# Topological bands and localized vibration modes in quasiperiodic beams

## TL;DR

This paper explores the topological properties of vibrational spectra in quasiperiodic elastic beams, revealing localized modes and their topological invariants, with potential applications in designing elastic structures with controlled vibration localization.

## Contribution

It introduces a novel framework linking quasiperiodic structures, topological invariants, and localized vibrational modes in elastic beams, expanding understanding of topological mechanics in continuous systems.

## Key findings

- Localized vibration modes can be induced and controlled via topological invariants.
- Phase modulation of ground springs causes edge-to-edge topological state transitions.
- Explicit formulas relate density of states changes to topological invariants.

## Abstract

We investigate a family of quasiperiodic continuous elastic beams, the topological properties of their vibrational spectra, and their relation to the existence of localized modes. We specifically consider beams featuring arrays of ground springs at locations determined by projecting from a circle onto an underlying periodic system. A family of periodic and quasiperiodic structures is obtained by smoothly varying a parameter defining such projection. Numerical simulations show the existence of vibration modes that first localize at a boundary, and then migrate into the bulk as the projection parameter is varied. Explicit expressions predicting the change in the density of states of the bulk define topological invariants that quantify the number of modes spanning a gap of a finite structure. We further demonstrate how modulating the phase of the ground springs distribution causes the topological states to undergo an edge-to-edge transition. The considered configurations and topological studies provide a framework for inducing localized modes in continuous elastic structural components through globally spanning, deterministic perturbations of periodic patterns defined by the considered projection operations.