The Flapwise Bending Free Vibration Analysis of Micro-rotating Timoshenko Beams Using the Differential Transform Method

Abstract

The flapwise bending free vibration analysis of isotropic rotating Timoshenko microbeams, including the size effects, is presented in this paper. A nonclassical theory, i.e. the Modified Couple Stress theory, has been employed to include the size effect in the presented formulation. By consideration of the Timoshenko beam assumptions the shear deformation and the rotary inertia effects are taken into account. The Hamilton’s principle is applied to the obtained strain and kinetic energy relations to derive the nonlinear equations of motion and the associated boundary conditions. After nondimensionalization of the equations of motion and the corresponding boundary conditions, the linearized form of the equations of motion and the accompanied boundary conditions are developed. A semi-analytical approach, i.e. the differential transform method, is implemented to achieve the flapping and axial frequencies. The achieved results are validated via comparison with the available results in the literature. The material length scale, shear deformation consideration, rotating speed and the slenderness ratio influences on the natural frequencies are examined. The results demonstrate that the slenderness ratio and the thickness to the material length scale parameter quotient are the dominant indicators in determining the usage of the nonclassical theories against classical theories. On the other hand, the precision in determination of the higher modes frequencies motivates us to implement the Timoshenko beam model instead of the Euler–Bernoulli beam model.