Numerical Analyses in the Nonlinear Dynamics and Control of Microcantilevers in Atomic Force Microscopy

It is well known in the literature that AFMs operating in dynamic mode can exhibit several nonlinear phenomena, such as bifurcations, in-well instability regions and eventually chaotic motion, that are common to many other dynamical systems and represent an undesirable behavior which restricts the operating range of many electronic and mechanical devices. The deep investigation of their dynamical bifurcation behavior as a function of the main system parameters is thus a topic of great theoretical and practical importance, not only to frame such systems in the literature scene, but also because its potentiality in enhancing performance, effectiveness, reliability and safety of systems is crucial to the aim of developing novel design criteria. In this perspective, the nonlinear response of a single-mode model of noncontact AFM has been analyzed by making use of several computational tools, in order to investigate the evolution of the main system periodic solutions and relevant basins of attraction under variations of the most significant system parameters. Different numerical simulations and continuation techniques have been employed (using Dynamics software and AUTO software) taking into account the presence of the horizontal parametric excitation and of the vertical external one, separately. Several bifurcation diagrams have been obtained in a large range of forcing frequencies which includes the fundamental (primary) and principal (subharmonic) parametric (external) resonances, whereby the main periodic solutions and local bifurcations have been detected thanks to the Floquet multipliers computation. The local bifurcation loci have been summarized in behavior charts, which report also the system stability threshold obtained as the envelope of local bifurcation escape thresholds in different parameter ranges. Moreover, erosion process of the basins of attraction of the various solutions, which is indeed a critical issue corresponding to system impending escape (corresponding to the unwanted jump-to-contact) and thus governing its practical safety, is investigated by applying the dynamical integrity concepts. Thanks to the analysis of basins of attraction evolution, and making use of specific computational tools such as the evaluation of different integrity measures (GIM and IF), several erosion profiles have been obtained as a function of the increasing excitation amplitude, with the aim to detect thresholds of residual integrity able to ensure acceptable safety targets established a priori according to the required system performances. The topic of controlling undesirable system dynamical responses is then addressed through the insertion in the AFM model of an external feedback control technique, with the aim to take the system response to a selected reference one. The periodic motion used as reference in the control procedure is chosen to be the response of the corresponding uncontrolled system, for which the previous analyses have already allowed to detect the main stability regions in various parameters planes. Upon checking the effectiveness of the procedure in the weakly nonlinear regime via a perturbation approach, the description of bifurcation/response scenarios of the controlled system under scan excitation up to the strongly nonlinear regime, and the critical comparison with the results related to the uncontrolled system permit to highlight the influence of the applied control on the overall dynamical behavior of the AFM system, and provide indications to refer to in practical applications.