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

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


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 & Continuous Dynamical Systems - B, 2003, 3 (2) : 285-298. doi: 10.3934/dcdsb.2003.3.285

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