# Qualitative Analysis of Sleeper Supporting Condition for Railway Ballasted Tracks Using Modal Test
## 4 Analysis and Discussion  
### Analysis of Natural Frequency Characteristics by Type of Sleeper Support Condition  
This study analyzed the natural frequencies according to the left and right under-sleeper support conditions using the FRFs measured in the field via modal testing, as shown in Figure 14. The measurement results are shown in Figure 15.  
Under normal conditions, as demonstrated in Figure 15, the first natural frequencies on both sides of the frequency response function were similar at approximately 250 Hz, which aligns with the expected values under normal support conditions. This similarity indicates that the under-sleeper support on both sides was functioning properly.  
Figure 12: Bending mode shape analysis results for gravel track sleepers. (**a**) First mode with ballast spring stiffness = 50 kN/mm (167.2 Hz); (**b**) first mode with ballast spring stiffness = 40 kN/mm (160.12 Hz).  
Figure 13: Relationship between ballast spring stiffness and natural frequencies.  
Furthermore, this study investigated instances where track settlement occurred, owing to subgrade deformation from adjacent excavation activities, resulting in compromised support conditions, as shown in Figure 16. In the case of poor support conditions on the lower side of the concrete sleeper, such as floating sleepers or gravel loosening, these conditions can be represented by a gap element, as shown in Figure 16. Owing to the gap element, the stiffness of the left and right sleepers is not the same (\(k_{2,L}\
eq k_{2,R}\)). The ballast spring stiffnesses on the left and right may differ, which indicates that a certain level of sleeper displacement must occur--as in the case of floating sleepers or gravel loosening--for the ballast to perform its role as a spring. The measurement results for concrete sleepers under such conditions are shown in Figure 17.  
Figure 16: Concrete sleeper with poor support conditions on one side.  
Figure 14: Concrete sleeper left and right under support conditions (normal).  
Figure 15: Concrete sleeper measurement results.  
In the case of poor under-sleeper support on the right side, as shown in Figure 17, the first natural frequency on the right side of the FRF was smaller than that on the left.  
The measurement results for poor under-sleeper support on the left side, shown in Figure 18, indicate a lower first natural frequency on the left side of the FRF than that on the right, confirming that sleeper support conditions, qualitatively evaluated via visual inspection, can be quantitatively assessed by analyzing the differences in the first natural frequency through modal testing.  
When under-sleeper support conditions on both sides of the concrete sleeper were poor, with gravel loosening occurring on both sides, gap elements occurred on both the left and right lower sides of the sleeper, as shown in Figure 19. When gravel loosening occurs on a gravel-ballasted track, a certain level of sleeper displacement must occur for the track gravel to function as a spring.  
In cases where both the left and right under-sleeper supports were poor, as shown in Figure 20, the first natural frequencies on both sides of the FRF were similar. However, the first natural frequencies were significantly lower than those in normal support conditions, suggesting that both sides of the sleeper may be floating or the ballast may have loosened, indicating a reduced spring stiffness of the ballast.  
Figure 17: Measurement results for poor under-sleeper support conditions (right side) in concrete sleepers.  
Figure 18: Measurement results for poor under-sleeper support (left side) in concrete sleepers.