# Calculation method of gear meshing stiffness based on tooth root dynamic strain measurement

According to the definition of mechanics of materials: under the action of external force, the energy stored in elastic body due to elastic deformation of elastic body is called elastic strain energy. In the mechanical analysis of gear, the gear tooth is usually simplified as a cantilever beam with variable cross-section. Therefore, the elastic strain energy stored in the gear tooth during gear meshing can be defined as:

In the formula, E is the elastic modulus and σ is the distribution of bending stress in gear teeth.

In the process of gear meshing, the bending stress σ of gear teeth is constantly changing along the direction of tooth height and tooth thickness. In order to facilitate the subsequent calculation, the average bending stress σ AV of gear teeth is used to replace the bending stress σ. In the process of gear meshing, σ AV is a constant for a certain position of gear meshing in the tooth profile. Therefore, the elastic strain energy u stored in gear teeth can be modified as follows:

In the current case, the gear tooth is regarded as a cantilever beam structure with variable cross-section, so the bending stress of different sections of gear tooth along the direction of tooth height can be expressed by formula

In the formula, M is the bending moment of gear meshing teeth, C is the half tooth thickness of different sections along the tooth height direction, and I is the area inertia moment of different sections.

The bending stress distribution along the tooth height direction drawn by the formula is shown in the figure. The normal force applied at the meshing position of the gear profile is 2.4633 × 10N, and the bending stress distribution at other meshing positions is not given. The average factor of bending stress at the meshing position can be calculated by using the bending stress distribution of the graph. The average factor of bending stress can be defined as the maximum bending stress and the average bending stress along the tooth height direction. The maximum bending stress occurs at the root of the tooth, and the average bending stress is defined as the sum of the maximum bending stresses in different sections along the tooth height direction than the total number of the sections along the tooth height direction. Therefore, the average factor of bending stress is constant for a certain meshing position during gear meshing. However, with the gear meshing and rotating, the normal load on the gear profile will also move on the tooth profile, so the distribution of gear bending stress will change. According to the distribution of bending stress in different positions of gear meshing, the average factor of bending stress in the whole gear meshing process can be calculated. With the change of gear meshing position, the average factor of bending stress will change. 