# Pattern Classification and Gear Design of Spatial Noncircular
## 3 Principle and Classification Design of Spatial Noncircular Gear CVT
### Classification Design and Analysis of Transmission Ratio  
The transmission ratio of spatial noncircular gear pair in Table 2 is the transmission ratio of the working area where motion and power are transmitted. In order to obtain a complete transmission ratio, a polynomial method can be used to construct the transmission ratio function of the transition area where only motion is transmitted but no power is transmitted, and the periodic transmission ratio was obtained. Taking noncircular gears 1 and 2 as examples, the transmission ratio equation is shown in Equation (1), and the transmission ratios of other spatial noncircular gear pairs can be obtained in the same way. \[i_{21}(\theta_{1})=\left\{ \begin{array}{l}i_{01}(\theta_{1}),\theta_{1}\in[0,\Phi]\\ i_{02}(\theta_{1})=\sum\limits_{j=0}^{6}\left(m_{j}\theta_{1}^{j}\right),\theta _{1}\in[\Phi,U]\end{array} \right., \tag{1}\]  
where, \(i_{01}(\theta_{1})\) is the transmission ratio in the working area. \(i_{02}(\theta_{1})\) is the transmission ratio in the transition area. \(m_{j}\) is the polynomial coefficient. \(\Phi\) is the maximum rotation angle of noncircular gear 1 in the working area. \(U=2\pi/n_{1}\) is the period of the transmission ratio, and the positive integer \(n_{1}\) is the order of noncircular gear 1. According to Equation (2), the unknown parameters in \(i_{21}(\theta_{1})\) can be solved. \[\left\{ \begin{array}{l}\frac{2\pi}{n_{2}}=\int_{0}^{\Phi}i_{01}(\theta_{1}) \mathrm{d}\theta_{1}+\int_{\Phi}^{U}i_{02}(\theta_{1})\mathrm{d}\theta_{1}\\ i_{01}(\Phi)=i_{02}(\Phi)\\ i^{\prime}{}_{01}(\Phi)=i^{\prime}{}_{02}(\Phi)\\ i_{01}(0)=i_{02}(U)\\ i^{\prime}{}_{01}(0)=i^{\prime}{}_{02}(U)\\ i^{\prime}{}_{01}(0)=i^{\prime}{}_{02}(U)\\ \end{array} \right., \tag{2}\]  
where the positive integer \(n_{2}\) is the order of noncircular gear 2. \(i^{\prime}{}_{01}(\theta_{1})\), \(i^{\prime\prime}{}_{01}(\theta_{1})\), \(i^{\prime}{}_{02}(\theta_{1})\), and \(i^{\prime\prime}{}_{02}(\theta_{1})\) are the first and second derivatives of \(i_{01}(\theta_{1})\) and \(i_{02}(\theta_{1})\) with respect to \(\theta_{1}\), respectively. According to the above analysis, take the parameters in Table 4 and analyze the transmission ratio of each pattern, as shown in Figures 7 and 8. \(n_{i}\) is the number of cycles of the input gear and \(n_{l}\) is the number of branches. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Parameters** & **\(\Phi\)/Rad** & \(\mathbf{a_{1}}\) & \(\mathbf{b_{1}}\) & \(\mathbf{a_{2}}\) & \(\mathbf{b_{2}}\) & \(\mathbf{\theta_{p}}\)/rad** & **K** & \(\mathbf{n_{i}}\) & \(\mathbf{n_{l}}\) & \(\mathbf{a}\) & \(\mathbf{n_{1}}\) & \(\mathbf{n_{2}}\) \\ \hline Class A & 0.09 & 0.9 & 0.24 & 0.45 & \(\pi/6\) & \(-0.6\) & 1 & 3 & & 1 & 1 \\ \hline Class B & 0.24 & 0.46 & 0.12 & 0.23 & & 2 & 1 & 3 & / & 1 & 1 \\ \hline Class C & 5\(\pi/9\) & 0.30 & 1.7 & 0.48 & 0.8 & \(\pi/18\) & 1.6 & 2 & 2 & & 2 & 1 \\ \hline Class D & 11\(\pi/18\) & & & / & & \(\pi/6\) & / & 2 & 3 & 1.03 & 2 & 2 \\ \hline Class E & 10\(\pi/9\) & & & / & & \(\pi/6\) & / & 1 & 3 & 1.04 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Structural parameters of each pattern. \begin{table}
\begin{tabular}{c c} \hline
**Class** & **Parameter Condition** \\ \hline A & \(Ka_{2}+(1-K)a_{1}=0\), \(Ka_{2}\
eq 0\) \\ \hline B & \(Ka_{2}-a_{1}=0\), \(Ka_{2}\
eq 0\), \(K\
eq 1\) \\ \hline C & \(Ka_{1}-a_{2}=0\), \(a_{2}\
eq 0\), \(K\
eq 1\) \\ \hline D & \(a>0\), \(a\
eq 1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter conditions for each pattern. Figure 7 is the variation law of the transmission ratio of class A. \(\theta_{e}=\Phi-\theta_{p}\) is the length of the effective working area, that is, the area where each branch outputs a constant rotation speed. To achieve a constant rotation speed output with an input rotation angle in the range of 0 to 360\({}^{\circ}\), Equation (3) must be satisfied.