## 2 Basic Dynamic Model Updating by Pairing Correlated Modes  
The equation of free motion for a complex undamped structure, modeled by FE, is given by  
\[\mathbf{M}\ddot{\mathbf{x}}(t)+\mathbf{K}\mathbf{x}(t)=0 \tag{1}\]  
where \(\mathbf{M}\) and \(\mathbf{K}\) are, respectively, the mass and stiffness matrices. \(\mathbf{x}(t)\) and \(\ddot{\mathbf{x}}(t)\) are displacement and acceleration vectors, respectively. The generalized eigenvalue problem is  
\[(\mathbf{K}-\lambda\mathbf{M})\boldsymbol{\varphi}=0 \tag{2}\]  
where \(\lambda\) and \(\boldsymbol{\varphi}\) are the eigenvalues and eigenvectors, respectively. The solution to the eigenvalue problem of Equation (2) gives the natural frequencies and mode shapes. Then the frequency relative error (FRE) \(FRE_{ik}\) for the pair of the \(i^{\text{th}}\) analytical mode natural frequency \(f_{A,i}\) and the \(k^{\text{th}}\) experimental mode natural frequency \(f_{E,k}\) can be obtained by  
\[FRE_{ik}=\frac{f_{A,i}-f_{E,k}}{f_{E,k}}\times 100\% \tag{3}\]  
where \"\(A\)\" and \"\(E\)\" as subscripts represent analytical mode and experimental mode, respectively. The corresponding \(MAC_{ik}\) value for the pair of the \(i^{\text{th}}\) analytical mode shape \(\boldsymbol{\varphi}_{A,i}\) and the \(k^{\text{th}}\) experimental mode shape \(\boldsymbol{\varphi}_{E,k}\) can be calculated by  
\[MAC_{ik}=\frac{\left(\boldsymbol{\varphi}_{A,i}^{T}\boldsymbol{\varphi}_{E,k} \right)^{2}}{\left(\boldsymbol{\varphi}_{A,i}^{T}\boldsymbol{\varphi}_{A,i} \right)\left(\boldsymbol{\varphi}_{E,k}^{T}\boldsymbol{\varphi}_{E,k}\right)} \tag{4}\]  
The \(MAC_{ik}\) close to 1 indicates a good correlation between \(\boldsymbol{\varphi}_{A,i}\) and \(\boldsymbol{\varphi}_{E,k}\), while when it is close to 0 it indicates a poor correlation between them. The problem of FE model updating in structural dynamics can be solved as a constrained optimization problem:  
\[ \begin{array}{ll}&\min J(\theta)\\ \text{\emph{subject}}&\text{\emph{to}}&a_{i}\leq\theta_{i}\leq b_{i}\quad(i= 1,2,\cdots,N_{\theta})\end{array}  \tag{5}\]  
where \(\theta=[ \begin{array}{cccc}\theta_{1}&\theta_{2}&\cdots&\theta_{N_{\theta}} \end{array} ]^{T}\) is the vector of updating parameters, \(N_{\theta}\) is the number of updating parameters, and \(a_{i}\) and \(b_{i}\) are the lower and upper boundary of the \(i^{\text{th}}\) updating parameter \(\theta_{i}\). The objective function \(J(\theta)\) often used can be expressed generally as  
\[ \begin{array}{ll}J(\theta)&=\sum\limits_{k=1}^{N}\left[w_{f,k}\left(\frac{f_ {A,k^{\prime}}(\theta)-f_{E,k}}{f_{E,k}}\right)^{2}+w_{\varphi,k}(1-MAC_{k^{ \prime}k}(\theta))^{2}\right]\\ &=\sum\limits_{k=1}^{N}\left[w_{f,k}(FRE_{k^{\prime}k}(\theta))^{2}+w_{ \varphi,k}(1-MAC_{k^{\prime}k}(\theta))^{2}\right]\\ &=\sum\limits_{k=1}^{N}h_{k}(\theta)\end{array}  \tag{6}\]  
where \(N\) is the number of correlated modes. \(w_{f,k}\) and \(w_{\varphi,k}\) are the weighting factors for the \(k^{\text{th}}\) experimental natural frequency and the MAC value for the \(k^{\text{th}}\) experimental mode shape, respectively. \(k^{\prime}\) is the analytical mode number correlated with the \(k^{\text{th}}\) experimental mode. \(I_{k^{\prime}k}(\theta)\) presents the subitem of the objective function \(J(\theta)\) for the pair of the \(k^{\prime\text{th}}\) analytical mode and the \(k^{\text{th}}\) experimental mode. For convenience, the updating parameters and corresponding parentheses, i.e., \((\theta)\), are omitted in the following paragraph. Thus, the primary prerequisite for dynamic model updating is to screen out the pair of the \(k^{\prime\text{th}}\) analytical mode and the \(k^{\text{th}}\) experimental mode in Equation (6). The situations described in Section 1 will be encountered when calculating the value of subitem \(J_{k^{\prime}k}\). When case 1\(\leftrightarrow\)1 occurs only, each subitem value \(I_{k^{\prime}k}\) can be calculated uniquely because the one-to-one CMPs can be determined based on the given frequencies or/and MAC thresholds.