Vibrational properties of a continuous self-similar structure

Abstract

We present an extensive investigation of the normal modes of vibration of a prototype hierarchical continuous system, consisting of a Cantor-like sequence of piezoelectric and resin elements. From a detailed analysis of the density of states, displacement profiles, and eigenvalue-spacing distributions, evidence is found for the existence of multiple fracton and phonon regimes. The role of resonant modes and the effect of disorder, which may be of primary importance in real systems, are also discussed in detail.