## 4 Verification and Variation Rules of the Line of Action
### Verification of the Equation of the Line of Action  
The values of roller pinion angle and rack moving distance for the five special points of Figure 8 according to the theoretical formula are calculated. The theoretical calculated results of the five points are shown in Table 3.  
The trochoid roller pinion rack is modelled in 3D according to the parameters data in Table 2, and kinematics is simulated by using Adams. The moving pair is applied to the rack, the rotating pair is applied to the roller pinion, the drive is applied to the rack moving pair with a speed of 20 mm/s, the contact force is set between the rack and the roller pins, and the simulation time is set to 5 s. The number of steps is set to 1000, and the simulation results are shown as following.  
Figure 9 shows the meshing situation at the beginning of the simulation, which corresponds to point 1 in Figure 8. The green roller pin is defined as \(a\), the black roller pin is defined as \(b\), and the black vertical line is defined as the line with the \(X\)-axis (horizontal axis) coordinate value of 0 in Figure 7, its position is fixed. At this time, the rolling pin \(a\) begins to engage.  
Figure 10 corresponds to the simulation result of point 2 in Figure 8. The number of simulation steps at this time is the 32 nd. Since the total number of steps is 1000, the moving distance of the rack is 3.2000 mm at this moment, and the corresponding roller pinion angle is 3.0558\({}^{\circ}\).  
Figure 11 corresponds to the simulation result of point 3 in Figure 8. The number of simulation steps at this time is the 65 nd. The moving distance of the rack is 6.5000 mm at this moment, and the corresponding roller pinion angle is 6.2070\({}^{\circ}\).  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{3}{*}{**Point Number**} & **Theoretical** & **Simulation** & **Error of the** & **Simulation** & **Theoretical** & **Simulation** & **Error of** & **Simulation** \\  & **Value of Rack** & **Value of Rack** & **Back** & **Uncertainty** & **Value of** & **Value of** & **Roller Pinion** & **Uncertainty** \\  & **Moving** & **Moving** & **Moving** & **Moving** & **Moving** & **Moving** & **Moving** & **Moving** \\  & **Distance/mm** & **Distance/mm** & **Distance** & & **Angle/\({}^{\circ}\)** & **Angle/\({}^{\circ}\)** & **Angle** & **Pinion** \\ \hline
1 & 0 & 0 & 0 & 0\% & & 0 & 0\% & \\
2 & 3.1800 & 3.2000 & 0.63\% & & 3.0367 & 3.0558 & 0.63\% & \\
3 & 6.5259 & 6.5000 & 0.40\% & & 6.2318 & 6.2070 & 0.40\% & \\
4 & 37.6991 & 38.0000 & 0.80\% & & 36.0000 & 36.2873 & 0.80\% & \\
5 & 52.9539 & 52.9000 & 0.10\% & & 50.5672 & 50.5158 & 0.10\% & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simulation errors of the line of action.  
Figure 9: Simulation result of point 1.  
Figure 12 corresponds to the simulation result of point 4 in Figure 8. The number of simulation steps at this time is the 380 nd. The moving distance of the rack is 38.000 mm at this moment, and the corresponding roller pinion angle is 36.2873\({}^{\circ}\).  
Figure 13 corresponds to the simulation result of point 5 in Figure 8. The number of simulation steps at this time is the 529 nd. The moving distance of the rack is 52.900 mm at this moment, and the corresponding roller pinion angle is 50.5158\({}^{\circ}\).  
According to the simulation results, the theoretical calculation values of the actual line of action and the errors of the simulation results for the five special points are calculated as shown in Table 3. It can be seen that the theoretical value of the line of action calculated by the formula and the value obtained by the software simulation have errors within 1%, so it can be proved that the derivation of the equation of the line of action of the trochoid roller pinion rack in this paper is correct. According to the variance calculation method, the rack simulation uncertainty is 0.0189 and the roller simulation uncertainty is 0.0172.