# Effect of Axial Modification on the Meshing Performance of Involute Developid Gear Pair
## 3 Meshing Model of the Beveloid Gear Pair with Axial Modification  
The finite element simulation meshing model of a beveloid gear pair is constructed by establishing a meshing coordinate system for the beveloid gear pair. As illustrated in Figure 4, during the meshing process of the parallel shaft leveloid gear pair, the two gears are axially aligned in the central section. The toe of the pinion and the heel of the gear are positioned on opposing sides of the central section, as are the heel of the pinion and the toe of the gear. In the figure, coordinate systems \(S_{1}(x_{1},y_{1},z_{1})\) and \(S_{2}(x_{2},y_{2},z_{2})\) represent the pinion and gear coordinate systems of the beveloid gear pair, respectively, while coordinate system \(S_{j}(x_{j},y_{j},z_{j})\) is a fixed coordinate system. The working pitch cones of the two gears are parallel, and contact occurs at point \(P\). \(E\) represents the shortest distance between the axes of the two thickened gears, while \(\gamma_{1}\) and \(\gamma_{2}\) represent the working pitch cone angles of the pinion and gear, respectively. Additionally, \(r_{w1}\) and \(r_{w2}\) represent the working pitch circle radius of the pinion. \(D\) represents the linear distance between the apexes of the two pitch cones along the axis line.  
Figure 3: Beveloid gear with segmental parabolic axial modification.  
In accordance with the specifications outlined in Equation (6), the optimal modification parameters have been identified, with a tentative maximum modified amount of 10 \(\upmu\)m. Additionally, the modified lengths for the heel and toe have been established at 20% of the tooth width. In accordance with the principles of parallel shaft beveloid gear design [24], the geometric parameters of the gear pair illustrated in Table 1 are calculated.  
The modified model of the beveloid gear pair was generated using mathematical modelling software and three-dimensional modelling software. The digital modelling process is illustrated in Figure 5.  
In order to enhance computational efficiency, simplify the simulation process, and circumvent the impact of rigid body boundary coupling constraints on the calculation outcomes; the grid model of the beveloid gear pair will be simplified to a five-tooth model. Additionally, the grid model will be locally refined using Hypermesh, and the results are illustrated in Figure 6.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Pinion** & **Gear** \\ \hline Normal module (mm) & 2.5 & 2.5 \\ Centre distance (mm) & 80 & 80 \\ Pressure angle (\({}^{\circ}\)) & 20 & 20 \\ Teeth number & 27 & 37 \\ Tooth width (mm) & 26 & 24 \\ Pitch angle (\({}^{\circ}\)) & 6 & 6 \\ Modified amount (\(\upmu\)m) & 10 & 10 \\ Modified length (mm) & 5.2 & 4.8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Basic parameters of the involute beveloid gear.  
Figure 4: Meshing coordinate system of the beveloid gear pair.  
Figure 5: Digital modelling process.  
The grid model was imported into ANSYS Workbench 2022 R1 for the purpose of conducting a meshing performance analysis of the paralleled beveloid gear pair. The density of the gear material is 7.85 \(\times\) 10\({}^{3}\) kg/m\({}^{3}\), the Young's modulus is 210 GPa, the Poisson's ratio is 0.3, and the meshing model of the beveloid gear is illustrated in Figure 7.  
The pinion is identified as the driving wheel, while the gear is identified as the driven wheel. The face-to-face contact type is selected between the pinion and gear, wherein the friction coefficient is 0.15. The degrees of freedom of movement and rotation of the \(X\) and \(Y\) axes are constrained at the axis position of the pinion and gear, respectively, in order to create the connecting pair. A rotation displacement is applied to the pinion, and a moment in the same direction of 500 Nm is applied to the gear in order to simulate the real situation of the transmission of the beveloid gear pair.