## 1 Introduction  
With the continuous development of advanced manufacturing technology, people have made a variety of electronic devices such as telephone watches, wireless headphones, and microelectronic systems. At present, most electronic devices need external batteries to provide power for normal operation. However, the energy supply cycle is limited and the battery needs to be replaced regularly. The extensive use and replacement of batteries have brought a series of problems such as environmental pollution. To find a way to overcome the difficulty, researchers have tried to harvest energy from the surrounding environment to replace the demand for batteries. For example, converting energy such as electromagnetic [1], piezoelectric [2], and electrostatic [3] into electrical energy. Piezoelectric energy harvesters are widely used because of their high energy density, no magnetic interference, and easy processing, miniaturization, and integration in structure. At present, many researchers have studied piezoelectric energy harvesters. For instance, [PERSON] and [PERSON] [4, 5] conducted theoretical and experimental studies on the electromechanical responses of cantilevered unimorph and bimorph piezoelectric energy harvesters based on the Euler-Bernoulli beam theory. An exact analytical solution for a cantilevered piezoelectric energy harvester was proposed, and the modal electromechanical coupling and the dependence of the electrical output on the locations of the electrode were discussed. [PERSON] and [PERSON] [6] developed an analytical model of a nanoscale ulimorph piezoelectric energy harvester containing the flexoelectric effect. An analytical model of a piezoelectric cantilevered energy harvester by through-width delamination was proposed by [PERSON] et al. [7]. [PERSON] et al. [8] experimentally investigated the performance of a film-type piezoelectric energy harvester under simultaneous pneumatic and base excitation. [PERSON] and [PERSON] [9] investigated the energy harvesting performance of a piezoelectric circular nanomembrane under human blood pressure with surface effects. In recent decades, a lot of research has been performed on piezoelectric energy harvesters. In most studies, the proposed piezoelectric energy harvesters are linear, and work only in a limited excitation frequency spectrum. In order to obtain a wider operating frequency band for the piezoelectric energy harvester, researchers try to consider the nonlinearity in the study of piezoelectric energy harvesters. At present, the research on the nonlinearity of piezoelectric energy harvesters has made great progress. For instance, [PERSON] et al. [10] studied the nonlinear dynamical parameters of the bistable vibration of a piezoelectric energy harvester. [PERSON] et al. [11] studied the nonlinear electromechanical behavior of a constrained bimorph piezoelectric energy harvester by using a coupled multimodal fully nonlinear model. It is not difficult to find that considering the nonlinearity cannot only improve the calculation accuracy but also improve the harvesting energy efficiency of energy harvester. In addition, researchers also found that the broadband behavior of piezoelectric energy harvester can be achieved by applying axial loads to a fixed beam. For example, [PERSON] and [PERSON] [12] established an electromechanical model of a clammed single-deformation piezoelectric energy harvester subjected to transverse excitation and static axial load, and made a comprehensive discussion of the energy harvesting of the loaded beam using an axial load. Their results show that the axial load can achieve tunable characteristics and power enhancement. [PERSON] and [PERSON] [13] proposed a nonlinear electromechanical model of an energy harvester based on an axial pre-loaded piezoelectric beam, and combined the flexoelectric effect, which can be operated in pre-buckling (monostable) and post-buckling (bistable) configurations.