## 3 Case Study  
So far, some types of train have installed on-board real-time monitoring system, based on which online vibration signals of wheel treads can be monitored. A complete set of on-board real-time monitoring system is shown in Figure 3. We could find that it includes shock vibration sensor, data pre-processor, diagnostic instrument, display terminal, etc. When the train is running, the system can output original vibration data to data analysis center via the wireless local area network (WLAN). To demonstrate the applicability of the proposed RUL framework, vibration signals of wheel treads of a high-speed train collected from the system were adopted as the data set. Figure 4 shows a schematic for the wheel tread with sensor. The marked point 11 and point 16 are installation positions of the sensors used for monitoring the wheel treads. Figure 5 presents the original vibration signals of two wheel treads. It is easy to find that the monitoring system recorded the running mileage of wheel treads and corresponding amplitude (dB) at each monitoring point. The reference value for measuring dB values was \(10^{-5}\) mm/s. Note that the recorded amplitude reflected the normalized strength of vibration shocks by eliminating the effects of running speed and axle diameter of the train. The recorded running mileage started from 402,590 km and ended at 447,870 km and 450,000 km for the  
Figure 3: Illustration of the structure of the on-board real-time monitoring system. first and the second wheel tread, respectively. In addition, because of human factors, the amplitudes at some monitoring points were missing and this could be reflected by the blank intervals in Figure 5.  
To understand the failure processes of two wheel treads more clearly, failure mode analysis were carried out based on physical technologies. Results show that the degradation of the first wheel tread was caused by out-of-round and the second one was caused by stripping. Engineering experience shows that out-of-round and stripping are two most common failure modes of the wheel tread of train. For an illustrative purpose, an illustration of the two failure modes was provided, as shown in Figure 6a,b respectively. In particular, out-of-round is always caused by uneven wear of wheel treads. During the running state, because of various factors such as possible scratches in traction and braking, foreign body bruising and so on, the wheel rolling circle loses its roundness and presents a polygon phenomenon. Vibration and noise caused by wheel polygon not only affect the comfort of the ride but also increase the vertical contact force of the wheel and rail, resulting in accelerated damage of the track and the components of train and shortening the service life. Different from this, stripping is always caused by fatigue stress, which is inevitable in all of rotary machinery systems. Tiny defects in the material will be continuously amplified under the effect of fatigue stress until part of the material is stripped from the entire wheel tread. Around the occurrence of stripping, degradation velocity of tread varies significantly because the balance of rotary system has been broken. This phenomenon will be verified in the following. Note that when the degradation caused by either out-of-round or stripping reaches a certain extent, appropriate and seasonable maintenance measures must be taken. Otherwise, the train may happen catastrophic failure during its high-speed running state. For the wheel treads provided in our case, turning maintenance was carried out at the ending  
Figure 4: A schematic for the wheel tread with a sensor in our monitoring program. Figure 5: Original vibration signals of two wheel treads collected via on-board real-time monitoring system. (**a**) Original vibration signals of the 1-st wheel tread and (**b**) original vibration signals of the 2-nd wheel tread. points of their monitoring processes, which means that the wheel treads have been considered to be failed in practical applications.