# Analytical calculation model of meshing stiffness of helical gear pair with tooth profile modification

In helical gear transmission system, internal excitation such as stiffness excitation and error excitation of helical gear is an important reason for the vibration and noise of helical gear transmission system. The research on the calculation method, modeling and coupling mechanism of helical gear stiffness excitation and error excitation is of great significance for the dynamics of helical gear system.

In terms of stiffness excitation, the calculation methods mainly include material mechanics method, finite element method, approximate substitution method and so on. According to different objects, it is divided into spur gear and helical gear. At present, the calculation methods of spur gear meshing stiffness mainly use material mechanics method and finite element method. Chaari et al. Calculated the meshing stiffness of spur gear by using the material mechanics method, considering the bending deformation, shear deformation, radial compression deformation, Hertz contact deformation and wheel body deformation of spur gear, and compared it with the finite element method. The error is within 5%. Because the material mechanics method is faster and more efficient than the finite element method, and the calculation results are consistent with the finite element method, the time-varying meshing stiffness of spur gear is generally calculated by the material mechanics method in the dynamic modeling of helical gear system. In the calculation method of helical gear meshing stiffness, Liu et al. Used the approximate substitution method, and used the time-varying of helical gear meshing line to replace the change of meshing stiffness. Due to the large error of the approximate substitution method, ajmi et al. Used the sheet method to discretize the helical gear into a sheet helical gear along the tooth width direction, ignored the axial potential energy of the sheet helical gear, used the material mechanics method to obtain the load distribution of each sheet helical gear, and then obtained the elastic deformation and meshing stiffness of each sheet helical gear according to the deformation coordination equation, The results are compared with those obtained by finite element method. Because the traditional material mechanics method to solve the meshing stiffness of helical gear ignores the axial deformation of helical gear, it is not widely used. With the development of computer technology, some scholars use the finite element method to solve the meshing stiffness of helical gears. Bu Zhonghong and others calculated the meshing stiffness of a group of internal and external helical gears with different helix angles by using the finite element method, and studied the variation law of time-varying meshing stiffness. Chang Lehao et al. Combined the finite element method with elastic contact theory to calculate the meshing stiffness of helical gear.

The error excitation of helical gear is mainly caused by gear machining and installation. In the early modeling process of helical gear dynamics, error excitation and stiffness excitation were substituted into the dynamic model for helical gear dynamics analysis. Ma Hui and others studied the influence of different tooth top trimming on the meshing stiffness of spur gear by using the finite element method, substituted it into the dynamics of gear system, and analyzed the influence of tooth top trimming on the vibration response of the system. Because the calculation efficiency of the finite element method is low and the number of calculation points is limited, Chen and others proposed a new nonlinear excitation analytical calculation model based on the material mechanics method, established the spur gear meshing stiffness model considering the influence of tooth error, and studied and analyzed the influence of different tooth profile modification on the comprehensive meshing stiffness. On the basis of Chen, Wang Qibin et al. Extended the model, established an analytical model of spur gear meshing stiffness considering tooth direction modification, and compared it with the finite element method. The results show that the relative error of meshing stiffness solved by the two methods is less than 5% under different modification, and the calculation efficiency is much higher than that of the finite element method.

Because the tooth error of helical gear is a three-dimensional space problem, the analytical calculation method of meshing stiffness after tooth profile modification is different from that of spur gear, and the traditional analytical calculation method does not consider the three-dimensional space position of helical gear meshing line and meshing position when calculating the meshing stiffness of helical gear, so the meshing stiffness of helical gear after tooth profile modification can not be calculated.

Based on the improved three-dimensional calculation method of helical gear meshing line and meshing position, considering the influence of axial deformation of helical gear, a general analytical calculation method of helical gear meshing stiffness is proposed, and a nonlinear excitation calculation model coupled with tooth error and helical gear meshing stiffness is established to verify the accuracy and rapidity of the model, The influence of different profile modification parameters on the meshing stiffness of helical gears is studied.

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