## 5 Machining of a Non-Circular Gear Using the WEDM Method  
An innovative method of modeling the machining of non-circular gears using the wire electrical discharge machining (WEDM) method was developed at the Warsaw University of Technology with the participation of the author of this article--[PERSON] [9,18]. [PERSON] et al. [19] also examined this problem. In Figure 11, the non-circular gear in design is represented as two roll lines P1 and P2 with rotation centers at points O1 and O2, respectively. Those lines work together, rolling without slipping. The shape P1 is described by a polar equation as a function of radius \(r_{1}(\varphi_{1})\). The gear ratio is described by the function:  
\[\
u(\varphi_{1})=\frac{\omega_{1}(\varphi_{1})}{\omega_{2}(\varphi_{1})}= \frac{r_{2}(\varphi_{1})}{r_{1}(\varphi_{1})} \tag{8}\]  
where \(\omega_{1}(\varphi_{1})\) and \(\omega_{2}(\varphi_{2})\) are angular velocity functions of the first and the second gears, respectively. The axle distance A is constant:  
\[\text{A}=r_{1}(\varphi_{1})+r_{2}(\varphi_{1})=\text{const} \tag{9}\]  
The roll lines work together without slipping:  
\[\text{d}\varphi_{2}=\frac{r_{1}(\varphi_{1})}{r_{2}(\varphi_{1})}\text{d} \varphi_{1} \tag{10}\]  
Taking Equations (8) to (10) into account and integrating  
\[\varphi_{2}=\int\limits_{0}^{\varphi_{1}}\frac{r_{1}(\varphi_{1})}{\text{A}- r_{1}(\varphi_{1})}\text{d}\varphi_{1}, \tag{11}\]  
Figure 11: Roll lines representing a non-circular gear; P1, P2—roll lines, O1, O2—rotation centers. the integrating constant was found from the following condition: if \(\varphi_{1}=0\), then \(\varphi_{2}=0\). Note that to continuously transfer the motion, the gear ratio function must be periodic and the gear's period T must be related to periods T1 and T2 of gears 1 and 2, respectively, as follows:  
\[\text{T}=\frac{\text{T}_{1}}{\text{n}_{2}}=\frac{\text{T}_{2}}{\text{n}_{1}} \tag{12}\]  
where n1 and n2 are natural numbers. In the case shown in Figure 11, the period \(T\) of the gear ratio function \(\
u(\varphi_{1})=\pi\), T1 = 2\(\pi\), and T2 = 3\(\pi\); therefore, when \(\varphi_{1}=\pi\), then \(\varphi_{2}=\) 2/3\(\pi\). This renders:  
\[\frac{2}{3}\pi=\int\limits_{0}^{\pi}\frac{\text{r}_{1}(\varphi_{1})}{\text{A}- \text{r}_{1}(\varphi_{1})}\text{d}\varphi_{1} \tag{13}\]  
The value of the parameter A can be calculated from Equation (13). The P2 = r2(\(\varphi_{1}\)) can be calculated from Equations (9) and (10). We assume there is roll line P1 for which a cooperating line P2 needs to be found that fulfils the requirements described above. The line P1 has been divided into \(i\) segments \(l_{1i}\) so that each of those segments can be represented as an Archimedes spiral with adequate precision (Figure 12). An Archimedes spiral is described by:  
\[\text{r}=\text{k}\alpha \tag{14}\]  
Therefore:  
\[ \begin{array}{c}\alpha_{1}=\frac{\text{r}_{1}}{\text{k}};\ \alpha_{2}=\frac{\text{r}_{2}}{\text{k}};\text{k}= \frac{\text{r}_{2}-\text{r}_{1}}{\alpha}\\ \text{d}\text{l}=\sqrt{\left(\text{d}\text{r}\right)^{2}+\left(\text{rd} \alpha\right)^{2}};\text{d}\text{r}=\text{d}(\text{k}\alpha)=\text{k}\text{d} \alpha\\ \text{d}\text{l}=\sqrt{\left(\text{k}\text{d}\alpha\right)^{2}+\left(\text{rd} \alpha\right)^{2}}=\sqrt{\left(\text{k}\text{d}\alpha\right)^{2}+\left(\text{ k}\alpha\text{d}\alpha\right)^{2}}=\text{k}\sqrt{1+\alpha^{2}}\text{d}\alpha\\ \int\text{d}l=\frac{\text{k}}{2}\Big{[}\alpha\sqrt{\alpha^{2}+1}+\text{ln}( \alpha+\sqrt{\alpha^{2}+1})\Big{]}\\ 1=\int\limits_{\alpha_{1}}^{\alpha_{2}}\text{d}\text{l}=\frac{\text{k}}{2} \Big{[}\alpha_{2}\sqrt{\alpha_{2}^{2}+1}+\text{ln}\Big{(}\alpha_{2}+\sqrt{ \alpha_{2}^{2}+1}\Big{)}-\alpha_{1}\sqrt{\alpha_{1}^{2}+1}-\text{ln}\Big{(} \alpha_{1}+\sqrt{\alpha_{1}^{2}+1}\Big{)}\Big{]}\end{array}