Nonlinear dynamics and control of the scan process in noncontacting atomic force microscopy

The focus of this paper is on the nonlinear dynamics and control of the scan process in noncontacting atomic force microscopy. An initial-boundary-value problem is consistently formulated to include both nonlinear dynamics of a microcantilever with a localized atomic interaction force for the surface it is mapping, and a horizontal boundary condition for a constant scan speed and its control. The model considered is obtained using the extended Hamilton's principle which yields two partial differential equations for the combined horizontal and vertical motions. Isolation of a Lagrange multiplier describing the microbeam fixed length enables construction of a modified equation of motion which is reduced to a single mode dynamical system via Galerkin's method. The analysis includes a numerical study of the strongly nonlinear system leading to a stability map describing an escape bifurcation threshold where the tip, at the free end of the microbeam, 'jumps-to-contact' with the sample. Results include periodic ultrasubharmonic and quasiperiodic solutions corresponding to primary and secondary resonances.