# Time-varying meshing stiffness of hypoid gears of automobile drive axle

The transmission error of gear motion is the main excitation of gear vibration energy. In most gear equipment, it will cause other components to vibrate and radiate outwards in the form of noise, which makes people feel upset. For the hypoid gear system in the main reducer of automobile drive axle, the previous research mainly focused on the simulation of the gear processing process to obtain the high-precision gear outer contour, through the gear geometric contact analysis to optimize the gear geometric parameters and machine processing parameters, and then calculated the gear root stress, load distribution, motion transmission error, etc, However, these results can not fully reflect the vibration and noise response of the gear system in the process of gear meshing. Many undesirable gear meshing noise problems still exist in the vehicle drive axle. For this reason, many researchers have established the dynamic response mathematical model of the main reducer gear system, studied the influence of the stiffness, damping and inertia mass of each component in the gear system on the gear dynamic system, and established the relationship between the static analysis of the gear and the dynamic response of the system through the meshing stiffness. The biggest difference between the hypoid gear and the spur gear in the three kinds of parameters is the meshing stiffness. The main reason is that the direction of meshing force is basically unchanged when the spur involute gear is meshed. There are more mature calculation methods and empirical formulas , but the geometry and meshing process of the hypoid gear are complex, The direction of meshing force and the position of meshing point in the process of gear meshing change with the change of gear rotation angle and loading force . Compared with the spur gear, the research on meshing stiffness of this kind of gear is less.

In foreign countries, teik et al. Established a multi degree of freedom model of quasi double faced gear mesh, and used a constant and trigonometric series to approximate the gear mesh stiffness. Mohamadpour et al. Obtained the time-varying meshing stiffness of hypoid gears through the professional finite element software calyx, simplified the meshing stiffness of gears in the form of trigonometric series, and finally obtained the empirical formula of meshing stiffness of gears. In China, Fang Zongde established the dynamic meshing model of hypoid gear according to the static stress state of gear meshing, but this model did not consider the time-varying characteristics of meshing stiffness in the process of gear meshing. Tang Jinyuan et al. Calculated the meshing stiffness of a single gear tooth from the basic theory of gear stiffness, and then obtained the meshing stiffness of multiple gear teeth when meshing at the same time from the superposition of a single gear tooth. This model ignored the influence of the direction of gear meshing force changing with the change of gear meshing position in the process of hypoid meshing.

It can be concluded from the above analysis that the time-varying meshing stiffness of hypoid gears has put forward corresponding calculation methods in foreign countries, and the meshing stiffness has been approximately processed by Fourier series, and the gear meshing stiffness expression is directly output by software, while the actual Fourier series can not fully represent the meshing stiffness characteristics of gears, and the specific calculation details have not been disclosed, However, there are few researches on the meshing stiffness of hypoid gears based on ABAQUS or ANSYS. In China, there are few researches on the real meshing characteristics of hypoid gears, and there are few applications of calyx software. It is difficult to establish accurate geometric model and finite element model of hypoid gears. Therefore, this paper proposes a complete calculation method of time-varying stiffness of hypoid gears, and uses the current mature Numerical Calculation software MATLAB to get the coordinate points of gear tooth surface, Then, the 3D model of hypoid gear is built in CATIA, and the finite element modeling process of hypoid gear is described in detail by using ABAQUS software, and the time-varying meshing stiffness of hypoid gear is obtained by postprocessing the finite element calculation results. This meshing stiffness can be applied to the dynamic analysis of the gear system without any other assumptions, so it can provide a basis for better prediction of the dynamic response of the hypoid gear drive system of the automobile drive axle. 