## 1 Introduction  
Vortex-induced vibration (VIV) is a dynamic phenomenon characterized by oscillatory forces generated on bluff bodies immersed in fluid flow, resulting from the periodic shedding of Karman vortices. Along with galloping, flutter, and buffeting, VIV belongs to the family of aeroelastic vibratory phenomena, which are critically important in mechanical engineering, aerospace engineering, and wind energy. VIV primarily manifests as oscillations of the object perpendicular to the flow direction. Galloping is an instability arising from the nonlinear dependence of aerodynamic force on displacement, while flutter involves the interaction of aerodynamic forces with multiple structural vibration modes (e.g.,coupled bending and torsion). The last phenomenon, buffeting, refers to random, chaotic fluctuations in aerodynamic forces induced by turbulent flow or jet interaction. Hybrid piezoelectric wind energy harvesting systems, which utilize appropriately designed bluff bodies combining circular and angular features and operating at optimal angles of attack, can significantly enhance energy harvesting efficiency across a wide range of wind speeds by synergistically leveraging VIV and galloping [1]. The exploitation of the interaction between VIV and galloping is an effective approach to improving the energy harvesting efficiency of harvesters operating at low wind speeds [2]. A mathematical model of a hybrid wind energy generator utilizing both VIV and galloping is presented in [3]. Analysis presented in [4] indicates a significant increase in research activity within the field of VIV-based energy harvesting, with a notable rise in publications in recent years. A comprehensive review of recent advancements in flow-induced vibration energy harvesting, which holds significant potential for powering microsensors and electronic devices, is presented in [5]. An extensive overview of modeling methods for flow-induced vibrations of bluff bodies, encompassing traditional (mathematical and numerical) and novel machine learning-based approaches, is provided in [6]. Reviews of previous studies on flow-induced vibrations (FIV) is presented in [7,8]. In vortex-induced vibration (VIV), the presence of bluff bodies is crucial. These are irregularly shaped objects (e.g., cylinders, prisms, plates) that cause significant flow separation. Research has been conducted on a rigid circular cylinder with two attached piezoelectric beams in a wind tunnel [9]. Experiments revealed an optimal load resistance for maximizing power in VIV and an optimal distance between cylinders in wake-induced vibrations (WIV) for maximizing output voltage and power. The implementation of a dual series system of piezoelectric VIV energy generators is an effective method for enhancing the efficiency of energy harvesting from wind flow, significantly broadening the wind speed range over which the system operates effectively [10]. Utilizing nonlinear magnetic forces is advantageous for designing efficient and broadband VIV-based piezoelectric energy generators [11]. Results from [12] indicate that buoyancy force influences the characteristics of vortices and flows around inverted D-shaped cylinders, which in turn impacts VIV. Flow separation can lead to significant oscillations transverse to the flow direction. VIV can serve as a passive energy source harvested from wind or water flow. The design and research related to a piezoelectric device for harvesting energy from vortex-induced vibrations, intended for low-speed water flow, are presented in [13]. Such a system is termed an energy harvester. This type of system converts mechanical vibrations into electrical energy using piezoelectric or electromagnetic elements. Piezoelectric materials generate voltage when mechanically deformed and, unlike electromagnetic generators, do not require magnets, coils, or other mechatronic components. Their advantage lies in their easy integration with small bluff body structures or mounting within the housing of a vibrating object.