Automated Tooth Bending Stress Calculation with the Consideration of Measured Tooth Gap Contours

Based on the geometry analysis carried out, the following section presents a procedure for the automated tooth root stress calculation, considering the measured tooth gap contours. The automation includes both the stress calculation according to ISO 6336 3:2019, hereafter referred to as the standard-based method, and the possibility of calculating tooth root stresses using the FE-based tooth contact analysis STIRAK (Ref. 14). The automation allows a reproducible and consistent calculation of the actual tooth root stresses when performing and evaluating tooth bending strength tests on the pulsator. A consistent stress calculation procedure is essential when investigating multiple variants to ensure comparability of results and avoid errors in the interpretation.

As input variables for the automatic tooth root stress calculation, the results of the gap contour measurements, an input file of the test gears for the FE simulation program, and the pulsator forces for the pulsator tests are required. All three input data are used for the stress calculation according to ISO 6336 3:2019 and for the FE simulation. Figure 5 shows the procedure for the standard-based tooth root stress calculation. In addition to the known macro geometric gear parameters, such as normal module, normal pressure angle, number of teeth, and some other parameters, the actual manufactured root diameter and tooth root radius, as well as the generating profile shift coefficient used to machine the tooth gap contour, are required to perform the stress calculation. Therefore, the first step is to create a hypothetical manufacturing tool that could theoretically produce a measured root gap contour in one operation. The data of the manufacturing tool in the input file is used as a starting value for the iteration of the hypothetical manufacturing tool. If the data is a protuberance tool, only information describing the basic rack profile of a simple production tool is considered, such as the reference profile angle, the addendum coefficient, the dedendum coefficient, and the tool tip corner rounding coefficient. The start value for the tool tip corner rounding factor is set to taP0,Start = 0.05, regardless of the used basic rack profile, to ensure that the basic rack profile of the hypothetical manufacturing tool is generated and to avoid a tool tip that is far from reality. The iteration then starts with a stepwise adjustment of the generating profile shift coefficient in order to fit the generated FE-based contour to the measured contour in the area of the tooth flank. The fitting quality is assessed using the median normal distance deviation ΔnDev,median between the two curves in the tooth flank observation area (see Figure 5). The addendum coefficient of the hypothetical finished gear tool is then adjusted using Equation 1.

The third adjustment to the basic rack profile of the manufacturing tool concerns the area of the tooth root that is machined by the tool tip corner rounding of the manufacturing tool. To achieve the best possible match between the FE-based contour and the actual contour, the tool tip corner rounding coefficient is changed step by step in an iterative process. As with the tooth flank iteration, the fitting quality is checked after each iteration step using the median normal deviation distance ΔnDev,Median between the two curves in the tooth root area. A maximum deviation of ΔDev,Median ≤ 1 μm was used as a stop criterion for the iteration of the flank area and the tooth root area.

Once the basic rack profile of a suitable hypothetical manufacturing tool has been fully determined, the second step is to calculate the actual tooth root stress using the equation shown in Figure 5 (Ref. 14). When calculating the required correction factors, it should be noted that the point of force application in the pulsator is often not at the outer point of single tooth contact and is dependent on the number of teeth clamped. The resulting geometric relationships are shown in Figure 5, bottom left, and must be taken into account when calculating the correction factors YF and YS. In addition, the generating profile shift value from the tool iteration must be used. The required tooth root radius results from the tooth root radius of the basic rack profile of the gear ρfP and corresponds to the tool tip corner rounding ρaP0 of the hypothetical manufacturing gear tool (Ref. 37). Considering the relationships above, the equations from ISO 6336 3:2019 are then used to calculate the actual tooth root stress of the tooth gap contour under consideration (Ref. 14).

For the tooth root stress calculation with the FE-based tooth contact analysis, the measured tooth gap contours are first smoothed using a spline approximation. The smoothed tooth gap contours can be read directly by the used tooth contact analysis program and considered when building the FE-model. The implemented FE meshing process of the program requires a smoothed tooth gap contour as the contour normally used to determine the meshing direction of the FE mesh. Any contour discontinuities in the measured and unsmoothed tooth gap contours would therefore lead to an unfavorable FE mesh in some cases and thus negatively affect the results. Furthermore, the rolling position of the FE-based simulation, which corresponds to the contact conditions in the pulsator test, must be determined, as the tooth contact analysis simulates the quasi-static tooth flank contact of a running test. The rolling position is determined by comparing the pulsator diameter dPuls, which represents the diameter of the force application in the pulsator test, with the contact line diameter of all rolling positions of the FE-based simulation. The pulsator diameter dPuls results from the base tangent length WK of the number of clamped teeth k and defines the position of the contact line on the tooth flank in the pulsator test. The rolling position with the best fit is used for further tooth root stress calculation. The required simulation torque is determined by iteratively adjusting the torque until the sum of the tooth normal forces of the selected contact line matches the desired pulsator force. Finally, the tooth root stresses occurring in the pulsator are calculated for each measured tooth gap contour, considering the simulation torque and the selected rolling position.