## 1 Introduction
An analytical model was derived for a parallel-connected macro-fiber composite (MFC) piezoelectric material in a bimorph configuration attached to a cantilever beam, and the model was solved numerically. [PERSON] and [PERSON] [31] presented a shunt with a nonlinear impedance equivalent to a positive or negative cubic inductor connected in parallel with a linear inductor and a linear resistor. This shunt, along with the inherent piezoelectric capacitance, behaves as a synthetic Duffing circuit, providing a digital analog of a hardening- or softening-type nonlinear stiffness, linear stiffness, linear damping, and mass. Numerical simulations were performed using single-term and multi-term harmonic balance solutions to predict the periodic response. These approximate analytical predictions were compared against time-domain numerical simulations and were experimentally validated across various mechanical excitation amplitudes and nonlinear coefficients. [PERSON] et al. [32] introduced a semi-active control method based on a piezoelectric smart cantilever beam with a passive shunt circuit for vibration control. The contribution is that the proposed semi-active control method is used to achieve energy dissipation switching by changing the piezoelectric electrical boundary conditions. A prototype was manufactured, and the measurement system was constructed. The correctness and feasibility of the semi-active vibration control method were verified through experiments. [PERSON] et al. [33] provide a hybrid nonlinear vibration attenuation method. To achieve a 2:1 internal resonance between the mechanical and electrical resonances, the shunt circuit was tuned using a quadratic nonlinearity. Above a threshold, this allowed the mechanical amplitude to be independent of the forcing amplitude by producing a nonlinear antiresonance and a saturation phenomenon. It has been researched and optimized to tune the shunt circuit's nonlinear properties. The objective of this work is to exploit the internal resonance to effectively suppress the nonlinear vibration of cantilever beams equipped with a nonlinear piezoelectric shunt circuit. The coupled system consists of a cantilever beam as the structure, a piezoelectric patch as the transducer, and a shunt circuit as the nonlinear absorber. A 2:1 internal resonance is created between the mechanical natural and electrical natural frequencies by tuning the elements of the electrical circuit. A parametric study is performed to find the optimal circuit parameters for the maximum vibration suppression while preserving the 2:1 internal resonance. The performance of the proposed absorber is investigated under resonant harmonic excitation of the structure, and it is found to significantly reduce the vibration amplitude over a substantial frequency range around resonance. Moreover, the absorber is found to function equally well for the linear vibration of the beam and the nonlinear vibration when inertia and geometric nonlinearities are taken into consideration. The distinctive contribution of this work is that it employs nonlinear capacitance in passive circuits to trigger internal resonance for the saturation phenomenon, performs a parametric tuning strategy, and lastly applies the control strategy on geometrically nonlinear beam dynamics. Challenges and limitations in real-world applications, e.g., aerospace applications, include the risk of affecting structural integrity, inefficient performance for systems that often vibrate across a wide spectrum, and control of multiple vibration modes at once. The rest of this paper is organized as follows: Section 2 presents the mathematical model of the nonlinear vibration of a cantilever beam coupled to a piezoelectric shunt circuit, Section 3 presents the results and discussions, Section 4 presents some conclusions, and finally outlines the references cited in this paper.