Publication Date

7-2022

Advisor(s) - Committee Chair

Ahmet Ozer (chair), Richard Schugart, Mikhail Khenner, Mark Robinson

Degree Program

Department of Mathematics

Degree Type

Master of Science

Abstract

Novel space-discretized Finite Differences-based model reductions are proposed for the partial differential equations (PDE) model of a multi-layer Mead-Marcus-type beam with (i) hinged-hinged and (ii) 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 boundary sensor, placed at the tip of the beam, to control the overall dynamics on the beam.

For (i), it is first shown that the PDE model is exactly observable by the so-called nonharmonic Fourier series approach. However, a "blind" model reduction by Finite Differences is not able to retain the exact observability uniformly as the discretization parameter h⇢0. This is mainly due to the spurious high-frequency eigenvalues preventing the sensor from being able to distinguish one vibrational frequency from another as h ⇢0. For a robust sensor design, the low-frequency components of the solutions are controlled in order to eliminate the high-frequency components of the solutions, the so-called direct Fourier filtering.

For (ii), the methodology of (i) does not work since the spectrum of the PDE can not be constructed analytically. Therefore, the so-called multipliers approach is first 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 can only be obtained a condition on the parameters.

To the best of our knowledge, this is the first work dedicated to develop robust model reductions and sensor designs of a Mead-Marcus beam. The main challenge in this project 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 energy estimates. This is sharply different from a single-layer (Euler-Bernoulli) beam.

Finally, novel Wolfram Demonstrations Projects are provided to allow the end user match the results from the robust sensor/feedback controller design with what is happening in the real world.

Disciplines

Applied Mathematics | Computer Sciences | Controls and Control Theory | Electrical and Computer Engineering | Engineering | Numerical Analysis and Scientific Computing | Physical Sciences and Mathematics

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