On a localization phenomenon in two types of bio-inspired hierarchically organized oscillatory systems

Abstract

This study presents two different groups of models of bio-inspired hierarchically organized oscillatory mechanical models exhibiting a localization phenomenon, when only certain parts of the system oscillate. The first group of models corresponds to an idealized sympodial tree-type branched structure consisting of branches represented by physical pendula attached mutually via hinges and linear rotational springs. The localization phenomenon is examined analytically and numerically in the structures with a different level of hierarchy, i.e. with a different level of branches, and the corresponding localized modes and their respective natural frequencies are found. The cases of small and large-amplitude vibration are considered. The second group of models has the form of a chain of hierarchically organized block masses attached mutually via tension–extension springs. The case of an arbitrary number of masses attached via linear springs is considered to show which localized modes exist at certain frequencies and the theorem related to the number of localized models is formulated and proven. It is pointed out that localized modes appear also in a system with nonlinear (pure cubic) springs. The results obtained can further be used in theoretical dynamical studies of these systems with different types of excitations, as well as for appropriate biomimetic applications, especially related to their utilization as vibration absorbers.