## 4 Modeling and Simulation of Non-Circular Gears  
### Determination of Basic Gear Geometric Parameters  
The double arc pitch curve is created, and the \(X\) values are determined based on the model for the normal meshing gear. The gear geometric parameters of non-circular gears are shown in Table 2. Following the selection of the gear settings, CAXA software is used to create the corresponding pitch curves and non-circular gear tooth profiles. Then, three engagement positions are verified to ensure that the non-circular gear pitch curves are tangential at every engagement position and that the designed gears are not shown at each limit position.  
The numerical geometry mapping of \(X=[x_{1}\,y_{1}\,r_{1}\,x_{2}\,y_{2}\,r_{2}]\) after the model for normal meshing gear shows no interference in the tooth profile's first and third limit position meshing positions, proving that the meshing gear requirements are met. This ensures the stability of the alternate meshing of the center wheel, planetary wheel, and inner gear ring, laying the foundation for the later modeling and simulation, as shown in Figure 9.  
CAXA and UG software were used to sketch, model, and assemble double arc non-circular gears, as shown in Figure 10. The non-circular gears include the inner gear ring and the center wheel, with only one planetary wheel completed last to facilitate numerical simulation calculations.  
In this paper, simulations are carried out using ADAMS software with the following setup:  
(1) Assuming that the inner gear ring is fixed and the center wheel rotates;  
(2) Considering that the end faces of the non-circular gears need the ground to be flat, the end faces of the planetary and center wheels are designed to be co-planarly constrained with the end faces of the internal gear ring;  
(3) The initial rotation speed of the center wheel was set to \(20^{\circ}/\mathrm{s}\);  
(4) The meshing contact of the gears between the planetary wheel, the center wheel and the inner gear ring is provided;  
Based on the above setup, the built model is imported into ADAMS to produce the simulation results of angular velocity and engagement force, as shown in Figure 11.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Gear Parameters** & **Values** \\ \hline \(M(mm)\) & 0.75 \\ \(Z_{1}\) & 44.00 \\ \(Z_{2}\) & 66.00 \\ \(Z_{3}\) & 10.00 \\ \(\alpha^{\circ}\) & 20.00 \\ \(lu*\) & 0.75 \\ \(c^{*}\) & 0.25 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Double arc non-circular gear geometric parameters.  
Figure 9: Tooth profile limit position diagram. (**a**) The first engagement positions; (**b**) The third engagement positions.  
As shown in Figure 11, the peak of engagement force is found at 1.5 s and 5.5 s in a stage, because the principal normal vector direction of the inner gear ring at the intersection points suddenly changed at this time. It is revealed that the engagement force of integer teeth decreases by about 17% and is more stable than that of non-integer teeth at two positions.  
The variation of the angular velocity of the center of the planetary wheel around the origin is shown in Figure 12.  
Figure 11: The engagement force between the planetary wheel and the inner gear ring.  
Figure 10: 3D model of a non-circular gear. (**a**) The inner gear; (**b**) The center wheel; (**c**) The planetary wheel; (**d**) The double arc non-circular gear.  
As shown in Figure 12, the maximum fluctuation of angle speed is found at 1.5 s and 5.5 s in a stage, because the principal normal vector direction of the inner gear ring at the intersection points suddenly changed at this time. The difference between the maximum value of the fluctuation range of angular velocity at the intersection point, which is the difference between the maximum and minimum values of angular velocity, is significantly reduced by 17% for integer teeth compared to non-integer teeth.