# Response Analysis of the Tristable Energy Harvester with an Uncertain Parameter
## 2 The Tistable Energy Harvester  
The schematic diagram of a piezoelectric cantilever vibration energy harvester [24] is shown in Figure 1. The mechanical part consists of a vertical cantilever beam with a tip magnet and two external magnets located symmetrically near the free of the metal substrate. Two piezoceramic layers are attached onto the root of cantilever beam. Under external excitations, the beam vibrates in the magnetic field created by the two external magnets and a tip magnet, which leads to the physical deformation of piezoelectric materials near the free end of cantilever beam. Due to the polarization phenomenon induced by piezoelectric effect, the opposite charges are generated on the upper and lower surfaces of piezoelectric materials, resulting in the output voltage, which realizes the conversion from ambient vibrations to the electrical energy. Adjusting the magnetoelastic interaction depending on the distance \(h\) between the tip magnet and external magnets, the distance \(d\) between the two external magnets and angle \(\alpha\) of the external magnets in a vertical direction, the energy harvester may have one, three (with one stable) or five (with three stable) equilibrium positions. Among these, the TEH can jump among three equilibrium points at low excitation level and can realize more frequently jumping to generate higher output voltage when the external excitation intensity is fairly large. The electromechanical equations of the TEH can be expressed as  
\[\left\{ \begin{array}{l}m\ddot{x}(t)+c\dot{x}(t)+\frac{\mathrm{d}U(\dot{x})}{ \mathrm{d}\dot{x}}-\theta\bar{v}(t)=F(t),\\ C_{p}\dot{v}(t)+\frac{1}{k_{1}}\bar{v}(t)+\dot{\theta}\dot{x}(t)=0,\end{array} \right. \tag{1}\]  
where \(\bar{x}(t)\), \(m\) and \(c\) are the tip displacement in the transverse direction, the equivalent mass and the equivalent damping, respectively; \(\theta\) is the electromechanical coupling coefficient; \(C_{p}\) is the capacitance of the piezoelectric energy harvester; \(\bar{v}(t)\) is the output voltage measured across the load resistance \(R_{i}\); \(F(t)\) is the external mechanical force as the excitation term. The potential function \(U(\dot{x})\), which depends on the tip displacement, has the following form  
\[U(\dot{x})=\frac{1}{2}k_{1}\bar{x}^{2}+\frac{1}{4}k_{3}\bar{x}^{4}+\frac{1}{6} k_{5}\bar{x}^{6}, \tag{2}\]  
where \(k_{1}\), \(k_{3}\) and \(k_{5}\) are the coefficients of nonlinear restoring forces. For convenience, the dimensionless electromechanical equations under the periodic force \(A\cos(\omega t)\)[24] can be recast as follows  
\[\left\{ \begin{array}{l}\ddot{x}+\zeta\dot{x}+x+\lambda x^{3}+\mu x^{5}-\alpha v =A\cos(\omega t),\\ \dot{v}+\beta v+\alpha\dot{x}=0,\end{array} \right. \tag{3}\]  
where \(\zeta\), \(\lambda\) and \(\mu\) represent the mechanical damping ratio, the coefficient of the dimensionless cubic nonlinearity and dimensionless quantic nonlinearity, respectively; \(\alpha\) represents the dimensionless electromechanical coupling coefficient; \(\beta\) represents the ratio between the period of the mechanical system to the time constant of the harvester. Different properties of the electromechanical model will be performed on account of the different values of \(\lambda\) and \(\mu\). When \(\mu>0\) and \(\lambda<-2\sqrt{\mu}\), the system (3) is a TEH. The tristable potential functions with different values of \(\lambda\) and \(\mu\) are shown in Figure 2, which have one middle potential well and two symmetric potential wells on both sides. Moreover, the potential well barrier of two symmetric potential wells becomes  
Figure 1: Schematics of a general piezoelectric energy harvester [24]. smaller with the increasing of the values of \(\lambda\) and \(\mu\), yet there are small variations for the depth and width of middle potential well.