Semi-analytical solution for free transverse vibrations of Euler–Bernoulli nanobeams with manifold concentrated masses

Abstract

In the present study, transverse vibrations of nanobeams with manifold concentrated masses, resting on Winkler elastic foundations, are investigated. The model is based on the theory of nonlocal elasticity in presence of concentrated masses applied to Euler-Bernoulli beams. A closed-form expression for the transverse vibration modes of Euler-Bernoulli beams is presented. The proposed expressions are provided explicitly as the function of two integrated constants which are determined by the standard boundary conditions. The utilization of the boundary conditions leads to definite terms of natural frequency equations. The natural frequencies and vibration modes of the concerned nanobeams with different numbers of concentrated masses in different positions under some typical boundary conditions (Simply Supported, Cantilevered and Clamped-Clamped) have been analyzed by means of the proposed closed-form expressions in order to show their efficiency. It's worth mentioning that the effect of Downloaded by [University of Nebraska, Lincoln] at 03:53 11 June 2016 ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT 2 various nonlocal length parameters and Winkler modulus on natural frequencies and vibration modes are also discussed. Finally the results are compared with those corresponding to classical local model.

Semi-exact solution for nonlinear dynamic analysis of nanobeams reinforced with functionally graded carbon nanotube located on a viscoelastic foundation

“…In fact, the issue of size-dependent behaviors which take place in nanosized structures cannot be explained by classical continuum theories. As a result of this, some theories related to higher order continuum, such as the classical couple stress theory, 1–4 strain gradient theory, 5 micropolar theory, 6 nonlocal elasticity theory, 7 and surface elasticity, were proposed to account for the existence of size effect in some structures with a small scale.…”