# Design and Analysis of Internal-External Composite Meshing Hy-Vo Chain
## 2 Basic Design
### Meshing Parameters  
In the silent chain meshing research area, the chain-sprocket-hob meshing system is always used to analyze the external meshing relations between the chain and the sprocket, as Figure 5 shows. The solid red line denotes the profile of the hob; the hob pitch is \(p_{2}\), and the hob half-tooth angle is \(\alpha_{2}\). The meshing parameters of the Hy-Vo chain and sprocket are listed in Table 2[22].  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Meanings** \\ \hline \(p\) & The chain pitch \\ \(A\) & The hole pitch \\ \(\alpha\) & The half angle of chain plate tooth profile \\ \(f\) & The standard apothem \\ \(f_{1}\) & The equivalent apothem when chain is straightened \\ \(\gamma\) & The positioning offsetting angle \\ \(r_{\rm r}\) & The contact radius of rocker pin \\ \hline \hline \end{tabular}
\end{table}
Table 1: Basic parameters of external meshing Hy-Vo chain.  
Figure 3: Parameters of the mixed assembly Hy-Vo chain: (**a**) external meshing chain plate; (**b**) internal–external meshing chain plate.  
In addition to these meshing parameters, there is another important parameter that should be discussed, which is the radius of the sprocket pitch circle \(r_{\rm j}\). In previous studies, the meaning of \(r_{\rm j}\) is the pitch circle on the sprocket when the Hy-Vo chain is meshing with it [4]. In this study, we hold that when a chain meshes with a sprocket, there is no pitch circle on the sprocket. The essence of the pitch circle is the set of instantaneous centers when the meshing happens between two gears. If a chain meshes with a sprocket, the set of instantaneous centers on the sprocket should be a kind of polygon that is the inevitable result of the system polygon action. Hence, we believe the meaning of \(r_{\rm j}\) should be the radius of the sprocket pitch circle when the hob meshes with it. The function of \(r_{\rm j}\) is as follows:  
\[r_{\rm j}=\frac{p_{2}z}{2\pi}, \tag{3}\]  
Figure 4: Hy-Vo chain at the rotating state.  
Figure 5: Meshing system of Hy-Vo chain.  
\begin{table}
\begin{tabular}{c c l l} \hline \hline
**Classes** & **Parameters** & **Meanings** & **Functions** \\ \hline Hy-Vo chain & \(p_{\rm P}\)\({}^{\prime\prime}\) & The equivalent positioning pitch & \(p_{\rm P}\)\({}^{\prime\prime}=p+2r_{\rm r}+(p-A+2r_{\rm r})\)-tan \(\gamma\)-tan \((\varphi/2)-2r_{\rm r}/\cos\,(\varphi/2)\) \\  & \(f_{\rm P}\)\({}^{\prime\prime}\) & The equivalent positioning apthom & \(f_{\rm P}\)\({}^{\prime\prime}=f+(A-p_{\rm P}\)\({}^{\prime\prime})\)-\(\cos\,\alpha/2\) \\  & \(r_{\rm P}\) & The radius of the positioning circle & \(r_{\rm P}=p_{\rm P}\)\({}^{\prime\prime}\)/[2 sin \((\pi/z)\)] \\ \hline  & \(p_{1}\) & The pitch & - \\  & \(\alpha_{1}\) & The pressure angle & - \\  & \(z\) & The tooth number & - \\  & \(\varphi\) & The pitch angle & \(\varphi=2\pi/z\) \\  & \(r\) & The radius of reference circle & \(r=p_{1}z/(2\pi)\) \\  & \(r_{\rm b}\) & The radius of base circle & \(r_{\rm b}=r\)-\(\cos\,\alpha_{1}\) \\  & \(r_{\rm a}\) & The radius of addendum circle & - \\  & \(r_{\rm f}\) & The radius of addendum circle & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: Meshing parameters of Hy-Vo chain system.