# New Optimal Design of Multimode Shunt-Damping Circuits for Enhanced Vibration Control
## 2 Methodology
### Finite-Element Model of the Laminated Composite Beam
#### 2.1.4 Displacement and Electric Field Discretization  
The finite-element formulation is based on the super-convergent FE approach developed by [PERSON] et al. [27]. The super-convergent element produces an exact elemental stiffness matrix by using higher-order interpolating polynomials that are obtained by solving the static part of the governing equations of motion. However, the calculated consistent mass matrix is only roughly accurate. This element predicts natural frequency with a better degree of precision and smaller discretization than any other traditional finite elements since the stiffness matrix is precise for static analysis.  
The beam element with length \(L_{e}\) has two nodes with three mechanical degrees of freedom (DoFs) per node: the axial and transverse displacement \(u_{0}\) and \(w_{0}\) and the rotation \(\psi_{x}\). Thus, the vector of mechanical DoFs is defined by:  
\[\mathbf{d}^{e}=\left\{u_{0}^{1},w_{0}^{1},\psi_{x}^{1},u_{0}^{2},w_{0}^{2},\psi_{x}^ {2}\right\}^{T} \tag{22}\]  
The axial displacement \(u_{0}\) and the rotation \(\psi_{x}\) are interpolated by quadratic polynomials, while the transverse displacement \(w_{0}\) is interpolated by cubic polynomials and is expressed in terms of the finite-element shape functions. Then, the generalized displacement vector is expressed as follows:  
\[\overline{u}=\left\{u_{0},w_{0},\psi_{x}\right\}^{T}=\mathbf{N}(x)\mathbf{d}^{e}(t)= \left\{\mathbf{N}_{u},\mathbf{N}_{w},\mathbf{N}_{\psi}\right\}^{T}\mathbf{d}^{e} \tag{23}\]  
where \(\mathbf{N}_{u},\mathbf{N}_{w}\) and \(\mathbf{N}_{\psi}\) are the super-convergence shape functions which are given in [27].  
Using Equation (23), the generalized stain field can be written in the following form:  
\[\overline{e}=\left\{\begin{array}{c}\epsilon_{x}^{0}\\ x_{x}\\ \gamma_{xz}^{0}\end{array}\right\}=\left\{\begin{array}{c}\frac{\partial\mathbf{ N}_{u}}{\partial x}\\ \frac{\partial\mathbf{N}_{\psi}}{\partial x}\\ \frac{\partial\mathbf{N}_{w}}{\partial x}+\mathbf{N}_{\psi}\end{array}\right\}\mathbf{d}^ {e}=\mathbf{B}\mathbf{d}^{e} \tag{24}\]  
Finally, to represent the electric potential difference at the top of the lower and upper piezoelectric patches, two extra electric DoFs per element are inserted, namely, \(\sigma_{1}^{e}\) and \(\sigma_{1}^{e}\). Note that, for every additional piezoelectric layer, an additional electric DoF is needed per element. Thus, the vector of electrical DoFs is defined by:  
\[\mathbf{v}^{e}=\left\{\sigma_{1}^{e},\sigma_{2}^{e}\right\}^{T} \tag{25}\]  
and the electric field distribution can be written as:  
\[\mathbf{E}^{e}=\left\{\begin{array}{c}E_{z}^{(1)}\\ E_{z}^{(2)}\end{array}\right\}^{e}=\left[\begin{array}{cc}-1/h_{p1}&0\\ 0&-1/h_{p2}\end{array}\right]\left\{\begin{array}{c}\
u_{1}^{e}\\ \
u_{2}^{e}\end{array}\right\}=\left[\begin{array}{cc}B_{v}^{(p_{1})}&0\\ 0&B_{v}^{(p_{2})}\end{array}\right]=\mathbf{B}_{v}\mathbf{v}^{e} \tag{26}\]  
where \(B_{v}^{(p_{1})}\) and \(B_{v}^{(p_{2})}\) are the electrical field gradient operator of the lower and upper piezoelectric layer, respectively.