Robust optimised design of 3D printed elastic metastructures: A trade-off between complexity and vibration attenuation

In this work, a strategy for optimal design of mechanical metastructure is proposed taken into account uncertainties arising from additive manufacturing. A locally resonant Pi-shaped beam with parallel plate-like insertions and two cantilever mass resonators at each unit cell is manufactured through a selective laser sintering process. The variability of the material properties introduced by the additive manufacturing procedure is experimentally obtained. Given that such manufacturing approaches are predominantly employed for producing complex metastructure architectures, it can significantly compromise the optimality of the design. A transfer matrix approach is employed to propagate variability at a structural level and predict the structural receptance due to a point harmonic force in the finite length metastructure. Then, the mass ratio of the metastructure is optimised for maximising vibration attenuation considering different numbers of added resonators and relative masses. A cost function is introduced in the classical robust design approach in order to favour designs with least complexity, represented by the number of added resonators. It is exhibited in several cases that the robustly optimal design is away from the deterministic optimal one, emphasising the relevance of the proposed approach in the optimisation of complex and locally resonant structures. Moreover, it is shown that the frequency range of interest plays a major role on the derived optimal design for each number of implemented resonators. The presented results show that even small variability in the Young's modulus of up to 3% and in the mass density of up to 1% can still affect the robustness of the optimal design for locally resonant metastructure as due to the consequent mistuning of the added resonators.