## 3 Results and Analysis  
### Influence of Different Forces on the Resonance Properties of Graphene with Vacancies  
In this communication, the total energy of the system is analyzed to determine whether a stable equilibrium state is obtained. In this simulation, the relative variation of the total energy is output. According to processing and counting output data, the relationship between the total energy and time can be obtained. As Figure 4 shows, the total energy of the system in the initial relaxation time changes to some degree. With the increase of the relaxation time, the total energy fluctuates on a small scale. After that, the system reached a stable equilibrium state. Figure 3: Simulation model of the graphene sheet with defects: (**a**) Single vacancy defect; (**b**) Edge defects. Figure 2: Model schematic of a graphene resonator. The graphene nano resonator is generally driven by an electrostatic capacitive force induced by the potential difference between the graphene sheet and the gate [6; 8]. So, in this simulation, the magnitude of the external force is calculated by the formula (1):  
\[F=\frac{\partial W}{\partial d_{0}}=-\frac{\varepsilon abU^{2}}{2d_{0}} \tag{1}\]  
where \(F\) is the electrostatic force between two plate electrodes; \(W\), \(d_{0}\) are the energy and distance between the two plate electrodes, respectively; \(U\) is the voltage loaded between the two plate electrodes; and \(a\), \(b\) are the length of the plate electrodes. Different voltages are selected to obtain different magnitudes of stress. A uniform force is loaded perpendicularly on the surface of the graphene, except the fixed ends (Figure 5). The electrostatic force is evenly distributed on each atom in the graphene nanosheet [29]. The total force is \(F=f\times N\), where \(f\) is the force applied to each atom of the graphene sheet, and \(N\) is the number of atoms in graphene's effective region. The applied indented forces were 1.77 nN, 7.07 nN, 15.92 nN, 28.29 nN, 44.20 nN, 63.65 nN, 86.64 nN, and 113.15 nN. In this communication, a Fourier transform was applied to obtain the resonant frequency of the graphene with two different vacancies by selecting the kinetic energy of the system as the time-domain signal [30]. When the graphene is subjected to a certain load, the potential energy of the system will change accordingly. After 20,000 time steps, the load applied to the graphene was removed. The graphene structure will recover to the initial state due to the C-C bonds inside the graphene structure. At this point, the potential energy of the graphene will be converted to kinetic energy, and the graphene structure will oscillate. When the kinetic energy and potential energy of graphene were converted to each other, the graphene structure undergoes free vibration. During oscillation, 250,000 time steps or 250 ps were carried out. Figure 6 shows the amplitude versus frequency curve of the graphene sheet with a single defect. Figure 7 shows the function of resonance frequency of external force with two different vacancy defects. Figure 4: Relationship between total energy and relaxation time of grapheme. Figure 5: Schematic diagram of stress loading on the graphene structure. Figure 6: Amplitude versus frequency for external forces of (**a**) 1.768 nN; (**b**) 7.072 nN; (**c**) 15.912 nN; (**d**) 28.288 nN; (**e**) 44.2 nN; (**f**) 63.648 nN; (**g**) 86.638 nN; and (**h**) 113.152 nN. Figure 7: Resonance frequency as a function of external force with two different vacancy defects. As shown in Figure 6, the resonance frequencies of both two types of graphene increase with increasing external stress loading. Additionally, the frequency shows a clear step-shaped variation, indicating that the dynamic behavior of graphene is dominated by nonlinear effects. The fundamental cause is the nonlinear deformation of the sheet between resonant processes [31].