Near-Grazing Dynamics of Macro-scale and Micro-scale Cantilevers with Nonlinear Tip Interaction Forces

In this article, the dynamics of base-excited elastic cantilevers with non-linear attractive and repulsive forces in macro-scale and micro-scale systems are studied through experimental and numerical means. The macro-scale set-up consists of a base-excited elastic cantilever with a long-range attractive force and a short-range repulsive force. The attractive force is generated by a combination of two magnets, one located at the cantilever structure’s tip and another on a movable translatory base. The repulsive force is generated through impacts between the cantilever tip and a compliant material that covers the magnet on the translatory stage. This macro-scale experimental system is motivated by micro-scale cantilevers used in tapping mode or dynamic mode atomic force microscopy (AFM). In tapping mode AFM, the micro-cantilever undergoes a long-range van der Waals attractive force and a short-range repulsive force as the cantilever tip approaches the sample. The authors study the macro-scale system and the micro-scale system, when the excitation frequency is away from the first natural frequency. For off-resonance excitations, period-doubling events are observed in these impacting systems. Areduced-order model is developed to numerically study these systems on the basis of a single mode assumption. In the numerical studies, similar nonlinear tip-sample forces are used to model the interaction forces on the cantilever’s tip in both macro-scale and micro-scale systems. In an effort to understand the effects of noise on the dynamics, the responses of the systems are studied when Gaussian white noise is introduced into the base excitation, along with a harmonic component. It is observed that the addition of Gaussian white noise facilitates contact between the tip and the sample, for low levels of a harmonic base excitation.