Free Vibration, Buckling and Dynamical Stability of Timoshenko Micro/Nano-Beam Supported on Winkler-Pasternak Foundation Under a Follower Axial Load

Abstract

Axially loaded micro/nano-beams supported on foundations are extensively applied in micro/nano-electro-mechanical systems. This paper deals with the problems of free vibration, buckling and dynamical stability of Timoshenko micro/nano-beam supported on Winkler–Pasternak foundation subjected to a follower axial periodic load. Based on Hamilton’s principle, the governing equations of the system are derived in conjunction with nonlocal strain gradient and Timoshenko theory. The transition parameter is introduced to describe the follower direction of axial load during the deformation of the micro/nano-beam. Employing the weighted residual method, the variational consistency boundary conditions (BCs) can be derived according to the governing equations. Using differential quadrature method (DQM), the governing equations are discretized and numerical solutions of the natural frequencies, critical buckling load and instability region are obtained. Numerical examples are performed to verify present solutions by those available in the literature. Significant effects of the transition parameter and variational consistency BCs are revealed on the free vibration, buckling and dynamical stability of the axially loaded micro/nano-beam. The present analysis is of significance to axially-loaded micro/nano-beam mechanical system, especially for determination of the direction of follower axial force.