On Nonlinear Fractional Dynamics Analysis of Atomic Force Microscopy System in Trolling Mode with Viscoelasticity Term

In this paper, we investigate a nonlinear dynamic model of the trolling mode in atomic force microscopy (AFM-TR) considering the fractional viscoelasticity. The term fractional viscoelasticity is an approximate representation of the medium in which the system is used for the analysis of biological samples. The fractional nonlinear dynamic model of the AFM-TR considers a nanoneedle coupled to a nanobeam that, when subjected to vibrations during the interaction with the sample surface, generates a three-dimensional image of its topology. These interactions between the nanoneedle and sample are Van Der Waals force type, and other interactions are considered biological environment. With these considerations, the fractional dynamics were analyzed considering the Riemann–Liouville fractional derivative operator. The results obtained the intervals in which the system has a chaotic or periodic behavior, such analyses corroborate to determine a set of parameters in which the term of fractional viscoelasticity may influence the dynamics of the AFM-TR.