### 1. Mathematical model of transmission error

Figure 1 shows the dynamic model of spiral bevel gear transmission system under elastic support. Due to the elastic deformation of transmission shaft and bearing, a bending torsion shaft pendulum coupling vibration analysis model is formed. In this model, a rectangular coordinate system is established. The axial direction of the pinion is x-axis and the axial direction of the big gear is y-axis. Each bevel gear has 6 degrees of freedom in space, and the whole gear transmission system has a total of 12 degrees of freedom, including the coupling vibration of various vibration forms such as transverse vibration, axial vibration, torsional vibration and torsional pendulum vibration of the gear along the X, y and Z axes. In each degree of freedom, the equivalent mass, equivalent damping and equivalent stiffness are determined by the specific structural dimensions of gear, transmission shaft and bearing. In the figure, K and C are stiffness coefficient and damping coefficient respectively, t θ They are torque and torsional vibration displacement respectively, and subscripts 1 and 2 represent small wheel and large wheel respectively.

In the figure: CJI and kji (I = 1, 2; J = x, y, z) are the translational damping and stiffness coefficients of the main and driven gears along the X, y and Z axes respectively.

Vibration equation of spiral bevel gear transmission system:

Relative displacement along the normal direction of the meshing point caused by vibration and error between the meshing points of two bevel gears:

Where: Xi, Yi, Zi (I = 1, 2) are the translational vibration displacement of the main driven gear shaft along the x-axis, Y-axis and z-axis respectively; θ 1x， θ 2X is the torsional vibration displacement of the driving gear around the x-axis and the driven gear around the y-axis respectively; θ 1y， θ 2Y is the swing vibration displacement of the driving gear around the x-axis and y-axis respectively; θ 2x， θ 2Z is the swing vibration displacement of the driven gear around the x-axis and z-axis respectively; δ 1， δ 2. Pitch cone angles of main and driven bevel gears respectively; α N is the normal pressure angle; β 1、 β 2. The helix angle at the midpoint of the main and driven bevel gears respectively; RP1 and RP2 are the meshing point radius of the main driven wheel; En (T) is the normal static transmission error.

### 2. The mathematical formula of transmission error is extracted by finite element analysis

The transmission error is defined as the deviation between the actual rotation angle and the theoretical rotation angle of the large wheel when the small wheel rotates a certain angle, which is expressed by the following formula:

Where: φ 1， φ 2 is the actual rotation angle of small wheel and large wheel; φ 01， φ 02 is the initial position of the two wheels; Z1 / Z2 is the theoretical transmission ratio of the gear pair.