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Research Papers

Boundary Backstepping Control of Flow-Induced Vibrations of a Membrane at High Mach Numbers

[+] Author and Article Information
Aziz Sezgin

Department of Mechanical Engineering,
Istanbul University Avcilar Kampusu,
Avcilar, Istanbul 34320, Turkey
e-mail: asezgin@istanbul.edu.tr

Miroslav Krstic

Professor
Department of Mechanical
and Aerospace Engineering,
University of California, San Diego,
La Jolla, CA 92093-0411
e-mail: krstic@ucsd.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received May 13, 2014; final manuscript received December 10, 2014; published online April 14, 2015. Assoc. Editor: YangQuan Chen.

J. Dyn. Sys., Meas., Control 137(8), 081003 (Aug 01, 2015) (8 pages) Paper No: DS-14-1203; doi: 10.1115/1.4029468 History: Received May 13, 2014; Revised December 10, 2014; Online April 14, 2015

We design a controller for flow-induced vibrations of an infinite-band membrane, with flow running across the band and only above it, and with actuation only on the trailing edge of the membrane. Due to the infinite length of the membrane, the dynamics of the membrane in the spanwise direction are neglected, namely, we employ a one-dimensional (1D) model that focuses on streamwise vibrations. This framework is inspired by a flow along an airplane wing with actuation on the trailing edge. The model of the flow-induced vibration is given by a wave partial differential equation (PDE) with an antidamping term throughout the 1D domain. Such a model is based on linear aeroelastic theory for Mach numbers above 0.8. To design a controller, we introduce a three-stage backstepping transformation. The first stage gets the system to a critically antidamped wave equation, changing the stiffness coefficient's value but not its sign. The second stage changes the system from a critically antidamped to a critically damped equation with an arbitrary damping coefficient. The third stage adjusts stiffness arbitrarily. The controller and backstepping transformation map the original system into a target system given by a wave equation with arbitrary positive damping and stiffness.

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Figures

Grahic Jump Location
Fig. 1

Infinite rectangular membrane with flow across and above the membrane

Grahic Jump Location
Fig. 2

System response for the systems (10)–(12) without control

Grahic Jump Location
Fig. 3

System response for the systems (10)–(12) with control

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