Nonlinear multimode dynamics and internal resonances of the scan process in noncontacting atomic force microscopy

The focus of this paper is on the nonlinear multimode dynamics of a moving microbeam for noncontacting atomic force microscopy (AFM). An initial-boundary-value problem is consistently formulated, which includes both nonlinear dynamics of a microcantilever with a localized atomic interaction force, and a horizontal boundary condition for a constant scan speed and its control. The model considered is obtained using the extended Hamiltons principle, which yields two partial differential equations for the combined horizontal and vertical motions. The model incorporates, for the first time to our knowledge, two independent time-varying terms that depict the vertical base excitation of the AFM and the horizontal forcing term depicts the periodic scanning motion of the cantilever. Manipulation of these equations via a Lagrange multiplier enables construction of a modified equation of motion, which is reduced, via Galerkins method, to a three-mode dynamical system, corresponding to finite amplitude AFM dynamics. The analysis includes a numerical study of the strongly nonlinear system culminating with 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, quasiperiodic, and non-stationary chaotic-like solutions corresponding to primary and secondary internal combination resonances, where the latter corresponds to energy balance between the cantilever modes.