## 3 Equations of Line of Action
 \tag{2}\]  
The length of \(\overline{AO}\) can be obtained from the triangle \(OAB\) according to the coordinates of point \(A\), and the length of \(\overline{AD}\) is the radius of the roller pin, so the Equation (1) can be changed to Equation (3). \[\frac{x_{D}}{-\frac{mz}{2}\sin\varphi}=\frac{y_{D}}{\frac{mz}{2}+mc_{y}-\frac {mz}{2}\cos\varphi}=\frac{\sqrt{{x_{A}}^{2}+{y_{A}}^{2}}-\frac{mc_{f}}{2}}{ \sqrt{{x_{A}}^{2}+{y_{A}}^{2}}} \tag{3}\]  
Figure 4: Schematic diagram of the formation of trochoid rack tooth profile. Figure 5: Schematic diagram of meshing point of trochoid roller pinion rack. According to Equation (3), the coordinate equations of the meshing point \(D\) are derived as Equations (4). \[\left\{ \begin{array}{c}x_{D}=-\frac{mz}{2}\sin\varphi\frac{\sqrt{\left(-\frac{mz} {2}\sin\varphi\right)^{2}+\left(\frac{mz}{2}+mc_{y}-\frac{mz}{2}\cos\varphi \right)^{2}}-\frac{mc_{f}}{2}}{\sqrt{\left(-\frac{mz}{2}\sin\varphi\right)^{2 }+\left(\frac{mz}{2}+mc_{y}-\frac{mz}{2}\cos\varphi\right)^{2}}}\\ y_{D}=\left(\frac{mz}{2}+mc_{y}-\frac{mz}{2}\cos\varphi\right)\frac{\sqrt{\left( -\frac{mz}{2}\sin\varphi\right)^{2}+\left(\frac{mz}{2}+mc_{y}-\frac{mz}{2} \cos\varphi\right)^{2}}-\frac{mc_{f}}{2}}{\sqrt{\left(-\frac{mz}{2}\sin \varphi\right)^{2}+\left(\frac{mz}{2}+mc_{y}-\frac{mz}{2}\cos\varphi\right)^{2 }}-\frac{mc_{f}}{2}}\\ \end{array} \right. \tag{4}\]  
The above process is the derivation of the theoretical equation of line of action, but the line of action has a certain length in the actual meshing process. During the meshing process of the roller pins and the rack, the roller pins engage from the root of the tooth and mesh out at the top of the tooth, so the calculation of the actual length of line of action is actually the calculation of the position of the meshing in and meshing out. In this paper, the meshing in point and meshing out point are defined by the rotation angle of the roller pinion, and the meshing in point at the root of the tooth is defined in this paper to be \(0^{\circ}\), and the meshing out point at the top of the tooth is \(\varphi_{h}\). Figure 6 is the schematic diagram of the meshing in point and meshing out point. The solid line is the meshing situation at the beginning of meshing. Point 1 is the meshing in point, at this time, the angle of the roller pinion is \(0^{\circ}\), and the dotted line is the meshing situation at the ending of meshing, and point 2 is the meshing out point, at this time, the rotation angle of the roller pinion is \(\varphi_{h}\). The calculation of the rotation angle of the roller pinion at the meshing point 2 also needs to be analysed from the formation principle of the trochoid profile. The meshing process of the roller pinion and the rack is actually the formation process of the trochoid rack tooth profile. Figure 7 shows the position of the roller pin when the roller pin is out of engagement. \(XOY\) is the coordinate system during meshing, which coincides with the \(X\)-axis in Figure 5. The difference is that the coordinate system is moving relative to the rack, but the coordinate value in the \(Y\)-axis direction is the same as the value when the tooth profile is formed. The value of \(Y\)-axis coordinate of the meshing out point is \(h+a\). Figure 6: Schematic diagram of the meshing in and out points of the trochoid roller pinion rack. Figure 7: Schematic diagram of the moment when the roller pin is out of engagement. The profile equations of the trochoid rack are Equations (5) [6]. According to the above analysis and the coordinate equations of the trochoid profile, the roller pinion angle \(\varphi_{h}\) can be obtained by Equation (6). \[\left\{