# Optimal Piezoelectric Potential Distribution for Controlling Multimode Vibrations
## 2 Governing Equations for a Piezoelectric Coupled Beam
Choosing a suitable spectrum load, it is possible to activate the piezoelectric plate to obtain \(w_{p}(t)\) opposite in phase with respect to \(w_{F}(t)\), so that, indicating with \(|w_{T}|,|w_{F}|\), and \(|w_{p}|\) the amplitudes of the relative displacements, \(|w_{T}|=|w_{F}|-|w_{p}|\). In this paper, the optimal potential distribution has been considered as the one that minimizes the amplitude of \(w_{T}\) so that the optimal distribution will be the one that, for the same V, maximizes \(|w_{p}|\). It can be demonstrated that ([22]), without taking into account the transient part,  
\[|\;\widetilde{w}_{p}\;(\delta(V_{i}),x_{i},x_{i-1})|=\sum_{j=1}^{N_{p}}\Big{|} \frac{B_{j}\big{(}\delta(V_{i}),x_{i},x_{i-1}\big{)})V_{j}\phi_{j}(L_{b})}{ \alpha\omega_{i}}\Big{|} \tag{9}\]where \(N_{s}\) are the excited modes. Choosing a load that excites two modes, indicating with \(i_{1}\) and \(i_{2}\) the excited modes and considering a parameter \(r\) that takes into account the distribution of the load between the two eigenfrequencies, the following can be written:  
\[\mathbf{F}(t)=\Big{[}F_{1}(1-r)cos(\omega_{i_{1}}t)+F_{2}cos(\omega_{i_{2}}t) \Big{]}\mathbf{e}_{2}. \tag{10}\]  
To achieve the maximum efficiency (or, as mentioned above, in order to obtain \(w_{p}(t)\) opposite in phase to \(w_{F}(t)\)), the spectrum of the potential distribution has been chosen with the same characteristics of the load ([22]) and parameterized by the same parameter:  
\[V(t)=V\Big{[}(1-r)cos(\omega_{i_{1}}t)+rcos(\omega_{i_{2}}t)\Big{]}. \tag{11}\]  
Substituting Equation (11) in Equation (9),  
\[\begin{split}\mid\widetilde{w}_{p}\ (\delta(V_{i}),x_{i},x_{i-1}) \mid=V\ \widetilde{M}\ \Big{\{}&\big{(}\frac{1-r}{\alpha\omega_{i_{1}}} \big{)}\Big{|}\sum_{i=1}^{N_{p}}\delta(V_{i})\big{[}\phi_{i_{1}}^{\prime}(x_{i })-\phi_{i_{1}}^{\prime}(x_{i-1})\big{]}\phi_{i_{1}}(L_{b})\Big{|}+\\ &\the points \(x_{j}\) coincide with the extrema of \(f_{i_{1},i_{2}}(r,x)\). In particular, considering that the function \(\phi_{i_{1}}^{\prime}(x)\phi_{i_{1}}(L_{b})\) assumes its maximum positive value in \(x=L_{b}\),  
\[0<\phi_{j}^{\prime}(L_{b})\phi_{j}(L_{b})>\phi_{j}^{\prime}(x)\phi_{j}(L_{b})\ \ \forall x\in[0,L_{b})\ \ \forall j, \tag{15}\]  
it is convenient to choose \(\delta(V_{N_{p}})=+1\) in Equation (13). Moreover, comparing \(f_{i_{1},i_{2}}(r,x)\) with Equation (13), it can be noticed that, to obtain the maximum value of \(\mid\widehat{w}_{p}\mid\), \(x_{N_{p}-1}\) must coincide with the first minimum of \(f_{i_{1},i_{2}}(r,x)\) closer to \(L_{b}\), choosing \(\delta(V_{N_{p}-1})=-1\), the point \(x_{N_{p}-2}\) must coincide with the first maximum of \(f_{i_{1},i_{2}}(r,x)\) closer to \(x_{N_{p}-1}\), choosing \(\delta(V_{N_{p}-2})=+1\), etc. In this manner, \(\mid\widehat{w}_{p}\mid\) will be  
\[ \begin{split}\mid\widehat{w}_{p}\mid=&\frac{V \overset{\sim}{\alpha}}{\alpha}\Big{\{}\big{|}\big{(}\frac{1-r}{\alpha\omega_{ i_{1}}}\big{)}\phi_{i_{1}}^{\prime}(L_{b})\phi_{i_{1}}(L_{b})-2\big{(}\frac{1-r}{ \omega_{i_{1}}}\big{)}\phi_{i_{1}}^{\prime}(x_{N_{p}-1})\phi_{i_{1}}(L_{b})+2 \big{(}\frac{1-r}{\omega_{i_{1}}}\big{)}\phi_{i_{1}}^{\prime}(x_{N_{p}-2}) \phi_{i_{1}}(L_{b})+\\ &+...-2(-1)^{N_{p}}\big{(}\frac{1-r}{\omega_{i_{1}}}\big{)}\phi_ {i_{1}}^{\prime}(x_{1})\phi_{i_{1}}(L_{b})\Big{|}+\\ &+\big{|}\big{(}\frac{r}{\alpha\omega_{i_{2}}}\big{)}\phi_{i_{2} }^{\prime}(L_{b})\phi_{i_{2}}(L_{b})-2\big{(}\frac{r}{\omega_{i_{2}}}\big{)} \phi_{i_{2}}^{\prime}(x_{N_{p}-1})\phi_{i_{2}}(L_{b})+2\big{(}\frac{r}{\omega_ {i_{2}}}\big{)}\phi_{i_{2}}^{\prime}(x_{N_{p}-2})\phi_{i_{2}}(L_{b})+\\ &+...-2(-1)^{N_{p}}\big{(}\frac{r}{\omega_{i_{2}}}\big{)}\phi_{i_ {2}}^{\prime}(x_{1})\phi_{i_{2}}(L_{b})\big{|}\Big{\}}\end{split} . \tag{16}\]  
In Figure 3, the functions \(f_{i_{1},i_{2}}\) for different combinations of the first four eigenmodes, and various values of \(r\), are reported.