# Flow-Induced Vibration Analysis by Simulating a High-Speed Train Pantograph
## 4 Establishment of the Pantograph-Induced Vibration Model  
The motion differential equations for the pantograph structure can be expressed as follows:  
\[[M]\big{\{}\ddot{x}\big{\}}+[C]\big{\{}\dot{x}\big{\}}+[K]\big{\{}x\big{\}}=\{f\} \tag{7}\]  
where \([M]\) is the mass matrix, \([C]\) is the damping matrix, and \([K]\) is the stiffness matrix, all of which are square matrices of order n, where n is the degrees of freedom of the system. \(\{f\}\) represents the generalized external force vector, indicating the external forces acting on each degree of freedom. \(\{x\}\) is the generalized displacement matrix of the system, with each parameter representing the generalized displacement of the corresponding degree of freedom. \(\big{\{}\dot{x}\big{\}}\) and \(\big{\{}\ddot{x}\big{\}}\), respectively, denote the generalized velocity vector and generalized acceleration vector of the system. Figure 17: Diagram of the 1 st to 5 th modal shapes of the overall structure of the pantograph. (**a**) The first modal shape diagram of the pantograph without preload. (**b**) The first modal shape diagram of the pantograph with preload. (**c**) The second modal shape diagram of the pantograph without preload. (**d**) The second modal shape diagram of the pantograph with preload. (**e**) The third modal shape diagram of the pantograph without preload. (**f**) The third modal shape diagram of the pantograph with preload. (**g**) The fourth modal shape diagram of the pantograph without preload. (**h**) The fourth modal shape diagram of the pantograph without preload. (**i**) The fifth modal shape diagram of the pantograph without preload. (**j**) The fifth modal shape diagram of the pantograph without preload. A coordinate transformation exists for \(\{x\}\):  
\[\{x\}=[U]\times\{y\} \tag{8}\]  
where \([u]\) is the coordinate transformation matrix. Substituting Equation (8) into Equation (7) transforms the system of motion differential equations into modal coordinates:  
\[[M][U]\{\ddot{y}\}+[C][U]\{\dot{y}\}+[K][U]\{y\}=\{f\} \tag{9}\]  
Multiplying both sides of the equation by the transpose of the modal matrix \([U]^{T}\):  
\[[U]^{T}[M][U]\{\ddot{y}\}+[U]^{T}[C][U]\{\dot{y}\}+[U]^{T}[K][U]\{y\}=[U]^{T} \{f\} \tag{10}\]  
Let us define the generalized mass matrix, generalized damping matrix, generalized stiffness matrix, and generalized excitation as follows:  
\[ \begin{array}{c}[M_{n}]=[U]^{T}[M][U]\\ [C_{n}]=[U]^{T}[C][U]\\ [K_{n}]=[U]^{T}[K][U]\\ \{f_{p}\}=[U]^{T}\{f\}\end{array}  \tag{11}\]  
The system of motion differential equations can be expressed as below:  
\[[M_{n}]\{y\}+[C_{n}]\{\dot{y}\}+[K_{n}]\{y\}=\{f_{p}\} \tag{12}\]  
The system of differential equations obtained after modal coordinate transformation, Equation (12), can be decomposed into \(n\) individual differential equations:  
\[m_{i}\ddot{y}_{i}+c_{i}\dot{y}_{i}+k_{i}y_{i}=f_{pi},(i=1,2,3,\ldots,n) \tag{13}\]  
dividing both sides of the equation by \(m_{i}\), we have the following:  
\[ \begin{array}{c}k_{i}=m_{i}\omega_{mi}{}^{2}\\ c_{i}=2m_{i}\ddot{c}_{i}\omega_{ni}\end{array}  \tag{14}\]  
thus, Equation (13) can be transformed into the following:  
\[\ddot{y}_{l}+2\zeta_{i}\omega_{i}\dot{y}_{l}+\omega_{i}^{2}y_{i}=\frac{f_{pi} }{m_{i}},(i=1,2,3,\ldots,n) \tag{15}\]  
Equation (15) represents the differential equation of motion for the \(i\) mode. For a multi-degree-of-freedom system, the system of motion differential Equation (15) can be represented in matrix form as follows:  
\[\begin{array}{c}\left\{ \begin{array}{c}\dot{Y}\\ \ddot{Y}\end{array} \right\}=\left[ \begin{array}{cc}0&1\\ -\omega^{2}&-2\delta\omega^{2}\end{array} \right]\left\{ \begin{array}{c}\dot {Y}\\ \dot{Y}\end{array} \right\}+\left\{ \begin{array}{c}0\\ F\end{array} \right\} \tag{16}\]  
further, it can be expressed as below:  
\[\dot{X}(t)=AX(t)+U(t) \tag{17}\]where  
\[X(t)=\left\{ \begin{array}{c}Y\\ \dot{Y}\end{array} \right\} \tag{18}\] \[A=\left[ \begin{array}{cc}0&1\\ -\omega^{2}&-2\delta\omega\end{array} \right]\] \[U(t)=\left\{ \begin{array}{c}0\\ F\end{array}