# Parametric Study on the Effect of Rail Dampers on Track Decay Rate
## 3 Numerical Application
At low frequencies, below the low resonance, the rail receptance is little influenced by the frequency because the vibration behavior is dominated by the elastic force, which is frequency independent. At high frequency, beyond the pinned-pinned resonance, the rail receptance decreases because the inertia force is prevalent and the rail has the tendency to vibrate independently. This happens until the next rail bending mode occurs, but its resonance frequency is much higher, and it is not captured in diagrams. Table 2 presents the main frequencies mentioned above for each case. The soft/stiff rail pad determines the high resonance around 265/562 Hz, independent of the ballast stiffness. The tampered/settled ballast gives the low resonance around 79.5/124.5 Hz, whatever the rail pad stiffness. The anti-resonance frequency depends on the position along the rail. The pinned-pinned resonance occurs around 1135.5 Hz for any case considered. Figure 3 shows the rail receptance for the sleeper bay of 0.595 m, considering the same cases, and Table 3 encompasses the main frequencies. Compared to the preceding case (sleeper bay of 0.544 m), the main frequencies are a little lower (the decrease is smaller than 4.10%) because the stiffness of the rail support per unit length decreases. The pinned-pinned frequency lowers with 13.5% because the wavelength of the bending wave increases from 2 \(\times\) 0.544 m to 2 \(\times\) 0.595 m; at pinned-pinned frequency, the wavelength is twice as long as sleeper bay. Rail receptance is higher when the sleeper bay is 0.595 m because the equivalent stiffness per unit length of the rail pads and ballast is smaller. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Low** & \multirow{2}{*}{**Anti-Resonance**} & **High** & **Pinned–Pinned** \\  & **Resonance** & & **btw. Sleepers/abv. Sleeper [Hz]** & **Resonance** & **Resonance** \\  & **[Hz]** & & **[Hz]** & **[Hz]** \\ \hline Soft rail pad–tampered ballast & 78 & 135/139 & 252 & 982 \\ \hline Soft rail pad–settled ballast & 121 & 173/176 & 258 & 982 \\ \hline Stiff rail pad–tampered ballast & 79 & 216/256 & 538 & 982 \\ \hline Stiff rail pad–settled ballast & 125 & 241/278 & 540 & 982 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Main frequencies characterizing the rail response for sleeper bay of 0.595 m.  
Figure 3: Rail receptance for the sleeper bay of 0.595 m: (**a**) \(k_{p}\) = 60 MN/m, \(k_{b}\) = 40 MN/m; (**b**) \(k_{p}\) = 60 MN/m, \(k_{b}\) = 100 MN/m; (**c**) \(k_{p}\) = 150 MN/m, \(k_{b}\) = 40 MN/m; and (**d**) \(k_{p}\) = 150 MN/m, \(k_{b}\) = 100 MN/m;  —, between sleepers; —, above sleeper. Second, the bending waves in rail are analyzed. Equation (9) shows that two kinds of bending waves propagate along the infinite rail without supports: a propagative wave with the wavenumber \(\beta_{2}\) and an evanescent wave with attenuation constant \(\beta_{1}\); both waves can be progressive and regressive. Figure 4 presents the propagate constants for the two kinds of waves. Both propagate constants increase monotonically within the frequency interval considered (0-2500 Hz), which means that the wavelength of the propagative wave lowers, and the attenuation constant of the evanescent wave increases continually as the frequency increases. The two wavenumber values displayed on the diagram are associated with the pinned-pinned frequencies; each wavelength equals the corresponding double sleeper bay. Figure 5 illustrates the bending waves through the rail without support at the frequency of 1000 Hz represented at the time moments \(t=0\), \(t=T/12\), \(t=2T/12\), \(t=3T/12\), \(t=4T/12\), \(t=5T/12\), and \(t=6T/12\), where \(T\) is the period (\(T=\omega/(2\pi)\)). In a propagative wave (Figure 5a), all points move with the same amplitude due to the lack of damping but with a delay as the wave reaches each point. In an evanescent wave (Figure 5b), all points move in phase, but the amplitude is exponentially decreasing.