## 2 Synchronized Assessment Method  
### Forward Problem  
VDM is a quick reanalysis method [27]. When the response of the original structure is given, the VDM allows the response of the modified structure to be quickly calculated without a time-consuming full structural simulation. Instead, this is conducted by adding the response caused by this modification-related virtual distortion. Theoretically, the modification in a finite element is represented by the equivalent pseudo-loads that are applied in its DOFs. In this way, a stiffness-related modification, such as structural damage, can be performed in the form of equivalent virtual forces or virtual distortions of the affected element. As a result, instead of considering a modified structure subjected to external excitation, the VDM considers the original structure subjected to certain virtual forces and the same external excitation, respectively, and both of these systems share the same dynamic responses. Assuming the damage as linear stiffness decreasing, we can define \(\zeta_{i}\) as the damage ratio of the \(i^{th}\) element:  
\[\zeta_{i}=\mathbf{k}_{i}^{d}/\mathbf{k}_{i}^{o} \tag{1}\]  
where \(\mathbf{k}_{i}^{o}\) and \(\mathbf{k}_{i}^{d}\) represent the stiffness matrixes of the \(i^{th}\) element before and after damage. Hence, using the nodal displacement vector of the \(i^{th}\) element \(v_{i}\), it can express the nodal force vectors before and after damage \(p_{i}^{o}\) and \(p_{i}^{d}\), respectively:  
\[p_{i}^{d}=\mathbf{k}_{i}^{d}v_{i}=\zeta_{i}\mathbf{k}_{i}^{o}v_{i}=\zeta_{i}p_{i}^{o} \tag{2}\]Therefore, the nodal force variation of the damaged and intact elements, namely the virtual force vector \(p_{i}^{e}\) yields:  
\[p_{i}^{e}=p_{i}^{o}-p_{i}^{d}=(1-\zeta_{i})\mathbf{k}_{i}^{o}v_{i}=(1-\zeta_{i})p_{i} ^{o} \tag{3}\]  
Assuming \(\phi_{ij}\) as the \(j^{th}\) basic distortion vector of the \(i^{th}\) element, there is \(p_{i}^{o}=\sum_{j}\mathbf{k}_{i}^{o}\phi_{ij}=\sum_{j}\lambda_{ij}^{o}\) according to the Eigen equation of \(\mathbf{k}_{i}^{o}\), where \(\lambda_{ij}^{o}\) is the basic nodal force vector of the intact element. Equation (3) is also performed to:  
\[p_{i}^{e}=\sum_{j}(1-\zeta_{i})\mathbf{k}_{i}^{o}\phi_{ij}=\sum_{j}\lambda_{ij}^{e} \tag{4}\]  
where \(\lambda_{ij}^{e}\) is the basic virtual force vector of the damaged element. This equation reveals the physical meaning of virtual force, which is determined by the basic distortion of the element. For example, a two-dimensional beam element has three situations of distortion: axial compression or tension, pure bending and bending, plus shearing. Accordingly, its virtual force contains axial force, shear force, and bending moment. Expanding Equation (4) to global coordinate, if the coordinate transformation matrix of the \(i^{th}\) element is named as \(\mathbf{T}_{i}\), the \(i^{th}\) virtual force vector \(p_{i}^{e}\) can be performed in the global coordinate as the product of \(p_{i}^{e}\) and the transpose of \(\mathbf{T}_{i}\). Hence the global virtual force vector containing multiple damages is obtained:  
\[P^{e}=\sum_{i}\mathbf{T}_{i}^{e}p_{i}^{e}=\sum_{i}\sum_{j}\mathbf{T}_{i}^{T}\lambda_{ij }^{e} \tag{5}\]  
Similarly, the global nodal force vectors of the damaged structure \(P^{d}\) and intact structures \(P^{o}\) are \(\sum_{i}\mathbf{T}_{i}^{T}p_{i}^{d}\) and \(\sum_{i}\mathbf{T}_{i}^{T}p_{i}^{o}\), respectively. It is not difficult to find the relationship between \(P^{d}\), \(P^{o}\) and \(P^{e}\):  
\[P^{d}=P^{o}-P^{e} \tag{6}\]  
Therefore, the motion equation of a damaged bridge subjected to a moving force is  
\[\mathbf{M}\ddot{V}+\mathbf{C}\dot{V}+\mathbf{K}^{d}V=F \tag{7}\]  
where \(\mathbf{M},\ \mathbf{C},\ \text{and}\ \mathbf{K}^{d}\) are the global mass, damping, and stiffness matrices, \(F\) is the excitation force vector, \(V,\dot{V},\ \text{and}\ \ddot{V}\) are the global displacement, velocity, and acceleration vectors, respectively. It is worth noting that the term \(K^{d}V\) actually presents the global nodal forces \(P^{d}\).