Tip trajectories of a smart micro-cantilever beam: Analysis and design

The tip trajectories of a smart micro-cantilever beam consisting of an atomic force microscope probe with an additional segment of piezoelectric material on top of the probe are studied. A precise model with inhomogeneous partial differential equations and compatible inhomogeneous boundary conditions is developed to describe the dynamics of the smart micro-cantilever beam. The forced vibration solution of this model with respect to two independent inputs from the piezoelectric actuator and the base excitation is derived. By using this solution and the geometric relationship, the trajectory of the end of the tip is obtained from the motion of the free end of the cantilever beam. On the basis of the resonant response to harmonic inputs at the second dynamic mode, nano-scale elliptical and linear tip trajectories are predicted. Within this paper, a smart micro-cantilever beam is shown to produce nano-scale curved tip trajectories for the first time. Analytical and simulation findings indicate that the characteristics of the resulting trajectories are influenced by the magnitudes of two inputs. Potential applications of the elliptical and linear trajectories for nanomanipulation are proposed.