Dynamic and electrostatic modeling for a piezoelectric smart composite and related stabilization results

TL;DR

This paper develops and compares dynamic and electrostatic models for a piezoelectric smart composite beam, analyzing their stability properties and demonstrating that electrostatic models achieve exponential stability with fewer controllers.

Contribution

It introduces fully dynamic and electrostatic Rao-Nakra and Mead-Marcus models for piezoelectric beams, providing stability analysis and controller design insights.

Findings

Electrostatic models are exponentially stable with fewer controllers.

Fully dynamic models may lack asymptotic stability under certain parameters.

Electrostatic modeling simplifies stability control of piezoelectric beams.

Abstract

A cantilevered piezoelectric smart composite beam, consisting of perfectly bonded elastic, viscoelastic and piezoelectric layers, is considered. The piezoelectric layer is actuated by a voltage source. Both fully dynamic and electrostatic approaches, based on Maxwell's equations, are used to model the piezoelectric layer. We obtain (i) fully-dynamic and electrostatic Rao-Nakra type models by assuming that the viscoelastic layer has a negligible weight and stiffness, (ii) fully-dynamic and electrostatic Mead-Marcus type models by neglecting the in-plane and rotational inertia terms. Each model is a perturbation of the corresponding classical smart composite beam model. These models are written in the state-space form, the existence and uniqueness of solutions are obtained in appropriate Hilbert spaces. Next, the stabilization problem for each closed-loop system, with a thorough analysis, is investigated for the natural $B^*-$type state feedback controllers. The fully dynamic Rao-Nakra model with four state feedback controllers is shown to be not asymptotically stable for certain choices of material parameters whereas the electrostatic model is exponentially stable with only three state feedback controllers (by the spectral multipliers method). Similarly, the fully dynamic Mead-Marcus model lacks of asymptotic stability for certain solutions whereas the electrostatic model is exponentially stable by only one state feedback controller.stable by one feedback controller applied to the bending moment only.