## 3 Transfer Function of the Measurement System
### Dynamics of the Wheel-Rail Coupled System
When the point mass wheel model is used, the axlebox acceleration is equal to the one of the wheel centres and can be calculated by using [PERSON]'s second law:  
\[\ddot{z}_{ax}=\frac{\Delta F_{z}}{M_{w}}, \tag{19}\]  
On the other hand, if the flexible wheelset model is considered, the cross-acceptance \(A_{Bz}(f)\) representing the axlebox vertical displacement per unit contact force is used. It is derived by the modal model of the wheelset. Thus, the axlebox acceleration takes the following form:  
Figure 5:0  
Thus, the transfer function \(H(f)\) of the axlebox measurement system is calculated with the point mass and flexible wheelset approaches by combining Equation (18) with Equations (19) and (20), respectively. The point mass wheelset model is expressed by the following relationship:  
Figure 5:1  
while for the flexible wheelset model it becomes  
Figure 5:2  
The transfer functions obtained with the two approaches are compared in Figure 5:3 When the wheel is modelled as a point mass the transfer function of the system is mainly  
Figure 5:4 Wheel, track, Hertzian spring and coupled wheel–rail system direct vertical receptances: (**a**) considering the wheelset as a point mass; (**b**) considering the deformable wheelset model. determined by the track dynamics, since the wheelset dynamics is completely neglected. On the other hand, the modal model of the wheelset allows to consider its flexibility in the calculation of the axlebox acceleration. In general, at frequencies below 50 Hz the two approaches lead to a similar behaviour of transfer response, since the overall dynamics is determined by the wheelset mass. As the frequency increases, the point mass wheel model leads to a global overestimation of the irregularity except at specific frequencies that correspond to the resonances of the wheelset, where an underestimation occurs. In particular, this happens in correspondence to the peaks at about 70 Hz (first bending mode of the wheelset) and at 190 Hz (second bending mode). The difference between the two approaches arises from the calculation of the axlebox acceleration. When the deformable wheelset is considered, the axlebox acceleration is derived considering the wheelset dynamics (term \(A_{\text{Bz}}(f)\) of Equation (20)). On the other hand, when the wheelset is modelled as a point mass, the axlebox acceleration is related to the vertical force fluctuation by [PERSON]'s second law (Equation (19)). The overestimation is due to the fact that part of the energy introduced by the irregularity is dissipated by the wheelset flexibility, which is completely neglected in the point mass wheel model. The assumption of the point mass wheel model is a good approximation if the travelling speed of the vehicle and the consequent excitation frequency induced by the rail irregularity (Equation (3)) are sufficiently low. As the speed increases, this assumption is less valid. It is also worth mentioning that the underestimation of the irregularity at frequencies corresponding to wheelset modes can lead to errors in monitoring rail corrugation. In fact, it is common that that corrugation tends to form at wavelengths related to the frequencies of wheelset modes. The transfer function is calculated considering the dynamics of a specific track section, but the approach is general. If the receptances of all the different trackforms present along the line are known, they can be substituted in Equation (18) to obtain the corresponding transfer function of the measurement system, allowing rail corrugation monitoring along the entire line.