Elastic Foundation Induced Wide Bandgaps for Actively-tuned Topologically Protected Wave Propagation in Phononic Crystal Beams

This work presents the active control effects on a piezoelectric phononic crystal beam resting on elastic foundation. The bending of passive periodic beams that rest on elastic foundation can exhibit distinguished applications for geotechnical and railroad structures. Generally, elastic waves propagate in these structures in a certain frequency range and these structures are able to block the other input waves outside this range. These two frequency ranges are usually called the pass band and the bandgap of structures. With periodic attachment of piezoelectric patches having negative capacities, the structural band properties can be easily tuned to a wider range. However, the beam system with negative capacities is always restricted by the presence of electrical boundary conditions in the structural stability response. It leads to tuning inconvenience and tuning disabilities of the structures. In view of these tuning shortcomings, this study proposes to capitalize the effects of elastic foundation or elastic supports to improve the active control of piezoelectric beams for bandgap generation in wave propagation. By artificially positioning periodic negative capacities and piezoelectric patches, the system sub-structure is designed as an A-B-A phononic crystal beam. Further, the plane wave expansion method is applied to the governing equation to derive analytical solutions and to compare with numerical solutions. The topologically protected phase transition is observed by analyzing mode shapes of the specific cases and the Zak phase across the Brillouin zone. The topologically protected interface modes are also captured in this system and the topologically protected manner provides a new paradigm for manipulating wave propagations in the bending of beam system resting on the elastic foundation. Finally, the efficiency of electrical parameters and foundation stiffness on system tuning is examined. The controllable system is further justified to allow operation in a wider frequency range.