## 3 Transfer Function of the Measurement System
### Wheel Dynamics  
The main difference between the proposed model and the HRSCA system presented in [11] is in the model used for the description of wheelset dynamics. In [11], the wheel is modelled as a point mass \(M_{w}\) running on the rail. The direct vertical receptance of the wheel, in this case, can be written as the following:  
\[A_{Wz}(f)=-\frac{1}{\left(2\pi f\right)^{2}M_{w}}, \tag{6}\]  
However, the vibration modes of the wheelset are involved in almost all the different mechanisms that define the wavelength of rail corrugation [4; 5], and therefore it is important to include them in the calculation of the wheel direct receptance. To this aim, the dynamics of the wheel is derived by means of a modal model of the wheelset on the primary suspension. The mode shapes and the natural frequencies are obtained through a Finite Element Analysis (FEA) under the following assumptions:  
* The axle is modelled using 2D Euler-Bernoulli beam elements;
* The wheels are considered as rigid bodies; due to the frequency range of interest for corrugation problems, the flexibility of the wheels does not have a strong influence on the different formation mechanisms [5];
* The wheels and the brake disks are modelled as concentrated masses in their centre, accounting also for their moment of inertia;
* The axleboxes are represented by concentrate masses placed in correspondence to their centre of gravity, while their rotational inertia is neglected;
* The wheelset gyroscopic effect is neglected. The gyroscopic effect becomes significant at high speed [16; 17; 18]. On the considered subway line, speeds above 90 km/h are never reached. For higher speeds, the gyroscopic effect of the wheelset must be included in the analysis. The data relative to the wheelset are summarised in Table 1.  
The equation of motion of the wheelset has the following form:  
\[[M]\underline{\underline{u}}+[C]\underline{\underline{u}}+[K]\underline{ \underline{u}}=\underline{E}, \tag{7}\]  
where \(\underline{E}\) is the vector of the contact forces at the wheel-rail interface and \(\underline{u}\) is the nodal displacements of the finite element model. The equation of motion is solved in modal coordinates by limiting to 800 Hz the mode shapes extracted from the FE model. This assumption is reasonable, since the frequency range of interest for the rail corrugation problem is limited to 300 Hz (in the present application). Due to the lack of experimental data on the structural damping of the wheelset, a modal damping ratio of 1% is assumed on all the bending modes. Damping ratios in this order of magnitude have been identified in an experimental modal analysis on a wheelset with similar characteristics. The contact point is not included in the nodes of the finite element model, but thanks to the rigid wheel assumption, its displacement, and thus the receptance, can be derived. The procedure for the computation of the contact point receptance is outlined hereafter:  
* A unit vertical contact force is considered varying harmonically and it is applied to the wheel centre, also accounting for the transport moments;
* The displacement vector of the wheel centre is defined by \[\underline{X}_{Wc}=\big{\{}x_{Wc}\quad y_{Wc}\quad z_{Wc}\quad\varphi_{xWc} \quad\varphi_{yWc}\quad\varphi_{zWc}\big{\}}^{T},\] (8)
* The first three elements of the displacement vector are linear displacements along longitudinal, lateral and vertical axes, respectively, while the others are the rotations around the same axes. Thanks to the modal model this vector can be computed;
* The linearised displacements of the contact point can be derived from the wheel centre ones by assuming rigid motion as \[ \begin{cases}x_{W}\\ y_{W}\\ z_{W}\end{cases} = \begin{cases}x_{Wc}\\ y_{Wc}\\ z_{Wc}\end{cases} +[\Lambda_{x}]\big{[}\Lambda_{y}]\big{[}\Lambda_{z}] \begin{cases} 0\\ 0\\ -R_{w}\end{cases} - \begin{cases}0\\ 0\\ -R_{w}\end{cases} ,\] (9) where \[[\Lambda_{x}]= \begin{bmatrix}1&0&0\\ 0&1&-\varphi_{xWa}\\ 0&\varphi_{xWa}&1\end{bmatrix} \quad