In this study, we propose a 'rainbow' metamaterial to achieve broadband multifrequency vibration attenuation. The rainbow metamaterial is constituted of a Π-shaped beam partitioned into substructures by parallel plates insertions with two attached cantilever-mass acting as local resonators. Both resonators inside each substructure can be non-symmetric such that the metamaterial can have multi-frequency bandgaps. Furthermore, these cantilever-mass resonators have a progressively variant design along the beam, namely rainbow-shaped, for the purpose of achieving broader energy stop bands. Π-shaped beams partitioned by parallel plate insertions can be extended to honeycomb sandwich composites, hence the proposed rainbow metamaterial can serve as a precursor for future honeycomb composites with superior vibration attenuation for more industrial applications. A mathematical model is first developed to estimate the frequency response functions of the metamaterial. Interaction forces between resonators and the backbone structure are calculated by solving the displacement of the cantilever-mass resonators. The plate insertions are modeled as attached masses with both their translational and rotational motion considered. Subsequently, the mathematical model is verified by comparison with experimental results. Metamaterials fabricated through an additive manufacturing technique are tested with a laser doppler receptance measuring system. After the validation of the mathematical model, a numerical study is conducted to explore the influences of the resonator spatial

In this study, we present an optimization scheme for the resonator distribution in rainbow metamaterials that are constitutive of a Π-shaped beam with parallel plate insertions and two sets of spatially varying cantilever-mass resonators. To improve the vibration attenuation of the rainbow metamaterials at frequencies of interest, two optimization strategies are proposed, aiming at minimizing the maximum and average receptance values respectively. Objective functions for both single and multiple frequency ranges optimization are set up with the frequency response functions predicted by an analytical model. The masses of the two sets of resonators clamped at different side walls of the Π-shaped beams constitute the set of design variables. Optimization functions are solved out with the employment of the Genetic Algorithm method. Dedicate case studies are subsequently conducted to show the feasibility of the proposed scheme. The receptance values are found greatly reduced within the single and multiple optimization frequency ranges. Moreover, it is found that, the maximum  Corresponding author.

Additive manufacturing has become a fundamental tool to fabricate and experimentally investigate mechanical metamaterials and phononic crystals. However, this manufacturing process produces spatially correlated variability that breaks the translational periodicity, which might compromise the wave propagation performance of metamaterials. We demonstrate that the vibration attenuation profile is strictly related to the spatial profile of the variability, and that there exists an optimal disorder degree below which the attenuation bandwidth widens; for high disorder levels, the band gap mistuning annihilates the overall attenuation. The variability also induces a spatially variant locally resonant band gap that progressively slow down the group velocity until an almost zero value, giving rise to wave trapping effect near the lower band gap boundary. Inspired by this wave trapping phenomenon, a rainbow metamaterial with linear spatial-frequency trapping is also proposed, which have potential applications in energy harvesting, spatial wave filtering and non-destructive evaluation at low frequency. This report provides a deeper understanding of the differences between numerical simulations using nominal designed properties and experimental analysis of metamaterials constructed in 3D printing. These analysis and results may extend to phononic crystals and other periodic systems to investigate their wave and dynamic performance as well as robustness under variability.

No abstract

Additive manufacturing has been used to propose several designs of phononic crystals and metamaterials due to the low cost to produce complex geometrical features. However, like any other manufacturing process, it can introduce material and geometrical variability in the nominal design and therefore affect the structural dynamic performance. Locally resonant metamaterials are typically designed such that the distributed resonators have the same natural frequency or, in the case of rainbow metastructures, a well-defined spatial profile. In this work, the effects of the break of periodicity caused by additive manufacturing variability on the attenuation performance of a multi-frequency metastructure is investigated. First, an experimental investigation on the manufacturing tolerances of test samples produced from a Selective Laser Sintering process are assessed and variability levels are used to propose a random field model for the metastructure. Subsequently, the stochastic model is used to investigate the vibration suppression performance of broadband multi-frequency metastructures. An analytical model based on a transfer matrix approach is used to calculate transfer receptance due to a point time harmonic force in a finite length metastructure, which is composed of evenly spaced nonsymmetric resonators attached to a beam with Π-shaped cross-section. This design creates a multifrequency metastructure, i.e. band gaps in more than one frequency band. Individual samples of the random fields are used to show that the mistuned resonators can change the vibration attenuation performance of the metastructure and that even small levels of variability, given by less than 1% for the masse and less than 3% for the Young's modulus can have a significant effect on the overall vibration attenuation performance of the metastructure when considered together. It is also shown that different spatial profiles can have a significant effect on the vibration attenuation performance in both band gaps. Therefore, the modelling of the uncertainty metastructures has to take into account the spatial correlation of the properties of the metastructure resonators. The obtained results are expected to be useful for further robust design in mass produced industrial applications.