# Experimental and Mathematical Study of Flexible-Rigid Rail Vehicle Riding Comfort and Safety
## 2 Rail Vehicle Mathematical Modelling
### Frequency Domain Analysis  
The linear system of equations that was discussed in the prior section may be rewritten in the form of a matrix as \(\dot{X}+\mathbf{C}\dot{X}+\mathbf{K}X=\mathbf{D}Z_{0}\). In addition, \(\mathbf{D}\) represents the stiffness matrix that is connected to the track excitation. Thus, the frequency response function matrix may be obtained using Equations (24)-(25).  
\[\begin{split}\mathbf{H}(\omega)=&\left(-\mathbf{M} \omega^{2}+j\omega\mathbf{C}+\mathbf{K}\right)^{-1}\\ &\quad\mathbf{H}_{d}(\omega)=\mathbf{H}(\omega)\mathbf{D}\end{split} \tag{24}\]  
Hence, the power spectral density (PSD) of the system's displacements may be calculated using Equation (25).  
\[S_{Oi}(\omega)=\sum_{r=1}^{8}\sum_{s=1}^{8}\mathbf{H}_{dir}^{s}(\omega) \mathbf{S}_{lrs}(\omega)\mathbf{H}_{dis}^{\text{T}}(\omega) \tag{25}\]  
where \(S_{I}(\omega)\) is the PSD matrix of the input excitation [19]. Equations (26)-(27) define the torsional irregularity (cross-level) and vertical irregularity of the track's center, respectively.  
\[\frac{Z_{r}+Z_{l}}{2}=Z_{v} \tag{26}\]  
\[\frac{Z_{r}-Z_{l}}{2}=Z_{c} \tag{27}\]  
The cross-correlation and autocorrelation of the aforementioned inputs may be derived under the following assumptions [3; 14; 18; 20; 21]:  
* Every kind of irregularity may be understood as a stationary random process that has no average.
* Gaussian distribution best describes the irregularity probability density.
* A range of wavelengths from 3 to 30 \(\upmu\)m is used to determine spectral density.
* There is no link at all among the many anomalies that have been found.  
By using autocorrelation, the system is able to calculate the autocorrelation and power spectral density (PSD) of the abnormalities perceived by each wheel, as shown in Equation (28):  
\[\begin{split} R_{i}(\tau)=E[Z_{i}(t)Z_{i}(t+\tau)]=E[Z_{v}(t)+Z_{ c}(t)Z_{v}(t+\tau)+Z_{c}(t+\tau)]=\\ E[Z_{v}(t)Z_{v}(t+\tau)]+E[Z_{v}(t)Z_{c}(t+\tau)]+E[Z_{c}(t)Z_{v}(t +\tau)]+E[Z_{c}(t)Z_{c}(t+\tau)]\end{split} \tag{28}\]  
As a result, Equation (29) will be used to define the autocorrelation of the abnormalities associated with each wheel.  
\[R_{i}(\tau)=R_{v}(\tau)+R_{c}(\tau) \tag{29}\]  
where \(R_{v}(\tau)\) and \(R_{c}(\tau)\) are the vertical and roll autocorrelations of track abnormalities, respectively. In a similar manner, the PSD of the irregularities associated with each wheel may be computed, as described by Equation (30):  
\[S_{i}(\omega)=S_{v}(\omega)+S_{c}(\omega) \tag{30}\]where \(S_{v}(\omega)\) and \(S_{c}(\omega)\) represent the spectral density function (SDF) of the track's roll and the vertical components of abnormalities, respectively. Moreover, cross-SDF and cross-correlation between the right and left wheels of the wheelset can be calculated as  
\[R_{RLi}(\tau) =R_{v}(\tau)-R_{c}(\tau) \tag{31}\] \[S_{RLi}(\omega) =S_{v}(\omega)-S_{c}(\omega)\]  
The input cross-SDF of the irregularity can be calculated by Equation (32):  
\[S_{ij}(\omega)=S_{i}(\omega)e^{-j\omega(x_{j}-x_{i})/v} \tag{32}\]  
The input cross-SDF is illustrated in Equation (33), when wheels are on different rails:  
\[S_{RLij}(\omega)=S_{RLi}(\omega)e^{-j\omega(x_{j}-x_{i})/v} \tag{33}\]  
Equation (34) is used to calculate the track irregularity for the roll and vertical components, as per RDDO guidelines:  
\[S_{v}(\Omega)=\frac{A_{v}\Omega_{0}^{2}\big{(}\Omega^{2}+\Omega_{1}^{2}\big{)} }{\Omega^{4}\big{(}\Omega^{2}+\Omega_{2}^{2}\big{)}} \tag{34}\] \[S_{c}(\Omega)=\frac{A_{c}\Omega_{0}^{2}}{\big{(}\Omega^{2}+ \Omega_{1}^{2}\big{)}\big{(}\Omega^{2}+\Omega_{2}^{2}\big{)}}\]