###### Abstract  
In this study, the natural frequencies and roots (Eigenvalues) of the transcendental equation in a cantilever steel beam for transverse vibration with clamped free (CF) boundary conditions are estimated using a long short-term memory-recurrent neural network (LSTM-RNN) approach. The finite element method (FEM) package ANSYS is used for dynamic analysis and, with the aid of simulated results, the Euler-Bernoulli beam theory is adopted for the generation of sample datasets. Then, a deep neural network (DNN)-based LSTM-RNN technique is implemented to approximate the roots of the transcendental equation. Datasets are mainly based on the cantilever beam geometry characteristics used for training and testing the proposed LSTM-RNN network. Furthermore, an algorithm using MATLAB platform for numerical solutions is used to cross-validate the dataset results. The network performance is evaluated using the mean square error (MSE) and mean absolute error (MAE). Finally, the numerical and simulated results are compared using the LSTM-RNN methodology to demonstrate the network validity. clamped free; finite element method; transcendental equation; roots (Eigen values); long short-term memory; recurrent neural network +
Footnote †: journal: Applied  
solve the vibration-induced engineering problems for natural frequencies and mode forms to analyze the vibration of structures [8; 9; 10; 11; 12; 13]. Similarly, the literature mentioned in this paragraph provides an analysis of beams with free vibration. For free vibrations with a large amplitude uniform beam, [PERSON] et al. applied a simplified finite element model [14]. [PERSON] et al. used a varied finite element model to perform a free vibration study on delaminated beams [15]. [PERSON] et al. used transverse normal stress and sinus finite elements to study the vibration of multilayered beams [16]. [PERSON] et al. performed an experimental analysis on the transverse vibration of clamped-pinned-free beam with mass at the free end to find the transcendental equation roots of the beam [17]. Although the methodologies mentioned above have been applied to the problems of structural analysis, design and optimization using conventional approaches involve complexity and time-consuming processes. Thus, during the past few decades, soft computing techniques have been widely used in various scientific and engineering fields, particularly in the study of vibration problems in engineering structures. Moreover, such approaches make it easier to perform a vibration analysis of complicated structural systems. Machine learning (ML), artificial neural networks (ANN) and evolutionary algorithms are some well-known soft computing methods used in computational mechanics, especially beam applications [18; 19; 20]. For instance, [PERSON] et al. presented the natural frequency of the beam using ANN and heuristic research [21]. [PERSON] et al. use an ANN approach for the transcendental equation of longitudinal vibration [22]. [PERSON] et al. apply an ANN strategy on the nonlinear vibrations of stepped beam [23]. In order to determine the depth of cracks on a cantilever beam, [PERSON] et al. used FEM to evaluate three natural frequencies of the cantilever beam and then further applied an ANN technique trained using particle swarm optimization [24]. Although ANNs and evolutionary techniques show promising performance for analyzing and predicting the beam's natural frequencies, the approaches are implemented with few datasets. Furthermore, the prediction performance evaluation is mainly performed for the natural frequencies of the structural beams. Several traditional methods, such as Newton-Raphson, Bisection, Iteration, etc., have been used to find the transcendental equation roots [25]. However, the problems associated with such conventional methods are as follows: (1) Techniques require numerical iterations for each frequency and root values consequently need a longer computational time; (2) convergence is often slow; and (3) the guess interval cannot be used if there are discontinuities.