# Effect of the Anti-Yaw Damper on Carbody Vertical Vibration and Ride Comfort of Railway Vehicle
## 5 Evaluation of Ride Comfort during Vertical Vibrations  
According to the UIC 513R leaflet [48] and the standard of railway applications EN 12299 [49], the quantification of the comfort to vibrations is performed using a parameter, called ride comfort index (\(N_{MV}\)), and with a conventional scale linking the values of this index and the comfort sensation (see Table 1).  
The ride comfort index is calculated with the general relation [48; 49]  
\[N_{MV}=6\cdot\sqrt{\Big{(}a_{X95}^{Wat}\Big{)}^{2}+\Big{(}a_{Y95}^{Wat}\Big{)} ^{2}+\Big{(}a_{Z95}^{Wat}\Big{)}^{2}}, \tag{31}\]  
where \(a_{X}\), \(a_{Y}\) and \(a_{Z}\) are the root mean square of the longitudinal, lateral and vertical carbody acceleration, 95 refers to the quantile of order 95%, and \(W_{ad}\), \(W_{ab}\) represent the weighting filters of the longitudinal, lateral, respectively vertical acceleration.  
To evaluate the ride comfort at vertical vibration, the partial comfort index is used, which is calculated with the general relation [49]  
\[N_{MVZ}=6a_{Z95}^{W_{ab}}, \tag{32}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Ride Comfort Index**\(N_{MV}\) & **Significance** \\ \hline \(N_{MV}<1\) & Very comfortable \\ \(1\leq N_{MV}<2\) & Comfortable \\ \(2\leq N_{MV}<4\) & Medium \\ \(4\leq N_{MV}<5\) & Uncomfortable \\ \(N_{MV}\geq 5\) & Very uncomfortable \\ \hline \hline \end{tabular}
\end{table}
Table 1: The significance of the ride comfort index \(N_{MV}\).  
where \(W_{ab}=W_{a}.W_{b}\) represents the weighting filter of the vertical acceleration.  
The filter \(W_{a}\) is a passband filter with the following frequency weighting function  
\[H_{a}(s)=\frac{s^{2}(2\pi f_{2})^{2}}{\left[s^{2}+\frac{2\pi f_{2}}{\mathcal{Q} _{\rm{J}}}s+(2\pi f_{1})^{2}\right]\left[s^{2}+\frac{2\pi f_{2}}{\mathcal{Q}_{ \rm{J}}}s+(2\pi f_{2})^{2}\right]}, \tag{33}\]  
where \(s=\mathrm{i}\omega\) (with \(\mathrm{i}^{2}=-1\)), \(f_{1}=0.4\) Hz, \(f_{2}=100\) Hz and \(Q_{1}=0.71\).  
The filter \(W_{b}\) takes into account the high human sensitivity to the vertical vibrations within the frequency interval of 4 Hz and 12 Hz and has the frequency weighting function in the form of  
\[H_{b}(s)=\frac{(s+2\pi f_{3})\cdot\left[s^{2}+\frac{2\pi f_{3}}{\mathcal{Q}_{ \rm{J}}}s+(2\pi f_{5})^{2}\right]2\pi Kf_{4}^{2}f_{6}^{2}}{\left[s^{2}+\frac{2 \pi f_{4}}{\mathcal{Q}_{\rm{J}}}s+(2\pi f_{4})^{2}\right]\left[s^{2}+\frac{2 \pi f_{6}}{\mathcal{Q}_{\rm{J}}}s+(2\pi f_{6})^{2}\right]f_{3}f_{5}^{2}}, \tag{34}\]  
where \(f_{3}=16\) Hz,\(f_{4}=16\) Hz,\(f_{5}=2.5\) Hz,\(f_{6}=4\) Hz, \(Q_{2}=0.63\), \(Q_{4}=0.8\) and \(K=0.4\).  
When adopting the hypothesis that the vertical accelerations have a Gaussian distribution with the null mean value, we will have the following relation to calculate the partial ride comfort index in the carbody reference points  
\[N_{MV_{C}}=6\Phi^{-1}(0.95)\sqrt{\frac{1}{\pi}\int G_{C}(\omega)\left|H_{ab}( \omega)\right|^{2}\mathrm{d}\omega}\, \tag{35}\]  
\[N_{MV_{B_{1,2}}}=6\Phi^{-1}(0.95)\sqrt{\frac{1}{\pi}\int G_{B_{1,2}}(\omega) \left|H_{ab}(\omega)\right|^{2}\mathrm{d}\omega}\, \tag{36}\]  
where \(\Phi^{-1}(0.95)\) represents the quantile of the standard Gaussian distribution with the probability of 95% and \(H_{ab}(\omega)=H_{a}(\omega)\cdot H_{b}(\omega)\), where \(H_{a}(\omega)\) and \(H_{b}(\omega)\) are the frequency weighting function corresponding to the filters \(W_{a}\) and \(W_{b}\).