The elastic deformation analysis and time varying meshing stiffness (TVMS) calculation of spur gears have long been important and difficult issues in the study of spur gear dynamics. Currently, the main methods for solving time-varying meshing stiffness are the potential energy method (also known as the energy method) and the finite element method. Due to its large computational complexity, the finite element method is often used as an auxiliary means for experimental verification.
In 1987, Yang et al. proposed an analytical method for determining the meshing stiffness of time-varying gears from the perspective of potential energy, with a calculation accuracy of 99. 5% After that, the potential energy method was widely accepted by scholars at home and abroad. Based on Yang’s theory, Tian considers the impact of shear action and divides meshing stiffness into four parts: Hertz contact stiffness, bending stiffness, radial compression stiffness, and shear stiffness. Sainsot et al. deduced a two-dimensional analytical formula for the circular ring of the spur gear base with a shaft hole considering that the actual wheel body is of an annular structure, and modified the calculation formula for the stiffness of the wheel body. Chaari et al. calculated the meshing stiffness of spur gears with crack faults considering the impact of energy on each part of the gear teeth. According to the objective fact that the base circle radius and the root circle radius of gear teeth are not completely equal, Wan Zhiguo et al. considered the size relationship between the base circle and the root circle under different gear tooth numbers, reducing the calculation error of TVMS. Yang Yun et al. provided formulas for calculating the meshing stiffness and the equivalent torsional stiffness of a single tooth at the meshing point location. Based on the proposed calculation model, the effects of parameters such as tooth profile eccentricity on the meshing stiffness and the equivalent torsional stiffness of a single tooth were analyzed. Sun Yalin replaced the transition curve of the tooth profile with a straight line segment and solved the meshing stiffness of spur gears under normal/crack fault conditions using the potential energy method. Based on the potential energy method, Zhang Tao et al. derived an analytical formula for the meshing stiffness of spur gear pairs, and analyzed the variation of the meshing stiffness of spur gears under different frictional forces. Wang Xiaopeng et al. combined the potential energy method to construct a calculation and analysis model for transforming TVMS from a physical model to a mathematical model, and accurately calculated the TVMS of spur gear pairs through numerical calculation methods.
Huang Jinfeng et al. solved the time-varying meshing stiffness of spur gears based on the general equation of spur gear tooth profile and the energy method, considering the radial meshing force deviating from the center of the spur gear. Meng Zong et al. considered the transition curve function of spur gears and calculated the stiffness changes for 10 different crack lengths by analyzing the complete tooth profile curve. Ma Hui et al. established a universal gear tooth model considering the real tooth profile equation to solve the time-varying meshing stiffness of spur gears. Chen et al. proposed a general analytical model for spur gear meshing based on tooth profile deformation, tooth contact deformation, and other tooth profile deviations. In view of the calculation of meshing stiffness of straight bevel gears, Chen Siyu et al. converted the tooth profile into the equivalent straight cylindrical tooth profile at the midpoint of the tooth width, analyzed the error source of the potential energy method, and compared it with the results of the finite Metacomputing calculation. Liu Yang et al. obtained a tooth profile equation with the tool rolling angle as a unified variable using the principle of generation method, and solved the time-varying meshing stiffness of spur gears based on the energy method. Dai et al. proposed an improved gear tooth analysis model for calculating the meshing stiffness of spur gears, taking into account tooth tip modification and variable load conditions. Chen et al. considered tooth profile deviation and tooth profile modification, optimized tooth side clearance and compared it with three traditional potential energy dynamic models, revealing the differences and applicability of the proposed methods. Wang et al. proposed two types of TVMS models for internal gears considering the effects of tooth backlash and external loads: the first method uses the potential energy method to establish an analytical model; The second method uses a hybrid finite element analysis method to calculate the meshing stiffness. Sun et al. divided spur gears into many separate slices along the tooth width, proposed a modified calculation model for spur gear pairs with tooth profile modification based on the relationship between deformation and the total stiffness of the meshing cycle, and discussed the errors of this method under different modification amounts.
On the basis of previous studies, the full tooth profile of spur gears is divided into root transition curve tooth profiles and involute tooth profiles, and the transition curve parameter equation is introduced. According to the size relationship among the initial circle, base circle, and root circle of the involute, a gear tooth model with different tooth numbers is established, making the gear tooth model more accurate; The upper integral limit of the involute tooth profile is modified to further reduce the error in calculating time-varying meshing stiffness using the potential energy method; The time-varying stiffness of meshing spur gear pairs with different tooth numbers is calculated, and the effectiveness of the method is verified by comparing the calculated results of the improved method and the finite element method.