## 4 Wear and Corrugation Simulation
### Analysis of Longitudinal and Vertical Rail Modes  
Figure 9 shows the power spectra of the calculated wear under different corrugation amplitudes. When A = 20 \(\upmu\)m, a major frequency component is observed at approximately 1296 Hz, approaching the corrugation passing frequency of 1297 Hz (estimated with a wheel passing at 38.9 m/s over the corrugation with a wavelength of 30 mm). When A is reduced, this major frequency also decreases. At A = 2.5 \(\upmu\)m, the frequency is 1290 Hz, which is 7 Hz lower. Thus, the frequency of the wear is initially (i.e., when \(A\) is small) lower than that of the modeled corrugation, and it is influenced by the corrugation. When A is increased to 20 \(\upmu\)m, there is a sudden PSD increase at 1185 Hz; see the second major peak in Figure 9. Modal analysis in Figure 10a,b shows that these two major frequencies of 1185 and 1296 Hz correspond to two natural frequencies of the system (1185 Hz of a vertical rail mode and 1291 Hz of a longitudinal rail mode), i.e., the two natural frequencies that are closest to and/or lower than the corrugation passing frequency of 1297 Hz (see in Figure 10 the characteristic modal shapes). The closest pin-pin resonance is at approximately 1110 Hz (Figure 10c), and it thus does not seem to have influence on the wear development in this case. Based on Figure 9, it is reasonable to assume that the development of the observed corrugation with a wavelength of 30 mm is mainly related to these two modes. Then, the track parameters are a stronger determinant of the corresponding wavelength of the calculated wear than is the modeled corrugation because the track parameters determine the longitudinal and vertical modes. Figure 9: Power spectral density (PSD) of wear \(\lambda\) = 30 mm, with a major peak at 1296 Hz and a secondary peak at 1185 Hz, when A = 20 \(\upmu\)m.  
When the corrugation amplitude is close to zero, e.g., A = 2.5 \(\upmu\)m or less, the major frequency of the wear is approximately 1290 Hz, which is 6 Hz lower than the 1296 Hz of the wear at A = 20 \(\upmu\)m, 1 Hz lower than the longitudinal modes, but 105 Hz higher than the vertical mode. This suggests that for the initiation of corrugation, the longitudinal mode should have played a major role compared with the vertical mode. Additionally, the periodic wear under the smooth rail condition has one peak at 1290 Hz. When the amplitude of corrugation is progressively increased, the differential wear at this frequency grows consistently. This phenomenon does not happen at the other frequencies. This implies a positive-feedback process taking place at around this frequency. Besides, in Figure 9, the series of peaks from A = 20 \(\upmu\)m and further to the smooth rail converge asymptotically to the smooth rail situation of A = 0 \(\upmu\)m. Based on the positive feedback and trend analysis, it is reasonable to infer that the longitudinal vibration modes are probably dominant for short pitch corrugation initiation. In the corrugation initiation, the longitudinal modes in the form of compression and rarefaction waves influence the local wheel-rail contact of forces and creepage and result in their fluctuation and thus the formation of the differential wear. With the growth of corrugation, the vertical mode plays an increasingly important role. This behavior is confirmed by Figure 9: when A is larger than 10 \(\upmu\)m, the PSD peak of the secondary frequency at 1185 Hz begins to make a significant contribution to the wear. This frequency corresponds to the vertical mode and is excited when the corrugation amplitude is sufficiently large. When the corrugation amplitude is further increased, it will incur wear that is of the same frequency as the corrugation. Therefore, when the corrugation amplitude is significant, the vertical modes will be dominant. For intermediate situations, the longitudinal and vertical modes together determine whether corrugation will grow or be suppressed by wear based on whether the wear is of the same frequency and correct phase.