Effects of Non-Ideal Boundary Conditions on the Vibrations of a Slightly Curved Micro Beam

摘要

Response of a slightly curved resonant microbeam having non-ideal boundary conditions is investigated. Non-ideal boundary conditions are formulated using perturbation theory. These non-ideal conditions allow for small deflection at the right end of the microbeam. The curvature is taken as a sinusoidal function of the spatial variable. The initial displacement is due to the geometry of the microbeam itself. They are produced intentionally to be curved or made curved by buckling straight beams through compressive axial loads. The model accounts for mid-plane stretching, an applied axial load and an AC harmonic force. The ends of the curved microbeam are on immovable simple supports. Immovable end conditions introduce integral type nonlinearity. The integro-differential equations of motion are solved analytically by means of direct application of the method of multiple scales (a perturbation method). The amplitude and phase modulation equations are derived for the case of primary resonances. The effect of curvature on the vibrations of the microbeam is examined. It is found that the effect of curvature is of softening type. The frequencies and mode shapes obtained are compared with the ideal boundary conditions case and the differences between them are contrasted on frequency response curves.