## 5 Conclusions  
This paper proposes the use of axle-box acceleration to detect railway-wheel polygons of high-speed trains. A dynamics model of a vehicle-track rigid-flexible coupled system was designed to capture the dynamic features of axle-box acceleration associated with polygonized wheels. The time-frequency representation of axle-box accelerations obtained by using the EEMD-WVD combined method clearly reveals the presence of polygonized wheels. The main conclusions are as follows:  
1. The quantitative relationship between the characteristics of axle-box acceleration and wheel polygon defects is determined based on a parameter-lumped simple wheel-track model that accounts for wheel out-of-roundness. The frequency of the axle-box acceleration coincides with the characteristic frequency of polygonized wheels, which underpins the detection of wheel polygon defects based on axle-box vibration.
2. A dynamic model of a vehicle-track coupled system with flexible wheelsets and tracks is presented to determine the relationship between wheel polygon defects and axle-box acceleration.  
Figure 11: Results of EEMD-WVD analysis of axle-box acceleration caused by real polygonized wheels [27]: (**a**) time-frequency spectrogram; (**b**) time-frequency-energy spectrogram.  
The results indicate that the maximum amplitude of the vertical axle-box acceleration is linear in both the harmonic order and in OOR amplitude of the polygonal wheel. Furthermore, the growth rate of acceleration increases monotonically with vehicle speed.
3. The EEMD-WVD combined method is used to make a time-frequency analysis of the axle-box acceleration. The frequency and corresponding amplitude of axle-box acceleration are quantitatively related to wheel polygon defects. It's concluded that EEMD-WVD can be applied for wheel polygons detection and severity assessment.
4. The analysis shows that axle-box acceleration measurements can be used to detect and assess polygon wheels. When the characteristic frequency \(f_{1}\) of axle-box acceleration and the maxima magnitude \(a_{\text{axlebox}}\) at a given train speed \(v_{1}\) are determined, the wheel polygonal defects can be detected, provided \(f_{1}\) does not coincide with the axle-box vibration frequency \(f_{2}\) excited by round wheels. If \(f_{1}\#f_{2}\), the harmonic order \(n\) of the polygon wheel can be determined by comparing \(f_{1}\) with the characteristic frequencies corresponding to the first 25 harmonics of a polygon wheel at the train speed \(v_{1}\).Based on the peak amplitude and the frequency of axle-box acceleration from the joint time-frequency analysis, the degradation due to wheel polygon defects can be evaluated.  
Conceptualization, Y.S.; Formal analysis, L.L.; Writing--review & editing, Y.D. and B.S.; funding acquisition, Y.S. All authors have read and agreed to the published version of the manuscript. This research was funded by National Natural Science Foundation of China, grant number11372199, Natural Science Foundation of Hebei Province, grant number E2019210152, and China Postdoctoral Science Foundation Funded Project, grant number 2018M643521.  
We acknowledge TopEdit LLC for the linguistic editing and proofreading during the preparation of this manuscript.  
The authors declare no conflict of interest.