## 4 Results and Discussions
### Time-Frequency Analysis of Axle-Box Accelerations based on EEMD and WVD  
In this section are presented some results of the calculated axle-box acceleration analyzed by using the EEMD-WVD combined method described in Section 3.2, and comparison with the results calculated from WVD time-frequency representation. Figure 7 shows the time domain acceleration of the axle-box induced by a 24 th-order wheel polygon with OOR amplitude of 0.02 mmand a train speed of 300 km\(\cdot\)h\({}^{-1}\). Figure 8 shows the time-frequency spectrograms of the acceleration. The result of the WVD time-frequency analysis in Figure 8a reveals a clear horizontal band at about 690 Hz, which corresponds to the theoretical characteristic frequency of 691.9 Hz caused by the 24 th-order polygonized wheel, as per the wavelength-fixing mechanism \(f=v/\lambda=nv/2\pi R=24\times(300/3600)/(2\pi\times 460)=691.9\)Hz. However, unwarranted characteristic information also appears at other frequencies, which is due to the WVD crossterm and leads to an incorrect estimation. Figure 8b shows the time-frequency analytic spectrogram produced by the EEMD-WVD combined method. The most prominent characteristic frequency is again at 690 Hz, which represents information about abnormal vibration caused by the 24 th-order wheel polygon. Furthermore, no other noticeable characteristic frequencies appear, unlike in Figure 8a. These analytic results show that the EEMD-WVD combined method suppresses the crossterm of the WVD but does not reduce the time-frequency resolution.  
Figure 6: Axle-box acceleration as a function of harmonic order of polygonized wheel (i.e., the number of wavelengths around the wheel) for several train speeds.  
Figure 7: Time-domain response of vertical axle-box acceleration due to 24 th-order wheel polygon with OOR amplitude of 0.02 mm at 300 km\(\cdot\)h\({}^{-1}\).  
Figure 9 shows partial EEMD-WVD time-frequency-energy spectrograms of calculated axle-box acceleration caused by a 24 th-order wheel polygon with OOR amplitudes varying from 0.01 to 0.12 mm. Note that the largest-power amplitudes in the spectrograms appear around 690 Hz, produced by the 24 th-order wheel polygons. The frequency band at zero amplitude is narrow (690-700 Hz), which reflects a high frequency resolution. In addition, the peak amplitude (i.e., peak acceleration) of the simulated axle-box increases with increasing OOR amplitude (see Figure 10). These results imply that the presence of wheel polygons is clearly reflected in the spectrogram as a narrow band whose frequency depends on the harmonic order of the polygonized wheels and on the train speed. Furthermore, the peak amplitudes of the EEMD-WVD time-frequency-energy spectrogram are approximately linear in the OOR amplitude of the polygonized wheels. Thus, given the power amplitude of the axle-box acceleration, the train speed, and the harmonic order, the OOR amplitude of a polygonized wheel can be determined.  
Figure 8: Spectrogram corresponding to the signal shown in Figure 7: (**a**) WVD time-frequency analytic spectrogram; (**b**) EEMD-WVD time-frequency analytic spectrogram.  
Figure 9: EEMD-WVD time-frequency-energy spectrograms corresponding to the axle-box acceleration caused by 24 th-order polygonized wheels and for different OOR amplitudes: (**a**) OOR amplitude of0.04 mm; (**b**) OOR amplitude of0.06 mm; (**c**) OOR amplitude of0.08 mm; (**d**) OOR amplitude of0.10 mm; (**e**) OOR amplitude of0.12 mm.