Abstract
In this paper, the vibration of nanobeams resting on the Winkler foundation is proposed using the modified couple stress theory. Hamilton’s principle is utilized to construct the governing equations. The size effect of the nanobeam cannot be captured by using classical Euler-Bernoulli beam theory, but the modified couple stress theory model can capture it because it includes material length scale parameter that a newly developed model has. Once the material length scale parameter is assumed to be zero, the classical Euler-Bernoulli beam theory equation is obtained. Multiple scale method is employed to obtain the result. Simply supported boundary condition is used to study natural frequencies. The influence of material length scale parameter and the Winkler elastic foundation parameter on the fundamental frequencies of the nanobeam is investigated and tabulated. Also, in the present study, Poisson’s ratio is taken as constant. Nanobeam resting on the Winkler foundation which is simply supported is analyzed to illustrate the size effects on the free vibration. Numerical results for the simply supported nanobeam indicate that the first fundamental frequency calculated by the presented model is higher than the classical one. Moreover, it is obtained that the size influence is more substantial for higher vibration modes. The results indicate that the significant importance of the size influences the analysis of nanobeams. The vibration of nanobeam exhibits a hardening spring behavior, and the newly developed models are the beams stiffer than according to the classical beam theory. Modified couple stress theory tends to be more helpful in describing the size-dependent mechanical properties of nanoelectromechanical systems (NEMS).
Highlights
In recent years, research on nanostructures, which are used as a fundamental part of a large number of micro and nanoelectromechanical systems (MEMS and NEMS), has received great interest from researchers
It is obvious from the table that the natural frequencies increase once the dimensionless Winkler foundation parameter (κ) increases
This is because increasing the Winkler foundation parameters increases the stiffness of the beam
Summary
Research on nanostructures, which are used as a fundamental part of a large number of micro and nanoelectromechanical systems (MEMS and NEMS), has received great interest from researchers. Double-walled carbon nanotubes (DWCNTs) that convey fluid are examined with the aid of the modified couple stress theory to analyze the vibration and instability characteristics The results of their studies showed that the effect of length scale parameter performs a significant role in the fluid conveying DWCNNTs frequency and critical flow velocity. Simsek [7] focused on the static bending of microbeams in large amplitude and free vibration of microbeams on the nonlinear elastic medium by employing Euler-Bernoulli beam theory and modified couple stress theory His results indicated that the increment of the dimensionless scale parameter decreases the nonlinear frequency ratio. Barooti and Ghadiri [8] studied the critical speed and free vibration analysis of spinning 3D SWCNTs rested on an elastic medium considering the modified couple stress theory They have concluded that material length scale parameter and the angular velocity have an important effect on the rotating SWCNTs fundamental frequencies. Considerable numerical data is presented in tabulated form for various values of the parameters; in the future, these results may be used as a reference
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