# A Numerical Gear Rolling Test Method for Face-Hobbed Hypoid Gears
## 4 Numerical Gear Rolling Test Method
### Calculation of Instant Meshing Point and Transmission Error  
The meshing point can be calculated by the following steps:  
Step 1: Tooth surface discretization. The gear tooth surface \(\Sigma 2\) is discretized into the grids, the pinion tooth surface \(\Sigma 1\) is discretized into \(15\times 15\) grids, and the three-dimensional coordinate of each grid point can be determined. Step 2: Adjustment of the position of the tooth surface. Adjusting the rotation angle \(\varphi_{2}\) or \(\varphi_{1}\), until the normal at a node on the gear tooth surface passes through the pinion tooth surface \(\Sigma 1\) (Figure 9a). In this study, the pinion is fixed, and \(\varphi_{2}\) is clockwise rotated to facilitate the calculation. Figure 8: Meshing coordinate systems. Figure 9: _Cont._Step 3: Intersection points determination. Because the pinion tooth surface \(\Sigma 1\) has 14 \(\times\) 14 surface patches, it is necessary to determine which surface patch the intersection point is located on. If the intersection point falls within the first surface patch or on the edge of the first surface patch, there are four lines from the intersection point to the four corner points of the surface patch, and the sum of the angles \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\), and \(\alpha_{4}\) between the four lines is 360\({}^{\circ}\) (Figure 9b). If this condition is not satisfied, the intersection point falls outside the area of the surface patch, and then surface patches will continue to be examined until the surface patch with the intersection point is discovered. The NURBS surface equation can accurately express the surface patch equation. The shortest distance \(d_{\min}\) from the grid point on \(\Sigma 2\) to \(\Sigma 1\) can be calculated using the formula for the shortest distance from the point to the surface. Step 4: Calculation of meshing points. When the shortest distance \(d_{\min}\) from each grid point of the gear to the pinion tooth surface is calculated, the minimum value \(D_{\min}\) of the shortest distance \(d_{\min}\) is determined, as shown in Figure 9c (\(d^{{}^{\prime}}_{\min}\) and \(d^{{}^{\prime}}_{\min}\) represent \(d_{\min}\) at different positions). The minimum value \(D_{\min}\) corresponds to the following three meshing conditions:  
1. If the smallest value \(D_{\min}\) is less than 0.01 \(\upmu\)m and other positions do not interfere, this point is considered to be the meshing point. Step 6 is performed. 2. If the smallest value \(D_{\min}\) is less than 0.01 \(\upmu\)m but there is interference, the rotary angle \(\varphi_{2}\) is too large and must be adjusted. Step 5 is done for detailed adjustment. 3. If the smallest value \(D_{\min}\) is not less than 0.01 \(\upmu\)m, the grid is too sparse and the rotary angle \(\varphi_{2}\) is too small. Local grid refinement near the minimum value \(D_{\min}\) is performed on the gear, and the shortest distance \(d_{\min}\) corresponding to all refined grid points and the minimum value \(D_{\min}\) are calculated again, as shown in Figure 9d. If the decrease in amplitude of \(D_{\min}\) is greater than 1%, the grid continues to be refined. If the decrease in amplitude of \(D_{\min}\) is lower than 1%, the rotary angle \(\varphi_{2}\) is rotated in a clockwise direction according to Equation (4), and Step 3 has to be carried out. The adjusted step length formula is \[h=k\cdot D_{\min}\] (4) where \(k\) is the step length coefficient, \(h\) is the adjusted step length, and \(D_{\min}\) is the minimum value of the shortest distance \(d_{\min}\) between the tooth surfaces of the pinion and gear. The initial step coefficient \(k\) is 0.5, and the subsequent \(k\) value is the \(h\) value obtained in the previous step. Step 5: Rotation angle adjustment.