## 4 Results and Discussion  
### The Influence of Rail Weld Geometry on ABA  
Time-course changes of ABA excited by rail welds with different wavelengths on a high-speed railway line (300 km/h) are presented in Figure 10a. The data has been processed by a 1500 Hz LPF and conforms to the measured data shown in Figure 4. Considering the typical size of the measured weld geometry [5] and the features of short wavelengths concerned in the paper, the wave depth was assumed as 0.2 mm and three wavelengths including 0.05 m, 0.1 m, and 0.2 m were selected. Taking as an example the working condition at a wavelength of 0.05 m, the acceleration of the axle box fluctuated within \(\pm\)20 m/s\({}^{2}\) as a result of elastic deformation between the wheel and rail. This value is less than the measured data (\(\pm\)50 m/s\({}^{2}\), see Figure 4) due to the geometry irregularities of the actual line not being considered. When passing the rail weld, it excites strong vibrations of ABA. The first wave crest ABA\({}_{\text{F}}\) was reached at 204.6 m/s\({}^{2}\) and was caused by the impact of the rail weld. The continuous fluctuation depends on coupled vibration of the vehicle-track system with a vibration wavelength of 72 mm. It can be observed that ABA\({}_{\text{V}}\) is significantly higher than the first wave crest excited by the rail weld. The phenomena can be found in the measured data of RW1 in Figure 5a.  
The ABA\({}_{\text{F}}\) decreased from 204.6 m/s\({}^{2}\) to 102.4 m/s\({}^{2}\) when the wavelength increased from 0.05 to 0.2 m. When the wavelength was 0.2 m, the resonance wavelength (238 mm) caused by the vehicle-track system was significantly greater than that of 0.05 m and 0.1 m. This is because rail welds with long wavelengths do not easily excite high-frequency vibrations in vehicle-track systems, as shown in Figure 10b. This figure shows the distribution of three signals in the frequency domain. There were two obvious dominant frequencies: 350 Hz and 1150 Hz marked as \(f_{1}\) and \(f_{2}\). The two frequencies corresponding to the wavelengths of 238 mm and 72 mm at a running speed of 300 km/h agree with the vibration wavelength shown in Figure 10a.  
The variations of ABA under more wavelengths are presented in Figure 11. Figure 11a indicates that ABA\({}_{\text{F}}\) decreased accordingly from 204.6 m/s\({}^{2}\) to 24.7 m/s\({}^{2}\) with increasing wavelengths from 0.05 m to 0.5 m. In Figure 11b, a corresponding amplitude of ABA at frequency\(f_{1}\) first increased and then decreased along with an increment of the wavelength, reaching a peak at 0.2 m. By contrast, the amplitude at\(f_{2}\) decreased continuously. When the wavelength reached 0.3 m, no dynamic response at\(f_{2}\) was excited. Note that\(f_{2}\) corresponding to vibration wavelengths of 72 mm and 65-80 mm are typical wavelengths of high-speed rail corrugations. Therefore, keeping the wavelength of a rail weld irregularity to no less than 0.3 m can help prevent the occurrence of short-wave rail corrugations, which might also explain why rail corrugations only appeared in certain sections of lines.  
Figure 10: The response of ABA under excitation of rail welds at different wavelengths in (**a**) time and (**b**) frequency domains.  
Wave depth is another important parameter of rail weld irregularities, and was also calculated here to know its influence on ABA; a typical wavelength was considered as 0.1 m. The frequency-domain responses of ABA under wave depths of 0.1 and 0.2 mm are presented in Figure 12a. It can be seen that the wave depth affects only the amplitude of ABA but with identical frequency under different wave depths. Thereby, \(\text{ABA}_{\text{F}}\) in Figure 12b grows almost linearly when the wave depth increased from 54.3 m/s\({}^{2}\) to 272.1 m/s\({}^{2}\).