Deep Neural Network-Driven Analysis of Free Vibrations in Tapered Beams

Most physical systems exhibit nonlinear behavior while in motion, making their resolution challenging due to nonlinearity, dynamic effects, and sensitivity to parameters such as frequency and amplitude. Traditional analytical and numerical approaches can address these challenges but offer high computational costs, particularly in solving the system of free vibrations produced by the tapered beam. Predicting the behavior of this model is complicated, due to its high sensitivity and nonlinearity. Previously, standard neural network models have been used to solve dynamical systems, but they lack efficiency in handling nonlinearity. In this paper, we propose a novel deep learning model that predicts the amplitude of vibrations of a tapered beam. The primary focus of this study is to address the nonlinearity of the model and accurately predict the amplitude of vibrations. To solve this issue, we introduce a deep neural network designed to manage both nonlinearity and dynamical effects, including amplitude. The approach is significant in terms of computational and time efficiency compared to traditional numerical methods. The proposed work provides comparative results generated by the deep neural network, the backward difference formula as an analytical technique, and the Adams–Bashforth–Moulton predictor–corrector method as a numerical approach. The results demonstrate that our model outperforms existing numerical and analytical techniques. With the help of mean square error, Thiel’s inequality coefficient, and mean absolute error, the accuracy of our model can be verified; the lower these values, the more accurate our model will be. In our proposed model, the values are 8.389× 10−9 for mean square error, 5.563×10−4 for Thiel’s inequality coefficient, and 0.347 for mean absolute error; all these values are close to zero, signifying the accuracy of our model. The conclusion confirms that our proposed approach, even with changeable hyperparameters, is more suitable and accurate than numerical and analytical methods.