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We investigate a family of quasiperiodic continuous elastic beams, thetopological properties of their vibrational spectra, and their relation to theexistence of localized modes. We specifically consider beams featuring arraysof ground springs at locations determined by projecting from a circle onto anunderlying periodic system. A family of periodic and quasiperiodic structuresis obtained by smoothly varying a parameter defining such projection. Numericalsimulations show the existence of vibration modes that first localize at aboundary, and then migrate into the bulk as the projection parameter is varied.Explicit expressions predicting the change in the density of states of the bulkdefine topological invariants that quantify the number of modes spanning a gapof a finite structure. We further demonstrate how modulating the phase of theground springs distribution causes the topological states to undergo anedge-to-edge transition. The considered configurations and topological studiesprovide a framework for inducing localized modes in continuous elasticstructural components through globally spanning, deterministic perturbations ofperiodic patterns defined by the considered projection operations.

Topological bands and localized vibration modes in quasiperiodic beams