## 3 Numerical Results and Discussions
### The Evaluation of Stiffeners' Locations  
Figure 5 shows the cantilevered laminated composite panel with two \(x\)-stiffeners. Piezoelectric ceramics are boded at the upper and lower surfaces of the structure. The material properties of the composite panel (T300/976 graphite-epoxy) and the piezoceramic (PZT G1195N) are already presented in Table 1. The structural model is consist of four layers (\(15^{\circ}/-15^{\circ}/-15^{\circ}/15^{\circ}\)) with (0.4/0.4/0.4/0.4) mm layers. The stiffeners with width 6 mm and depth 12 mm are made of the same materials. The thickness of each piezoceramic layer is 0.2 mm. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Natural Frequency (Hz)** & **Ref. [16]** & **Ref. [23]** & **Present** \\ \hline
1 st & 68.61 & 68.47 & 68.13 \\
2 nd & 71.20 & 68.66 & 68.77 \\
3 rd & 124.70 & 119.59 & 117.55 \\
4 th & 150.40 & 162.16 & 160.128 \\
5 th & 183.20 & 177.11 & 177.45 \\
6 th & 184.80 & 177.39 & 178.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Natural frequencies of stiffened models. Figure 4: Displacement-time responses of the center of the square model subjected to air blast. As mentioned in advance, the upper piezo layer acts as a sensor and the lower layer is assumed as an actuator. [PERSON] et al. [11] investigated the experimental and computational analysis of a panel under blast loads, and the data are used in this research. The LQR optimal control algorithm is adopted to the structure. Modal superposition method is applied to the analysis. In addition, the initial damping ratio is assumed to be 0.7%. The transient responses are calculated with the Newmark-direct integration method. At first, the effect of the stiffener's location on the suppression of the model is studied. The two kinds of cases are presented in Figure 5b,c. The stiffeners are placed at both end sides in Case 1, while the stiffeners are located in the middle part in Case 2. Then, the first six natural frequencies of the stiffened panel with different stiffener location listed in Table 3. It can be seen from Table 3 that the natural frequencies of the Stiffener Case 1 are higher than that of the Stiffener Case 2. This means that the stiffened panel in Case 1 is stiffer than Case 2. Figure 5: Cantilevered piezoelectric composite model and two cases of the stiffener’s location: (**a**) Stiffened composite model; (**b**) Stiffener Case 1; (**c**) Stiffener Case 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Natural Frequency (Hz)** & **Stiffener Case 1** & **Stiffener Case 2** \\ \hline
1 st & 67.6098 & 39.6451 \\
2 nd & 117.7273 & 78.6680 \\
3 rd & 155.3218 & 133.6110 \\
4 th & 212.1017 & 178.2795 \\
5 th & 290.4140 & 214.3758 \\
6 th & 379.4205 & 273.2984 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Natural frequencies of different stiffener cases. Figure 6 shows magnitude-frequency responses for the two cases of stiffened panel using LQR method. The weighting matrices \(\mathbf{Q}\) and \(\mathbf{R}\) are as follows:  
\[\mathbf{Q}=\frac{1}{7\times 10^{35}}\left[ \begin{array}{cc}\mathbf{\Omega}^{2}&0 \\ 0&\mathbf{I}\end{array} \right]\text{, \ }\mathbf{R}=1\times 10^{-2}\text{ and }1\times 10^{-3} \tag{27}\]  
where \(\mathbf{\Omega}^{2}\) is a diagonal matrix that stores the squares of the natural frequencies \(\omega_{i}^{2}\) and \(\mathbf{I}\) is the unit matrix with suitable size. When the location of the stiffeners is centered (Case 2), the magnitude of the model is the higher than in the other case (Case 1). In addition, the peak point appeared earlier than Case 1. Figure 7 describes phase-frequency responses for the two cases of stiffened panel using LQR method. The results represent that controlled responses have the lower magnitude of the first, second, and third bending modes. When the stiffened panel is under uniformly distributed blast pressures, the vertical displacement (\(W_{2}\)) at the tip (0.4, 0.1 m), the transverse bending and the lateral twisting for various \(\mathbf{R}\) are given in Figures 8-10.