## 2 Materials and Methods
\begin{pmatrix}\mathbf{K}&\mathbf{P}_{e}&\mathbf{P}_{n}\\ \mathbf{P}_{\mathrm{T}}^{\mathrm{T}}&\mathbf{E}_{e}&\mathbf{P}_{\mathrm{en}} \\ \mathbf{P}_{\mathrm{n}}^{\mathrm{T}}&\mathbf{P}_{\mathrm{en}}^{\mathrm{T}}& \mathbf{E}_{\mathrm{n}}\end{pmatrix}\begin{pmatrix}\mathrm{U}\\ \Phi_{e}\\ \Phi_{n}\end{pmatrix} = \begin{pmatrix}\mathrm{F}\\ \mathrm{Q}_{n}\end{pmatrix} , \tag{12}\]  
where \(\mathrm{Q}_{n}\) is the vector of electric charges in the nodes belonging to the ungrounded electrode, and \(\Phi_{e}\) is the vector of electric potentials in the nodes outside the electrode. By eliminating \(\Phi_{e}\) from the second equation of the system (12) we obtain  
\[ \begin{pmatrix}\mathbf{M}&0\\ 0&0\end{pmatrix}\begin{pmatrix}\mathrm{U}\\ 0\end{pmatrix} + \begin{pmatrix}\overline{\mathbf{K}}&\overline{\mathbf{P}}\\ \overline{\mathbf{P}}^{\mathrm{T}}&\overline{\mathbf{E}}\end{pmatrix}\begin{pmatrix} \mathrm{U}\\ \Phi_{n}\end{pmatrix} = \begin{pmatrix}\mathrm{F}\\ \mathrm{Q}_{n}\end{pmatrix} , \tag{13}\]where  
\[\overline{\mathbf{K}} = \mathbf{K}-\mathbf{P}_{\mathrm{e}}\mathbf{E}_{\mathrm{e}}^{-1} \mathbf{P}_{\mathrm{e}}^{\mathrm{T}}, \tag{14}\] \[\overline{\mathbf{E}} = \mathbf{E}_{\mathrm{n}}-\mathbf{P}_{\mathrm{en}}^{\mathrm{T}} \mathbf{E}_{\mathrm{e}}^{-1}\mathbf{P}_{\mathrm{en}},\] (15) \[\overline{\mathbf{P}} = \mathbf{P}_{\mathrm{n}}-\mathbf{P}_{\mathrm{e}}\mathbf{E}_{ \mathrm{e}}^{-1}\mathbf{P}_{\mathrm{en}}. \tag{16}\]  
At resonance, we assume [16, 17], that the electrical impedance takes the minimum value and the electrodes are short circuited (that is, both electrodes have the same electrical potential), i.e., \(\Phi_{n}=0\). In resonance we can regard \(\mathrm{F}=0\), the system (13) can be modified to the form  
\[ \begin{pmatrix}\mathbf{M}&0\\ 0&0\end{pmatrix}\begin{pmatrix}\dot{\mathrm{U}}\\ 0\end{pmatrix} + \begin{pmatrix}\overline{\mathbf{K}}&\overline{\mathbf{P}}\\ \overline{\mathbf{P}}^{\mathrm{T}}&\overline{\mathbf{E}}\end{pmatrix} \begin{pmatrix}\mathrm{U}\\ 0\end{pmatrix} = \begin{pmatrix}0\\ \mathrm{Q}_{\mathrm{n}}\end{pmatrix} . \tag{17}\]  
The resonance frequencies are then obtained as solutions of a generalized eigenvalue problem in the form  
\[\big{(}\overline{\mathbf{K}}-\lambda\mathbf{M}\big{)}\mathrm{U}=0. \tag{18}\]  
When calculating the antiresonance frequency, we use the system of Equation (13) again. In accordance with [18, 19], we consider disconnected electrodes, thus the electric charge on the ungrounded electrode meets the condition  
\[\mathrm{Q}_{\mathrm{n}}=0. \tag{19}\]  
However, the part of the system of equations related to the grounded electrode was eliminated due to the condition (11), see the transition from (10) to (12). At the same time, \(\mathrm{F}=0\) will apply; the system (13) can be written as  
\[ \begin{pmatrix}\mathbf{M}&0\\ 0&0\end{pmatrix}\begin{pmatrix}\dot{\mathrm{U}}\\ 0\end{pmatrix} + \begin{pmatrix}\overline{\mathbf{K}}&\overline{\mathbf{P}}\\ \overline{\mathbf{P}}^{\mathrm{T}}&\overline{\mathbf{E}}\end{pmatrix} \begin{pmatrix}\mathrm{U}\\ \Phi_{n}\end{pmatrix} = \begin{pmatrix}0\\ 0\end{pmatrix} . \tag{20}\]  
From the second equation we express  
\[\Phi_{n}=\overline{\mathbf{E}}^{-1}\overline{\mathbf{P}}^{\mathrm{T}}\mathrm{ U}. \tag{21}\]  
We then obtain the antiresonance frequency as a solution to the generalized problem of eigenvalues in the form  
\[\big{(}\hat{\mathbf{K}}-\lambda\mathbf{M}\big{)}\mathrm{U}=0, \tag{22}\]  
where  
\[\hat{\mathbf{K}}=\overline{\mathbf{K}}-\overline{\mathbf{P}}\,\overline{ \mathbf{E}}^{-1}\overline{\mathbf{P}}^{\mathrm{T}}. \tag{23}\]  
Knowing the values of the resonance frequencies, calculated according to the relation (22), and the values of the antiresonance frequencies according to the relation (22), the values of the effective electromechanical coupling coefficients can be obtained by substituting in the relation (1).