## 1 Introduction  
With the development of low-power and low-cost electronic devices, these devices are extensively used in civil, industrial, and military areas. Furthermore, these devices are primarily powered by batteries, which have the weaknesses of high replacement cost, limited life time, and environmental pollution. Thus energy supply becomes a prominent problem that needs to be solved urgently. Energy harvesting from ambient environment is regarded as a promising means to replace conventional batteries or extend the lifetime of batteries for powering low-power electronic devices. A variety of energy harvesting devices have been proposed to harness mechanical energy including vibration, airflow, ocean waves, rotation, and human motion, which is widely distributed in environment. Piezoelectric [1; 2], electromagnetic [3; 4], electrostatic [5], and triboelectric [6] are the main transaction mechanisms used to convert mechanical energy into electrical energy. Considering the traditional linear energy harvesters are unable to harvest energy from broadband excitations or excitations with time-varying frequency. Nonlinear energy harvesters have been developed to overcome the above shortcomings of the linear harvesters. Monostable and bistable harvesters are the typical structures for nonlinear energy harvesting. For monostable energy harvester, [PERSON] et al. [7] developed a monostable piezoelectric energy harvester by applying a symmetric magnetic attraction and a pair of stoppers, which can harvest vibration energy from low-level excitations. [PERSON] et al. [8] proposed an efficient and broadband vortex-induced vibrations energy harvester by using nonlinear attractive magnetic forces. As another class of nonlinear harvesters, bistable energy harvesters have also been investigated extensively. Double potential wells of the potential function is the critical feature of these devices, which can realize the inter-well oscillation to dramatically improve the vibration displacement and velocity. Magnetic forces [9; 10], residual stresses that are produced by fabrication process [11], or axial preloaded [12; 13; 14] are the main methods to achieve bistablity. [PERSON] et al. [15] investigated a piezomagnetoelastic energy harvester using a piezoelectric ferromagnetic beam and external magnets to enhance the operation bandwidth and the output power. A bistable inertial harvester comprised of a piezoelectric cantilever with a tip magnet and a fixed magnet is proposed by [PERSON] et al. [9]. [PERSON] et al. [16] designed a piezoelectric bistable plate harvester utilizing a bistable composite plate with attached piezoelectric patches to realize large amplitude vibrations and a wide response bandwidth. For axial preload bistable energy harvesters, [PERSON] and [PERSON] [17] proposed a tunable-resonance vibration energy harvester using an axial preload to the piezoelectric bimorph for adjusting its resonance frequency. A comprehensive understanding of the axial preload piezoelectric harvester was developed by [PERSON] and [PERSON] [18]. In order to enhance the output voltage-amplitude and broaden the frequency response bandwidth at low frequencies and small axial excitations, a magnet-coupled buckled-beam piezoelectric harvester was developed by [PERSON] and [PERSON] [19]. Further, an elastic clamp boundary with axial compressive effects and compliant suspensions were developed in order to improve the average power and bandwidth of the buckled piezoelectric beam harvesters [14; 20]. [PERSON] et al. [21] designed a wind energy harvester using the buckled beam to enhance the output performance. [PERSON] et al. [22] proposed a multi-step buckled beam piezoelectric energy harvester to obtain a wide operational bandwidth. [PERSON] et al. [23] presented a bridge shaped nonlinear energy harvester to widen the frequency bandwidth significantly. In consideration of stochastic, multi-frequencies, and time-varying motions, frequency up-conversion using direct impact or non-contact magnetic plucking can be another solution.