# Design and Analysis of Internal-External Composite Meshing Hy-Vo Chain
## 3 Internal Meshing Profile
### Proper Internal Meshing Profile  
In actuality, the Hy-Vo chain cannot be a rigid body as a rack, and the distance between the chain pitch line and the sprocket rotation center \(s_{\rm j}\) will change with the change in the sprocket rotation angle if the sprocket rotates from its initial position. As Figure 8 demonstrates, when the pitch of chain plate 2 is fully within the sprocket positioning circle, \(s_{\rm j}\) will be at its maximum:  
\[s_{\rm jmax}=r_{\rm p}=\frac{p_{\rm p^{\prime\prime}}}{2}\csc\frac{\varphi}{2}, \tag{18}\]  
Figure 8: Meshing state when the chain pitch line is furthest from the sprocket rotation center.  
Equations (5), (17) and (18) mean that if there is only an external meshing mechanism for the Hy-Vo chain, the distance between the chain pitch line and the sprocket rotation center \(s_{\text{j}}\) will fluctuate from \(s_{\text{jmin}}\) to \(s_{\text{jmax}}\). The fluctuation of \(s_{\text{j}}\) shows that the meshing between the external meshing Hy-Vo chain and the sprocket is totally different from the meshing between the rack and the short addendum gear, and the fluctuation of \(s_{\text{j}}\) is the real cause of the system polygon action.  
From the above analysis, if we let the internal meshing profile of the Hy-Vo chain coincide with the profile of the rigid Hy-Vo chain, the Hy-Vo chain would continuously mesh with the sprocket; this would be the internal-external composite meshing Hy-Vo chain. In this case, the internal meshing profile of the internal-external meshing Hy-Vo chain should be a straight line; according to Equations (5), (17) and (18), the value of the convexity of the internal meshing profile should satisfy the following:  
\[\delta=s_{\text{jmax}}-s_{\text{jmin}}=\frac{p_{\text{p}}{}^{\prime\prime}}{2} (\csc\frac{\pi}{z}-\cot\frac{\pi}{z})\sin\alpha, \tag{19}\]  
As a result, when \(p_{1}=p\) and \(\alpha_{1}=\alpha\), the internal meshing profile under the internal-external composite meshing mechanism exists in the Hy-Vo chain transmission system. Figure 9 displays the design scheme of the internal-external composite meshing Hy-Vo chain; the shape of the internal meshing profile is a straight line, and the convexity of the internal meshing profile should satisfy Equation (19). Therefore, we call \(p_{1}=p\) and \(\alpha_{1}=\alpha\) necessary conditions for the existence of the internal-external composite meshing Hy-Vo chain.  
As we can see from Equation (19) and Figure 9, the internal meshing profile of the internal-external composite meshing Hy-Vo chain is related to the sprocket tooth number. When the tooth number of the drive sprocket is not equal to that of the driven sprocket, we can obtain two or more internal meshing profiles. Therefore, based on the mathematical characteristic of Equation (19), we can guess that if the tooth number of the drive sprocket is near that of the driven sprocket, the corresponding internal meshing profiles should be very similar. Moreover, we can also guess that if the drive ratio of the system is not equal to one, the internal-external composite meshing mechanism may increase the meshing impacts between the chain plates and the sprocket, because there are two appropriate internal meshing profiles in theory.