Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory

In this article, the nonlinear free vibration behavior of Timoshenko nanobeams subject to different types of end conditions is investigated. The Gurtin–Murdoch continuum elasticity is incorporated into the Timoshenko beam theory in order to capture surface stress effects. The nonlinear governing equations and corresponding boundary conditions are derived using Hamiltonʹs principle. A numerical approach is used to solve the problem in which the generalized differential quadrature method is applied to discretize the governing equations and boundary conditions. Then, a Galerkin-based method is numerically employed with the aim of reducing the set of partial differential governing equations into a set of time-dependent ordinary differential equations. Discretization on time domain is also done via periodic time differential operators that are defined on the basis of the derivatives of a periodic base function. The resulting nonlinear algebraic parameterized equations are finally solved by means of the pseudo arc-length continuation algorithm through treating the time period as a parameter. Numerical results are given to study the geometrical and surface properties on the nonlinear free vibration of nanobeams.