# Idealisations of Dynamic Modelling for Railway Ballast in Flood Conditions
## 2 Materials and Methods
### Methods
#### 2.2.3 Model Built Using Fourier Transformation  
Dynamic characteristics of ballast represented by an elastic spring and dashpot component in the vertical direction can be obtained from the motion and frequency relationship of the object using the Fourier transformation formula. Fourier transformation is the frequency domain representation of the vibration signal. The simplest form of dynamic characteristics in the vertical direction, which  
Figure 8: The flooding ballast at different levels.  
Figure 7: Capture data using the instrumented hammer.  
is commonly used in coupled train-track interaction models, can be described by the well-known equation of motion [38].  
\[M\ddot{X}+C\dot{X}+KX=F(t) \tag{1}\]  
The displacement function \(X(t)\) can be written as:  
\[X=Ae^{-i\omega t} \tag{2}\]  
\[\dot{X}=-i\omega Ae^{-i\omega t} \tag{3}\]  
\[\ddot{X}=-\omega^{2}Ae^{-i\omega t} \tag{4}\]  
\[Ae^{-i\omega t}(-\omega^{2}M-iC\omega+K)=F(t) \tag{5}\]  
According to the well-known Euler formula in complex analysis (illustrated in Figure 9), we can obtain:  
\[e^{i\varphi}=\cos\varphi+i\sin\varphi \tag{6}\]  
Note that:  
\[A(cos\varphi-isin\varphi)(-\omega^{2}M-iC\omega+K)=F(t) \tag{7}\]  
\[\frac{A}{F}=\frac{1}{(K-\omega^{2}M)\cos\varphi-C\omega\sin\varphi-i(K-\omega ^{2}M)\sin\varphi-iC\omega\cos\varphi)} \tag{8}\]  
Then,  
\[tan\varphi=-\frac{C\omega}{K-\omega^{2}M}=\frac{-2\zeta\omega/\omega_{n}}{1- \omega^{2}/\omega_{n}{}^{2}} \tag{9}\]  
\[\omega_{n}=\sqrt{\frac{k}{m}} \tag{10}\]  
\[\zeta=\frac{c}{2\sqrt{km}} \tag{11}\]  
\[\omega=2\pi f \tag{12}\]  
where \(M\), \(C\), and \(K\) generally represent the effective mass, damping, and stiffness of railway ballast respectively \(\omega_{n}\), \(\omega\) represent the natural frequency and radial vibration frequency, and \(\zeta\) is the damping ratio.  
Figure 9: [PERSON]’s formula.  
Therefore, the magnitude of FRF is given by:  
\[H(\omega)=\frac{A}{F}=\frac{1}{M}\times\frac{1}{\sqrt{\left(2\zeta\omega/\omega_{ n}\right)^{2}+\left(1-\omega^{2}/\omega_{n}{}^{2}\right)^{2}}} \tag{13}\]  
On this basis, [PERSON] and [PERSON] further propose an independent model, which adds a frequency-dependent spring to the original basic model equivalent to the action system of two springs [39].  
The dynamic expression of the dynamic characteristics in the vertical direction as shown in Figure 10 can be reformulated as:  
\[M\ddot{\mathrm{X}}+\frac{1}{\frac{1}{\mathcal{L}\omega}+\frac{1}{\mathcal{K} _{1}}}X+K_{2}X=F(t) \tag{14}\]  
\[H(\omega)=\frac{A}{F}=\frac{1}{M}\times\frac{1}{\sqrt{\left(\frac{\frac{1}{ \mathcal{L}\omega}+\frac{M}{\mathcal{K}_{1}}}\right)^{2}+\left(\frac{K_{2}}{M }-\omega^{2}\right)^{2}}} \tag{15}\]  
where,  
\(K_{1}=\) frequency-dependent stiffness (N/m)  
\(K_{2}=\) frequency-independent stiffness (N/m)  
\(C_{1}=\) frequency-dependent viscous damping (Ns/m)  
\(\alpha=\frac{\omega^{2}}{z^{2}+\omega^{2}}\) (16)  
\(\beta=\) damping fractional coefficient  
\[\beta=\frac{z^{2}}{\omega^{2}} \tag{17}\]  
where \(\omega\) is the radial frequency (s \(-\) 1), \(\omega=2\pi f\); and \(z\) is the partial inverse loss value (s \(-\) 1), \(z=K_{1}/C_{1}\)[39].  
According to these coefficients, the integration formula is shown in Table 2.  
\begin{table}
\begin{tabular}{c c c} \hline
**Property** & **Kelvin Model** & **Poyning-Thomson Model** \\ \hline K\({}_{\mathrm{t}}\) & K & \(=\alpha K_{1}+K_{2}\) \\ C\({}_{\mathrm{t}}\) & C & \(=\alpha\beta C_{1}\) \\ \hline \end{tabular}
\end{table}
Table 2: The Kelvin and Poynting-Thomson mechanical model definition.  
Figure 10: The state dependent model.