# Response Analysis of Asymmetric Monostable Harvesters Driven by Color Noise and Band-Limited Noise
## 3 Response of the Asymmetric MEHs under Color Noise
### Moment Differential Equation Analysis  
For the convenience of calculation, we define \(x_{1}=x,x_{2}=\dot{x},x_{3}=v,x_{4}=\eta(t)\) and then the model of the MEH excited by color noise can be expressed as follows:  
\[\left\{ \begin{array}{l}\frac{dx_{1}}{dt}=x_{2}\\ \frac{dx_{2}}{dt}=-2\xi x_{2}-x_{1}-\beta{x_{1}}^{2}-\delta{x_{1}}^{3}+\kappa^ {2}x_{3}+x_{4}\\ \frac{dx_{3}}{dt}=-\alpha x_{3}-x_{2}\\ \frac{dx_{4}}{dt}=-\frac{1}{\tau}x_{4}+\frac{1}{\tau}n(t)\end{array} \right. \tag{6}\]  
Based on the theory of moment differential equation [33], the first-order statistical moments can be obtained as follows:  
\[\frac{dE(X_{1})}{dt}=E(X_{2}) \tag{7}\]  
\[\frac{dE(X_{2})}{dt}=-2\xi E(X_{2})-E(X_{1})-\beta E\Big{(}{X_{1}}^{2}\Big{)} -\delta E\Big{(}{X_{1}}^{3}\Big{)}+\kappa^{2}E(X_{3})+E(X_{4}) \tag{8}\]  
\[\frac{dE(X_{3})}{dt}=-\alpha E(X_{3})-E(X_{2}) \tag{9}\]  
\[\frac{dE(X_{4})}{dt}=-\frac{1}{\tau}E(X_{4}) \tag{10}\]  
of response displacement will increase with the increase in noise intensity. Thus, the absolute value of mean displacement increases with the increase in the absolute value of the quadratic coefficient. The mean square voltage and output power exhibit the same increasing trend with an increase in the noise intensity. When it comes to the damping ratio, Figure 2b demonstrates that its increase results in the decrease in the absolute value of mean displacement, thus reducing the mean square voltage and average power output of the asymmetric MEH. Furthermore, it should be noticed the simulated data agree well with the theoretical outcomes. The influences of the time constant ratio and electromechanical coupling coefficient on the response of asymmetric MEH are depicted in Figure 3a,b, respectively. It is observed that the variation of the time constant ratio does not affect the response of displacement. The mean square value of output voltage decreases with the increase in the time constant ratio, while the average output power first increases and then decreases with an increase in the value of the time constant ratio. Therefore, there is an optimum time constant ratio for the asymmetric MEH to produce maximum output power. From Figure 3b, one can notice that that the increase in the electromechanical coupling coefficient has a negative effect on the output performance of the asymmetric MEH, decreasing the absolute mean value of response displacement, mean square voltage, and average power output. Regarding the influence of quadratic and cubic coefficients on the mean displacement, mean square voltage, and power output of the asymmetric MEH, the numerical and theoretical results are illustrated in Figure 4a,b. It can be seen that the quadratic and cubic coefficients have a reverse effect on the output performance. For the quadratic coefficient, its increase results in a rise in the absolute value of mean displacement; thus, the mean square voltage output and average output power are all increasing with the increasing trend in the quadratic coefficient. Therefore, the appearance of asymmetric potential in MEH enhances the output performance, and the increase in the degree of asymmetry makes the average power much higher. The reason for this phenomenon is that the increase in the degree of asymmetry makes the nonlinear restoring force have a smaller value on the left of the equilibrium position. On the contrary, the increase in the cubic coefficient leads to higher values of the nonlinear restoring forces on both sides of the equilibrium position, thus decreasing the mean displacement, mean square voltage, and power output. Figure 2: Influence of (**a**) noise intensity and (**b**) damping ratio on the mean displacement, mean square voltage, and power output. Furthermore, Figure 5 explains the influence of correlation time on the response of asymmetric potential MEH. It is seen that the mean displacement increases with an increase in the correlation time, while the mean square voltage and average power output decreases.