## 3 Results and Discussion
### Selection of the Design Parameters  
In this section, a parametric study is performed to assess the influence of each shunt-circuit parameter on the performance of the proposed absorber. Namely, the parameters of the circuit, damping, and the piezoelectric patch are considered. The objective is to determine the optimum set of parameters for maximum vibration suppression. First, the coupled system's response is studied with variation in the circuit resistance, \(R\). Figure 5 presents the force-response plot for the beam's vibration with three values of the circuit resistance: 3, 3.5, and 4 k\(\Omega\). The figure indicates that decreasing resistance results in decreased saturation threshold force and decreased vibration amplitude. Namely, the force saturation threshold reduces from 0.18 N to 0.14 N, and the vibration modal amplitude reduced from 0.05 to 0.04 as the resistance reduced from 4 k\(\Omega\) to 3 k\(\Omega\), respectively, as shown in the figure. This result agrees with the results reported in [39]. Figure 4: Uncoupled and coupled beam’s response: (**a**) time response at an excitation amplitude of 0.5 N and (**b**) the amplitude–force response. Figure 5: Force response plot of the beam’s vibration for various circuit resistances. The influence of the gain, the constant \(N\) in the model, is considered. We examined the effect of the nonlinear gain on the absorber's performance at three values: \(10^{10}\), \(2\times 10^{10}\), \(3\times 10^{10}\)\(V/C^{2}\), as shown in Figure 6. As the figure shows, as the gain increases, the vibration amplitude and the force saturation threshold decrease. The governing equations, Equations (30) and (31), show that increasing the gain \(N\) strengthens the coupling between the electrical and mechanical modes, and hence increases energy transfer. The piezoelectric patch, which couples the beam with the absorber, the electric circuit, plays an important role in vibration suppression. The influence of the piezoelectric patch parameters, including its location on the beam and the geometrical dimensions (surface area and thickness), is investigated. First, we consider three key locations for the piezoelectric patch: at the beam's root, middle, and tip, as shown in Figure 7.  
According to Equations (29) and (32), the location of the piezoelectric patch on the beam matters when it comes to the piezoelectric material coefficient, \(K_{c}\), and the electromechanical coupling factor (EMCF), \(k_{1}\). The EMC values for the piezoelectric patch  
Figure 6: Force response of the beam’s vibration for various nonlinear gain \(N\). Figure 7: Piezoelectric patch location on the cantilever beam’s (**a**) root, (**b**) middle, and (**c**) tip. locations shown in Figure 7a-c are calculated as 6.31%, 2.97%, and 0.06%, respectively. It is worth mentioning that the 2:1 internal resonance ratio has been preserved throughout parametric study. The force-response plot for the beam's vibration with the three PZT patch locations is shown in Figure 8. As the figure illustrates, placing the PZT patch at the beam's root is the most efficient in terms of vibration suppression due to the maximum strain generated at the beam's root. The PZT-patch location not only reduces the steady-state vibration amplitude, but it also reduces the force saturation threshold, as Figure 8 implies. These results emphasize that the location of the PZT patch strongly influences vibration suppression performance due to variation in strain energy distribution along the beam. Next, we consider the significance of the size of the piezoelectric patch on the performance of the absorber. Two parameters depend on the PZT patch size: the electromechanical coupling coefficient (EMCF) and the capacitance of the patch, as shown in Table 2. Figure 9 presents the force-response plot for the beam's vibration for various patch sizes. The figure shows that as the patch size increases, the coupling coefficient increases, and the capacitance increases.