SUMMARY

In the study of periodic structures, the architecture of a unit cell is designed to control the properties of the global system. Considering dynamic applications, the unit cell design controls the flow of wave energy, affecting its speed and direction. Research has been conducted on reconfigurable magneto-elastic lattices, which can change their internal and global geometry. Reconfigurations afford the lattices the ability to have their properties modified. This has been demonstrated for wave propagation properties and for equivalent continuum properties. Dynamic reconfiguration has also been investigated as a means of changing a structure’s geometry. The next step in the extension of this research in periodic structures is the investigation of topologically protected boundary modes (TPBMs). Though ``topology'' sometimes refers to connectivity in mechanical systems, in this context the word is much more broad and refers to a measure that remains constant with large continuous changes in a system, such as the Chern number. As long as the topology of the structure remains the same, a system may undergo changes to its geometry and other properties that will not harm the existence of modes that are associated with that topology. Thus, those modes are topologically protected. Perhaps the most notable property exhibited by such systems is the ability of some to support wave modes that propagate only in one direction on a boundary. In such cases the topology is not affected by geometric defects in the boundary, which allows energy transmission without back-scattering and is of great interest to the scientific and engineering community.
The understanding of TPBMs in quantum-mechanical systems is currently ahead of TPBMs in mechanical systems, so ongoing efforts will utilize analogies to quantum-mechanical eigenvalue problems where possible. This is expected to lead to mathematical analyses that will simply describe characteristics that the mass and stiffness matrices of a system must possess to exhibit TPBMs. Then example structures can be designed, and previous work regarding structural reconfiguration can be employed. Successful completion of the proposed research is expected to produce structures that will be able to switch between “perfect” wave transmission and no wave transmission.