Control of Limit Cycle Oscillations of a Two-Dimensional … · 2020. 10. 8. · Control of Limit Cycle Oscillations of a Two-Dimensional Aeroelastic System M. Ghommem, A. H. Nayfeh,

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2010, Article ID 782457, 13 pagesdoi:10.1155/2010/782457

Research ArticleControl of Limit Cycle Oscillations ofa Two-Dimensional Aeroelastic System

M. Ghommem, A. H. Nayfeh, and M. R. Hajj

Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University,Blacksburg, VA 24061, USA

Correspondence should be addressed to M. R. Hajj, [email protected]

Received 19 August 2009; Accepted 3 November 2009

Academic Editor: Jose Balthazar

Copyright q 2010 M. Ghommem et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Linear and nonlinear static feedback controls are implemented on a nonlinear aeroelastic systemthat consists of a rigid airfoil supported by nonlinear springs in the pitch and plunge directionsand subjected to nonlinear aerodynamic loads. The normal form is used to investigate the Hopfbifurcation that occurs as the freestream velocity is increased and to analytically predict theamplitude and frequency of the ensuing limit cycle oscillations (LCO). It is shown that linearcontrol can be used to delay the flutter onset and reduce the LCO amplitude. Yet, its required gainsremain a function of the speed. On the other hand, nonlinear control can be effciently implementedto convert any subcritical Hopf bifurcation into a supercritical one and to significantly reduce theLCO amplitude.

1. Introduction

The response of an aeroelastic system is governed by a combination of linear and nonlineardynamics. When combined, the nonlinearities (geometric, inertia, free-play, damping, and/oraerodynamics) lead to different behavior [1–3], including multiple equilibria, bifurcations,limit cycles, chaos, and various types of resonances (internal and super/subharmonic) [4]. Ageneric nonlinear system that has been used to characterize aeroelastic behavior and dynamicinstabilities is a two-dimensional rigid airfoil undergoing pitch and plunge motions [5, 6]. Asthe parameters of this system (e.g., freestream velocity) are varied, changes may occur inits behavior. Of particular interest is its response around a bifurcation point. Depending onthe relative magnitude and type of nonlinearity, the bifurcation can be of the subcritical orsupercritical type. Hence, one needs to consider the combined effects of all nonlinearitiesto predict the system’s response. Furthermore, the nonlinearities provide an opportunity toimplement a combination of linear and nonlinear control strategies to delay the occurrence

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Copyright by the Hindawi Publishing Corporation. M. Ghommem, A. H. Nayfeh, and M. R. Hajj, "Control of Limit Cycle Oscillations of a Two-Dimensional Aeroelastic System,” Mathematical Problems in Engineering, vol. 2010, Article ID 782457, 13 pages, 2010. doi:10.1155/2010/782457

2 Mathematical Problems in Engineering

of the bifurcation (i.e., increase the allowable flight speed) and avoid catastrophic behaviorof the system (subcritical Hopf bifurcation) by suppressing or even alleviating the large-amplitude LCO and eliminating LCO that may take place at speeds lower than the nominalflutter speed.

Different methods have been proposed to control bifurcations and achieve desirablenonlinear effects in complex systems. Abed and Fu [7, 8] proposed a nonlinear feedbackcontrol to suppress discontinuous bifurcations of fixed points, such as subcritical Hopfbifurcations, which can result in loss of synchronism or voltage collapse in power systems.For the pitch-plunge airfoil, Strganac et al. [9] used a trailing edge flap to control a two-dimensional nonlinear aeroelastic system. They showed that linear control strategies maynot be appropriate to suppress large-amplitude LCO and proposed a nonlinear controllerbased on partial feedback linearization to stabilize the LCO above the nominal flutter velocity.Librescu et al. [10] implemented an active flap control for 2D wing-flap systems operating inan incompressible flow field and exposed to a blast pulse and demonstrated its performancesin suppressing flutter and reducing the vibration level in the subcritical flight speed range.Kang [11] developed a mathematical framework for the analysis and control of bifurcationsand used an approach based on the normal form to develop a feedback design for delayingand stabilizing bifurcations. His approach involves a preliminary state transformation andcenter manifold reduction.

In this work, we present a methodology to convert subcritical bifurcations ofaeroelastic systems into supercritical bifurcations. This methodology involves the followingsteps: (i) reduction of the dynamics of the system into a one-dimensional dynamical systemusing the method of multiple scales and then (ii) designing a nonlinear feedback controllerto convert subcritical to supercritical bifurcations and reduce the amplitude of any ensuingLCO.

2. Representation of the Aeroelastic System

The aeroelastic system, considered in this work, is modeled as a rigid wing undergoing two-degree-of-freedom motions, as presented in Figure 1. The wing is free to rotate about theelastic axis (pitch motion) and translate vertically (plunge motion). Denoting by h and αthe plunge deflection and pitch angle, respectively, we write the governing equations of thissystem as [4, 9]

(mT mWxαb

mWxαb Iα

)(h

α

)+

(ch 0

0 cα

)(h

α

)+

(kh(h) 0

0 kα(α)

)(h

α

)=

(−LM

), (2.1)

where mT is the total mass of the wing and its support structure, mW is the wing massalone, Iα is the mass moment of inertia about the elastic axis, b is the half chord length,xα = rcg/b is the nondimensionalized distance between the center of mass and the elasticaxis, ch and cα are the plunge and pitch structural damping coefficients, respectively, L andM are the aerodynamic lift and moment about the elastic axis, and kh and kα are the structuralstiffnesses for the plunge and pitch motions, respectively. These stiffnesses are approximated

Mathematical Problems in Engineering 3

U α

M

L

kh kα

cg

xα

h

Figure 1: Sketch of a two-dimensional airfoil.

in polynomial form by

kα(α) = kα0 + kα1α + kα2α2 + · · · ,

kh(h) = kh0 + kh1h + kh2h2 + · · · .

(2.2)

The aerodynamic loads are evaluated using a quasi-steady approximation with a stall model[9] and written as

L = ρU2bclα

(αeff − csα3

eff

),

M = ρU2b2cmα

(αeff − csα3

eff

),

(2.3)

where U is the freestream velocity, clα and cmα are the aerodynamic lift and momentcoefficients, and cs is a nonlinear parameter associated with stall. The effective angle of attackdue to the instantaneous motion of the airfoil is given by [9]

αeff = α +h

U+(

12− a)bα

U, (2.4)

where a is the nondimensionalized distance from the midchord to the elastic axis.For the sake of simplicity, we define the state variables

Y =

⎛⎜⎜⎜⎜⎜⎝

Y1

Y2

Y3

Y4

⎞⎟⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎜⎝

h

α

h

α

⎞⎟⎟⎟⎟⎟⎠, (2.5)

and write the equations of motion in the form

Y = F(Y, U), (2.6)


Table 1: System variables.

d = mTIα −m2Wx

2αb

2

k1 = (Iαρbclα +mWxαρb3cmα)/d

k2 = −(mWxαρb2clα +mTρb

2cmα)/d

c1 = [Iα(ch + ρUbclα) +mWxαρUb3cmα

]/d

c2 = [IαρUb2clα(1/2 − a) −mWxαbclα +mWxαρUb4cmα

(1/2 − a)]/dc3 = [−mWxαb(ch + ρUbclα) −mTxαρUb

2cmα]/d

c4 = [mT (cα − ρUb3cmα(1/2 − a)) −mWxαρUb

3clα(1/2 − a)]/dpα(Y) = −mWxαbkα(Y)/d

qα(Y) = mTkα(Y)/d

ph(Y) = Iαkh(Y)/d

qh(Y) = −mWxαbkh(Y)/d

gNL1(Y) = (csρU2b)(clα Iα +mWxαb2cmα

)α3eff(Y)/d

gNL2(Y) = −(csρU2b2)(clαmWxα +mTcmα)α3

eff(Y)/d

where

F(Y, U) =

⎛⎜⎜⎜⎜⎜⎝

Y3

Y4

−ph(Y1)Y1 −(k1U

2 + pα(Y2))Y2 − c1Y3 − c2Y4 + gNL1(Y )

−qh(Y1)Y1 −(k2U

2 + qα(Y2))Y2 − c3Y3 − c4Y4 + gNL2(Y )

⎞⎟⎟⎟⎟⎟⎠. (2.7)

The set of new variables that are used in (2.7) in terms of physical parameters is provided inTable 1. The original system, (2.6), is then rewritten as

Y = A(U)Y +Q(Y,Y) + C(Y,Y,Y), (2.8)

whereQ(Y,Y) andC(Y,Y,Y) are, respectively, the quadratic and cubic vector functions of thestate variables collected in the vector Y.

To determine the system’s stability, we consider the linearized governing equations,which are written in a first-order differential form as

Y = A(U)Y, (2.9)

where

A(U) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 0

0 0 0 1

−Iαkh0

d−(k1U

2 − mWxαbkα0

d

)−c1 −c2

mWxαbkh0

d−(k2U

2 +mTkα0

d

)−c3 −c4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (2.10)


−3

−2

−1

0

1R

eal(λj)

(s−1)

0 2 4 6 8 10 12

U (m/s)

(a)

−15

−10

−5

0

5

10

15

Imag

(λj)

(Hz)

0 2 4 6 8 10 12

U (m/s)

(b)

Figure 2: Variations of (a) damping (Real(λj)) and (b) frequencies (Imag(λj)) with the freestream velocityU.

The 4 × 4 matrix A(U) has a set of four eigenvalues, {λj , j = 1, 2, . . . , 4}. These eigenvaluesdetermine the stability of the trivial solution of (2.6). If the real parts of all of the λj arenegative, the trivial solution is asymptotically stable. On the other hand, if the real part of oneor more eigenvalues is positive, the trivial solution is unstable. The flutter speedUf , for whichone or more eigenvalues have zero real parts, corresponds to the onset of linear instability.For the specific values given in [9], Figures 2(a) and 2(b) show, respectively, variations of thereal and imaginary parts of the λj with U, which, respectively, correspond to the dampingand frequencies of the plunge and pitch motions. We note that the damping of two modesbecomes positive at Uf = 9.1242 m/s, which corresponds to the flutter speed at which theaeroelastic system undergoes a Hopf bifurcation.

3. Static Feedback Control

To manage the Hopf bifurcation and achieve desirable nonlinear dynamics, we follow Nayfehand Balachandran [12] and use a static feedback control. To the system given by (2.6), we adda static feedback u(Y), which includes linear, LuY, quadratic Qu(Y,Y), and cubic Cu(Y,Y,Y)components; that is,

u(Y) = LuY +Qu(Y,Y) + Cu(Y,Y,Y). (3.1)

Hence, the controlled system takes the form

Y = F(Y, U) + u(Y). (3.2)

3.1. Normal Form of Hopf Bifurcation

To compute the normal form of the Hopf bifurcation of (3.2) near U = Uf , we follow Nayfehand Balachandran [12] and introduce a small nondimensional parameter ε as a book keeping


parameter. Defining the velocity perturbation as a ratio of the flutter speed σUUf , we writeU = Uf + ε2σUUf and seek a third-order approximate solution of (3.2) in the form

Y(t, σU, σα, σh) = εY1(T0, T2) + ε2Y2(T0, T2) + ε3Y3(T0, T2) + · · · , (3.3)

where the time scales Tm = εmt. In terms of these scales, the time derivative d/dt is written as

d

dt=

∂

∂T0+ ε2 ∂

∂T2+ · · · = D0 + ε2D2 + · · · . (3.4)

Scaling Lu as ε2Lu, substituting (3.3) and (3.4) into (3.2), and equating coefficients of likepowers of ε, we obtain

Order (ε),

D0Y1 −A(Uf

)Y1 = 0, (3.5)

Order (ε2),

D0Y2 −A(Uf

)Y2 = Q(Y1,Y1) +Qu(Y1,Y1), (3.6)

Order (ε3),

D0Y3 −A(Uf

)Y3 = −D2Y1 + σUBY1 + LuY1 + 2[Q(Y1,Y2) +Qu(Y1,Y2)]

+ C(Y1,Y1,Y1) + Cu(Y1,Y1,Y1),(3.7)

where

B = −2k1U2f I1 − 2k2U

2f I2, I1 =

⎛⎜⎜⎜⎜⎜⎝

0 0 0 0

0 0 0 0

0 1 0 0

0 0 0 0

⎞⎟⎟⎟⎟⎟⎠, I2 =

⎛⎜⎜⎜⎜⎜⎝

0 0 0 0

0 0 0 0

0 0 0 0

0 1 0 0

⎞⎟⎟⎟⎟⎟⎠. (3.8)

The general solution of (3.5) is the superposition of four linearly independent solutionscorresponding to the four eigenvalues: two of these eigenvalues have negative real parts andthe other two are purely imaginary (±iω). Because the two solutions corresponding to thetwo eigenvalues with negative real parts decay as T0 → ∞, we retain only the nondecayingsolutions and express the general solution of the first-order problem as

Y1(T0, T2) = η(T2)peiωT0 + η(T2)pe−iωT0 , (3.9)

where η(T2) is determined by imposing the solvability condition at the third-order level andp is the eigenvector of A(Uf) corresponding to the eigenvalue iω; that is,

A(Uf

)p = iωp. (3.10)


Substituting (3.9) into (3.6) yields

D0Y2 −A(Uf

)Y2 = [Q(p,p) +Qu(p,p)]η2e2iωT0 + 2[Q(p,p) +Qu(p,p)]ηη

+ [Q(p,p) +Qu(p,p)]η2e−2iωT0 .

(3.11)

The solution of (3.11) can be written as

Y2 = (ζ2 + ζ2u)η2e2iωT0 + 2(ζ0 + ζ0u)ηη +(ζ2 + ζ2u

)η2e−2iωT0 , (3.12)

where

[2iωI −A

(Uf

)]ζ2 = Q(p,p),

[2iωI −A

(Uf

)]ζ2u = Qu(p,p),

A(Uf

)ζ0 = −Qu(p,p), A

(Uf

)ζ0u = −Qu(p,p).

(3.13)

Substituting (3.9) and (3.12) into (3.7), we obtain

D0Y3 −A(Uf

)Y3 = −

[D2ηp − (σUB + Lu)ηp

− (4Q(p, ζ0) + 2Q(p, ζ2)

+ 3C(p,p,p) + 4Qu(p, ζ0) + 2Qu(p, ζ2)

+3Cu(p,p,p) + 4Q(p, ζ0u) + 2Q(p, ζ2u))η2η]eiωT0 + cc + NST,

(3.14)

where cc stands for the complex conjugate of the preceding terms and NST stands for termsthat do not produce secular terms. We let q be the left eigenvector of A(Uf) corresponding tothe eigenvalue iω; that is,

A(Uf

)Tq = iωq. (3.15)

We normalize it so that qTp = 1. Then, the solvability condition requires that termsproportional to eiωT0 in (3.14) be orthogonal to q. Imposing this condition, we obtain thefollowing normal of the Hopf bifurcation:

D2η = βη + Λη2η, (3.16)

where

β = β + βu, (3.17)


with

β = qTσUBp, βu = qTLup,

Λ = Λ + Λu,(3.18)

with

Λ = 4qTQ(p, ζ0) + 2qTQ(p, ζ2) + 3qTC(p,p,p),

Λu = 4qTQu(p, ζ0) + 2qTQu(p, ζ2) + 3qTCu(p,p,p)

+ 4qTQ(p, ζ0u) + 2qTQ(p, ζ2u).

(3.19)

Letting η = (1/2)a exp(iθ) and separating the real and imaginary parts in (3.16), we obtainthe following alternate normal form of the Hopf bifurcation:

a = βra +14Λra

3, (3.20)

θ = βi +14Λia

2, (3.21)

where (·)r and (·)i stand for the real and imaginary parts, respectively, a is the amplitude andθ is the frequency of the oscillatory motion associated with the Hopf bifurcation.

We note that, because the a component is independent of θ, the system’s stability isreduced to a one-dimensional dynamical system given by (3.20). Assuming that Λr /= 0, aadmits three steady-state solutions, namely,

a = 0, a = ±

√√√√−4βrΛr

. (3.22)

The trivial fixed point of (3.20) corresponds to the fixed point (0, 0) of (3.2), and a nontrivialfixed point (i.e., a/= 0) of (3.20) corresponds to a periodic solution of (3.2). The origin isasymptotically stable when βr < 0, unstable when βr > 0, unstable when βr = 0 and Λr > 0,and asymptotically stable when βr = 0 and Λr < 0. On the other hand, the nontrivial fixedpoints exist when −βrΛr > 0. They are stable when βr > 0 and Λr < 0 (supercritical Hopfbifurcation) and unstable when βr < 0 and Λr > 0 (subcritical Hopf bifurcation). We notethat a stable nontrivial fixed point of (3.20) corresponds to a stable periodic solution of (3.2).Likewise, an unstable nontrivial fixed point of (3.20) corresponds to an unstable periodicsolution of (3.2).

Therefore, to delay the occurrence of Hopf bifurcation (i.e., stabilize the aeroelasticsystem at speeds higher than the flutter speed), one needs to set the real part of β to anegative value by appropriately managing the linear control represented by Lu in (3.2). Toeliminate subcritical instabilities and limit LCO amplitudes to small values at speeds higherthan the flutter speed (supercritical Hopf bifurcation which is a favorable instability for such


systems), the nonlinear feedback control given by Qu(Y,Y) +Cu(Y,Y,Y) should be chosen sothat Real(Λ + Λu) < 0.

3.2. Case Study

To demonstrate the linear and nonlinear control strategies, we consider an uncontrolled case(i.e., u(Y) = 0) in which only the pitch structural nonlinearity is taken into account; thatis, kh1 = kh2 = cs = 0, kα1 = 9.9967, and kα2 = 167.685. The hysteretic response as afunction of the freestream velocity, obtained through the numerical integration of (2.6) forthese parameters, is presented in Figure 3. The onset of flutter takes place at Uf = 9.1242 m/sand is characterized by a jump to a large-amplitude LCO when transitioning through theHopf bifurcation. As the speed is increased beyond the flutter speed, the LCO amplitudes ofboth of the pitch and plunge motions increase. Furthermore, LCO take place at speeds lowerthan Uf if the disturbances to the system are sufficiently large. Clearly, this configurationexhibits a subcritical instability (Real(Λ) > 0).

For linear control, we consider the matrix Lu defined in (3.2) in the form of

Lu =

⎛⎜⎜⎜⎜⎜⎝

−kl 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

⎞⎟⎟⎟⎟⎟⎠, (3.23)

where kl is the linear feedback control gain. Then, for the specific values of the systemparameters given in [9], we obtain

β =(0.433891σUUf − 0.110363kl

)+ ı(0.323942σUUf − 0.303804kl

). (3.24)

To guarantee damped oscillations of the airfoil at speeds higher than the flutter speed, oneneeds to set kl to a value such that Real(β) < 0. Using a gain of 10, we plot in Figure 4 theplunge and pitch displacements for a freestream velocity of U = 10 m/s with and withoutlinear control. Clearly, linear control damps the LCO of the uncontrolled system. We notethat, by increasing the linear feedback control gain kl, the amplitudes of pitch and plungedecay more rapidly.

Although linear control is capable of delaying the onset of flutter in terms of speedand reducing the LCO amplitude, the system would require higher gains at higher speeds.Furthermore, it maintains its subcritical response. To overcome these difficulties and convertthe subcritical instability to a supercritical one, we introduce the following nonlinear feedbackcontrol law:

uT = −(knl1 knl2 knl3 knl4

)α3, (3.25)

where the knli are the nonlinear feedback control gains. For the specific airfoil’s geometrygiven in [9], we obtain

Λr = 0.866899 − (132.844knl1 + 13.3643knl2 + 2.9858knl3 + 1.32415knl4). (3.26)


0

0.005

0.01

0.015

0.02

0.025

0.03

5 6 7 8 9 10 11 12 13

Max

(am

p)−h

U (m/s)

(a) Plunge motion

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

5 6 7 8 9 10 11 12 13

Max

(am

p)−α

U (m/s)

Low ICHigh IC

(b) Pitch motion

Figure 3: Hysteretic response of the aeroelastic system (subcritical instability). The steady-state amplitudesare plotted as a function of U.

This equation shows that applying gain to the plunge displacement is more effective thanapplying it to the pitch displacement or plunge velocity or pitch velocity.

The subcritical instability takes place for positive values of Λr . As such it can beeliminated by forcing Λr to be negative. This can be achieved by using nonlinear controlgains knl1 = 0.02 (knl2 = knl3 = knl4 = 0). The results are presented in Figure 5. The subcriticalHopf bifurcation at Uf observed in Figure 3 has been transformed into the supercriticalHopf bifurcation of Figure 5. A comparison of the two figures shows that the unstable limit


−0.03

−0.02

−0.01

0

0.01

0.02

0.03h(m

)

0 5 10 15 20 25 30

Time (s)

(a) Plunge motion

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

α(r

ad)

0 5 10 15 20 25 30

Time (s)

(b) Pitch motion

Figure 4: Measured pitch and plunge responses:: −, without linear control, ∗, with linear control.

0

1

2

3

4

5

6

7

8×10−3

8 8.5 9 9.5 10 10.5

Max

(am

p)−h

U (m/s)

(a) Plunge motion

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

8 8.5 9 9.5 10 10.5

Max

(am

p)−α

U (m/s)

(b) Pitch motion

Figure 5: LCO amplitudes of plunge and pitch motions (controlled configuration): −, analytical prediction,∗, numerical integration.

cycles observed over the freestream speed between U = 6.5 m/s and U = 9.12 m/s havebeen eliminated. Furthermore, the exponentially growing oscillations predicted by the linearmodel are limited to a periodic solution whose amplitude increases slowly with increasingfreestream velocity. Moreover, increasing the value of knl1 reduces the amplitude of the limitcycles created due to Hopf bifurcation.

In order to check the accuracy of the analytical formulation given by the normal formin predicting the amplitude of LCO near the Hopf bifurcation, we consider the first-ordersolution given by (3.9). The amplitude of plunge and pitch LCO, Ah and Aα, respectively, aregiven by

Ah = a√p12

r + p12i ,

Aα = a√p22

r + p22i ,

(3.27)


where pjr and pji denote the real and imaginary parts of the jth component of the vector p,respectively. In Figures 5(a) and 5(b), we plot the LCO amplitudes for both pitch and plungemotions obtained by integrating the original system and those predicted by the normal form.The results show good agreement in the LCO amplitudes only near the bifurcation.

4. Conclusions

Linear and nonlinear controls are implemented on a rigid airfoil undergoing pitch and plungemotions. The method of multiple scales is applied to the governing system of equationsto derive the normal form of the Hopf bifurcation near the flutter onset. The linear andnonlinear parameters of the normal form are used to determine the stability characteristicsof the bifurcation and efficiency of the linear and nonlinear control components. The resultsshow that linear control can be used to delay the flutter onset and dampen LCO. Yet, itsrequired gains remain a function of the speed. On the other hand, nonlinear control canbe efficiently implemented to convert subcritical to supercritical Hopf bifurcations and tosignificantly reduce LCO amplitudes.

Acknowledgment

M. Ghommem is grateful for the support from the Virginia Tech Institute for CriticalTechnology and Applied Science (ICTAS) Doctoral Scholars Program.

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