# 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 MCM
All of them can be diagonally assembled from \(\mathbf{M_{\mathrm{ve}}}\), \(\mathbf{C_{\mathrm{ve}}}\), \(\mathbf{K_{\mathrm{ve}}}\), \(\mathbf{T_{\mathrm{sve}}}\), and \(\mathbf{T_{\mathrm{dve}}}\) of the vehicle elements, as given in Equation (2), where \(\mathbf{u_{\mathrm{t}}}\) is the vibratory displacement vector containing the DOFs of the car body and the bogies in Table 1, while \(\mathbf{u}_{\mathrm{w}}\) is the vibratory displacement vector containing the DOFs of the wheel-sets in Table 1. \(\mathbf{K_{\mathrm{ve}}}\) in Equation (3) contains the auto-coupling stiffness matrices of the car body and bogies, namely \(\mathbf{K_{\mathrm{cc}}}\) and \(\mathbf{K_{\mathrm{bb}}}\) expressed in Equations (5) and (6), as well as the cross-coupling stiffness matrix \(\mathbf{K_{\mathrm{bc}}}(\eta)\), which represents the car-bogie interaction and differentiates the front and rear bogies using \(\eta=1\) and \(-1\), respectively. \(\mathbf{M_{\mathrm{ve}}}\) is diagonally assembled using the mass of different components and their moments of inertia around different axes. \(\mathbf{T_{\mathrm{sve}}}\) in Equation (8) is a stiffness projector matrix multiplied by the displacement vector \(\mathbf{u}_{\mathrm{w}}\) of wheel-track contact points to form the force vector of the train. The expression of \(\mathbf{C_{\mathrm{ve}}}\) and \(\mathbf{T_{\mathrm{sve}}}\), respectively similar to Equations (3) and (8), can be obtained simply by replacing the spring factor \(k\) with the corresponding damping factor \(c\). \[\mathbf{K_{\mathrm{t}}}=\mathrm{diag}\big{[}\mathbf{K_{\mathrm{ve}}}\quad\mathbf{K_{ \mathrm{ve}}}\quad\cdots\quad\mathbf{K_{\mathrm{ve}}}\big{]} \tag{2}\]  
\[\mathbf{K_{\mathrm{ve}}}= \begin{bmatrix}\mathbf{K_{\mathrm{cc}}}&\mathbf{K_{\mathrm{bc}}} (1)^{\mathrm{T}}&\mathbf{K_{\mathrm{bc}}}(-1)^{\mathrm{T}}\\ \mathbf{K_{\mathrm{bc}}}(1)&\mathbf{K_{\mathrm{bb}}}&0\\ \mathbf{K_{\mathrm{bc}}}(-1)&0&\mathbf{K_{\mathrm{bb}}}\end{bmatrix}  \tag{3}\]  
\[\mathbf{M_{\mathrm{ve}}}=\mathrm{diag}(m_{\mathrm{c}},m_{\mathrm{c}},I_{\mathrm{ scc}},I_{\mathrm{yc}},I_{\mathrm{zc}},m_{\mathrm{b}},m_{\mathrm{b}},I_{\mathrm{sb}},I_{ \mathrm{xb}},m_{\mathrm{b}},m_{\mathrm{b}},m_{\mathrm{b}},I_{\mathrm{xb}},I_{ \mathrm{xb}},I_{\mathrm{xb}},I_{\mathrm{xb}}  
sets in Equation (10). In addition, the symbol \"\(\circ\)\" in Equation (9) represents the Hadamard product, also known as the element-wise product between two matrices.  
\[\mathbf{u}_{\text{w}}(t)=\left[\sqrt{2}\sum_{k=1}^{N}\sqrt{\mathcal{S}_{\text{ir}}(n _{k})\text{d}n_{k}}\circ\cos(n_{k}v(t-t_{\text{d}})+\theta_{k})\right] \tag{9}\]  
\[\mathbf{t}_{\text{d}}=\left[t_{\text{d}}^{\text{ve}}(1)\quad\mathbf{t}_{\text{d}}^{ \text{ve}}(2)\quad\cdots\quad\mathbf{t}_{\text{d}}^{\text{ve}}(n_{\text{v}})\right] ^{\text{T}} \tag{10}\]  
\[\mathbf{t}_{\text{d}}^{\text{ve}}(i)=\mathbf{1}_{1\times 12}\text{diag}[(i-1)l_{\text{v}} \mathbf{I}_{3},\frac{2d_{1}+(i-1)l_{\text{v}}}{V}\mathbf{I}_{3},\frac{2d_{2}+(i-1)l_{ \text{v}}}{V}\mathbf{I}_{3},\frac{2(d_{1}+d_{2})+(i-1)l_{\text{v}}}{V}\mathbf{I}_{3}] \tag{11}\]  
\[\mathbf{S}_{\text{ir}}(n)=\Big{\{}\left[S_{\text{a}}(n)\quad S_{\text{v}}(n) \quad S_{\text{c}}(n)\right]\left[\mathbf{I}_{3}\quad\mathbf{I}_{3}\quad\cdots\quad \mathbf{I}_{3}\right]_{3\times 12n_{\text{v}}}\Big{\}}^{\text{T}} \tag{12}\]  
where \(n_{\text{v}}\) is the total number of vehicle elements; \(\mathbf{S}_{\text{ir}}\) is the given vertical track irregularity PSD matrix in Equation (12) composed of the alignment, vertical, and cross-level PSDs; \(n_{k}\) is an angular wave number sample; \(\text{d}n\) is the bandwidth; and \(\theta_{k}\) is a random phase angle obeying the uniform distribution U(0,2\(\pi\)). The integer order of \(k\) ranges from 1 to \(N\).