On the use of effective mass in modeling undamped mass-in-mass metamaterials

The theoretical foundations of the effective mass concept in undamped mass-in-mass metamaterials are revisited to reassess its validity and physical interpretation. Within a dynamical systems framework, analytical derivations reveal that the effective mass is not an intrinsic property but a dynamic quantity linking the motion of the internal mass to the observable response of the primary mass under specific initial conditions. An infinite family of periodic orbits is identified using Poincaré maps, each characterized by an identical initial potential energy, elucidating the mathematical source of ambiguity in defining a unique effective mass. The analysis is extended to a discrete lattice of internal resonators, where a single periodic solution satisfies the periodicity constraints, confirming that the effective mass concept holds only under specific dynamic conditions.