# Effect of Axial Modification on the Meshing Performance of Involute Developid Gear Pair
## 2 Geometric Design of the Beveloid Gear with Axial Modification  
The theory proposed by [PERSON] posits that the tooth profiles of the beveloid gear, both left and right, can be considered to be generated under the envelope of the tooth profile of a usual rack cutter [23]. The equations for the teeth of the beveloid gear are derived by forming the rack cutter normal plane coordinate system \(S_{n}(x_{n},y_{n},z_{n})\), the rack cutter end plane coordinate system \(S_{o}(x_{o},y_{o},z_{o})\), and the rack cutter pitch plane coordinate system \(S_{c}(x_{c},y_{c},z_{c})\), as illustrated in Figure 1. In the figure, the angle formed by the normal plane coordinate system \(S_{n}\) and the end plane coordinate system \(S_{o}\) of the rack cutter is the helix angle \(\beta\), and the angle formed by the end plane coordinate system \(S_{o}\) and the pitch plane coordinate system \(S_{c}\) is the pitch angle \(\gamma\). Coordinate system \(S_{j}(x_{j},y_{j},z_{j})\) is a movable coordinate system, which is fixedly connected to the blank of the involute beveloid gear. Coordinate system \(S_{b}(x_{b},y_{b},z_{b})\) is a fixed coordinate system. The origin is situated at the centre of the end plane of the beveloid gear blank. The location \(O_{n}\) of the origin of the \(S_{n}\) is determined by the variable \(u=O_{p}O_{n}\) and the helix angle \(\beta\) jointly determined. The angular velocity of rotation of the beveloid gear blank is \(\omega\), the radius of the beveloid gear indexing is \(r\), the speed of translation of the rack cutter is \(v=\omega\cdot r\), and the rotation angle of the beveloid gear during the machining process is \(\varphi_{j}\). In the normal plane coordinate system \(S_{n}\) of the rack cutter, the normal tooth profile vector of the rack cutter is represented as follows:  
\[\mathbf{r}_{n}=[x_{n}\,y_{n}\,z_{n}\,1]^{\mathrm{T}}. \tag{1}\]  
The rack cutter plane in the pitch coordinate system, designated \(S_{c}\), can be obtained by combining the normal tooth profile vectors and applying the following coordinate transformations:  
\[r_{c}=M_{cn}\mathbf{r}_{n}=[x_{c}\,y_{c}\,z_{c}\,1]^{\mathrm{T}}, \tag{2}\]  
where \(\mathbf{r}_{c}\) is the position vector of the rack cutter surface in the coordinate system \(S_{n}\); the transformation matrix, designated as \(M_{cn}\), is employed for the purpose of effecting a change  
Figure 1: Coordinate system for the generation of the beveloid gear. from the coordinate system Sn to the coordinate system \(S_{c}\); and the transformation matrix is given by the following:  
\[M_{cn}=\left[ \begin{array}{cccc}\cos\gamma&-\sin\beta\sin\gamma&\cos\beta\sin \gamma&u\cos\beta\sin\gamma\\ 0&\cos\beta&\sin\beta&u\sin\beta\\ -\sin\gamma&-\cos\gamma&\cos\beta\cos\gamma&u\cos\beta\cos\gamma\\ 0&0&0&1\end{array} \right]. \tag{3}\]  
The position vector \(\mathbf{r}_{c}\) of the tooth face of the rack cutter in the section coordinate system \(S_{c}\) is combined with the relative motion relationship between the rack cutter and the gear blank during the machining process. Consequently, the equation representing the tooth face of the beveloid gear in coordinate system \(S_{j}\) is derived as follows:  
\[\mathbf{r}_{j}:\left\{ \begin{array}{l}x_{j}=x_{c}\cos\varphi_{j}-y_{c}\sin\varphi _{j}+r(\cos\varphi_{j}+\varphi_{j}\sin\varphi_{j})\\ y_{j}=x_{c}\sin\varphi_{j}-y_{c}\cos\varphi_{j}+r(\sin\varphi_{j}+\varphi_{j} \cos\varphi_{j})\\ z_{j}=z_{c}\\ \varphi_{j}=\mathbf{n}_{xx}y_{c}-\mathbf{n}_{yy}y_{c}/\mathbf{n}_{xc}r\\ \end{array} \right., \tag{4}\]  
\[\left\{ \begin{array}{l}\mathbf{n}_{xc}=-\cos\alpha_{n}\sin\beta\sin\gamma\mp \cos\alpha_{n}\cos\gamma\\ \mathbf{n}_{yc}=\cos\alpha_{n}\cos\beta\\ \mathbf{n}_{zc}=-\cos\alpha_{n}\sin\beta\cos\gamma\pm\sin\alpha_{n}\sin\gamma\\ \end{array} \right., \tag{5}\]  
where \(x_{j},y_{j}\), and \(z_{j}\) represent the position vectors of the tooth surface of the beveloid gear, and \(\mathbf{n}_{xc}\), \(\mathbf{n}_{yc}\), and \(\mathbf{n}_{zc}\) denote the unit normal vectors of the rack cutter surface.