According to the Hertz contact analysis theory, the geometric model of the contact end face of two cylinders as shown in the figure, considering the elastic deformation range of the material, the Hertz normal contact half width a under the action of normal load f can be expressed as:
A-contact half width, m;
Z-contact length of cylinder, m;
V1, V2 – Poisson’s ratio of two cylinder materials;
E1, E2 – elastic modulus of two cylinder materials, N / m2;
R1, R2 – radius of curvature of two cylinders, M.
If two cylinders use the same material elastic properties, that is, v = V1 = V2 and E = E1 = E2, the formula can be further simplified:
According to the local enlarged part of the end contact shown in the figure, the Hertz normal contact deformation δ can be expressed by the following formula from the geometric relative relationship after the contact deformation:
Therefore, the formula can be further simplified, and the simplified results are shown in the formula. By substituting the formula into the derivation, the mathematical relationship between the contact deformation and the normal load f can be obtained. According to the linear elastic stiffness solution method, the mathematical model of the contact stiffness can be derived. See the formula.
According to the formula, the results show that the contact stiffness is independent of the load, and only related to the material properties and contact length of the cylinder. This derivation model is also the Yang sun contact constant model used by scholars at home and abroad in the early contact stiffness analysis of time-varying meshing stiffness. With the further research on the interface contact and the simplified Hertz contact deformation formula derived from palmgreen’s theoretical derivation and experimental summary for the Hertz contact problem of bearings, Comell’s research is based on the simplified semi empirical formula of contact rolling bearings developed by palmgren, assuming that the two contact materials are the same, that is, there are v = V1 = V2 and E = E1= E2, the local deformation equation along the tooth thickness is derived, as shown in the formula. According to the solution method of linear elastic stiffness, the mathematical model of contact stiffness can be derived. See the formula. It can be seen that the empirical mathematical model is related to load and material. In order to distinguish and apply the two models in the future, the formulas are defined as Yang sun model and semi empirical formula respectively.