## 3 Calculation Results and Analysis  
### Validation  
In order to verify the validity of the model in this paper and the accuracy of the calculation, we first calculated the free vibration numerical results of the non-rotating circular plates and compared this with the results of [27]. The governing equations derived from the Reddy plate theory are complicated, and the omission of related parameters can simplify the Reddy plate governing equations to the Kirchhoff plate governing equations and Mindlin plate governing equations. Therefore, this paper selects the numerical results under the Reddy plate theory for comparison. According to the data in Table 1, it can be found that the numerical results in this paper are in good agreement with those in the literature. Therefore, the validity of the model and the accuracy of the calculation can be verified.  
Next, based on three different plate theories, the free vibration behaviors of the rotating functionally graded nano-annular plates under two boundary conditions are analyzed, and the effects of various parameters such as the inner and outer radius ratio, rotational velocity, nonlocal parameter, functional gradient index, and temperature change on the vibration frequencies under two boundary conditions are examined. The two boundary conditions of the nano-annular plates are inner and outer rings clamped (C-C), inner rings clamped and outer rings free (C-F). Take the thickness \(h\) of the annular plates as 1 nm and the outer radius \(r_{2}\) as 50 nm. The upper surface of the annular plates is made of ceramic (Si\({}_{3}\)N\({}_{4}\)) with density of \(\rho_{c}=2707\) kg/m\({}^{3}\), and the lower surface material is made of metal (SUS304) with density of \(\rho_{m}=3800\) kg/m\({}^{3}\). Assume the initial ambient temperature \(T_{0}=300\) K at which the annular plates do not undergo initial stress and deformation, and the modulus of elasticity and the coefficient of thermal expansion change with the change in the ambient temperature. The specific temperature-related coefficients are shown in Table 2. Since Poisson's ratio does not change much, the constant is taken as 0.3.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Mode** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \multirow{2}{*}{C-C} & This paper & 10.215 & 39.771 & 89.102 & 158.18 & 246.99 & 355.54 \\  & Literature [27] & 10.215 & 39.771 & 89.102 & 158.18 & 246.99 & 355.54 \\ \hline \multirow{2}{*}{C-F} & This paper & 9.0026 & 38.443 & 87.749 & 156.81 & 245.62 & 354.17 \\  & Literature [27] & 9.0026 & 38.443 & 87.749 & 156.81 & 245.62 & 354.17 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of natural frequencies for Reddy circular plates (\(\overline{\Omega}=0\), \(k\rightarrow\infty\), \(s=0\)).  
In order to ensure the numerical results have universal applications, the physical quantities are made dimensionless  
\[\begin{array}{l}\overline{u}=\frac{\bar{u}}{r_{2}},\overline{w}=\frac{\bar{w}}{ r_{2}},\ \overline{r}=\frac{r}{r_{2}},s=\frac{r_{1}}{r_{2}},\delta=\frac{\hbar}{r_{2}}, \tau=\frac{e_{0}q}{\tau_{2}},\overline{l}=\frac{t}{r_{2}^{2}}\sqrt{\frac{E_{m} \hbar^{3}}{12\rho_{m}h(1-\mu^{2})}}\\ \overline{\Omega}^{2}=\frac{12\rho_{m}h\mu_{2}^{2}\left(1-\mu^{2}\right)}{E_{m }\hbar^{3}}\Omega^{2},\overline{\omega}^{2}=\frac{12\rho_{m}h\mu_{2}^{2}\left(1 -\mu^{2}\right)}{E_{m}\hbar^{3}}\omega^{2},\overline{N}^{RT}=\frac{12\pi^{RT} r_{2}^{2}\left(1-\mu^{2}\right)}{E_{m}\hbar^{3}}\end{array} \tag{35}\]  
The research contents of the Sections 3.1-3.6 are the influence of various physical parameters on the natural frequencies of the rotating nano-annular plates. The results obtained through [PERSON]'s theory are the same as the conclusions of the literature [28].