## 2 Theoretical Analysis
### Bridge Dynamics of a Simple VBI System
The displacement mode shape function is  
\[\overline{m}\ddot{u}+EIu^{mr}=p(x,t) \tag{2a}\]1  
which is the same for the low- and high-frequency responses. Assuming that three cross-sections were selected as \(x_{1}\), \(x_{2}\), and \(x_{3}\) and only the first two driving-force modes are considered, one can obtain the following matrix (Equation (7)) from Equation (5a)):\[ \begin{bmatrix}sin\frac{\pi x_{1}}{L}&sin\frac{2\pi x_{1}}{L}\\ sin\frac{\pi x_{2}}{L}&sin\frac{2\pi x_{2}}{L}\\ sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{3}}{L}\\ sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{3}}{L}\\ \end{bmatrix}\begin{bmatrix}\frac{\Delta_{it,1}}{1-S_{1}^{2}}sin\frac{\pi vt_{ 0}}{L}&\frac{\Delta_{it,1}}{1-S_{1}^{2}}sin\frac{\pi vt_{1}}{L}&\cdots&\frac{ \Delta_{it,1}}{1-S_{1}^{2}}sin\frac{\pi vt_{N}}{L}\\ \frac{\Delta_{it,2}}{1-S_{2}^{2}}sin\frac{2\pi vt_{0}}{L}&\frac{\Delta_{it,2} }{1-S_{2}^{2}}sin\frac{2\pi vt_{1}}{L}&\cdots&\frac{\Delta_{it,2}}{1-S_{2}^{2} }sin\frac{2\pi vt_{N}}{L}\\ \vdots&\vdots&\vdots&\vdots\\ \frac{\Delta_{it,it}}{1-S_{2}^{2}}sin\frac{\pi vt_{0}}{L}&\frac{\Delta_{it,2 }}{1-S_{2}^{2}}sin\frac{\pi vt_{1}}{L}&\cdots&\frac{\Delta_{it,2}}{1-S_{2}^{2 }}sin\frac{\pi vt_{N}}{L}\\ \end{bmatrix}  \tag{7}\] \[=\mathbf{\Phi}(\mathbf{x}).\mathbf{Y}(t)= \begin{bmatrix}u_{x_{1},t_{0} }&u_{x_{1},t_{1}}&\cdots&u_{x_{1},t_{N}}\\ u_{x_{2},t_{0}}&u_{x_{2},t_{1}}&\cdots&u_{x_{2},t_{N}}\\ u_{x_{3},t_{0}}&u_{x_{3},t_{1}}&\cdots&u_{x_{3},t_{N}}\\ \end{bmatrix} \] where \(t_{0}\), \(t_{1}\), \(\ldots t_{N}\) are the \(N+1\) time points. Further, two sub-equations (Equation (8a,b)) can be generated from Equation (7). \[ \begin{bmatrix}sin\frac{\pi x_{1}}{L}&sin\frac{2\pi x_{1}}{L}\\ sin\frac{\pi x_{2}}{L}&sin\frac{2\pi x_{2}}{L}\\ sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{3}}{L}\\ \end{bmatrix}\begin{bmatrix}\frac{\Delta_{it,1}}{1-S_{1}^{2}}sin\frac{\pi vt_ {0}}{L}&\frac{\Delta_{it,1}}{1-S_{1}^{2}}sin\frac{\pi vt_{1}}{L}&\cdots&\frac {\Delta_{it,1}}{1-S_{1}^{2}}sin\frac{\pi vt_{N}}{L}\\ \frac{\Delta_{it,2}}{1-S_{2}^{2}}sin\frac{2\pi vt_{1}}{L}&\frac{\Delta_{it,2} }{1-S_{2}^{2}}sin\frac{2\pi vt_{1}}{L}&\cdots&\frac{\Delta_{it,2}}{1-S_{2}^{2 }}sin\frac{2\pi vt_{N}}{L}\\ \end{bmatrix} = \begin{bmatrix}u_{x_{1},t_{0}}&u_{x_{1},t_{1}}&\cdots&u_{x_{1},t _{N}}\\ u_{x_{2},t_{0}}&u_{x_{2},t_{1}}&\cdots&u_{x_{2},t_{N}}\\ \end{bmatrix} \] (8a) \[ \begin{bmatrix}sin\frac{\pi x_{2}}{L}&sin\frac{2\pi x_{2}}{L}\\ sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{3}}{L}\\ \end{bmatrix}\begin{bmatrix}\frac{\Delta_{it,1}}{1-S_{1}^{2}}sin\frac{\pi vt_ {1}}{L}&\frac{\Delta_{it,1}}{1-S_{1}^{2}}sin\frac{\pi vt_{1}}{L}&\cdots&\frac {\Delta_{it,1}}{1-S_{1}^{2}}sin\frac{\pi vt_{N}}{L}\\ \frac{\Delta_{it,2}}{1-S_{2}^{2}}sin\frac{2\pi vt_{1}}{L}&\cdots&\frac{\Delta_{it, 2}}{1-S_{2}^{2}}sin\frac{2\pi vt_{N}}{L}\\ \end{bmatrix} = \begin{bmatrix}u_{x_{2},t_{0}}&u_{x_{2},t_{1}}&\cdots&u_{x_{2},t_{N} }\\ u_{x_{3},t_{0}}&u_{x_{3},t_{1}}&\cdots&u_{x_{3},t_{N}}\\ \end{bmatrix}  \tag{8b}\]  
The responses at \(x_{3}\) can then be derived from Equation (8a,b) as  
\[ \begin{bmatrix}u_{x_{2},t_{0}}&u_{x_{2},t_{1}}&\cdots&u_{x_{2},t_{N}}\\ u_{x_{3},t_{0}}&u_{x_{3},t_{1}}&\cdots&u_{x_{3},t_{N}}\\ \end{bmatrix} =T_{dis} \begin{bmatrix}u_{x_{1},t_{0}}&u_{x_{1},t_{1}}&\cdots&u_{x _{1},t_{N}}\\ u_{x_{2},t_{0}}&u_{x_{2},t_{1}}&\cdots&u_{x_{2},t_{N}}\\ \end{bmatrix}  \tag{9}\]  
where \(T_{dis}\) is the inter-section transfer matrix of low-frequency displacement from \(x_{1}\) and \(x_{2}\) to \(x_{3}\), and  
\[T_{dis}= \begin{bmatrix}sin\frac{\pi x_{2}}{L}&sin\frac{2\pi x_{2}}{L}\\ sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{3}}{L}\\ \end{bmatrix}\begin{bmatrix}sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{3}}{L}\\ sin\frac{\pi x_{2}}{L}&sin\frac{2\pi x_{3}}{L}\\ \end{bmatrix}