## 4 Conclusions  
The nonlinear vibration suppression of cantilever beams using shunted piezoelectric elements is investigated. The piezoelectric shunt circuit exhibits a quadratic nonlinear capacitance that is tuned with the beam's natural frequency to achieve a two-to-one internal resonance. The internal resonance coupling is exploited to trigger the saturation phenomenon, where the beam's vibration reaches an almost constant value beyond a force excitation threshold. A mathematical model for the nonlinear vibration of a cantilever beam coupled with the piezoelectric shunt circuit considering the inertia and geometric nonlinearity was developed. A reduced-order model based on [PERSON]'s discretization method was developed and used to numerically simulate the performance of coupled and uncoupled systems. First, the response of the linear structure to resonant harmonic excitations was examined with and without the shunt circuit. It was found that the shunt circuit performs effectively as a lightweight, easy-to-use, absorber around the resonant frequency. The vibration intensity of the beam is reduced significantly due to the saturation phenomenon after the force amplitude reaches its threshold. A parametric analysis was performed to serve as a design guideline for optimizing system parameters including the circuit parameters and piezoelectric patch size and location. It was found that the larger the piezoelectric patch surface area, the lower the vibration amplitude after the saturation. It was also found that attaching the patch at the beam's root is more effective in vibration suppression than moving it toward the tip. The cantilever beam's nonlinearity was examined, and the geometric cubic nonlinearity was found dominant over the inertia nonlinearity for the primary resonance considered in this study. The shunt circuit absorber was tested against resonant harmonic excitation for the nonlinear structure, and its performance was found to be robust and highly effective. Future research directions include, but are not limited to, exploring multi-mode internal resonances, experimental prototyping, and nonlinear vibration suppression in wide-band applications.  
Figure 13: Frequency-response plots of the nonlinear response for the uncoupled and coupled beam with a force amplitude of (**a**) 0.1 N, (**b**) 0.3 N, and (**c**) 0.5 N.  
**Author Contributions:** Conceptualization, K.A.-S. and S.E.; methodology, K.A.-S. and S.E.; software, K.A.-S. and K.K.; validation, K.A.-S. and S.E.; formal analysis, K.A.-S. and S.E.; investigation, K.A.-S.; resources, K.A.-S. and S.E.; data curation, K.A.-S.; writing--original draft preparation, K.A.-S. and K.K.; writing--review and editing, S.E.; visualization, K.A.-S.; supervision, S.E.; project administration, S.E.; funding acquisition, S.E. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.  
**Conflicts of Interest:** The authors declare no conflict of interest.