When calculating time-varying meshing stiffness of gear pair by energy method, the gear base is assumed to be rigid, while the elastic flange deformation at the root of the gear teeth is neglected, so that the calculated time-varying meshing stiffness of gear has a large error.To compensate for this defect, Sainsot studied the energy stored in the deformation of the transition fillet at the root of the gear according to the theory of elastic ring. The displacement, stress and external load were expressed by Fourier series. It was assumed that the gear was rigid and the base of the gear was an elastic ring with the same width as the teeth, the outer diameter was the root circle, the inner diameter was the shaft bore of the gear, and the teeth were fixed on the root circle.When the external load acts on the tooth surface along the meshing line, the elastic deformation of the ring causes the displacement of the tooth. By fitting the calculated deformation accurately, the rigidity calculation formula corresponding to the flexible deformation of the gear base under the meshing state is obtained:

Where f denotes the rigidity corresponding to the flexible deformation of the gear base; f denotes the thickness of the root circle; f

Represents the minimum distance from the point of intersection between the meshing line of the gear and the symmetrical axis of the gear teeth to the base circle; the parameters,, _and_can be calculated from the following polynomials:

In the formula: the values of coefficient i, i, i, i, I and i; fi denotes the ratio of the radius of the root circle to the radius of the shaft bore of the gear; f denotes half of the span angle of a tooth on the root circle.Their calculation formulas are as follows:

Formula: f denotes the radius of the root circle; I denotes the radius of the shaft bore of the gear; I denotes the number of teeth of the gear; alpha denotes the coefficient of the top height of the gear, which is generally taken as standard value 1; 0 denotes the pressure angle on the indexing circle of the gear; _denotes the coefficient of the radius of the gear cutter, which is calculated by:

Where C denotes the radius of the cutter fillet; and denotes the modulus of the gear.