## 3 Transfer Function of the Measurement System
### Track Dynamics
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Rail mass (unit length) & \(m_{r}\) & 49.9 kg/m \\ Rail cross-section & \(A_{r}\) & 63.62 cm\({}^{2}\) \\ Rail moment of inertia (y-y) & \(I_{r,yy}\) & 1844 cm\({}^{4}\) \\ Rail Young Modulus & \(E_{r}\) & 2.06 \(\times\) 10\({}^{5}\) MPa \\ Poisson ratio & \(\
u_{r}\) & 0.28 \\ Rail shear factor & \(\kappa_{r}\) & 0.34 \\ Rail loss factor & \(\eta_{r}\) & 0.02 \\ Support spacing & \(L_{s}\) & 0.75 m \\ Railroad stiffness & \(K_{rp,z}\) & 150 kN/mm \\ Railpad loss factor & \(\eta_{rp,z}\) & 0.13 \\ Railpad viscous damping & \(C_{rp,z}\) & 1.5 kNs/m \\ Tieplate mass & \(M_{s}\) & 15 kg \\ Ins. plate stiffness & \(K_{ip,z}\) & 30 kN/mm \\ Ins. plate loss factor & \(\eta_{ip,z}\) & 0.20 \\ Ins. plate viscous damping & \(C_{ip,z}\) & 2.8 kNs/m \\ \hline \hline \end{tabular}   
* The main assumptions of the analytical model are reported in the following:
* Displacements are infinitesimally small, to ensure the linearity;
* The centroid of the beam cross-sections lays on the \(x\)-axis, while \(y\)- and \(z\)-axes are the principal axes of the beam cross-section;
* The two rails are considered dynamically decoupled;
* The load applied on the rail is considered fixed in space at \(x=0\), since the effect of a moving load on the beam dynamics is negligible [21], especially considering the low maximum speeds reached on metro lines;
* The inclination of the rail is neglected. The load is considered applied in correspondence to the rail neutral axis. This hypothesis allows to decouple the lateral and vertical track dynamics. This is a strong simplification, since the coupling effect in  
\begin{table} \begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Rail mass (unit length) & \(m_{r}\) & 49.9 kg/m \\ Rail cross-section & \(A_{r}\) & 63.62 cm\({}^{2}\) \\ Rail moment of inertia (y-y) & \(I_{r,yy}\) & 1844 cm\({}^{4}\) \\ Rail Young Modulus & \(E_{r}\) & 2.06 \(\times\) 10\({}^{5}\) MPa \\ Poisson ratio & \(\
u_{r}\) & 0.28 \\ Rail shear factor & \(\kappa_{r}\) & 0.34 \\ Rail loss factor & \(\eta_{r}\) & 0.02 \\ Support spacing & \(L_{s}\) & 0.75 m \\ Railpad stiffness & \(K_{rp,z}\) & 150 kN/mm \\ Railpad loss factor & \(\eta_{rp,z}\) & 0.13 \\ Railpad viscous damping & \(C_{rp,z}\) & 1.5 kNs/m \\ Tieplate mass & \(M_{s}\) & 15 kg \\ Ins. plate stiffness & \(K_{ip,z}\) & 30 kN/mm \\ Ins. plate loss factor & \(\eta_{ip,z}\) & 0.20 \\ Ins. plate viscous damping & \(C_{ip,z}\) & 2.8 kNs/m \\ \hline \hline \end{tabular}   
* The main assumptions of the analytical model are reported in the following:
* Displacements are infinitesimally small, to ensure the linearity;
* The centroid of the beam cross-sections lays on the \(x\)-axis, while \(y\)- and \(z\)-axes are the principal axes of the beam cross-section;
* The two rails are considered dynamically decoupled;
* The load applied on the rail is considered fixed in space at \(x=0\), since the effect of a moving load on the beam dynamics is negligible [21], especially creases as the distance along \(y\)-axis of the load's point of application from the neutral axis increases due to the rotation of the rail cross-section. The system has three degrees of freedom: the rail displacement \(z_{R}\), the rotation of the rail cross-section around the neutral axis \(\phi\) and the tieplate displacement \(z_{S}\). The equations of motion of the system are  
\[\left\{ \begin{array}{l}m_{r}\ddot{z}_{R}+k_{\alpha,\beta z}^{*}z_{R}+G_{r}^{* }A_{r}\kappa_{r}\frac{\partial}{\partial x}\left(\phi-\frac{\partial z_{R}}{ \partial x}\right)=F_{z}(t)\delta(x)\\ \rho_{r}I_{r,yy}\dot{\phi}-E_{r}^{*}I_{yy}\frac{\partial^{2}\phi}{\partial x^{ 2}}+G_{r}^{*}A_{r}\kappa_{r}\left(\phi-\frac{\partial z_{R}}{\partial x}\right)= 0\end{array}