A Novel Sensor Design for a Cantilevered Mead-Marcus-type Sandwich Beam Model by the Order-reduction Technique

TL;DR

This paper develops a novel finite difference-based model reduction for a multi-layer sandwich beam PDE, enabling effective sensor placement for vibration control despite complex coupling and spectral analysis challenges.

Contribution

It extends a space-discretized model reduction technique to complex multi-layer PDEs, proving observability and enabling sensor design for advanced beam structures.

Findings

PDE model is exactly observable with sub-optimal observation time.

Order-reduction technique effectively simplifies complex coupled PDEs.

Sensor design enables control of beam dynamics despite coupling complexities.

Abstract

A novel space-discretized Finite Differences-based model reduction, introduced in (Liu,Guo,2020) 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 can not 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 $h\to 0$ 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.