## 2 Synchronized Assessment Method
### Inverse Problem  
Considering the bridge as a linear system, the dynamic response of the bridge can be generated from the Duhamel Integral if the initial condition is zero [28]. Using the discrete form of the Duhamel Integral, the response vector \(Y\) caused by the moving force and global virtual force is:  
\[Y=\mathbf{I_{\text{F}}}F+\mathbf{I_{e}}P^{e}=\mathbf{I}\left[ \begin{array}{c}F\\ P^{e}\end{array} \right] \tag{10}\]  
The first term refers to the response of the intact bridge subjected to the moving force, \(\mathbf{I_{\text{F}}}\) is the impulse response matrix of \(F\). The second term is the response of the intact bridge subjected to the global virtual force, \(\mathbf{I_{e}}\) is the impulse response matrix caused by \(P^{e}\). Substituting Equation (5), Equation (10) is elaborated as:  
\[\left[ \begin{array}{c}Y^{1}\\ \vdots\\ \vdots\\ Y^{n}\end{array} \right]=\left[ \begin{array}{cccc}\mathbf{I_{\text{F}}}&\mathbf{I_{ \text{11}}}&\dots&\mathbf{I_{ij}}^{1}\\ \mathbf{I_{\text{F}}^{2}}&\mathbf{I_{\text{11}}}&\dots&\mathbf{I_{ij}^{2}}\\ \dots&\dots&\dots&\dots\\ \mathbf{I_{\text{F}}^{n}}&\dots&\dots&\mathbf{I_{ij}^{n}}\end{array} \right]\left[ \begin{array}{c}F\\ \lambda_{11}^{e}\\ \vdots\\ \lambda_{ij}^{e}\end{array} \right] \tag{11}\]  
where \(Y=\left[ \begin{array}{cc}Y^{1}&\dots&Y^{n}\end{array} \right]^{T}\) is the response vector of the damaged structure, including \(n\) measured \(Y^{n}\). \(I\) is a partitioned matrix constituted by Toeplitz matrices \(\mathbf{I_{\text{F}}}\) and \(\mathbf{I_{ij}^{n}}\), which represent the impulse response of the \(n^{th}\) measure point of the intact structure subjected to the impulse force of \(F\) and \(\lambda_{ij}^{e}\), respectively. To guarantee the unique solution of this equation, \(n\geq i\times j+1\). Therefore, the unknowns can be calculated from the inversion of Equation (11). \[\left[ \begin{array}{c}F\\ \lambda_{11}^{e}\\ \vdots\\ \lambda_{ij}^{e}\end{array} \right]=\mathbf{I}^{-1}Y \tag{12}\]  
However, in practical cases \(\mathbf{I}\) is very sensitive to noise, the metering noise is likely to cause an ill-conditioned problem in Equation (12). In addition, the dimension of \(\mathbf{I}\) is huge when dealing high sampling rate or long-term measurement, which makes it difficult to be calculated. Introducing a truncated load shape function (TLSF) can significantly restrain the influences caused by the problems above. TLSF compares the time history of excitation force to the span of a 'finite element beam'. Consulting the concept of the shape function in the finite element method, the amplitude of the moving force \(F\) is regarded as the displacement of the beam, which can be calculated via the load shape function. \[F=\mathbf{N}\gamma_{F} \tag{13}\]  
where \(N\) is the load shape function matrix and \(\gamma_{F}\) is the fitting coefficient vector. Similarly, \(\lambda_{ij}^{e}\) could also be expressed as:  
\[\lambda_{ij}^{e}=\mathbf{N}\gamma_{ij} \tag{14}\]  
Substituting Equations (13) and (14) to Equation (12), the identification of an unknown moving force and virtual forces is turned into the identification of a fitting coefficient:  
\[Y=\mathbf{I}\left[ \begin{array}{ccc}\mathbf{N}&\dots&\mathbf{N}\\ \vdots&\ddots&\vdots\\ \mathbf{N}&\dots&\mathbf{N}\end{array} \right]\left[ \begin{array}{c}F\\ \lambda_{11}^{e}\\ \vdots\\ \lambda_{ij}^{e}\end{array} \right]=\left[ \begin{array}{cccc}\mathbf{R_{\text{F} }}&\mathbf{R_{\text{11}}}&\dots&\mathbf{R_{ij}^{1}}\\ \mathbf{R_{\text{F}}^{2}}&\mathbf{R_{\text{11}}^{2}}&\dots&\mathbf{R_{ij}^{2}}\\ \dots&\dots&\dots&\dots\\ \mathbf{R_{\text{F}}^{n}}&\dots&\dots&\mathbf{R_{ij}^{n}}\end{array} \right]\left[ \begin{array}{c}\gamma_{F}\\ \gamma_{11}\\ \vdots\\ \gamma_{ij}\end{array} \right]=R\Gamma \tag{15}\]  
The product of the impulse response matrix \(\mathbf{I}\) as well as the load shape function matrix \(\mathbf{N}\) could be collectively called influence matrix \(\mathbf{R}\). Its physical meaning is defined as the impulse response of intact structure caused by the shape function.