A variational approach for free vibrating micro-rods with classical and non-classical new boundary conditions accounting for nonlocal strengthening and temperature effects

Abstract

Transverse vibration of a circular cross sectional micro-rod subjected to a new kind of boundary constraints with elastic torsional springs is presented based on nonlocal elasticity. A nonlocal strengthening beam model is utilized and the effect of temperature changing is taken into consideration. The variational method and Hamilton’s principle are applied to derive the governing equation of motion and corresponding boundary conditions. A higher-order partial differential equation that is a typical characteristic of nonlocal strengthening model is developed, and the boundary conditions contain not only classical conditions but also non-classical higher-order conditions. Unlike previous studies which were only concerned with some conventional boundary constraints, we consider more general boundary conditions named elastic torsional spring supports. Such boundary conditions are between the simply supported and clamped ones, and they are closer to the actual constraints of existing engineering structures. Natural frequencies of micro-rods with new boundary constraints are determined via an eigenvalue method and compared with other results in the literature. It is shown that the nonlocal scale factor, thermal parameter, rigidity parameter and torsional spring coefficient play significant roles in free vibration of micro-rods. The research can provide a reference for a large class of boundary conditions ranging from simply supported to clamped micro-rods.