## 2 Materials and Methods
Where \(f_{k}(t)\) is given by:  
\[f_{k}(t)=F_{o}e^{\omega t}\delta\bigg{(}x-\frac{l}{2}\bigg{)}. \tag{16}\]  
Equation (15) is a nonhomogeneous second order differential equation which can be assumed as the following equation:  
\[\eta_{k}(t)=A_{1}e^{(-\xi\omega_{k}+\omega_{k}\sqrt{\zeta^{2}-1})t}+A_{2}e^{( -\xi\omega_{k}-\omega_{k}\sqrt{\zeta^{2}-1})t}+\frac{F_{o}}{-2\xi\omega_{k}^{ 2}}\text{cos}\omega t. \tag{17}\]  
where the constants \(A_{1},\ A_{2}\) are obtained from the boundary conditions, \(F_{o}\) is the amplitude of the external load. Now using superposition technique, the deflection at \(x\) at instant time, in another word the deflection with the function of \(x\) and \(t\), is given by:  
\[u(x,t)=\sum\
olimits_{k}\text{sin}\bigg{(}\frac{k\pi x}{L}\bigg{)}\bigg{[}A_{ 1}e^{(-\xi\omega_{k}+\omega_{k}\sqrt{\zeta^{2}-1})t}+A_{2}e^{(-\xi\omega_{k}- \omega_{k}\sqrt{\zeta^{2}-1})t}+\frac{F_{o}}{-2\xi\omega_{k}^{2}}\text{cos} \omega t\bigg{]} \tag{18}\]  
The temperature and the applied stress both influence the beam's degree of strain. Because of this, it is possible to express strain as a function of stress and temperature using the following equations:  
\[S=g(\sigma,T), \tag{19}\]  
\[dS=\bigg{(}\frac{\partial S}{d\sigma}\bigg{)}_{T}d\sigma+\bigg{(}\frac{ \partial S}{\partial T}\bigg{)}_{\sigma}dT, \tag{20}\]  
where \(E=\bigg{(}\frac{\partial\sigma}{\partial S}\bigg{)}_{T}\) and \(\alpha=\bigg{(}\frac{\partial\pi}{\partial T}\bigg{)}_{\sigma}\). While \(\bigg{(}\frac{\partial\pi}{\partial T}\bigg{)}_{\sigma}\) and \(\bigg{(}\frac{\partial\sigma}{\partial S}\bigg{)}_{T}\) are the gradients in temperature and stresses, respectively. It is possible to determine the beam's direct strain by integrating Equation (21), assuming that the temperature is uniform, and the thermal expansion (\(\alpha\)) is linear:  
\[dS=\frac{d\sigma}{E}+\alpha Tl \tag{21}\]  
where \(l\) is the length of the beam. Electromechanical ModellingPiezoelectricity is a substance's ability to generate electrical charges in response to mechanical strain. The inverse piezoelectric effect refers to the opposite effect of deformation caused by an external electric field. A set of constitutive equations is typically used to describe the aforementioned phenomena. Given that the total strain in the transducer is the sum of mechanical strain caused by mechanical stress and controllable actuation strain caused by the applied electric voltage, their strain-charge form is as follows:  
\[D_{m}=\varepsilon_{ik}^{e}\psi_{k}+d_{mi}\sigma_{i}, \tag{22}\]  
\[S_{i}=s_{ij}^{\psi}\sigma_{j}+d_{mi}\psi_{m}, \tag{23}\]  
where \(\sigma\) is the stress vector; \(S\) is the strain vector; \(\psi\) is the vector of applied electric field; \(D\) is the vector of electric displacement; \(s\) is the compliance tensor coefficients; \(d\) is the matrix of piezoelectric coupling coefficients; and \(\varepsilon\) is the matrix of electric permittivity. The superscripts in the compliance matrix indicate that the electric field (\(\psi\)) is constant and in electric permittivity it indicates that the stress (\(\sigma\)) is constant. The indices \(m,k=1,2,3\) and \(i\), \(j=1,2,3\ldots 6\) refer to the direction's relationship with the transducer stress in the different directions. By employing the approach of linear electrical circuit analysis, we were able to derive the equation for the piezoelectric transducer, complete with the attached resistor. The piezo component was analyzed as if it were an electric charge source, and its electrical capacity was calculated accordingly. It was determined that the entire system could be modeled as an RC electric circuit with an initial state of zero (since the capacitor had not been pre-charged) as indicated in Figure 1b. A transient state is produced in the system whenever the harmonic voltage \(V_{p}\) is generated as a result of the deformation of the piezo element in the system.