# Response Analysis of Asymmetric Monostable Harvesters Driven by Color Noise and Band-Limited Noise
## 2 Model of the Asymmetric Monostable Energy Harvester (MEH)  
Figure 1 depicts the schematic plot of the piezoelectric monostable energy harvester (MEH) composed of a stainless steel layer, two piezoelectric layers at the roots, a tip magnet, and two external rotational magnets. By adjusting the magnet polarity and system parameters \(d,h\), and \(\theta\), a monostable energy harvesting system could be achieved. In the experiments, an MEH with a perfectly symmetric potential energy function is difficult or even impossible to obtain due to the asymmetry in materials and structures. Therefore, the MEH obtained is always with an asymmetric potential energy function. Based on the Hamilton principle and Lagrange equation, the dynamic model of the asymmetric potential MEH can be expressed as follows:  
\[\left\{\begin{array}{l}m\Xi^{\prime\prime}\left(\overline{t}\right)+c\Xi^{ \prime}\left(\overline{t}\right)+k_{1}\overline{z}+k_{2}\Xi^{2}+k_{3}\Xi^{3}- \theta\overline{v}\left(\overline{t}\right)=\overline{F}\left(\overline{t} \right)\\ C_{p}\overline{v}^{\prime}\left(\overline{t}\right)+\frac{\overline{v}\left( \overline{t}\right)}{\overline{R}}+\theta\overline{z}^{\prime}\left(\overline{ t}\right)=0\end{array}\right. \tag{1}\]  
in which \(m\) is the equivalent mass, \(c\) is the equivalent damping, \(\theta\) is the equivalent electromechanical coupling coefficient, \(C_{p}\) is the equivalent capacitance, and \(R\) is the load resistance. \(k_{1},k_{2}\), and \(k_{3}\) are the coefficients of linear, quadratic, and cubic forces, respectively. \(\overline{z}\), \(\overline{v}\), and \(\overline{t}\) are the tip displacement of the cantilever, the output voltage across the load resistance, and the time, respectively. In addition, \(\left(\ \right)^{\prime}\) refers the derivate with respect to time \(\overline{t}\).  
Figure 1: Schematic plot of the asymmetric monostable energy harvester (MEH).  
In order to simplify the analysis process, we introduce the variables \(x=\frac{\pi}{l_{0}},t=\overline{t}\omega_{0},v=\frac{C_{0}\overline{\eta}}{\beta l _{0}}\) to obtain the following non-dimensional model.  
\[\left\{\begin{array}{l}\ddot{x}(t)+2\xi_{x}^{x}(t)+x(t)+\beta x^{2}(t)+\delta x (t)^{3}-\kappa^{2}v(t)=\eta(t)\\ \dot{v}(t)+\alpha v(t)+\dot{x}(t)=0\end{array}\right. \tag{2}\]  
where \(l_{0}\) and \(\omega_{0}\) are the length scale and resonance frequency. In Equation (2), \(x\), \(v\), and \(\eta(t)\) refer to the dimensionless displacement, voltage, and ambient random excitations, respectively, which can be modeled by color noise and band-limited noise. Additionally, \(\zeta\) is the damping ratio, \(\alpha\) is the time constant, and \(\kappa^{2}\) is the dimensionless electromechanical coupling coefficient. \(\beta\) and \(\delta\) are the coefficients of dimensionless quadratic and cubic nonlinear forces, respectively. Furthermore, the upper dot is the derivate of dimensionless time \(t\). For the non-dimensional model, the average output power in a certain time range could be calculated as:  
\[P=\alpha E\Big{[}v^{2}\Big{]} \tag{3}\]