# Damage Identification Method Using Additional Virtual Mass Based on Damage Sparsity
## 3 Improved OMP Damage Identification Method Based on Sparsity
### Sensitivity Correlation Criterion  
The residual vector corresponding to each damage-factor variation was calculated using Equation (17) to form the residual matrix \(\varepsilon=(\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{n})\) and obtain the correlation coefficient between each residual vector and its corresponding sensitivity column vector:  
\[A_{i}=\frac{\mathbf{r}_{i}^{T}\varepsilon_{i}}{\|\mathbf{r}_{i}\|_{2}} \tag{21}\]  
\[A=[A_{1},\ldots,A_{i},\ldots,A_{n}]^{T}\]  
where \(\mathbf{r}_{i}\) is the ith column element of \(\mathbf{R}\). Each element of the correlation vector \(\mathbf{A}\) is sorted from the largest to the smallest, and the sparse degree of damage -factor variation is determined to be \(N\) by setting the threshold value \(p_{0}\). The \(n\)-\(N\) column vectors corresponding to the smaller correlation coefficient in the sensitivity matrix \(\mathbf{R}\) are eliminated to obtain \(\widetilde{\mathbf{R}}_{0,1}\). The residual vector \(\varepsilon_{0,1}\) corresponding to \(\widetilde{\mathbf{R}}_{0,1}\) is computed using Equation (17).  
\[p_{0}\leq\frac{\sum_{i=1}^{N}A_{i}}{\sum_{i=1}^{n}A_{i}} \tag{22}\]  
Let the residual vector corresponding to the remaining \(N\) damage factors form the residual matrix \(\widetilde{\varepsilon}_{0,1}\). The correlation vector \(\widetilde{A}_{0,1}\) is calculated and sorted to obtain the sensitivity matrix \(\widetilde{\mathbf{R}}_{0,2}\) and its residual vector \(\varepsilon_{0,2}\) by removing the column vector \(\mathbf{r}_{s}\) corresponding to the minimum correlation coefficient \(A_{j}\) from matrix \(\widetilde{\mathbf{R}}_{0,1}\). The final residual matrix \(\varepsilon_{0}=(\varepsilon_{0,1},\varepsilon_{0,2},\ldots\ldots,\varepsilon _{0,N})\) is determined by repeating the above step to determine the number and location of damage substructures using the principal component analysis method and obtain the specific values of the possible structural damage factors using objective function (5).  
The damage to structure mainly occurs in the local position, which exhibits a strong sparseness. The main principle of the principal component analysis method is to reflect most variables using a small amount of variable information, and the information contained in few variables is not repeated. This principle is consistent with the actual structural damage identification, in which a few damaged substructures, instead of all substructures, may be analyzed. Therefore, the principal component analysis method was used in this study to analyze the residual matrix and determine the number of damaged substructures. The specific steps are as follows:  
1. The mean value of each row of the residual matrix \(\varepsilon_{0}\) was determined, and all elements were subtracted from their rows mean value to form matrix \(\varepsilon_{0,m}\).
2. The covariance matrix \((\varepsilon_{0,m})^{T}\varepsilon_{0,m}\) of \(\varepsilon_{0,m}\) was calculated, and the eigenvalues of this covariance matrix were determined and arranged in descending order to form \(\mathbf{\Lambda}=(\lambda_{1},\lambda_{2},\ldots\ldots,\lambda_{N})\).  
3. The ratio, \(p=\frac{\sum_{i=1}^{i}\lambda_{j}}{\sum_{i=1}^{N}\lambda_{j}}\), of the first \(i\) substructures eigenvalues to all eigenvalues was calculated. When \(p\) reached a particular threshold, it was assumed that the first \(i\) substructures were damaged while the other parts of the structures were undamaged.  
By combining the additional virtual mass method and the IOMP method, the frequency vector \(\hat{\mathbf{\omega}}\) and sensitivity matrix \(\mathbf{R}\) of the virtual structure can be assembled to increase the amount of modal data for structural damage identification and to improve the accuracy. In addition, the IOMP method overcomes the disadvantage of non-sparse to achieve optimization results that satisfy the initial sparsity conditions consistent with actual engineering.