# Response Analysis of Asymmetric Monostable Harvesters Driven by Color Noise and Band-Limited Noise
## 1 Introduction  
Vibrational energy harvesting is a green energy technology that can transduce the vibrational energy existing in the surroundings into available electrical energy. This technology has great potential application value as it has a promising ability to supply electricity for low-power consumption wireless sensors and portable electronics. There are three transmission mechanisms applied to convert vibrational energy into electricity, namely, the piezoelectric [1], electromagnetic [2], and electrostatic [3] energy harvesters. Due to the merits of high energy density and easy fabrication in miniaturization, piezoelectric energy harvesters have garnered a great deal of attention. Initially, linear energy harvesters were designed as different structures and then optimized for enhancing energy harvesting efficiency. However, linear energy harvesters only perform well near their resonance frequencies, and a slight change in the excitation frequency will decrease their performance. To address these problems, a large number of studies have focused on broadening the frequency band of response by applying the mechanical and magnetic coupling approaches [4]. Particularly, the investigations of monostable [5], bistable [6], and tristable [7] energy harvesters have attracted a lot of attention. As an example, [PERSON] et al. [8] proposed a nonlinear monostable energy harvester (MEH) by applying a permanent magnet end mass that interacted with the field of oppositely poled stationary magnets. By adjusting the nonlinear magnetic forces between the magnets, both hardening and softeningresponses can be obtained for enhancing the energy harvesting performance. [PERSON] et al. [9] presented an MEH by introducing symmetric magnetic attraction to a piezoelectric cantilever beam and a pair of stoppers to confine the maximum deflection of the beam. Under excitation with an amplitude of 3 m/s\({}^{2}\), a 54% increase in the operating bandwidth and a 253% increase in the magnitude of output power were obtained compared to its linear counterpart. Due to the snap-through characteristics, bistable oscillators were widely applied in energy harvesting for the reason that they could generate considerably more energy in a wider frequency range. By utilizing the device proposed by [PERSON] and [PERSON] [10], [PERSON] et al. [6,11] proposed a bistable energy harvester (BEH) and established the model by introducing piezoelectric coupling and Kirchhoff laws. Numerical, theoretical, and experimental data all indicated that the BEH can achieve chaotic and large-amplitude inter-well oscillation, thus resulting in an 800% increase in the power amplitude. By introducing two rotatable external magnets to the energy harvesting systems, [PERSON] et al. [12] proposed a BEH and pointed out that the system could cover a broad low-frequency range of 4-22 Hz by changing the magnet orientation. [PERSON] et al. [13] proposed a new magnetically coupled bistable piezoelectric energy harvesting approach for underwater applications. The bistable harvesters were also designed and optimized by applying the post-buckled beam [14], flexure hinge mechanism [15], inner stress of composite materials [16], as well as multiple degrees of freedom (DOF) systems [17]. To further improve the energy harvesting performance, the tristable [18], quad-stable [19], and quin-stable [20] piezoelectric energy harvesters have also been investigated extensively. In the environment, the vibration has the characteristic of randomness, and the energy always distributes in a wide frequency range. Therefore, the investigation of piezoelectric energy harvesters subjected to random excitations has great significance. Numerical, experimental, and theoretical methods have all been applied to evaluate the output of the nonlinear harvesters. For example, [PERSON] et al. [21] numerically studied the response of a BEH excited by stationary Gaussian white noise.