## 3 The Active Design of the Pitch Circle of the Integer Gear Model Building
### Pitch Circle Model of Double Arc Non-Circular Gears
The number of teeth on the arcs of the inner ring \(r_{1}\) and \(r^{\prime}_{1}\) is actively rounded based on the value of the nonlinear solution. A part of an integer gear is used as a double arc pitch curve teeth profile, which requires rescaling based on known \(X\) values and satisfying the constraints of the geometry of non-circular gear pitch curve design, as shown in Figure 7.  
Figure 7 depicts three parameters of the inner ring \(x_{1},y_{1}\), and \(r_{1}\), redesigned by the pitch circle model, where \(r_{11}\), \(r_{11}\), and \(x_{11}\) become the core data of the internal ring design. \[\Delta d=r_{1}-r_{11} \tag{10}\]  
\[x_{11}=x_{1}+\Delta d \tag{11}\]  
\(\Delta d\)--The difference after taking an integer number of teeth for the double arc pitch curve; \(x_{11}\)--The \(x_{1}\) of the pitch circle model; \(r_{11}\)--The \(r_{1}\) of the pitch circle model; \(r^{\prime}_{11}\)--The \(r^{\prime}_{1}\) of the pitch circle model;  
Calculating the pitch circle model parameters depends on the geometric and transmission relationships. The latter \(x_{11}\), \(r_{11}\), and \(r^{\prime}_{11}\) are replaced by \(x_{1}\), \(r_{1}\), and \(r^{\prime}_{1}\) to facilitate subsequent calculations and reading. First, the ray with an angle of \(30^{\circ}\) is made from the origin. Then, a circle is drawn with the connection point between \(x_{1}\) and \(r_{1}\) as the center, \(O_{1}O^{\prime}_{1}\) is the sum of the radii of the double arcs \(r_{1}\) and \(r^{\prime}_{1}\), as the radius makes an intersection with the ray, whose distance from the intersection point to the origin is the length of \(y_{1}\). The meeting and source are connected from Figure 7a to Figure 7b. The three basic parameters \(x_{1},y_{1}\), and \(r_{1}\) of the internal gear ring can be obtained through the pitch circle model, which \begin{table}
\begin{tabular}{c c} \hline \(y_{2}\)/mm & **Amount of Interference/mm** \\ \hline
979.1195 & 0.0411 \\
972.8100 & 0.0396 \\
854.7425 & 0.0350 \\
724.2570 & 0.0281 \\
698.3684 & 0.0264 \\ \hline \end{tabular}
\end{table} 
Table 1: The relationship between the \(y_{2}\) value and the interference quantity. Figure 7: Schematic diagram of the solution of the inner gear ring of the pitch circle model. (**a**) The start step; (**b**) The final step. can be used as the basis to achieve the design requirements of a smooth transition of the intersection point of the double arc pitch curve and the number of teeth to be rounded. Similarly, the center wheel can be handled in the same way. The final determination of \(X=[x_{1}\ r^{\prime}_{1}\ r_{1}\ x_{2}\ r^{\prime}_{2}\ r_{2}]\) parameters through the model for the normal meshing gear pitch curve tangent and using the geometric constraints can be found for all the values of \(X=[x_{1}\ y_{1}\ r_{1}\ x_{2}\ y_{2}\ r_{2}]\). In the following procedure of the integer teeth based on the model for normal meshing gear, the problem of rounding up or down for non-integer teeth is verified using orthogonal experiments. The number of integer teeth for each section arc is determined by the orthogonal experiments, which will be addressed in the next section. When compared to the original data, the largest change in the pitch circle model's \(X\) value is 1.7%, while the minimum is 0.0332%. This assures that the nonlinear solution is correct. The tangency of the non-circular gear pitch curve is verified by Equation (9).