# Linear and non-linear free vibration of nano beams based on a new fractional non-local theory

Zaher Rahimi (Department of Mechanical Engineering, Urmia University, Urmia, Iran)
Wojciech Sumelka (Institute of Structural Engineering, Politechnika Poznanska, Poznan, Poland)
Xiao-Jun Yang (School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou, China)

ISSN: 0264-4401

Article publication date: 3 July 2017

271

## Abstract

### Purpose

Recently, a new formulation has been introduced for non-local mechanics in terms of fractional calculus. Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and is used successfully in various fields such as mathematics, science and engineering. The purpose of this paper is to introduce a new fractional non-local theory which may be applicable in various simple or complex mechanical problems.

### Design/methodology/approach

In this paper (by using fractional calculus), a fractional non-local theory based on the conformable fractional derivative (CFD) definition is presented, which is a generalized form of the Eringen non-local theory (ENT). The theory contains two free parameters: the fractional parameter which controls the stress gradient order in the constitutive relation and could be an integer and a non-integer and the non-local parameter to consider the small-scale effect in the micron and the sub-micron scales. The non-linear governing equation is solved by the Galerkin and the parameter expansion methods. The non-linearity of the governing equation is due to the presence of von-Kármán non-linearity and CFD definition.

### Findings

The theory has been used to study linear and non-linear free vibration of the simply-supported (S-S) and the clamped-free (C-F) nano beams and then the influence of the fractional and the non-local parameters has been shown on the linear and non-linear frequency ratio.

### Originality/value

A new parameter of the theory (the fractional parameter) makes the modeling more fixable – this model can conclude all of integer and non-integer operators and is not limited to special operators such as ENT. In other words, it allows us to use more sophisticated mathematics to model physical phenomena. On the other hand, in the comparison of classic fractional non-local theory, the theory applicable in various simple or complex mechanical problems may be used because of simpler forms of the governing equation owing to the use of CFD definition.

## Citation

Rahimi, Z., Sumelka, W. and Yang, X.-J. (2017), "Linear and non-linear free vibration of nano beams based on a new fractional non-local theory", Engineering Computations, Vol. 34 No. 5, pp. 1754-1770. https://doi.org/10.1108/EC-07-2016-0262

## Publisher

:

Emerald Publishing Limited