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Nonlinear Dynamics of Tapping Mode Atomic Force Microscopy
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A mathematical model is developed to investigate the grazing dynamics of tapping mode atomic force microscopes (AFM) subjected to a base harmonic excitation. The nonlinear dynamics of the AFM microcantilever are studied in both of the monostable and bistable phases with the microcantilever tip being, respectively, located in the monostable and bistable regions of the static bifurcation diagram in the reference configuration. Free-vibration responses of the AFM probes, including the microcantilever natural frequencies and mode shapes, are determined. It is found that, for the parameters used in a practical operation of an AFM, the natural frequencies and mode shapes of the AFM microcantilever are almost the same as those of a free-end microcantilever with the same geometry and made of an identical material. A multimode Galerkin approximation is utilized to discretize the nonlinear partial-differential equation of motion and associated boundary conditions governing the cantilever response and obtain a set of nonlinearly coupled ordinary-differential equations (ODE) governing the time evolution of the system dynamics. The corresponding nonlinear ODE set is then solved using numerical integration schemes. A comprehensive numerical analysis is performed for a wide range of the excitation amplitude and frequency. The tip oscillations are examined using nonlinear dynamic tools through several examples. The non-smoothness in the tip/sample interaction model is treated rigorously. A higher-mode Galerkin analysis indicates that period doubling bifurcations and chaotic vibrations are possible in tapping mode microscopy for certain operating parameters. It is also found that a single-mode Galerkin approximation, which accurately predicts the tip nonlinear responses far from the sample, is not adequate for predicting all of the nonlinear phenomena exhibited by an AFM, such as grazing bifurcations, and leads to both quantitative and qualitative errors. A point-mass model is also developed based on the single-mode Galerkin procedure to compare with the present distributed-parameter model. In addition, a reduced-order model based on a differential quadrature method (DQM) is employed to explore the dynamics of the AFM probe in the bistable phase where the multimode Galerkin procedure is computationally expensive. We found that the DQM with a few grid points accurately predicts the static bifurcation diagram. Moreover, we found that the DQM is capable of precise prediction of the lowest natural frequencies of the microcantilever with only a few grid points. For the higher natural frequencies, however, a large number of grid points is required. We also found that the natural frequencies and mode shapes of the microcantilever about non-contact equilibrium positions are almost the same as those of the free-end microcantilever. On the other hand, free-vibration responses of the microcantilever about contact equilibrium positions are quite different from those of the free-end microcantilever. Moreover, we used the DQM to discretize the partial-differential equation governing the microcantilever motion and a finite-difference method (FDM) to calculate limit-cycle responses of the AFM tip. It is shown that a combination of the DQM and FDM applied, respectively, to discretize the spatial and temporal derivatives provides an efficient, accurate procedure to address the complicated dynamic behavior exhibited by the AFM probe. The procedure was, therefore, utilized to study the response of the microcantilever to a base harmonic excitation through several numerical examples. We found that the dynamics of the AFM probe in the bistable region is totally different from those in the monostable region.
- Doctoral Dissertations