# Free Vibration Analysis of Triclinic Nanobeams Based on the Differential Quadrature Method
## 4 Numerical Results  
In this section a triclinic nanobeam is considered with length \(L=36.8\) nm, and thickness depending on its length. A preliminary convergence analysis is performed between our model and predictions from Ref. [7] based on the application of the DQM (see Table 1). Then, using the present size-dependent model for anisotropic materials, the convergence of the model is studied for triclinic nanobeams with different boundary conditions (see Table 2). Based on these two tables, a fast convergence of the results can be observed, even with a reduced number of grid points. This justifies the selection of the limited number of grid points \(N=19\), as done henceforth within the numerical investigation. To validate the numerical size-dependent methodology of the present work, the first-two dimensionless natural frequencies of the nanostructure are compared to predictions by [PERSON] [59] for different values of the nonlocal parameter, based on the Euler Bernoulli beam theory (EBBT), see Table 3. A systematic study is performed to check for the sensitivity of the response for different boundary conditions, with a clear good agreement between the two different approaches, and a general increase of the natural frequencies while moving to a clamped nanostructure at both sides. More specifically, in Table 4 the first four non-dimensional frequencies of a simply supported triclinic nanobeam are summarized for a different length-to-thickness ratio (\(L\)/\(h\)), nonlocal parameter \(\mu\), and strain gradient \(l\). The same results are also represented in the 3D plots of Figure 2. An increased mode number enables higher values of the frequency, which, in turn reduce for increasing nonlocal parameters, and increase with the strain gradient parameters. In addition, the small-scale parameter seems to affect the response especially for higher frequencies rather than the lower ones. For lower mode numbers, any meaningful impact can be observed for a varying length-to-thickness ratio, whereas a visible increase of the natural frequency can be observed by changing the length-to-thickness ratio, for higher modes of vibration (see e.g., the results associated to the forth mode of vibration), while keeping fixed the strain gradient and the nonlocal parameter. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{6}{c}{**Node**} \\ \hline DQM [7] & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 19 & 21 & 23 \\
3.1123 & 3.0875 & 3.1129 & 3.1123 & 3.1123 & 3.1123 & 3.1123 & 3.1123 & 3.1123 & 3.1123 & 3.1123 \\
6.0676 & 8.9282 & 6.0430 & 6.0702 & 6.0675 & 6.0676 & 6.0676 & 6.0676 & 6.0676 & 6.0676 & 6.0676 \\
8.7784 & 12.8690 & 8.6806 & 8.7981 & 8.7771 & 8.7784 & 8.7784 & 8.7784 & 8.7784 & 8.7784 & 8.7784 \\
11.229 & 24.0798 & 15.288 & 11.377 & 11.248 & 11.228 & 11.229 & 11.229 & 11.229 & 11.229 \\
13.441 & 24.3004 & 17.893 & 13.477 & 13.509 & 13.433 & 13.443 & 13.442 & 13.441 & 13.441 & 13.441 \\
15.449 & 24.7858 & 24.080 & 21.134 & 16.092 & 15.527 & 15.449 & 15.450 & 15.449 & 15.449 & 15.449 \\
17.285 & & 24.307 & 22.975 & 17.684 & 17.416 & 17.261 & 17.289 & 17.284 & 17.285 & 17.285 \\
18.975 & & 25.859 & 24.080 & 24.080 & 20.430 & 19.224 & 18.991 & 18.977 & 18.975 & 18.975 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Convergence analysis of the size-independent Timoshenko beam model.