# Mathematical model of involute straight bevel gear transmission dynamics

Establishing an accurate mathematical model of gear transmission dynamics is the basis for analyzing and studying the vibration characteristics and dynamic behavior of gear system. Through the efforts of many scholars, the research on dynamic analysis model has achieved rich results. With the in-depth study of mechanical dynamics, the dynamic analysis model of gear transmission has also been developed, such as linear to nonlinear, time invariant and time-varying, single gear to the whole gear transmission system.

If the specific requirements of dynamic analysis are different, the types of mathematical models will be different. The vibration of straight bevel gear transmission system is complex. The existence of clearance and time-varying stiffness makes many vibrations of straight bevel gear system nonlinear. Based on the gear nonlinear vibration theory and the dynamic analysis model of general gear transmission, this chapter establishes the multi degree of freedom vibration model of involute straight bevel gear transmission, The nonlinear factors in the model are discussed and determined.

Based on the theory of gear nonlinear dynamics, the multi degree of freedom vibration model of involute straight bevel gear system is established. According to Newton’s second law and the balance condition of force, the dynamic equation of straight bevel gear transmission is obtained. Then the motion equation of straight bevel gear is dimensionless, and the nonlinear factors of straight bevel gear system are determined. The nonlinear factors in the dynamic model of straight bevel gear will affect the transmission stability of gear system. The determination of nonlinear factors will also make the dynamic model of straight bevel gear more in line with the actual situation of straight bevel gear transmission. In this chapter, using the curve fitting function of MATLAB software and through multiple polynomial fitting, it is concluded that with the higher and higher degree of polynomial, the fitting curve is closer and closer to the original gap function curve, but the fitting accuracy begins to decrease when the degree of fitting curve is higher than 7 times. Therefore, in order to facilitate the analysis and calculation of vibration equation, The tooth side clearance function can be approximated by cubic polynomial. Because there is no step change in the meshing stiffness curve of straight bevel gear and the overall change range of stiffness is small, the dynamic equation of straight bevel gear system can be solved by referring to the method of linear differential equation, and the meshing damping of straight bevel gear can be obtained with the help of the empirical formula given by relevant scholars.

In addition, the solution method of the dynamic equation of straight bevel gear is also discussed, including analytical method and numerical method. The response expressions of speed and acceleration of straight bevel gear transmission are deduced and solved by using the direct integration method. According to the relevant parameters of the straight bevel gear studied, the displacement, velocity and acceleration response curves of the axial vibration, bending vibration and torsional vibration of the straight bevel gear can be obtained.

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