# Vibration and Stress Response of High-Speed Train Gearboxes under Different Excitations
## 3 Influence of Ideal Excitations on Dynamic Response of Gearbox Housing
### Effect of Wheel Polygonization  
According to measurement on the wheel polygonization, it can be concluded that the 20 th order of the wheel polygonization dominates in most of the wheels of the high-speed train [17]. In order to study the effect of the wheel polygonization on the dynamic stress of the gearbox housing, the ideal 20 th order wheel polygonization shown in Figure 10 was considered in the rigid-flexible coupling model of the multi-body system, while the wheel polygons on the left and right sides of the wheelset in the model were considered in the same phase of wear. The amplitudes of both wheels' polygonal wear were 0.01 mm. The wheelset was considered as a rigid body and the traction torque of the gearbox set to 0 Nm. Figure 11 shows the relationship between the wheel-rail contact force and the running speed of the high-speed train. When the velocity is in the range of 100-360 km/h, with the increase in speed, the wheel-rail normal force amplitude at the main frequency shows a parabolic trend, reaching the peak value at the 360 km/h speed. Nevertheless, when the running speed is higher than 360 km/h, the amplitude of the main frequency decreases gradually while the speed increases. The reason for this phenomenon is that the wheel-rail system has an eigenmode around 750 Hz, and when the excitation frequency of the wheel polygonization is coupled with the eigenfrequency, the resulting wheel-rail impact energy reaches its peak. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Value** & **Unit** \\ \hline Sample frequency & 10,000 & Hz \\ Order of the wheel & & \\ polygonization & 20 & \\ Wheel rolling circle radius & 0.43 & m \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation parameters. The spectrum analysis of the vertical vibrate acceleration at each speed level is shown in Figure 12. With the increase in the running speed, the excitation caused by the wheel polygonization leads to an increase in the vertical vibration acceleration of the gearbox. However, unlike the wheel-rail force law, the amplitude of the gearbox vibration suddenly increases in the two speed ranges around 200 km/h and 280 km/h. According to the running speed and wheel radius, the 20 th wheel polygonal wear excitation frequency is calculated to be 411 Hz and 576 Hz at speeds of 200 km/h and 280 km/h, respectively, which is in the resonance range of the gearbox eigenfrequencies (405 Hz and 578 Hz) in Section 2.2. It indicates that the gearbox is resonating in these speed ranges and causing abnormal vibration acceleration. When the speed exceeds 360 km/h, the wheel-rail force decreases with increasing speed in Figure 11. Nevertheless, the vibration acceleration of the gearbox increases with speed, which is also an abnormal vibration caused by the resonance of the gearbox, since the gearbox has an eigenfrequency of 927 Hz. Figure 11: The relationship between the wheel–rail force and the running speed under 20 th wheel wear. Figure 10: Wheel polygonization used in simulation. As shown in Figure 13, dynamic stress and vibration acceleration have the same frequency domain characteristics and reach the peak at 411 Hz and 576 Hz due to resonance. This shows that when the high-speed train wheel polygon wear causes modal resonance of the gearbox, the dynamic stress of the gearbox will increase sharply. If it is operated under the wheel polygon wear condition for a long time, it will lead to the acceleration of fatigue damage accumulation in the gearbox. With the continuous increase in the vehicle operating mileage, the wear on the wheel will also increase. In order to study the influence of the polygonal wear amplitude of the wheel on the dynamic stress of the gearbox, the dynamic stress of the gearbox is calculated when the vehicle runs at the speed of 280 km/h under the condition that the amplitude of the 20 th order polygonal wear amplitude range from 0.01 mm to 0.08 mm. It can be seen from Figure 14 that the maximum and average values of dynamic stresses in the gearbox increase as the polygon amplitude increases.