Feedback control of noise in a 2-D nonlinear structural acoustics model
type: Item
instance of: scholarly article
title: Feedback control of noise in a 2-D nonlinear structural acoustics model
zbMATH Open document ID: 0872.93035
published in: Discrete and Continuous Dynamical Systems
publication date: 1997-10-13T00:00:00Z
review text: The problem considered by the authors is complex. The cavity (it could be the interior of an aircraft) responds to vibration of one of its walls, which is regarded as a beam. The acoustic response of the cavity is coupled with the vibrations of the structure. Piezo-electric patches are bounded to the vibrating beam. These patches exert bending moments on the vibrating beam. Only a two-dimensional cross-section of the cavity is considered in this paper. The purpose of the control is to minimize the noise. While this project deals with the fast growing study of ``smar materials'', it has a more modest purpose of noise control, through action of piezo-electric ceramic actuators. Acoustics are modeled by a simple wave equation \(\phi_{tt}=\Delta\phi\), where \(\phi\) is the velocity potential. The Euler-Bernoulli beam with a Voigt-Kelvin damping is one of the equations of the interacting system. The moment exerted by a piezo-electric patch is assumed to be proportional to the voltage applied to it. The nonlinearities arise because of the surface action of the patches. The coupling between the beam and the aboustic response is also nonlinear. All this is considered in the abstract setting, where the system of state equations is written in a weak form in an appropriate Hilbert space. The well-posedness of similar system was considered in a previous paper by the same authors, and is again discussed here. Gains are first computed in the LQR setting then are inserted numerically into the nonlinear system. Several control inputs are considered, and a reasonable success in reducing the noise is claimed.
Mathematics Subject Classification ID: 93B52
Mathematics Subject Classification ID: 93C25
Mathematics Subject Classification ID: 93C95
Mathematics Subject Classification ID: 74K10
Mathematics Subject Classification ID: 74F10
zbMATH DE Number: 1010961
zbMATH Keywords: structural-acoustic interaction
zbMATH Keywords: piezo-electric patches
zbMATH Keywords: vibrating beam
zbMATH Keywords: noise control
zbMATH Keywords: wave equation
zbMATH Keywords: acoustic response
zbMATH Keywords: Hilbert space
author: H. Thomas Banks
author: Ralph C. Smith
reviewed by: Vadim Komkov
MaRDI profile type: Publication
full work available at URL: dcds.1995.1.119
OpenAlex ID: W1978221522
DOI: 10.3934/DCDS.1995.1.119
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description: scientific article; zbMATH DE number 1010961