Bandgap formation and dual rainbow trapping in graded ladder-shunted piezoelectric metamaterial beams

A theoretical framework is developed for graded multi-resonant piezoelectric metastructures with two independently tunable electromechanical resonances per unit cell, realized by a second-order ladder shunt—the electromechanical analog of a two-degree-of-freedom resonator—which opens two locally resonant bandgaps. Using Euler–Bernoulli beam theory and an assumed-modes formulation, closed-form expressions for the bandgap edge frequencies are derived in the infinite-electrode limit and confirmed by plane-wave expansion and finite-element simulations. An exact bandwidth-conservation relation shows that the total bandgap width is fixed by the primary resonance and the electromechanical coupling alone, while the secondary resonance and capacitance ratio only redistribute it between the two gaps. Spatially grading both resonances substantially widens the attenuation band: at a grading parameter ε = 0.07 the two-resonance metastructure reaches an attenuation bandwidth of 263.6 Hz versus 180.1 Hz for its single-resonance counterpart. The graded system also exhibits dual rainbow trapping—two simultaneous wave-localization branches in the frequency–position plane—so that each region traps two distinct frequencies. This enhanced localization increases the distributed harvesting potential by up to 9.9% under idealized cell-wise electrical loading at ε = 0.05 , and by about 2% under a fixed passive load.