# Design and Analysis of Internal-External Composite Meshing Hy-Vo Chain
## 3 Internal Meshing Profile  
### Instantaneous Meshing Condition  
When there is only an external meshing mechanism for the Hy-Vo chain, the meshing between the chain and the sprocket is instantaneous. In Figure 6, the origin of the absolute coordinate system \(S(xOy)\) coincides with the rotation center of the sprocket. \(S_{2}(x_{2}O_{2}y_{2})\) is the reference coordinate system to design the internal meshing profile of the internal-external composite meshing Hy-Vo chain plate. The sprocket rotates anticlockwise at velocity \(\omega\), and the sprocket is symmetrical along the \(y\)-axis. In this case, chain plate 1 is positioned on the sprocket, chain plate 3 is straightened horizontally, and chain plate 2 meshes with the sprocket while it is also horizontally straightened. Supposing that the right-side external meshing profile of chain plate 2 is T\({}_{0}\)-T\({}_{0}\), the relative sprocket tooth involute profile is \(\Gamma_{0}\), the relative rotation angle of involute \(\Gamma_{0}\) on the base circle of the sprocket from the \(y\)-axis to its practical position is \(\theta\), the corresponding angle of the space width of the sprocket on the base circle is \(\varphi_{\text{b}}\). For the above parameters, the following is posed:  
\[\varphi_{\text{b}}=2(\theta-\varphi/2), \tag{4}\]  
At this time, based on the meshing theory, the distance between the chain pitch line and the sprocket rotation center is the shortest. If we suppose the value of this distance is \(s_{\text{jmin}}\), we have the following:  
\[s_{\text{jmin}}=r_{\text{p}}\cos\frac{\varphi}{2}=\frac{{p_{\text{p}}}^{\prime \prime}}{2}\cot\frac{\varphi}{2}, \tag{5}\]  
According to the meshing relations in Figure 6, the function of line T\({}_{0}\)-T\({}_{0}\) in \(S_{2}(x_{2}O_{2}y_{2})\) is as follows:  
\[y_{2}=x_{2}\cot\alpha-f_{1}\csc\alpha, \tag{6}\]  
Based on the variable pitch characteristic of the Hy-Vo chain, the coordinates of \(O_{2}\) in \(S(xOy)\) are \((p/2,{p_{\text{p}}}^{\prime\prime}\cot(\varphi/2))\). When plugging the coordinates of \(O_{2}\) and Equation (1) into Equation (6), we can yield the function of line T\({}_{0}\)-T\({}_{0}\) in \(S(xOy)\) as follows:  
\[y=x\cot\alpha-f_{1}\csc\alpha-\frac{p}{2}\cot\alpha+\frac{p}{2}\cot\frac{ \varphi}{2}, \tag{7}\]  
Figure 6: Meshing between the external meshing Hy-Vo chain and the involute sprocket.  
Thus, the function of the tangent line of involute \(\Gamma_{0}\) in \(S(xOy)\) is as follows:  
\[y=x\cot(\Phi+\theta)+\frac{r_{\rm b}\Phi}{\sin(\Phi+\theta)}, \tag{8}\]  
where \(\Phi\) is the rolling angle on the involute, and it is a variate.  
Because line T\({}_{0}\)-T\({}_{0}\) also tangents to involute \(\Gamma_{0}\), line T\({}_{0}\)-T\({}_{0}\) is the tangent line of involute \(\Gamma_{0}\). Based on Equations (7) and (8), we can obtain the following:  
\[\begin{cases}\cot\alpha=\cot(\Phi+\theta)\\ -f_{1}\csc\alpha-\frac{p}{2}\cot\alpha+\frac{p}{2}\cot\frac{\varphi}{2}=\frac {r_{\rm b}\Phi}{\sin(\Phi+\theta)}\end{cases}, \tag{9}\]  
The solution of Equation (9) is as follows:  
\[\theta=\alpha+\frac{f_{1}}{r_{\rm b}}+\frac{p\cos\alpha}{2r_{\rm b}}-\frac{p} {2r_{\rm b}}\cot\frac{\varphi}{2}\sin\alpha, \tag{10}\]  
If the sprocket pitch \(p_{1}\) is equal to the theoretical chain pitch \(p\), and the sprocket pressure angle \(\alpha_{1}\) is equal to the half angle of chain plate tooth profile \(\alpha\), Equation (10) will be simplified as follows:  
\[\theta=\alpha+\frac{f_{1}}{r_{\rm b}}+\frac{\pi}{z}-\frac{\pi}{z}\cot\frac{ \varphi}{2}\tan\alpha, \tag{11}\]