# Pattern Classification and Gear Design of Spatial Noncircular
## 3 Principle and Classification Design of Spatial Noncircular Gear CVT
### Transmission Ratio Parameter Equation  
According to the analysis in Section 3.1, the transmission ratio parameter equations of spatial noncircular gear pairs were designed, respectively, and the intermediate transmission ratio and overall transmission ratio of each pattern were calculated. See Table 2 for each parameter equation.  
In Table 2, \(\theta_{1}\) and \(\theta_{3}=\theta_{1}+\theta_{p}\) are the rotation angles of gears 1 and 3 under each pattern, respectively. \(\theta_{p}\) is the phase angle. \(i_{21}\) and \(i_{43}\) are the transmission ratios of the corresponding gear pairs, respectively. \(I_{oi}\) is the overall transmission ratio between the output shaft and input shaft. \(a_{1}\), \(b_{1}\), \(a_{2}\), and \(b_{2}\) are the parameters of the transmission ratios \(i_{21}\) and \(i_{43}\) of type I. \(K\) is the characteristic parameter of the differential mechanism of type I. \(a\) is the parameter of the transmission ratios \(i_{21}\) and \(i_{43}\) of type II.  
For class A, \(z_{5}\), \(z_{6}\), and \(z_{7}\) are the teeth number of bevel gears 5, 6, and 7, respectively. \(i_{7i}\) is the transmission ratio between the bevel gear 7 and the input shaft. Constant \(i_{07}\) is the transmission ratio between the output shaft and the bevel gear 7. For class B, \(z_{5}\) and \(z_{7}\) are the teeth number of the ring gear 5 and center gear 7, respectively. \(z_{6i}\) and \(z_{60}\) are the number of internal teeth and external teeth of planetary gear 6, respectively. \(I_{Hi}\) is the transmission ratio between the planet carrier \(H\) and input shaft. Constant \(i_{OH}\) is the transmission ratio between the output shaft and the planet carrier \(H\). For class C, \(z_{9}\), \(z_{11}\), and \(z_{13}\) are the teeth number of the corresponding gears, respectively. \(z_{10i}\), \(z_{10v}\), \(z_{12i}\), and \(z_{12o}\) are the teeth number of the inner and outer rings of the corresponding bilateral gears. \(i_{Hi}\) is the transmission ratio between the planet carrier \(H\) and input shaft. Constant \(i_{OH}\) is the transmission ratio between the output shaft and the planet carrier \(H\). For classes D and E, \(i_{42}\) is the transmission ratio between gear 4 and gear 2. Constant \(i_{04}\) is the transmission ratio between the output shaft and gear 4.  
As can be seen from Table 2, when the coefficient related to the rotation angle \(\theta_{1}\) is 0, the overall transmission ratio \(i_{oi}\) of each pattern is a function related only to the variable phase angle \(\theta_{p}\). By continuously changing \(\theta_{p}\), combined with the transmission selection mechanisms of multiple branches, the CVT with an input rotation angle in the range of 0 to 360\({}^{\circ}\) can be realized. The parameter conditions that each pattern need to meet were shown in Table 3.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **Class** & **Transmission Ratio** & **Characteristic Parameter** \\ \hline \multirow{4}{*}{I} & \multirow{4}{*}{A} & \multirow{4}{*}{B} & \multirow{4}{*}{\(\left\{\begin{array}{l}i_{21}=-(a_{1}\theta_{1}+b_{1})\\ i_{43}=-(a_{2}\theta_{3}+b_{2})\\ i_{7i}=Ki_{43}+(1-K)i_{21}\\ i_{oi}=i_{7i}i_{07}=-\{[Ka_{2}+(1-K)a_{1}]\theta_{1}+Ka_{2}\theta_{p}+Kb_{2}+(1-K)b _{1}\}i_{07}\end{array}\right.\)} \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \hline \multirow{4}{*}{II} & \multirow{4}{*}{D} & \multirow{4}{*}{E} & \multirow{4}{*}{\(\left\{\begin{array}{l}i_{21}=a^{\theta_{1}}\\ i_{43}=a^{\theta_{3}}\\ i_{oi}=i_{42}i_{04}=-a^{\theta_{p}}i_{04}\end{array}\right.\)} \\ \cline{2-4}  & &