## 5 Results and Discussion
### MATLAB Simulation Analysis  
This section aims to cross-validate the accuracy of the ANSYS-generated datasets and numerically computed natural frequency (\(f_{n}\)) and root (\(\beta_{n}l\)) values. Although the trigonometric frequency equation is derived analytically in section II, where the values of the root need to be evaluated initially to find the values of natural frequencies (\(f_{n}\)), the development of an algorithm to solve the frequency equations based on a trial-and-error method is proposed. The objective was to find out the roots of the transcendental equation numerically with the help of the programming environment offered in MATLAB.  
For this purpose, an approximate estimation of the lower and upper limits for an interval comprising the first few roots, i.e., natural frequencies, can be obtained by visualizing the frequency equation. The roots can then be identified using an incremental search strategy. The frequency equation can then be solved and the algorithm is derived based on the change of sign of a function in the vicinity of a possible root. Finally, frequency Equation (19) is utilized in the MATLAB code containing the root values and the beam properties mentioned in Table 1; \(f_{n}\) in Hz and rad/se are calculated. Moreover, root values  
Figure 4: Defining analysis choices for cantilever prismatic beam.  
are utilized for the corresponding theoretical mode shapes for the first six modes based on Equation (21), as plotted in Figure 6.  
Calculated based on the technique mentioned above, the first ten\(f_{n}\) and \(\beta_{n}l\) values of the prismatic beam are mentioned in Table 3. The results show that the natural frequency (\(f_{n}\)) and root (\(\beta_{n}l\)) values are consistent with ANSYS datasets, thereby cross-validating the results of the numerical approach and generated dataset. In fact, for evaluating the proposed model performance, valid datasets are essential for testing and training the model. Numerical results for roots (\(\beta_{n}l\)) are also valid based on the theoretical values of the transverse vibration of the CF beam case mentioned in [3].  
Figure 5: First six mode shapes (\(n=1\)–\(6\)) of transverse vibration of a cantilever beam.  
The generalized percentage (%) absolute error equation used for comparison purposes is presented below:  
\[\%error=\left|\frac{X_{nSim}-X_{nNum}}{X_{nNum}}\right|\times 100 \tag{24}\]  
where \(X_{nNum}\) represents the MATLAB numerical values of \(f_{n}\) and \(\beta_{n}l\) parameters. Similarly, \(X_{nSim}\) represents the simulated values of \(f_{n}\) and \(\beta_{n}l\) from the ANSYS and LSTM-RNN model, respectively.  
Table 4 presents the comparison results with an absolute % error of the first ten natural frequencies between MATLAB (numerical) and ANSYS (datasets) simulated results. Figure 7 shows the comparison results using graphical analysis. The results overlap between each MATLAB numerical analysis and ANSYS datasets, thus justifying the cross validation of the results.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Nodes (\(n\))** & \(\beta_{n}l\) & **Freq (rad/s)** & **Freq (Hz)** \\ \hline
1 & 1.8751 & 157.71 & 25.1 \\
2 & 4.6941 & 987.09 & 157.1 \\
3 & 7.8548 & 2763.34 & 439.8 \\
4 & 10.9955 & 5415.47 & 861.9 \\
5 & 14.1372 & 8952.27 & 1424.8 \\
6 & 17.2788 & 13,373.12 & 2128.4 \\
7 & 20.4203 & 18.678 \(\times 10^{3}\) & 2.9727 \(\times 10^{3}\) \\
8 & 23.5619 & 24.867 \(\times 10^{3}\) & 3.9577 \(\times 10^{3}\) \\
9 & 26.7035 & 31.941 \(\times 10^{3}\) & 5.0835 \(\times 10^{3}\) \\
10 & 29.8451 & 39.898 \(\times 10^{3}\) & 6.3500 \(\times 10^{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Numerical results using MATLAB.  
Figure 6: Theoretical mode shapes for first six natural frequencies of a beam.