# An Attempt to Establish a Mathematical Model for an Unconventional Worm Gear with Bearings
## 2 Materials and Methods
The position equations of point \(M\) related to the worm (system \(O_{1 xyz}\)) are:  
\[\left\{ \begin{matrix}x=\frac{d_{2}}{2}\cdot\sin\varphi_{2}\\ y=\left[\left(a-\frac{d_{2}}{2}\right)+\frac{d_{2}}{2}\cdot\left(1-\cos\varphi_ {2}\right)\right]\cdot\cos\varphi_{1}\\ z=-\left[\left(a-\frac{d_{2}}{2}\right)+\frac{d_{2}}{2}\cdot\left(1-\cos \varphi_{2}\right)\right]\cdot\sin\varphi_{1}\end{matrix} \right.\left\{ \begin{matrix}x=\frac{d_{2}}{2}\cdot\sin\varphi_{2}\\ y=\left(a-\frac{d_{2}}{2}\cdot\cos\varphi_{2}\right)\cdot\cos\varphi_{1}\\ z=-\left(a-\frac{d_{2}}{2}\cdot\cos\varphi_{2}\right)\cdot\sin\varphi_{1} \end{matrix} \right. \tag{2}\]  
where  
\(d_{2}\) is the worm wheel pitch diameter, mm;  
\(a\) represents the center distance, mm;  
\(\varphi_{1}\) is the angle of the position for the worm wheel, \({}^{\circ}\) (degrees);  
\(\varphi_{2}\) is the angle of the position for the worm wheel, \({}^{\circ}\) (degrees). \[\varphi_{1}=u_{12}\cdot\varphi_{2} \tag{3}\]  
where  
\(u_{12}\) represents the gear ratio  
\[u_{12}=\frac{z_{2}}{z_{1}}=\frac{n_{1}}{n_{2}} \tag{4}\]  
Figure 2: Worm gear with a single row of bearings per roller. where  
\(z_{2}\) is the worm wheel number of radial ball bearings, i.e., the number of teeth (in a classic worm gearing);  
\(z_{1}\) represents the thread number of the worm;  
\(n_{1}\) and \(n_{2}\) represent the rotational speed of the worm and wheel, respectively. Relative to the worm, the trajectory of the \(M\) point is a helix wound on a toroidal surface (an inner circular torus). Starting from Equation (2) of point \(M\), the position equations of characteristic points \(A\), \(B\), \(D\), and \(E\) (related to the coordinate system \(O_{1}xyz\)) are shown:  
Equations of point \(A\) (system \(O_{1 xyz}\)) are as follows:  
\[\left\{ \begin{array}{c}x=\frac{d}{2}\cdot\cos\varphi_{2}+\left(\frac{d_{2}}{ 2}-h\right)\cdot\sin\varphi_{2}\\ y=\left[a+\frac{d}{2}\cdot\sin\varphi_{2}-\left(\frac{d_{2}}{2}-h\right)\cdot \cos\varphi_{2}\right]\cdot\cos\varphi_{1}\\ z=-\left[a+\frac{d}{2}\cdot\sin\varphi_{2}-\left(\frac{d_{2}}{2}-h\right) \cdot\cos\varphi_{2}\right]\cdot\sin\varphi_{1}\end{array} \right. \tag{5}\]  
where  
\(d\) is the outer diameter of the bearing. Figure 3: The geometry of the worm gear with bearings. \[h=\frac{B_{r}}{2} \tag{6}\]  
where  
\(B_{r}\) is the bearing width. Equations of point \(B\) (system \(O_{1 xyz}\)) are as follows:  
\[\left\{ \begin{array}{c}x=\frac{d}{2}\cdot\cos\varphi_{2}+\left(\frac{d_{2}}{2} +h\right)\cdot\sin\varphi_{2}\\ y=\left[a+\frac{d}{2}\cdot\sin\varphi_{2}-\left(\frac{d_{2}}{2}+h\right)\cdot \cos\varphi_{2}\right]\cdot\cos\varphi_{1}\\ z=-\left[a+\frac{d}{2}\cdot\sin\varphi_{2}-\left(\frac{d_{2}}{2}+h\right) \cdot\cos\varphi_{2}\right]\cdot\sin\varphi_{1}\end{array} \right. \tag{7}\]  
The parametric equations of the line \(AB\) (\(O_{1 xyz}\) system) are as follows:  
\[\left\{ \begin{array}{c}x\cdot\cos\varphi_{2}\cos\varphi_{1}+y\cdot\sin \varphi_{2}-\left[a\cdot\sin\varphi_{2}+\frac{d}{2}\right]\cdot\cos\varphi_{1 }=0\\ y\cdot\sin\varphi_{1}+z\cdot\cos\varphi_{1}=0\end{array} \right. \tag{8}\]  
For point \(D\), the equations are:  
\[\left\{ \begin{array}{c}x=-\frac{d}{2}\cdot\cos\varphi_{2}+\left(\frac{d_{2} }{2}-h\right)\cdot\sin\varphi_{2}\\ y=\left[a-\frac{d}{2}\cdot\sin\varphi_{2}-\left(\frac{d_{2}}{2}-h\right)\cdot \cos\varphi_{2}\right]\cdot\cos\varphi_{1}\\ z=-\left[a-\frac{d}{2}\cdot\sin\varphi_{2}-\left(\frac{d_{2}}{2}-h\right) \cdot\cos\varphi_{2}\right]\cdot\sin\varphi_{1}\end{array} \right. \tag{9}\]  
For point \(E\), the equations are:  
\[\left\{ \begin{array}{c}x=-\frac{d}{2}\cdot\cos\varphi_{2}+\left(\frac{d_{2} }{2}+h\right)\cdot\sin\varphi_{2}\\ y=\left[a-\frac{d}{2}\cdot\sin\varphi_{2}-\left(\frac{d_{2}}{2}+h\right) \cdot\cos\varphi_{2}\right]\cdot\cos\varphi_{1}\\ z=-\left[a-\frac{d}{2}\cdot\sin\varphi_{2}-\left(\frac{d_{2}}{2}+h\right) \cdot\cos\varphi_{2}\right]\cdot\sin\varphi_{1}\end{array} \right.