# Analysis of Train Car-Body Comfort Zonal Distribution by Random Vibration Method
## 3 Sperling Index Based on Vibratory Acceleration History and PSD  
### Zonal Distribution Deduction of Sperling Index Based on Vibratory Acceleration History  
The Sperling index, an indicator of stability, is a measurement method to determine the comfort of passengers and occupants on the rolling stock, as well as the status of transported goods. The evaluation is based on the measurement of the vehicle body's vibration acceleration. The less comfort the passengers experience, the higher the Sperling index will be [21]. To derive the Sperling index at a given location, the zonal acceleration should first be determined. In Figure 2, the vibration acceleration is linearly distributed in the space of the car body according to the principle of rigid dynamics. The vertical vibration acceleration at any zonal location in the horizontal plane of the car-body center can be determined by considering the horizontal coordinate (\(x\),\(y\)), vertical, roll, and pitch acceleration of the car-body center. In this case, the coordinate origin was at the car-body center. Figure 2: Linear zonal relation model of car-body motion between the center and a certain point. (**a**) front view of car-body motion of the vertical levitation and the pitch; (**b**) lateral view of car-body motion of the vertical levitation and the roll; (**c**) top view of car-body motion of the lateral sway and the yaw. Therefore, the zonal distribution of the vertical acceleration can be derived using Equation (19). In the case of horizontal acceleration, the horizontal coordinate (\(x\),\(y\)), lateral sway, and yaw acceleration of the car-body center are considered. Thus, the zonal distribution of the horizontal acceleration can be expressed as Equation (20). \[\ddot{z}_{ci}(x,y,t)=\ddot{z}_{ci}(0,0,t)+x\ddot{\varphi}_{ci}(0,0,t)+y\ddot{ \theta}_{ci}(0,0,t) \tag{19}\]  
\[\ddot{y}_{ci}(x,y,t)=\ddot{y}_{ci}(0,0,t)+x\ddot{\psi}_{ci}(0,0,t) \tag{20}\]  
Secondly, the corresponding acceleration amplitude spectrum within the prescribed interval of frequency must be calculated using fast Fourier transform (FFT). In this case, the single-frequency component for the Sperling index can be defined as Equation (21), while the 10 th power root of the sum of the 10 th power is described in Equation (22) for the Sperling index of multiple-frequency component vibrations for the car body of vehicle element \(i\). \[\mathbf{W}_{ki}(x,y)=3.57\sqrt[10]{\frac{\mathbf{A}_{ki}^{3}}{f_{k}}\circ\mathbf{F}(f_{k})} =3.57\sqrt[10]{\frac{\left|\text{FFT}_{[\ddot{y}_{ci}\,\ddot{z}_{ci}]}(f_{k}) \right|^{3}}{f_{k}}\circ\left[\mathbf{F}_{\text{v}}(f_{k})\quad\mathbf{F}_{\text{h}}( f_{k})\right]} \tag{21}\]  
\[\mathbf{W}_{i}=\sqrt[10]{\sum_{k=1}^{N}\mathbf{W}_{ki}^{10}} \tag{22}\]  
where \(\mathbf{A}_{k}\) (unit: m/s\({}^{2}\)) represents the amplitude vector of the vibration acceleration at the frequency point \(f_{k}\) (unit: Hz) after Fourier transform, \(\mathbf{F}(f_{k})\) represents the frequency correction coefficient vector whose vertical and horizontal elements are sectionally expressed in Equations (23) and (24), and the number of frequency \(N\) is strongly related to the interval limit of frequency and the measurement duration. In accordance with the code GB/T5599-2019, the standard measurement duration is 5 s, the reciprocal of which is the bandwidth frequency d\(f\) of the fast Fourier transform. \[\mathbf{F}_{\text{v}}(f_{k})=\left\{ \begin{array}{ccc}0.325f_{k}^{2}&(f_{k}=0.5 \,\sim 5.9\text{Hz})\\ 400/f_{k}^{2}&(f_{k}=5.9\,\sim 20\text{Hz})\\ 1&(f_{k}=20\,\sim 40\text{Hz})\end{array} \right. \tag{23}\]  
\[\mathbf{F}_{\text{h}}(f_{k})=\left\{ \begin{array}{ccc}0.8f_{k}^{2}&(f_{k}=0.5 \,\sim 5.4\text{Hz})\\ 650/f_{k}^{2}&(f_{k}=5.4\,\sim 26\text{Hz})\\ 1&(f_{k}=26\,\sim 40\text{Hz})\end{array} \right. \tag{24}\]  
In general, the derivation of the zonal distribution of the Sperling index based on the course of the vibration acceleration is random, as the phase angles of the course of the track irregularities generated in each MCM vary.