## 3 Governing Equations
### Inclusion of the Passive Shunt Circuit  
In the event that a \(R-L\) circuit is connected to the piezoelectric patches (as depicted in Figure 1), the total output charge is equivalent to the individual output charge produced by each piezoelectric layer, namely \(Q_{1}=Q_{2}=q\). Similarly, the total output voltage is the aggregate of each output voltage, i.e., \(v=v_{1}+v_{2}\). Additionally, as the two piezoelectric layers are identical in terms of material and dimensions and are poled in opposite directions, \(C_{p_{1}}=C_{p_{2}}=C_{p}\). Summing up the Equations (21) and (22), we obtain  
\[C_{p}v-2q-\left(\mathbf{\Theta_{1}}^{T}+\mathbf{\Theta_{2}}^{T}\right)\mathbf{d}=0 \tag{29}\]  
By employing [PERSON]'s voltage law, the electrical dynamics is represented by a second-order equation:  
\[v=-R\dot{q}-L\ddot{q} \tag{30}\]  
Solving the Equations (21) and (22) with respect to \(v_{1}\) and \(v_{2}\), respectively, substituting into (20) and using (29) and (30) we obtain the governing equations of the shunted piezoelectric damping structural system:  
\[M\ddot{\mathbf{d}}+\mathbf{K_{o}}\mathbf{d}+C_{p}^{-1}(\mathbf{\Theta_{1}}+\bm {\Theta_{2}})q =\mathbf{F}_{m} \tag{31}\] \[LC_{p}\ddot{q}+RC_{p}\dot{q}+2q+\left(\mathbf{\Theta_{1}}^{T}+\mathbf{ \Theta_{2}}^{T}\right)\mathbf{d} =0 \tag{32}\]  
In order to derive the frequency response equation of the above system, a harmonic input \(\mathbf{F}_{m}=\mathbf{F}_{0}e^{j\omega t}\) can be assumed. Therefore, solutions of the mechanical displacement and electric charge responses can be formulated as \(\mathbf{d}=\mathbf{d}_{0}e^{j\omega t}\) and \(q=q_{0}e^{j\omega t}\), where \(\mathbf{d}_{0}\) and \(q_{0}\) are the amplitudes of the displacement and the charge, respectively, and is the driving frequency. Substituting the assumed solutions in the system Equations (31) and (32) and doing some mathematical manipulations, the frequency response equation is obtained as:  
\[\mathbf{H}^{sh}(\omega)\mathbf{d}_{0}=\mathbf{f}_{0} \tag{33}\]  
where the analytical frequency response matrix is given by  
\[\mathbf{H}^{sh}(\omega)=\left[-\omega^{2}\mathbf{M}+\mathbf{K}_{o}-\frac{1}{C_{p}(-\omega ^{2}LC_{p}+j\omega RC_{p}+2)}(\mathbf{\Theta_{1}}+\mathbf{\Theta_{2}})\Big{(}\mathbf{ \Theta_{1}}^{T}+\mathbf{\Theta_{2}}^{T}\Big{)}\right]^{-1} \tag{34}\]  
Open- and closed-loop behaviour of the laminated GPLFR beams is studied by the frequency response function (FRF) for the transverse displacement at the free end of the laminated GPLFR beam. Since the study focuses on the passive vibration damping of the second mode, a range of frequencies only around the second mode of vibrations is used. Thus, the performance of the shunt damping control for the laminated GPLFR beam can be evaluated as the percentage reduction between the maximum amplitudes of the FRF (resonance peak) for short-circuit and shunt-circuit conditions as  
\[PI=\frac{max_{\omega}|H^{o}_{il}(\omega)|-max_{\omega}|H^{sh}_{il}(\omega)|}{max _{\omega}|H^{o}_{il}(\omega)|} \tag{35}\]  
where \(H^{o}_{il}(\omega)\) and \(H^{sh}_{il}(\omega)\) denote, respectively, the frequency response of the open-loop (short circuit) and closed-loop (shunt circuit) system at the tip (_i_ node) for excitation at the same point (_i_ node).