May  2003, 3(2): 285-298. doi: 10.3934/dcdsb.2003.3.285

The coupled PDE system of a composite (sandwich) beam revisited

1. 

Department of Mathematics, University of Virginia, P.O. Box 400137, Charlottesville, VA 22904

Received  April 2002 Revised  February 2003 Published  February 2003

In this paper we consider the coupled PDE system which describes a composite (sandwich) beam, as recently proposed in [H.1], [H-S.1]: it couples the transverse displacement $w$ and the effective rotation angle $\xi$ of the beam. We show that by introducing a suitable new variable $\theta$, the original model in the original variables $\{w,\xi\}$ of the sandwich beam is transformed into a canonical thermoelastic system in the new variables $\{w,\theta\}$, modulo lower-order terms. This reduction then allows us to re-obtain recently established results on the sandwich beam--which had been proved by a direct, ad hoc technical analysis [H-L.1]--simply as corollaries of previously established corresponding results [A-L.1], [A-L.2], [L-T.1]--[L-T.5] on thermoelastic systems. These include the following known results [H-L.1] for sandwich beams: (i) well-posedness in the semigroup sense; (ii) analyticity of the semigroup when rotational forces are not accounted for; (iii) structural decomposition of the semigroup when rotational forces are accounted for; and (iv) uniform stability.
In addition, however, through the aforementioned reduction to thermoelastic problems, we here establish new results for sandwich beams, when rotational forces are accounted for. They include: (i) a backward uniqueness property (Section 4), and (ii) a suitable singular estimate, critical in control theory (Section 5). Finally, we obtain a new backward uniqueness property, this time for a structural acoustic chamber having a composite (sandwich) beam as its flexible wall (Section 6).
Citation: Roberto Triggiani. The coupled PDE system of a composite (sandwich) beam revisited. Discrete and Continuous Dynamical Systems - B, 2003, 3 (2) : 285-298. doi: 10.3934/dcdsb.2003.3.285
[1]

Kokum R. De Silva, Tuoc V. Phan, Suzanne Lenhart. Advection control in parabolic PDE systems for competitive populations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1049-1072. doi: 10.3934/dcdsb.2017052

[2]

Martin Schechter. Monotonicity methods for infinite dimensional sandwich systems. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 455-468. doi: 10.3934/dcds.2010.28.455

[3]

David L. Russell. Modeling and control of hybrid beam systems with rotating tip component. Evolution Equations and Control Theory, 2014, 3 (2) : 305-329. doi: 10.3934/eect.2014.3.305

[4]

Roberto Triggiani. Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: the clamped case. Conference Publications, 2007, 2007 (Special) : 993-1004. doi: 10.3934/proc.2007.2007.993

[5]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations and Control Theory, 2022, 11 (1) : 199-224. doi: 10.3934/eect.2020108

[6]

Enrique Fernández-Cara, Diego A. Souza. On the control of some coupled systems of the Boussinesq kind with few controls. Mathematical Control and Related Fields, 2012, 2 (2) : 121-140. doi: 10.3934/mcrf.2012.2.121

[7]

Mohammad Akil, Ibtissam Issa, Ali Wehbe. Energy decay of some boundary coupled systems involving wave\ Euler-Bernoulli beam with one locally singular fractional Kelvin-Voigt damping. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021059

[8]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185

[9]

Reinhard Racke. Instability of coupled systems with delay. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1753-1773. doi: 10.3934/cpaa.2012.11.1753

[10]

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

[11]

Aaron A. Allen, Scott W. Hansen. Analyticity and optimal damping for a multilayer Mead-Markus sandwich beam. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1279-1292. doi: 10.3934/dcdsb.2010.14.1279

[12]

A. Özkan Özer, Scott W. Hansen. Uniform stabilization of a multilayer Rao-Nakra sandwich beam. Evolution Equations and Control Theory, 2013, 2 (4) : 695-710. doi: 10.3934/eect.2013.2.695

[13]

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

[14]

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

[15]

Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento, Karina Schiabel. Pullback attractors for a class of non-autonomous thermoelastic plate systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3553-3571. doi: 10.3934/dcdsb.2017214

[16]

Salim A. Messaoudi, Abdelfeteh Fareh. Exponential decay for linear damped porous thermoelastic systems with second sound. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 599-612. doi: 10.3934/dcdsb.2015.20.599

[17]

Pedro Roberto de Lima, Hugo D. Fernández Sare. General condition for exponential stability of thermoelastic Bresse systems with Cattaneo's law. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3575-3596. doi: 10.3934/cpaa.2020156

[18]

R. Demarque, J. Límaco, L. Viana. Local null controllability of coupled degenerate systems with nonlocal terms and one control force. Evolution Equations and Control Theory, 2020, 9 (3) : 605-634. doi: 10.3934/eect.2020026

[19]

Matthias Gerdts, Sven-Joachim Kimmerle. Numerical optimal control of a coupled ODE-PDE model of a truck with a fluid basin. Conference Publications, 2015, 2015 (special) : 515-524. doi: 10.3934/proc.2015.0515

[20]

Marc Puche, Timo Reis, Felix L. Schwenninger. Funnel control for boundary control systems. Evolution Equations and Control Theory, 2021, 10 (3) : 519-544. doi: 10.3934/eect.2020079

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (74)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]