Spatio-temporal dynamics of microcantilevers tapping on samples observed under an atomic force microscope integrated with a scanning laser Doppler vibrometer: applications to proper orthogonal decomposition and model reduction

As the use of microsystems has become more widespread, the need for reduced order models that accurately and efficiently predict their dynamic behavior has also grown. Proper orthogonal decomposition (POD) has proven to be a beneficial model-reduction tool in structural dynamics; however, its use in the development of reduced order models of microsystems from experimental data has not received much attention. In this paper, we combine a scanning laser vibrometer with a fully functional atomic force microscope (AFM) to measure the vibration field of AFM microcantilevers tapping on stiff and soft samples. Both stiff and soft AFM microcantilevers with internal resonance are studied. This represents a wide class of vibrating microsystems with spatially confined nonlinearity at the sharp probe tip arising from short and long range surface forces. We apply POD to the measured vibration field of AFM cantilevers and show that a small number of experimentally extracted proper orthogonal modes (POMs) can accurately represent the nonlinear microcantilever dynamics under a variety of conditions. Moreover only one or two sets of POMs are needed to represent the dynamics under a wide range of operating conditions including in the attractive and repulsive regimes of oscillation.