Non-smooth dynamics and control of tapping mode atomic force microscopy

Interaction of an oscillating micro-cantilever with near-field surface potentials in a tapping-mode atomic force microscope results in several complex nonlinear dynamical phenomena. However, existing analytical tools are inadequate for the understanding of these phenomena as the force field governing the dynamics of the system is both nonlinear and non-smooth.;Here, a discontinuity-mapping analysis is presented to study the nonlinear non-smooth dynamics of a tapping-mode atomic force microscope. Two different cases of tangential (grazing) contact of a solution trajectory with the discontinuity surface are considered. In the simpler co-dimension one case, a close connection is established between the contact event and the transition of the stable low amplitude oscillation state to a stable high amplitude state, through a saddle-node bifurcation. An event-driven, discrete feedback strategy which seeks to avoid transitions between the low and high amplitude stable solution states using the amplitude and the phase information of the oscillation states is presented. This strategy allows for mapping out the force-distance curve of an atomic force microscope experimentally, making it possible to use an unstable solution state as a set-point for imaging.;The case of co-dimension two grazing characterized by simultaneous tangential contact of a solution trajectory as well as a solution branch with the discontinuity surface is also presented. Analytical conditions are formulated to locate the co-dimension two grazing point and discontinuity-mapping technique is used to obtain a normal form description of the near-grazing dynamics. In addition to the non-smooth vector field, the relative strength of the vector fields on either side of the discontinuity surface is shown to be critical in determining the occurrence of discontinuity-induced isola and branch-point bifurcations. This result motivates a feedback strategy which modifies the near-grazing dynamics to ensure the persistence of the grazing solution trajectory beyond grazing. The feedback scheme applied to a linear impacting oscillator, and a dynamic model of tapping-mode atomic force microscope ensures persistence of the grazing solution trajectory by preventing loss of stability through a discontinuity induced saddle-node bifurcation.