Comparative analysis of beam responses via Hencky and fractional models under different mass distributions

Abstract

This study presents a comparative analysis of the mechanical responses of beams modeled using Hencky and fractional approaches under various mass distribution conditions. The beam models considered in this study are composed of rigid segments connected by rotational springs, with three distinct mass distribution schemes analyzed: masses concentrated at the joints, masses located at the midpoints of each rigid part, and masses uniformly distributed along each segment. The developed model converges to the Euler–Bernoulli beam theory in the continuum limit, as the number of rigid segments tends to infinity. Closed-form expressions for natural frequencies are derived for simply supported boundary conditions. To approximate the dynamic response of these discrete models and capture their scale-dependent effects, corresponding nonlocal fractional continuum models are formulated using the symmetric Caputo derivative. These models enable precise fractional parameter calibration based on discrete systems’ dispersion relations. The analysis reveals a strong influence of the mass distribution on the dynamic behavior, with stiffening or softening effects emerging depending on the mass arrangement. The proposed fractional framework successfully replicates the dispersion characteristics of each discrete Hencky model and confirms that the accurate selection of fractional derivative parameters effectively models the vibrational behavior of microstructured beams.