# New Optimal Design of Multimode Shunt-Damping Circuits for Enhanced Vibration Control
## 1 Introduction
Initial studies on passive vibration control through shunted piezoelectric transducers considered a simplified structural model with one degree of freedom [4, 10, 11]. In addition, [PERSON] and his coauthor [13, 15] considered analytical methods to model the dynamic of simple structural elements (beams, plates) with piezoelectric shunted transducers. These models were based on classical beam/plate theory and the modal analysis technique. Finite-element modeling, on the other hand, is regarded as the most efficient modeling technique for more complicated structures, particularly those of industrial interest, and has been used in a few research investigations [7, 23, 24]. Motivated by these observations, this work presents a new approach for the design a multimode shunt-damping circuit using a single piezoelectric element. A modification \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{**References**} & **Single-Mode** & **Multi-Mode Control** & **Single Piezo** & **Multiple Piezo** & **Other** & **Current-Flowing Approach** & **Other** \\  & **Control** & **Mode Control** & **Patch** & **Patches** & **PSO** & **Optimization Approach** & **Method** & **Shunt Method** \\ \hline [PERSON] (1979) [3] & ✓ & & & & & & & & ✓ \\ [PERSON] et al. (1991) [4] & ✓ & & & & & & & ✓ \\ [PERSON] (2003) [5] & ✓ & & & & & & & ✓ \\ [PERSON] et al. (2011) [6] & ✓ & & & & & & & ✓ \\ [PERSON] et al. (2019) & ✓ & & & & ✓ & & & ✓ \\ [PERSON] et al. (2022) [8] & ✓ & & & ✓ & & ✓ & & ✓ \\ [PERSON] et al. (2022) [9] & ✓ & & & ✓ & & & ✓ & \\ [PERSON] (1994) [10] & & ✓ & ✓ & & & ✓ & & ✓ \\ [PERSON] (1998) [11] & ✓ & & ✓ & & & & ✓ & ✓ \\ [PERSON] et al. (2000) [12] & ✓ & & & & & ✓ & ✓ & ✓ \\ [PERSON] et al. (2002) [13, 14, 15] & ✓ & ✓ & & & ✓ & ✓ & ✓ \\ [PERSON] et al. (2003) [16, 17] & ✓ & ✓ & & & ✓ & & ✓ \\ [PERSON] (2009) [18] & ✓ & ✓ & ✓ & ✓ & & & ✓ \\ [PERSON] et al. (2022) [19] & ✓ & ✓ & ✓ & & & & ✓ \\ [PERSON] et al. (2020) [20] & ✓ & ✓ & ✓ & & & ✓ & ✓ \\ [PERSON] et al. (2020) [21] & ✓ & & ✓ & & ✓ & & ✓ \\ [PERSON] (1996) [22] & ✓ & & ✓ & & & ✓ & ✓ \\ \hline \end{tabular}
\end{table} 
Table 1: Summary of the various passive damping methods investigated. of the \"current-flowing\" shunt circuit [15] for controlling multiple vibration modes of a piezoelectric laminate beam is used. To overcome the negative impact on the performance of the shunt-damping system due to the arbitrary selection of the capacitance values, as well as the closed-form solutions of resonant shunt branches, this paper considers the capacitances and the shunting branch inductors as new design variables along with the resistor damping components. The proposed optimization approach uses particle swarm optimization (PSO) algorithm to minimize the \(H_{\infty}\) norm of the damped system. To investigate in detail the effect of these variables on the damping performance, as well as for comparison with the other methods proposed previously in the literature [15, 17, 18], two additional numerical examples with fewer design variables are addressed. To simulate the dynamic behavior of the piezoelectric composite beam, a finite-element model is developed, which accounts for the electromechanical coupling and the presence of the shunt circuit. The formulation is based on Timoshenko beam theory and super-convergent finite elements. In summary, the main contributions of the present work lie in the following main areas: the development of a finite-element model more efficiently describing the dynamics of the piezoelectric composite beam structure; the creation of a state-space representation of the shunt-damping system, which enables us to simulate the frequency response functions and treat the damping method in the MATLAB environment; and, finally, the proposed optimal design of a multimode \"current-flowing\" shunt-damping circuit that is capable of delivering higher levels of performance in terms of adding damping to the system and, at the same time, reducing the required values of the shunt inductances. The rest of the paper is organized as follows.