## 3 Independent Modal Resonant Shunt Circuit  
There is a two-step frequency-selecting mechanism of energy in independent modal resonant shunt circuit method. For each mode, first, a piezoelectric transducer is allocated to convert the mechanical energy of this mode into electric energy. Then a resonant shunt is connected to this transducer to dissipate the electric energy of this mode. Due to the fine frequency-selecting property of the _RLC_ resonant circuit, the second step plays a major role. Because of this frequency-selecting process, when the frequencies of different modes are well separated, each shunt is almost independent. Moreover, because only a small part of the rods is replaced by transducers, the dynamical characteristics of the structure are almost invariant. Thus, adding, changing or removing a shunt does not affect the effects of other shunts. The mechanism of this method is illustrated in Figure 5.  
Figure 5: Energy dissipation mechanism of mode \(i\).  
In order to enhance the performance of the shunt circuit, first, the location of the modal transducer must be optimized to convert the energy of this mode into electric energy as more as possible. Truncate Equation (11) using only the \(i\)th mechanical mode (mode of Kagome structure without shunt circuits), the following system can be obtained:  
\[\begin{array}{l}\ddot{x}_{si}+\omega_{oi}x_{si}-\frac{nd_{3}k_{a}}{C(1-k^{2} )}\lambda_{i}q=0\\ L\ddot{q}+R\dot{q}+\frac{1}{C(1-k^{2})}q+\frac{nd_{3}k_{a}}{C(1-k^{2})}\lambda _{i}x_{si}=0\end{array} \tag{13}\]  
where \(x_{si}\) is the coordinate of the \(i\)th mode, \(\omega_{oi}\) is open-electrode frequency of the \(i\)th mode with open-electrode transducer stiffness \(K_{\text{ao}}\):  
\[K_{\text{ao}}=\frac{K_{\text{a}}}{1-k^{2}} \tag{14}\]  
and \(\lambda_{i}\) is defined as:  
\[\lambda_{i}=\{b\}_{i}\{\varphi\}_{i}^{T} \tag{15}\]  
where \(\{\varphi\}_{i}\) represents modal shape of the \(i\)th mode of the mechanical structure. Obviously, \(\lambda_{i}\) reflects the coupling strength of piezoelectric transducer with the \(i\)th mode. Thus, for mode \(i\), the transducer must be located to replace the rod with maximal \(\lambda_{i}\).  
Because the mathematical model of a resonant shunt is similar to a tuned mass damper, the key problem is tuning the values of inductance and resistance in every shunt to achieve optimal performance. There have been several methods to optimize the \(RL\) value of the resonant shunt. [PERSON] [21] provided the optimal \(RL\) value based on root locus plot while [PERSON] [22] used pole placement and transfer function criterion. But these tuning methods were based on one degree-of-freedom (DOF) systems. In [15] it is shown that when the number of mechanical DOF is far greater than the electrical DOF, the optimal \(RL\) value based on single-mode truncation can lose effectiveness, so a full-order model must be used in the optimization process. In the independent modal resonant shunt circuit method, supposing there are \(n\) modes to control, owing to the independence between the shunts, the \(2n\)-dimensional optimization problem can be reduced to \(2n\) one-dimensional problems while each can be resolved by a simple heuristic search method summarized below:  
1. Choose initial values of \(RL\) for the shunt circuit of the first mode to control. The optimal \(RL\) values obtained using one DOF system in [21] or [22] can be set as initial values. Add electrical DOF to the finite element model of the structure using Equation (12) to generate \([M]\), \([D]\) and \([K]\).
2. Calculate the eigenvalues of the structure coupled with the first shunt by solving complex eigenvalue problem of the full-order model. Through changing \(L\) value heuristically, make the frequency of the electrical mode close to its corresponding mechanical mode.
3. Make the damping ratio of the first mode as high as possible through changing \(R\) heuristically.
4. Move to the shunt of the next mode.