加速度極心位於反曲點圓上，因此，若以機構某位置之加速度極心為耦桿點，當此耦桿點移動至此加速度極心之鄰近位置時，其運動除了是有限長度的近似直線位移外，又具備在此加速度極心位置為瞬時等速的特性。因此，其運動特性可以解釋為：在一直線上無窮接近的三個等距點。在實務上，需要在工作點附近為一有限長度近似直線的等速運動時，加速度極心是此類應用之工作點的最佳選擇。
本論文之目的，在於推導兩型平面連桿機構耦桿的加速度極心位置之參數式，及其軌跡之多項式方程式，並且分別提出設計程序，藉以合成耦桿點通過兩個加速度極心之機構。本研究所探討之平面連桿機構，以鄰接固定桿之曲柄為輸入桿，並且此輸入桿為等速轉動，包括曲柄滑件機構及四連桿機構。
首先，利用加速度極心為耦桿上加速度值為零之點的條件，分別推導出兩型平面連桿機構的加速度極心位置參數式，並將加速度極心軌跡分別描繪於固定平面及耦桿平面上加以觀察。接著，利用耦桿平面之加速度極心軌跡的交點，作為機構中可通過兩個加速度極心的耦桿點，並且配合已推導出來之參數式，合成通過兩個加速度極心的機構。
其次，針對曲柄滑件機構，當已知曲柄滑件機構尺寸，且曲柄為等速轉動時，先求出耦桿的加速度角之解析式，再根據加速度極心和耦桿上任何一點的連線與該點之加速度向量所夾的角度均等於加速度角的觀念，推導曲柄滑件機構之加速度極心軌跡，分別推導出描繪在固定平面及耦桿平面上之多項式方程式。
最後，針對四連桿機構，當四連桿機構尺寸為已知，且曲柄為等速轉動時，曲柄動軸樞之加速度方向指向曲柄定軸樞，因此可以求得曲柄動軸樞之加速度角方程式。另外，由於加速度極心位於反曲點圓上，加速度極心到反曲點中心與加速度極心到速度極心之夾角為 。由上述兩個條件，加上Freudenstein方程式，可以推導出四連桿機構之加速度極心軌跡，此為描繪在固定平面之多項式方程式。再利用坐標轉換原理，可以得到描繪在耦桿平面之加速度極心軌跡的多項式方程式。
由上述推導方程式之結果顯示，曲柄滑件機構描繪在固定平面及耦桿平面之加速度極心軌跡，分別為二十四次及二十八次的多項式方程式；而四連桿機構描繪在固定平面及耦桿平面之加速度極心軌跡，均為三十次的多項式方程式。以上之多項式方程式的圖形均是全圓點曲線。除此之外，本論文並探討各加速度極心軌跡曲線與任意直線之交點數目。

Since the acceleration pole of a coupler is located on the inflection circle for a specific position, if it is chosen as the coupler point, the path of the coupler point in the vicinity of the acceleration pole has the properties of an approximate straight line within a limited distance and a constant velocity in the vicinity of the acceleration pole. As a result, the kinematic characteristics of an acceleration pole can be stated that it can be treated as three infinitesimally separated and equally spaced points on an approximate straight line. In a practical application, the acceleration pole is the best choice if it is required to design path generators for tracing an approximate straight line with a constant velocity in the vicinity of a working point.
This work presents the parametric equations of acceleration poles and the polynomial equations for the loci of acceleration poles of the couplers of two planar linkages. Moreover, the design procedures for synthesizing the mechanisms with a coupler point tracing through two acceleration poles are proposed. The planar linkages discussed in this study are the slider crank and four-bar mechanisms, in which the input crank with a constant rotaion speed is adjacent to the fixed link.
First of all, the parametric equations of acceleration poles of the couplers of the two planar linkages are derived by utilizing the concept that an acceleration pole is a point having zero acceleration on the coupler. Then, the loci of acceleration poles described on the fixed and coupler planes, respectively, are examined. For a given mechanism, the coupler points passing through two acceleration poles are the double points on the loci of acceleration poles described on the coupler plane. Furthermore, the two mechanisms with a coupler point tracing through two acceleration poles are synthesized by utilizing the derived parametric equations.
Secondly, for the slider crank mechanism with given dimensions and having a constant crank speed, the analytical form of the acceleration angle of the coupler is developed. Then, according to the theorem that the angle between the acceleration vector of any point on the coupler plane and the line joining the point to the acceleration pole is equal the acceleration angle, the polynomial equations of the loci of acceleration poles described on the fixed and coupler planes are derived, respectively.
Finally, for the four-bar linkage with given dimensions and having a constant crank speed, since the acceleration of the moving pivot of the crank points toward its fixed pivot, the equation of the acceleration angle of the moving pivot can be obtained. In addition, the angle between the acceleration pole to the inflection pole and to the velocity pole is . The polynomial equation of the loci of the acceleration pole described on the fixed plane for a four-bar linkage is derived by utilizing the above two conditions and the Freudenstein equation of a four-bar linkage. Furthermore, the polynomial equation of the loci of the acceleration pole described on the coupler plane for a four-bar linkage can be obtained by using the coordinates transformation.
As a result, the degrees of the polynomial equations of the loci of acceleration poles described on the fixed and coupler planes for a slider crank mechanism are twenty-four and twenty-eight, respectively. The degrees of the polynomial equations of the loci of acceleration poles described on the fixed and coupler planes for a four-bar linkage are both the same as thirty. All of the above polynomial equations have full circularity. In addition, the intersections of the loci of acceleration poles with an arbitrary line are also discussed in this work.

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