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

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.2010 Mathematics Subject Classification. Primary: 35Q60, 35Q93, 93D15; Secondary: 74F15, 93C20. AHMETÖZKANÖZERThe models of piezoelectric smart composites in the literature all assume the electrostatic assumption for its piezoelectric layer. These models differ by the assumptions for the viscoelastic layer and the geometry of the composite, i.e. see [36]. The models are either a Mead-Marcus (M-M) type or a Rao-Nakra (R-N) type as obtained in [3,15], respectively. The M-M model only describes the bending motion, and the R-N model describes bending and longitudinal motions all together. These models reduce to the classical counterparts once the piezoelectric strain is taken to be zero [19,31]. The active boundary feedback stabilization of the classical R-N model (having no piezoelectric layer) is only investigated for hinged [28] and clamped-free [40] boundary conditions. The exact controllability of the M-M and R-N models are shown for the fully clamped, fully hinged, and clamped-hinged models [12,29]. The exponential stability in the existence of the passive distributed "shear" damping term is investigated for the R-N and M-M models ([1, 42]) .Asymptotic stabilization of piezoelectric smart composite models are investigated in [3,15] for various PID-type feedback controllers and a shear-type distributed damping. The exponential stability of the electrostatic M-M and R-N models for clamped-free boundary conditions has been open problems for more than a decade. Recently, exponential stability of the electrostatic R-N model is shown by using four type feedback controllers [24], two for the...

The Analysis and Design of a High Efficiency Piezoelectric Harvesting Floor with Impacting Force Mechanism

“…However, the FE model only analyzes the dynamic mechanical responses with a single dimension, and the ability of performance design optimization was hard to be achieved. The other way was proposed to formulate the governing equations of the Euler-Bernoulli beam [22,23]. According to the assumption of boundary condition, the deformed distribution with rectangular bimorph or unimorph cantilever can be estimated with tip mass under base excitation reported in Reference [24].…”