According to the energy method, the bending potential energy, shear deformation energy and radial compression deformation energy are also stored during gear meshing, thus forming Hertzian stiffness, bending stiffness, shear stiffness and radial compression stiffness.Therefore:
In the equation, is the meshing force,, s and alpha respectively represent the bending potential energy, shear deformation energy and radial compression deformation energy, and b, s and alpha respectively represent the corresponding bending rigidity, shear rigidity and radial compression rigidity.The energy method simplifies the teeth of involute spur gear into a cantilever beam with variable cross-section, in which the diagram of meshing forces of individual teeth is shown.
According to the deformation theory of cantilever beam, it can be seen that:
Formula indicates the horizontal distance between the engagement point and the base circle; indicates the distance between the engagement point and the symmetrical line of the teeth; and respectively represent the elastic modulus and shear modulus of the gear. The relationship between them is:
Ix and x represent the moment of inertia and the cross-section area of the gear teeth at the distance from the base circle respectively, and their calculation formulas are:
The radial and tangential components of alpha and B are respectively represented by the following formulas:
In order to describe the parameter attributes of the driving and driven wheels at different angular displacements, the engagement pressure angle 1 is used as the independent variable. According to the geometric characteristics of the gear involute,,, X and DCan be expressed as follows:
In the formula, B is the radius of the base circle of the gear; 1 and 2 represent the engagement pressure angle of the gear system and the curvature of the half teeth on the base circle, respectively, as shown in the figure, and their calculation formulas are as follows:
Formula 1, 2 and 11 respectively represent the number of active teeth, driven teeth and gear teeth; 0 represents the pressure angle on the indexing circle of the gear; and represents the crest height factor.
The expressions of bending stiffness b, shear stiffness s and radial compression deformation stiffness are:
It can be found from the formula that the bending rigidity, shear rigidity and radial compression deformation rigidity of the gear only relate to the angular variable 1, which determines the position of the engagement point on the profile curve.