## 2 Materials and Methods
The longitudinal elongation of a fiber at a distance \(y\) that is considered to be the neutral fiber enables one to express the strain as:  
\[S(x,y,t)=-yk=-y\frac{\partial^{2}u(x,t)}{\partial x^{2}}. \tag{5}\]  
Employing an ElectroPuls wave generator, a dynamic cyclic load was applied while the displacement of Equation (6) was kept under control:  
\[A_{0}\text{sin}(\omega t)\delta\bigg{(}x-\frac{l}{2}\bigg{)}, \tag{6}\]  
where \(A_{0}\)--amplitude, \(\omega\)--excitation frequency, \(l\)--length of the laminated MFC piezoelectric beam and \(t\)--time (s). Based on the above Equations (1) and (2), the dynamic equation of the beam can be written as:  
\[EI(x)\frac{\partial^{4}u}{\partial x^{4}}(x,t)+(\rho A)\frac{\partial^{2}u}{ \partial t^{2}}(x,t)=\delta\bigg{(}x-\frac{l}{2}\bigg{)}F_{o}(t). \tag{7}\]  
Equation (7) is a partial differential equation for which an analytical solution is difficult to find. In this instance, the problem is effectively solved using the separation approach (superposition). Therefore:  
\[\frac{\partial^{2}M}{\partial x^{2}}(x,t)+(\rho A)\frac{\partial^{2}u}{ \partial t^{2}}(x,t)=\delta\bigg{(}x-\frac{l}{2}\bigg{)}F_{o}(t). \tag{1}\]0  
where \(\varphi_{k}(x)\) defines the natural frequency and eigenfunction of the \(k\)-th vibration mode which can be analytically solved considering the homogenous solution of the beam vibration equation given in the Equation (6), and \(\eta_{k}(t)\) is the modal mechanical response of the \(k\)-th vibration mode. To solve the natural frequency and mode shapes, we consider the undamped mode in the bending vibration of the beam with uniform sectional property, the response is given by:  
\[\frac{\partial^{2}M}{\partial x^{2}}(x,t)+(\rho A)\frac{\partial^{2}u}{ \partial t^{2}}(x,t)=\delta\bigg{(}x-\frac{l}{2}\bigg{)}F_{o}(t). \tag{1}\]1  
In which \(\omega\) is the natural frequency of the beam. Substituting Equation (9) into Equation (7), one has:  
\[\frac{\partial^{2}M}{\partial x^{2}}(x,t)+(\rho A)\frac{\partial^{2}u}{ \partial t^{2}}(x,t)=\delta\bigg{(}x-\frac{l}{2}\bigg{)}F_{o}(t). \tag{1}\]2  
where: \(\beta^{2}=\omega\sqrt{\frac{\rho A}{EI}}\). The general solution of the Equation (10) is:  
\[\frac{\partial^{2}M}{\partial x^{2}}(x,t)+(\rho A)\frac{\partial^{2}u}{ \partial t^{2}}(x,t)=\delta\bigg{(}x-\frac{l}{2}\bigg{)}F_{o}(t). \tag{1}\]3  
where \(A\), \(B\), \(C\), and \(D\) are integration constants to be evaluated using the boundary conditions. Assuming the beam is a simply supported beam, the boundary conditions at the ends of the support imply the following conditions on mode shape functions:  
\[ \begin{array}{ll}\varphi(0)=\varphi(l)=0,\\ \frac{d^{2}\varphi(0)}{dx^{2}}=\frac{d^{2}\varphi(l)}{dx^{2}}=0.\end{array}  \tag{12}\]  
The coefficient of the modal function is given by: B = D = 0  
\[ \begin{array}{ll}\begin{bmatrix}\text{sin}h(\beta L)&\text{sin}(\beta L)\\ \text{sin}h(\beta L)&-\text{sin}(\beta L)\end{bmatrix}\begin{bmatrix}A\\ C\end{bmatrix}=\begin{bmatrix}0\\ 0\end{bmatrix}\end{array}  \tag{13}\]Substituting the boundary conditions in Equation (11), the natural frequency and mode shape function in the \(k\)-th mode is given by:  
\[\omega_{k}=(k\pi)^{2}\sqrt{\frac{EI}{\rho L^{4}}}\cdot\varphi_{k}(x)=\text{sin}k \bigg{(}\frac{k\pi}{L}x\bigg{)}. \tag{14}\]  
By employing the method of mode superimposition, Equation (7) is replaced to the mechanical equation of motion expressed in modal coordinates, which is as follows:  
\[\frac{d^{2}\eta_{k}(t)}{dt^{2}}+2\xi\omega_{k}\frac{d\eta_{k}(t)}{dt}+\omega_{k }^{2}\eta_{k}(t)=f_{k}(t). \tag{15}\]  
\(\omega_{k}\) is the modal structural damping ratio derived from the proportional damping assumption, and \(\xi\) is the undamped natural frequency of the \(k\)-th vibration mode.