Wasserstein distributionally robust co-design of a fractional-order dual-resonator elastic metamaterial for broadband vibration attenuation

Abstract

Locally resonant elastic metamaterials provide subwavelength attenuation but remain sensitive to uncertainty in stiffness, damping, and fractional order. This study develops a Wasserstein distributionally robust co-design framework that jointly selects the mass split, resonance tuning, loss scales, and fractional orders of a dual-resonator periodic cell. Bloch dispersion is condensed into a complex effective-mass model, and a feasibility-aware loss balances integrated attenuation, threshold coverage, internal-motion amplification, and resonance separation. Five bounded uncertainties in host stiffness, resonator stiffnesses, common loss scale, and fractional-order drift are represented in a finite 1-Wasserstein ambiguity set trained on 64 scrambled Sobol scenarios with 256 adversarial candidates. The proposed design is independently compared with equal-mass, nominal, and conditional-value-at-risk alternatives over 4,096 unseen scenarios. Across this validation set, the Wasserstein design reduced worst loss by 73.68% and increased fifth-percentile integrated attenuation by 32.50% relative to the equal-mass baseline while satisfying the internal-motion constraint. It allocated 56.03% of the resonator mass to the lower branch and selected tuning frequencies of 27.20 and 40.31 Hz. Exact antiresonance verification, uncertainty envelopes, finite-chain attenuation-equivalent insertion loss, and local sensitivity analysis support the numerical results. The framework provides a reproducible basis for robust broadband attenuation and subsequent specimen-level calibration and experimental validation.