## 4 Modeling and Simulation of Non-Circular Gears
### Orthogonal Experiments of Non-Circular Gears Based on the Pitch Circle Model  
The challenge of rounding non-integer teeth on different pitch curves requires 24 trials based on the model for normal meshing gear under a nonlinear solution. The orthogonal experimental design is a strategy for studying numerous components and levels that relies on orthogonality to select some representative points from a thorough test that are uniformly dispersed and comparable.  
In the process of a trial rotation, the double arc non-circular gear jamming phenomenon is expected in non-circular tooth assembly because the planetary wheel force suddenly increases in the operation to a particular position. Building on a quantitative criterion to the peak forces on planetary wheels for non-circular gear tooth rounding is necessary. To select the best combination of teeth, the planetary rotation is allowed to run for one week, and the meshing force is measured at any moment. At the same time, the increase in the sum of two peak meshing forces of the planetary wheels in cycles is used as an indicator. Equation (18) is used as the evaluation index for the optimization results of this simulation.  
Evaluation indicators for this orthogonal experiment:  
\[a=\frac{\left|s_{i}-s_{i-1}\right|}{s_{i}}\ \ (i=2\ldots 6) \tag{17}\]  
\[P=\sum_{i=2}^{i=6}a \tag{18}\]  
\(s_{i}\)--The sum of the two maximum forces in one cycle through the tooth at the point of intersection of the double arc  
\(a\)--Increment of adjacent cycles.  
\(P\)--Engagement force coefficient--the ratio of incremental engagement force of adjacent cycles.  
Based on the nonlinear solution, the final result was \(X=\) [14.8790, 32.7749, 10.2336, 11.1665, 468.5791, 6.3664]. The non-circular gear hydraulic motor has four arcs to draw the corresponding tooth shape, whose results on any angles are \(z_{1}=\) 27.2896, \(z_{1}^{\prime}=\) 30.3008, \(z_{2}=\) 16.98, and \(z_{2}^{\prime}=\) 1115.39, respectively. Because the number of teeth of \(z_{2}\) is 16.98,  
Figure 12: Angular velocity of planetary wheel center.  
incredibly close to 17, the teeth of \(z_{2}\) are directly rounded. In the subsequent design process, there are three arcs on which the number of teeth needs to be rounded, with the number of teeth on the pitch curve of each hook rounded up or down. The factors and levels of the non-circular gear orthogonal experiments are shown in Table 3.  
The experiment has three factors and two levels; the orthogonal experiment table was chosen to be L4(2\({}^{3}\)), requiring four experiments to perform simulation experiments for four different combinations of tooth numbers on the pitch curve. The coefficient P of the engagement force for each group of measurement schemes can be obtained, as shown in Table 4.  
Table 4 shows that the engagement force coefficient of experiment 4 is the smallest, which indicates that the stability of the engagement force of the planetary wheel of experiment 4 is optimal. The difference in the peak force per cycle is relatively small.  
The range value R reflects the effect of the tooth number factor on the non-circular gear-meshing force. The larger R-value range indicates that this factor has a more significant influence on the engagement force than others. Table 5 shows that the change in the number of \(z^{\prime}{}_{2}\) teeth has the most significant effect. The change in the number of \(z_{1}\) teeth has a smaller impact. The number of \(z^{\prime}{}_{1}\) teeth has a minor effect. The optimal combination of better levels corresponding to the number of teeth is shown in Table 5.  
The most optimal combination of tooth numbers is obtained with \(z_{1}=28\), \(z^{\prime}{}_{1}=30\), \(z_{2}=17\), and \(z^{\prime}{}_{2}=1115\). By analyzing the R values, the degree of influence on the engagement force is from the largest to the smallest: \(z^{\prime}{}_{2}\), \(z^{\prime}{}_{1}\), \(z_{1}\).