## 5 Conclusions  
In this study, the vibration characteristics of ballasted tracks with and without a ballast mat are analysed using numerical simulations and field experiments with wheelset drop tests. The accelerations of the rail, sleeper and ballast in both the time and frequency domains prove that the installation of the ballast mat can enhance the resilience and elasticity of the ballasted track. The ballast mat can effectively reduce vibration at the mid- and high-frequency range of 40-3000 Hz with a maximum vibration reduction of 15-17 dB for different track components. In the low-frequency range, the ballast mat slightly amplifies the system vibrations by 5-15 dB for different track components. This is noteworthy because human activities are easily disturbed by vibrations in such low-frequency ranges and also because low-frequency vibrations can propagate for a long distance in soils and tunnels with little attenuation.  
The wheelset drop test performed in this study indicates that it is an effective method to obtain the dynamic characteristics of track structures. The accelerations of the rail and sleeper are obtained through the test and the insertion losses of the rail and sleeper are obtained through the one-third octave analysis. The numerical insertion losses are compared with the experimental results, and a good agreement is found in the frequency range of 0 to 600 Hz. A slight difference exists between the predicted insertion losses and the measured data. The possible reason for the difference between the field test and numerical calculation results can be attributed to the intrinsic nature of the discreteness and randomness of the ballasted track bed [31; 32]. Essentially, the ballasted track bed is consisted of granular particles of different sizes and is subjected to moving vehicle loads and maintenance loads such as tamping [33]. Thus, the system parameters cannot be easily taken as a uniform determined value and need further calibration by trial-and-error tests. The methodology proposed in this paper assumes that the ballasted track bed is regarded as the lumped-mass system [34] and certainly behaves slightly different from the discrete dynamical systems. However, the comparison results showed that even with this slight difference, the model established in this paper could grasp the intrinsic dynamic properties of the ballasted track system and showed good agreement with the test results. Future possible solutions can be achieved by carrying out simulations based on the discrete element method [35; 36].  
The spatial coupled wheel-track model defined in this study can simulate the wheelset drop test phenomenon, which opens perspectives for future developments. In particular, the track model can be extended to other aspects such as vehicle-track coupling dynamics.  
Figure 21: One–third octave analysis of rail and sleeper insertion loss.  
**Author Contributions:** B.H.: writing--original draft, writing--review and editing, methodology and validation. D.W.: writing--original draft, software. B.W.: writing--review and editing. X.C.: investigation. J.P.: writing--review and editing. All authors have read and agreed to the published version of the manuscript.  
**Funding:** The paper is supported by open project of the national key laboratory of the China Academy of Railway Sciences Co., Ltd. (2021 YJ051), the China Railway Corporation Science and Technology Research and Development Program (N2018G071) and the National Natural Science Foundation of China (51708021).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data used to support the findings of this study are available from the corresponding author upon request.  
**Acknowledgments:** The contribution of [PERSON] to this work was supported by FCT, through IDMEC, under LAETA, project UIDB/50022/2020.  
**Conflicts of Interest:** The authors declare no conflict of interest.