## 1 Introduction  
In recent years, wireless sensor technology has been widely used in various industries [1, 2], but the energy supply problem of nodes has always restricted its development. The use of a chemical battery power supply is easy to cause pollution to the environment, and the cost of replacing the battery is high. There is a large amount of vibrational energy in the environment [3, 4, 5], and harvesting vibration energy from the environment to power wireless sensors is becoming a popular research activity. At present, vibration energy harvesting technologies include electrostatic [6], electromagnetic [7], and piezoelectric. Among them, piezoelectric energy harvesting technology has the characteristics of easy application and high output voltage and has become a reliable means of energy supply [8]. The early piezoelectric energy harvesting technology was mainly researched on linear systems, but the linear piezoelectric energy harvester is affected by the natural frequency, and it is difficult to adapt to the environment of wide frequency. To solve the problem of the narrow operating frequency bandwidth of the linear piezoelectric energy harvester, the array method [9,10] and the special-shaped beam method [11] have become popular methods for the linear piezoelectric energy harvester to broaden the operating frequency band. However, these energy harvesters have disadvantages such as large structure size and low energy density. Compared with linear piezoelectric energy harvesters, nonlinear piezoelectric energy harvesters can harvest energy in a wider frequency range. [PERSON] [12] investigated an energy harvester based on bistable laminated plates by utilizing material properties, and experimental results show that this piezoelectric energy harvester can output higher voltages at multiple excitation frequencies. However, special carbon fiber materials can only be completed after complex processing. [PERSON] [13] designed a piezoelectric energy harvester with variable axial stress, which effectively broadened the operating frequency bandwidth of the system. [PERSON] [14] developed a cantilever piezoelectric energy harvester with a limiter. The numerical simulation results show that the nonlinear behavior of the system can broaden the working frequency bandwidth of the energy harvester. [PERSON] [15] fabricated a piezoelectric energy harvester with an elastic amplifier on the base, and simulated and analyzed the effect of springs on the performance of the piezoelectric energy harvester. The results show that this structure can improve harvesting efficiency. However, the spring will swing during the vibration process, which will damage the stability of the system operation. The above-mentioned researchers have broadened the operating frequency bandwidth of piezoelectric energy harvesters by changing material properties or using spring structures, but these methods pose challenges to the fabrication of materials and the stability of system operation. In the current research on nonlinear piezoelectric energy harvesters, the introduction of magnetic force into piezoelectric energy harvesters is considered an effective way to broaden the operating frequency bandwidth and improve energy output [16,17]. Among various stable piezoelectric energy harvesters, bistable piezoelectric energy harvesters have become one of the most popular research areas [18,19]. [PERSON] [20] constructed a common bistable piezoelectric energy harvester by using magnets with opposite axes, and this structure has a wide operating frequency bandwidth. [PERSON] [21] improved the output efficiency of the piezoelectric energy harvester by properly adjusting the position of the fixed-end magnet in the vertical direction. On this basis, [PERSON] [22] placed the magnet at the fixed end on top of the magnet at the end of the cantilever beam and established a system model based on the Euler-Bernoulli beam theory. Electric energy harvesters are capable of energy harvesting in a wide operating frequency bandwidth.