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

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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Uzal, E., and Kapkin, S., 2010, “Vibrations of an Infinite Plate Placed in a Circular Channel Containing Fluid Flow,” Aircr. Eng. Aerosp. Technol., 81(6), pp. 533–535. [CrossRef]
Vedeneev, V. V., 2012, “Vibrations of an Infinite Plate Placed in a Circular Channel Containing Fluid Flow,” J. Fluids Struct., 29, pp. 79–96. [CrossRef]
Doare, O., Sauzade, M., and Eloy, C., 2011, “Flutter of an Elastic Plate in a Channel Flow: Confinement and Finite-Size Effects,” J. Fluids Struct., 27(1), pp. 76–88. [CrossRef]
Zhang, W. W., Ye, Z. Y., and Zhang, C. A., 2009, “Supersonic Flutter Analysis Based on a Local Piston Theory,” AIAA J., 47(10), pp. 2321–2328. [CrossRef]
Uzal, E., and Korbahti, B., 2010, “Vibration Control of an Elastic Strip by a Singular Force,” Sadhana, 35(2), pp. 233–240. [CrossRef]
Singh, K., Michelin, S., and Langre, E., 2012, “Energy Harvesting From Axial Fluid-Elastic Instabilities of a Cylinder,” J. Fluids Struct., 30, pp. 159–172. [CrossRef]
Korbahti, B., 2010, “Specially Orthotropic Panel Flutter Control Using PID Controller,” Acta Mech., 212(3–4), pp. 191–197. [CrossRef]
Dowell, E. H., and Hall, K. C., 2001, “Modeling of Fluid-Structure Iteration,” Annu. Rev. Fluid Mech., 33, pp. 445–490. [CrossRef]
Kumhaar, H., 1963, “The Accuracy of Applying Linear Piston Theory to Cylindrical Shells,” AIAA Journal, pp. 1448–1449. [CrossRef]
Dowell, E. H., Crawley, E. F., Curtiss, H. C., Peters, D. A., Scanlan, R. H., and Sisto, F., 1995, A Modern Course in Aeroelasticity, Kluwer Academic Publisher, Dordrecht/Boston.
Epureanu, B. I., and Yin, S. H., 2004, “Identification of Damage in an Aeroelastic System Based on Attractor Deformations,” Comput. Struct., 82(31–32), pp. 2743–2751. [CrossRef]
Freitas, P., and Zuazua, E., 1996, “Stability Results for the Wave Equation With Indefinite Damping,” J. Differ. Equations, 132(2), pp. 338–352. [CrossRef]
Menz, G., 2007, “Exponential Stability of Wave Equations With Potential and Indefinite Damping,” J. Differ. Equations, 242(1), pp. 171–191. [CrossRef]
Rivera, J. E. M., and Racke, R., 2008, “Exponential Stability for Wave Equations With Non-Dissipative Damping,” Nonlinear Anal., 68(9), pp. 2531–2551. [CrossRef]
Cox, S., and Zuazua, E., 1994, “The Rate at Which Energy Decays in a String Damped at One End,” Commun. Partial Differ. Equ., 19(1--2), pp. 213–243. [CrossRef]
Tebou, L., 2007, “Stabilization of the Elastodynamic Equations With a Degenerate Locally Distributed Dissipation,” Syst. Control Lett., 56(7–8), pp. 538–545. [CrossRef]
Smyshlyaev, A., Cerpa, E., and Krstic, M., 2010, “Boundary Stabilization of a 1-D Wave Equation With In-Domain Antidamping,” SIAM J. Control Optim., 48(6), pp. 4014–4031. [CrossRef]
Krstic, M., 2011, “Dead-Time Compensation for Wave/String PDEs,” ASME J. Dyn. Syst. Meas. Control, 133(3), p. 031004. [CrossRef]
Krstic, M., Guo, B. Z., Balogh, A., and Smyshlyaev, A., 2008, “Output-Feedback Stabilization of an Unstable Wave Equation,” Automatica, 44(1), pp. 63–74. [CrossRef]
Krstic, M., and Smyshlyaev, A., 2008, “Adaptive Control of PDEs,” Annu. Rev. Control, 32(2), pp. 149–160. [CrossRef]
Krstic, M., 2009, “American Control Conference,” (ACC'09), St. Louis, MO, June 10–12, pp. 1505–1510. [CrossRef]
Krstic, M., and Smyshlyaev, A., 2008, Boundary Control of PDEs: A Course on Backstepping Designs, SIAM, Philadelphia.
Brake, M. R., and Segalman, D. J., 2010, “Nonlinear Model Reduction of von Kármán Plates Under Quasi-Steady Fluid Flow,” AIAA J., 48(10), pp. 2339–2347. [CrossRef]
Queiroz, M. S., and Rahn, C. D., 2002, “Boundary Control of Vibration and Noise in Distributed Parameter Systems: An Overview,” Mech. Syst. Signal Process., 16(1), pp. 19–38. [CrossRef]
Knight, J. J., Lucey, A. D., and Shaw, C. T., 2010, “Fluid–Structure Interaction of a Two–Dimensional Membrane in a Flow With a Pressure Gradient With Application to Convertible Car Roofs,” J. Wind Eng. Ind. Aerodyn., 98(2), pp. 65–72. [CrossRef]
Lucey, A. D., Cafolla, G. J., Carpenter, P. W., and Yang, M., 1997, “The Nonlinear Hydroelastic Behaviour of Flexible Walls,” J. Fluids Struct., 11(7), pp. 717–744. [CrossRef]
Hansen, S., 2001, “Exact Controllability of an Elastic Membrane Coupled With a Potential Fluid,” Int. J. Appl. Math. Comput. Sci., 11(6), pp. 1231–1248.
Yang, F., and Yao, R., 1996, “The Solution for Mixed Boundary Value Problems of Two-Dimensional Potential Theory,” Indian J. Pure Appl. Math., 27(3), pp. 313–322.
Smyshlyaev, A., and Krstic, M., 2005, “Backstepping Observers for a Class of Parabolic PDEs,” Syst. Control Lett., 54(7), pp. 613–625. [CrossRef]
Krstic, M., Kanellakopoulos, I., and Kokotovic, P., 1995, Nonlinear and Adaptive Control Design, Wiley, New York.
Serrani, A., and Isidori, A., 2000, “Global Robust Output Regulation for a Class of Nonlinear Systems,” Syst. Control Lett., 39(2), pp. 133–139. [CrossRef]
Bisplinghoff, R. L., and Ashley, H., 1962, Principals of Aeroelasticity, Dover Publication, New York.


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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