# Experiment Investigation of Bistable Vibration Energy Harvesting with Random Wave Environment
## 4 Discussion
### Applicable Conditions for Vibration Power Generation  
An inverted cantilever beam bistable model was used in this study. When a cantilever beam with a mass block at its tip is in a stationary state, the cantilever beam is bent laterally by the gravity of the mass block, the position is stopped at the position as shown in Figure 22, and the position becomes the vibration center point of the single stable vibration. In addition, since the vibration model is a left-right symmetrical structure, and there is a single stable vibration state on both sides of the central symmetry axis, it is easy to judge that the model can form a double stable vibration system. When the mass of the mass block of Figure 22 is relatively small, the cantilever beam in the stationary state moves along the vertical direction. The mass block stops at the symmetry axis position, and the vibration model becomes a single stable vibration centered on the symmetry axis. In other words, it is necessary to consider the conditions necessary for the bistable oscillation. According to Reference [38], maximum deflection equation for lateral loading of cantilever beams subjected to lateral loading, as follows:  
\[\delta=\frac{PL^{3}}{3 EI}. \tag{34}\]  
From Equation (34), bending stiffness of cantilever beams can be calculated by the following equation:  
\[K_{Beam}=\frac{P}{\delta}=\frac{3 EI}{L^{3}}. \tag{35}\]  
In the equation, \(K_{beam}\) is the flexural rigidity of the cantilever beam, \(L\) is the length of the beam, \(E\) is the elastic Young's modulus of the material, and \(I\) is the second moment of the beam. According to Reference [38], the angle of inclination of the center point of the mass block can be calculated by the following equation. \[\theta=\frac{PL^{2}}{2 EI}. \tag{36}\]  
The distance from the center point to the intersection point O of the concentrated mass block is expressed by the following, Equations (34) and (35):  
\[L_{e}=\frac{\delta}{\theta}=\frac{2L}{3}. \tag{37}\]  
Figure 22: Bi-stable vibration model of inverted cantilever with lumped mass block at its tip. (**a**) Original vibration model. (**b**) Virtual vibration model. By Equation (37), in the course of vibration, since the distance Le is always constant in relation to the length L of the beam, it is possible to apply the virtual model to the virtual model of the rigid link with the mass block at the tip of Figure 22b, and to consider the vibration characteristic using the virtual model. In the equivalent model shown in Figure 22b, the bending rigidity \(K_{Mass}\) for deformation of the mass block by gravity \(mg\) and returning it to the center symmetry axis is approximately expressed by the following equation. \[K_{Mass}=\frac{mg\sin\theta}{L_{e}\theta}=\frac{mg}{L_{e}}. \tag{38}\]  
Equation (37) is substituted for Equation (38) to obtain the following equation:  
\[K_{Mass}=\frac{3 mg}{2L}. \tag{39}\]  
By comparing the Equations (35) and (39), if \(K_{Ram}>K_{Mass}\), the cantilever beam is kept in the vertical position in the stationary state because of its strong flexural rigidity, the vibration of a mass block becomes a single stable vibration centering on the symmetry axis. Conversely, if \(K_{Mass}\geq K_{Ram}\), the equation can be obtained by Equations (35) and (39), as follows:  
\[mg\geq\frac{2 EI}{L^{2}}. \tag{40}\]  
Since the cantilever beam's flexural rigidity is relatively weak, the cantilever beam is bent laterally by the gravity of the mass block, and the mass block is initially displaced. The vibration state is bistable vibration centered on the two initial displacement positions on the left and right sides. Therefore, it can be seen from Equation (40) that the vibration state of the inverted cantilever beam vibration model with mass block at the front end is a necessary condition for the double stable vibration.