Free vibration of shear beams with finite rotational inertia

Free vibration of shear beams is studied when rotational motion is taken into account, while classical shear beams do not consider rotational motion. From a single governing equation of Timoshenko beams, we analytically derive Rayleigh beams and shear beams as two limiting cases of the ratio of reduced shear stiffness to bending stiffness being sufficiently large and small, respectively. Emphasis is placed on the analysis of free vibration of nonclassical shear beams without damping effect. Under the condition of general end restraints, a characteristic equation for nonclassical shear beams with finite rotational inertia is derived in explicit form. A condition that the nonclassical shear beams reduce to the classical ones is found, and classical shear beams may be understood as nonclassical ones with infinite large rotational inertia. Nonclassical natural frequencies and mode shapes are calculated for a standing shear beam on an elastic foundation. Previous results of pinned-free, and free–free shear beams can be taken as special cases of the present analysis. The effects of finite rotational inertia, material properties, geometrical conditions and end restraints on the natural frequencies of shear beams are discussed.

Research Highlights
► Timoshenko beams can analytically reduce to nonclassical shear beams.
► A closed-form characteristic equation of nonclassical shear beams is derived.
► Rotary inertia has a strong effect on the natural frequencies of shear beams.