# Parametric Study on the Effect of Rail Dampers on Track Decay Rate
## 3 Numerical Application
Higher ballast stiffness limits the sleepers' motion and then the propagation of the rail's bending waves. All diagrams in Figure 9 show that TDR is lower when the harmonic force acts above sleeper. This can be explained by the fact that the distance between the section of the harmonic force and the first sleeper is twice as high and the bending waves travel 'freely' more time until the first point of reflection. Higher coupling between the rail and sleepers due to stiff rail pad increases the difference between the two cases (Figure 9c,d). Figure 10 presents TDR calculated in the same conditions as above, excepting the sleeper bay that is 0.595 m in this case. TDR diagrams look like the ones above, showing the same main characteristics. The highest TDR peak is located within the second range of the standing wave: within the one-third octave of 160 Hz for the track with soft rail pads-tampered ballast, within the one-third octave of 200 Hz for the track with soft rail pads-settled ballast, and within the one of 315 Hz for the track with stiff rail pads and for any condition of the ballast. The second TDR peak is signaled within the one-third octave of 1 kHz that corresponds to the pinned-pinned resonance from 982 Hz. Also, TDR decreases within the frequency ranges where the bending waves become propagative waves. However, the comparison between Figures 9 and 10 shows that the larger sleeper bay lowers TDR because the bending waves travel 'freely' a longer distance from one reflection point to another. Finally, the influence of the rail dampers is analyzed. Figure 11 displays the transfer functions of the rail damper (moduli and phases with/without damping) calculated as a ratio between the displacement of the masses 1 and 2, respectively, and the rail displacement, considered the following harmonic function:  
\[ \begin{array}{l}\overline{H}_{1}(\omega)=\frac{\overline{z}_{1}(\omega)}{ \overline{w}(k_{p},\omega)}=\frac{\overline{k}_{1}\big{(}-\omega^{2}M_{2}+ \overline{k}_{2}\big{)}}{M_{1}M_{2}\omega^{4}-\big{[}\big{(}\overline{k}_{1} +\overline{k}_{2}\big{)}M_{2}+\overline{k}_{2}M_{1}\big{]}\omega^{2}+ \overline{k}_{1}\overline{k}_{2}},\\ \overline{k}_{1}\overline{k}_{2}\overline{k}_{1}\overline{k}_{2}\overline{k}_ {1}\overline{k}_{2}\\ \end{array}  \tag{21}\]Rail damper has two resonance frequencies, at 190 Hz and 628 Hz, and mass 1 exhibits anti-resonance behavior at 201 Hz. When the rail frequency is around the first resonance of the rail damper, the vibration of mass 1 is lower than the mass 2 vibration, which is out of phase with the rail displacement. Consequently, the second elastic element of the rail damper dissipates more energy. When the rail frequency is around the second resonance and beyond this, the vibration of the mass 1 becomes higher, and it is out of phase with the rail displacement and the mass 2 of the rail damper. This time, the first elastic element dissipates more energy than the second one. Figure 12 shows the comparison between the track without rail dampers and the track with rail dampers in terms of the rail receptance calculated between sleepers (Figure 12a,b) and above sleepers (Figure 12c,d), considering the sleeper bay of 0.544 m and all four cases regarding the stiffness of the rail pad and ballast. Rail damper modifies the rail receptance and becomes effective beyond the anti-resonance frequency for soft rail pads (Figure 12a,c) and beyond the high resonance for stiff rail pads (Figure 12b,d); it works  
Figure 11: Frequency response function of the rail damper’s masses: (**a**) modulus and (**b**) phase. —, mass 1 with damping: \(\cdots\), mass 1 without damping; —, mass 2 with damping; \(\cdots\), mass 2 without damping. practically independently by the ballast condition. Rail damper reduces the rail receptance around the high resonance and within the second range of the propagative wave and narrows the difference between the pinned-pinned peak and the depth of the receptance calculated above sleeper, especially when the rail pads are soft (Figure 12a,c).