## 5 Conclusions  
Many scholars, including the author, have been studying the dynamic characteristics of the transition section of high-speed railways for a long time, mainly focusing on the track-subgrade coupling system. The train action is normally simulated by moving load, measured acceleration and other methods, without considering its vibration function. As a result, this leads to inadequate research in vibration and interaction between the vehicle, rail and subgrade. To fully consider the vibration and interaction, a coupled train-track-subgrade system was established with train-track coupling theory and D'Alembert's principle. The proposed model was verified by in-site measurements and focused on the study of the influencing factors that dominate the dynamics of the coupled system. Specifically, the following conclusions were obtained.  
(1) It is confirmed that differential settlement and rail bending angle in the transition section are the main factors that cause dynamic irregularity in the transition section. There are some other indirect controlling factors such as rigidity ratio and length of the transition section, which may reduce the differential settlement and rail bending, leading to less dynamic responses of the coupled system.  
(2) Under the operation of a high-speed train with the speed of 350 km/h, the above four factors are recommended to meet the following range: the differential settlement shall not exceed 5 mm, the limit value of the rail bending angle of the bridge-embankment transition shall be less than 1/1000, the rigidity ratio shall be controlled within 1:6, and the length of the transition section shall be 25 m.  
(3) With regard to satisfying the conditions in (2), our research found that stress and displacement have the following distribution characteristics. On the one hand, in the longitudinal direction of the roadbed, the maximum dynamic stress and dynamic displacement of the transition section appear between the rigid bridge and the flexible roadbed. For example, the dynamic stress peaks at the roadbed surface 18 m away from the tail of the abutment, while the dynamic displacement peaks at 25-30 m away from the tail. On the other hand, in the depth direction of the subgrade, the bed layer resists the main dynamic response, so the dynamic stress and dynamic displacement attenuate sharply in the depth direction. Based on the above analysis, in order to reduce the dynamic stress and dynamic displacement response of the transition section, it is recommended to reinforce the subgrade at a distance of 18 m and 25-30 m longitudinally from the abutment, and to strengthen the bed layer in the depth of the subgrade.  
(4) This paper also preliminarily concludes that under the condition of power spectral density (PSD) of track irregularity, a flexible infrastructure is much safer than a rigid infrastructure. Nevertheless, under different working conditions, the changes of the train acceleration, wheel load reduction rate, track subgrade dynamic stress, dynamic displacement, and other indicators on the flexible and rigid subgrades need further study.  
Writing--original draft preparation, P.H.; writing--review and editing, C.Z. and W.G.; project administration, Y.W.; funding acquisition, P.H. and Y.W. All authors have read and agreed to the published version of the manuscript.  
Financial support from Hunan Provincial Natural Science Foundation (No. 2020 JJ5488) and National Natural Science Foundation of China (No. 50678177) are gratefully acknowledged.  
The authors declare that they have no conflict of interest.