Size dependent dynamics of a bi-directional functionally graded nanobeam via machine learning methods

Abstract

This study explores the lateral vibration behavior of bi-directional functionally graded nanobeams using a combination of semi-analytical and machine learning approaches. The semi-analytical method uses the Fourier sine series and Stokes' transform for the deflection function of a bi-directional functionally graded nanobeam constrained by elastic springs at both ends and considers nonlocal modified couple stress theory to account for size effects. In the last step of the method, an eigenvalue problem is derived and the resulting frequency values are then used to train machine learning models, including extreme gradient boosting (XGB), artificial neural networks (ANN) and decision tree regression (DTR). The models' ability to predict the nanobeam's natural frequencies is evaluated using metrics like R-2, MAE, MAPE, RMSE, and the A20-index, alongside visual tools such as scatter plots, radar plots, and Taylor diagrams. The results indicate that ML models can accurately predict the natural frequencies of a bi-directional functionally graded nanobeams when provided with sufficient training data. In particular, ANN demonstrated exceptional generalization capability by achieving the highest R-2 and the lowest MAE, MAPE, and RMSE on both the training and testing datasets. The impact of various effects on vibration frequencies is detailed through a series of graphs and tables.