The requirements for gearboxes are constantly growing. Among other factors, driven by the concept of achieving sustainability (Ref. 1) and the increasing requirements of e-Mobility, gearboxes are being designed increasingly for maximum performance. The load distribution in the gear contact is a fundamental quantity in the design phase, which is used to design micro-modifications and to determine lifetime. The load distribution can be calculated numerically with full-contact finite element analysis (FE) or with (semi-) analytical methods. Full FE contact investigations are complex and time-consuming. The tooth contact simulation of a gearbox can easily take several hours or even days. This is the reason why alternative methods have been developed and are worthy of further research. Methods deviating from the full FE contact analysis are often summarized in the literature under the term loaded tooth contact analysis (LTCA). With these, analyses are often possible within seconds to a few minutes. In LTCA, stiffnesses of the gear components (housing, bearings, shafts, gearing, etc.) are determined and assembled in a global system stiffness matrix (Refs. 2–6). Tooth stiffness is an essential component in LTCA. Typically, the tooth stiffness is divided into three parts as shown in Figure 1: a) Tooth deformation (normal, shear, and bending deformation); b) Gear body deformation (Tooth tilting deformation); and c) Contact deformation.

A comprehensive review of analytical, hybrid and pure FE methods for tooth stiffness calculation is provided by Marafona et al. (Ref. 9) and Natali et al. (Ref. 10). A basic method for calculating gear body stiffness analytically goes back to Weber and Banaschek (W/B) from 1953 (W/B) (Ref. 7). The theory is widely used (Refs. 9,11–14) and even found its way in standards indirectly through the series development of the tooth stiffness (Refs. 15–17). For the gear body deformation/stiffness, W/B assumes that the tooth is rigid and that the connecting parts of the gear body deform elastically when force is applied. Figure 3a shows the applied contact force P on the tooth at the contact point.

The gear body is represented as a half-plane on which boundary stresses/line loads along the tooth root thickness b are specified. Even though the tooth is stiff, the base surface where the line loads are applied can deform freely. The boundary stresses are intended to reproduce the stress state at the tooth root. Figure 3 shows the boundary loads for bending (b), normal (c), and shear load (d). The loads in Figure 3b–d are used to determine the deformation of the half-plane and the partial work integrals. Summing up and equating the partial work integrals with the work done by the external load P results in the deformation uWB in the direction of engagement. 

Face gear drives are a special type of angular gear unit, in which an involute pinion meshes with a face gear wheel (Refs. 24, 25). The pairing can be helical and with or without center offset. Figure 2 shows an exemplary spur face gear drive configuration without center offset. The pressure angle of the face gear wheel is variable across the tooth width and increases towards the outside radius (Refs. 26, 27). One advantage of face gear drives is that both gearings can be manufactured on conventional cylindrical gear machines, and the contact pattern in the axial direction of the pinion does not need to be adjusted (Ref. 25).

However, the comparison of the gear geometries in Figure 2 leads to the conclusion that the approaches for calculating the gear body stiffness of cylindrical gears cannot represent the deformation state of face gear wheels. Due to the plane shape of the face gear wheel, it is to be expected that the deformation increases in the direction of the outer radius. As a result, the stiffness of the gear body decreases toward the outer radius. This work aims to develop a basic model for calculating the gear body stiffness of face gear wheels based on mechanical analytical approaches.

This article is an abridged version of the presentation of the same name given at the AGMA Fall Technical Meeting 2024 (Ref. 28). For more in-depth information, please refer to the corresponding article.