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在连杆机构的运动分析和综合中,为了建立输入杆同输出杆的位移方程,常常把机构简图画成矢量多边形,并进行以下的演算。先列出多边形上i、j二个顶点在某直角坐标系上的六个坐标(x_i,y_i,z_i;x_j,y_j,z_j)的代数式,再将它们代入二点距离(l_(ij))的平方公式l_(ij)~2=(x_i-x_j)~2+(y_i-y_j)~2+(z_i-z_j)~2;最后展开上式并作整理和化简。演算工作,因多边形的边数和各矢量的指向不同而逐一进行,既复杂而又容易出差错。因此,避免上述的列式和演算,用一个直接公式迅速写出所需要得到的结果,是一个值得研究的问题。本文提出的多边形余弦公式(简称余弦公式)可作为解决这个问题的普遍公式。文中附有若干个用余弦公式直接求连杆机构连架杆的位移方程的例题。
In the motion analysis and synthesis of the link mechanism, in order to establish the displacement equation between the input rod and the output rod, the mechanism is usually drawn as a vector polygon and the following calculation is performed. The algebraic expressions of the six coordinates (x_i, y_i, z_i; x_j, y_j, z_j) of the two vertices of i and j on a rectangular coordinate system are listed first and then they are substituted into the two-point distance (l_ (ij)) The square formula l_ (ij) ~ 2 = (x_i-x_j) ~ 2 + (y_i-y_j) ~ 2 + (z_i-z_j) ~ 2; Finally, expand the above formula and make up and simplify. The calculation work is carried out one by one because of the difference between the number of sides of the polygons and the orientation of the vectors, which is complicated and error-prone. Therefore, to avoid the above formulation and calculation, a direct formula to quickly write down the desired result is a question worthy of study. The polygonal cosine formula (cosine formula) proposed in this paper can be used as a general formula to solve this problem. The paper attached to the cosine formula with a number of links directly to the rod rack body displacement equation example.