# Idealisations of Dynamic Modelling for Railway Ballast in Flood Conditions
## 3 Results
### Modal Idealisation Concepts
#### 3.2.1.3 Discussion of Model-1  
The remarks and suitability of the dynamic attributes of ballast can be discussed as follows:  
* The Shift of Resonance Frequency  
By varying different masses, the frequency interval of the resonant frequency is shifted. Before reaching the full flooding condition (i.e., critical flooding level: 35 cm), the respective frequency  
Figure 14: Curve fitting.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Types of Test** & **Flooding** & **Damping** & **Stiffness** & **Frequency** & **Curve** & **Correlation** \\  & **Level** & **(kN-s/m)** & **(MN/m)** & **(Hz)** & **Fitting Error** & **Coefficient** \\ \hline  & 00 cm & 1.1626 & 14.6742 & 58 & 0.74\% & 99.26\% \\ Concrete Sleeper & 10 cm & 1.7753 & 13.7405 & 56 & 0.57\% & 99.43\% \\ on natural ballast & 20 cm & 1.8268 & 13.4096 & 56 & 0.45\% & 99.55\% \\ (23 \({}^{\circ}\)C-26 \({}^{\circ}\)C) & 30 cm & 2.3174 & 11.9497 & 52 & 1.56\% & 98.44\% \\  & 35 cm & 2.3701 & 11.329 & 51 & 0.17\% & 99.83\% \\  & 40 cm & 3.3087 & 6.5838 & 39 & 1.25\% & 98.75\% \\ \hline  & 00 cm & 1.3239 & 15.0783 & 129 & 6.96\% & 93.04\% \\ FFU Sleeper on & 10 cm & 1.3123 & 15.3719 & 130 & 6.15\% & 93.85\% \\ natural ballast (21 & 20 cm & 1.3859 & 14.8682 & 128 & 0.05\% & 99.95\% \\ \({}^{\circ}\)C–23 \({}^{\circ}\)C) & 30 cm & 1.1585 & 15.0044 & 129 & 2.67\% & 97.33\% \\  & 35 cm & 1.4073 & 15.0508 & 129 & 0.60\% & 99.40\% \\  & 40 cm & 6.3001 & 13.1991 & 125 & 0.44\% & 99.56\% \\ \hline  & 00 cm & 1.0815 & 14.2493 & 215 & 3.16\% & 96.84\% \\ Concrete block on & 10 cm & 1.156 & 13.51 & 210 & 5.00\% & 95.00\% \\ natural ballast (24 & 20 cm & 1.1975 & 12.5404 & 202 & 1.47\% & 98.53\% \\ \({}^{\circ}\)C–25 \({}^{\circ}\)C) & 30 cm & 1.3944 & 13.1293 & 207 & 0.93\% & 99.07\% \\ \({}^{\circ}\)C–25 \({}^{\circ}\)C) & 35 cm & 1.6208 & 11.2456 & 192 & 2.62\% & 97.38\% \\  & 40 cm & 1.2125 & 9.4513 & 176 & 5.13\% & 94.87\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: The dynamic properties’ changes with flooding level.  
segments are concentrated in the fixed change interval and the deviation is relatively small. After being completely submerged, a significant 'jump' of natural frequencies occurs, which is caused by the bottom surface of the mass coming into contact with flood water, changing the original resonant frequency band irregularly. The second factor is the influence of buoyancy (which will be analysed in a later section): the ballast is soaked in the water and is subjected to upward buoyancy, which induces a significant change in the vibration frequency. Simultaneously, the effective contribution of the dynamic mass will also change. The frequency changes can be illustrated in Figure 15.  
* The Change of Dynamic Stiffness  
Overall, the change of dynamic stiffness is considerable. With the increase in the flood water level, the stiffness will maintain continuous downward trend.  
In the dry condition, the stiffness values of the same ballast material are shown to be approximately constant, but are negligibly affected by water temperature (as shown in Table 3). The dynamic stiffness of natural ballast is generally stable at around 15 MN/m.  
After experiencing the flood conditions, the change in stiffness depends significantly on the mass level difference. It is clear that the FFU and concrete sleepers exhibit relatively stable dynamic frequencies [39]. Figure 16 illustrates that the FFU sleeper tends to yield slightly higher stiffness of ballast than the others. Note that the ballast specimen is identical. This clearly implies that the track mass plays a key role on the ballast stiffness when flooding conditions occur. It is also clear that, under dry conditions, the track mass does not play an influential role on ballast properties and the discrepancy of dynamic stiffness derived from various test setups is less than 3%.