Design of tunable hierarchical waveguides based on Fibonacci-like microstructure

A novel class of Fibonacci-based high performance tunable hierarchical waveguides is here proposed and special focus is devoted to the design of adaptive-passive control systems for the Bloch wave propagation. By exploiting the features of periodic approximants, the metamaterial is conceived by the proper repetition of an elementary cell along a fixed direction. This elementary cell is made up of two building blocks repeated according to the Fibonacci sequence, to form a quasi-periodic finite microstructured system The former building block is made of a homogeneous elastic material, while the latter is a microstructured two-phase laminate. One of these phases is piezoelectric shunted by a suitably conceived electrical circuit, such that the constitutive properties of the piezoelectric phase are tuned by adjusting its equivalent impedance/admittance. Moreover, the overall constitutive properties of the microstructured layer are determined, exploiting the scale separation, via an asymptotic homogenization scheme. Then, it is possible to determine the frequency band structure of the tunable waveguide by exploiting the transfer matrix approach. With the aim of providing broad design directions, attention is paid to characterizing the dispersion waves properties of the tunable Fibonacci like superlattices described by several generations of the Fibonacci sequence and for different values of the orientation angle of the two-phase laminate.