. In gear mechanisms, enhanced understanding of the generation process and fundamental laws of gears can be achieved by utilizing the ESE concept. In cam mechanisms, the contact point between the cam profile and the follower is regarded as a moving point on a plane, thus, the radius of curvature of the unknown cam profile can be located through the relative motion between the cam and the follower. The challenge in applying the ESE for determining the center of curvature of the disk cam profile arose from the burdensome accessibility of the inflection circle describing the relative motion between the cam and the follower. In this paper, we transform the relative motion in planar mechanisms into the relative rolling between the centrodes, and then define the inflection circle between them. Afterward, the ESE can be advantageously applied to locate the center of the curvature of the direct contact mechanism profile. Hence, the radius of curvature of the gear or cam profile can be found and synthesized simultaneously. This method is beneficial for understanding the relative motion of direct contact mechanisms and simplifying the calculation and synthesis of the profile curvature.