A theory for transverse vibrations of the timoshenko beam

Abstract

A second-order variational formalism is presented for deriving the Timoshenko equation and boundary conditions consistent with it. Properties of the second high-frequency mode of vibrations predicted in the Timoshenko theory are investigated. It is shown that the frequencies of this mode depend non-analytically on a small parameter describing the influence of shear deformation on the transverse vibrations of the beam. When this parameter vanishes the frequencies of the second series do not return to the unperturbed values, but become infinite. Hence they cannot be predicted exactly, but the fact that they are being taken into account corrects and improves the values of the frequencies of the fundamental mode of vibrations. For these frequencies the Ostrogradskii energy of the Timoshenko beam turns out to be negative. The part played by the second mode of vibrations in the Timoshenko theory is discussed. A simple method for taking into account the effect of the deformation of the crosssection of the beam during the vibrations on its natural frequencies is suggested.