# Experiment Investigation of Bistable Vibration Energy Harvesting with Random Wave Environment
## 2 Materials
### Bistable Vibration Characteristics  
For the purpose of the examination, the cantilever beam and the concentrated mass block on the vibration model body shown in Figure 3 are separated from each other, and the other part of the vibration model body is fixed to the end of the analytical model. Noise-induced by random waves and periodic excitation force by a small shaker is applied to the fixed end. In reference [29], a detailed theoretical analysis of an inverted cantilever beam with a mass block at the front end is carried out. Based on the research results, the motion equation of the motion model shown in Figure 3 is expressed as follows. \[ \begin{split}&\left[I_{t}N_{5}^{2}+m+\rho AN_{1}+\left(\rho AN_{3}+mN_ {4}^{2}+I_{t}N_{5}^{4}\right)x^{2}\right]\ddot{x}+\left[\rho AN_{3}+mN_{4}^{2} +I_{t}N_{5}^{4}\right]x\dot{x}^{2}\\ &\qquad\qquad+\left[EN_{6}-\rho AgN_{8}-mgN_{4}+2 EIN_{7}x^{2} \right]x=-[\rho AN_{2}+m]\ddot{x}_{t}.\end{split}  \tag{1}\]  
However, \(m\) is the mass of the mass block, \(L\) is the length of the beam, \(E\) is the elastic Young's modulus of the beam, \(I\) is the secondary moment of the cross section of the beam, \(\rho\) is the density of the beam, \(A\) is the cross sectional area of the beam, \(g\) is the gravitational acceleration, and \(I_{t}\) is the rotational inertia moment of the mass block, \(\ddot{x_{t}}\) is the accelerate. Symbols \(N_{1}\)\(\sim\)\(N_{8}\) are represented by the following equations:  
\[N_{1}=\frac{(3\pi-8)L}{2\pi}, \tag{2}\]  
\[N_{2}=\frac{(\pi-2)L}{\pi}, \tag{3}\]  
Figure 3: The bistable vibration model of a cantilever with a mass block at the tip. \[N_{3}=\frac{\pi^{2}(2\pi^{2}-9)}{384L}, \tag{4}\]  
\[N_{4}=\frac{\pi^{2}}{8L}, \tag{5}\]  
\[N_{5}=\frac{\pi}{2L}, \tag{6}\]  
\[N_{6}=\frac{\pi^{4}}{32L^{3}}, \tag{7}\]  
\[N_{7}=\frac{\pi^{6}}{2^{9}L^{5}}, \tag{8}\]  
\[N_{8}=\frac{\pi^{2}-4}{16}. \tag{9}\]  
Equations (2) to (9) are substituted for the Equation (1), and the equations of motion are expressed as follows. \[\Big{(}M_{1}+J_{1}x^{2}\Big{)}\ddot{x}+J_{1}x\dot{x}^{2}+\Big{(}K_{1}+J_{2}x^{ 2}\Big{)}x=-M_{2}\ddot{x}_{t}. \tag{10}\]  
Then,  
\[N_{2}=\frac{(\pi-2)L}{\pi}, \tag{3}\]0  
\[N_{2}=\frac{(\pi-2)L}{\pi}, \tag{3}\]1  
\[N_{2}=\frac{(\pi-2)L}{\pi}, \tag{3}\]2  
\[N_{2}=\frac{(\pi-2)L}{\pi}, \tag{3}\]3  
\[N_{2}=\frac{(\pi-2)L}{\pi}, \tag{3}\]4  
From the equation of motion (10), the potential energy of the proposed vibration model is expressed by the following equation. \[N_{2}=\frac{(\pi-2)L}{\pi}, \tag{3}\]5  
As an actual structural parameter of the vibration model shown in Figure 1, the mass of the mass block is 0.15 kg, the length of the beam is 380 mm, the width is 30 mm, the thickness of the plate is 0.5 mm, the young's modulus of the beam material is 210 GPa, the density is 7850 kg/m\({}^{3}\), and the gravitational acceleration is 9.8 m/s\({}^{2}\), and the result of the potential energy distribution shown in Figure 4 is obtained. \[N_{2}=\frac{(\pi-2)L}{\pi}, \tag{3}\]6 The potential energy distribution of bistable vibration model. In Figure 4, the abscissa is the transverse displacement of the mass block, and the ordinate is the potential energy, it is found from the figure that the potential energy distribution has two-well characteristics (There is a valley of potential energy at each side of the graph, the valley bottom is the equilibrium position of the mass block at rest). At the center of the graph, there is a mountain potential energy peak, which corresponds to the potential energy at which the mass block passes the vibration model's central position. In general, the mass receiving relatively weak random excitation signal vibrates singly on the left or right side, as shown in Figure 5a,b. In order to generate stochastic resonance, a periodic excitation signal is needed to be applied to the mass block to cross the central position and generate a bistable vibration state across both sides, as shown in Figure 5c.