Dynamic stability of a size-dependent micro-beam

The effect of variation of the geometrical dimensions on dynamic instability regions (DIRs) of a rectangular cross-sectioned micro-beam with simply supports is discussed in this study based on modified coupled stress theory. A set of linear equations are derived on basis of the Lagrange method and trial series expansions for vertical displacement and rotation of the Timoshenko micro-beam model while longitudinal displacement is neglected due to the stretching effect of the micro-beams mid-plane. The first approximation of the dynamic stability analysis is done by the application of the Bolotin method besides obtaining the Mathieu-Hill equations. The numerical results demonstrate how the cross section's height and micro-beams length are the only effective parameters on DIRs somehow that when the geometrical dimensions are increased the DIRs are going to coincide.