A fully consistent linearized model for vibration analysis of rotating beams in the framework of geometrically exact theory

Abstract

The equations of motions governing the free vibrations of prismatic slender beams rotating in a plane at constant angular velocity are derived according to a geometrically exact approach. Compared to other modeling methods, additional stiffening terms induced by pre-stress are found in the dynamic equations after fully consistent linearization about the deformed equilibrium configuration. These terms include axial, bending and torsional stiffening effects which arise when second-order generalized strains are retained. It is shown that their contribution becomes relevant at moderate to high angular speeds, where high means that the equilibrium state is subject to strains close to the limit where a physically linear constitutive law still applies. In particular, the importance of the axial stiffening is specifically investigated. The natural frequencies as a function of the angular velocity and other system parameters are computed and compared with benchmark cases available in the literature. Finally, the error on the modal characteristics of the rotating beam is evaluated when the linearization is carried out about the undeformed configuration.