1. Physical model

Take 2K-H Planetary gear transmission system as the analysis object, as shown in Fig. 1 (a). The sun gear and planet carrier are set as input end and output end respectively, and the sun gear rotates counterclockwise to drive the three planet gears to rotate clockwise. The three-dimensional model of the gear system is shown in Fig. 1 (b). The torsional spring and hinge are introduced at the root of the sun gear, and the failure factors are simulated by changing the stiffness of the torsional spring.

2. Dynamics of planetary gear system

The dynamic model of the planetary gear transmission system is established, as shown in Figure 2. Spring is used to simulate the torsion and meshing stiffness of the parts in the transmission system. Taking the center of the sun wheel as the coordinate origin, OS, X and Y represent the horizontal direction and vertical direction respectively. The tooth will deform when it is subjected to contact force. The deformation is simulated by the change of torsional spring stiffness. Change the torsion spring stiffness K σ To simulate the whole process of the sun gear tooth from crack generation to tooth fracture.

The contact model of each tooth is shown in Figure 2. The relative displacement between the planetary gear and the sun gear is expressed by the relative deformation relationship between the planetary gear and the inner gear ring

Where: δ RPI is the relative displacement of inner ring gear and planetary gear I (I = 1,2,3) in the direction of meshing line; Ur is the position vector of inner gear ring; UPI is the position vector of the ith planetary gear; YPI is the projection of the displacement of the ith planetary gear in the Y direction; Yr is the projection of displacement of inner gear ring in Y direction; X R is the projection of the displacement of the inner gear ring in X direction; φ RPI is the phase angle of the ith planetary gear; φ C is the introduced spring phase angle; Erpi is the meshing error; ω C is the rotation speed of the torsion spring; φ PI is the phase angle. When the sun gear is meshed with the planetary gear, the meshing force and elastic deformation between the two teeth are expressed by the spring, the compression is positive and the tension is negative. Considering the material damping, taking the contact between the sun gear and the planet gear as an example, the generalized Hertz contact formula can be expressed as follows:

Where: δ SPI is the normal relative elastic deformation of the contact surface of two gear teeth; δ SPI is the relative velocity between the teeth of the gauge; CSPI is the damping coefficient, CSPI= λ xn， λ Is the lag damping coefficient; Kspi is Hertz contact stiffness, which is determined by material properties and radius of curvature 22-24. R1 and R2 are used to represent the curvature radius of the tooth profile of the sun gear and planetary gear respectively, EI is used to represent the elastic modulus ν If I is Poisson’s ratio, kspi can be expressed as:

The relationship between hysteresis damping coefficient and recovery coefficient is as follows

Therefore, fspi can be expressed as:

The above formula explains the relationship between the contact force and the restitution coefficient and velocity of the two masses before collision. Under the action of the contact stress fspi, the gear teeth will deform. In order to calculate this kind of deformation, the gear teeth are simplified as a cantilever beam with variable cross-section on the elastic foundation, as shown in Fig. 3 (a).

Fig. 3 (b) shows that after the contact stress fpsi is applied, the tooth takes the root as the center of the circle, and the tooth takes the root as the center τ Is the displacement in the direction. For a pair of meshing teeth, the meshing stiffness changes with the change of meshing point, LS is the effective distance from the tooth root to the meshing point, σ J is the comprehensive deformation of the meshing point, σ CJ is the contact deformation under Hertz contact theory, σ BJ is the J-point deformation caused by the bending and shearing of the tooth, σ Bij is the deformation at the load point J caused by the contact force fspi of a small segment I separated from the gear teeth, σ MJ is the additional deformation of meshing point caused by foundation elasticity. Take different values according to the width of gear teeth (B / HP ＜ 5 is narrow gear, otherwise it is wide gear)

The stiffness K of torsion spring can be obtained

According to the balance relationship, the dynamic equation of gear can be obtained by analyzing the force relationship among the parts

Where m is the mass matrix; C is the damping matrix; K is the stiffness matrix; F is the generalized force vector.