# Qualitative Analysis of Sleeper Supporting Condition for Railway Ballasted Tracks Using Modal Test
## 5 Conclusions  
This study proposed a novel modal testing technique for assessing ballast loosening and the presence of floating sleepers resulting from gravel track deformation. The key conclusions can be summarized as follows:  
1. A field survey was conducted on gravel tracks adjacent to excavation sites, selecting sections with both good and poor conditions. Modal testing was employed to measure frequency response functions in areas where changes in sleeper support conditions were suspected within the ballast track. The first natural frequency at locations with floating sleepers was observed to be over 50 Hz lower than that of sleepers in good condition. Similarly, areas with insufficient ballast compaction also showed lower frequencies than those in normal conditions.  
Figure 27: Spring stiffness analysis results before and during construction (down track). (**a**) Left side; (**b**) right side.  
2. The study found that sleeper support conditions on both the left and right sides, previously assessed qualitatively through visual inspections, can be quantitatively evaluated using modal testing by analyzing frequency differences. Additionally, it experimentally confirmed that disparities in support conditions beneath the left and right sides of a sleeper--where two rails are affixed to a single sleeper, as typically seen with concrete sleepers on gravel tracks--can be effectively analyzed using this modal testing technique. Consequently, the frequency response function results from modal testing are instrumental in evaluating variations in sleeper support conditions and assessing differences in spring stiffness at rail support points.
3. This study treated the spring stiffness of the ballast as a variable and confirmed the natural frequency of the track through finite element analysis, corresponding to changes in this parameter. The finite element analysis results showed a maximum deviation of approximately 0.91% compared to the field measurement values. This close agreement validates the use of the finite element model for simulating the track behavior under varying ballast support conditions. Therefore, the spring stiffness of the ballast and finite element modeling can be used to predict damage to gravel-ballasted track sleepers.
4. A predictive formula was developed based on the correlation analysis between ballast spring stiffness and natural frequency, enabling the calculation of ballast spring stiffness at measurement sites based on observed natural frequencies. The analysis of the damping ratio shows that, in most cases, the damping ratio decreased during construction compared to before construction. Therefore, the modal testing technique developed in this study offers a robust quantitative method for assessing the condition of gravel ballast beneath the left and right sides of concrete sleepers in gravel tracks.  
Conceptualization, J.-Y.C. and T.J.Y.; methodology, J.-Y.C.; software, T.J.Y. and S.-H.K.; formal analysis, T.J.Y.; investigation, J.-Y.C. and T.J.Y.; resources, T.J.Y.; data curation, S.-H.K.; writing--original draft preparation, T.J.Y. and S.-H.K.; writing--review and editing, S.-H.K. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.