Low-frequency spillover phenomenon in NC-piezoelectric shunt damping

This paper discusses a spillover phenomenon on the lower frequency side in the multimode piezoelectric NC (Negative Capacitor)-shunt damping problem. In contrast to standard spillover phenomena, this phenomenon is quite interesting because the instability is caused by neglecting lower frequency vibration modes in controller design. First, an example of spillover phenomenon on the lower frequency side is illustrated. When the NC-shunt damping problem is interpreted as the H∞-control problem, the combination of mechanical and electrical systems is recognized as a stable generalized plant and the admittance of the NC circuit is recognized as an unstable controller. The circuit parameters of NC-circuits are tuned for each vibration mode, and then, it is demonstrated that the closed loop system becomes unstable when the first vibration mode is neglected in parameter tuning. Then, the graphical instability analyses are carried out by investigating the Bode and Nyquist diagrams. According to the Nyquist stability criterion, the closed loop system between the stable generalized plant and the unstable controller becomes stable if and only if the Nyquist diagram encircles −1+ j0 once in the counter-clockwise direction on the complex plane. In the example of unstable closed loop system, it is confirmed that the Nyquist diagram encircles −1+ j0 once in the clockwise direction and that both gain and phase intersections occur at the anti-resonance frequency in the Bode diagram. Finally, the approximate instability analysis is carried out to prove that the instability is caused by simultaneous gain and phase intersections at the anti-resonance frequency indeed. From the instability analysis, it is clarified that spillover phenomenon on the lower frequency side is not a special phenomenon limited to a specific experimental setup but can generally occur in the piezoelectric NC-shunt damping problem.