Stephenson-Ⅲ六杆机构死点位置的结式消元法识别

Identification the Dead-center Positions of Stephenson-Ⅲ Six-bar Linkages by Sylvester Resultant

提出了一种基于结式消元的Stephenson-Ⅲ型六杆机构死点位置识别的新方法。Stephenson-Ⅲ六杆机构输入-输出方程的重根和机构的死点与双点位置是一一对应的。将Stephenson-Ⅲ六杆机构看作由一个四杆链和一个五杆链组成,首先分别建立这两个运动链的闭环矢量方程,然后利用消元法和正切半角替换推导出机构的输入-输出多项式方程和关于机构双点的一元六次方程;基于Sylvester结式定理求得机构输入-输出方程的重根和双点方程的实数根;从输入-输出方程的重根中删除机构的双点位置即得到机构的死点位置构型。

In this paper,a new method based on Sylvester resultant for discerning the dead-center positions of the Stephenson-Ⅲ six-bar linkages is presented.The repeated roots of the input-output polynomial equation of a linkage have one-one correspondence with its dead-center positions and the double positions.A Stephenson-Ⅲ six-bar linkage consists of a four-bar chain and a five-bar chain.Firstly,the two-loop closure equations of the linkages are established.Then,the input-output polynomial equation and the six-order polynomial equation about the double points of the linkage are derived by using the Sylvester resultant and the half-angle substituting.Based on the Sylvester resultant the repeated roots of the input-output polynomial equation and the solutions of the six-order polynomial equation are obtained.Lastly,the true dead-center positions of the Stephenson-Ⅲ six-bar linkages can be gained throuhh wiping off the double positions from the Sylvester resultant and the half-angle substituting.