## 4 Numerical Result
### Flutter Suppression  
In this subsection, the aeroelastic flutter and the vibration suppression were further studied for a GPLs reinforced composite plate attached with a piezoelectric patch under supersonic flow. As shown in Figure 1, the piezoelectric actuator and sensor were located at the center of the rectangular base plate. The weight fraction of the GPLs was fixed at 0.6%. The parameters of the piezoelectric layer PZT-4 were given as elastic constants \(c_{11}=139\) GPa, \(c_{12}=77.8\) GPa, \(c_{66}=30.6\) GPa, mass density \(\rho=7500\) kg/m\({}^{3}\), and piezoelectric constant \(e_{31}=6.98\) C/m\({}^{2}\). The length, width, and thickness of the piezoelectric layer were given as 0.2 m, 0.2 m and 0.001 m, respectively. The aeroelastic flutter of the GPLs reinforced composite plate under supersonic flow without control was investigated first. Under this condition, the control gains \(K_{a}\) and \(K_{d}\) were equal to zero. Figure 5 shows the real parts and imaginary parts of the first two order eigenvalues of the plate in the supersonic airflow. It is found from Figure 5a that the real parts of all the eigenvalues keep negative when the Mach number is small. The real part of the eigenvalue changes from negative to positive as the Mach number increase to the critical Mach number \(M_{a}^{c}=3.09\), which means the aeroelastic flutter of the plate occurs. Meanwhile, the first two frequencies represented by the imaginary parts of the eigenvalues coalesce at the critical Mach number, as shown in Figure 5b. Figure 6 presents the responses of the functional graded GPLs plates under three different Mach numbers, which are both lower than the critical Mach number. It shows that the response of the GPLs plate will decay to zero after the initial transient response, if the Mach number is smaller than the critical Mach number \(M_{a}^{c}=3.09\). Besides, the plate under the lower flow speed will decay faster. Figure 5: The eigenvalues of the system with the Mach number, (**a**) the real parts of the eigenvalues, (**b**) the imaginary parts of the eigenvalues. Figure 6: Responses of the GPLs plate under different Mach numbers before the critical flutter number. The response of the GPLs reinforced composite plate near the critical Mach number \(M_{a}^{c}=3.09\) is shown in Figure 7. It indicates that the response of the plate no longer decays as time. However, the response of the plate becomes stable, which means the plate can continually absorb energy from the flow and then results in the occurrence of the flutter [40]. Then, the influnces of the active stiffness \(\mathbf{K_{P}}\), which is related to the displacement feedback gain \(K_{d}\), on the aeroelastic flutter behaviors of the GPLs plate attached with piezoelectric layer were considered for three cases of control gains, \(\left(K_{a}=K_{d}=0\right)\), \(\left(K_{a}=0,\ K_{d}=1.5\times 10^{5}\right)\), and \(\left(K_{a}=0,\ K_{d}=3\times 10^{5}\right)\). Figure 8 depicts the real parts and the imaginary parts of the eigenvalues of the GPLs plate with piezoelectric layer varying with Mach number. It is seen from Figure 8a that the point at which the real part of the eigenvalues varies from negative to positive increases from \(M_{a}=3.1\) to \(M_{a}=3.9\) as the displacement feedback gain \(K_{d}\) increases from \(K_{d}=0\) to \(K_{d}=3\times 10^{5}\). It implies that the occurrence of the aeroelastic flutter of the plate can be suppressed and delayed if displacement feedback control is introduced. Thereby, the active stiffness \(K_{d}\) can effectually enhance the aeroelastic flutter characteristics of the GPLs plate under supersonic flow. The same conclusion can also be obtained from Figure 8b, that the coalescence point of the imaginary parts of the eigenvalues moves towards the right as the displacement feedback gain \(K_{d}\) increases.