# New Optimal Design of Multimode Shunt-Damping Circuits for Enhanced Vibration Control
## 2 Methodology
### Finite-Element Model of the Laminated Composite Beam
#### 2.1.3 Variational Formulation  
The extended [PERSON]'s principle shown in Equation (10) is utilized to obtain the coupled electromechanical equations of motion for the piezocomposite beam:  
\[\int\limits_{t_{1}}^{t_{2}}(\delta(T-U_{m}-U_{E})+\delta W)dt=0 \tag{10}\]  
where \(t_{1}\) and \(t_{2}\) denote arbitrary time moments. Additionally, \(T\) stands for the kinetic energy, \(U_{m}\) for the mechanical potential energy and \(U_{E}\) for the electrical potential energy, and \(W\) stands for the virtual work carried out by the external forces, which are given by the following relations:  
\[T=\frac{1}{2}\int\limits_{\Omega_{b}}\rho^{(b)}\vec{u}^{T}\vec{u}\;d\Omega+ \frac{1}{2}\int\limits_{\Omega_{p1}}\rho^{(p_{1})}\vec{u}^{T}\vec{u}\;d\Omega+ \frac{1}{2}\int\limits_{\Omega_{p2}}\rho^{(p_{2})}\vec{u}^{T}\vec{u}\;d\Omega \tag{11}\]\[U_{m}=\frac{1}{2}\int\limits_{\Omega_{b}}\varepsilon^{T}\sigma^{(b)}d\Omega+ \frac{1}{2}\int\limits_{\Omega_{p_{1}}}\varepsilon^{T}\sigma^{(p_{1})}d\Omega+ \frac{1}{2}\int\limits_{\Omega_{p_{2}}}\varepsilon^{T}\sigma^{(p_{2})}d\Omega \tag{12}\]  
\[U_{E}=\frac{1}{2}\int\limits_{\Omega_{p_{1}}}E_{2}^{(1)}D_{z}^{(p_{1})}d\Omega+ \frac{1}{2}\int\limits_{\Omega_{p_{2}}}E_{2}^{(2)}D_{z}^{(p_{2})}d\Omega \tag{13}\]  
\[\delta W^{e}=\delta\overline{u}^{T}\!f_{L}+\sum\limits_{i}\delta v_{i}q_{i} \tag{14}\]  
In the above equations, a dot denotes a partial derivative with respect to time \(t\), \(\Omega\) denotes the volume, \(\rho\) denotes the mass density and the subscripts \(b\), \(p_{1}\) and \(p_{2}\) stand for the host beam structure and the lower (1) and the upper (2) piezoelectric layer, respectively. Finally, \(f_{L}\) denotes concentrated forces and \(q_{i}\) concentrated electric charges. The composite beam is divided into a finite number of elements \(\Omega_{e}\) according to the finite-element method (FEM); therefore, each element's own energy terms must be calculated before they can be assembled to represent the entire structure. Using the displacement Equation (1) and performing some mathematical manipulations, the first variation of the kinetic energy of an element with a surface-bonded piezoelectric layer can be written as:  
\[ \begin{array}{l}\delta T^{e}=\int\limits_{0}^{L_{e}}\int\limits_{A_{b}}^{ \rho(b)}\{\delta\dot{u}_{0}+z\delta\dot{\psi}_{x}\}(\dot{u}_{0}+z\dot{\psi}_{x })+\delta\dot{w}_{0}\dot{w}_{0}]dAdx\\ +\sum\limits_{i=1}^{2}\int\limits_{0}^{L_{e}}\int\limits_{A_{p_{i}}}^{\rho(p_ {i})}\{\rho(\dot{u}_{0}+z\delta\dot{\psi}_{x})(\dot{u}_{0}+z\dot{\psi}_{x})+ \delta\dot{w}_{0}\dot{w}_{0}]dAdx\\ =\int\limits_{0}^{L_{e}}[\delta\dot{u}_{0}I_{1}\dot{u}_{0}+\delta\dot{u}_{0}I _{2}\dot{\psi}_{x}+\delta\dot{\psi}_{x}I_{2}\dot{u}_{0}+\delta\dot{\psi}_{x}I _{3}\dot{\psi}_{x}+\delta\dot{w}_{0}I_{1}\dot{w}_{0}]dx\\ =\int\limits_{0}^{L_{e}}\left\{\begin{array}{c}\delta\dot{u}_{0}\\ \delta\dot{w}_{0}\\ \delta\dot{\psi}_{x}\end{array}\right\}^{T}\left[\begin{array}{ccc}I_{1}&0 &I_{2}\\ 0&I_{1}&0\\ I_{2}&0&I_{3}\end{array}\right]\left\{\begin{array}{c}i_{0}\\ \dot{w}_{0}\\ \dot{\psi}_{x}\end{array}\right\}dx\equiv\int\limits_{0}^{L_{e}}\left(\delta \dot{\overline{u}}\right)^{T}\!\overline{\mathcal{I}}\,\dot{\overline{u}}dx \end{array}  \tag{15}\]  
where \(L_{e}\) is the length of the element and \(A_{(a)}\), \(a=b,p_{1}\) and \(p_{2}\) is the cross-sectional area of the beam and the lower and upper piezoelectric layer, respectively. In addition, the elements of the inertia matrix \(\mathcal{I}\) are given by:  
\[I_{i}=I_{i}^{(b)}+\sum\limits_{k=1}^{2}I_{i}^{(p_{k})},\;\mathrm{i}=1,2,3\;\; \left\{I_{1}^{(a)}\;\;I_{2}^{(a)}\;\;I_{3}^{(a)}\right\}=b\int\limits_{z}\rho^ {(a)}\Big{\{}1\;\;z\;\;z^{2}\Big{\}}dz.,\;\;a=b,p_{1},p_{2} \tag{16}\]  
Notice that integration with respect to \(z\) in Equation (16) is performed either on the region occupied by the elastic beam or on the region occupied by the piezoelectric elements.