# Free Vibration Analysis of Triclinic Nanobeams Based on the Differential Quadrature Method
## 4 Numerical Results
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline _L/h_ & \(l\) & _μ_ & _ω_\_1 & _ω_\_2 & _ω_\_3 & _ω_\_4 \\ \hline
20 & 0 & 0 & 3.8937 & 6.9151 & 9.8060 & 12.5432 \\  & & 1 & 3.8855 & 6.8620 & 9.6453 & 12.1940 \\  & & 2 & 3.8774 & 6.8109 & 9.4968 & 11.8885 \\  & & 3 & 3.8694 & 6.7616 & 9.3591 & 11.6178 \\ \hline
1 & 0 & 3.9018 & 6.9680 & 9.9669 & 12.8993 \\  & & 1 & 3.8937 & 6.9151 & 9.8060 & 12.5476 \\  & & 2 & 3.8856 & 6.8641 & 9.6573 & 12.2390 \\  & & 3 & 3.8776 & 6.8150 & 9.5192 & 11.9651 \\ \hline
2 & 0 & 3.9098 & 7.0185 & 10.1150 & 13.2682 \\  & & 1 & 3.9017 & 6.9658 & 9.9547 & 12.9287 \\  & & 2 & 3.8937 & 6.9151 & 9.8062 & 12.6298 \\  & & 3 & 3.8857 & 6.8661 & 9.6680 & 12.3634 \\ \hline
100 & 0 & 0 & 3.9253 & 7.0621 & 10.1921 & 13.3133 \\  & & 1 & 3.9170 & 7.0073 & 10.0226 & 12.9366 \\  & & 2 & 3.9087 & 6.9545 & 9.8663 & 12.6078 \\  & & 3 & 3.9006 & 6.9036 & 9.7214 & 12.3170 \\ \hline
1 & 0 & 3.9335 & 7.1167 & 10.3617 & 13.6911 \\  & 1 & 3.9253 & 7.0621 & 10.1922 & 13.3125 \\  & & 2 & 3.9171 & 7.0095 & 10.0357 & 12.9807 \\  & & 3 & 3.9090 & 6.9588 & 9.8905 & 12.6864 \\ \hline
2 & 0 & 3.9416 & 7.1687 & 10.5173 & 14.0230 \\  & 1 & 3.9334 & 7.1144 & 10.3490 & 13.6518 \\  & 2 & 3.9253 & 7.0621 & 10.1930 & 13.3228 \\  & 3 & 3.9172 & 7.0116 & 10.0480 & 13.0288 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Effect of the _L/h_ and small-scale parameters on the natural frequencies for a clamped-simply supported triclinic nanobeam. Figure 3: Variation of the first four natural frequencies for different _L/h_ and small-scale parameters (clamped-simply supported nanobeam). As a further boundary condition, a fully clamped triclinic nanobeam is analyzed under the same geometry and mechanical assumptions. The results are summarized in Table 6 along with the plots in Figure 4. As expected, an overall increase in stiffness is observed, because of the clamped boundary condition at both sides of the structure. Note also that an increase in the strain gradient parameter \(l\), and nonlocal parameter \(\mu\), cause a general increase and decrease of the fundamental frequencies, respectively, in line with the previous examples. As also reported in the pioneering work on the topic [60], the fundamental frequency computed according to the MD is always lower than predictions based on the classical continuum elasticity theory. This behavior is consistent with our findings for nonlocal clamped nanobeams. The last combination of boundary conditions analyzed herein, accounts for a clamped-free triclinic nanobeam, whose parametric vibration response is listed in Table 7 and represented in Figure 5, in terms of the first natural frequencies, while varying the strain gradient parameter \(l\), the nonlocal parameter \(\mu\), and the geometrical ratio \(L\)/\(h\). Based on the results, note that the clamped-free nanobeam exhibits a different behavior compared to the structural response for the other boundary conditions. Except for the first frequency, the other frequencies reduce for increasing values of \(\mu\), and increase for an increasing value of \(l\). The contrary occurs for the first frequency, which decreases for an increasing strain gradient parameter \(l\), and increases by increasing the nonlocal parameter \(\mu\). Remarkably, these results are perfectly in line with the findings of [PERSON] et al. [59] for a nonlocal cantilever beam. Due to the higher flexibility of the free structure at one side, the lowest values of natural frequencies are registered and compared to all the other examples previously discussed. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(L\)/\(h\) & \(l\) & \(\mu\) & \(\omega_{1}\) & \(\omega_{2}\) & \  
Figure 4: Variation of the first four natural frequencies for different \(L\)/\(h\) and small-scale parameters (clamped-clamped nanobeam). Figure 5: Variation of the first four natural frequencies for different \(L\)/\(h\) and small-scale parameters (clamped-free cantilever nanobeam).