On the forced vibration analysis of Timoshenko nanobeams based on the surface stress elasticity theory

Abstract

This paper aims to present nonlinear forced vibration characteristics of nanobeams including surface stress effect. By considering the local geometrical nonlinearity based on von Karman relation, a new formulation of the Timoshenko beam model is developed through the Gurtin–Murdoch elasticity theory in which the effect of surface stress is incorporated. By using a variational approach on the basis of Hamilton’s principle, the size-dependent equations of motion and associated boundary conditions are obtained. The generalized differential quadrature (GDQ) method is employed to discretize the non-classical governing differential equations over the spatial domain by using the shifted Chebyshev–Gauss–Lobatto grid points. Subsequently, a Galerkin-based numerical approach is put to use in order to reduce the set of nonlinear equations into a time-varying set of ordinary differential equations of Duffing-type. In the next step, the time domain is discretized via spectral differentiation matrix operators which are defined based on the derivatives of a periodic base function. Finally, the pseudo arc-length method is employed to solve the resulting nonlinear parameterized algebraic equations. The frequency–response curves for forced vibration behavior of nanobeams including the effect of surface stress are predicted corresponding to various values of beam thickness, length to thickness ratio and surface elastic constants. It is revealed that by incorporating the surface stress effect, the maximum amplitude occurs at lower excitation frequencies and the wide of region of the response tends to decrease.