Abstract
Vibration dynamics of elastic beams that are used in nanotechnology, such as atomic force microscope modeling and carbon nanotubes, are considered in terms of a fundamental response within a matrix framework. The modeling equations with piezoelectric and surface scale effects are written as a matrix differential equation subject to tip-sample general boundary conditions and to compatibility conditions for the case of multispan beams. We considered a quadratic and a cubic eigenvalue problem related to the inclusion of smart materials and surface effects. Simulations were performed for a two stepped beam with a piezoelectric patch subject to pulse forcing terms. Results with Timoshenko models that include surface effects are presented for micro- and nanoscale. It was observed that the effects are significant just in nanoscale. We also simulate the frequency effects of a double-span beam in which one segment includes rotatory inertia and shear deformation and the other one neglects both phenomena. The proposed analytical methodology can be useful in the design of micro- and nanoresonator structures that involve deformable flexural models for detecting and imaging of physical and biochemical quantities.
Highlights
Atomic force microscopy (AFM) is a scanning probe microscopy (SPM) technique to obtain images of surface topography at the atomic scale, in a noninvasive manner, from a wide variety of samples on a scale from angstroms to 100 microns [1]
We considered a quadratic and a cubic eigenvalue problem related to the inclusion of smart materials and surface effects
∗Experimentally obtained that the spatial amplitude diminishes with the inclusion of a piezoelectric patch when the pulse is positioned in the piezoelectric layer segment; this effect does not appear when the pulse is positioned in the third segment (Figure 3)
Summary
Atomic force microscopy (AFM) is a scanning probe microscopy (SPM) technique to obtain images of surface topography at the atomic scale, in a noninvasive manner, from a wide variety of samples on a scale from angstroms to 100 microns [1]. We seek to develop a vibration dynamics framework for beams that include smart materials and subject to surface effects This framework is considered in the case of a two-span beam in which the first segment is governed by the Timoshenko model and the second segment is an Euler-Bernoulli beam model [3]. It has been observed that for beam length on the order of nanometer to microns, the difference between natural frequencies is apparent, and by increasing the length of the microbeam, the results tend to Timoshenko classical theory; that is, the surface effects are significant only in nanoscale. AFMbased nanoscale processing with continuum surrounding media such as that found in biology and nanomachining applications [20,21,22] suggests observing frequency effects that arise with an academic two span beam model in which one segment includes rotatory inertia and shear deformation and the other one neglects both effects
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