# Parametric Study on the Effect of Rail Dampers on Track Decay Rate
## 3 Numerical Application  
This section studies how rail dampers influence the dynamic response of the rail and the track decay rate based on the results obtained with the mechanical model shown above. Table 1 contains the parameter values of the track and rail dampers model. It is about a ballasted track provided with a type 49 rail and concrete sleepers with the sleeper bay between 544 and 595 mm in accordance with the CFR regulations. Two values of the stiffness of the rail pads for soft and stiff rail pads are considered, respectively, as well as two values of the stiffness of the ballast for tampered (low stiffness) and settled ballast (high stiffness). Rail damper parameters have been obtained based on the experimental test performed using the impact hammer technique to determine the rail receptance of a setup consisting of a rail of finite length with rail damper and on the numerical results derived from the model of the setup [20]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Notation** & **Value** \\ \hline Rail mass per unit length & \(m\) & 49.4 kg/m \\ \hline Rail density & \(\rho\) & 7850 kg/m\({}^{3}\) \\ \hline Young’s modulus of elasticity & \(E\) & 210 GPa \\ \hline Shear modulus of elasticity & \(G\) & 81 GPa \\ \hline Rail loss factor & \(\eta\) & 0.01 \\ \hline Cross-section area & \(S\) & \(62.92\times 10^{-4}\) m\({}^{2}\) \\ \hline Area moment of inertia & \(I\) & 18.16 \(\times\) 10\({}^{-6}\) m\({}^{4}\) \\ \hline Shear coefficient & \(\kappa\) & 0.40 \\ \hline Sleeper linear mass (half) & \(M_{t}\) & 131 kg \\ \hline Rail pad stiffness & \(k_{p}\) & Soft & 60 MN/m \\ \cline{2-3}  & \multirow{2}{*}{\(k_{p}\)} & Stiff & 300 MN/m \\ \cline{2-3}  & & 0.35 \\ \hline Rail pad loss factor & \(\eta_{p}\) & 0.35 \\ \hline \multirow{2}{*}{Ballast stiffness} & \multirow{2}{*}{\(k_{b}\)} & Tampered & 40 MN/m \\ \cline{2-3}  & & Settled & 100 MN/m \\ \hline Ballast loss factor & \(\eta_{b}\) & 0.60 \\ \hline Sleeper bay & \(l_{k}-l_{k-1}\) & Short & 0.544 m \\ \cline{2-3}  & \multirow{2}{*}{Long} & 0.595 m \\ \hline First body mass & \(M_{1}\) & 2 \(\times\) 1.825 kg \\ \hline Second body mass & \(M_{2}\) & 2 \(\times\) 1.757 kg \\ \hline First elastic element stiffness & \(k_{1}\) & 50.62 MN/m \\ \hline First elastic element loss factor & \(\eta_{1}\) & 0.35 \\ \hline Second elastic element stiffness & \(k_{2}\) & 5.617 MN/m \\ \hline Second elastic element loss factor & \(\eta_{2}\) & 0.25 \\ \hline \end{tabular}
\end{table} 
Table 1: Track and rail dampers model parameters. First, the main characteristic of the rail response is analyzed using the receptance diagrams as depicted in Figure 2. There are four combinations between the rail pad stiffness and ballast stiffness: soft/stiff rail pad with tampered/settled ballast; the sleeper bay is 0.544 m. Rail receptance is calculated either halfway between the sleepers or above a sleeper on the interval between 10 Hz and 2500 Hz with the step of 1 Hz. The section of the harmonic force coincides with the calculation section in both situations. At low and medium frequencies, the response of the rail resembles that of an elastic system with two degrees of freedom. Two local maxima of rail receptance are observed between which there is a local minimum. The two maxima correspond to the low and high resonance frequencies of the rail and the sleepers vibrating together on the rail pads and on the ballast prism. The local minimum of the rail receptance shows an anti-resonance behavior due to the dynamic absorb effect that the sleepers play in relation to the rail. At high frequency, there is a peak in the rail receptance diagram calculated between sleepers due to the resonance of the pinned-pinned mode of the rail and a deep (anti-resonance) peak when the rail receptance is calculated above a sleeper. The low frequency of resonance is mainly influenced by ballast stiffness, and the high frequency of resonance is determined by the rail pad stiffness. The frequency of pinned-pinned mode is determined by the rail features and the sleeper bay.