# New Optimal Design of Multimode Shunt-Damping Circuits for Enhanced Vibration Control
## 3 Results and Discussion
### Solution of Optimization Problem: Model 1  
The optimization problem in Equation (54) is addressed at this stage by considering the resistance values \(R_{i}\), the capacitances \(C_{i}\) and the inductance values of the shunt-damping sub-branch \(\widetilde{L}_{i}\) as the design variables. The inductance values \(\hat{L}_{i}\) of the current-flowing branch are given by Equation (31), while the inductances of the simplified shunt circuit in Figure 3b are obtained as \(L_{i}=\hat{L}_{i}+\widetilde{L}_{i}\). The problem is solved with the limit values given in Table 3. The optimal values of the electrical parameters of the simplified shunt circuit are given in Table 5.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Nodes**} & \multicolumn{3}{c}{**Short-Circuit Frequencies (Hz)**} & \multicolumn{3}{c}{**Open-Circuit Frequencies (Hz)**} \\ \cline{2-7}  & FE [24] & Exp. [24] & Present FE & FE [24] & Exp. [24] & Present FE \\ \hline
1 & 48.96 & 51.64 & 48.96 & 49.42 & 52.17 & 49.2 \\
2 & 337.1 & 337.0 & 336.9 & 340.7 & 340.2 & 338.68 \\
3 & 951.8 & 936.3 & 950.23 & 960.6 & 940.0 & 954.64 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The first three natural frequencies (Hz) of the piezoelectric composite beam.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **Minimal Value** & **Maximum Value** \\ \hline \(R_{2}\) (k\(\Omega\)) & 0 & 30 \\ \(R_{3}\) (k\(\Omega\)) & 0 & 30 \\ \(\tilde{C}_{2}\) (\(\mu\)F) & \(10^{-8}\) & \(1\) \\ \(C_{3}\) (\(\mu\)F) & \(10^{-8}\) & \(1\) \\ \(\widetilde{L}_{2}\) (H) & 1 & 20 \\ \(\widetilde{L}_{3}\)(H) & 1 & 20 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The limits of the design variables.  
To justify the \(H_{\infty}\) norm optimization technique, the magnitude frequency response of the tip displacement, before and after shunt damping, is presented in Figure 6. It is observed that the damped FRF exhibits two peaks of equal amplitude around each of the second and the third natural frequencies, which are comparatively low with respect to the uncontrolled structure (when the transducer is either short or open circuited). Indeed, the resonant amplitudes for the second and third modes are reduced by 39.61 dB and 55.92 dB, respectively.  
Figure 5: Beam tip FRF with RL shunt tuned on mode 2: undamped response (blue --), damped—present FE (red), damped FE of [PERSON] et al., 2009 [24] (black).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(R_{2}\) (k\(\Omega\)) & 1.40 \\ \(R_{3}\) (k\(\Omega\)) & 2.36 \\ \(C_{2}\) (μ) & 1 \\ \(C_{3}\) (μF) & 0.0083 \\ \(\widetilde{L}_{2}\) (\(H\)) & 7.43 \\ \(\widetilde{L}_{3}\) (\(H\)) & 1.68 \\ \(L_{2}\) (\(H\)) & 7.68 \\ \(L_{3}\) (\(H\)) & 5.06 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Shunt parameters for model 1.  
Figure 6: Simulated frequency response: undamped response (blue --), damped response—model 1 (red).  
Figure 7 shows the convergence of the particle swarm optimization algorithm for model 1. As can be seen from Figure 7, the process converges after generation 45, when the fitness function remains quasi constant. It takes about 1 h to carry out the 105 iterations. In addition, the algorithm stops before the maximum iteration is reached, since the relative changes in objective function are less than the tolerance (\(10^{-6}\)).