Nonlocal vibration and stability of a multiple-nanobeam system coupled by the Winkler elastic medium

Abstract

The nonlocal Euler–Bernoulli beam theory is proposed for the free vibration and stability analyses of a multiple-nanobeam system (MNBS) using the Eringen nonlocal continuum theory. It is assumed that every nanobeam in the system of multiple-nanobeams is simply supported and under the influence of axial load. The MNBS is embedded in the Winkler elastic medium. The motion of the system is described by a set of m homogeneous partial differential equations, which are derived by using D'Alembert's principle. Analytical solutions of free vibrations and buckling for a multiple-nanobeam system are obtained by using the classical Bernoulli–Fourier method and trigonometric method, for the case of the identical nanobeam. To validate the accuracy of application of trigonometric methods for a determined natural frequency and buckling load of MNBS, numerical solution of the characteristic polynomial are also conducted for a different number of nanobeams. Moreover, the finite difference method is employed to solve the system of partial differential equations of motion of MNBS, where good agreement with the analytical results is achieved. Numerical studies show the effect of the nonlocal parameter, stiffness of Winkler elastic medium and number of nanobeams for the free transversal vibration and buckling of MNBS. The presented analyses are very useful for the study and design of the nanoelectromechanical systems such as nano-resonators.