Influence of Tooth Root Contour Deviations on the Tooth Bending Strength

To meet the increasing demands for high-performance applications in power density, new material alloys and heat treatment processes for gears are constantly being developed (Refs. 1–3). The investigation and evaluation of different materials and heat treatments for gears is carried out using S-N curves (Refs. 2,4,5). However, when generating S-N curves for tooth bending strength, the influence of deviations around the tooth root contour is often not considered or recorded with insufficient accuracy, so that an identical tooth root contour is assumed for all variants (Refs. 2,4,5). Due to the lack of a uniform evaluation standard for the tooth root area, the quality of test gears is usually evaluated only for the tooth flank area using ISO 1328 1:2013 (Refs. 6–10). Stress-increasing effects, such as changes in the tooth root radius due to deformations from the heat treatment, are therefore not considered (Refs. 2,4). In particular, when testing gears with an unground tooth root area, form deviations from the heat treatment and manufacturing deviations from gear hobbing around the tooth root have a direct effect on the test results. These deviations lead to a change in the occurring tooth root stresses and influence the tooth bending strength (Refs. 8,11–16). In the following study, the influence of manufacturing-related geometry deviations around the tooth root on the tooth bending strength is investigated. State of the Art Sufficient load carrying capacity of the tooth flank and the tooth root is a basic requirement for gear design as gear damage may lead to a total failure of the gearbox in the worst case (Ref. 17). For this reason, extensive knowledge of the material strength, the heat treatment combination and the occurring stresses in the application are of utmost importance, especially in the design process of high-performance gears. Factors Influencing the Tooth Bending Strength The tooth bending strength of gears is obtained by comparing the stress and load capacity around the tooth root (Refs. 14,17). Both stress and load capacity are influenced by various parameters, which, depending on their characteristics, can have a positive or negative effect on the tooth bending strength (Ref. 15). Factors Influencing the Tooth Root Stress In addition to the application and the associated load, the stress on a gear is determined by the design of the macro and micro geometry as well as the manufacturing process (Ref. 14). The normal pressure angle, as a macro geometric gear parameter, influences both the length of the bending moment arm and the length of the tooth root chord (Ref. 14). An increase of the normal pressure angle leads to a reduction of the bending moment arm as well as to an increase of the tooth root chord (Ref. 14). Based on the bending beam theory, which can be used to calculate the tooth root stress in a simplified way, a reduction of the bending moment arm leads to a reduction of the bending moment and therefore to a lower tooth root stress (Ref. 17). Increasing the tooth root chord also reduces the tooth root stress because it increases the area moment of inertia of the critical tooth root section (Ref. 18). The area of the critical tooth root section can be further increased by increasing other macro geometric gear parameters, such as the tooth width and the normal modulus, to reduce the tooth root stress (Refs. 14,18). Both the increase of the normal module, with otherwise constant gear parameters, and the increase of the gear width led to a larger area of the critical tooth root section and thus to a reduction of the tooth root stress (Ref. 14,18). Another geometric factor influencing the tooth root stress is the tooth root contour (Ref. 14). The smaller the existing tooth root radius at the critical section, the greater the stress-increasing notch effect and thus the stress on the tooth root (Refs. 14,15). Taking this relationship into account, various studies have shown that by optimizing the tooth root geometry and leaving the tooth geometry otherwise unchanged, an increase in tooth bending strength of 10–30 percent can be achieved (Refs. 11,12,15,16,19). The optimization potential depends on the initial state of the tooth root geometry and the optimization method used (Refs. 11,12,15,16,19). Conversely, deviations from the specified tooth root geometry due to process variations or manufacturing errors can lead to a reduction in the tooth bending strength (Refs. 13,20).