# Calculation and Dressing Simulation of the Profile of the Form Grinding Wheel for Modified ZI Worms
## 2 Form Grinding Theory of Modified ZI worms
### Mathematical Model of the Grinding Process of Modified ZI Worm  
Figure 5 shows the structural diagram of the form grinding machine. In the process of form grinding, the grinding wheel revolves around its axis, while simultaneously feeding along the radius direction of the grinding wheel. The ZI worm also revolves around its axis and moves linearly along its own axial direction at the same time. As shown in Figure 6, is the machining coordinate system of the form grinding of the ZI worm. The coordinate system \(S_{1}(O_{1}-X_{1}Y_{1}Z_{1})\) is fixed to the grinding wheel; the coordinate system \(S_{2}(O_{2}-X_{2}Y_{2}Z_{2})\) is fixed to the axial movement worktable of the ZI worm; the coordinate system \(S_{1g}(O_{1g}-X_{1g}Y_{1g}Z_{1g})\) is fixed to the base of the swivel table of the grinding wheel frame; the coordinate system \(S_{2w}(O_{2w}-X_{2w}Y_{2w}Z_{2w})\) is fixed to the machined ZI worm. \(\theta_{2}\) is the rotation angle of the ZI worm, \(p\theta_{2}\) is the movement distance along \(Z_{2}\), p is the helix parameter, and \(L\) is the installation center distance between the grinding wheel and ZI worm. The value of the center distance \(L\) is constantly changing due to the radial feeding of the grinding wheel during the grinding, and \(\Delta L\) is its change amount. In addition, the installation angle of the grinding wheel may be changed when installing the grinding wheel or during the grinding, and \(\Delta L\) is its change amount.  
According to the analysis of the spatial positional relationship among the fixed coordinate system of the grinding wheel and the fixed and machining coordinate systems of the ZI worm in Figure 6, the transformation matrix from the fixed coordinate system \(S_{1g}\) of the grinding wheel to the fixed coordinate system \(S_{2}\) of the ZI worm can be expressed as follows:  
Figure 5: Form grinding machining coordinate system.  
Figure 6: Form grinding coordinate system.  
\[\text{M}_{21g}=\begin{pmatrix}-1&0&0&L+\Delta L\\ 0&-1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix} \tag{19}\]  
Similarly, the transformation matrix from the fixed coordinate system \(S_{2}\) of the ZI worm to the machining coordinate system \(S_{2w}\) of the ZI worm is expressed as follows:  
\[\text{M}_{2w2}=\begin{pmatrix}\cos\theta_{2}&\sin\theta_{2}&0&0\\ -\sin\theta_{2}&\cos\theta_{2}&0&0\\ 0&0&1&-\text{p}\theta_{2}\\ 0&0&0&1\end{pmatrix} \tag{20}\]  
Similarly, the transformation matrix from the coordinate system \(S_{1}\) of the grinding wheel to the fixed coordinate system \(S_{1g}\) of the grinding wheel is expressed as follows:  
\[\text{M}_{1g1}=\begin{pmatrix}1&0&0&0\\ 0&\cos(\Sigma+\Delta\Sigma)&\sin(\Sigma+\Delta\Sigma)&0\\ 0&-\sin(\Sigma+\Delta\Sigma)&\cos(\Sigma+\Delta\Sigma)&0\\ 0&0&0&1\end{pmatrix} \tag{21}\]  
From Equations (19)-(21), the transformation matrix from the coordinate system \(S_{1}\) of the grinding wheel to the machining coordinate system \(S_{2w}\) of the ZI worm can be deduced as follows:  
\[\text{M}_{2w1}=\text{M}_{2w2}\cdot\text{M}_{21g}\cdot\text{M}_{1g1} \tag{22}\]  
Therefore, during the form grinding, the transformation relationship between an arbitrary point \(P(X_{1},Y_{1},Z_{1})\) in the coordinate system \(S_{1}\) of the grinding wheel and the corresponding point \(P^{\prime}(X_{2w},Y_{2w},Z_{2w})\) in the machining coordinate system \(S_{2w}\) of the ZI worm is expressed as follows:  
\[\begin{pmatrix}X_{2w}&Y_{2w}&Z_{2w}&1\end{pmatrix}^{T}=\text{M}_{2w1}\Big{(}X_ {1}&Y_{1}&Z_{1}&1\Big{)}^{T} \tag{23}\]  
The following expression is obtained:  
\[\begin{pmatrix}X_{1}&Y_{1}&Z_{1}&1\end{pmatrix}^{T}=\text{M}_{12w}\Big{(}X_{2w }&Y_{2w}&Z_{2w}&1\Big{)}^{T} \tag{24}\]  
By means of Formulas (23) and (24), the mutual transformation is realized between a certain point on the grinding wheel profile and the worm-tooth surface. Suppose the parameters of the worm are given; the profile of the form grinding wheel can be obtained through the coordinate transformation on the contact surface.