## 3 Equations of Line of Action  
Some scholars have studied the line of action of gears for a long time. The research results show that the line of action of the gear is a straight line, which is the internal common tangent of the base circle of the two gears. There is no research on the line of action of the trochoidal roller pinion rack, so this section first deduces the equation of line of action. The determination of the line of action of the trochoidal roller pinion rack needs to be analyzed from the formation process of its tooth profile. Reference [6] carried out a detailed study of the trochoid rack tooth profile, and its meshing principle and tooth profile equation have been studied. This paper further studies the line of action of the trochoid roller pinion rack on the basis of reference [6]. The meshing principle is used to deduce the equations of the line of action, but since no scholars have studied the line of action of the trochoid roller pinion rack before, in order to prove the correctness of the derivation of the equation of the line of action in this paper, in Section 4 of this paper, kinematics simulation of the trochoid roller pinion rack is carried out to prove the correctness of the theoretical derivation. Figure 4 is the schematic diagram of the formation of the trochoid rack tooth profile [6]. As shown in the Figure 4, the rack pitch line is the \(x\)-axis, and the tooth profile symmetry axis is the \(y\)-axis to establish a coordinate system. \(r\) is the radius of the reference circle of the roller pinion, \(R\) is the radius of the pitch circle of the roller, \(O_{c}\) is the center of the two circles, and the initial position of \(O_{c}\) is on the \(y\)-axis. The pitch circle of the roller rolls along the \(x\)-axis, then the path traversed by a point \(M^{\prime}\) on the reference circle of the roller pinion is the theoretical tooth profile of the trochoid, the curve formed by a point \(N^{\prime}\) that is apart from the normal direction of the theoretical tooth profile by \(r_{f}\) is the actual tooth profile of the trochoid rack. The point \(N^{\prime}\) is also the meshing point at a certain moment of meshing. When considering the movement of the roller pinion and rack, the line formed by the meshing point \(N^{\prime}\) at each moment during the movement of the trochoid roller pinion rack is the line of action. According to the meshing characteristic that the meshing point must be on the straight line formed by the center of the meshing roller pin and the pitch node, the equation of line of action is deduced in this paper. Figure 3: Schematic diagram of meshing of trochoidal roller pinion rack. The coordinate system is established as shown in Figure 5, the origin \(O\) of the coordinate system is the pitch node, the \(X\)-axis is the moving direction of the rack, and the \(Y\)-axis passes through the center of the roller pinion. When the rotation angle of the roller pinion is \(\varphi\), its meshing state is shown in Figure 5. Point \(A\) is the center of the meshing roller pin, point \(B\) is the intersection of point \(A\) and the \(X\)-axis, point \(D\) is the meshing point, and point \(C\) is the intersection of point \(D\) and the \(X\)-axis. The triangle \(ODC\) is similar to the triangle \(OAB\), so these line segments satisfy the relationship of Equation (1). \[\frac{\overline{OC}}{\overline{OB}}=\frac{\overline{DC}}{\overline{AB}}=\frac{ \overline{OD}}{\overline{OA}} \tag{1}\]  
It can be seen from Figure 5 that the length of \(\overline{OB}\) is the absolute value of abscissa of point \(A\), the length of \(\overline{AB}\) is the value of ordinate of point \(A\), the length of \(\overline{OC}\) is the absolute value of abscissa of point \(D\), and the length of \(\overline{DC}\) is the value of ordinate of point \(D\). Since point \(A\) is on the reference circle of the roller pinion, the coordinate value of point \(A\) can be expressed by Equations (2). \[ \begin{cases}x_{A}=-\frac{mz}{2}\sin\varphi\\ y_{A}=\frac{mz}{2}+mc_{y}+\frac{mz}{2}\cos\varphi\end{cases}