Vibration Analaysis for Nano Beam Resting on Winkler Foundation via Modified Couple Stress Theory

Abstract

In this study, vibration of nano beams resting on Winkler foundation is investigated based on modified couple stress theory. The governing equation of the system is obtained on the base of Hamilton’s principle. The size effect of the nano beam cannot be captured by using classical Euler-Bernoulli beam theory, but the modified couple stress theory model can be captured it. Because, the newly developed model contains a material length scale parameter. Material length scale parameter are set to be zero corresponds to the equation of the classical Euler-Bernoulli beam theory. The method of multiple scale is employed to obtain result. Simply supported boundary condition is used to study natural frequencies. The effect of Winkler elastic foundation parameter and material length scale parameter on the natural frequencies of the nano beam is investigated and tabulated. In addition, Poisson’s ratio is assumed constant in the presented model. The free vibration of the simply supported nanobeam resting on Winkler foundation is analyzed to illustrate the size effects. The numerical results for simply supported nanobeam show that the fundamental natural frequency estimated by presented model is higher than classical one. Moreover, it is observed that size effect is more significant for higher vibration modes. The results show that the significant importance of the size effects in analyzing of nanobeams. Hardening spring behavior can be seen in the vibration of nano beam, and the beams stiffer than does the classical beam theory in the newly developed models. This study may be useful to describe the dependent of size of mechanical properties of nano electromechanical systems (NEMS).