abstract
- This work investigates the complex impact dynamics of low-density polyethylene (LDPE) foams through a multifaceted mathematical approach that couples fractal and fractional operators. We propose a hybrid fractal¿fractional Bagley¿Torvik oscillator to describe the dynamic response of polymeric cushioning materials, where a fractal derivative models the multiscale inertial behavior and a Caputo-type fractional derivative of order ¿ ¿ (1, 2) captures viscoelastic energy dissipation. This formulation is rigorously compared with a generalized conformable derivative framework, in which a sensitivity analysis of the conformable order ¿ reveals a characteristic material response that remains consistent across varying impact energies. A key innovation of this study is the mapping of experimental acceleration profiles onto a Gaussian distribution. The results show strong linear correlations with the initial drop height (H0) and the characteristic frequency ¿, yielding predictive scaling laws. The fractal¿fractional Bagley¿Torvik oscillator achieves superior agreement with experimental data, particularly in reproducing transient damping and peak acceleration values.