Transverse-longitudinal Coupling Effect in the Free Vibration of Eccentrically Supported Functionally Graded Beams

Abstract This paper takes the functionally graded beam as the research object and explores the coupling between transverse and longitudinal vibrations in functionally graded beams with eccentric support boundary conditions. Considering the case where the boundary simply supported at the beam ends is not aligned with the neutral plane, the displacement of the supporting section due to the constraint causes a change in the beam’s neutral plane length during longitudinal bending. That is, during dynamic vibration, the bending mode and the axial mode couple with each other. The functionally graded beam is simplified using the Euler beam model, while considering the coupled response of bending and axial vibrations at the boundary supports. Using Hamilton’s principle and the variational principle, a system of coupled vibration equations is derived to solve for the beam’s natural frequencies. Through comparison with finite element results, the predicted natural frequencies from the model show consistent trends with the simulation results as the support position changes, with theoretical results slightly exceeding the simulation outcomes. The main cause of this difference is that the theoretical model disregards the transverse shear deformation of the beam, resulting in an overestimation of the beam’s stiffness. The numerical results further substantiate that the interaction between horizontal and vertical modes, due to the eccentric support constraints, causes notable changes in the natural frequencies.