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<p>This study presents a comprehensive mathematical model for the free vibration analysis of axially functionally graded tapered nanobeams with perforations. The formulation incorporates both axial material gradation and geometric tapering, while explicitly accounting for the presence of uniformly spaced square holes. The bending stiffness, mass per unit length, and rotary inertia are modified to reflect these effects. The nonlocal elasticity theory is adopted to account for size-dependent behavior at the nanoscale. The governing equations are solved using the Galerkin method with orthonormal polynomials to obtain the frequency parameters and associated mode shapes. Convergence study and validation against established solutions in specific cases are presented to demonstrate the accuracy and reliability of the proposed model. A detailed parametric investigation is carried out to assess the influence of key factors, including the number of holes, filling ratio, axial power-law index, nonlocal parameter, taper ratio, and slenderness ratio, under various boundary conditions. Results indicate that perforation and material gradation significantly affect the vibrational characteristics, with notable sensitivity to the nonlocal scale and geometric configuration. The findings offer valuable insights for the optimal design of advanced nanostructured components.</p>

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Coupled effects of axial material gradation and tapered geometry on nanobeam dynamics with uniform square perforations

  • 期刊论文

  • 发表日期：

    2026年1月6日

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  • Akash Kumar Gartia,
  • Ramanath Garai,
  • S. Chakraverty

摘要

This study presents a comprehensive mathematical model for the free vibration analysis of axially functionally graded tapered nanobeams with perforations. The formulation incorporates both axial material gradation and geometric tapering, while explicitly accounting for the presence of uniformly spaced square holes. The bending stiffness, mass per unit length, and rotary inertia are modified to reflect these effects. The nonlocal elasticity theory is adopted to account for size-dependent behavior at the nanoscale. The governing equations are solved using the Galerkin method with orthonormal polynomials to obtain the frequency parameters and associated mode shapes. Convergence study and validation against established solutions in specific cases are presented to demonstrate the accuracy and reliability of the proposed model. A detailed parametric investigation is carried out to assess the influence of key factors, including the number of holes, filling ratio, axial power-law index, nonlocal parameter, taper ratio, and slenderness ratio, under various boundary conditions. Results indicate that perforation and material gradation significantly affect the vibrational characteristics, with notable sensitivity to the nonlocal scale and geometric configuration. The findings offer valuable insights for the optimal design of advanced nanostructured components.