## 1. Simulation of planetary gear transmission system under fault free condition

Using the established planetary gear dynamic model, when all the teeth of the sun gear torsion spring K σ When it is consistent, it is no fault state. The input speed of the system is 720 R / min and the output load is 500 n. The time domain diagram of meshing force between planetary gear and inner gear ring under fault free condition is shown in Fig. 1 (a). It can be seen that under normal conditions, the meshing force fluctuates periodically due to the alternate meshing of single and double teeth, and the fluctuation amplitude is stable. The minimum value of meshing force is Fmin = 488 n, the maximum value is Fmax = 956 n, and the average value is FN = 742 n. The frequency domain diagram of meshing force is shown in Fig. 1 (b). The meshing frequency appears at 124 Hz, and the amplitude is the largest, which is consistent with the theoretical value. The amplitudes of second and fourth harmonic generation are relatively prominent, and the amplitudes of high harmonic generation are relatively small.

## 2. Simulation of planetary gear transmission system under crack condition

As the crack depth of the sun gear tooth deepens, the torsional spring stiffness KF is lower than that of other torsional springs. Under the same working condition, the time domain diagram of the engagement force is shown in Fig. 2 (a), which shows that at 0.149 3 s, 0.489 s and 0.83 s, the engagement force has a smaller one increasing impact and two decreasing impact respectively. The cracks of the sun gear mesh with each planetary gear in turn, which is a failure cycle TZ, TZ = 0.34 s in the figure. Fmax = 963 n, Fmin = 472 n, compared with the simulation results of planetary gear transmission system under no fault condition, there is a slight change, FN = 747 n has no obvious change. Let FA and FB be the increasing and decreasing amplitudes respectively, then FA = Fmax FN and FB = FN Fmin. The frequency domain diagram of the meshing force is shown in Fig. 2 (b). Compared with the normal situation, the amplitude of the fundamental frequency is reduced, the amplitude of the octave frequency is unchanged, and a small number of low-frequency bands appear in the low-frequency region.

## 3. The change of gear rotation angle difference with torsion spring stiffness

With the change of torsional spring stiffness, the gear teeth will have elastic deformation due to the influence of flexibility, which leads to the change of rotation angle of the sun gear teeth. As shown in Figure 3, as the stiffness of torsion spring decreases from 10 to 0, that is, the crack depth increases from 0 to 100%, the root fracture occurs, and the torsion angle of gear tooth increases from 0 to 1.6 °。

## 4. Variation of crack depth with torsion spring size

The root crack is shown in Figure 4, and Q1 is the crack depth.

In the established model, by changing the stiffness of the torsion spring of the tooth root, the crack gradually deepens until the tooth root fracture occurs, that is, with the decrease (increase) of the stiffness KF of the torsion spring of the sun wheel, Q1 also increases (decreases) in equal proportion.

With the deepening of the crack depth, the contact area between the sun gear teeth and the planetary gear teeth decreases, the meshing force decreases with the decrease of the meshing area, and the meshing force between the planetary gear and the inner gear ring also decreases. At the same time, due to the reduction of contact area and the softening of tooth root, the position trajectory of the sun wheel is offset, and the offset increases with the deepening of crack depth.

As shown in Figure 5, with the deepening of crack depth, the variation of each variable can be divided into three stages. The first stage is from 0 to 20%. Because of the small crack depth, the difference of meshing force impact and the deviation of sun wheel position trajectory change little. When the crack depth deepens from 20% to 90%, with the further deepening of the crack depth, the difference of meshing force impact and the offset of sun wheel position trajectory begin to increase gradually. When the crack depth deepens from 90% to 100%, that is, the root fracture occurs, the impact of meshing force is large instantaneously, and the position trajectory of the sun wheel also deviates a lot instantaneously.

## 5. Simulation of planetary gear transmission system under broken tooth condition

When the sun gear is broken, that is, the torsional spring stiffness of the sun gear KF is 0, the contact area between the sun gear and the planetary gear decreases in the meshing process, resulting in the reduction of the meshing force; At the same time, the meshing force between the planetary gear and the inner ring gear increases. Under the same working conditions, when the broken tooth of the sun gear is meshed with the measured planetary gear, the impact of meshing force between the planetary gear and the inner gear ring is greatly reduced, and the impact of meshing force between the other two planetary gears and the inner gear ring is increased, and the reduction value is the sum of the two increasing values. At the same time, as the meshing force between the sun gear and the planetary gear decreases (increases), the meshing force between the planetary gear and the inner gear ring also decreases (increases). As shown in Figure 6 (a), the failure period TZ = 0.34 s. The frequency domain diagram of meshing force is shown in Fig. 6 (b). Compared with no fault condition, when the sun gear breaks, the amplitude of fundamental frequency decreases, and the amplitude of multiple frequency has no obvious change, but there are side bands around, and a large number of low-frequency bands appear in the low-frequency area.

## 6. Simulation analysis of planetary gear transmission system at different speeds

The load at the input end of the planetary gear is fixed at 500 N, and the meshing forces between the planetary gear and the inner ring gear at different speeds are extracted respectively when the gear is broken. According to the transmission dynamics of the system, the impact on the system increases with the increase of the speed.

Figure 7 shows the amplitudes of three fault conditions at different speeds. It can be seen that: when the speed increases from 720 R / min to 4200 R / min, the amplitude of broken tooth, crack and no fault increases with the increase of speed, and increases by 124 n, 176 N and 151 n respectively.

## 7. Simulation analysis of transmission system under different loads

When the input speed of the planetary gear is fixed at 1 440 R / min, the load of the output planetary carrier increases by 200, 500, 800 and 1 000 n in turn. The relationship between the meshing force impact amount and the amplitude under different loads is shown in Figure 8. In the process of increasing the load on the output end, the meshing force between the inner ring gear and the planetary gear increases continuously, and the amplitude curve increases linearly under the three fault conditions. It can be seen from figure 8 that under different loads, the increase of broken tooth fault is obvious after each load increase, and the amplitude increase is 399 n; Under the condition of crack and no fault, the amplitude increases obviously after the load is 500 N, which are 143 N and 71 n respectively.