Free Vibration of Axially Loaded Shear Beams Carrying Elastically Restrained Lumped-Tip Masses via Asymptotic Timoshenko Beam Theory

Classical shear beam theory does not consider the rotation of cross sections. This paper studies the free vibration of axially loaded shear beams carrying lumped masses at elastically supported ends where rotational motion of the cross section is taken into account. By using asymptotic analysis of the Timoshenko beam theory, a unified analytical approach for dealing with free-vibration problems of nonclassical shear beams subjected to axial compressive or tensile force according to Engesser’s model is presented. A simple characteristic equation is derived for axially loaded shear beams with translational and rotational springs and with attached lumped end masses. The resulting frequency equation is compared with the classical one. A condition causing the nonclassical shear beams to collapse to the classical ones is found. Natural frequencies are evaluated and mode shapes are given explicitly. The influences of the spring coefficients, axial loads, and rotational inertia on the natural frequencies are expounded. The frequency equations of shear beams with typical ends such as free-free and free-pinned ends can be recovered from the present study as special cases.