Active Feedback-Driven Defect-Band Steering in Phononic Crystals with Piezoelectric Defects: A Mathematical Approach

Defective phononic crystals (PnCs) have garnered significant attention for their ability to localize and amplify elastic wave energy within defect sites or to perform narrowband filtering at defect-band frequencies. The necessity for continuously tunable defect characteristics is driven by the variable excitation frequencies encountered in rotating machinery. Conventional tuning methodologies, including synthetic negative capacitors or inductors integrated with piezoelectric defects, are constrained to fixed, offline, and incremental adjustments. To address these limitations, the present study proposes an active feedback approach that facilitates online, wide-range steering of defect bands in a one-dimensional PnC. Each defect is equipped with a pair of piezoelectric sensors and actuators, governed by three independently tunable feedback gains: displacement, velocity, and acceleration. Real-time sensor signals are transmitted to a multivariable proportional controller, which dynamically modulates local electroelastic stiffness via the actuators. This results in continuous defect-band frequency shifts across the entire band gap, along with on-demand sensitivity modulation. The analytical model that incorporates these feedback gains has been demonstrated to achieve a level of agreement with COMSOL benchmarks that exceeds 99%, while concurrently reducing computation time from hours to seconds. Displacement- and acceleration-controlled gains yield predictable, monotonic up- or down-shifts in defect-band frequency, whereas the velocity-controlled gain permits sensitivity adjustment without frequency drifts. Furthermore, the combined-gain operation enables the concurrent tuning of both the center frequency and the filtering sensitivity, thereby facilitating an instantaneous remote reconfiguration of bandpass filters. This framework establishes a new class of agile, adaptive ultrasonic devices with applications in ultrasonic imaging, structural health monitoring, and prognostics and health management.