Deep neural network-based surrogate modeling for nonlinear vibrations of functionally graded stepped beams informed by a discrete model

Abstract A discrete mechanical model for large-amplitude free vibrations of two-stepped functionally graded beams is developed in this study. The beam’s material properties vary through the thickness according to a power-law distribution between the metallic and ceramic phases. The continuous beam is replaced by an N-degree-of-freedom system of lumped masses, longitudinal, and torsional springs. Using Hamilton’s principle, the governing nonlinear algebraic equations are derived and solved through the single-mode approach (SMA) to obtain the nonlinear frequency-amplitude relationships. In addition, an Artificial Neural Network (ANN)-based surrogate model is proposed to provide fast and accurate predictions of the nonlinear-to-linear frequency ratio as a function of key parameters such as step ratio, step position, boundary conditions, and the power-law index. Trained on data generated by the discrete formulation, the surrogate attains excellent generalization with a drastic reduction in computation time. The combined discrete-ANN framework offers both physical interpretability and computational efficiency, making it suitable for rapid design and optimization of complex FGM beam structures.