## 5 Simulation and Discussion  
In this section, the time-domain response of the Kagome structure is simulated to validate the efficiency of the proposed method. The Rayleigh damping model is applied to the structure so the structure damping matrix can be written as:  
\[\left[D\right]_{\mathrm{s}}=\alpha\left[M\right]_{\mathrm{s}}+\beta\left[K \right]_{\mathrm{s}} \tag{26}\]  
where \(\alpha=2.35\) and \(\beta=3.96\times 10^{-7}\) to set the modal damping ratios of mode 1 and 2 at \(1\%_{\mathrm{c}}\). The modal damping ratio of mode \(i\) can be calculated using:  
\[\xi_{i}=\frac{1}{2}\left(\frac{\alpha}{\omega_{i}}+\beta\omega_{i}\right) \tag{27}\]  
To validate the efficiency of the proposed method on the random response suppression, the response of the Kagome structure under the excitation of a vertically distributed band-limited white noise load on finite element nodes of the whole face sheet is simulated. The noise is with uniform PSD 0.2 N\({}^{2}\)/Hz in 0-800 Hz. The random forces on different nodes are uncorrelated. The purpose is to control the RMS response of vertical vibration of the whole face sheet, so the output DOF is also vertically DOF of the nodes of the whole solid face sheet. Because the upper bound of the excitation is 800 Hz, in Equation (22) only the first five modes are considered. The locations of the five transducers are decided as in Figure 6a through calculating \(\lambda_{i}\), and their fraction of MSE are shown in Figure 6b. The modal criteria \(\eta_{i}\) of mode 1-5 are drawn in Figure 7, which shows that the modal criterion decreases with increasing mode number. Two transducers are used to control modes 1 and 2 because they possess higher values of modal criterion. For comparison, another strategy with mode 3 and 4 under control are also computed. The tuned \(RL\) values and added damping ratios calculated by complex modal analysis of the full mechano-electric finite element model (the size of the matrices \([M]\), \([D]\), \([K]\) in (12) is 3770 \(\times\) 3770) are listed in Tables 2 and 3. The added damping ratios of different modes are distinct, because fraction of MSE of the rods in Figure 6a are distinct, as is shown in Figure 6b. After generating the random excitation using inverse FFT technique, the Newmark-\(\beta\) method based on MATLAB code with time step \(\Delta t=1/2^{13}\) is employed to simulate the time-domain response. In order to observe the decrease of the resonance peaks, an observation point is set which is shown in Figure 2. The PSD of vertical response of this point in three conditions are illustrated in Figure 8. Obviously, when modes 1 and 2 are under control, the resonance peaks of modes 1 and 2 are declined considerably while other peaks remain unchanged. When modes 3 and 4 are under control, the only decreased peaks are 3 and 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode** & **Resistance** & **Inductance** & **Added Damping Ratios** \\ \hline
1 & 515 \(\Omega\) & 2.34 H & 0.029 \\
2 & 266 \(\Omega\) & 0.80 H & 0.024 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Tuned RL values and added damping ratios for mode 1–2 when controlling mode 1–2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode** & **Resistance** & **Inductance** & **Added Damping Ratios** \\ \hline
3 & 204 \(\Omega\) & 0.64 H & 0.021 \\
4 & 145 \(\Omega\) & 0.37 H & 0.019 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Tuned RL values and added damping ratios for mode 3–4 when controlling mode 3–4. Figure 6: (a) Locations of transducers for modes 1–5; (b) fraction of MSE in the replaced rods. Figure 7: Modal criterion of mode 1–5. Finally, in order to validate the modal criterion proposed in this paper to suppress the overall transverse vibration of the face sheet, RMS vertical displacements of all the nodes in the face sheet, say, the \(H_{2}\) norm of vector \(\{y\}\) in expression (16), in different sampling time points are calculated in three conditions. The same realization of random excitation is used for convenience of comparison.