# A Numerical Gear Rolling Test Method for Face-Hobbed Hypoid Gears
###### Abstract  
A numerical gear rolling test (NGRT) method for actual tooth surfaces was proposed. The non-uniform rational B-spline (NURBS) fitting method was adapted to reconstruct the actual tooth surface, and the instant meshing point as well as the transmission error of two meshing gears were solved. The instant contact ellipse boundary of each meshing point can be accurately searched, and then the contact pattern formed by a series of contact ellipses on the actual tooth surface can be obtained. Finally, the rolling test experiment for a pair of hypoid gears was conducted to verify the feasibility of the proposed method. The result shows that the contact pattern obtained is consistent with the actual rolling test result, the relative errors of the transmission error amplitudes were only 4.5% and 5.3% for the drive and coast sides, respectively, and the contact ellipse of the actual tooth surface is not necessarily a standard ellipse and the instant contact path is not necessarily a straight line. This NGRT method helps to reduce equipment and time costs in closed-loop gear manufacturing, which is of great significance for mass production.  
hypoid gear; measuring tooth; numerical gear rolling test; non-uniform rational b-spline; actual tooth surface +
Footnote †: journal: Applied sciences  
0000-0002-288-8000-0000-0000-0000-0000-0000-0000-0000-00000-0000-0000-00000-0000-0000-00000-0000-0  
through computational tooth surface grids coinciding with the original tooth surface grids. The tooth surface deviation was taken as the objective function for optimization for obtaining the actual tooth surface of the spiral bevel gear.  
The numerical actual tooth surface modeling method based on measuring data is the foundation for the actual tooth surface contact simulation technology. [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) established an actual tooth surface with tooth surface deviation and conducted RTCA of hypoid gears. [PERSON] et al. [PERSON] et al. (2018) used a series of continuously changing cylindrical surfaces to intercept the meshing tooth surface for RTCA. Utilizing the least rotation angle method and the improved quad-tree search algorithm, [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) calculated the continuous transmission error and contact pattern in the numerical tooth contact analysis. [PERSON] et al. [PERSON] et al. (2018) developed an algorithm to approximate the actual tooth surface and calculated the contact pattern. [PERSON] et al. [PERSON] et al. (2018) used the interference of the tooth surface to solve the contact spot for an actual tooth surface.  
The above research enables us to compare and verify RTCA results with rolling test results to solve the problem of inconsistent evaluation standards. For this method to be feasible, the RTCA result must accurately reflect the meshing conditions of the actual tooth surface, and the contact pattern and transmission error must be determined with high accuracy. However, the solution of the contact pattern relies on the quadratic approximation near the meshing point, the complicated main curvature, and the principal direction derivation in the RTCA method [PERSON] et al. (2018); [PERSON] et al. (2018), which is inaccurate because the instant contact ellipse is not the actual contact ellipse. [PERSON] et al. [PERSON] et al. (2018) proposed a new TCA method based on numerical iteration that produces accurate contact ellipse boundaries and instant contact lines through the TCA of theoretical tooth surfaces. However, this method has not been used for the TCA of the actual tooth surface.  
Therefore, the NGRT method is proposed to replace traditional rolling tests, improve gear production efficiency, and reduce equipment costs.