# Calculation of Hertz contact stiffness of gears

When the energy method is used to calculate the time-varying meshing stiffness of gears, it is assumed that the meshing gears are isotropic elastic materials. According to Hertz elastic contact theory, the elastic compression deformation of the contact part of two isotropic elastic bodies is basically the same as the contact between paraboloids of two objects, and the error is within 5% The theory shows that the Hertz contact stiffness of a pair of gears made of the same material without manufacturing error on the meshing line is a fixed value and will not change with the change of gear meshing position. The expression formula of Hertz contact stiffness is as follows:

Where 𝐸 is the elastic modulus of the gear material, 𝐿 is the tooth width, and 𝜐 is the Poisson’s ratio of the gear material.

It can be seen from the formula that when the gear material is fixed, the Hertz contact stiffness is only related to the width of the tooth, but the tooth width will not change with the gear rotation, so the Hertz contact stiffness is constant in the process of gear meshing. The energy of Hertz contact potential energy generated by tooth contact degeneration is stored near the contact point of the tooth The calculation expression of is as follows:

Where 𝐹 is the meshing force and kh is the effective Hertz contact stiffness in the same direction as the meshing force. 