## 2 Observation of Time-Frequency Vibration of the Measured Axle Box Acceleration  
High-frequency contact vibration generated on the wheel/rail surface under excitation of all kinds of irregularities can be easily transferred to the axle box due to large contact stiffness between wheel and rail. A trial measurement at a speed of 300 km/h on a certain Chinese high-speed railway was conducted by the State Key Laboratory of Traction Power of Southwest Jiaotong University. The track is non-ballast track containing rail, fastening, slab, and mortar and the car was CRH380. The ABA measurement (four accelerometer sensors) was installed on the axle box of a bogie (see Figure 3). The data acquisition unit was connected with the sensor mounted on the tested components of the vehicle. The sampling frequency of the axle box acceleration sensor was set to 16.4 kHz. Figure 2: Installation of the acceleration sensor on the axle box. Figure 4 shows the measured ABA changing along with the running time. Note that a 1500 Hz LPF (low pass filter) has been imposed on the signals with consideration of typical frequency range of wheel/rail interaction and noises containing high-frequency signals. The ABA amplitude fluctuated within the range of \(\pm\)50 m/s\({}^{2}\) under 300 km/h on account of geometry irregularities on the surface and elastic deformation of the wheel/rail contact interface. Rails of non-ballast tracks are welded every 100 m. Short-wave irregularities occurred easily in the welds as mentioned in the introduction, and a pulse impulsion occurs every 1.2 s (it is about 100 m as multiplied by the speed of 300 km/h) in Figure 4, whose amplitude reaches 150 to 200 m/s\({}^{2}\), as shown in RW1. By contrast, the amplitude at RW2 is rather gentle, possibly because there were no serious irregularities. To obtain a more detailed analysis and display the vibration of the ABA signal in the time-frequency domain (i.e., the changes of frequency components of acceleration at different times during the process of a vehicle passing the weld), the S transform method was used. This method was selected because it inherits and develops localized ideas of continuous wavelet transform and short-time Fourier transform, and adopts a time window of Gaussian function that presents a good resolution in the time and frequency domains [24]. The S transform can be described by Equations (1) and (2):  
\[S(\tau,f)=\int_{-\infty}^{\infty}x(t)\omega(\tau-t,f)\exp(-\mathrm{j}2\pi ft) \mathrm{d}t \tag{1}\]  
\[\omega(\tau-t,f)=\frac{|f|}{\sqrt{2\pi}}\exp(-\frac{f^{2}(\tau-\mathrm{t})^{2 }}{2}) \tag{2}\]  
Figure 4: Time-course response of measured ABA under 300 km/h. RW1 and RW2 are two examples of rail welds. Figure 3: A typical bogie and axle box acceleration (ABA) measurement on an axle box. where \(f\) is frequency; \(\omega(\tau-t,f)\) is the Gaussian window function; \(\tau\) is a parameter controlling the Gaussian window in a position of time \(t\). Time-frequency vibration characteristics of the ABA at rail welds are presented in Figure 5 by taking RW1 in Figure 4 as an example. Figure 5a shows the time-domain variation of the ABA between 11.1 and 11.2 s. As we can see, an obvious impact was generated with a high-frequency vibration when passing RW1. The time-frequency dynamic response through the S transform is presented in Figure 5b; the contour has been normalized \(S(\tau,f)\) and the mean vibration energy changed from small to large in blue and red. By this figure, we know that the vibration of ABA at a speed of 300 km/h covers a wide frequency domain. Energy excited by weld geometry irregularity around 11.14 s gathers mainly in two frequency bands: from 350 to 500 Hz and from 1000 to 1200 Hz. High-frequency vibration lasting about 2 to 5 ms is dissipated by the wheel/rail contact interface, which is a major cause of fatigue generation for all kinds of rolling contacts.