New Stability Results for Piezoelectric Beams with a Dynamic Tip Load: Partial Damping and the Interplay of Lower- and Higher-Order Effects

Abstract This study extends previous work (Özer in 63rd IEEE conference on decision and control, pp 8498–8503, 2024) on the stabilization of piezoelectric beam systems with a dynamic tip mass, emphasizing collocated partial damping designs. Unlike earlier approaches based on noncollocated controllers and Lyapunov methods, this work investigates the performance of collocated controllers under reduced damping configurations. A distinction is made between higher order damping, associated with the strain rate, and lower order damping, associated with the tip velocity and the accumulated electrical current at the electrodes. It is shown that higher order damping provides stronger stabilization compared to tip velocity feedback. Conditions for exponential and polynomial decay rates are identified in relation to the damping configuration and material properties. A rigorous analytical framework is introduced to characterize these decay behaviors without relying on Lyapunov techniques or spectral analysis. Numerical simulations support the theoretical findings, highlighting the essential role of the higher-order feedback mechanism in achieving exponential stability. The results offer practical insights for applications in energy harvesting, acoustic wave control, and high-precision sensing.