Size-dependent three-dimensional free vibration of rotating functionally graded microbeams based on a modified couple stress theory

Abstract

A size-dependent three-dimensional dynamic model of rotating functionally graded (FG) microbeams is developed based on the Euler–Bernoulli beam theory. A two-constituent material varying along the thickness is considered following the power law. In addition, Poisson's ratio is assumed constant in the present model. Hamilton's principle is adopted in conjunction with a modified couple stress theory to derive the governing equations with the von Kármán geometric nonlinearity incorporated. The Galerkin method is utilized to solve these equations in which the coupling of the axial, chordwise and flapwise deformations is included. The centrifugal stiffening effect of the rotating FG microbeam is captured by the nonlinear coupling deformations. The convergence, accuracy and validity of the present method are confirmed by several examples. The influence of the size-dependency on modal characteristics is investigated combined with the material gradient index, dimensionless rotational speed and hub radius ratio. Finally, the achieved results of dynamic responses indicate that the deformations of rotating FG microbeams are greatly affected by the size-dependency and material gradient variation.