# Damage Identification Method Using Additional Virtual Mass Based on Damage Sparsity
## 2 Damage Identification Method Based on Additional Virtual Mass and Damage Sparsity  
### Additional Virtual Mass Method  
Actual engineering structures are typically large, and there is a tendency for their sizes to increase. Therefore, the corresponding modal information is also numerous, so usually, we can only apply a few lower-order modal information. However, incomplete modal information leads to deviations when identifying the structural damage location and extent. The additional virtual mass method can conveniently and effectively expand the modal data used for structural damage identification in actual complex cases. The structure is divided into \(n\) substructures, and \(\mu_{l}\) is set as the damage factor of the lth substructure, which is the ratio of damaged substructural stiffness \(\widetilde{\mathbf{k}}_{l}\) to undamaged substructural stiffness \(\mathbf{k}_{l}\), that is, \(\widetilde{\mathbf{k}}_{l}=\mu_{l}\mathbf{k}_{l}\). The whole structure damage-factor is expressed as \(\mathbf{\mu}=\left[\mu_{1},\dots,\mu_{l},\dots,\mu_{n}\right]^{T}\), and the damage structural stiffness matrix is expressed as follows:  
\[\mathbf{k}(\mathbf{\mu})=\sum_{l=1}^{n}\mu_{l}\mathbf{k}_{l} \tag{1}\]  
A single-point excitation \(F_{i}(\omega)\) is applied to the \(i\)th substructure, and the acceleration response \(A_{i}(\omega)\) is measured at the same position and direction as \(F_{i}(\omega)\). Next, the frequency domain response function \(h_{ii}^{V}(\omega,m)\) at the \(i\)th substructure with additional virtual mass \(m\) is calculated using Equation (2). The total \(p\)-order frequency with additional virtual mass \(m\) on the \(i\)th substructure is identified based on the constructed frequency domain response \(h_{ii}^{V}(\omega,m)\). Let the \(j\)th order recognized frequency be \(\omega_{ji}(m)\), and \(j=1,2,\dots,p\):  
\[h_{ii}^{V}(\omega,m)=\frac{A_{i}(\omega)}{F_{i}(\omega)+mA_{i}(\omega)} \tag{2}\]  
\(\mathbf{\hat{\omega}}\) is the experimental frequency constructed using the additional virtual mass method based on the experimental modal information of the actual damaged structure. \[\left\{ \begin{array}{c}\hat{\omega}_{j}(m)=\left[\omega_{j1}(m),\dots, \omega_{ji}(m),\dots,\omega_{jn}(m)\right]\\ \hat{\omega}=\left[\hat{\omega}_{1}(m),\dots,\hat{\omega}_{j}(m),\dots,\hat{ \omega}_{p}(m)\right]^{T}\end{array} \right. \tag{3}\]  
\(\overline{\omega}(\mathbf{\mu})\) is the model frequency established using the additional virtual mass method based on the modal information of the structural finite element model when the damage-factor is \(\mathbf{\mu}\). \[\left\{ \begin{array}{c}\overline{\omega}_{j}(\mathbf{\mu},m)=\left[\omega_{j1} (\mathbf{\mu},m),\dots,\omega_{ji}(\mathbf{\mu},m),\dots,\omega_{jn}(\mathbf{\mu},m)\right] \\ \overline{\omega}(\mathbf{\mu})=\left[\overline{\omega}_{1}(\mathbf{\mu},m),\dots, \overline{\omega}_{j}(\mathbf{\mu},m),\dots,\overline{\omega}_{p}(\mathbf{\mu},m) \right]\end{array} \right. \tag{4}\]Considering the noise error in experimental measurement data, which means \(\hat{\mathbf{\omega}}=\overline{\omega}(\mathbf{\mu})+\mathbf{e}_{noise}\), the objective function \(g(\mathbf{\mu})\) is constructed, as expressed in Equation (5). The optimal value of \(\mathbf{\mu}\) is solved using the objective function as actual damage structural factors. \[\min g(\mathbf{\mu})=\left\|\hat{\mathbf{\omega}}-\overline{\mathbf{\omega}}(\mathbf{\mu}) \right\|_{2}^{2} \tag{5}\]  
The sensitivity of the frequency to the damage factors is introduced to improve optimization efficiency. When the structural damage-factor is \(\mathbf{\mu}\), and the additional virtual mass in the \(i\)th substructure is \(m\), the sensitivity \(R_{ji,l}\) of the \(j\)th order frequency \(\omega_{ji}(\mathbf{\mu},m)\) to the \(l\)th substructure is calculated as follows.