## 1 Introduction
[PERSON] and [PERSON] [10] were the first to use a piezoelectric transducer connected to a passive electrical circuit to dissipate mechanical energy. Moreover, they added a series resistor-inductor circuit and adjusted the electric circuit's impedance frequency to match the structural resonant frequency to mitigate the maximum vibration amplitude. However, large inductors are needed to achieve the low electrical resonance needed to diminish low-frequency vibration. This drawback was addressed in the work of [11]. [PERSON] [12] used a multi-layer piezoelectric transducer connected to an (RL) circuit. The purpose of multi-layering the piezoelectric element was to increase its capacitance, reducing the requirement for a high inductance value at low frequencies. [PERSON] et al. [13] investigated how to improve the inductance values using closed magnetic cores composed of materials with high permeability. They showed that customized designs can enable the use of passive resonant shunt techniques at lower frequencies. Researchers have explored multimode piezoelectric shunt circuit vibration control. A method for suppressing several modes with a single piezoelectric device was created by [PERSON] [14]. To create a single piezoelectric component, additional LRC circuit branches are connected in parallel to an RL branch. There were as many RLC branches as there were modes to take into account. In order to regulate the branches, alternative circuit layouts were also suggested. Using a current-blocking circuit, [PERSON] [15; 16] created a multi-mode shunt. At other branch frequencies, the LC anti-resonant circuits' electrical resistance is intended to reach infinity and function as an open circuit, while the targeted branch that cor responds to the structure's resonance frequency functions and reduces vibration. However, [PERSON] et al. [17; 18] employed shunt circuits with current-flowing branches. The current-flowing shunt and the current-blocking shunt serve the same purpose. Instead of halting the current at its normal frequency, the current-flowing shunt allows it to continue. Since current blocking necessitates an equal number of inductors for each mode, this approach is superior [19]. A novel tuning technique for a simpler current-blocking shunt circuit was presented by [PERSON] et al. [20] and verified both experimentally and numerically. Their novel approach seeks to enhance multimode damping without the use of any optimization algorithms by concentrating on the effects of nonresonant modes and the electromechanical coupling between the structure and the shunt branches. This strategy differs from others that employ optimization algorithms to adjust circuit parameters, including ant colony optimization [21], particle swarm optimization [22; 23], and other tuning methods [24; 25]. The use of internal resonances for vibration suppression was first proposed in [26] by introducing a nonlinear auto-parametric absorber that is based on internal resonance. Their approach relies on connecting a single-degree-of-freedom system to a cantilever beam with a tip mass. After forcing the system to resonate, they discovered that the absorber could efficiently limit and create a threshold on the amplitude of oscillations of the main mass by tuning the beam and other system frequencies. Moreover, most of the researchers have used internal resonance in active control systems [27; 28]. [PERSON] et al. [29] introduced a new active nonlinear vibration absorber for flexible constructions that makes use of quadratic nonlinearities and two-to-one internal resonances. The technique connects second-order controllers to a cantilever beam via sensors and actuators. [PERSON] and [PERSON] [30] suggested a passive nonlinear piezoelectric vibration isolation, and the induced structural nonlinearity was demonstrated experimentally. A passive nonlinear component is suggested to be a hardening capacitor obtained using a P-N junction.