## 4 Results and Discussions  
### Results of Axle-Box Vibration Acceleration Caused by Wheel Polygons in Time-Domain  
#### 4.1.1 Effect of Wheel Polygon  
In the section, we present some numerical results showing how the axle-box vibration is related to types of wheel polygons, and examine the time dependence of the vertical acceleration of the axle-box induced by wheel polygons, as derived from the numerical model in Section 3.1.  
Figure 4 shows the time dependence of the vertical axle-box acceleration caused by wheel polygons of various harmonic orders and OOR amplitudes at a vehicle speed of 350 km\(\cdot\)h\({}^{-1}\). The harmonic orders of the polygonal wheels range from 1 and 25 (i.e., the polygonized wheels have 1 to 25 wavelengths around the circumference of the wheel). The OOR amplitudes range from 0.01 mm to 0.12 mm [27].  
Figure 4: Vertical acceleration of axle-box as a function of (**a**) out-of-roundness (OOR) amplitude and (**b**) harmonic order of polygonized wheel at 350 km\(\cdot\)h\({}^{-1}\).  
Figure 4a shows that the axle-box acceleration increases approximately linearly with OOR amplitude, with the slope for the highest-order polygonized wheel being several times greater than that for the lowest-order polygonized wheel. Table 2 shows the increase of axle-box acceleration per unit OOR amplitude (m-s\({}^{-2}\)-mm\({}^{-1}\)) for the first 25 orders of polygonized wheels. For example, when the velocity is 200 km-h\({}^{-1}\), the growth in axle-box acceleration per unit OOR amplitude for the 25 th-order polygonized wheel is 1421 m-s\({}^{-2}\)-mm\({}^{-1}\), which is 7.2 times the growth per unit OOR amplitude for a 4 th-order polygonized wheel. The growth rate increases gradually as the speed increases, reaching 285 m-s\({}^{-2}\)-mm\({}^{-1}\) and 2115 m-s\({}^{-2}\)-mm\({}^{-1}\) for the 4 th and 25 th orders, respectively, at a speed of 350 km-h\({}^{-1}\), which is 1.44 and 1.49 times greater than at 200 km-h\({}^{-1}\). When the vehicle speed is 350 km-h\({}^{-1}\), the growth per unit OOR amplitude of the axle-box acceleration with OOR amplitude for the 23 rd-order polygonized wheels is 2034 m-s\({}^{-2}\)-mm\({}^{-1}\), which is 4.9 times that for a 6 th-order polygonized wheel. Thus, high-order polygonized wheels more strongly affect the axle-box vibration than do low-order polygonized wheels.  
Figure 4b shows that the axle-box acceleration increases with increasing harmonic order and that the growth per harmonic order (m-s\({}^{-2}\)) increases substantially with increasing OOR amplitude. Table 3 lists the growth per harmonic order of axle-box acceleration for different OOR amplitudes. These results reveal that, for anOOR amplitude in the range 0.01 to 0.12 mm, the growth per harmonic order increases from 0.38 to 6.12 m-s\({}^{-2}\), at 200 km-h\({}^{-1}\), and from 1.05 to 10.22 m-s\({}^{-2}\) at 350 km-h\({}^{-1}\). It's concluded that the growth per harmonic order of axle-box acceleration due to polygonized wheels increases gradually with increasing velocity at all OOR amplitudes.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{6}{c}{**Train Speed(km\(\cdot\)h\({}^{-1}\))**} \\ \cline{2-7}  & **200** & **225** & **250** & **275** & **300** & **325** & **350** \\
**Harmonic Order** & & & & **Growth Rate (m-s\({}^{-2}\)-mm\({}^{-1}\))** & & & \\ \hline
1 & 108 & 122 & 156 & 170 & 192 & 233 & 242 \\
2 & 123 & 155 & 171 & 184 & 224 & 248 & 261 \\
3 & 173 & 199 & 214 & 225 & 246 & 255 & 275 \\
4 & 198 & 206 & 216 & 229 &The above analysis indicates that the axle-box vibrations caused by polygonized wheels are almost the largest in the entire vehicle system, a result attributed to the axle-box being directly connected without damping to the axle through a bearing. On Chinese high-speed trains, the axle-box and its related parts commonly suffer from vibration fatigue, which destroys end covers and loosens bolts.