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September & December  2018, 8(3&4): 789-808. doi: 10.3934/mcrf.2018035

## Weak stability of a laminated beam

 1 College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China 2 Department of Mathematics and Statistics, University of Minnesota, Duluth, MN 55812, USA 3 College of Information Science and Technology, Donghua University, School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, China

* Corresponding authorr: Yang Wan

Received  October 2017 Revised  May 2018 Published  September 2018

In this paper, we consider the stability of a laminated beam equation, derived by Liu, Trogdon, and Yong [6], subject to viscous or Kelvin-Voigt damping. The model is a coupled system of two wave equations and one Euler-Bernoulli beam equation, which describes the longitudinal motion of the top and bottom layers of the beam and the transverse motion of the beam. We first show that the system is unstable if one damping is only imposed on the beam equation. On the other hand, it is easy to see that the system is exponentially stable if direct damping are imposed on all three equations. Hence, we investigate the system stability when two of the three equations are directly damped. There are a total of seven cases from the combination of damping locations and types. Polynomial stability of different orders and their optimality are proved. Several interesting properties are revealed.

Citation: Yanfang Li, Zhuangyi Liu, Yang Wang. Weak stability of a laminated beam. Mathematical Control and Related Fields, 2018, 8 (3&4) : 789-808. doi: 10.3934/mcrf.2018035
##### References:
 [1] A. A. Allen and S. W. Hansen, Analyticity of a multilayer Mead-Markus plate, Nonliear Anal., 71 (2009), e1835-e1842.  doi: 10.1016/j.na.2009.02.063. [2] A. A. Allen and S. W. Hansen, Analyticity and optimal damping for a multilayer Mead-Markus sandwich beam, Discrete Contin. Dyn. Syst. Ser. B., 14 (2010), 1279-1292.  doi: 10.3934/dcdsb.2010.14.1279. [3] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0. [4] S. W. Hansen and Z. Liu, Analyticity of semigroup associated with a laminated composite beam, Control of Distributed Parameter and Stochastic Systems (Hangzhou, 1998), Kluwer Acad. Publ., Boston, MA, 1999, 47–54. [5] S. W. Hansen and R. Spies, Structural damping in a laminated beam due to interfacial slip, J. Sound and Vibration, 204 (1997), 183-202.  doi: 10.1006/jsvi.1996.0913. [6] Z. Liu, S. A. Trogdon and J. Yong, Modeling and analysis of a laminated beam, Math. Comput. Modeling, 30 (1999), 149-167.  doi: 10.1016/S0895-7177(99)00122-3. [7] Z. Liu and S. Zheng, Semigroup Associated with Dissipative System, Res. Notes Math., Vol 394, Chapman & Hall/CRC, Boca Raton, 1999. [8] D. J. Mead and S. Markus, The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions, J. Sound Vibr.(2), 10 (1969), 163-175.  doi: 10.1016/0022-460X(69)90193-X. [9] A. Özkan Özer and S. W. Hansen, Uniform stabilization of a multilayer Rao-Nakra sandwich beam, Evol. Equ. Control Theorey, 2 (2013), 695-710.  doi: 10.3934/eect.2013.2.695. [10] Y. V. K. S Rao and B. C. Nakra, Vibrations of unsymmetrical sanwich beams and plates with viscoelastic cores, J. Sound Vibr.(3), 34 (1974), 309-326. [11] C. A. Raposo, Exponential stability of a structure with interfacial slip and frictional damping, Applied Math. Letter, 53 (2016), 85-91.  doi: 10.1016/j.aml.2015.10.005. [12] J. M. Wang, G. Q. Xu and S. P. Yung, Stabilization of laminated beams with structural damping by boundary feedback controls, SIAM Control Optim., 44 (2005), 1575-1597.  doi: 10.1137/040610003. [13] M. J. Yan and E. H. Dowell, Governing equations for vibratory constrained-layer damping sandwich plates and beams, J. Appl. Mech.(4), 39 (1972), 1041-1046.

show all references

##### References:
 [1] A. A. Allen and S. W. Hansen, Analyticity of a multilayer Mead-Markus plate, Nonliear Anal., 71 (2009), e1835-e1842.  doi: 10.1016/j.na.2009.02.063. [2] A. A. Allen and S. W. Hansen, Analyticity and optimal damping for a multilayer Mead-Markus sandwich beam, Discrete Contin. Dyn. Syst. Ser. B., 14 (2010), 1279-1292.  doi: 10.3934/dcdsb.2010.14.1279. [3] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0. [4] S. W. Hansen and Z. Liu, Analyticity of semigroup associated with a laminated composite beam, Control of Distributed Parameter and Stochastic Systems (Hangzhou, 1998), Kluwer Acad. Publ., Boston, MA, 1999, 47–54. [5] S. W. Hansen and R. Spies, Structural damping in a laminated beam due to interfacial slip, J. Sound and Vibration, 204 (1997), 183-202.  doi: 10.1006/jsvi.1996.0913. [6] Z. Liu, S. A. Trogdon and J. Yong, Modeling and analysis of a laminated beam, Math. Comput. Modeling, 30 (1999), 149-167.  doi: 10.1016/S0895-7177(99)00122-3. [7] Z. Liu and S. Zheng, Semigroup Associated with Dissipative System, Res. Notes Math., Vol 394, Chapman & Hall/CRC, Boca Raton, 1999. [8] D. J. Mead and S. Markus, The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions, J. Sound Vibr.(2), 10 (1969), 163-175.  doi: 10.1016/0022-460X(69)90193-X. [9] A. Özkan Özer and S. W. Hansen, Uniform stabilization of a multilayer Rao-Nakra sandwich beam, Evol. Equ. Control Theorey, 2 (2013), 695-710.  doi: 10.3934/eect.2013.2.695. [10] Y. V. K. S Rao and B. C. Nakra, Vibrations of unsymmetrical sanwich beams and plates with viscoelastic cores, J. Sound Vibr.(3), 34 (1974), 309-326. [11] C. A. Raposo, Exponential stability of a structure with interfacial slip and frictional damping, Applied Math. Letter, 53 (2016), 85-91.  doi: 10.1016/j.aml.2015.10.005. [12] J. M. Wang, G. Q. Xu and S. P. Yung, Stabilization of laminated beams with structural damping by boundary feedback controls, SIAM Control Optim., 44 (2005), 1575-1597.  doi: 10.1137/040610003. [13] M. J. Yan and E. H. Dowell, Governing equations for vibratory constrained-layer damping sandwich plates and beams, J. Appl. Mech.(4), 39 (1972), 1041-1046.
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