## 1 Introduction  
Vibrations are a common occurrence in our daily lives, ranging from the vibrations of minute mechanical components to the substantial vibrations of aerospace engineering structures like airplanes and rockets. The potential to convert this energy into reusable energy holds the key to realizing sustainable management practices. With global attention increasingly focused on energy issues in recent decades, Vibration Energy Harvesting Systems (VEH) have emerged as a noteworthy green energy technology, garnering significant interest from scholars. Efficiently recycling and repurposing the constant vibrations in our surroundings could establish a robust electrical energy conversion method with substantial environmental benefits. [PERSON] et al. [1] attribute object vibrations to factors like uneven system quality or surface irregularities resulting from material wear and tear. [PERSON] et al. [2] assert that mechanical vibration represents a dependable energy source, and its conversion to electrical energy could enable electronic devices to operate without conventional batteries. Piezoelectric devices have become a focal point in the analysis and research of power generation due to their effectiveness in converting vibration energy into electrical energy, as demonstrated by [PERSON] et al. 's research [3]. Moreover, piezoelectric patches (PZTs), known for their compact size and independence from additional power supplies, prove suitable for installation in various sensors, buildings, facilities, and household appliances--areas frequently subjected to tactile interactions--yielding significant energy-saving effects. In the pursuit of compact and lightweight electronic products, [PERSON] and [PERSON] [4] designed a power generation device incorporating a double-layer bendable rod and a piezoelectric patch (PZT). They anticipate that future electronic devices will trend toward smaller and lighter designs. Emphasizing the pervasive nature of vibration energy in daily life, [PERSON] et al. [5] installed a vibration energy recovery device beneath a car's hood, utilizing natural forces such as magnets and gravity to convert vibrations into electrical energy and store it effectively. [PERSON] et al. [6] investigated a vibration energy harvesting system employing, during flight, piezoelectric devices installed on aircraft wings. Their study delved into the impact of different flight altitudes on aircraft vibration, underscoring the significance of aerodynamics in vibration energy recovery systems. [PERSON] et al. [7] devised a bistable piezoelectric energy harvester capable of recovering vibrations from general machinery, a concept gaining popularity in recent years. [PERSON] et al. [8] creatively utilized PZTs on simply supported beams, tapping into environmental vibrations to power small electronic devices. [PERSON] et al. [9] proposed an innovative energy harvester employing multiple PZTs to convert vibration energy into electrical energy, storing it in an electrical storage device. [PERSON] and [PERSON] [10] highlighted the amplification of power generation in a bistable system during transitions between stable states, with many bistable energy harvesters employing magnetic forces for driving. [PERSON] et al. [11] contributed to the understanding of bistable oscillators' power generation devices, employing magnets and cantilever beams. Their practical design considerations, including the impact of varying magnet spacing on power generation efficiency, contribute to the evolving landscape of bistable oscillator technology. Regarding piezoelectric material technology, [PERSON] et al. [1] have outlined that these materials typically encompass piezoelectric polycrystals, piezoelectric single crystals, and piezoelectric films. Notably, Polyvinylidene Difluoride (PVDF) stands out as a common material exhibiting a robust piezoelectric effect. [PERSON] et al.