# Parametric Study on the Effect of Rail Dampers on Track Decay Rate
## 2 Mechanical Model of the Track Featured with Rail Dampers  
The model of the ballasted track featured with rail dampers, reduced to a rail with rail dampers due to track longitudinal symmetry, is shown in Figure 1.  
The rail is assimilated with an infinite uniform Timoshenko beam placed on punctual equidistant support consisting of a viscoelastic element (rail pad), a rigid body in vertical translation (sleeper), and another viscoelastic element attached to the rigid base to model the ballast. The rail dampers are represented using oscillating systems with two degrees of freedom that are connected to the rail in the middle of the distance between the sleepers. Parameters for the track model are as follows:  
* For the rail: \(E\) and \(G\)--the Young and shear moduli of elasticity; \(m\) and \(\rho\)--the mass per unit length and density; \(S\) and \(I\)--the cross-section area and moment of inertia; and \(\kappa\)--the shear coefficient. * For the rail support: \(M_{\mathrm{s}}\)-the semi-sleeper mass; \(k_{p,b}\)--the stiffness of the rail pad and ballast; and \(l_{k}\)--the position of the \(k\) rail support. * For the rail damper (a pair for each sleeper bay): \(M_{1,2}\)--the mass of the rail damper bodies; \(k_{1,2}\)--the stiffness of rubber elements; and \(a_{k}\)--the position of the \(k\) rail damper. Considering the steady-state harmonic behavior of angular frequency \(\omega\), equations of motion can be written as follows:  
* For the rail: \[ \begin{array}{l}\overline{G}\kappa\left[\frac{\mathrm{d}^{2}\overline{w}(x, \omega)}{\mathrm{d}x^{2}}-\frac{\mathrm{d}\,\overline{\theta}(x,\omega)}{ \mathrm{d}x}\right]+\omega^{2}m\overline{w}(x,\omega)=-\overline{F}\delta(x-x _{F})+\\ \overline{k}_{p}\sum\limits_{k=-\infty}^{\infty}(\overline{w}(l_{k},\omega)- \overline{z}_{tk}(\omega))\delta(x-l_{k})+\overline{k}_{1}\sum\limits_{k=- \infty}^{\infty}(\overline{w}(a_{k},\omega)-\overline{z}_{1k}(\omega)\delta(x- a_{k}),\\ \overline{E}I\frac{\mathrm{d}^{2}\overline{\theta}(x,\omega)}{\mathrm{d}x^{2}}+ \overline{G}\kappa\left[\frac{\mathrm{d}\overline{w}(x,\omega)}{\mathrm{d}x}- \overline{\theta}(x,\omega)\right]+\omega^{2}\rho\overline{\theta}(x,\omega)=0,\end{array} \] (1)
* For the sleeper '\(K\): \[\left(-\omega^{2}M_{t}+\overline{k}_{p}+\overline{k}_{b}\right)\overline{z}_{ tk}(\omega)-\overline{k}_{p}\overline{w}(l_{k},\omega)=0,\] (2)
* For the rail damper '\(K\):  
Figure 1: The model of the rail featuring rail dampers. \[ \begin{array}{l}\left(-\omega^{2}M_{1}+\overline{k}_{1}+\overline{k}_{2}\right) \overline{z}_{1k}(\omega)-\overline{k}_{1}\overline{w}(a_{k},\omega)- \overline{k}_{2}\overline{z}_{2k}(\omega)=0,\\ \left(-\omega^{2}M_{2}+\overline{k}_{2}\right)\overline{z}_{2k}(\omega)- \overline{k}_{2}\overline{z}_{1k}(\omega)=0\end{array}  \tag{3}\]  
where \(\overline{w}(x,\omega)\) and \(\overline{\delta}(x,\omega)\) are the complex amplitudes of the rail displacement and cross-section rotation in the \(x\) section with respect to the \(Oxz\) reference; \(\overline{z}_{tk}(\omega)\) is the complex amplitude of the \(k\) sleeper displacement; \(\overline{z}_{1,2k}(\omega)\) is the complex amplitude of the \(k\) rail damper internal displacements; and \(\overline{F}\) is the complex amplitude of the harmonic force of angular frequency \(\omega\) applied in section \(x_{F}\). Using the complex stiffnesses in Equations (1)-(3), the hysteretic damping is introduced into the model as follows:  
\[\overline{k}_{p,b}=k_{p,b}(1+i\eta_{p,b}),\ \ \ \overline{k}_{1,2}=k_{1,2}(1+i\eta_{1,2}), \tag{4}\]  
where \(i^{2}=-1\), \(\eta_{p,b}\), and \(\eta_{1,2}\) are the loss factors for the rail pad, ballast, and rail damper, respectively.