Free vibration of Euler and Timoshenko nanobeams using boundary characteristic orthogonal polynomials

Laxmi Behera,S Chakraverty

doi:10.1007/s13204-013-0202-4

Laxmi Behera, S Chakraverty

Open Access

https://doi.org/10.1007/s13204-013-0202-4

Copy DOI

Journal: Applied nanoscience	Publication Date: Feb 19, 2013
Citations: 37	License type: CC BY 2.0

Affiliation: National Institute of Technology Rourkela

Abstract

Vibration analysis of nonlocal nanobeams based on Euler–Bernoulli and Timoshenko beam theories is considered. Nonlocal nanobeams are important in the bending, buckling and vibration analyses of beam-like elements in microelectromechanical or nanoelectromechanical devices. Expressions for free vibration of Euler–Bernoulli and Timoshenko nanobeams are established within the framework of Eringen’s nonlocal elasticity theory. The problem has been solved previously using finite element method, Chebyshev polynomials in Rayleigh–Ritz method and using other numerical methods. In this study, numerical results for free vibration of nanobeams have been presented using simple polynomials and orthonormal polynomials in the Rayleigh–Ritz method. The advantage of the method is that one can easily handle the specified boundary conditions at the edges. To validate the present analysis, a comparison study is carried out with the results of the existing literature. The proposed method is also validated by convergence studies. Frequency parameters are found for different scaling effect parameters and boundary conditions. The study highlights that small scale effects considerably influence the free vibration of nanobeams. Nonlocal frequency parameters of nanobeams are smaller when compared to the corresponding local ones. Deflection shapes of nonlocal clamped Euler–Bernoulli nanobeams are also incorporated for different scaling effect parameters, which are affected by the small scale effect. Obtained numerical solutions provide a better representation of the vibration behavior of short and stubby micro/nanobeams where the effects of small scale, transverse shear deformation and rotary inertia are significant.

Highlights

Nanomaterials have encouraged the interest of the scientific researchers in physics, chemistry and engineering
Deflection shapes of nonlocal clamped Euler–Bernoulli nanobeams are incorporated for different scaling effect parameters, which are affected by the small scale effect
This paper mainly focuses on solving the governing differential equations of Euler–Bernoulli and Timoshenko nanobeams by an efficient way

Summary

Introduction

Nanomaterials have encouraged the interest of the scientific researchers in physics, chemistry and engineering. Free vibration of Euler–Bernoulli and Timoshenko nanobeams based on nonlocal continuum mechanics has been solved analytically by Wang et al (2007). This paper mainly focuses on solving the governing differential equations of Euler–Bernoulli and Timoshenko nanobeams by an efficient way As such Rayleigh–Ritz method with simple polynomials and orthonormal polynomials has been used in this investigation. It may be noted that the kinetic and potential energy expressions used in the Rayleigh–Ritz method are as such not simple as compared to classical beams and plates This is due to the fact that governing differential equations of nanobeams should be handled considering the nonlocal theory as mentioned above. Investigation is carried out to understand the small scale effects in the free vibration of nonlocal nanobeams based on Euler–Bernoulli and Timoshenko beam theories. Substituting Eq (21) into Eqs. (3) and (5) and differentiating partially with respect to unknown coefficients cj, a generalized Eigen value problem will be obtained as 1⁄2KfYg 1⁄4 k21⁄2MfYg ð24Þ where fYg 1⁄4 1⁄2c1c2. . .cnT , and the matrices [K] and [M] are given in ‘‘Appendix’’

Solution methodology

Method of solution using orthonormal polynomials