Evaluation of the time-varying mesh stiffness for gears with tooth spalls with curved-bottom features

Gear tooth spalling is one of the most common defects in gear transmission. The loss of surface materials due to tooth spall reduces the Time-Varying Mesh Stiffness (TVMS) of the gear pair, and thus modifies the vibration response of the gear transmission. The evaluation of the TVMS of the gear tooth pair under gear tooth spalling conditions plays an important role in gear dynamic simulation and the corresponding fault feature analysis. Common approaches assume that the tooth spall has a flat bottom with a constant tooth spall depth. This assumption implies an abrupt change in dent depth and cliff-like material loss. However, in practice, the flake of the surface material usually results in a gradually changing dent depth with a curve-shaped bottom. Thus, the modelling of tooth spalls with a flat bottom may result in mesh stiffness differences. To address this shortcoming, this paper proposes a curved-bottom shaped tooth spall to model the tooth spalling geometric features as observed in practice. The proposed method is constructed based on an ellipsoid geometry which is capable of varying radii in three dimensions to best fit the shape of the tooth spall. The foundation stiffness within the double tooth contact area in the proposed method is corrected and the non-linearity of Hertzian contact stiffness is considered. The effectiveness of the proposed method on modelling single and multiple tooth spalls with different shapes and severity conditions is then validated.