# American Institute of Mathematical Sciences

• Previous Article
Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method
• EECT Home
• This Issue
• Next Article
Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations
December  2018, 7(4): 639-668. doi: 10.3934/eect.2018031

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

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

Received  July 2017 Revised  May 2018 Published  September 2018

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

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.

Citation: Ahmet Özkan Özer. Dynamic and electrostatic modeling for a piezoelectric smart composite and related stabilization results. Evolution Equations & Control Theory, 2018, 7 (4) : 639-668. doi: 10.3934/eect.2018031
##### References:

show all references

##### References:
,5,20]">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]
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
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 $U_1^3(x, z)=v^3(x)- (z-\hat z_3)w_x, ~~U_3(x, z)=w(x)$ $S_{11}=\frac{\partial v^3}{\partial x}+\frac{1}{2}(w_x)^2- (z-\hat z_3) \frac{\partial^2 w}{\partial x^2}, ~~S_{13}=0$ $T_{11}=\alpha^3 S_{11}-\gamma\beta D_3, ~~T_{13}=T_{12}=T_{23}=0$ $E_1=\beta_{1}D_1, ~~E_3=-\gamma\beta S_{11}+\beta D_3$
 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 $U_1^3(x, z)=v^3(x)- (z-\hat z_3)w_x, ~~U_3(x, z)=w(x)$ $S_{11}=\frac{\partial v^3}{\partial x}+\frac{1}{2}(w_x)^2- (z-\hat z_3) \frac{\partial^2 w}{\partial x^2}, ~~S_{13}=0$ $T_{11}=\alpha^3 S_{11}-\gamma\beta D_3, ~~T_{13}=T_{12}=T_{23}=0$ $E_1=\beta_{1}D_1, ~~E_3=-\gamma\beta S_{11}+\beta D_3$
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 Induced current A.S. E-static Mead-Marcus Angular velocity (bending) + shear velocity (middle layer) E. S. F. Dynamic 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.
 Assumption Model $B^*-$measurement for $V(t)$ at $x=L$ Stability E-static Rao-Nakra Stretching & compressing velocity E.S. F. Dynamic Induced current A.S. E-static Mead-Marcus Angular velocity (bending) + shear velocity (middle layer) E. S. F. Dynamic 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.
 [1] A. Özkan Özer, Scott W. Hansen. Uniform stabilization of a multilayer Rao-Nakra sandwich beam. Evolution Equations & Control Theory, 2013, 2 (4) : 695-710. doi: 10.3934/eect.2013.2.695 [2] Scott W. Hansen, Rajeev Rajaram. Riesz basis property and related results for a Rao-Nakra sandwich beam. Conference Publications, 2005, 2005 (Special) : 365-375. doi: 10.3934/proc.2005.2005.365 [3] Aliki D. Muradova, Georgios K. Tairidis, Georgios E. Stavroulakis. Adaptive Neuro-Fuzzy vibration control of a smart plate. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 251-271. doi: 10.3934/naco.2017017 [4] Scott W. Hansen, Oleg Yu Imanuvilov. Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions. Mathematical Control & Related Fields, 2011, 1 (2) : 189-230. doi: 10.3934/mcrf.2011.1.189 [5] Manoel J. Dos Santos, João C. P. Fortes, Marcos L. Cardoso. Exponential stability for a piezoelectric beam with a magnetic effect and past history. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021283 [6] Chuong Van Nguyen, Phuong Huu Hoang, Hyo-Sung Ahn. Distributed optimization algorithms for game of power generation in smart grid. Numerical Algebra, Control & Optimization, 2019, 9 (3) : 327-348. doi: 10.3934/naco.2019022 [7] Jian Mao, Qixiao Lin, Jingdong Bian. Application of learning algorithms in smart home IoT system security. Mathematical Foundations of Computing, 2018, 1 (1) : 63-76. doi: 10.3934/mfc.2018004 [8] Shunsuke Matsuzawa, Satoru Harada, Kazuya Monden, Yukihiro Takatani, Yutaka Takahashi. Effect of mobility of smart meters on performance of advanced metering infrastructure. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1495-1510. doi: 10.3934/jimo.2017004 [9] Chunqiang Hu, Jiguo Yu, Xiuzhen Cheng, Zhi Tian, Kemal Akkaya, and Limin Sun. CP_ABSC: An attribute-based signcryption scheme to secure multicast communications in smart grids. Mathematical Foundations of Computing, 2018, 1 (1) : 77-100. doi: 10.3934/mfc.2018005 [10] Yeming Dai, Yan Gao, Hongwei Gao, Hongbo Zhu, Lu Li. A real-time pricing scheme considering load uncertainty and price competition in smart grid market. Journal of Industrial & Management Optimization, 2020, 16 (2) : 777-793. doi: 10.3934/jimo.2018178 [11] Aaron A. Allen, Scott W. Hansen. Analyticity and optimal damping for a multilayer Mead-Markus sandwich beam. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1279-1292. doi: 10.3934/dcdsb.2010.14.1279 [12] Rajeev Rajaram, Scott W. Hansen. Null controllability of a damped Mead-Markus sandwich beam. Conference Publications, 2005, 2005 (Special) : 746-755. doi: 10.3934/proc.2005.2005.746 [13] Roberto Triggiani. The coupled PDE system of a composite (sandwich) beam revisited. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 285-298. doi: 10.3934/dcdsb.2003.3.285 [14] Mirelson M. Freitas, Anderson J. A. Ramos, Manoel J. Dos Santos, Jamille L.L. Almeida. Dynamics of piezoelectric beams with magnetic effects and delay term. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021015 [15] Maya Bassam, Denis Mercier, Ali Wehbe. Optimal energy decay rate of Rayleigh beam equation with only one boundary control force. Evolution Equations & Control Theory, 2015, 4 (1) : 21-38. doi: 10.3934/eect.2015.4.21 [16] Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721 [17] Jitka Machalová, Horymír Netuka. Optimal control of system governed by the Gao beam equation. Conference Publications, 2015, 2015 (special) : 783-792. doi: 10.3934/proc.2015.0783 [18] Marcio A. Jorge Silva, Vando Narciso, André Vicente. On a beam model related to flight structures with nonlocal energy damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3281-3298. doi: 10.3934/dcdsb.2018320 [19] Laetitia Paoli. Vibrations of a beam between stops: Collision events and energy balance properties. Evolution Equations & Control Theory, 2020, 9 (4) : 1133-1151. doi: 10.3934/eect.2020057 [20] Yue Sun, Zhijian Yang. Strong attractors and their robustness for an extensible beam model with energy damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021175

2020 Impact Factor: 1.081