## 2 Stiffened Laminated Panel with Piezoelectric Materials
### Finite Element Formulations
\(\mathbf{M}_{p}^{e}\) is the element mass matrix of plate and \(\mathbf{M}_{xs}^{e}\) is the element mass matrix of the stiffeners. The element electrical energy is calculated as follows:  
\[ \begin{array}{ll}W_{E}^{e}&=\frac{1}{2}\int_{V_{p}}\mathbf{E}^{T}\Big{(} \mathbf{e}\mathbf{\varepsilon}_{p}+\mathbf{\varepsilon E}\Big{)}dV\\ &=-\frac{1}{2}\phi^{eT}\int_{A}\Big{(}\mathbf{B}_{0}^{T}\mathbf{E}_{1}\mathbf{ B}_{m}+\mathbf{B}_{0}^{T}\mathbf{E}_{2}\mathbf{B}_{b}\Big{)}dA\mathbf{\xi}_{p}^{e}+ \frac{1}{2}\phi^{eT}\int_{A}\mathbf{B}_{0}^{T}\mathbf{E}_{3}\mathbf{B}_{0}dA \phi^{e}\\ &=-\frac{1}{2}\phi^{eT}\overline{\mathbf{K}}_{\phi u}\mathbf{\xi}_{p}^{e}+ \frac{1}{2}\phi^{eT}\mathbf{K}_{\phi\phi}\phi^{e}\end{array}  \tag{12}\]  
where \(\mathbf{K}_{\phi u}\) is element elastic-electric coupling stiffness matrix and \(\mathbf{K}_{\phi\phi}\) is element electric stiffness matrix. \(\mathbf{E}_{3}=\int_{z_{p}}\mathbf{\varepsilon}_{ij}dz\) and \(V_{p}\) is the volume of the piezoelectric layer. The virtual work and the applied electrical charge density are derived as follows:  
\[ \begin{array}{ll}\Delta W_{S}^{e}&=\Delta\mathbf{\xi}_{p}^{e}\int_{S_{1}} \mathbf{B}_{f}^{T}\mathbf{f}_{s}dA-\Delta\phi^{eT}\int_{S_{2}}\mathbf{B}_{0}^{ T}\mathbf{q}_{s}dA\\ &=\Delta\mathbf{\xi}_{p}^{e}\mathbf{F}_{s}^{e}+\Delta\phi^{eT}\mathbf{F}_{q}^{e }\end{array}  \tag{13}\]  
where \(S_{1}\) and \(S_{2}\) are the surface areas that the surface forces and electrical charges are applied, respectively. \(\mathbf{F}_{s}^{e}\) is the applied mechanical force due to surface forces, and \(\mathbf{F}_{q}^{e}\) is the applied electrical charge. The governing equation can be described using the [PERSON]'s principle. \[\mathbf{M}_{uu}^{e}\ddot{\mathbf{\xi}}_{p}+\mathbf{K}_{uu}^{e}\mathbf{\xi}_{p }^{e}+\mathbf{K}_{u\phi}^{e}\phi^{e}=\mathbf{F}_{s}^{e} \tag{14}\]  
\[\mathbf{K}_{\phi u}^{e}\mathbf{\xi}_{p}^{e}+\mathbf{K}_{\phi\phi}^{e}\phi^{e}= \mathbf{F}_{q}^{e} \tag{15}\]  
where \(\mathbf{M}_{uu}^{e}=\mathbf{M}_{p}^{e}+\mathbf{M}_{xs}^{e}\), \(\mathbf{K}_{uu}^{e}=\mathbf{K}_{p}^{e}+\mathbf{K}_{xs}^{e}\). Then, the global dynamic equation is obtained as follows:  
\[\mathbf{M}_{uu}\ddot{\mathbf{\xi}}_{p}+\mathbf{C}_{uu}\dot{\mathbf{\xi}}_{p}+ \Big{(}\mathbf{K}_{uu}+\mathbf{K}_{u\phi}\mathbf{K}_{\phi\phi}{}^{-1}\mathbf{ K}_{\phi u}\Big{)}\mathbf{\xi}_{p}=\mathbf{F}_{s}+\mathbf{K}_{u\phi}\phi_{a} \tag{16}\]  
where damping matrix \(\mathbf{C}_{uu}\) is assumed as \(\mathbf{C}_{uu}=\alpha\mathbf{M}_{uu}+\beta\mathbf{K}_{uu}\). Next, modal transformations and state-space formulations are derived. The nodal displacement is represented by  
\[\mathbf{\xi}_{p}=\mathbf{\Phi}\mathbf{\eta} \tag{17}\]  
The global dynamic equation can be transformed to the reduced modal space form as follows:  
\[\mathbf{\Phi}^{T}\mathbf{M}_{uu}\mathbf{\Phi}\mathbf{\eta}+\mathbf{\Phi}^{T} \mathbf{C}_{uu}\mathbf{\Phi}\dot{\mathbf{\eta}}+\mathbf{\Phi}^{T}\Big{(} \mathbf{K}_{uu}+\mathbf{K}_{u\phi}\mathbf{K}_{\phi\phi}{}^{-1}\mathbf{K}_{ \phi u}\Big{)}\mathbf{\Phi}\mathbf{\eta}=\mathbf{\Phi}^{T}\mathbf{F}_{s}+ \mathbf{\Phi}^{T}\mathbf{K}_{u\phi}\phi_{a} \tag{18}\]  
\[\overline{\mathbf{M}}\ddot{\mathbf{\eta}}+\overline{\mathbf{C}}\ddot{\mathbf{ \eta}}+\overline{\mathbf{K}}\mathbf{\eta}=\overline{\mathbf{F}}_{s}+\overline {\mathbf{K}}_{u\phi}\phi_{a} \tag{19}\]  
Introducing the state-space variable \(\mathbf{\xi}=\Big{\{}\ \mathbf{\eta}\ \ \ \dot{\mathbf{\eta}}\ \ \dot{\mathbf{\eta}}\ \ \dot{\mathbf{\eta}}\ \ \dot{\mathbf{\eta}}\ \ \dot{\mathbf{\eta}}\ \ \dot{\mathbf{\eta}}\ \ \dot{\mathbf{\eta}}\ \ \dot{\mathbf{\The output equations can be written in the physical coordinates and state space coordinates as  
\[\mathbf{y}=\mathbf{C}_{o}\mathbf{\xi}_{p},\mathbf{\overline{y}}=\left\{ \begin{array} []{c}\mathbf{C}_{o}\mathbf{\Phi}\\ 0\end{array} \right\}\mathbf{\xi}=\overline{\mathbf{C}}_{o}\mathbf{\xi}, \tag{22}\]  
where \(\mathbf{C}_{o}\) is the output matrix. Finally, the linear quadratic regulator (LQR) control algorithm [4; 18] is applied. The control gain can be obtained.