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.

References

1.
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
.10.1108/00022660910997847
2.
Vedeneev
,
V. V.
,
2012
, “
Vibrations of an Infinite Plate Placed in a Circular Channel Containing Fluid Flow
,”
J. Fluids Struct.
,
29
, pp.
79
96
.10.1016/j.jfluidstructs.2011.12.011
3.
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
.10.1016/j.jfluidstructs.2010.09.002
4.
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
.10.2514/1.37750
5.
Uzal
,
E.
, and
Korbahti
,
B.
,
2010
, “
Vibration Control of an Elastic Strip by a Singular Force
,”
Sadhana
,
35
(
2
), pp.
233
240
.10.1007/s12046-010-0020-2
6.
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
.10.1016/j.jfluidstructs.2012.01.008
7.
Korbahti
,
B.
,
2010
, “
Specially Orthotropic Panel Flutter Control Using PID Controller
,”
Acta Mech.
,
212
(3–4), pp.
191
197
.10.1007/s00707-009-0255-3
8.
Dowell
,
E. H.
, and
Hall
,
K. C.
,
2001
, “
Modeling of Fluid-Structure Iteration
,”
Annu. Rev. Fluid Mech.
,
33
, pp.
445
490
.10.1146/annurev.fluid.33.1.445
9.
Kumhaar
,
H.
,
1963
, “
The Accuracy of Applying Linear Piston Theory to Cylindrical Shells
,”
AIAA Journal
, pp. 1448–1449.10.2514/3.1832
10.
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
.
11.
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
.10.1016/j.compstruc.2004.03.079
12.
Freitas
,
P.
, and
Zuazua
,
E.
,
1996
, “
Stability Results for the Wave Equation With Indefinite Damping
,”
J. Differ. Equations
,
132
(2), pp.
338
352
.10.1006/jdeq.1996.0183
13.
Menz
,
G.
,
2007
, “
Exponential Stability of Wave Equations With Potential and Indefinite Damping
,”
J. Differ. Equations
,
242
(
1
), pp.
171
191
.10.1016/j.jde.2007.04.002
14.
Rivera
,
J. E. M.
, and
Racke
,
R.
,
2008
, “
Exponential Stability for Wave Equations With Non-Dissipative Damping
,”
Nonlinear Anal.
,
68
(9), pp.
2531
2551
.10.1016/j.na.2007.02.022
15.
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
.10.1080/03605309408821015
16.
Tebou
,
L.
,
2007
, “
Stabilization of the Elastodynamic Equations With a Degenerate Locally Distributed Dissipation
,”
Syst. Control Lett.
,
56
(7–8), pp.
538
545
.10.1016/j.sysconle.2007.03.003
17.
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
.10.1137/080742646
18.
Krstic
,
M.
,
2011
, “
Dead-Time Compensation for Wave/String PDEs
,”
ASME J. Dyn. Syst. Meas. Control
,
133
(
3
), p.
031004
.10.1115/1.4003638
19.
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
.10.1016/j.automatica.2007.05.012
20.
Krstic
,
M.
, and
Smyshlyaev
,
A.
,
2008
, “
Adaptive Control of PDEs
,”
Annu. Rev. Control
,
32
(2), pp.
149
160
.10.1016/j.arcontrol.2008.05.001
21.
Krstic
,
M.
,
2009
, “
American Control Conference
,” (
ACC'09
), St. Louis, MO, June 10–12, pp. 1505–1510.10.1109/ACC.2009.5159799
22.
Krstic
,
M.
, and
Smyshlyaev
,
A.
,
2008
,
Boundary Control of PDEs: A Course on Backstepping Designs
,
SIAM
, Philadelphia.
23.
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
.10.2514/1.J050357
24.
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
.10.1006/mssp.2001.1438
25.
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
.10.1016/j.jweia.2009.09.003
26.
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
.10.1006/jfls.1997.0107
27.
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
.
28.
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
.
29.
Smyshlyaev
,
A.
, and
Krstic
,
M.
,
2005
, “
Backstepping Observers for a Class of Parabolic PDEs
,”
Syst. Control Lett.
,
54
(7), pp.
613
625
.10.1016/j.sysconle.2004.11.001
30.
Krstic
,
M.
,
Kanellakopoulos
,
I.
, and
Kokotovic
,
P.
,
1995
,
Nonlinear and Adaptive Control Design
,
Wiley
, New York.
31.
Serrani
,
A.
, and
Isidori
,
A.
,
2000
, “
Global Robust Output Regulation for a Class of Nonlinear Systems
,”
Syst. Control Lett.
,
39
(2), pp.
133
139
.10.1016/S0167-6911(99)00099-7
32.
Bisplinghoff
,
R. L.
, and
Ashley
,
H.
,
1962
,
Principals of Aeroelasticity
,
Dover Publication
, New York.
You do not currently have access to this content.