Abstract
In this study, the non-local Euler-Bernoulli beam theory was employed in the nonlinear free and forced vibration analysis of a nanobeam resting on an elastic foundation of the Pasternak type. The analysis considered the effects of the small-scale of the nanobeam on the frequency. By utilizing Hamilton’s principle, the nonlinear equations of motion, including stretching of the neutral axis, are derived. Forcing and damping effects are considered in the analysis. The linear part of the problem is solved by using the first equation of the perturbation series to obtain the natural frequencies. The multiple scale method, a perturbation technique, is applied in order to obtain the approximate closed solution of the nonlinear governing equation. The effects of the various non-local parameters, Winkler and Pasternak parameters, as well as effects of the simple-simple and clamped-clamped boundary conditions on the vibrations, are determined and presented numerically and graphically. The non-local parameter alters the frequency of the nanobeam. Frequency-response curves are drawn.
Highlights
Nanotechnology is the manipulation of matter on a supramolecular, molecular, and atomic scale
We examine the literature presented in the above, and it can be seen clearly that an elastic medium surrounded by a Pasternak-type model is limited in literature
The small scale and damping effects are taken into account and nonlinear vibration behaviors of the nanobeam are illustrated
Summary
Nanotechnology is the manipulation of matter on a supramolecular, molecular, and atomic scale. 2016, 21, 3 consideration of the von Kármán geometric nonlinearity and the nonlinear van der Waals forces [26], forced vibration of an elastically-connected DWCNT carrying a moving nanoparticle [27], nonlinear vibration of embedded multiwalled carbon nanotubes (MWCNT) in thermal environments [28], vibration analysis of embedded MWCNT at an elevated temperature with considering the small-scale effect on the large amplitude [29], free transverse vibration of SWCNT embedded in elastic matrix under various boundary conditions [30], thermal vibration of SWCNT embedded in an elastic medium [31], thermal-mechanical vibration and buckling instability of a SWCNT conveying fluid and resting on an elastic medium [32], electro-thermo-mechanical vibration analysis of non-uniform and non-homogeneous boron nitride nanorod embedded in elastic medium [33], buckling behavior of SWCNT on a Winkler foundation under various boundary conditions [34], critical buckling temperature of SWCNT embedded in a one parameter elastic medium [35], and buckling analysis of SWCNT including the effect of temperature change and surrounding elastic medium [36] were studied with the aid of nonlocal elasticity theory. The small scale and damping effects are taken into account and nonlinear vibration behaviors of the nanobeam are illustrated
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