## 4 Dynamic Response of the Carbody to Vertical Vibrations  
The effect of frequency weighting according to the Sperling index method on the dynamic response of the carbody to vertical vibrations is calculated based on the mechanical model previously presented and analyzed. The vehicle's dimensions are specific to the type X coach equipped with 250 km/h bogie, in accordance with [30]. The dynamic response of the vehicle carbody is analyzed by considering the PSD of the acceleration, both unweighted and frequency-weighted, using the two weighting functions (\(B_{\eta}\) and \(B_{rc}\)) corresponding to RQ and RC, respectively. The results obtained are correlated with the results of the Wz indices for the RQ evaluation and the RC evaluation. Table 1 contains the values of the mechanical model parameters. The vehicle is running on a low-quality track. Figure 5 presents the frequency response functions of the acceleration at the relevant points of the carbody--points \(M\), \(B_{1}\), and \(B_{2}\), obtained for the undamped case and for \(c_{zc}=0\), \(c_{xx}=0\), \(c_{zb}=0\), \(c_{m}=0\), and \(c_{zs}=0\), respectively. The response function curves show several peaks corresponding to the resonance frequencies of the vehicle and several minima corresponding to the antiresonance frequencies. On diagram (a), the peak of the response functions at the resonance frequency of the wheelset is marked at 43 Hz. The frequency range, in which the peaks of the resonance frequencies of the vibration modes of the carbody and bogies are found, is selected on diagram (a) and marked in gray; these are highlighted in diagram (a'). The following eigenfrequencies are marked on diagram (a'): 1.17 Hz--resonance frequency of the carbody bounce; 1.56 Hz--resonance frequency of the carbody pitch; 6.65 Hz--resonance frequency of the bogie bounce; 8.00 Hz--resonance frequency of the carbody vertical bending; 9.63 Hz--resonance frequency of the bogie pitch. Figure 6 displays the frequency response functions of the carbody acceleration at the relevant points of the carbody, considering the damping constants from Table 1. In diagram (a), the frequency range 0-50 Hz was considered, in which the resonance frequency of the vertical vibrations of the wheelset is found, at 43 Hz, and in diagram (a'), only the frequency range 0-10 Hz was selected, which includes all the resonance frequencies of the carbody. In the detail in diagram (a'), the peaks of the response functions corresponding to the resonance frequencies of the vibration modes of the carbody are marked; the peaks of the frequency response functions corresponding to the resonance frequencies of the vibration modes of the bogies are faded due to the damping of the system. \begin{table}
\begin{tabular}{c c} \hline \(m_{c}=\) 34,000 kg & \(2k_{zc}=\) 1.2 MN/m \\ \hline \(m_{b}=\) 3200 kg & \(2k_{xc}=\) 4 MN/m \\ \hline \(m_{w}=\) 1650 kg & \(2c_{zc}=\) 34.28 kNs/m \\ \hline \(m_{r}=\) 175 kg & \(2c_{xc}=\) 50 kNs/m \\ \hline \(J_{c}=\) 1,963,840 kg\(\cdot\)m\({}^{2}\) & \(4k_{zb}=\) 4.4 MN/m \\ \hline \(J_{b}=\) 2048 kg\(\cdot\)m\({}^{2}\) & \(4c_{zb}=\) 52.21 kNs/m \\ \hline \(L_{c}=\) 26.4 m & \(2k_{zr}=\) 1.4 MN/m \\ \hline \(2a_{b}=\) 19 m & \(2c_{zr}=\) 40 kNs/m \\ \hline \(2a_{b}=\) 2.56 m & \(2k_{H}=\) 3000 MN/m \\ \hline \(h_{c}=\) 1.3 m & \(m_{mc}=\) 35,224 kg \\ \hline \(h_{b}=\) 0.2 m & \(c_{mc}=\) 53.117 kNm/s \\ \hline \(F_{c}J_{c}=\) 3.158 \(\times\) 10\({}^{9}\) Nm\({}^{2}\) & \(k_{mc}=\) 88.998 MN/m \\ \hline \end{tabular}
\end{table} 
Table 1: Mechanical model parameters. The antiresonance frequencies, which are identified in Figures 5 and 6 on the frequency response curves of the carbody in the form of minima, are due to the geometric filtering effect. This effect represents a distinctive feature of the vertical vibration of railway vehicles, which has been analyzed in several studies [31; 32; 33; 34]. The minima of the PSD of the acceleration correspond to the frequencies of geometric filtering.