## 3 Transfer Function of the Measurement System
### Dynamics of the Wheel-Rail Coupled System  
The vertical dynamics of the wheel-rail interaction can be studied using the scheme represented in Figure 7.  
Figure 5: Analytical and experimental direct vertical receptance of the tested track in correspondence with a support. Figure 6: Analytical direct receptance calculated with discretely and continuously supported track. When discrete supports are considered, the receptance is calculated at different positions along the span between two supports. As already said, the irregularity of the rail surface is considered as a harmonic excitation which causes fluctuations in the contact parameters around their reference value, indicated with \(\Delta\). The displacements are taken positive when upwards, while the force exchanged between the wheel and the rail is taken positive, as represented in Figure 7. The Hertzian contact stiffness \(k_{H}\) allows to model the local deformability of the wheel and the rail. Its value is considered constant with frequency and equal to 1.2 \(\mathrm{GN}/\mathrm{m}\). The compression of the contact spring \(\delta\) can be expressed as follows:  
\[\delta=\Delta z_{R}+z_{irr}-\Delta z_{W}, \tag{15}\]  
where \(\Delta z_{W}\) and \(\Delta z_{R}\) are the fluctuations in the displacement of the contact point on the wheel and the rail, respectively, induced from the rail irregularity \(z_{irr}\). The compression of the Hertzian contact spring is always positive since it works only in compression. It can be expressed as follows:  
\[\delta=\frac{\Delta F_{z}}{k_{H}}, \tag{16}\]  
The fluctuations in the contact point displacements are calculated by considering wheel and rail direct receptances:  
\[\Delta z_{W}=A_{Wz}(f)\Delta F_{z}\quad\Delta z_{R}=A_{Rz}(f)(-\Delta F_{z}), \tag{17}\]  
The minus in the expression of \(\Delta z_{R}\) is related to the fact that the receptance of the rail is calculated considering force and displacement positive when directed in the same direction, while in the notation of Figure 7 these are opposite. By substituting Equations (16) and (17) into Equation (15), the irregularity on the rail can be written as  
\[z_{irr}=\bigg{(}A_{Wz}(f)+A_{Rz}(f)+\frac{1}{k_{H}}\bigg{)}\Delta F_{Z}=T(f) \Delta F_{z}, \tag{18}\]  
The transfer function \(T(f)\) represents the ratio between the rail irregularity and the induced fluctuation in the vertical contact force, i.e., the vertical receptance of the coupled wheel-rail system. It is obtained as the sum of three components according to Equation (18): the direct receptances of wheel, rail and Hertzian spring. The transfer function \(T(f)\) derived with the point mass and deformable wheelset model are shown in Figure 8a,b, respectively. In the figures, the single components are also shown. At low frequencies,  
Figure 7: Scheme of the vertical wheel–rail interaction. the dynamics of the wheel-rail interaction is mainly determined by the dynamics of the wheelset, while the track dynamics governs the high-frequency behaviour. The receptances of the coupled system derived with the two wheelset models present a similar trend since they are in the 20-50 Hz frequency range, where the wheelset dynamics is dominant. In the frequency range of interest for this problem, the effect of the contact stiffness is negligible. When higher frequencies are involved, the effect of the contact stiffness becomes more relevant, and also the effect of the contact filter effect should be considered [22; 23]. The equations written above are valid for both point mass and flexible wheelset models. They differ only in the analytical expression of \(A_{Wz}(f)\). The main difference between the two approaches is in the calculation of the transfer function between the vertical axlebox acceleration and the fluctuation in the vertical contact force. It can be calculated as the double derivative of the axlebox displacement with respect to time.