## 3. Sperling Index Method  
The Sperling index method, also known as the Wz index method [17; 18], is used both for the evaluation of RQ and RC in the carbody. By applying this method, the Wz index is obtained, which has a precise meaning, based on which the performance of the railway vehicle, in terms of RQ and RC, can be evaluated. The scale of values and significance of the Wz indices for RQ is presented in Figure 2, and Figure 3 displays the scale of values of the Wz indices indicating passengers' perception of railway vehicle vibrations. The calculation relation of the comfort index Wz is of the form  
\[W_{z}=\sqrt{\int\limits_{0.5}^{30}a^{3}(f)\cdot B^{3}(f)df}, \tag{39}\]  
where \(a\) stands for the vertical acceleration of the carbody, expressed in cm/s\({}^{2}\), \(B\) stands for the frequency weighting function, and \(f\) is the frequency (\(f=\omega/2\pi\)). Figure 2. Scale for the RQ index Wz. Relation (39) is valid on the assumption that the acceleration spectrum is a continuous function of frequency between 0.5 and 30 Hz. The weighting function \(B\) is computed differently for the evaluation of the RQ and the RC (see Figure 4), respectively, according to the relations below. * for the evaluation the RQ we use the following relation: \[B_{\eta}(f)=q_{0}\sqrt{\frac{\left[\left(1-q_{1}f^{2}\right)^{2}+\left(q_{2}f \right)^{2}\right]\cdot q_{3}f^{2}}{\left[\left(1-q_{4}f^{2}\right)^{2}+\left(q _{5}f-q_{6}f^{3}\right)^{2}\right]\cdot\left(1+q_{7}f^{2}\right)}};\] (40)
* for evaluation the RC to vertical vibrations, we use the following one: \[B_{rc}(f)=c_{0}\sqrt{\frac{c_{1}f^{2}+\left(c_{2}f^{2}\right)^{2}}{\left(1-c_ {3}f^{2}\right)^{2}+\left(c_{4}f-c_{5}f^{3}\right)^{2}}},\] (41) where \(q_{0}=1.14\), \(q_{1}=0.056\), \(q_{2}=0.645\), \(q_{3}=3.55\), \(q_{4}=0.252\), \(q_{5}=1.547\), \(q_{6}=0.00444\), \(q_{7}=3.55\), and \(c_{0}=0.588\), \(c_{1}=1.911\), \(c_{2}=0.25\), \(c_{3}=0.277\), \(c_{4}=1.563\) and \(c_{5}=0.0368\). The RMS acceleration of the carbody is computed in the three relevant points of the vibration behavior of the carbody, respectively, points \(M\), \(B_{1}\), and \(B_{2}\) (see Figure 1), based on the PSD of the following acceleration:  
Figure 4: The frequency weighting function \(B\): (**a**) for evaluation the RQ; (**b**) for evaluation the RC. Figure 3: Scale for the RC index Wz. the RMS of the acceleration at the middle of the carbody (point \(M\)),  
\[a_{M}=\sqrt{\frac{1}{\pi}\!\int G_{aM}(\omega)\mathrm{d}\omega}\,=\sqrt{\frac{1}{ \pi}\!\int G(\omega)\big{|}\overline{H}_{aM}(\omega)\big{|}^{2}\mathrm{d}\omega}; \tag{42}\]  
and the RMS of the acceleration above the two bogies (points \(B_{1}\) and \(B_{2}\)),  
\[a_{B_{1,2}}=\sqrt{\frac{1}{\pi}\!\int G_{aB_{1,2}}(\omega)\mathrm{d}\omega}\,= \sqrt{\frac{1}{\pi}\!\int G(\omega)\big{|}\overline{H}_{aB_{1,2}}(\omega)\big{|} ^{2}\mathrm{d}\omega}. \tag{43}\]  
where \(G_{aM}(\omega)\) and \(G_{aB_{1,2}}(\omega)\) represents the PSD of the acceleration at the carbody relevant points, which are calculated as a function of the PSD of the vertical irregularities of the track G(\(\omega\)) and the frequency response functions of the carbody acceleration, \(\overline{H}_{aM}(\omega)\) and \(\overline{H}_{aB_{1,2}}(\omega)\). The theoretical curve of the PSD of the track irregularities can be described by relation [29]:  
\[G(\omega)=\frac{A\Omega_{c}^{2}\mathrm{v}^{3}}{[\omega^{2}+(V\Omega_{c})^{2} ]\big{[}\omega^{2}+(V\Omega_{r})^{2}\big{]}}, \tag{44}\]  
where \(\Omega_{c}\) = 0.8246 rad/m, \(\Omega_{r}\) = 0.0206 rad/m, and the constant \(A\) depends on the quality of the track. For a track of good quality, it is considered that \(A\) = 2119 \(\times\) 10\({}^{-7}\) radm, and in the case of a track with low quality, the constant \(A\) is assigned a value of 6.124 \(\times\) 10\({}^{-7}\) radm.