Natural dynamics of biaxially graded nonlocal stress-driven beam

Abstract

This study investigates the dynamic behaviour of Rayleigh nanobeams, considering the effects of nonlocal elasticity and bi-directional material gradation. An analytical model is developed using the Laplace transformation method to solve the governing ordinary differential equations. Both Helmholtz exponential and bi-exponential kernels are employed to derive the corresponding governing equations, and their impacts on the nanobeam's dynamic response are compared. The natural frequencies are computed under various boundary conditions, values of the nonlocal parameter, material inhomogeneity constants, and slenderness ratios. The findings indicate that nonlocal elasticity increases the beam’s stiffness, leading to higher frequencies, whereas material gradation causes a softening effect, reducing the frequencies. Additionally, the beam geometry and boundary conditions significantly influence the frequencies, with thicker beams displaying lower frequencies due to increased flexibility and clamped beams showing higher frequencies due to stronger constraints. The results are validated through comparisons with existing literature, demonstrating excellent agreement with published data. These insights are crucial for the design and analysis of nano-electromechanical systems (NEMS), nano-robotics, and other nanostructures, where controlling vibrational behaviour is essential for performance optimization. The proposed model offers a robust tool for future research in designing nanostructures with optimized dynamic properties.