Wave propagation in quasiperiodic structures: stop/pass band distribution and prestress effects

The band structure of dispersion diagrams for axial and flexural waves of quasiperiodic infinite beams is investigated. Every structure is composed of a repeated elementary cell generated adopting the Fibonacci sequence (different terms of this sequence are investigated); the problem is then solved evaluating the transmission matrix of the cell and applying the Floquet–Bloch technique. In the case of axial vibrations, a homogeneous rod where the quasiperiodicity is given by the particular distribution of external springs is considered, while for the flexural problem the quasiperiodicity is defined in terms of relative distance between the simple supports that sustain the beam. In both cases it is shown that, for different Fibonacci sequences, the number of stop/pass bands within a defined range of frequencies changes and follows the Fibonacci recursion rule showing a self-similar pattern. In addition, the overall dispersion characteristic can be interpreted in terms of an invariant function of the circular frequency, independent of the sequence generating the elementary cell. For flexural waves, the effects of the axial prestress on dispersion diagrams is highlighted and the frequency-shift of the stop/pass band positions is quantified. It is noted that a tensile axial prestress promotes length reduction of pass bands while leaves almost unchanged the length of stop-band intervals.