A numerical model to predict the transverse free vibration of stepped beams

Abstract This work presents a modified numerical model for predicting the transverse free vibration of stepped beams with improved accuracy over the classical Timoshenko beam theory (TIM model). While TIM model performs well for beams with constant cross-sections, it fails to capture the dynamic response near abrupt changes in geometry, as is typical in stepped beams. To overcome these limitations, a new formulation is proposed (MOD model) that introduces smooth transitions in cross-sectional properties via a parametrized sigmoid function applied to stiffness and mass matrices. A systematic optimization of the transition parameters is conducted using a genetic algorithm to best approximate high-fidelity finite element results (REF model). The MOD model is validated across 35 beam configurations and outperforms the TIM model in most cases, especially for beams with high geometric discontinuities. Additional case studies involving asymmetric and multi-step beams demonstrate the generalization capability of the proposed approach. The MOD model thus offers a computationally efficient yet accurate tool for dynamic analysis of complex beam geometries.