## 4 Numerical Result
### Vibration Characteristics  
The effects of graphene material parameters, including GPLs distribution pattern and GPLs weight fraction, on the vibration characteristics of the functionally graded GPLs reinforced composite plate are investigated. The geometrical and material parameters of the plate are given as \(a=0.6\) m, \(b=0.4\) m, \(h=0.006\) m, \(l_{CPL}=2.5\)\(\mu\)m, \(w_{GPL}=1.5\)\(\mu\)m, \(h_{GPL}=1.5\)\(\mu\)m, \(v_{M}=0.34\), \(v_{GPL}=0.186\), \(\rho_{M}=1200\) kg/m\({}^{3}\), \(\rho_{GPL}=1062\) kg/m\({}^{3}\), \(E_{M}=3\) GPa, \(N_{L}=10\), \(E_{GPL}=1010\) GPa. Figure 3 shows the changes of the dimensionless frequency with the increase of the weight fraction of GPLs for two kinds of GPLs distribution patterns. Herein, the abscissa denotes the GPLs weight fraction, while the ordinate denotes the relative frequency change rate, which is expressed as \((\omega-\omega_{M})/\omega_{M}\times 100\%\). It was introduced to present the variation of the frequency of plate after adding GPLs reinforcement, where \(\omega_{M}\) is the dimensionless frequency of pure epoxy polymer, \(\omega\) represents the frequency of the plate after adding GPLs reinforcement. It can be found from Figure 3 that with the increment of the GPLs weight fraction, the relative frequency change rate augments and keep positive for both the X-GPLs distribution pattern and the O-GPLs distribution pattern. This indicates that introducing GPLs reinforcement can raise the plate's frequencies. Besides, the GPLs distribution pattern also can affect the vibration characteristics of the plate. Compared with the O-GPLs distribution patterns, the X-GPLs distribution pattern leads to a more dramatic improvement of the relative frequency change. It implies that the X-GPLs distribution pattern has a better effect on the increase of the plate' frequencies than the O-GPLs distribution patterns. It is attributed to the fact that the X-GPLs distribution patterns can make better use of the high Young's modulus GPLs reinforcement, hence, lead to a greater increment in the stiffness of the plate [37]. It is known that plates with greater stiffness often have better aerodynamic performance when they are subjected to aerodynamic load [38]. Figure 4 shows the effect of GPLs weight fraction on the critical Mach number for the occurrence of flutter of the GPLs reinforced plate. It illustrates that increasing the GPLs' weight fraction will increase the critical flutter Mach number of the plate, which means that adding graphene can effectively \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Pattern**} & \multirow{2}{*}{**Source**} & \multicolumn{5}{c}{_p, q_} \\ \cline{3-6}  & & **1,1** & **2,1** & **2,2** & **3,1** & **3,2** \\ \hline \multirow{3}{*}{Epoxy polymer} & [PERSON] et al. & 0.0584 & 0.1391 & 0.2132 & 0.2595 & 0.3251 \\  & ANSYS & 0.0563 & 0.1361 & 0.2064 & 0.2569 & 0.3173 \\  & Present & 0.0587 & 0.1406 & 0.2164 & 0.2641 & 0.3321 \\ \hline \multirow{3}{*}{X-GPLs} & [PERSON] et al. & 0.1378 & 0.3249 & 0.4939 & 0.5984 & 0.7454 \\  & ANSYS & 0.1319 & 0.3043 & 0.4424 & 0.5657 & 0.6475 \\  & Present & 0.1385 & 0.3289 & 0.5028 & 0.6112 & 0.7645 \\ \hline \multirow{3}{*}{O-GPLs} & [PERSON] et al. & 0.102 & 0.2456 & 0.3796 & 0.4645 & 0.586 \\  & ANSYS & 0.0881 & 0.2162 & 0.3336 & 0.4166 & 0.5658 \\ \cline{1-1}  & Present & 0.1023 & 0.2472 & 0.3835 & 0.4702 & 0.5949 \\ \hline \hline \end{tabular} Note: \(p\) and \(q\) mean the number of axial half-wave and circumferential wave in the modal functions for a simply supported GPLs reinforced composite plate.  
\end{table} 
Table 1: Comparisons of the frequency parameters \(\overline{\omega}=\omega h\sqrt{\rho_{M}/E_{M}}\) of the simply supported GPLs reinforced composite plate with references [37; 44]. suspend the flutter of plates happening. In Figure 4, the critical Mach number that is less than or equal to 1.5, is presented as equal to 1.5, as the first-order piston theory is only applicable as the speed of the airflow is above the Mach number of 1.5 [45].