# New Optimal Design of Multimode Shunt-Damping Circuits for Enhanced Vibration Control
## 4 Conclusions  
This work provides a new design optimization approach for enhancing multimode vibration damping of composite beam structures. A modification of the \"current-flowing\" shunt circuit [15] for controlling multimode vibration in a piezoelectric laminate beam is used. The proposed method is based on considering the capacitor and inductor values of each shunting branch as new design variables along with the resistor values. The optimization is performed for the minimum \(H_{\infty}\) norm of the piezoelectric composite beam using a PSO algorithm. A two-mode shunt circuit is designed and simulated for a cantilever  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of** & **Best Fitness** & **Time (s)** & \
\begin{tabular}{c} **Total** \\ **Iterations** \\ \end{tabular} &
\begin{tabular}{c} **Function** \\ **Evaluations** \\ \end{tabular} \\ \hline
1 & \(-\)70.19 & 4277.75 & 112 & 11,300 \\
2 & \(-\)86.34 & 3897.51 & 105 & 10,600 \\
3 & \(-\)86.23 & 6653.82 & 230 & 23,100 \\
4 & \(-\)86.23 & 3428.34 & 132 & 13,300 \\
5 & \(-\)70.19 & 2256.21 & 90 & 9100 \\
6 & \(-\)86.18 & 8431.59 & 250 & 25,100 \\
7 & \(-\)86.21 & 5426.75 & 173 & 17,400 \\
8 & \(-\)86.23 & 8045.17 & 250 & 25,100 \\
9 & \(-\)70.19 & 3946.70 & 142 & 14,300 \\
10 & \(-\)86 & 7613.49 & 250 & 25,100 \\ \hline
**Avg Total** & \(\mathbf{-81.399}\) & \(\mathbf{5397.73}\) & \(\mathbf{173.4}\) & \(\mathbf{17/440}\) \\ \hline \hline \end{tabular}
\end{table}
Table 10: Fitness, computation time and iterations of PSO.  
Figure 13: Convergence of the best fitness for PSO and GA optimization techniques.  
piezo laminated beam. Two additional models with different combinations of variables are investigated; model 2 considers only the resistances of the shunt circuit as design variable, and model 3 considers both resistances and capacitances. Frequency response analysis is conducted by the finite-element method using super-convergent beam elements. The obtained optimal solutions are presented and discussed. The overall results show that the proposed approach of model 1 improves the multimode shunt damping and gives the highest peak amplitude reduction of 39.61 dB for the second mode and 55.92 dB for the third mode. Another benefit of the approach introduced in model 1 is that of the reduction of the required values for the inductors without any expense in the damping performance. Finally, a comparison of the proposed PSO method with the GA algorithm is performed, which demonstrates the capability of PSO to reach better values for this kind of problem and exert less computational effort to reach the \"optimum\".  
Conceptualization, K.M., G.K.T. and G.E.S.; methodology, G.A.F. and N.A.A.; software, K.M., G.K.T., G.A.F., N.A.A. and G.E.S.; validation, K.M. and G.K.T.; formal analysis, N.A.A.; investigation, K.M., G.A.F. and G.E.S.; resources K.M., G.A.F. and G.E.S.; data curation, K.M. and G.A.F.; writing--original draft preparation, G.A.F. and N.A.A.; writing--review and editing, K.M., N.A.A. and G.E.S.; visualization, G.A.F. and N.A.A.; supervision, G.E.S. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.