## 3 The Dynamic Response of the Railway Vehicle to the Track Vertical Irregularities  
The track vertical irregularities are considered to be represented as a stationary stochastic process, which can be described via the power spectral density. The theoretical curve of the power spectral density is representative for the average statistical properties of the European railway, as in the relation [49]  
\[S(\Omega)=\frac{A\Omega_{c}^{2}}{(\Omega^{2}+\Omega_{c}^{2})(\Omega^{2}+\Omega _{c}^{2})} \tag{13}\]  
where \(\Omega\) is the wavelength, \(\Omega_{c}=0.8246\) rad/m, \(\Omega_{c}=0.0206\) rad/m, and \(A\) is a coefficient depending on the track quality. For a high-quality track, \(A=4.032\times\) 10\({}^{-7}\)rad\(\cdot\)m, whereas for a low-quality, the coefficient \(A\) is 1.080 \(\times\) 10\({}^{-6}\)rad\(\cdot\)m.  
As a function of the angular frequency \(\omega=V\Omega\), where \(V\) is the vehicle velocity, the power spectral density of the track irregularities can be written in the below form  
\[G(\omega)=\frac{S(\omega/V)}{V}=\frac{A\Omega_{c}^{2}V^{3}}{[\omega^{2}+(V \Omega_{c})^{2}][\omega^{2}+(V\Omega_{r})^{2}]} \tag{14}\]  
In terms of ride comfort, what interests us is the dynamic response in three carbody reference points, located at the carbody center and against the two bogies. These points are shown in Figure 1 as C--at the carbody center, \(B_{1}\) and \(B_{2}\)--above the bogies, against the leaning points of the carbody on the secondary suspension.  
The frequency response functions of the vertical acceleration at the carbody center and in the points above the two bogies, respectively, are expressed in the below equations  
\[\overline{H}_{C}(\omega)=-\omega^{2}\big{[}\overline{H}_{zc}(\omega)+X_{c}( L_{c}/2)\overline{H}_{T_{c}}(\omega)\big{]} \tag{15}\]  
\[\overline{H}_{B_{1,2}}(\omega)=-\omega^{2}\big{[}\overline{H}_{z_{c}}(\omega) \pm a_{c}\overline{H}_{\theta_{c}}(\omega)+X_{c}(l_{1,2})\overline{H}_{T_{c}}( \omega)\big{]} \tag{16}\]  
where \(\overline{H}_{z_{c}}(\omega)\), \(\overline{H}_{\theta_{c}}(\omega)\), \(\overline{H}_{T_{c}}(\omega)\) are the frequency response functions corresponding to the rigid vibration modes of bounce and pitch, and to the vertical bending of the carbody.  
Starting from the frequency response functions of the vertical acceleration of the carbody and the power spectral density of the track irregularities, the power spectral density of the carbody vertical acceleration in three carbody reference points can be calculated as  
\[G_{C}(\omega)=\omega^{4}G(\omega)\big{|}\overline{H}_{zc}(\omega)+X_{c}(L_{c} /2)\overline{H}_{T_{c}}(\omega)\big{|}^{2} \tag{17}\]\[G_{B_{1,2}}(\omega)=\omega^{4}G(\omega)\big{|}\overline{H}_{z_{c}}(\omega)\pm a_{ c}\overline{H}_{\theta_{c}}(\omega)+X_{c}(l_{1,2})\overline{H}_{T_{c}}(\omega) \big{|}^{2} \tag{18}\]  
To determine the ride comfort indices, it is required to calculate the root mean square of the carbody vertical acceleration. This is calculated based on the dynamic response of the vehicle carbody, expressed as the power spectral density of acceleration.  
\[a_{C}=\sqrt{\frac{1}{\pi}\!\!\int\limits_{f_{1}}^{f_{2}}G_{C}(\omega)\mathrm{d} \omega},\ a_{B_{1,2}}=\sqrt{\frac{1}{\pi}\!\!\int\limits_{f_{1}}^{f_{2}}G_{B_{ 1,2}}(\omega)\mathrm{d}\omega} \tag{19}\]  
where the frequencies \(f_{1}\) and \(f_{2}\) depend on the comfort assessment method. For the mean comfort method, \(f_{1}\) = 0.4 Hz and \(f_{2}\) = 80 Hz, while \(f_{1}\) = 0.5 Hz and \(f_{2}\) = 30 Hz are considered for the index method \(W_{z}\).