Tapping modes in the Atomic Force Microscope model with Lennard-Jones force and slow-fast base motion

Based on the idea of slow-fast dynamics, two tapping modes are investigated in the Atomic Force Microscope (AFM) model with Lennard-Jones force and slow-fast base motion. According to the approximation derived from Taylor expansion in the vicinity of static equilibrium of the linearized system, we give a prediction about harmonic responses of the fast sub-system (FSS). Then a truncated version of harmonic balancing method (HBM) is combined with the Floquet theory to give the distribution and stability of the harmonic responses, which indicates the contact and non-contact modes dominated by cycle manifolds. By directly integrating the FSS, the numerical bifurcation diagrams and the frequency spectrums corresponding to the periodic solutions are obtained to verify the results given by the half-analytic process. Moreover, distribution of basins of attraction shows that, when the tri-stability is destroyed by fold bifurcation of cycle, due to the highly fragmented fractal structures in basins, transition between different mode manifolds may present an weak irregular way. On the other hand, when the slow excitation is introduced, the regular and irregular "cycle-cycle" type mixed mode oscillations (MMOs) triggered by "fold of cycle/fold of cycle" hysteresis can be observed, which results in two tapping mode, .i.e., the slow tapping mode and the "compound of contact/snap tapping" mode. Besides that, our results show that, for the AFM under slow and near-resonant excitation, the beneficial effects of constant contact mode and snap tapping mode can be utilized.