# Optimal Piezoelectric Potential Distribution for Controlling Multimode Vibrations
## 2 Governing Equations for a Piezoelectric Coupled Beam  
In Figure 1, a scheme of the piezoelectric coupled beam is reported. Where \(L_{e}\), \(L_{in}\), \(L_{a}\), and \(L_{p}\) indicate the work of external, inertial, applied, and piezoelectric forces, respectively, the virtual work principle can be written as  
\[\delta L_{e}=\delta L_{in}+\delta L_{a}+\delta L_{p}. \tag{1}\]  
The calculus of \(\delta L_{p}\) has been confirmed by many studies (one of the first studies in this regard was conducted by [PERSON] et al. ([7]) in 1987), and experimental evidence shows that, in the case of perfectly bonded material interfaces, the stresses that a plate applies to the beam are concentrated at the ends of the plate. Therefore, by applying two harmonic voltages opposite in the phase between the upper plate and the lower plate, such action can be modeled by a couple of flexural moments M concentrated at their ends (Figure 2). Figure 1: The piezoelectric coupled beam. Figure 2: Actions of the piezoelectric plate on the beam. If  
\[\left\{ \begin{array}{l}\Lambda(t)=\frac{d_{3}}{T_{a}}V(t)\\ \psi=\frac{E_{a}T_{b}}{E_{a}T_{a}}\end{array} \right., \tag{2}\]  
the flexural moment can be written as ([7])  
\[M(t)=\frac{\psi}{6+\psi}E_{a}cT_{a}T_{b}\Lambda(t) \tag{3}\]  
and \(\delta L_{p}\) will be  
\[\delta L_{p}=M\sum_{i=1}^{N_{p}}\Big{[}\Big{(}\frac{\partial\,\widetilde{ \widetilde{w}}}{\partial x}|_{x=x_{i}}-\frac{\partial\,\widetilde{\widetilde{ w}}}{\partial x}|_{x=x_{i-1}}\Big{)}\delta(V_{i})\Big{]} \tag{4}\]  
where \(x_{i}\) represents the points where the potential changes its sign, so that \(\delta(V_{i})\) will be \(+1\) or \(-1\) depending on the potential distribution on the plate \(i\) (Figure 2). The optimization problem can be traced back to the research of the optimal distribution of the \(\delta(V_{i})\) or, in other words, to the research of the optimal distribution of the points \(x_{i}\). By introducing the inertial, internal, and external forces in Equation (1) and applying the modal analysis technique, the following equilibrium equation is obtained  
\[\mathbf{M}\;\tilde{\mathbf{X}}(t)+\mathbf{K}\mathbf{X}(t)=\mathbf{B}\big{(} \delta(V_{i})\big{)}V(t)+\mathbf{F}(t) \tag{5}\]  
where \(\mathbf{M}\) and \(\mathbf{K}\) are, respectively, the mass and the stiffness matrix, \(\mathbf{X}\) is the vector of the modal amplitudes, and the vector \(\mathbf{B}\) is  
\[\mathbf{B}\big{(}\delta(V_{i}),x_{i},x_{i-1}\big{)}= \widetilde{M}\sum_{i=1}^{N_{p}}\delta(V_{i})\Big{[}\phi_{1}^{ \prime}(x_{i})-\phi_{1}^{\prime}(x_{i-1}),\phi_{2}^{\prime}(x_{i})-\phi_{2}^{ \prime}(x_{i-1}),...,\phi_{N}^{\prime}(x_{i})-\phi_{N}^{\prime}(x_{i-1})\Big{]} \tag{6}\]  
where \(\widetilde{M}\)\(=\)\(\frac{\psi}{6+\psi}E_{a}cT_{a}T_{b}\frac{d_{3}}{T_{a}}\). If the viscous damping is added, Equation (5) becomes:  
\[\mathbf{M}\;\tilde{\mathbf{X}}(t)+\mathbf{C}\;\dot{\mathbf{X}}(t)+\mathbf{K} \mathbf{X}(t)=\mathbf{B}\big{(}\delta(V_{i})\big{)}V(t)+\mathbf{F}(t). \tag{7}\]  
Assuming that \(\phi_{i}\) are the orthonormal modes, it follows that \(\mathbf{M}\) is the identity matrix \(\mathbf{I}\) and that \(\mathbf{K}\) is the natural frequencies matrix \(\boldsymbol{\omega}\). If \(\mathbf{C}\) is related to the Rayleigh damping (\(\mathbf{C}=\alpha\mathbf{M}+\beta\mathbf{K}\)) where \(\beta=0\), Equation (7) can be written as  
\[\tilde{\mathbf{X}}(t)+\alpha\dot{\mathbf{X}}(t)+\boldsymbol{\omega}^{2} \mathbf{X}(t)=\mathbf{B}\big{(}\delta(V_{i})\big{)}V(t)+\mathbf{F}(t). \tag{8}\]  
Since the flexural eigenmode of the cantilever beam achieves its maximum displacement amplitude at the free end, this point has been chosen as a reference point for the vibrations of the whole cantilever beam. Indicating with \(w_{F}(t)\) and \(w_{p}(t)\) the free end displacement due to, respectively, the load F and the piezoelectric load, the total displacement can be written as \(w_{T}(t)=w_{F}(t)+w_{p}(t)\) as a consequence of the linearity of the system.