## 3 Comprehensive Frequency Parameter and Mode Shape Results  
Aimed at testifying the capability of the proposed approach in predicting free vibration response for rectangular thin plates with classic/non-classic type edge conditions, new precise comprehensive results including frequency parameters and the associated mode shapes are in comparison with FEM results and the existing analytical solutions [16,43]. It is worth noting that the reliable FEM results for comparison are acquired by using ABAQUS 6.13 (2013), where 4-node thin shell element S4R is employed. To guarantee the accuracy of the FEM results, we testify the convergence for the FEM results of CCFF plate with aspect ratio b/a = 1, as is shown in Table 1. It is found that most of the obtained FEM results for \(\phi\) converge when the mesh size is 1/400 of the plate length. Thus, the uniform mesh size with short edge length of 1/400 is adopted throughout the present study. It is notable that the investigation on classic boundaries is attained through changing the coefficient \(r\), in which \(r=0.999999\) means clamped and \(r=0.000001\) means simply-supported. Meanwhile, effects on free vibration of plates aroused by aspect ratio b/a and different degrees of boundary constraint are also conducted, in the present study Poisson's ratio \(\mu\) equals 0.3, the aspect ratio \(\chi=b/a\) of the CCFF, CSFF and SFF rectangular plate ranges from 0.5 to 5 and the coefficient \(r\) of the RRTF square plate ranges from 0.1 to 0.9. We first carry out vibration analysis on evaluating the non-dimensional frequency parameter, \(\phi=\omega a^{2}\sqrt{\rho h/D}\), for CSFF, SSFF and CCFF plates under 10 different aspect ratios such that 100 results are present for each case, which are tableted in Tables 1-3. The corresponding vibration mode shapes for the CSFF, CCFF and SSFF square plates are shown in Figures 2-4. As to the RRFF square plates, two different cases are considered: (1) the coefficient \(r_{x0}=r_{y0}=r\) ranges from 0.1 to 0.9, new exact results for \(\phi\) are given in Table 4; (2) the coefficient \(r_{x0}=r\) ranging from 0.1 to 0.9 with \(r_{y0}=0.5\), and new benchmark results for \(\phi\) are shown in Table 5. The first five vibration modes for the second case of RRFF square plates are given in Figure 5, where the coefficient \(r_{x0}\)\(=0.1,\ 0.3,\ 0.5\) and 0.8, respectively. As is shown in Tables 2-4, the non-dimensional frequency parameter decreases as the aspect ratio \(\chi=b/a\) increases and vice versa. Therefore, the minimum values of \(\phi\) corresponds to the aspect ratio (5.0). The non-dimensional frequency parameter decreases rapidly when the aspect ratio is 1 or less than one. Additionally, it can be observed that the values of \(\phi\) changes very little when \(\chi=b/a\) changes from 2.5 to 5. It can be concluded that the aspect ratio from 0.5 to 2.0 has more influence on the natural frequencies as compared to the aspect ratio from 2.5 to 5. Through Tables 2-4, it also found that boundary conditions have great influence on the non-dimensional frequency parameter. The acquired \(\phi\) for all boundary conditions is higher as the aspect ratio gets smaller than one. The magnitude of \(\phi\) is always greater for CCFF than those under CSFF and SSFF when aspect ratio \(\chi=b/a\) is same. Moreover, the non-dimensional frequency parameter for CSFF is higher than the SSFF for all aspect ratio. It can be concluded that the plate with more clamped edges requires more energy to vibrate. Through the comparisons in Tables 2-4, it is clearly seen that all the results obtained are in good agreement with ones solved numerically or analytically, especially with those in Ref. [43]. Similarly, through Tables 5 and 6, it is obvious that the obtained \(\phi\) increases with the increase of the rotating fixed coefficient r, which indicates that the RRFF plate with stricter boundary constraints needs more energy to vibrate.