# Analysis of Train Car-Body Comfort Zonal Distribution by Random Vibration Method
## 2 Establishment and Solution of the Motion Equation
### Motion Equation of Train System Based on the PEM  
The PEM proposed by LIN helps to establish a series of input pseudo simple harmonic excitation by the given random excitation PSD in order to obtain the structural response PSD and the variance [20]. In the PEM, the stationary pseudo solution can be calculated through the multiplication of the frequency response function matrix (FRF). Based on the same assumptions as those presented in Section 2.1, using the D'Alembert principle, the motion equation of the train can be expressed as:  
\[\mathbf{\tilde{U}}_{\text{t}}(\omega,t)= \begin{bmatrix}\mathbf{u}_{\text{t}}^{\text{a} }&\mathbf{u}_{\text{t}}^{\text{v}}&\mathbf{u}_{\text{t}}^{\text{c}}\end{bmatrix} = \left(-\omega^{2}\mathbf{M}_{\text{t}}+j\omega\mathbf{C}_{\text{t}}+\mathbf{K}_{\text{t}} \right)^{-1}(\mathbf{T}_{\text{s}}+j\omega\mathbf{T}_{\text{d}})\mathbf{\tilde{U}}_{\text {w}}(\omega,t) \tag{13}\]  
where \(\mathbf{\tilde{U}}\)t with the overhead tilde \"\(\sim\)\" is the displacement vibratory response of the train to represent the variables in the pseudo form as the function of angular frequency \(\omega\) and time \(t\). In addition, in Equation (13), it is composed of three types of displacement response subvectors respectively controlled by three independent directional types of track irregularity PSDs, namely the alignment \(S_{\text{a}}\), the vertical \(S_{\text{v}}\), and the cross-level \(S_{\text{c}}\). This is because the PEM does not allow the linear superposition of different pseudo inputs defined by mutually independent PSDs, and only their PSDs obey the linear superposition principle. Meanwhile, the pseudo velocity \(\mathbf{\dot{U}}\)t and acceleration \(\mathbf{\tilde{U}}\)t vibratory response vectors can be expressed as Equations (14) and (15). The pseudo displacement of wheel sets can be defined in Equation (16) by converting the angular wave number domain to the angular frequency domain; it contains three types of pseudo track irregularity motions, similar to the definition of Equation (13):  
\[\mathbf{\tilde{U}}_{\text{t}}(\omega,t)=j\omega\mathbf{\tilde{U}}_{\text{t}}(\omega,t) \tag{14}\]  
\[\mathbf{\tilde{U}}_{\text{t}}(\omega,t)=-\omega^{2}\mathbf{\tilde{U}}_{\text{t}}( \omega,t) \tag{15}\]  
\[\mathbf{\tilde{U}}_{\text{w}}(\omega,t)=\Big{[} \begin{array}{cccc}\mathbf{\tilde{u}} _{\text{w}}^{\text{a}}&\mathbf{u}_{\text{w}}^{\text{v}}&\mathbf{\tilde{u}}_{\text{w}}^{ \text{c}}\end{array} \Big{]}=\Big{\{}\big{[} \begin{array}{cccc}\mathbf{I}_{3}& \mathbf{I}_{3}&\cdots&\mathbf{I}_{3}\end{array} \big{]}_{3\times 12n_{\text{v}}}\Big{\}}^{\text{T}} \circ\sqrt{\frac{\mathcal{S}_{\text{ir}}(n)}{V}}\circ\exp[jnV(t-\mathbf{t}_{\text{d}})] \tag{16}\]  
With the pseudo displacement vibratory response of the train in Equation (16), the PSD matrix of the displacement vibratory response can be calculated through the sum of the multiplication of the conjugate and the transpose of the corresponding pseudo vectors,as shown in Equation (17). Accordingly, the PSD matrix of acceleration vibratory response can be deduced through the multiplication of the PSD matrix of the displacement vibratory response and the fourth power of angular frequency \(\omega\). The exponential term with the variable of time \(t\) will be eliminated due to the multiplication of the conjugate and the transpose, which makes the PSD independent of time; namely, stationary. Accordingly, in Equation (18), the principal diagonal elements represents the auto-PSDs of the corresponding DOFs, while the remaining elements represent the cross-PSDs of two different DOFs. \[\mathcal{S}_{U}(\omega)=\vec{\boldsymbol{U}}_{w^{*}}\vec{\boldsymbol{U}}_{w^{ \mathrm{T}}}^{\mathrm{T}}=\vec{\boldsymbol{u}}_{u}^{*}\vec{\boldsymbol{u}}_{ u}^{\mathrm{T}}+\vec{\boldsymbol{u}}_{u}^{*}\vec{\boldsymbol{u}}_{u}^{*}+\vec{ \boldsymbol{u}}_{u}^{*}-\vec{\boldsymbol{u}}_{u}^{*}+\vec{\boldsymbol{u}}_{u}^{ *}-\vec{\boldsymbol{u}}_{u}^{*}\vec{\boldsymbol{u}}_{u}^{*} \tag{17}\]  
\[\mathcal{S}_{\vec{U}}(\omega)=\omega^{4}\mathcal{S}_{U}(\omega)=\omega^{4} \left[