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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:
[1]

A. A. Allen and S. W. Hansen, Analyticity and optimal damping for a multilayer MeadMarkus sandwich beam, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1279-1292. doi: 10.3934/dcdsb.2010.14.1279.

[2]

H. T. Banks, K. Ito and C. Wang, Exponentially stable approximations of weakly damped wave equations, Estimation and Control of Distributed Parameter Systems (Vorau, 1990), 1–33, Internat. Ser. Numer. Math., 100, Birkhauser, Basel, 1991. doi: 10.1007/978-3-0348-6418-3_1.

[3]

A. Baz, Boundary control of beams using active constrained layer damping, J. Vib. Acoust., 119 (1997), 166-172.

[4]

Y. Cao and X. B. Chen, A Survey of Modeling and Control Issues for Piezo-electric Actuators, Journal of Dynamic Systems, Measurement, and Control, 137 (2014), 014001, 13pp. doi: 10.1115/1.4028055.

[5]

C. Y. K. CheeL. Tong and G. P. Steven, A review on the modelling of piezoelectric sensors and actuators incorporated in intelligent structures, J. Intell. Mater. Syst. Struct., 9 (1998), 3-19. doi: 10.1177/1045389X9800900101.

[6]

G. ChenM. C. DelfourA. M. Krall and G. Payre, Modeling, stabilization and control of serially connected beams, SIAM J. Contr. Opt., 25 (1987), 526-546. doi: 10.1137/0325029.

[7]

S. B. Choi and Y. M. Han, Chapter II: Piezoelectric Actuators: Control Applications of Smart Materials, CRC Press, 2010.

[8]

S. DevasiaE. Eleftheriou and S. O. R. Moheimani, A survey of control issues in nanopositioning, IEEE Transactions on Control Systems Technology, 15 (2007), 802-823. doi: 10.1109/TCST.2007.903345.

[9]

A. Erturk and D. J. Inman, An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations, Smart Materials and Structures, 18 (2008), 025009. doi: 10.1088/0964-1726/18/2/025009.

[10]

R. H. Fabiano and S. W. Hansen, Modeling and analysis of a three–layer damped sandwich beam, Dynamical systems and differential equations, Discrete Contin. Dynam. Systems, 2001 (2001), 143-155.

[11]

S. W. Hansen, Several related models for multilayer sandwich plates, Mathematical Models & Methods in Applied Sciences, 14 (2004), 1103-1132. doi: 10.1142/S0218202504003568.

[12]

S. W. Hansen and A. Ö. Özer, Exact boundary controllability of an abstract Mead-Marcus Sandwich beam model, The Proceedings of 49th IEEE Conf. on Decision & Control, Atlanta, USA (2010), 2578–2583 doi: 10.1109/CDC.2010.5717319.

[13]

M. A. Horn, Uniform decay rates for the solutions to the Euler-Bernoulli plate equation with boundary feedback acting via bending moments, Differ. Integral Equ., 5 (1992), 1121-1150.

[14]

J. E. Lagnese and J.-L. Lions, Modeling Analysis and Control of Thin Plates, Masson, Paris, 1988.

[15]

M. J. LamD. Inman and W. R. Saunders, Vibration control through passive constrained layer damping and active control, Journal of Intelligent Material Systems and Structures, 8 (1997), 663-677. doi: 10.1177/1045389X9700800804.

[16]

I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli equations with boundary dissipation occurring in the moments only, J. Differential Equations, 95 (1992), 169-182. doi: 10.1016/0022-0396(92)90048-R.

[17]

K. Liu and Z. Liu, Boundary stabilization of a non- homogenous beam by the frequency domain multiplier method, Computation and Applied Mathematics, 21 (2002), 299-313.

[18]

M. H. Malakooti and H. A. Sodano, Piezoelectric energy harvesting through shear mode operation, Smart Materials and Structures, 24 (2015), 055005. doi: 10.1088/0964-1726/24/5/055005.

[19]

D. J. Mead and S. Markus, The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions, J. Sound Vibr., 10 (1969), 163-175. doi: 10.1016/0022-460X(69)90193-X.

[20]

S. Miller and J. Jr. Hubbard, Observability of a Bernoulli - Euler Beam using PVF2 as a Distributive Sensor, The Seventh Conference on Dynamics & Control of Large Structures, VPI & SU, Blacksburg, VA, (1987), 375–390.

[21]

K. A. Morris and A. Ö. Özer, Modeling and stabilizability of voltage-actuated piezoelectric beams with magnetic effects, SIAM J. Control Optim., 52 (2014), 2371-2398. doi: 10.1137/130918319.

[22]

A. Ö. Özer, Further stabilization and exact observability results for voltage-actuated piezoelectric beams with magnetic effects, Mathematics of Control, Signals, and Systems, 27 (2015), 219-244. doi: 10.1007/s00498-015-0139-0.

[23]

A. Ö. Özer, Semigroup well-posedness of a voltage controlled active constrained layered (ACL) beam with magnetic effects, The Proceedings of the American Control Conference, Boston, MA, (2016), 4580-4585. doi: 10.1109/ACC.2016.7526074.

[24]

A. Ö. Özer, Modeling and controlling an active constrained layered (ACL) beam actuated by two voltage sources with/without magnetic effects, IEEE Transactions of Automatic Control, 62 (2017), 6445-6450. doi: 10.1109/TAC.2017.2653361.

[25]

A. Ö. Özer, Potential formulation for charge or current-controlled piezoelectric smart composites and stabilization results: electrostatic vs. quasi-static vs. fully-dynamic approaches IEEE Transactions of Automatic Control, in press, (2018), p1. doi: 10.1109/TAC.2018.2836864.

[26]

A. Ö. Özer, Exponential stabilization of the smart piezoelectric composite beam with only one boundary controller, The Proceedings of the International Federation of Automatic Control (IFAC) Conference on Lagrangian and Hamiltonian Methods for Nonlinear Control, Valparaiso, Chile, 51-3 (2018), 80–85. doi: 10.1016/j.ifacol.2018.06.019.

[27]

A. Ö. Özer, Nonlinear modeling and preliminary stabilization results for a class of piezoelectric smart composite beams, SPIE Proceedings on Smart Structures & Nondestructive Evaluation, 10595, Active and Passive Smart Structures and Integrated Systems XII, Denver CO, (2018), 105952C. doi: 10.1117/12.2296878.

[28]

A. Ö. Özer and S. W. Hansen, Uniform stabilization of a multi-layer Rao-Nakra sandwich beam, Evolution Equations and Control Theory, 2 (2013), 695-710. doi: 10.3934/eect.2013.2.695.

[29]

A. Ö. Özer and S. W. Hansen, Exact boundary controllability results for a multilayer Rao-Nakra sandwich beam, SIAM J. Cont. Optim., 52 (2014), 1314-1337. doi: 10.1137/120892994.

[30]

B. Rao, A compact perturbation method for the boundary stabilization of the Ragleigh beam equation, Appl. Math. Optim., 33 (1996), 253-264. doi: 10.1007/BF01204704.

[31]

Y. V. K. S. Rao and B. C. Nakra, Vibrations of unsymmetrical sandwich beams and plates with viscoelastic cores, J. Sound Vibr., 34 (1974), 309-326.

[32]

M. A. Shubov, Spectral analysis of a non-selfadjoint operator generated by an energy harvesting model and application to an exact controllability problem, Asymptotic Analysis., 102 (2017), 119-156. doi: 10.3233/ASY-171413.

[33]

R. C. Smith, Smart Material Systems, Society for Industrial and Applied Mathematics, 2005. doi: 10.1137/1.9780898717471.

[34]

H. F. Tiersten, Linear Piezoelectric Plate Vibrations, New York: Plenum Press, 1969.

[35]

R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation, Proc. Amer. Math. Soc., 105 (1989), 375-383. doi: 10.1090/S0002-9939-1989-0953013-0.

[36]

M. Trindade and A. Benjendou, Hybrid Active-Passive Damping Treatments Using Viscoelastic and Piezoelectric Materials:Review and Assessment, Journal of Vibration and Control, 8 (2002), 699-745.

[37]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Basel: Birkäuser Verlag, 2009. doi: 10.1007/978-3-7643-8994-9.

[38]

T. Voss and J. M. A. Scherpen, Port-hamiltonian modeling of a nonlinear timoshenko beam with piezo actuation, SIAM J. Control Optim., 52 (2014), 493-519. doi: 10.1137/090774598.

[39]

T. Voss and J. M. A. Scherpen, Stabilization and shape control of a 1-D piezoelectric Timoshenko beam, Automatica, 47 (2011), 2780-2785. doi: 10.1016/j.automatica.2011.09.026.

[40]

J. M. WangB. Z. Guo and B. Chentouf, Boundary feedback stabilization of a three-layer sandwich beam: Riesz basis approach, ESAIM Control Optim. Calc. Var., 12 (2006), 12-34. doi: 10.1051/cocv:2005030.

[41]

J. M. Wang and B. Z. Guo, Analyticity and dynamic behavior of a damped three-layer sandwich beam, J. Optim. Theory Appl., 137 (2008), 675-689. doi: 10.1007/s10957-007-9341-7.

[42]

C. Yang and J. M. Wang, Exponential stability of an active constrained layer beam actuated by a voltage source without magnetic effects, Journal of Mathematical Analysis and Applications, 448 (2017), 1204-1227. doi: 10.1016/j.jmaa.2016.11.067.

[43]

Y. L. Zhang and J. M. Wang, Moment approach to the boundary exact controllability of an active constrained layer beam, Journal of Mathematical Analysis and Applications, 465 (2018), 643-657. doi: 10.1016/j.jmaa.2018.05.032.

show all references

References:
[1]

A. A. Allen and S. W. Hansen, Analyticity and optimal damping for a multilayer MeadMarkus sandwich beam, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1279-1292. doi: 10.3934/dcdsb.2010.14.1279.

[2]

H. T. Banks, K. Ito and C. Wang, Exponentially stable approximations of weakly damped wave equations, Estimation and Control of Distributed Parameter Systems (Vorau, 1990), 1–33, Internat. Ser. Numer. Math., 100, Birkhauser, Basel, 1991. doi: 10.1007/978-3-0348-6418-3_1.

[3]

A. Baz, Boundary control of beams using active constrained layer damping, J. Vib. Acoust., 119 (1997), 166-172.

[4]

Y. Cao and X. B. Chen, A Survey of Modeling and Control Issues for Piezo-electric Actuators, Journal of Dynamic Systems, Measurement, and Control, 137 (2014), 014001, 13pp. doi: 10.1115/1.4028055.

[5]

C. Y. K. CheeL. Tong and G. P. Steven, A review on the modelling of piezoelectric sensors and actuators incorporated in intelligent structures, J. Intell. Mater. Syst. Struct., 9 (1998), 3-19. doi: 10.1177/1045389X9800900101.

[6]

G. ChenM. C. DelfourA. M. Krall and G. Payre, Modeling, stabilization and control of serially connected beams, SIAM J. Contr. Opt., 25 (1987), 526-546. doi: 10.1137/0325029.

[7]

S. B. Choi and Y. M. Han, Chapter II: Piezoelectric Actuators: Control Applications of Smart Materials, CRC Press, 2010.

[8]

S. DevasiaE. Eleftheriou and S. O. R. Moheimani, A survey of control issues in nanopositioning, IEEE Transactions on Control Systems Technology, 15 (2007), 802-823. doi: 10.1109/TCST.2007.903345.

[9]

A. Erturk and D. J. Inman, An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations, Smart Materials and Structures, 18 (2008), 025009. doi: 10.1088/0964-1726/18/2/025009.

[10]

R. H. Fabiano and S. W. Hansen, Modeling and analysis of a three–layer damped sandwich beam, Dynamical systems and differential equations, Discrete Contin. Dynam. Systems, 2001 (2001), 143-155.

[11]

S. W. Hansen, Several related models for multilayer sandwich plates, Mathematical Models & Methods in Applied Sciences, 14 (2004), 1103-1132. doi: 10.1142/S0218202504003568.

[12]

S. W. Hansen and A. Ö. Özer, Exact boundary controllability of an abstract Mead-Marcus Sandwich beam model, The Proceedings of 49th IEEE Conf. on Decision & Control, Atlanta, USA (2010), 2578–2583 doi: 10.1109/CDC.2010.5717319.

[13]

M. A. Horn, Uniform decay rates for the solutions to the Euler-Bernoulli plate equation with boundary feedback acting via bending moments, Differ. Integral Equ., 5 (1992), 1121-1150.

[14]

J. E. Lagnese and J.-L. Lions, Modeling Analysis and Control of Thin Plates, Masson, Paris, 1988.

[15]

M. J. LamD. Inman and W. R. Saunders, Vibration control through passive constrained layer damping and active control, Journal of Intelligent Material Systems and Structures, 8 (1997), 663-677. doi: 10.1177/1045389X9700800804.

[16]

I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli equations with boundary dissipation occurring in the moments only, J. Differential Equations, 95 (1992), 169-182. doi: 10.1016/0022-0396(92)90048-R.

[17]

K. Liu and Z. Liu, Boundary stabilization of a non- homogenous beam by the frequency domain multiplier method, Computation and Applied Mathematics, 21 (2002), 299-313.

[18]

M. H. Malakooti and H. A. Sodano, Piezoelectric energy harvesting through shear mode operation, Smart Materials and Structures, 24 (2015), 055005. doi: 10.1088/0964-1726/24/5/055005.

[19]

D. J. Mead and S. Markus, The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions, J. Sound Vibr., 10 (1969), 163-175. doi: 10.1016/0022-460X(69)90193-X.

[20]

S. Miller and J. Jr. Hubbard, Observability of a Bernoulli - Euler Beam using PVF2 as a Distributive Sensor, The Seventh Conference on Dynamics & Control of Large Structures, VPI & SU, Blacksburg, VA, (1987), 375–390.

[21]

K. A. Morris and A. Ö. Özer, Modeling and stabilizability of voltage-actuated piezoelectric beams with magnetic effects, SIAM J. Control Optim., 52 (2014), 2371-2398. doi: 10.1137/130918319.

[22]

A. Ö. Özer, Further stabilization and exact observability results for voltage-actuated piezoelectric beams with magnetic effects, Mathematics of Control, Signals, and Systems, 27 (2015), 219-244. doi: 10.1007/s00498-015-0139-0.

[23]

A. Ö. Özer, Semigroup well-posedness of a voltage controlled active constrained layered (ACL) beam with magnetic effects, The Proceedings of the American Control Conference, Boston, MA, (2016), 4580-4585. doi: 10.1109/ACC.2016.7526074.

[24]

A. Ö. Özer, Modeling and controlling an active constrained layered (ACL) beam actuated by two voltage sources with/without magnetic effects, IEEE Transactions of Automatic Control, 62 (2017), 6445-6450. doi: 10.1109/TAC.2017.2653361.

[25]

A. Ö. Özer, Potential formulation for charge or current-controlled piezoelectric smart composites and stabilization results: electrostatic vs. quasi-static vs. fully-dynamic approaches IEEE Transactions of Automatic Control, in press, (2018), p1. doi: 10.1109/TAC.2018.2836864.

[26]

A. Ö. Özer, Exponential stabilization of the smart piezoelectric composite beam with only one boundary controller, The Proceedings of the International Federation of Automatic Control (IFAC) Conference on Lagrangian and Hamiltonian Methods for Nonlinear Control, Valparaiso, Chile, 51-3 (2018), 80–85. doi: 10.1016/j.ifacol.2018.06.019.

[27]

A. Ö. Özer, Nonlinear modeling and preliminary stabilization results for a class of piezoelectric smart composite beams, SPIE Proceedings on Smart Structures & Nondestructive Evaluation, 10595, Active and Passive Smart Structures and Integrated Systems XII, Denver CO, (2018), 105952C. doi: 10.1117/12.2296878.

[28]

A. Ö. Özer and S. W. Hansen, Uniform stabilization of a multi-layer Rao-Nakra sandwich beam, Evolution Equations and Control Theory, 2 (2013), 695-710. doi: 10.3934/eect.2013.2.695.

[29]

A. Ö. Özer and S. W. Hansen, Exact boundary controllability results for a multilayer Rao-Nakra sandwich beam, SIAM J. Cont. Optim., 52 (2014), 1314-1337. doi: 10.1137/120892994.

[30]

B. Rao, A compact perturbation method for the boundary stabilization of the Ragleigh beam equation, Appl. Math. Optim., 33 (1996), 253-264. doi: 10.1007/BF01204704.

[31]

Y. V. K. S. Rao and B. C. Nakra, Vibrations of unsymmetrical sandwich beams and plates with viscoelastic cores, J. Sound Vibr., 34 (1974), 309-326.

[32]

M. A. Shubov, Spectral analysis of a non-selfadjoint operator generated by an energy harvesting model and application to an exact controllability problem, Asymptotic Analysis., 102 (2017), 119-156. doi: 10.3233/ASY-171413.

[33]

R. C. Smith, Smart Material Systems, Society for Industrial and Applied Mathematics, 2005. doi: 10.1137/1.9780898717471.

[34]

H. F. Tiersten, Linear Piezoelectric Plate Vibrations, New York: Plenum Press, 1969.

[35]

R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation, Proc. Amer. Math. Soc., 105 (1989), 375-383. doi: 10.1090/S0002-9939-1989-0953013-0.

[36]

M. Trindade and A. Benjendou, Hybrid Active-Passive Damping Treatments Using Viscoelastic and Piezoelectric Materials:Review and Assessment, Journal of Vibration and Control, 8 (2002), 699-745.

[37]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Basel: Birkäuser Verlag, 2009. doi: 10.1007/978-3-7643-8994-9.

[38]

T. Voss and J. M. A. Scherpen, Port-hamiltonian modeling of a nonlinear timoshenko beam with piezo actuation, SIAM J. Control Optim., 52 (2014), 493-519. doi: 10.1137/090774598.

[39]

T. Voss and J. M. A. Scherpen, Stabilization and shape control of a 1-D piezoelectric Timoshenko beam, Automatica, 47 (2011), 2780-2785. doi: 10.1016/j.automatica.2011.09.026.

[40]

J. M. WangB. Z. Guo and B. Chentouf, Boundary feedback stabilization of a three-layer sandwich beam: Riesz basis approach, ESAIM Control Optim. Calc. Var., 12 (2006), 12-34. doi: 10.1051/cocv:2005030.

[41]

J. M. Wang and B. Z. Guo, Analyticity and dynamic behavior of a damped three-layer sandwich beam, J. Optim. Theory Appl., 137 (2008), 675-689. doi: 10.1007/s10957-007-9341-7.

[42]

C. Yang and J. M. Wang, Exponential stability of an active constrained layer beam actuated by a voltage source without magnetic effects, Journal of Mathematical Analysis and Applications, 448 (2017), 1204-1227. doi: 10.1016/j.jmaa.2016.11.067.

[43]

Y. L. Zhang and J. M. Wang, Moment approach to the boundary exact controllability of an active constrained layer beam, Journal of Mathematical Analysis and Applications, 465 (2018), 643-657. doi: 10.1016/j.jmaa.2018.05.032.

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]
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
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 $i, j = 1, 2, 3.$
LayersDisplacements, 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$
$E_1=\beta_{1}D_1, ~~E_3=-\gamma\beta S_{11}+\beta D_3$
LayersDisplacements, 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$
$E_1=\beta_{1}D_1, ~~E_3=-\gamma\beta S_{11}+\beta D_3$
Table 2.  Stability results for the closed-loop system with the $B^*-$feedback controller corresponding to the control $V(t)$ of the piezoelectric layer
AssumptionModel $B^*-$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.
AssumptionModel $B^*-$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]

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