The effect of liquid medium on vibration and control of the AFM piezoelectric microcantilever

This article investigates the vibration motion and control of the piezoelectric microcantilever (MC) of the atomic force microscope in the amplitude mode in a liquid environment for both free and forced vibrations. The modeled MC includes two electrode layers, a piezoelectric layer, and the geometric discontinuities as a result of these layers and the change in the MC cross section at the probe-MC contact point is modeled. The equations of motion are derived using Hamilton's principle and then discretized with the aid of the finite element method. The system frequency and time response in the free vibration mode when placed in a liquid medium are compared with experimental results. Also, the sample surface topography in the noncontact mode, when passing through the surface roughness, has been modeled as rectangular and wedge-shaped. Furthermore, the control of piezoelectric MC in two cases of near and far from the sample surface is examined. In the case of far from the surface, the system is controlled using the piezoelectric layer installed on the beam, whereas in the case of near the surface, the piezo is turned off due to the existence of nonlinear forces such as Van der Waals and Derjaguin, Muller, and Toporov contact forces and the control is achieved through the base excitation. For system control modeling, the robust fuzzy sliding mode control (FSMC) approach is utilized and the results are compared with those of the simple sliding mode control (SMC) as well as proportional, integral, and derivative control. The presented FSMC approach can remarkably reduce the chattering phenomenon at the SMC input in the liquid medium and accordingly improve the results compared to the other two methods.