# Response Analysis of the Tristable Energy Harvester with an Uncertain Parameter
## 3 The Approximation of the TEH with an Uncertain Parameter
### Equivalent Deterministic System  
There is no doubt that the errors in manufacturing and installation of TEHs cannot be totally eliminated, particularly for the distance between the tip magnet and external magnets, the distance between two external magnets and the angle of external magnets. These uncertain variables are closely related to the potential function, and therefore we set the coefficient \(\lambda\) of the potential function as an uncertain parameter, which is expressed as  
\[\lambda=\bar{\lambda}+\sigma\xi, \tag{10}\]  
where \(\bar{\lambda}\) is the mean value of \(\lambda\); \(\xi\) is a bounded random variable defined on \([-1,1]\) with the probability density function \(p(\xi)\), which has been given in Equation (4); \(\sigma\) is the intensity of \(\lambda\). Then, the dimensionless electromechanical equations of TEH with an uncertain parameter can be written as follows  
\[\left\{ \begin{array}{l}\bar{x}(t)+\zeta\dot{x}(t)+x(t)+\left(\bar{\lambda} \!+\!\sigma\xi\right)x(t)^{3}+\mu x(t)^{5}-\alpha v(t)=A\cos(\omega t),\\ \bar{v}(t)+\beta v(t)\!+\!\alpha\dot{x}(t)=0.\end{array} \right. \tag{11}\]  
According to the theory of orthogonal polynomial approximation above, the responses of Equation (11) can be expressed by the following series  
\[\left\{ \begin{array}{l}x(t,\xi)=\sum\limits_{i=0}^{N}x_{i}(t)H_{i}(\xi),\\ v(t,\xi)=\sum\limits_{i=0}^{N}v_{i}(t)H_{i}(\xi),\end{array} \right. \tag{12}\]  
where \(x_{i}(t)\) and \(v_{i}(t)\) can be obtained by Equation (9). While \(N\rightarrow\infty\), \(\sum\limits_{i=0}^{N}x_{i}(t)H_{i}(\xi)\) and \(\sum\limits_{i=0}^{N}v_{i}(t)H_{i}(\xi)\) are strictly equivalent to \(x(t,\xi)\) and \(v(t,\xi)\). To meet the needs of the calculation efficiency, we take \(N=4\) in Equation (12), then  
\[\left\{ \begin{array}{l}x(t,\xi)\approx\sum\limits_{i=0}^{4}x_{i}(t)H_{i}(\xi ),\\ v(t,\xi)\approx\sum\limits_{i=0}^{4}v_{i}(t)H_{i}(\xi),\end{array} \right. \tag{13}\]  
which are approximate solutions with a minimal mean square residual error. Substituting Equation (13) into Equation (11), yields\[\left\{ \begin{array}{c}\sum\limits_{i=0}^{4}\dot{x}_{i}(t)H_{i}(\xi)+\zeta\sum \limits_{i=0}^{4}\dot{x}_{i}(t)H_{i}(\xi)+\sum\limits_{i=0}^{4}x_{i}(t)H_{i}( \xi)+\bar{\lambda}\left[\sum\limits_{i=0}^{4}x_{i}(t)H_{i}(\xi)\right]^{3}+ \sigma\xi\left[\sum\limits_{i=0}^{4}x_{i}(t)H_{i}(\xi)\right]^{3}\\ \hskip 56.905512 pt+\mu\left[\sum\limits_{i=0}^{4}x_{i}(t)H_{i}(\xi)\right]^{5} -\alpha\sum\limits_{i=0}^{4}v_{i}(t)H_{i}(\xi)=A\cos(\omega t),\\ \sum\limits_{i=0}^{4}\dot{v}_{i}(t)H_{i}(\xi)+\beta\sum\limits_{i=0}^{4}v_{i}( t)H_{i}(\xi)+\alpha\sum\limits_{i=0}^{4}\dot{x}_{i}(t)H_{i}(\xi)=0.\end{array} \right. \tag{14}\]  
Since the product of any two Chebyshev polynomials can be transformed into a linear combination of related single Chebyshev polynomials, the cubic term and quantic term can be reduced as  
\[\left[\sum\limits_{i=0}^{4}x_{i}(t)H_{i}(\xi)\right]^{3}=\sum\limits_{i=0}^{ 12}X_{i}(t)H_{i}(\xi), \tag{15}\]  
\[\left[\sum\limits_{i=0}^{4}x_{i}(t)H_{i}(\xi)\right]^{5}=\sum\limits_{i=0}^{ 20}Y_{i}(t)H_{i}(\xi). \tag{16}\]  
Moreover, according to recurrent formulas (6) of the Chebyshev polynomials,  
\(\sigma\xi\left[\sum\limits_{i=0}^{4}x_{i}(t)H_{i}(\xi)\right]^{3}\) in Equation (14) can be reduced to  
\[\sigma\xi\left[\sum\limits_{i=0}^{4}x_{i}(t)H_{i}(\xi)\right]^{3}=\sigma\xi \sum\limits_{i=0}^{12}X_{i}(t)H_{i}(\xi) =\frac{1}{2}\sigma\sum\limits_{i=0}^{12}X_{i}(t)[H_{i-1}(\xi)+H_ {i+1}(\xi)] \tag{17}\] \[=\frac{1}{2}\sigma\sum\limits_{i=0}^{12}\left[X_{i-1}(t)+X_{i+1} (t)\right]H_{i}(\xi),\]  
where \(X_{-1}(t)\) and \(X_{\xi}(t)\) are supposed to be zero. Substituting expressions (15), (16) and (17) into Equation (14), we have  
\[\left\{