Effect of negative poisson's ratio on vibrations of truncated conical meta-shells: A GDQ-based study with potential applications in sports engineering

This study investigates the free vibration behavior of truncated conical sandwich meta-shells composed of graphene platelet (GPL)-reinforced nanocomposite facesheets and a re-entrant auxetic core. The structural formulation is developed within the framework of first-order shear deformation theory (FSDT), and the governing equations are derived using Hamilton's principle and solved efficiently using the generalized differential quadrature (GDQ) method. The accuracy of the formulation is assessed by comparison with published benchmark results, while numerical convergence is systematically examined to ensure grid-independent predictions. A comprehensive parametric study is then conducted to quantify the effects of GPL weight fraction and distribution pattern, auxetic-cell geometry, shell geometry, and boundary conditions on the natural frequencies. The results demonstrate that increasing the GPL content generally enhances structural stiffness and increases the natural frequencies, with the AV distribution providing the strongest stiffening effect among the considered patterns. The auxetic architecture also exhibits a pronounced influence on the dynamic response: increasing the cell-wall thickness ratio β 1 increases the frequency, whereas increasing the cell-height ratio β 2 produces a softening effect. In addition, the re-entrant angle exhibits a nonlinear influence, indicating the existence of favorable auxetic configurations for maximizing vibration stiffness. The results further reveal strong interactions among material gradation, auxetic geometry, and edge constraints. These findings provide a mechanics-based framework for tailoring lightweight conical meta-shells with controllable vibration characteristics and suggest potential applications in sports-engineering components requiring low mass, vibration resistance, impact tolerance, and enhanced energy-absorption capability.