# Meshing stiffness of variable hyperbolic arc gear based on finite element method

In the process of gear meshing, the periodic changes of the number of meshing teeth and meshing position lead to the periodic changes of gear meshing stiffness, which leads to gear vibration and noise. In order to determine the meshing stiffness of gear, it is necessary to accurately calculate the comprehensive elastic deformation and normal contact force of the tooth surface in the meshing area at each moment, which has always been an important problem in the study of gear dynamics. In the aspect of gear stiffness calculation, scholars at home and abroad have done a lot of research. Wang Longbao analyzed the calculation formula of gear meshing stiffness and determined the influencing factors and regularity of spur gear meshing stiffness; Li Yapeng and others put forward an improved Ishikawa formula based on the analytical Ishikawa formula, and obtained a new calculation method of gear time-varying meshing stiffness; Bu Zhonghong and others used linear programming The meshing stiffness of the involute cylindrical gear is calculated by the finite element method, and the change law of the meshing stiffness in one meshing cycle is obtained. Tang Jinyuan et al. Studied the meshing stiffness calculation method of the spiral bevel gear by using the finite element method, and obtained the single tooth meshing stiffness and multi tooth comprehensive meshing stiffness. However, it is difficult to deduce and determine the meshing stiffness of a new type of variable hyperbolic arc gear whose tooth surface is a hyperboloid with constant curvature.

(1) Based on the finite element method, the meshing stiffness finite element calculation model and meshing stiffness calculation method of variable hyperbolic arc gear are given, which lays a foundation for dynamic modeling and analysis of variable hyperbolic arc gear.

(2) Taking the three tooth model of a pair of variable hyperbolic arc gear as the analysis object, the specific process of calculating the meshing stiffness is given, and finally the single tooth meshing stiffness curve and the double tooth comprehensive meshing stiffness curve are obtained. The results show that, with the increase of the driving wheel angle, the single tooth meshing stiffness curve presents an arch change trend of first increasing and then decreasing; the double tooth meshing stiffness curve shows that the stiffness value in the double tooth contact area basically remains unchanged, while in the single tooth contact area, the stiffness value becomes half of that in the double tooth meshing area.

(3) The variation of meshing stiffness curve of variable hyperbolic arc gear under different loads in the same meshing cycle is analyzed. The results show that the increase of the load will increase the meshing stiffness, because the increase of the load will increase the comprehensive deformation of the tooth surface, thus increasing the contact area of the meshing area; compared with the single tooth meshing stiffness, the double tooth comprehensive meshing stiffness is more affected by the load, and the peak value of the double tooth comprehensive meshing stiffness is about twice of the single tooth meshing stiffness, so the load should be considered when calculating the stiffness curve The effect on the stiffness.

(4) The influence of different tooth line radius on the meshing stiffness curve of variable hyperbolic arc tooth line gear in one meshing cycle is analyzed. The results show that the peak value of the stiffness curve decreases with the increase of the tooth line radius, and the influence of the tooth line radius on the meshing stiffness is greater.

Based on the finite element analysis of load contact of variable hyperbolic arc gear, the finite element calculation method of meshing stiffness of variable hyperbolic arc gear is studied. The principle of parameter selection in the process of finite element calculation is analyzed, and the comprehensive elastic deformation and maximum normal contact force of a pair of variable hyperbolic arc gear pair in the meshing area are analyzed by using the finite element software ABAQUS, and the single tooth meshing stiffness and double tooth meshing stiffness are obtained. Finally, the influence of different loads on the meshing stiffness curve is studied, and the influence law of load on the meshing stiffness of variable hyperbolic arc gear is obtained, which lays the foundation for the dynamic analysis of variable hyperbolic arc gear.

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