# 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 PSD  
To derive the zonal distribution of the Sperling index based on the PSD, the linearity of the PSD of the car-body acceleration must first be determined to form the auto-PSDs and cross-PSDs between the different DOFs. The linearity of the PSD can be derived from the linearity of the correlation function of random signals. In signal processing, cross-correlation is a measure of the similarity of two series as a function of the displacement of one series relative to the other, while autocorrelation is the cross-correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the time lag between them. Therefore, autocorrelation and cross-correlation can be defined as Equations (25) and (26), where \(\tau\) is the time lag, \(T\) is the period of the signal, and \(v\) and \(h\) are two signals [22]. \[R_{vv}(\tau)=\lim_{T\to\infty}\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}v(t)v(t+ \tau)dt \tag{25}\]  
\[R_{vh}(\tau)=\lim_{T\to\infty}\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}v(t)h (t+\tau)dt \tag{26}\]  
There exists a signal \(p\) that is equal to the linear additional relation as \(av+bh\), where \(a\) and \(b\) are two constants. Therefore, the auto-correlation of signal \(p\) can be deduced using Equation (27), from which it can be seen that the auto-correlation of the signal \(p\) is linked to the auto-correlation and cross-correlation of signals \(v\) and \(h\). \[ \begin{array}{l}R_{pp}(\tau)=R_{(av+bh)(av+bh)}(\tau)\\ =\lim_{T\to\infty}\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}a^{2}v(t)v(t+ \tau)+abv(t)h(t+\tau)+abh(t)v(t+\tau)+b^{2}h(t)h(t+\tau)dt\\ =a^{2}R_{vv}(\tau)+b^{2}R_{hh}(\tau)+ab[R_{hp}(\tau)+R_{vh}(\tau)]\end{array}  \tag{27}\]  
The Wiener-Khinchin theorem illustrates that the Fourier transform of auto-correlation of a signal is equal to its PSD [23]. Therefore, the PSD of signal \(p\) can be deduced by the inverse Fourier transform given in Equation (28), namely the linearity of PSD. \[S_{pp}(\tau)=\int_{-\infty}^{\infty}R_{pp}(\omega)e^{i\omega\tau}d\omega=a^{2 }S_{vv}(\tau)+b^{2}S_{hh}(\tau)+ab[S_{hp}(\tau)+S_{vh}(\tau)] \tag{28}\]  
Accordingly, the zonal relationship of the vertical and horizontal vibration acceleration between the center of the car body and a specific location on the same plane is given in Equations (19) and (20) in Section 3.1. Therefore, for the PSD of a vertical vibration acceleration at coordinate (\(x\),\(y\)), not only the auto-PSDs for the vertical levitation, roll, and pitch of the car-body center, but also the cross-PSDs for the vertical levitation, roll, and pitch of the car-body center are multiplied by the corresponding coordinate and superimposed shortly thereafter in Equation (29). Similarly, the PSD of the horizontal vibration acceleration at coordinate (\(x\),\(y\)) can be calculated using (30), which includes auto-PSDs for the lateral sway and yaw of the car-body center and cross-PSDs for the lateral sway and yaw of the car-body center. \[S_{\tilde{x}[i]}(\omega,x,y)=y^{2}S_{\tilde{\theta}_{\tilde{\alpha}}}+x^{2}S _{\tilde{\varphi}_{\tilde{\alpha}}}+x(S_{\tilde{x}_{\tilde{\varphi}_{\tilde{ \alpha}}}}+S_{\tilde{\varphi}_{\tilde{\varphi}_{\tilde{\alpha}}}\tilde{x}_{ \tilde{\gamma}}})+y(S_{\tilde{\theta}_{\tilde{\alpha}}\tilde{z}_{\tilde{ \alpha}}}+S_{\tilde{z}_{\tilde{\omega}}\tilde{\theta}_{\tilde{\alpha}}})+xy(S_ {\tilde{\theta}_{\tilde{\alpha}}\tilde{\varphi}_{\tilde{\alpha}}}+S_{\tilde{ \varphi}_{\tilde{\alpha}}\tilde{\theta}_{\tilde{\alpha}}}) \tag{29}\]  
\[S_{\tilde{y}[i]}(\omega,x,y)=x^{2}S_{\tilde{y}_{\tilde{\alpha}i}}+S_{\tilde{y} _{\tilde{\alpha}}}+x(S_{\tilde{y}_{\tilde{\alpha}}\tilde{\varphi}_{\tilde{ \alpha}}}+S_{\tilde{y}_{\tilde{\alpha}}\tilde{y}_{\tilde{\alpha}}}) \tag{30}\]  
The auto-PSDs and the cross-PSDs above can be respectively withdrawn from the main and the counter diagonal elements of the vibratory acceleration PSD matrix in Equation (18).