Advanced calculation of the deflection of worm shafts with FEM

Abstract

Worm gears are gear units with a shaft angle of mostly 90°. They are used in a variety of industrial applications due to large gear ratios and high transmittable torques. The high sliding ratio in the tooth contact requires a hard/soft material combination that is insensitive to scuffing. Worm shafts usually are made of case-hardened steel and worm wheels of a copper-tin-bronze alloy. Since damage predominantly occurs at the wheel (pitting, wear, tooth fracture, etc.), a wide range of calculation methods for load capacity and service life were developed. Regarding the worm shaft, only the deflection is examined, because high displacements of the shaft shifts the contact pattern, which leads to transmission errors and may lead to increased wear. Common calculation methods, e.g. according to DIN 3996 and ISO/TS 14521 provide a good approximation of the deflection. However, simplifications are made in favour of the manageability of the calculation method. Geometric modifications on the worm such as reduced tooth thickness or different lead angles cannot be taken into account with common methods. In order to close this gap, a new approach was presented by Norgauer in 2021, however, significantly overestimating the stiffening effect of the gearing in some cases by neglecting the helical winding of the worm teeth. For efficiency-optimal design of minimally thin worm shafts, approaches that go beyond the deflection are also missing. Investigations by the authors on worm gears using the finite element method (FEM) show a deviation of the bending line of worm shafts from the common calculation methods. The FEM calculations simulate the tooth meshing under load by a driving torque on the shaft, whereas the standard calculations simplify the load distribution on the worm shaft as a radial force introduced at a point. In addition to the magnitude of the radial forces, the axial tooth force component was identified as an influence factor on the bending line since it causes a displacement of the maximum bending location. Especially for increasing diameter factors the displacement of the bearings under load must be taken into account since they may be in the same range as the deflection. Furthermore, the helical winding of the teeth around the shaft leads to an average stiffening effect of the gearing. The cross-section depends on the gear meshing position and causes a periodic fluctuation of the area moment of inertia and a wobbling motion of the shaft. The changing load distribution on several teeth causes a periodic change of the lever arms and an additional dependence of the deflection from the mesh position. An analytical calculation method for the deflection of worm shafts is proposed. It aims at the precise calculation of the spatial expression of the bending line, extends the commonly used simplified model and takes the newly identified factors into account. The bending lines in both radial directions are considered separately. The values of the new introduced factors are derived from the FEM results. Finally, the calculated bending lines are compared with the FEM results and verified.