Analysis of hysteresis effect on the vibration motion of a bimodal non-uniform micro-cantilever using MCS theory

Nowadays, to enhance the performance of atomic force microscopy (AFM) micro-cantilevers (MCs) during imaging, reduce costs and increase the surface topography precision, advanced MCs equipped with piezoelectric layers are utilized. Using the modified couple stress (MCS) theory not only makes the modeling more exhaustive, but also increases the accuracy of prediction of the vibration behavior of the system. In this paper, Hamilton’s principle by consideration of the MCS theory has been used to extract the equations. In addition, to discretize the equations, differential quadrature method has been adopted. Analysis of the hysteresis effect on the vibration behavior of the AFM MC is of significant importance. Thus, to model the hysteresis effect, Bouc–Wen method, which is solved simultaneously with the vibration equations of non-uniform Timoshenko beam, has been utilized. Furthermore, a bimodal excitation of the MC has been considered. The results reveal that the hysteresis effect appears as a phase difference in the time response. Finally, the effect of the geometric parameters on the vibration frequency of the system which is excited by combination of the first two vibration modes of the non-uniform piezoelectric MC has been examined. The results indicate the considerable effect of the MC length in comparison with other geometric parameters such as the MC width and thickness.