Free vibration analysis of micro rotating beams based on the strain gradient theory using the differential transform method: Timoshenko versus Euler-Bernoulli beam models

Abstract

Abstract In this paper, a novel formulation for the micro rotating beams based on the strain gradient theory and the assumptions of Timoshenko and Euler-Bernoulli beam models is developed. By introducing the prestressed configuration induced by the centrifugal forces, the obtained equations of motion are linearized about the prestressed configuration. Some dimensionless parameters are assigned to achieve the nondimensionalized equations of motion. Differential transform method is applied on the equations of motion and the associated boundary conditions to obtain the flapping and axial natural frequencies. The current outcomes are validated by those available in the literature. After validation of the obtained results, some case studies are prepared to define the material length scale, the rotation speed and the slenderness ratio influences on the flapping and axial frequencies. The outcomes illustrated the natural frequencies are very sensitive to the case study parameters. Also, employing the Euler-Bernoulli beam model leads to the incorrect results in calculation of the flapping frequencies for a wide range of slenderness ratio even when the material length scale is close to the micro beam thickness.