Volume 29, 2023
|Number of page(s)||35|
|Published online||27 September 2023|
A novel sensor design for a cantilevered Mead-Marcus-type sandwich beam model by the order-reduction technique
Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA
2 Department of Mathematics, University of Maryland, Baltimore County, Baltimore, MD 21250, USA
* Corresponding author: firstname.lastname@example.org
Accepted: 14 August 2023
red A novel space-discretized Finite Differences-based model reduction introduced in [J. Liu and B.Z. Guo, SIAM J. Control Optim. 58 (2020) 2256-228] is extended to the partial differential equations (PDE) model of a multi- layer Mead-Marcus-type sandwich beam with clamped-free boundary conditions. The PDE model describes transverse vibrations for a sandwich beam whose alternating outer elastic layers constrain viscoelastic core layers, which allow transverse shear. The major goal of this project is to design a single tip velocity sensor to control the overall dynamics on the beam. Since the spectrum of the PDE cannot be constructed analytically, the so-called multipliers approach is adopted to prove that the PDE model is exactly observable with sub-optimal observation time. Next, the PDE model is reduced by the “order-reduced” Finite-Differences technique. This method does not require any type of filtering though the exact observability as h → 0 is achieved by a constraint on the material constants. The main challenge here is the strong coupling of the shear dynamics of the middle layer with overall bending dynamics. This complicates the absorption of coupling terms in the discrete energy estimates. This is sharply different from a single-layer (Euler-Bernoulli) beam.
Mathematics Subject Classification: 35B40 / 35B41 / 37B55 / 37L30
Key words: Mead-Marcus sandwich beam / model reductions / multilayer beam / observability and sensor design / finite differences / order reduction
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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