## 6 Conclusions  
In this study, the root (Eigenvalues) of the transcendental equation of the prismatic steel beams under the clamped free boundary condition are approximated through the proposed LSTM-RNN network. The conclusion of the paper is summarized below:  
1. Based on the beam parameters and boundary conditions, natural frequencies (\(f_{n}\)) datasets are initially generated using the finite element method (FEM) ANSYS software. Then, using the fundamental Euler-Bernoulli theory, transcendental root (\(\beta_{n}l\)) datasets are produced for implementation in the LSTM-RNN model.
2. An algorithm has been developed to cross-validate the datasets which uses the MATLAB programming platform to find the roots and natural frequencies numerically. The \(\beta_{n}l\) parameter results show consistency with conventional theoretical values. Moreover, the modes shapes for the first six natural frequencies, as an example, have been plotted using ANSYS and MATLAB simulation software.
3. Finally, a comparative analysis between ANSYS (datasets), MATLAB (numerical) and the proposed LSTM-RNN methodology has been performed in terms of percentage (%) MAE and MSE. Although the result shows that the % error is slightly higher than the numerical solution, the performance can be improved by increasing the number of datasets and system resources.
4. The proposed LSTM-RNN method aims to develop an alternative way to find the roots of transcendental or frequency equation based on datasets and beam parameters,  
\begin{table}
\begin{tabular}{c} \% Error** \\ **(RNN)** \\ \end{tabular} \\ \hline
1 & 1.8751 & 1.8764 & 0.07 & 1.8751 & 1.8505 & 1.31 & 1.8764 & 1.8505 & 1.38 \\
2 & 4.6941 & 4.6869 & 0.15 & 4.6941 & 4.6265 & 1.44 & 4.6869 & 4.6265 & 1.29 \\
3 & 7.8548 & 7.8213 & 0.43 & 7.8548 & 7.7406 & 1.45 & 7.8213 & 7.7406 & 1.03 \\
4 & 10.9955 & 10.9039 & 0.83 & 10.9955 & 10.8565 & 1.26 & 10.9039 & 10.8565 & 0.43 \\
5 & 14.1372 & 13.9479 & 1.31 & 14.1372 & 13.9674 & 1.20 & 13.9479 & 13.9674 & 0.14 \\
6 & 17.2788 & 17.3303 & 0.30 & 17.2788 & 17.3113 & 0.19 & 17.3303 & 17.3113 & 0.11 \\
7 & 20.4203 & 20.6002 & 0.88 & 20.4203 & 20.4141 & 0.47 & 20.6002 & 20.4141 & 0.85 \\
8 & 23.5619 & 23.4959 & 0.28 & 23.5619 & 23.3825 & 0.34 & 23.4959 & 23.3825 & 0.48 \\
9 & 26.7035 & 26.4404 & 0.99 & 26.7035 & 26.6837 & 0.82 & 26.4404 & 26.6837 & 0.92 \\
10 & 29.8451 & 30.0118 & 0.56 & 29.8451 & 29.9538 & 0.36 & 30.0118 & 29.9538 & 0.19 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparative analysis of the transcendental roots (\(\beta_{n}l\)).  
thereby avoiding complexity and time loss in evaluating the beam system with a large number of transverse modes \(n\).  
In future research, we will consider the case of finding the roots for the transcendental equation under different beam characteristics and with different boundary conditions.  
Conceptualization, M.B. and W.C.; methodology, M.B. and M.S.A.; software, M.B., K.S. and A.J.; validation, M.S.A., A.A. and M.U.A.; formal analysis, M.B. and M.S.A.; investigation, M.B., M.S.A. and A.J.; resources, M.U.A. and W.C.; data curation, M.B. and M.U.A.; writing--original draft preparation, M.B. and M.S.A.; writing--review and editing, M.B. and M.S.A.; visualization, K.S.; supervision, M.S.A. and W.C.; project administration, A.A. and W.C.; funding acquisition, A.A. and K.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Shenzhen Scientific and Technological Fund for Research and Development of China (No. 2021 Szvup152). This work was supported in the part by the Deanship of Scientific Research (DSR), Majmaah University, Al-Majmaah.  
Not applicable.  
The authors declare no conflict of interest.