Role of Symmetric Periodic Perforations in the Dynamics of Axially Graded Nanobeams

This study presents a mathematical framework for investigating the free vibration behavior of axially graded perforated nanobeams resting on elastic foundations. The governing differential equation is formulated based on the Euler-Bernoulli beam theory within the framework of nonlocal elasticity to account for small-scale effects that become significant at the nanoscale. Particular attention is given to the influence of symmetry arising from periodic perforation patterns and axial material gradation on the dynamic response of the nanobeam. Modified equivalent models are employed to represent the effects of regularly distributed perforations while preserving the geometric symmetry of the structure. The resulting governing equations are solved using the Galerkin method to determine the natural frequencies and corresponding mode shapes. The accuracy and reliability of the proposed formulation are validated through comparisons with available results reported in the literature. Numerical investigations reveal that the vibrational characteristics are strongly affected by the filling ratio, perforation geometry, material gradation, and elastic foundation parameters. Furthermore, the interaction between nonlocal effects and structural symmetry plays a significant role in governing the dynamic behavior under various boundary conditions. The results demonstrate that symmetric periodic perforations can substantially alter the stiffness and mass distribution of the nanobeam, thereby influencing its vibration characteristics. The findings provide useful insights for the design and development of advanced nanostructures with periodic perforated configurations.