Static and Dynamic Properties of Branched Self-Similar Structures: The Trichotomous Lattice

Abstract

The static and dynamic properties of self-similar branched structures (lattices) in which each subsequent cell replicates the structure of the previous one on some scale are investigated. The trichotomous lattices in which each element in each subsequent line divides into three self-similar ones are analyzed. The analytical conditions of static and dynamic self-similarity are found. The conditions of the same static deformation of each element are obtained. It is proved that the lattice is a band-pass filter, and its pass band depends on the similarity factor. It is shown that in the lattice there are multiple natural frequencies equal to the partial frequency of the generating element.