A nonlocal Timoshenko beam theory for vibration analysis of thick nanobeams using differential transform method

Abstract

This article presents the solution for free vibration of nanobeams based on Eringen nonlocal elasticity theory and Timoshenko beam theory. The small scale effect is considered in the first theory, and the transverse shear deformation effects as well as rotary inertia are taken into account in the latter one. Through variational formulation and the Hamilton principle, the governing differential equations of free vibration of the nonlocal Timoshenko beam and the boundary conditions are derived. The obtained equations are solved by the differential transformation method (DTM) for various frequency modes of the beams with different end conditions. In addition, the effects of slenderness and on vibration behavior are presented. It is revealed that the slenderness affects the vibration characteristics slightly whilst the small scale plays a significant role in the vibration behavior of the nanobeam.

The cantilever beams analysis by the means of the first-order shear deformation and the Euler-Bernoulli theory

“…Tessler et al have done the refinement of the Timoshenko Beam theory for sandwich beams using Zigzag Kinematics [8]. The Timoshenko theory was used with Eringen nonlocal elasticity theory to form differential transformation method for the analysis of the thick nano-beams vibrations [9].…”

Buckling analysis of nonlocal strain gradient axially functionally graded nanobeams resting on variable elastic medium

“…Following this, Lots of studies have been performed to investigate the size-dependent response of structural systems based on Eringen's nonlocal elasticity theory. [12][13][14][15][16][17][18][19][20] Also, Eltaher et al 21,22 examined static buckling and vibrational response of small-scale FG beams according to the nonlocal classical beam model. Ebrahimi and Salari 23 examined the application of differential transform method to vibration analysis of graded nanosize beams.…”

Influence of neutral surface position on dynamic characteristics of in-homogeneous piezo-magnetically actuated nanoscale plates

“…It is reported by various researchers that Eringen’s nonlocal elasticity theory 16,17 is an efficient tool to incorporate small scale impacts. 18–21 Recently, nonlocal elasticity theory is employed for analysis of FG nanostructures. 22–24 Among them, Şimşek and Yurtcu 25 investigated static and stability of simply supported Timoshenko FG nanobeams.…”