Nonlinear geometry effects investigation on free vibrations of beams using shear deformation theory

Abstract

This article presents an analytical procedure to predict the beams nonlinear natural frequencies using the first-order shear deformation theory. The nonlinear kinematics assumptions include the moderately large deformation of the mid-plane stretching and transverse deflection that are defined by using the von Kármán relations. The strains are small and Hooke’s law is used as the constitutive equation. A coupled nonlinear longitudinal-transverse set of motion equations are derived utilizing Hamilton’s principle. The equations are solved analytically using the powerful multiple scales method of the perturbation technique. At first, the linear natural frequencies and mode shapes are determined. Then the nonlinear frequencies which contain the corrections on the linear frequencies are calculated. The corrected parts of the nonlinear frequencies are functions of the axial and transverse amplitudes of vibrations. The influences of the axial load and aspect ratio on the linear and nonlinear frequencies are studied too. The results of the special cases are compared with the available results in the literature and the finite element analysis. The results show the noticeable effect of the axial amplitude as well as the transverse amplitude of vibrations on the nonlinear frequencies.