# Experiment Investigation of Bistable Vibration Energy Harvesting with Random Wave Environment
## 2 Materials
### Prediction of Periodic Signals Subject to Stochastic Resonance  
In order to generate the stochastic resonance, periodic signals with an appropriate frequency from outside are required. In Reference [8], a predictive equation is proposed for  
Figure 5: Three vibration states of the bistable vibration model. (**a**) The single stable vibration state on the left. (**b**) The single stable vibration state on the right. (**c**) The bistable vibration state. Figure 6: The time series displacement of random signal and its frequency component distribution. (**a**) Vibration displacement. (**b**) Power spectrum. the frequency of periodic signals in which stochastic resonance is most likely to occur in the bistable oscillation model. \[f_{K}=\frac{\omega_{0}\omega_{b}}{4\pi q}\exp\biggl{(}-\frac{\Delta U}{mD} \biggr{)}. \tag{17}\]  
Here, \(\omega_{0}\) and \(\omega_{b}\) are the natural angular frequencies of the stable point \(x_{b}\) and the unstable point \(x_{0}\) on both sides of the potential energy of the bistable vibration model, respectively, \(q\) is the attenuation parameter, \(\Delta U\) is the potential difference expressed by the difference between the maximum and minimum potential energy, and \(D\) is the random vibration intensity expressed by the white noise signal, which can be calculated according to the following equation, respectively. \[\omega_{0}=\sqrt{\frac{\left|U^{\prime\prime}\left(x_{0}\right)\right|}{m}}, \tag{18}\]  
\[\omega_{0}=\sqrt{\frac{\left|U^{\prime\prime}\left(x_{b}\right)\right|}{m}}, \tag{19}\]  
\[\mathrm{q}=\frac{c}{m}, \tag{20}\]  
\[\Delta U=U(x_{0})-U(x_{b}), \tag{21}\]  
\[D=\frac{1}{2N}\sum_{i=0}^{N}\bigl{(}\dot{x}_{i}-\dot{x}_{a\pi\sigma\sigma} \bigr{)}^{2}. \tag{22}\]  
Here, \(c\) is the attenuation coefficient, \(\dot{x}_{i}\) and \(\dot{x}_{a\pi\sigma\sigma}\) are the measured value and the average value of the velocity of the moving body when the random signal is used for excitation, respectively, and \(N\) is the sample point of the measurement experiment. When designing the bistable vibration model, the random vibration intensity \(D\) represented by Equation (22) can be measured firstly, which makes the \(\Delta U\) represented by Equation (21) slightly larger than that of mean kinetic energy of random vibration \(mD\). In the case of only random excitation, the mass block vibrates at a slightly larger amplitude on the left or right side of the central axis. When the periodic excitation signal is given, it is ensured that even the amplitude with a small periodicity can produce probability resonance. The symmetry condition determines the position of the unstable point \(x_{0}=0\) at the center of the potential energy. The position of the stable point \(x_{b}\) on both sides of the potential energy can be calculated by the following. \[x_{b}=\pm\sqrt{\frac{\rho AgN_{8}+mgN_{4}-EIN_{6}}{2 EIN_{7}}}. \tag{23}\]  
Equations (5), (7)-(9) are substituted by Equation (23) to obtain the following equation,  
\[x_{b}=\pm\sqrt{\frac{16\rho AgL^{5}(\pi^{2}-4)+32\pi^{2}mgL^{4}-8\pi^{4}EIL^{ 2}}{\pi^{6}EI}}. \tag{24}\]  
The configuration parameter of the proposed bistable oscillation model is substituted for Equation (24) to obtain \(x_{b}=0.2236m\). The following equation is obtained by differentiating twice the potential energy Equation (16). Figure 6:0The values \(x_{0}\) and \(x_{b}\) are substituted into Equations (25), (18) and (19), respectively, and the respective angular frequencies are obtained as follows. Figure 6:1  
Figure 6:2  
The corresponding natural frequencies corresponding to them are as follows. Figure 6:3  
Figure 6:4  
According to the measured results of actual vibration experiment, \(c=0.077\;\mathrm{Ns/m}\) and \(D=0.635\;\mathrm{J/kg}\) are obtained. Further, by substituting Equations (20), (26) and (27) into Equation (17), the predicted value of the exciting frequency which is most likely to occur is obtained as follows.