## 3 The Dynamic Response of the Railway Vehicle to the Track Vertical Irregularities  
To analyse the vertical vibrations behaviour of the vehicle carbody, we utilised the power spectral density of acceleration in three important points considered critical points of the vibrations level of the carbody and the ride comfort in the railway vehicles [22, 41]. These reference points are marked in Figure 1 and found at the centre of the carbody--point \(m\)--and against the support points of the carbody onto the secondary suspension, above the bogie 1--point \(b_{1}\)--and above the bogie 2--point \(b_{2}\).  
To calculate the power spectral density of the vertical acceleration of the carbody, the track vertical irregularities are considered to represent a stochastic stationary process, which can be described via the power spectral density. For the average statistical properties of the European railways, the theoretical curve of the power spectral density described by the relation [43]  
\[S(\Omega)\ =\ \frac{A\Omega_{c}^{2}}{(\Omega^{2}+\Omega_{r}^{2})(\Omega^{2}+ \Omega_{c}^{2})}, \tag{27}\]  
is considered representative, where \(\Omega\) is the wavenumber, \(\Omega_{c}=0.8246\ \mathrm{rad/m}\), \(\Omega_{r}=0.0206\ \mathrm{rad/m}\) and \(A\) is a constant depending on the track quality. For a high-quality track, \(A=2.119\times 10^{-7}\ \mathrm{radm}\), and \(A\) is \(6.124\times 10^{-7}\ \mathrm{radm}\) for a low-quality track.  
As a function of the angular frequency \(\omega=V\Omega\), the power spectral density of the track irregularities can be expressed as in the general equation  
\[G(\omega)\ =\ \frac{A\Omega_{c}^{2}V^{3}}{[\omega^{2}+(V\Omega_{c})^{2}][ \omega^{2}+(V\Omega_{r})^{2}]}. \tag{28}\]  
The power spectral density of the carbody vertical acceleration \(G_{c}(\omega)\) is calculated differently for the two models of the vehicle, starting from the power spectral density of the track vertical irregularities \(G(\omega)\) and from the response functions corresponding to the vibration modes of the carbody--bounce and pitch \(\overline{H}_{z_{c}}(\omega)\), \(\overline{H}_{\theta_{c}}(\omega)\)--and the vertical bending \(\overline{H}_{T_{c}}(\omega)\).  
In the point \(m\) located at the centre of the carbody, the power spectral density of the acceleration is calculated with the following equations:  
* for the, rigid carbody' model, \[G_{cm}(\omega)\ =\ G(\omega)\Big{|}\omega^{2}\overline{H}_{zc}(\omega)\Big{|}^{2}.\] (29)
* for the, flexible carbody' model, \[G_{cm}(\omega)\ =\ G(\omega)\Big{|}\omega^{2}\bigg{[}\overline{H}_{zc}(\omega)+X_{c} \bigg{(}\frac{L_{c}}{2}\bigg{)}\overline{H}_{T_{c}}(\omega)\bigg{]}\bigg{|}^{2}.\] (30)In the points \(b_{1}\) and \(b_{2}\), located against the two bogies, the power spectral density of the vertical acceleration is given by the equations below:  
* for the, rigid carbody' model, \[G_{cb1,2}(\omega)\ =\ G(\omega)\Big{|}\omega^{2}[\overline{H}_{z_{c}}(\omega) \pm a_{c}\overline{H}_{\theta_{c}}(\omega)]\Big{|}^{2}.\] (31)
* for the, flexible carbody' model, \[G_{cb1,2}(\omega)\ =\ G(\omega)\Big{|}\omega^{2}[\overline{H}_{z_{c}}(\omega) \pm a_{c}\overline{H}_{\theta_{c}}(\omega)+X_{c}(l_{1,2})\overline{H}_{T_{c}}( \omega)]\Big{|}^{2}.\] (32)  
To study the influence of the vertical bending upon the level of vertical vibrations in the vehicle carbody, the root mean square of the vertical acceleration in the three reference points of the carbody will be utilised. This is calculated based on the dynamic response of the vehicle expressed in the form of the power spectral density of the carbody acceleration  
* at the centre of the carbody, \[a_{cm}\ =\ \sqrt{\frac{1}{\pi}\int\limits_{0}^{\infty}G_{cm}(\omega)\mathrm{d} \omega}\ ;\] (33)
* above the two bogies, \[a_{cb1,2}\ =\ \sqrt{\frac{1}{\pi}\int\limits_{0}^{\infty}G_{cb1,2}(\omega) \mathrm{d}\omega}.\] (34)