# Finite Element Approaches to Model Electromechanical, Periodic Beams
## 2 Numerical Model  
The structure under consideration, presented in Figure 1, is a sequence of 50 unit cells, consisting of an aluminium beam with piezoelectric rectangular strips (from APC International, Ltd., Cat.No. 70-1000, item 721) attached on both sides. Each pair of piezoelectrics is connected to the RLC resonant circuit with a controlled inductance. The material parameters taken into calculations were as follows: for aluminium E = 67.5 GPa, \(\rho=2700\) kg/m\({}^{3}\) and \(\
u=0.33\) and for piezoelectric material E = 63 GPa, \(\rho=7800\) kg/m\({}^{3}\), \(\
u=0.33\) and piezoelectric electro-mechanical coupling coefficient \(k_{31}=0.35\). The geometry of the analysed beam was as follows: the length \(L=1\) m, the width \(b=0.02\) m, the height \(h=0.01\) m, the single RLC element length was 0.01 m.  
Numerical modelling of the piezoelectric material properties was based on the approach proposed in the literature [9; 10]. The authors presented there a formulae for calculation of the effective Young's modulus of the piezoelectric element being an element of a resonant circuit. The piezoelectric material has frequency-dependent stiffness and damping, and the frequency itself depends on the parameters of the resonance circuit. Therefore, the effective Young's modulus of the piezoelectric material in the resonant circuit can be described by the equation:  
\[E_{p}^{SU}(\omega)=E_{p}^{D}\left(1-\frac{k_{31}^{2}}{1+i\omega C_{p}^{c}Z^{ SU}(\omega)}\right), \tag{1}\]  
where \(E_{p}^{SU}\) is the effective Young's modulus of the piezoelectric material in a closed circuit mode, \(E_{p}^{D}\) is the effective Young's modulus of the piezoelectric material in an open circuit mode, \(k_{31}\) is the electro-mechanical coupling coefficient of thee piezoelectric material, \(C_{p}^{c}\) is the capacitance of the piezoelectric element, and \(Z^{SU}\) is the impedance of a resonant circuit. In the carried out numerical calculations, PZT impedance has been taken into account as an electrical circuit parameter. The impedance of the aluminium beam itself has been neglected because the aluminium beam is not a part of a controlled electrical circuit. In the presented paper, changes in the vibration characteristics of the aluminium beam with attached, actively controlled PZT elements have been analysed. The displacement and deformation fields of the analysed beam structure have been assumed according to the Timoshenko theory. The mathematical formulae can be expressed by [11; 12]:  
\[ \begin{cases}u(x)=z\phi(x)\\ w(x)=w_{0}(x),\end{cases}  \tag{2}\]  
Figure 1: A concept of an electromechanical periodic structure (**a**), unit cell (**b**). \[ \begin{cases}\epsilon_{x}=\frac{\partial u(x)}{\partial x}=z\frac{d\phi(x)}{dx}\\ \gamma_{xx}=\frac{\partial w(x)}{\partial x}+\frac{\partial u(x)}{\partial z}= \frac{dw_{0}(x)}{dx}+\phi(x),\end{cases}  \tag{3}\]  
where \(u(x)\) and \(w(x)\) are respectively the longitudinal and transverse components of the element displacements expressed in the global coordinate system, while the independent rotation \(\phi(x)\) around the \(y\) axis and the lateral displacement \(w_{0}(x)\) are nodal displacements, defined in neutral element axis. Following the standard FEM procedures, the inertia matrix \(M\) and the stiffness matrix \(K\) were evaluated:  
\[\mathbf{M}=\rho\iiint_{V}\mathbf{N}^{t}\mathbf{N}dV,\quad\mathbf{K}=\rho\iiint _{V}\mathbf{B}^{t}\mathbf{D}\mathbf{B}dV, \tag{4}\]  
where \(\rho\) is the density of the material, \(\mathbf{D}\) is the matrix of elasticity coefficients, and \(\mathbf{N}\) and \(\mathbf{B}\) are the shape function and strain-displacement matrices, respectively. The presented numerical simulations have been obtained by the use of the classical Finite Element Method (FEM), the Frequency Domain Spectral Element Method (FDSFEM, details have been widely presented by [PERSON] in [13], the interested Reader is encouraged to follow the source) or the Time Domain Spectral Finite element Method (TDSFEM) approach.