## 3 Results
### Case 2--Effect of Harmonic Order on Track Dynamics  
The time histories of vertical rail acceleration due to polygonal wheels with an OOR amplitude of 0.03 mm at 300 km/h for different harmonic orders are displayed in Figure 10. The relationship between the acceleration and the harmonic order exhibits the same trend as in Case 1. The numerically calculated dynamic responses of the rail in Case 2 are displayed in Figure 11. From Figure 11a, we see that the vibration acceleration of the rail increases linearly with the increase in the harmonic order of the wheel polygon. There are no significant differences in vibration acceleration between different parts of the rail cross-section. The influence of a higher-order polygonal wheel on rail acceleration is greater than that of a lower-order wheel. For instance, the acceleration caused by a 21 st-order polygonal wheel is 18 times greater than that caused by a third-order wheel. The simulation results show a high degree of congruence compared to the acceleration response of the track system reported in Ref. [14]. Figure 9: Effects of OOR amplitude on dynamics in rail-supporting structural components: (**a**) displacement and (**b**) von Mises equivalent stress. It can be observed from Figure 11b that the same trend appears at each of the different positions. The vibration displacement in the rail increases as the harmonic order increases from three to nine, and increases most rapidly as the harmonic order increases from three to six. When a ninth-order polygonized wheel is travelling along the track at 300 km/h, the passing frequency is given by \(f_{\rm n}=v/\lambda_{\rm n}\) = \(v/(2\pi R/n)\). When \(R\) = 0.46 m,\(f_{\rm n}\) = 259.5 Hz. According to the simulation results from tests of the slab track resonance mode, the low resonant frequency of the rail and the slab occurs at 286 Hz. This is very close to the passing frequency of 259.5 Hz. Hence, there is a significant increase in the ninth-order displacement, possibly because of an excitation of resonances in the rail and the slab. We conclude that modal properties of the track correlate with system parameters such as speed, polygon order, and wheel size. By contrast, the rail vibration displacement caused by wheel polygons with orders of 9 to 21 decreases with an increase in harmonic order. For a given harmonic order, the maximum vibration displacement is found in the rail head. In addition, there is little difference in vibration displacement between the rail web and rail foot. It can be seen from Figure 11c that the von Mises equivalent stress increases linearly with an increase in harmonic order at node A. The von Mises stress of the other nodes remains almost constant. The maximum stress occurs in the rail head, followed by that in the rail web, and the minimum stress  
Figure 10: Time histories of vertical vibration acceleration at points A to E for various harmonic orders. is found at point E for a given harmonic order. This implies that an increase in harmonic order has a greater influence on the stress in the rail head. The corresponding dynamic responses of the rail-supporting structural components--namely, the concrete slab layer, CA layer, and support layer--are displayed in Figure 12. The results illustrate the effects of the harmonic order of a polygonal wheel on the dynamics in the supporting track components. The influence of harmonic order on the vibration displacement and von Mises equivalent stress in the concrete slab layer is significant. Both the displacement and the stress increase with an increase in harmonic order. For instance, the vibration displacement and von Mises equivalent stress in the concrete slab layer due to a third-order polygonal wheel are 0.0037 mm and 0.7603 MPa, respectively, whereas those resulting from a 21 st-order polygonal wheel are 0.0064 mm and 1.1501 MPa, respectively. By contrast, the dynamic responses of the CA layer and support layer caused by polygonal wheels are far more subdued than those of the concrete slab layer.