Dynamic transverse vibration characteristics of nonuniform nonlocal strain gradient beams using the generalized differential quadrature method

Abstract

In this study, vibration response of small-scale tapered beams is investigated. Size-dependent effects are modeled in the framework of the nonlocal strain gradient theory and three different types of linearly cross-section variation are proposed by having width variation, thickness variation and a combination of them both. The small-scale beam is formulated using Euler-Bernoulli beam theory, nonlocal strain gradient theory and Hamilton’s principle. Equations of motion for all three types of nonuniformity are solved using the generalized differential quadrature method (GDQM). Results are presented and compared to those achieved for simplified models and effects of having different slenderness ratio are presented. Moreover, a comprehensive parametric study is proposed and the effects of varying nonuniformity terms, nonlocal and strain gradient parameters are precisely studied. Accordingly, with the vast application of tapered small-scale beams in many devices, especially scanning tunneling microscopes (STM) and atomic force microscopes (AFM), this study could be a step forward in understanding, predicting and controlling such kind of behaviors.