# New Optimal Design of Multimode Shunt-Damping Circuits for Enhanced Vibration Control
## 2 Methodology
### Finite-Element Model of the Laminated Composite Beam
#### 2.1.5 Coupled Electromechanical System  
Substituting Equations (23)-(26) into the energy expressions of Equations (14)-(18) and the [PERSON]'s principle of Equation (10) leads to:  
\[\begin{array}{c}\int\limits_{I_{1}}^{I_{2}}\Biggl{\{}\ \ (\delta\mathbf{d}^{e})^{T} \Biggl{[}\int\limits_{0}^{L_{e}}\mathbf{N}^{T}\mathbf{\mathcal{I}N}d\mathbf{x}\mathbf{\bar{d}} +\int\limits_{0}^{L_{e}}\mathbf{B}^{T}\mathbf{\mathcal{D}B}dx\mathbf{d}^{e}+\int\limits_{ 0}^{L_{e}}\mathbf{B}^{T}\mathbf{\mathcal{E}}^{(p)}B_{v}dx\mathbf{v}^{e}-\mathbf{F}_{m}^{e} \Biggr{]}\\ +(\delta\mathbf{v}^{e})^{T}\Biggl{[}-\int\limits_{0}^{L_{e}  
definition of the shape functions \(\mathbf{N}_{u},\ \mathbf{N}_{w}\) and \(\mathbf{N}_{\psi}\) and integrating, we can obtain the explicit form of the above matrices.  
The global electromechanical coupled equations can be obtained by assembling the elemental Equations (28) and (29). It should be noted that, during the assembly process, a special numbering scheme should be utilized to identify elements with a piezoelectric layer, such as 1 for elements with a piezoelectric patch and 0 for the rest elements. Furthermore, a single electrical degree of freedom, \(v_{i}\), should be assigned to each piezoelectric layer since they are completely covered by uniform electrodes in order to take into account the equipotential state on the electrodes. By partitioning the electric potential into sensory (\(v_{1}\)) and active (\(v_{2}\)) components, the global electromechanical system equations are given by:  
\[M_{u}\ddot{d}+C_{d}\dot{d}+\mathbf{K}_{u}\mathbf{d}+\mathbf{\Theta}_{1}v_{1}+\mathbf{\Theta}_{2} v_{2}=\mathbf{F}_{m} \tag{30}\]  
\[-\mathbf{\Theta}_{1}^{T}\mathbf{d}+C_{p}v_{1}=\mathcal{Q}_{1} \tag{31}\]  
\[-\mathbf{\Theta}_{2}^{T}\mathbf{d}+C_{p}v_{2}=\mathcal{Q}_{2} \tag{32}\]  
In the above equations, \(\mathbf{d}\) denotes the global vector of mechanical degrees of freedom, \(\mathbf{M}_{u}\) denotes the global mass matrix, \(\mathbf{K}_{u}\) the stiffness matrix, \(\mathbf{F}_{m}\) the mechanical force terms, \(\mathbf{\Theta}=[\mathbf{\Theta}_{1},\mathbf{\Theta}_{2}]\) denotes the global electromechanical coupling matrix and \(C_{p}\) denotes the piezoelectric capacitance, identical for both patches. Finally, \((v_{1},\mathcal{Q}_{1})\) and \((v_{2},\mathcal{Q}_{2})\) are the voltage/charge pairs associated with the top and the bottom piezoelectric patch, respectively. The finite-element model of Equations (30)-(32) can be used for a wide range of applications relating to piezoelectric smart structures, such as active vibration control, energy harvesting, etc. In the following section, it is suitably adapted to a case where the top piezoelectric patch is'shunted', that is, it stays connected to a passive electrical network for passive vibration control applications. In a case where both piezoelectric patches are shorted, the difference of the electric potential between its electrodes vanishes (\(v_{1}=v_{2}=0\)). Therefore, Equation (30) becomes:  
\[\mathbf{M}_{u}\ddot{\mathbf{d}}+\mathbf{K}_{u}\mathbf{d}=\mathbf{F}_{m} \tag{33}\]  
It is noted that the matrices \(\mathbf{M}_{u}\) and \(\mathbf{K}_{u}\) contain the contributions of the piezoelectric patch. Since we are interested in vibration control, a powerful representation of the system is the state-space representation, which allows the determination of both the frequency and impulse responses of various shunt configurations and electrical components. Thus, short-circuit Equation (33) is transformed into state-space form, as follows:  
\[ \begin{array}{l}\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{w}\\ \mathbf{y}=\mathbf{C}\mathbf{x}+\mathbf{D}\mathbf{w}\end{array}  \tag{34}\]  
where  
\[\mathbf{A}= \begin{bmatrix}\mathbf{0}&\mathbf{I}\\ -\mathbf{M}_{u}^{-1}\mathbf{K}_{u}&\mathbf{0}\end{bmatrix} ,\ \ \mathbf{B}= \begin{bmatrix}\mathbf{0} \\ \mathbf{M}_{u}^{-1}\end{bmatrix} ,\ \ \mathbf{x}= \begin{cases}\mathbf{d}\\ \mathbf{d}\end{cases}