Transverse vibration of beams application of a modified theory

Abstract

In the analysis of transverse vibration of beams the classical Euler-Bernoulli theory is insufficient to describe the behaviour of the beam in the higher modes. Including the effect of rotary inertia and shear deflection of the beam results in the so called Timoshenko theory which extends the applicability of the theory to higher modes. This study is based on a work by Levinson [9]. The theory developed takes warping of the cross-section into account. In this theory the arbitrariness in the shear coefficient appearing in the equations of motion is removed. The shear coefficient is shown to be equal to 5/6. Based on this theory, a theoretical analysis of the vibration of beams with simple and homogeneous b.oundary conditions is presented. The method of separation of variables is used to obtain the solutions to the Euler-Bernoulli beam theory, Timoshenko beam theory and the modified theory. The eigenvalue problem is formulated for each theory and both the eigenvalues (natural frequencies) and the eigenfunctions (normal modes) are determined for the clamped-free, clamped-clamped, hinged-hinged, free-free, and clamped-hinged boundary conditions. TheresiIlts of these theories are compared.