Nonlinear Vibrations of Filled Re-Entrant Hexagonal Units: Coupled Geometric–Inertial Effects

This work solves the problem associated with the lack of analytical models capable of describing the nonlinear vibration behavior of re-entrant hexagonal units when geometric nonlinearity and structural modifications, such as soft filling, are taken into account. The purpose of this study is to develop an analytical framework that enables prediction and control of vibration characteristics, with particular emphasis on achieving low-frequency response and enhanced energy storage and redistribution within the structure. The proposed approach is based on Lagrangian modeling of the unit cell, leading to a nonlinear equation of motion of the Liénard type that admits a first integral. By exploiting the existence of this integral, an approximate analytical expression for the oscillation period is derived using energy-based methods. The analysis is performed for two configurations: an empty unit and a unit filled with a soft material, allowing direct comparison of their dynamic responses. The analytical results are validated through comparison with numerical simulations and available experimental data. A parametric study is conducted to evaluate the influence of the mass ratio and the re-entrant angle on the oscillation period and frequency. Furthermore, the effects of filling mass, stiffness, and degree of filling are systematically investigated, revealing distinct inertia-dominated and stiffness-dominated regimes. The obtained results provide clear design guidelines for tailoring the dynamic response of re-entrant hexagonal units. It is shown that low-frequency vibration and increased capacity for energy storage can be achieved through appropriate selection of geometric parameters and filling properties, with potential applications in vibration control and structural design.