## 2 Dynamic Equations for Vehicle Group-Bridge Coupled Motion
In the case of multi-span continuous bridges or other intricate structures, such as suspension and cable-stayed bridges, the corresponding modes are highly complex, necessitating the use of the finite element method for obtaining exact mode shapes and frequencies. Conversely, for simply supported beams, the equation can be analytically solved by the following method. The \(n\)th-order mode shape of a simply supported beam is denoted by \(\phi_{n}=\sin\frac{n\pi\pi}{L}(t-\frac{l_{p}}{2c})\). By integrating it along the bridge length, the subitem integration result of \(p(x,t)\) is expressed as follows:  
\[p_{G}(x,t)=\int_{vt-l_{p}}^{vt}(\overline{\mu}_{1}+\overline{\mu}_{2})g\sin \frac{n\pi x}{L}dx=\frac{2(\overline{\mu}_{0}+\overline{\mu}_{1})gL}{n\pi}\sin \frac{n\pi l_{p}}{2L}\sin\frac{n\pi v}{L}(t-\frac{l_{p}}{2v}) \tag{5}\]  
\[p_{I}(x,t)=-\sum\limits_{i=1}^{\infty}\overline{\mu}_{0}\ddot{q}_{i}(t)\int_{vt -l_{p}}^{vt}\sin\frac{i\pi x}{L}\sin\frac{n\pi x}{L}dx=-\frac{\overline{\mu}_{ 0}L}{\pi}\sum\limits_{i=1}^{\infty}\ddot{q}_{i}(t)\Phi_{in} \tag{6}\]  
where \(\Phi_{in}=\frac{1}{t-n}\Big{[}\cos\frac{(i-n)\pi v}{L}(t-\frac{l_{p}}{2v}) \sin\frac{(i-n)\pi l_{p}}{2L}\Big{]}-\frac{1}{t+n}\Big{[}\cos\frac{(i+n)\pi v }{L}(t-\frac{l_{p}}{2v})\sin\frac{(i+n)\pi l_{p}}{2L}\Big{]}\). \[p_{S}(x,t)=\frac{2\overline{k}_{1}L(t)}{n\pi}\sin\frac{n\pi l_{p}}{2L}\sin \frac{n\pi v}{L}(t-\frac{l_{p}}{2v})-\frac{\overline{k}_{1}L}{\pi}\sum\limits _{i=1}^{\infty}q_{i}(t)\Phi_{in} \tag{7}\]  
\[p_{D}(x,t)=\frac{2\overline{c}_{1}L\dot{z}(t)}{n\pi}\sin\frac{n\pi l_{p}}{2L} \sin\frac{n\pi v}{L}(t-\frac{l_{p}}{2v})-\frac{\overline{c}_{1}L}{\pi}\sum \limits_{i=1}^{\infty}\dot{q}_{i}(t)\Phi_{in} \tag{8}\]  
Let us define \(\omega_{n}=n^{2}\pi^{2}\sqrt{EI/\overline{m}}/L^{2}\) as the \(n\)th-order circular frequencies of the structure and \(c_{n}=2\overline{m}_{n}\omega_{n}\) as the corresponding damping. By dividing both sides of Equation (2) by \(\overline{m}L/2\) and rearranging the equation, the dynamic equilibrium equation for the \(n\)th-order mode beam-slab structures under vehicle group excitation can be expressed in standard form as:  
\[ \begin{split}&[\ddot{q}_{n}(t)+\frac{2\overline{\mu}_{0}}{ \overline{m}\pi}\sum\limits_{i=1}^{\infty}\ddot{q}_{i}(t)\Phi_{in}]+\Big{[}2 \overline{c}_{n}\omega_{n}\dot{q}_{n}(t)+\frac{2\overline{c}_{1}}{\overline{m }\pi}\sum\limits_{i=1}^{\infty}\dot{q}_{i}(t)\Phi_{in}\Big{]}+\Big{[}\omega_{n }^{2}q_{n}(t)+\frac{2\overline{k}_{1}}{\overline{m}\pi}\sum\limits_{i=1}^{ \infty}q_{i}(t)\Phi_{in}\Big{]}\\ &-\frac{4}{n\overline{m}\pi}\sin\frac{n\pi l_{p}}{2L}\phi_{in} \Big{[}\overline{k}_{1}(t)+\overline{c}_{1}\dot{z}(t)\Big{]}=\frac{4( \overline{\mu}_{0}+\overline{\mu}_{1})g}{n\overline{m}\pi}\sin\frac{n\pi l_{p} }{2L}\phi_{n}\end{split}  \tag{9}\]  
For a simply supported girder bridge, if the generalized coordinates are taken as \(n\) items, the overall degree of freedom is reduced from an infinite number to \(n\). Consequently, the dynamic equation for the \(n\)th order system is transformed from the expression of Equation (1) to the following:  
\[\overline{\mu}_{1}\ddot{z}(t)+\overline{c}_{1}\dot{z}(t)+\overline{k}_{1}z(t )-\overline{c}_{1}\sum\limits_{i=1}^{n}\dot{q}_{i}(t)\cdot\phi_{i}(x)-\overline {k}_{1}\phi_{n}\sum\limits_{i=1}^{n}q_{i}(t)\cdot\phi_{i}(x)=0\]where the generalized displacement vector is denoted as \(U=[q_{1},\,q_{2},\,...,\,q_{n},\,z]^{T}\). The corresponding generalized mass matrix \(M\), damping matrix \(C\), stiffness matrix \(K\), and force vector \(F(\textbf{t})\) are given as Equation (12) to Equation (15), respectively:  
\[M= \begin{bmatrix}1+\rho_{M}\Phi_{11}&\rho_{M}\Phi_{12}&\cdots&\rho_{M}\Phi_{1n} &0\\ \rho_{M}\Phi_{21}&1+\rho_{M}\Phi_{22}&\cdots&\rho_{M}\Phi_{2n}&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ \rho_{M}\Phi_{n1}&\rho_{M}\Phi_{n2}&\cdots&1+\rho_{M}\Phi_{nn}&0\\ 0&0&\cdots&0&\overline{\mu}_{1}\end{bmatrix}  \tag{12}\]  
\[C=