## 5 The Meshing Stiffness of the Trochoid Roller Pinion Rack
### Calculation of Time-Varying Meshing Stiffness  
As the deformation flexibility and stiffness are reciprocal, the normal meshing stiffness of the trochoidal roller pinion rack with double tooth meshing is expressed in Equation (32). \[k=\sum_{i=1}^{2}\frac{1}{\delta_{h}+\delta_{b1,i}+\delta_{s1,i}+\delta_{a1,i}+ \delta_{b2,i}+\delta_{s2,i}+\delta_{a2,i}} \tag{32}\]  
Figure 22 shows the meshing situation of trochoidal roller pinion rack with double tooth meshing. The meanings of each parameter are the same as those in Figure 19. Because the force direction of the meshing point is not consistent with the tangential direction, so the meshing stiffness in the normal direction needs to be converted to the tangential direction of the roller pinion, and the meshing stiffness in the tangential direction of the roller pinion is calculated by Equation (33). \[k_{\mathrm{t}}=\sum_{j=1}^{2}\frac{\cos\theta_{i}}{\delta_{h}+\delta_{b1,j}+ \delta_{s1,i}+\delta_{a1,i}+\delta_{b2,j}+\delta_{s2,i}+\delta_{a2,i}} \tag{33}\]The triangle \(D_{2}OO^{\prime}\) in Figure 18, \(D_{2}\) is the meshing point, its coordinate equation is Equation (5). \(O\) is the origin (0, 0), The \(O^{\prime}\) coordinate is (0, \(R\)). Angle \(OD_{2}O^{\prime}\) can be calculated by using the cosine theorem in the triangle, and the calculation method of angle \(\theta_{i}\) can be calculated by Equation (34). \[\theta_{i}=\left|\frac{\pi}{2}-\arccos\left(\frac{\left(x_{D}^{2}+y_{D}^{2} \right)+\left(x_{D}^{2}+\left(R-y_{D}\right)^{2}\right)-R^{2}}{2\sqrt{x_{D}^{ 2}+y_{D}^{2}}\sqrt{x_{D}^{2}+\left(R-y_{D}\right)^{2}}}\right)\right| \tag{34}\]  
The distance between the meshing point and the center of the roller pinion changes at different meshing moments, as shown in Figure 21. The meshing radius \(R_{1}\) is \(D_{1}O^{\prime}\) at point \(D_{1}\), and the meshing radius \(R_{2}\) is \(D_{2}O^{\prime}\) at point \(D_{2}\). The meshing stiffness in the tangential direction of the meshing point \(k_{i}\) is transformed into the meshing stiffness of the trochoidal roller pinion rack \(k_{rp}\) by Equation (35). \[k_{rp}=\sum_{i=1}^{2}\frac{\cos(\theta_{i})R_{i}^{2}}{(\delta_{h}+\delta_{b1,i }+\delta_{s1,i}+\delta_{a1,i}+\delta_{b2,i}+\delta_{s2,i}+\delta_{a2,i})} \tag{35}\]  
The calculation method of \(R_{i}\) is shown in Equation (36). \[R_{i}=\sqrt{x_{D}^{2}+\left(R-y_{D}^{2}\right)} \tag{36}\]  
The meshing stiffness per tooth width of the trochoidal roller pinion rack can be obtained by calculating the parameters of the trochoidal roller pinion rack shown in Table 4, as shown in Figure 23. The meshing stiffness curve is the time-varying meshing stiffness curve of a roller pin in the whole process from entering meshing to leaving meshing. The meshing stiffness in Figure 23 shows the dividing line of single tooth and double tooth meshing area. That is double tooth meshing area at the beginning of meshing, at this point the second roller pin has just started to enter the mesh, the meshing stiffness of double tooth meshing area increases first and then decreases. The reason for the low meshing stiffness at the starting point is that the angle \(\theta_{1}\) is large at the beginning of double tooth meshing area, as shown in Figure 22, therefore, the value of meshing point \(D_{1}\) stiffness converted to tangent direction is very small, so the meshing point does not \begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Value** \\ \hline Number of roller pins \(z\) & 10 \\ Module \(m/\)mm & 7.5 \\ Rack tooth profile offset coefficient \(c_{y}\) & 0.305 \\ Addendum coefficient of rack \(c_{h}\) & 1.233 \\ Diameter coefficient of roller pin \(c_{f}\) & 1.867 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters used in meshing stiffness calculation. Figure 22: Schematic diagram of double tooth meshing of the trochoidal roller pinion rack. play an important role in meshing, and the force is mainly borne by \(D_{2}\).