## 1 Introduction  
In the latest years, with the fast development of wireless monitoring and microelectronics technology, wireless monitoring systems have shown strong application potential in monitoring fields such as ecological environment [1], equipment status [2], and traffic safety [3]. Most of the current wireless monitoring nodes use battery power; however, chemical batteries are not only difficult to recycle but also need to be replaced regularly and have high maintenance costs. Vibration energy harvesting technology can achieve the transformation of vibration energy into electric energy, which provides new ideas and methods to handle the endurance problem of wireless monitoring nodes [4; 5]. At present, the vibration energy capture method by using the piezoelectric principle has become the research hotspot due to its advantages of easy miniaturization, simple structure and high energy density [6; 7; 8]. The researchers initially mainly studied the linear piezoelectric energy capture system and characterized the system's motion characteristics by establishing a linear mathematical model. However, as far as the linear piezoelectric energy harvesting system [9] is concerned, only when the vibration frequency is near the intrinsic frequency of the system structure can the system produce an efficient output. The excitation frequency of the real external environment is often random, has a certain range, and cannot always be near the intrinsic frequency of the system. As a result, the electricity generation efficiency of the linear piezoelectric energy capture system is low, and it is difficult to be widely used. In order to broaden the effective operating frequency band of the piezoelectric energy capture system and improve the energy capture performance of the system, researchers introduced nonlinear factors on the basis of the linear energy capture system and began to explore and study the nonlinear piezoelectric energy harvesting system [10; 11]. The nonlinear energy harvesting systems currently studied are mainly realized by introducing nonlinear stiffness. There are three main methods for introducing nonlinear stiffness, which are external force coupling [12; 13], piecewise linear [14; 15], and nonlinear strain [16; 17]. Compared with the other two methods, external force coupling has the advantages of strong reliability and easy realization and is the current key research direction [18]. As the most common way of external force coupling, the introduction of nonlinear magnetism can make the system achieve a large response in monostable [19; 20], bistable [21; 22], and multistable [23; 24]. Researchers have conducted a great deal of research on the piezoelectric energy capture system using this way [25; 26; 27]. [PERSON] [28] proposed a monostable piezoelectric energy capture by introducing a set of symmetrical attracting magnets outside the cantilever beam. The research results indicated that the energy harvesting bandwidth and output voltage of the system are greatly improved compared with the linear system. [PERSON] [29] proposed an asymmetric double-beam monostable piezoelectric energy harvester and found that the asymmetric structure has higher output performance and harvesting efficiency than the symmetric structure. [PERSON] [30] arranged two mutually repelling magnets at the tip and outside to form a bistable piezoelectric energy harvester. Through experimental tests, it was found that the introduction of magnetism significantly enhanced the harvesting ability of the bistable energy harvester. [PERSON] [31] established a nonlinear bistable model by using the attraction of two fixed magnets to the ferromagnetic beam and proved that the bistable system has higher output voltage and power than the linear system through theoretical and experimental studies. [PERSON] [32] designed an improved bistable energy capture by placing a miniature magnet in the middle of two external magnets, and the analysis results showed that the improved bistable system can achieve high energy output at lower excitation intensity.