## 3 Results and Discussion
### Response of the Nonlinear Structure Coupled with the Shunt Circuit  
The results of the previous section showed that the suggested vibration control methodology efficiently suppresses the beam's linear vibration via the 2:1 internal resonance between the electrical and mechanical natural frequencies. In this section, the beam's cubic geometric nonlinearity is taken into consideration. The goal is to examine the effectiveness of the absorber in suppressing the nonlinear vibration of the beam. Figure 11 presents the frequency response curves near resonance of the uncoupled system, the beam's response without the absorber, for various values of the excitation amplitudes. Each frequency-response plot has two segments that result from forward and backward sweeps around the resonant frequency, which is the frequency of the first mode, to capture the upper and lower stable branches. The beam shows a hardening nonlinear response and ranges of frequency of multiple solutions and the possibility of jumping from one stable branch to the other.  
The steady-state response of the nonlinear vibration of the uncoupled beam, without the shunt circuit, and the coupled beam, with the shunt circuit, at an excitation amplitude of 0.5 N is shown in Figure 12a. The PZT patch was fixed at the beam's root, and the nonlinear gain \(N\) was set to \(3\times 10^{10}\)\(V/C^{2}\). The rest of the circuit parameters are given in row C in Table 1 for the patch surface area, and row B in Table 2 for the patch thickness. The force-response plot of the beam's response is shown in Figure 12b where the saturation phenomena holds and the nonlinear response of the beam without the shunt is evident. Frequency-response plots of the nonlinear vibration for the baseline beam without the absorber and the coupled beam with the shunt circuit are shown in Figure 13 for a variety of force amplitudes. The absorber shows an excellent ability to mitigate the nonlinear vibration and achieve significant reductions of about 63%, 73%, and 75% for forcing amplitudes  
Figure 11: Frequency-response plots for the nonlinear beam’s vibration under different excitation amplitudes.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Patch** & **Thickness: \(t_{p}\) (mm)** & \
\begin{tabular}{c} **Electromechanical** \\ **Factor: \(k_{1}\)** \\ \end{tabular} &
\begin{tabular}{c} **PZT Capacitance:** \\ \(C_{p}\) (nF)** \\ \end{tabular} \\ \hline A & 0.3 & 2.00\% & 161.0 \\ \hline B & 0.5 & 2.97\% & 96.58 \\ \hline C & 0.7 & 3.98\% & 68.98 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Thickness variation and properties for three different piezoelectric patches.  
of 0.1 N, 0.3 N, and 0.5 N, respectively. The performance of the proposed absorber proves to be robust for both the linear and nonlinear vibration of the cantilever beam. Moreover, Figure 13 shows that the absorber's performance is effective over a significant bandwidth of excitation frequencies. The beam shows a strong hardening nonlinear behavior due to geometric nonlinearity. The implication of this behavior is that the nonlinear natural frequency shifts from the linear one, which may impact the robustness of the proposed control strategy that is based on employing the saturation phenomenon due to the 2:1 internal resonance. Nevertheless, the proposed suppression methodology shows an excellent ability to transfer the energy from the beam to the circuit at relatively high forcing amplitude, as shown in Figures 12 and 13.  
Figure 12: Nonlinear response of the uncoupled and coupled beam’s vibration under an excitation of 0.5 N: (**a**) time history and (**b**) the force response.  
Figure 13: Cont.