# American Institute of Mathematical Sciences

November  2010, 14(4): 1279-1292. doi: 10.3934/dcdsb.2010.14.1279

## Analyticity and optimal damping for a multilayer Mead-Markus sandwich beam

 1 Area of Scientific Learning, Milligan College, Milligan College, TN 37682, United States 2 Department of Mathematics, Iowa State University, Ames, IA 50011, United States

Received  October 2009 Revised  March 2010 Published  August 2010

The classical Mead-Markus sandwich beam consists of two stiff outer layers modeled under Euler-Bernoulli beam assumptions and a compliant "core layer" that is elastic in shear. In this article we consider a multilayer analog consisting of $n = 2m + 1$ layers of alternating stiff and compliant beam layers ($m+1$ stiff and $m$ compliant) with viscous damping proportional to the shear in the compliant layers. We prove that the associated semigroup is analytic and describe the sector of analyticity. We also consider the problem of how to choose the damping parameters to optimize the angle of analyticity. We obtain an analytical solution to the optimization problem.
Citation: 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
 [1] Zayd Hajjej, Mohammad Al-Gharabli, Salim Messaoudi. Stability of a suspension bridge with a localized structural damping. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021089 [2] 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 [3] Qiong Zhang. Exponential stability of a joint-leg-beam system with memory damping. Mathematical Control & Related Fields, 2015, 5 (2) : 321-333. doi: 10.3934/mcrf.2015.5.321 [4] Maja Miletić, Dominik Stürzer, Anton Arnold. An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3029-3055. doi: 10.3934/dcdsb.2015.20.3029 [5] 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 [6] 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 [7] Irena Lasiecka, Roberto Triggiani. Heat--structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1515-1543. doi: 10.3934/cpaa.2016001 [8] Bochao Chen, Yixian Gao. Quasi-periodic travelling waves for beam equations with damping on 3-dimensional rectangular tori. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021075 [9] Yanan Li, Zhijian Yang, Fang Da. Robust attractors for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5975-6000. doi: 10.3934/dcds.2019261 [10] Yi Cheng, Zhihui Dong, Donal O' Regan. Exponential stability of axially moving Kirchhoff-beam systems with nonlinear boundary damping and disturbance. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021230 [11] Ryo Ikehata, Shingo Kitazaki. Optimal energy decay rates for some wave equations with double damping terms. Evolution Equations & Control Theory, 2019, 8 (4) : 825-846. doi: 10.3934/eect.2019040 [12] R.H. Fabiano, Scott W. Hansen. Modeling and analysis of a three-layer damped sandwich beam. Conference Publications, 2001, 2001 (Special) : 143-155. doi: 10.3934/proc.2001.2001.143 [13] 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 [14] 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 [15] 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 [16] Tuan Anh Dao, Michael Reissig. $L^1$ estimates for oscillating integrals and their applications to semi-linear models with $\sigma$-evolution like structural damping. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5431-5463. doi: 10.3934/dcds.2019222 [17] Jean-Paul Chehab, Georges Sadaka. On damping rates of dissipative KdV equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1487-1506. doi: 10.3934/dcdss.2013.6.1487 [18] Miroslav Grmela, Michal Pavelka. Landau damping in the multiscale Vlasov theory. Kinetic & Related Models, 2018, 11 (3) : 521-545. doi: 10.3934/krm.2018023 [19] M'hamed Kesri. Structural stability of optimal control problems. Communications on Pure & Applied Analysis, 2005, 4 (4) : 743-756. doi: 10.3934/cpaa.2005.4.743 [20] János Karsai, John R. Graef. Attractivity properties of oscillator equations with superlinear damping. Conference Publications, 2005, 2005 (Special) : 497-504. doi: 10.3934/proc.2005.2005.497

2020 Impact Factor: 1.327