## 3 Mode Pairing Using the Energy-Based Modal Assurance Criterion (EMAC)
### Real Modes  
In the case of real vibration modes, the relative modal strain energy uses the physical information of the stiffness matrix. Its calculation involves rearranging the modal vectors and dividing the stiffness matrix into submatrices that relate the different clusters [12].  
Assuming that the modal matrix (\(\mathbf{\Phi}\)) is normalized in relation to the mass matrix, the modal stiffness matrix, which constitutes the orthogonality condition in relation to the stiffness matrix, is as follows:  
\[\mathbf{\Phi}^{T}\mathbf{K}\mathbf{\Phi}=\begin{bmatrix}\ddots&&\\ &\omega_{j}^{2}&\\ &&\ddots\end{bmatrix} \tag{6}\]  
in which \(\omega_{j}\) corresponds to the angular natural frequency of mode \(j\). The total strain energy (Modal Strain Energy) associated with each vibration mode \(j\) (MSE\({}_{j}\)) is equal to \(1/2\times\omega_{j}^{2}\).  
The vector that contains the modal information of the numerical mode \(j\) can be rearranged by separating the degrees of freedom of the numerical model into \(n\) clusters:  
\[\mathbf{\Phi}_{j}^{T}=\begin{bmatrix}\mathbf{\Phi}_{j1}^{T}\mathbf{\Phi}_{j2 }^{T}\cdots\mathbf{\Phi}_{jn}^{T}\end{bmatrix}^{T} \tag{7}\]  
In turn, the stiffness matrix is also divided into submatrices (\(\mathbf{K}_{kl}\)) that relate the degrees of freedom of clusters \(k\) and \(l\), that is:  
\[\mathbf{K}=\begin{bmatrix}\mathbf{K}_{11}&\mathbf{K}_{12}&\cdots&\mathbf{K}_ {1n}\\ \mathbf{K}_{21}&\mathbf{K}_{22}&\cdots&\mathbf{K}_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{K}_{n1}&\mathbf{K}_{n2}&\cdots&\mathbf{K}_{nn}\end{bmatrix} \tag{8}\]  
where \(k\) and \(l\) take values equal to \(1,\ 2,\ \ldots,\ n\), and \(n\) is the total number of clusters.  
The modal strain energy of vibration mode \(j\) with respect to cluster \(k\) (MSE\({}_{jk}\)) is calculated based on the following expression:  
\[\text{MSE}_{jk}=\frac{1}{2}\sum_{l=1}^{n}\mathbf{\Phi}_{jk}^{T}\mathbf{K}_{kl }\mathbf{\Phi}_{jl} \tag{9}\]  
where \(\mathbf{\Phi}_{jk}\) is the matrix that contains the modal information of the numerical mode \(j\), corresponding to the degrees of freedom of cluster \(k\); \(\mathbf{K}_{kl}\) is the stiffness submatrix that relates the degrees of freedom of clusters \(k\) and \(l\); and \(\mathbf{\Phi}_{jl}\) is the matrix that contains the modal information of numerical mode \(j\), corresponding to the degrees of freedom of cluster \(l\).  
The total strain energy of vibration mode \(j\) is given by:  
\[\text{MSE}_{j}=\frac{1}{2}\sum_{k=1}^{n}\sum_{l=1}^{n}\mathbf{\[PERSON]}_{jk}^{T} \mathbf{K}_{kl}\mathbf{\[PERSON]}_{jl}=\frac{1}{2}\mathbf{\[PERSON]}_{j}^{T}\mathbf{K} \mathbf{\[PERSON]}_{j}=\frac{1}{2}\omega_{j}^{2} \tag{10}\]The relative strain energy (\(\prod_{jk}\)) represents the portion of the total energy mobilized by the vibration mode \(j\) considering only the degrees of freedom of cluster \(k\). It can be calculated based on Equations (9) and (10), that is:  
\[\prod_{jk}=\frac{\text{MSE}_{jk}}{\text{MSE}_{j}}=\frac{\sum_{l=1}^{n}\mathbf{\Phi}_{ jk}^{T}\mathbf{K}_{kl}\mathbf{\Phi}_{jl}}{\mathbf{\Phi}_{j}^{T}\mathbf{K}\mathbf{\Phi}_{j}} \tag{11}\]  
with \(\text{MSE}_{j}\
eq 0\). This parameter is a scalar that takes values in the range between 0 and 1. Finally, the EMAC\({}_{ijk}\) is calculated through Equation (5).