# Parametric Study on the Effect of Rail Dampers on Track Decay Rate
## 2 Mechanical Model of the Track Featured with Rail Dampers
\begin{split}\mathbf{A}_{11}&=k_{t}\big{(}g_{\mathcal{T}}(l_{i}, l_{i},\omega)\big{)}_{1\leq i,j\leq N}-\mathbf{I}_{N}\\ \mathbf{A}_{12}&=k_{d}\big{(}g_{\mathcal{T}}(l_{i},a_{j}, \omega)\big{)}_{1\leq i\leq N,\ 1\leq j\leq N-1}\\ \mathbf{A}_{21}&=k_{t}\big{(}g_{\mathcal{T}}(a_{i},l_{j}, \omega)\big{)}_{1\leq i\leq N-1,1\leq j\leq N}\\ \mathbf{A}_{22}&=k_{d}\big{(}g_{\mathcal{T}}(a_{i},a_{j}, \omega)\big{)}_{1\leq i,j\leq N-1}-\mathbf{I}_{N-1}\end{split}  \tag{15}\]  
and  
\[ \begin{split}\mathbf{X}_{1}&=\big{[}\ \overline{w}(l_{1}, \omega)\ \ \ \overline{w}(l_{2},\omega)\ \ \ \ldots\ \ \overline{w}(l_{N},\omega)\ \ \big{]}^{T},\\ \mathbf{X}_{2}&=\big{[}\ \overline{w}(a_{1}, \omega)\ \ \ \overline{w}(a_{2},\omega)\ \ \ \ldots\ \ \overline{w}(a_{N-1}, \omega)\ \big{]}^{T},\\ \mathbf{B}_{1}&=\overline{F}\big{[}\ g_{t}(l_{1},x_{F}, \omega)\ \ \ g_{t}(l_{2},x_{F},\omega)\ \ \ \ldots\ \ \ g_{t}(l_{N},x_{F},\omega)\ \ \big{]}^{T},\\ \mathbf{B}_{2}&=\overline{F}\big{[}\ g_{t}(a_{1},x_{F}, \omega)\ \ \ g_{t}(a_{2},x_{F},\omega)\ \ \ \ldots\ \ \ g_{t}(a_{N-1},x_{F},\omega)\ \big{]}^{T}.\end{split}  \tag{16}\]  
By solving Equation (14) numerically, the complex amplitude of the rail at \(l_{k,\ 1\leq k\leq N}\) and \(a_{k,\ 1\leq k\leq N-1}\) is obtained, and then the complex amplitude of the rail in the \(x\) section can be calculated:  
\[\overline{w}(x,\omega)=-\overline{F}_{\mathcal{GT}}(x,x_{F}, \omega)+\overline{k}_{t}\sum\limits_{k=1}^{N}\overline{w}(l_{k})_{\mathcal{GT }}(x,l_{k},\omega)+\overline{k}_{d}\sum\limits_{k=1}^{N-1}\overline{w}(a_{k}) _{\mathcal{GT}}(x,a_{k},\omega). \tag{17}\]  
The above approach is like the one applied in ref. [20], where a free-free Timoshenko beam with two degrees of freedom oscillator was used to build the model of a finite-length rail with rail damper discretely elastically supported at both ends that has been subject to experimental tests. Parameters of the rail damper model (two degrees of freedom oscillator) have been determined by comparing the theoretical and experimental results in terms of the rail receptance. To calculate TDR for a particular one-third octave band of the central angular frequency \(\omega_{c}\), a spatial partition, \(x_{0}\), \(x_{1}\), \(\ldots\), \(x_{S}\), of constant step, \(\Delta x=x_{i+1}-x_{i}\), where \(i=0\div S-1\), is needed, and then the following formula should be applied:  
\[TDR(x_{0},\omega_{c})=4.343\frac{w^{2}(x_{0},\omega_{c})}{\Delta x\sum\limits_ {i=0}^{S-1}w^{2}(x_{i},\omega_{c})} \tag{18}\]  
where \(w^{2}(x_{i}w_{c})\) is the mean square rail amplitude associated with the angular frequency \(\omega_{c}\)  
\[w^{2}(x_{i},\omega_{c})=\frac{1}{2\Delta\omega}\int_{\omega_{i}}^{\omega_{s}} \overline{w}(x_{i},\omega)\overline{w}^{*}(x_{i},\omega)\mathrm{d}\omega= \frac{1}{2\Delta\omega}\int_{\omega_{i}}^{\omega_{s}}|\overline{w}(x_{i}, \omega)|^{2}\mathrm{d}\omega \tag{19}\]  
where \(\omega_{is}\) are the limits of the one-third octave band and \(\Delta\omega\) is its width (\(\Delta\omega=\omega_{s}-\omega_{i}\)), and \(\overline{w}^{*}(x_{i},\omega)\) is the complex conjugate of the complex amplitude \(\overline{w}(x_{i},\omega)\). Numerically, the mean square rail amplitude can be calculated applying the trapezoidal method:  
\[w^{2}(x_{i},\omega_{c})\approx\frac{1}{2N}\left[\frac{1}{2}\left(\left|\overline{ w}(x_{i},\omega_{0})\right|^{2}+\left|\overline{w}(x_{i},\omega_{N})\right|^{2} \right)+\sum_{j=1}^{N-1}\left|\overline{w}(x_{i},\omega_{j})\right|^{2}\right] \tag{20}\]  
where \(\omega_{0}=\omega_{i}\), \(\omega_{N}=\omega_{s}\), and \(\omega_{j+1}-\omega_{j}=\Delta\omega/N\) with \(0\leq j\leq N-1\).