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

Ahmet Özkan Özer

doi:10.3934/eect.2018031

Article Contents

2018, Volume 7, Issue 4: 639-668. Doi: 10.3934/eect.2018031

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

Ahmet Özkan Özer

Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA

Received: June 30, 2017

Revised: April 30, 2018

Published: September 2018

The author is supported by the Western Kentucky University startup research grant.

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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.

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Mathematics Subject Classification: Primary: 35Q60, 35Q93, 93D15; Secondary: 74F15, 93C20.

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Figure 1. A voltage-actuated piezoelectric smart composite of length $L$ with thicknesses $h_1, h_2, h_3$ for its layers ①, ②, ③, respectively. The longitudinal motions of top and bottom layers ① and ③ are controlled by $g^1(t), g^3(t), V(t), $ and the bending motions (of the whole composite) are controlled by $M(t), g(t).$ For the fully-dynamic models, written in the state-space formulation $\dot\varphi = \mathcal{A} \varphi + B u(t)$, the $B^*-$type observation for the piezoelectric layer naturally corresponds to the total induced current at its electrodes. It is more physical in terms of practical applications. As well, measuring the total induced current at the electrodes of the piezoelectric layer is easier than measuring displacements or the velocity of the composite at one end of the beam, i.e. see [3,5,20]

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Figure 2. The equations of motion describing the overall "small" vibrations on the composite beam are dictated by the variables $v^1(x, t), v^3(x, t), w(x, t), \phi^2(x, t)$ which correspond to the longitudinal vibrations of Layers ① and ③, bending of the composite ①-②-③, and shear of Layer ②

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Table 1. Linear constitutive relationships for each layer. $U_i^j, T_{ij}, $ $ S_{ij}, $ $D_i, $ and $E_i$ denote displacements, the stress tensor, strain tensor, electrical displacement, and electric field for $i, j = 1, 2, 3.$

Layers	Displacements, Stresses, Strains, Electric fields, and Electric displacements
Layer ① - Elastic	$U_1^1(x, z)=v^1(x)- (z-\hat z_1)w_x, ~~U_3(x, z)=w(x)$
	$S_{11}=\frac{\partial v^1}{\partial x}+\frac{1}{2}(w_x)^2- (z-\hat z_1) \frac{\partial^2 w}{\partial x^2}, ~~S_{13}=0$
	$ T_{11}=\alpha_1S_{11}, ~~T_{13}=T_{12}=T_{23}=0$
Layer ② - Viscoelastic	$U_1^2(x, z)=v^2(x)+ (z-\hat z_2)\psi^2(x), ~~U_3(x, z)=w(x)$
	$ S_{11}=\frac{\partial v^2}{\partial x}- (z-\hat z_i) \frac{\partial \psi^2}{\partial x}, ~~S_{13}=\frac{1}{2}\phi^2$
	$ T_{11}=\alpha_1^2S_{11}, ~~T_{13}= 2G_{2} S_{13}, $ $T_{12}=T_{23}=0$
Layer ③ - Piezoelectric	$U_1^3(x, z)=v^3(x)- (z-\hat z_3)w_x, ~~U_3(x, z)=w(x)$
	$ S_{11}=\frac{\partial v^3}{\partial x}+\frac{1}{2}(w_x)^2- (z-\hat z_3) \frac{\partial^2 w}{\partial x^2}, ~~S_{13}=0$
	$ T_{11}=\alpha^3 S_{11}-\gamma\beta D_3, ~~T_{13}=T_{12}=T_{23}=0$
	$E_1=\beta_{1}D_1, ~~E_3=-\gamma\beta S_{11}+\beta D_3$

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Table 2. Stability results for the closed-loop system with the $B^*-$feedback controller corresponding to the control $V(t)$ of the piezoelectric layer

Assumption	Model	$B^*-$measurement for $V(t)$ at $x=L$	Stability
E-static	Rao-Nakra	Stretching & compressing velocity	E.S.
F. Dynamic	Rao-Nakra	Induced current	A.S.
E-static	Mead-Marcus	Angular velocity (bending) + shear velocity (middle layer)	E. S.
F. Dynamic	Mead-Marcus	Induced current	Not A.S.
Different electro-magnetic assumptions (due to Maxwell's equations) for the smart R-N and M-M models. Here E.S.=Exponentially Stability for all modes, A.S.= Asymptotically Stability for inertial sliding solutions, Not A.S.=Not Asymptotically Stability for inertial sliding solutions.

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