The study of metamaterials is an emerging discipline that stands at the crossroad of acoustics and vibration engineering. The periodicity of such structures has proven to cause bandgaps in the relevant band-structures, hence waves with a particular frequency are blocked by the nature of the structure itself. The application of metamaterial concepts within the field of civil engineering can be seen as a novel application of passive isolation of structures. This is our stimulating and arduous challenge.
Fig. 1 – Metafoundation – tank system: (a) 3D vision, (b) layout of the unit cells and (c) cross section of the Metafoundation. Dimensions in m
The metafoundation depicted in Fig. 1 is the subject of a forthcoming publication  in the journal Scientific Report of Nature. The metafoundation is composed of massive vibrating concrete blocks that are coupled to the slab by means of fully nonlinear hysteretic devices, i.e., wire ropes depicted in Fig. 2.
Fig. 2 – (a) Configuration of a single unit cell equipped with steel wire ropes; (b) details of a single wire rope. Dimensions are in cm
We have demonstrated the favourable response of the metafoundation to natural seismic records. Moreover, we have been investigating the periodic configuration of a nonlinear metamaterial with the dynamical characteristics of the metafoundation, as depicted in Fig. 3.
Fig. 3 – Dynamical systems: (a) coupled finite lattice metafoundation-tank; (b) metafoundation modelled as a periodic system
The nonlinearity is caused by the presence of the wire ropes linking the resonators to the foundation’s slab. We successfully applied the stochastic linearization technique to perform the dispersion analysis. In fact, the linearized band structure is analyzed – see Fig. 4 – and it turns out that the nonlinearity leads the dispersion properties of the structure to be input-dependent. The input is a stationary Gaussian filtered white noise with zero mean and spectral intensity S0. The dispersion analysis reveals that any range of frequencies gives rise to complex wavenumbers, due to damping provided by the hysteretic devices.
Fig. 4 – Periodic Metafoundation dispersion curves: (a) imaginary component of μ; (b) real component of μ
Furthermore, we investigated nonlinear response of the periodic metafoundation by means of numerical Frequency Response Functions. The results confirm the reliability of the dispersion analysis subsequent to applying the equivalent linearization technique. A keen reader will find in the paper many considerations in this regard. To name one, a strong attenuation zone in wave transmittances is located in the frequency range where dispersion curves indicate very high values of the imaginary propagation constant μi of Fig. 4.
. Basone, F., Bursi, O. S., Aloschi, F. & Fischbach, G. Vibration mitigation of an MDoF system by means of hysteretic nonlinear resonant metamaterials. Scientific Reports (2021), in print.