Vibrational modes of Timoshenko beams at small scales

Abstract

This letter presents a theoretical treatment of Timoshenko [S. Timoshenko, Philos. Mag. 41, 744 (1921)] beams, in which the influences of shear deformation, rotary inertia, and scale coefficient are taken into account. Based on the nonlocal elasticity theory, coupled equations for transverse deflection and rotation of cross section are derived. Free vibration of several typical beams is analyzed. Explicit expressions for modal shapes of vibration are presented. Natural frequencies are evaluated for free vibration of simply supported beams, clamped beams, cantilever beams, and clamped-hinged beams. The effects of the nonlocal parameter on natural frequencies and modal shapes are discussed in detail. © 2009 American Institute of Physics.