# Idealisations of Dynamic Modelling for Railway Ballast in Flood Conditions
## 3 Results
### Model-3 'Kc-K'
#### 3.4.2 Test Results in Model-3 and Justification  
The modal parameters based on Model-3 are shown in Table 5:  
An example of the best curve fitting can be illustrated in Figure 20.  
* The results of all four experiments show that the curve fitting error based on Model-1 is the least compared with those of other models, and the errors are especially higher in smaller-scale experiments.
* Based on Model-3, there is no significant change in frequency-independent stiffness. It can be seen from Figure 21a that the dynamic stiffness traits of the two models are very consistent.
* Figure 21b shows that the variation of the damping in Model-3 displays the same trend as Model-1. This model does not significantly change the original damping, but there is an almost constant gap.
* The frequency-dependent stiffness K\({}_{1}\) presents a similar variation of damping as shown in Figure 22. The trend is basically consistent with the trend of damping. The combination of spring with the dashpot is a subtractive relationship, so the combination with frequency-independent stiffness will reduce the total stiffness value (K\({}_{\text{t}}\)) of the system.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Types**} & **Flooding** & **Damping** & **Stiffness** & **Stiffness** & **Frequency** & **R\({}^{2}\)** & **C\({}_{\text{t}}\)** & **K\({}_{\text{t}}\)** \\  & **Level** & **C\({}_{\text{t}}\)****(LN-s/m)** & **K\({}_{\text{t}}\)****(MN/m)** & **K\({}_{\text{t}}\)****(MN/m)** & **(Hz)** & & **R\({}^{2}\)** & **C\({}_{\text{t}}\)** & **K\({}_{\text{t}}\)** \\ \hline  & 00 cm & 1.1626 & / & 14.6742 & 58 & 99.259\% & 1.16 & 14.67 \\ Concrete sleeper & 10 cm & 1.5842 & 5.1906 & 13.7423 & 56 & 99.600\% & 1.76 & 13.80 \\ on natural ballast & 20 cm & 1.6329 & 5.3604 & 13.4114 & 56 & 99.695\% & 1.81 & 13.47 \\ (23 °C–26 °C) & 30 cm & 2.0663 & 6.1864 & 11.9530 & 52 & 98.981\% & 2.29 & 12.03 \\  & 35 cm & 2.1142 & 6.3562 & 11.3321 & 51 & 99.884\% & 2.34 & 11.40 \\  & 40 cm & 3.0594 & 9.1594 & 6.5888 & 39 & 99.307\% & 3.29 & 6.65 \\ \hline  & 00 cm & 1.3239 & / & 15.0783 & 129 & 93.040\% & 1.32 & 15.08 \\ FFU sleeper on & 10 cm & 0.9748 & 3.0967 & 15.3856 & 130 & 95.694\% & 1.23 & 15.58 \\ natural ballast & 20 cm & 1.0355 & 3.2988 & 14.8826 & 128 & 99.886\% & 1.30 & 15.08 \\ (21 °C–23 °C) & 30 cm & 0.8643 & 2.7483 & 15.0147 & 129 & 98.099\% & 1.09 & 15.18 \\
35 cm & 1.0489 & 3.3429 & 15.0655 & 129 & 99.396\% & 1.32 & 15.27 \\  & 40 cm & 4.8084 & 16.0920 & 13.5564 & 125 & 99.680\% & 5.97 & 14.40 \\ \hline  & 00 cm & 1.0815 & / & 14.2493 & 215 & 96.840\% & 1.08 & 14.25 \\ concrete block on & 10 cm & 0.6797 & 2.1809 & 13.5743 & 210 & 93.230\% & 0.99 & 13.89 \\ natural ballast & 20 cm & 0.7222 & 2.3210 & 12.6049 & 202 & 94.220\% & 1.04 & 12.92 \\ (24 °C–25 °C) & 30 cm & 0.8292 & 2.6763 & 13.2210 & 207 & 93.275\% & 1.20 & 13.59 \\  & 35 cm & 1.0136 & 3.2896 & 11.3589 & 192 & 92.866\% & 1.42 & 11.76 \\  & 40 cm & 0.7944 & 2.5542 & 9.5055 & 176 & 93.639\% & 1.08 & 9.78 \\ \hline \hline \end{tabular}
\end{table}
Table 5: The dynamic properties of ballast in Model-3.  
Figure 20: The best curves fitting in Model-3 (Y axis: FRF; X axis: frequency).