Theoretical Analysis of Free Vibration of Microbeams under Different Boundary Conditions Using Stress-Driven Nonlocal Integral Model

Abstract

Previous studies have shown that Eringen’s differential nonlocal model leads to some inconsistencies for both the Euler–Bernoulli and Timoshenko beams subjected to different boundary conditions. In this paper, the free vibration analysis of the Euler–Bernoulli and Timoshenko beams is performed theoretically using the stress-driven nonlocal integral model. The Fredholm-type integral constitutive equations of the first kind are transformed to Volterra integral equations of the first kind by simply adjusting the limit of integrals. Also, the general solutions to the deflection, bending moment and so on are derived by solving the integro-differential governing equations by the Laplace transformation, of which the unknown constants are determined by the boundary conditions and extra constraint equations related to the constitutive relationship. Then the characteristic equations for free vibration of the Euler–Bernoulli and Timoshenko beams are derived, from which the vibration frequency for different boundary conditions can be determined. The effects of nonlocal parameter and vibration order on the natural frequency of the Euler–Bernoulli and Timoshenko beams are investigated numerically. The results from the present model are validated against those existing in the literature, and demonstrated to be theoretically consistent.