# 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 Modeling of the Modified ZI Worm  
As shown in Figure 2, the end-face profile of the ZI worm is involute. The radius of the base circle is \(r_{b}\), the right hand involute is ef, and the starting point is e. The angle between Oe and the \(X\)-axis is \(\sigma_{0}\). The left hand involute is symmetrical to the right hand involute about the \(X\)-axis. The normal line at any point M on the involute of the base circle is tangent at point a. The coordinates of the point M are \((x_{0},y_{0})\). Let the spreading angle of the involute be \(\mu\). At point M, let \(\angle\)eoa = \(\mu\), then \(M_{a}=r_{b}\mu\). Figure 1: Three-dimensional structural diagram of the workpiece and the grinding wheel. Figure 2: End-face section of involute ZI worm. Involute modification is usually performed near the tooth root and tooth tip areas. The area near the pitch circle is generally not modified. The modification schematic diagram of the ef section is shown in Figure 2. To meet these modification requirements, the involute is modified by the second-order parabola method, as shown in Figure 3. The BC section and the AB and \(CD\) sections are the pitch circle region, the tooth root region, and the tooth tip region, respectively. The relationship between the modification amount \(\Delta E\) and the spreading angle \(\mu\) is as follows:  
\[\Delta E=\left\{ \begin{array}{ll}a_{1}(\mu-\mu_{B})^{2}&\mu_{A}\leq\mu\leq \mu_{B}\\ 0&\mu_{B}\leq\mu\leq\mu_{C}\\ a_{1}(\mu-\mu_{C})^{2}&\mu_{C}\leq\mu\leq\mu_{D}\\ \end{array} \right., \tag{1}\]  
Among them, \(a_{1}\) is the tooth profile modification coefficient; \(u_{i}\) is the partial unwrapping angle of the end-face involute \((i=A,B,C,D)\). According to the above conditions, the involute of the ef section can be expressed as:  
\[\left\{ \begin{array}{ll}x_{0}=r_{\text{b}}\cos(\sigma 0+\mu)+(r_{\text{b}}\mu+ \Delta E)\sin(\sigma 0+\mu)\\ y_{0}=r_{\text{b}}\sin(\sigma 0+\mu)-(r_{\text{b}}\mu+\Delta E)\cos(\sigma 0+\mu)\\ z_{0}=0\\ \end{array} \right. \tag{2}\]  
The ef section is rotated in a helical motion around the \(z\)-axis. Assuming the rotation angle around the \(z\)-axis is \(\theta\), the equation of the involute helicoid on the right hand side of the tooth groove can be obtained as:  
\[\left( \begin{array}{c}x\\ y\\ z\\ 1\end{array} \right)=\left( \begin{array}{cccc}\cos\theta&-\sin\theta&0&0\\ \sin\theta&\cos\theta&0&0\\ 0&0&1&\text{p}\theta\\ 0&0&0&1\end{array} \right)\left( \begin{array}{c}x_{0}\\ y_{0}\\ z_{0}\\ 1\end{array} \right), \tag{3}\]  
The surface Equation (3) can be expressed as:  
\[\mathbf{r}=\mathbf{r}(\mu,\theta)=x(\mu,\theta)\mathbf{i}+y(\mu,\theta) \mathbf{j}+z(\mu,\theta)\mathbf{k} \tag{4}\]  
Among them, \(\mu\) and \(\theta\) are the two parameter variables of the surface. The normal vector \(\mathbf{n}\) at any point \(M(x,y,z)\) on the surface is:  
\[\mathbf{n}=\frac{\partial\mathbf{r}}{\partial\mu}\times\frac{\partial\mathbf{ r}}{\partial\theta} \tag{5}\]  
Figure 3: Tooth profile modification curve. If the helicoid of the workpiece is known, then Formula (10) can be transformed into:  
\[(\mathbf{k}^{\prime}\times\mathbf{R})\cdot\mathbf{n}=0 \tag{11}\]  
According to the vector relationship, \(\mathbf{R}\) can be expressed as:  
\[\mathbf{R}=-L\mathbf{i}+\mathbf{r}=(x-L)\mathbf{i}+y\mathbf{j}+z\mathbf{k} \tag{12}\]  
Among them, \(\mathbf{i}\) is the unit vector in the \(x\)-direction in the workpiece coordinate system and \(L\) is the center distance between the grinding wheel and the worm.