## 3 Mode Pairing Using the Energy-Based Modal Assurance Criterion (EMAC)
### Complex Modes  
In cases where the damping matrix is not proportional to the mass and stiffness matrices, the so-called complex modes of vibration are present. These are characterized by complex numbers, i.e., they encompass both magnitude and phase information. In a complex vibration mode, the movements of the points occur with a time delay proportional to the phase difference, situated between \(0^{\circ}\) and \(180^{\circ}\). The representation of the deformed structure is usually performed with animations that present the values of the amplitudes of the various components of the modes at different instants of time [17]. In the case of complex vibration modes, the previously presented formulation cannot be directly applied since the orthogonality conditions with respect to the stiffness matrix (Equation (6)) are not met. Therefore, to obtain an expression for the modal strain energy it is necessary to resort to a formulation based on state-space equations. In a state-space formulation, the system of second-order differential equilibrium equations, with dimension \(z\), is transformed into a system of \(2z\) first-order differential equations. To this end, the state vector \(\mathbf{x}(t)\) is defined with \(z\) lines, which are constituted by the displacements and velocities of the \(z\) degrees of freedom of the structure:  
\[\mathbf{x}(t)= \begin{bmatrix}\mathbf{q}(t)\\ \dot{\mathbf{q}}(t)\end{bmatrix}  \tag{12}\]  
Based on the state-space formulation, the equation of motion (\(\mathbf{M}\ddot{\mathbf{q}}+\text{C}\dot{\mathbf{q}}+\mathbf{K}\mathbf{q}= \mathbf{p}(t)\)) can be rewritten as:  
\[\mathbf{P}\dot{\mathbf{x}}(t)+\mathbf{Q}\mathbf{x}(t)= \begin{cases}\mathbf{p} (t)\\ 0\end{cases}  \tag{13}\]  
\(\mathbf{P}\) and \(\mathbf{Q}\) are defined in Equations (14) and (15), respectively, in terms of the stiffness, damping and mass matrices [35; 36]:  
\[\mathbf{P}= \begin{bmatrix}\mathbf{C}&\mathbf{M}\\ \mathbf{M}&\mathbf{0}\end{bmatrix}  \tag{14}\]  
\[\mathbf{Q}= \begin{bmatrix}\mathbf{K}&\mathbf{0}\\ \mathbf{0}&-\mathbf{M}\end{bmatrix}  \tag{15}\]  
Assuming \(q(t)=\varphi_{j}e^{\lambda_{j}t}\) as a solution for homogeneous differential equations results in the following eigenvalue problem:  
\[\mathbf{Q}\cdot\mathbf{\Psi}=-\mathbf{P}\cdot\mathbf{\Psi}\cdot\mathbf{\Lambda}_{\mathbf{C}} \tag{16}\]  
The eigenvalues (\(\mathbf{\Lambda}\)) and the eigenvectors (\(\mathbf{\Psi}\)) can be related to matrices that contain the vibration modes (\(\mathbf{\Theta}\)) and \(\lambda_{j}\), which characterize the dynamic behavior of the structure through the following expressions:  
\[\mathbf{\Lambda}_{\mathbf{C}}= \begin{bmatrix}\mathbf{\Lambda}&\mathbf{0}\\ \mathbf{0}&\mathbf{\Lambda}^{*}\end{bmatrix}  \tag{17}\]\[\boldsymbol{\Psi}= \begin{bmatrix}\boldsymbol{\Theta}&\boldsymbol{\Theta}^{*}\\ \boldsymbol{\Theta}\boldsymbol{\Lambda}&\boldsymbol{\Theta}^{*}\boldsymbol{ \Lambda}^{*}\end{bmatrix}  \tag{18}\]  
where  
\[\boldsymbol{\Lambda}= \begin{bmatrix}\ddots&&\\ &\lambda_{j}&\\ &&\ddots\end{bmatrix} \qquad j=1,\ldots,n \tag{19}\]  
\[\boldsymbol{\Theta}= \begin{bmatrix}\ldots&\boldsymbol{\Phi}_{j}&\ldots\end{bmatrix}  \qquad j=1,\ldots,n \tag{20}\]  
in which \(\boldsymbol{\Phi}_{j}\) is a vector containing the modal ordinates of mode \(i\), and \(\lambda_{i}\) is the associated eigenvalue [35; 36]. Based on the previously presented formulation, the following orthogonality conditions can be derived:  
\[\boldsymbol{\Psi}^{\mathrm{T}}\mathbf{P}\boldsymbol{\Psi}= \begin{bmatrix} \ddots&&\\ &a_{j}&\\ &&\ddots\end{bmatrix}  \tag{21}\]  
\[\boldsymbol{\Psi}^{\mathrm{T}}\mathbf{Q}\boldsymbol{\Psi}= \begin{bmatrix} \ddots&&\\ &b_{j}&\\ &&\ddots\end{bmatrix}  \tag{22}\]  
in which \(a_{j}\) and \(b_{j}\) play a role similar to the modal mass and modal stiffness for undamped vibration systems but are usually complex numbers [36].