## 1 Introduction  
Structural vibration suppression has been of common interest as an essential component in the design of systems for performance and integrity. In many engineering applications, vibration is undesirable and even harmful, which can eventually lead to failure [1]. As a result, efficiently mitigating vibration has always been a goal across a wide range of engineering applications. Traditional methods such as dynamic vibration absorbers (DVAs) and tuned mass dampers (TMDs) [2] have long been reliable instruments for reducing vibration and providing effective solutions to practical problems. However, they often exhibit limitations in terms of adaptability, accuracy, and efficiency. Nonetheless, innovation always intends to push boundaries in the dynamic realm of vibration control. Piezoelectric shunt circuits are one of the innovative treatments that have drawn growing interest as vibration absorbers. By utilizing piezoelectric transducers coupled to an electrical impedance, [PERSON] [2] suggested a vibration control method by transforming strain energy from the structure into electrical energy. Based on their methods of control, researchers categorize piezoelectric shunt circuits into three groups: hybrid, active, and passive. The electrical components in the shunt circuit and the existence of any external energy source determine which of those types is used. Using the circuit's intrinsic resistance to absorb vibrational energy, the passive approach dissipates energy passively without the need for an external power source. Conversely, active techniques use sensors and actuators to detect vibration and activate real-time vibration suppression [3]. From straightforward beams to more intricate space constructions, the active control approach has effectively decreased vibration in a variety of applications. In order to operate actuators, an active control system usually needs high-performance digital signal processors and power amplifiers, which are not suitable for space applications. Passive control techniques are less resilient to system disruptions even though they are easier to build than active control systems [4]. Consequently, the possibility of hybrid control systems to effectively provide vibration attenuation through piezoelectric shunt circuits has raised interest in them. These systems offer a flexible method of lowering vibration in a range of applications by combining the advantages of energy efficiency and customization. It is important to remember that passive techniques can be quite successful in reducing high-frequency excitations, whilst active techniques can be utilized to restrict low-frequency vibrations. Furthermore, hybrid approaches combine the best features of active and passive methods, making them appropriate for wide-frequency bands [5; 6]. When two modes of a nonlinear system are mutually synchronized [7], meaning that their frequencies are in a simple integer ratio, internal resonances occur. Saturation phenomena, a special interaction between these modes, can be produced by using this condition. When an external stimulation is applied to the first mode, the saturation phenomenon, which was initially identified by [PERSON] et al. [8], creates a threshold for one of the modes and transfers the energy to the other mode, resulting in an energy exchange. The system behavior and vibration amplitude can be greatly affected by this method. Internal resonance type in the system is determined by setting the frequency ratio between the modes and the type of nonlinearity coupling between them. A quadratic nonlinearity in the system is necessary to attain a 2:1 internal resonance if the frequency ratio between the modes is doubled. Moreover, cubic nonlinearity results in a different kind of internal resonance, known as 3:1, where the frequency ratio of the two modes is three [9]. Passive vibration techniques using piezoelectric transducers have been used by many researchers.