Effect of residual surface stress on parametrically excited nonlinear dynamics and instability of double-walled nanobeams: an analytical study

Abstract

A class of nonlinear Mathieu–Hill equation is established to determine the bifurcations and the regions of nonlinear dynamic instability of a short double-walled nanobeam, while the emphasis is placed on investigating the effect of residual surface stress on instability. To achieve this goal, first, a short double-walled nanobeam is modeled and embedded on a viscoelastic foundation and subjected to an axial parametric force. Second, based on the nonlocal elasticity and nonlinear von Karman beam theories, the nonlinear governing equation of motion is derived. Finally, Galerkin technique and multiple time scales method are used to solve the equation. Numerical examples are treated which show various discontinuous bifurcations. Also, infinitely stable and unstable solutions are addressed.