# Parametric Study on the Effect of Rail Dampers on Track Decay Rate
## 2 Mechanical Model of the Track Featured with Rail Dampers
The rail loss factor, \(\eta\), is considered in Equation (1) using the following notations:  
\[\overline{E}=E(1+i\eta),\ \ \ \overline{G}=G(1+i\eta). \tag{5}\]  
The boundary conditions should be associated with the rail equations of motion as follows:  
\[\lim_{|x|\to\infty}\overline{w}(x)=0. \tag{6}\]  
Applying the convolution theorem, the rail displacement in section \(x\) due to the harmonic force of angular frequency \(\omega\) applied in section \(x_{F}\) results in the following:  
\[\overline{w}(x,\omega)=\int_{-\infty}^{\infty}g_{T}(x,\xi,\omega)\left(- \overline{F}\delta(x_{F}-\xi)+\sum_{k\in\mathbb{Z}}\overline{k}_{t}\overline{ w}(l_{k})\delta(l_{k}-\xi)+\overline{k}_{d}\overline{w}(a_{k})\delta(a_{k}-\xi) \right)\mathrm{d}\xi, \tag{7}\]  
where \(g_{T}(x,\xi,\omega)\) is the frequency-domain Green function associated with the rail equations  
\[g_{T}(x,\xi,\omega)=\left(\frac{p\mathrm{I}\omega}{\kappa\overline{S}\overline {G}}-1\right)g(x,\xi,\omega)+\frac{\overline{E}I}{\kappa\overline{S}\overline {G}}\frac{\mathrm{d}^{2}g(x,\xi,\omega)}{\mathrm{d}\xi^{2}}, \tag{8}\]  
with \(g(x,\xi,\omega)\) being the frequency-domain Green function associated with the infinite Timoshenko beam  
\[g(x,\xi,\omega)=-\frac{\beta_{2}e^{-\beta_{1}|x-\xi|}+i\beta_{1}e^{-i\beta_{2 }|x-\xi|}}{2\beta_{1}\beta_{2}\left(\beta_{1}^{2}+\beta_{2}^{2}\right)\overline {E}I} \tag{9}\]  
where \(\beta_{1,2}\) depend on the angular frequency as follows:  
\[\beta_{1,2}=\sqrt{\frac{\omega\omega^{2}}{2\overline{E}}\left[\sqrt{\left( \frac{\overline{E}}{\overline{G}\kappa}-1\right)^{2}+\frac{4\overline{S} \overline{E}}{\omega\omega^{2}I}}\mp\left(\frac{\overline{E}}{\overline{G} \kappa}+1\right)\right]}. \tag{10}\]  
The complex stiffnesses in Equation (7) are as follows:  
\[\overline{k}_{t}=\overline{k}_{p}\left(1-\frac{\overline{k}_{p}}{-\omega^{2 }M_{t}+\overline{k}_{p}+\overline{k}_{b}}\right),\ \overline{k}_{d}=\overline{k}_{1}\left(1-\frac{\overline{k}_{1}\left(- \omega^{2}M_{2}+\overline{k}_{2}\right)}{\omega^{4}M_{1}M_{2}-\omega^{2}\left[ \overline{k}_{2}M_{1}+\left(\overline{k}_{1}+\overline{k}_{2}\right)M_{2} \right]+\overline{k}_{1}\overline{k}_{2}}\right). \tag{11}\]  
By performing the integral in Equation (7), it obtains the following:  
\[\overline{w}(x,\omega)=-\overline{F}_{ST}(x,x_{F},\omega)+\sum_{k\in\mathbb{Z }}\overlineComplex amplitude of the rail in \(l_{k}\) and \(a_{k}\) is needed to calculate the complex amplitude of the rail in the \(x\) section. To this end, it writes the following:  
\[ \begin{split}\overline{w}(l_{k},\omega)&=-\overline{ F}_{\mathcal{GT}}(l_{k},x_{F},\omega)+\sum\limits_{n\in\mathbb{Z}}\overline{k}_{t} \overline{w}(l_{n},\omega)_{\mathcal{GT}}(l_{k},l_{n},\omega)+\overline{k}_{d }\overline{w}(a_{n},\omega)_{\mathcal{GT}}(l_{k},a_{n},\omega),\\ \overline{w}(a_{k},\omega)&=-\overline{F}_{\mathcal{GT }}(a_{k},x_{F},\omega)+\sum\limits_{n\in\mathbb{Z}}\overline{k}_{t}\overline{ w}(l_{n},\omega)_{\mathcal{GT}}(a_{k},l_{n},\omega)+\overline{k}_{d} \overline{w}(a_{n},\omega)_{\mathcal{GT}}(a_{k},a_{n},\omega),\end{split}  \tag{13}\]  
where the index \(k\) has been replaced by \(n\) in sums. Considering the boundary condition (5), the complex amplitude of the rail is negligible when the distance between the support \(k\) and the \(x_{F}\) section of the harmonic force is large. Consequently, Equation (13) can be reduced to a set of \(2N-1\) equations and reorganized using the following matrix form:  
\[ \begin{bmatrix}\mathbf{A}_{11}&\mathbf{A}_{12}\\ \mathbf{A}_{21}&\mathbf{A}_{22}\end{bmatrix}\begin{bmatrix}\mathbf{X}_{1}\\ \mathbf{X}_{2}\end{bmatrix} = \begin{bmatrix}\mathbf{B}_{1}\\ \mathbf{B}_{2}\end{bmatrix}  \tag{14}\]  
where  
\[