# Control of Limit Cycle Oscillations of a Two-Dimensional Aeroelastic System

Authors: M. Ghommem; A. H. Nayfeh; M. R. Hajj
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

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


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## Body

## 1. Introduction

The response of an aeroelastic system is governed by a combination of linear and nonlinear dynamics. When combined,  the nonlinearities (geometric, inertia, free-play, damping, and/or aerodynamics) lead to different behavior [1–3], including multiple equilibria, bifurcations, limit cycles, chaos, and various types of resonances (internal and super/subharmonic) [4]. A generic nonlinear system that has been used to characterize aeroelastic behavior and dynamic instabilities is a two-dimensional rigid airfoil undergoing pitch and plunge motions [5, 6]. As the parameters of this system (e.g., freestream velocity) are varied, changes may occur in its behavior. Of particular interest is its response around a bifurcation point. Depending on the relative magnitude and type of nonlinearity, the bifurcation can be of the subcritical or supercritical type. Hence, one needs to consider the combined effects of all nonlinearities to predict the system's response. Furthermore, the nonlinearities provide an opportunity to implement a combination of linear and nonlinear control strategies to delay the occurrence of the bifurcation (i.e., increase the allowable flight speed) and avoid catastrophic behavior of 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 nominal flutter speed.Different methods have been proposed to control bifurcations and achieve desirable nonlinear effects in complex systems. Abed and Fu [7, 8] proposed a nonlinear feedback control to suppress discontinuous bifurcations of fixed points, such as subcritical Hopf bifurcations, 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 may not be appropriate to suppress large-amplitude LCO and proposed a nonlinear controller based 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 in an incompressible flow field and exposed to a blast pulse and demonstrated its performances in 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 bifurcations and used an approach based on the normal form to develop a feedback design for delaying and stabilizing bifurcations. His approach involves a preliminary state transformation and center manifold reduction.In this work, we present a methodology to convert subcritical bifurcations of aeroelastic systems into supercritical bifurcations. This methodology involves the following steps: (i) reduction of the dynamics of the system into a one-dimensional dynamical system using the method of multiple scales and then (ii) designing a nonlinear feedback controller to convert subcritical to supercritical bifurcations and reduce the amplitude of any ensuing LCO.


## 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 Figure1. The wing is free to rotate about the elastic axis (pitch motion) and translate vertically (plunge motion). Denoting by h and α the plunge deflection and pitch angle, respectively, we write the governing equations of this system as [4, 9]
(2.1)(mTmWxαbmWxαbIα)(ḧα̈)+(ch00cα)(ḣα̇)+(kh(h)00kα(α))(hα)=(-LM),
where mT is the total mass of the wing and its support structure, mW is the wing mass alone, 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 elastic axis, ch and cα are the plunge and pitch structural damping coefficients, respectively, L and M are the aerodynamic lift and moment about the elastic axis, and kh and kα are the structural stiffnesses for the plunge and pitch motions, respectively. These stiffnesses are approximated in polynomial form by
(2.2)kα(α)=kα0+kα1α+kα2α2+⋯,kh(h)=kh0+kh1h+kh2h2+⋯.
The aerodynamic loads are evaluated using a quasi-steady approximation with a stall model [9] and written as (2.3)L=ρU2bclα(αeff-csαeff3),M=ρU2b2cmα(αeff-csαeff3),
where U is the freestream velocity, clα and cmα are the aerodynamic lift and moment coefficients, and cs is a nonlinear parameter associated with stall. The effective angle of attack due to the instantaneous motion of the airfoil is given by [9]
(2.4)αeff=α+ḣU+(12-a)bα̇U,
where a is the nondimensionalized distance from the midchord to the elastic axis.Figure 1
Sketch of a two-dimensional airfoil.For the sake of simplicity, we define the state variables(2.5)Y=(Y1Y2Y3Y4)=(hαḣα̇),
and write the equations of motion in the form
(2.6)Ẏ=F(Y,U),
where
(2.7)F(Y,U)=(Y3Y4-ph(Y1)Y1-(k1U2+pα(Y2))Y2-c1Y3-c2Y4+gNL1(Y)-qh(Y1)Y1-(k2U2+qα(Y2))Y2-c3Y3-c4Y4+gNL2(Y)).
The set of new variables that are used in (2.7) in terms of physical parameters is provided in Table 1. The original system, (2.6), is then rewritten as
(2.8)Ẏ=A(U)Y+Q(Y,Y)+C(Y,Y,Y),
where Q(Y,Y) and C(Y,Y,Y) are, respectively, the quadratic and cubic vector functions of the state variables collected in the vector Y.Table 1
System variables.
d=mTIα-mW2xα2b2k1=(Iαρbclα+mWxαρb3cmα)/dk2=-(mWxαρb2clα+mTρb2cmα)/dc1=[Iα(ch+ρUbclα)+mWxαρUb3cmα]/dc2=[IαρUb2clα(1/2-a)-mWxαbclα+mWxαρUb4cmα(1/2-a)]/dc3=[-mWxαb(ch+ρUbclα)-mTxαρUb2cmα]/dc4=[mT(cα-ρUb3cmα(1/2-a))-mWxαρUb3clα(1/2-a)]/dpα(Y)=-mWxαbkα(Y)/dqα(Y)=mTkα(Y)/dph(Y)=Iαkh(Y)/dqh(Y)=-mWxαbkh(Y)/dgNL1(Y)=(csρU2b)(clαIα+mWxαb2cmα)αeff3(Y)/dgNL2(Y)=-(csρU2b2)(clαmWxα+mTcmα)αeff3(Y)/dTo determine the system's stability, we consider the linearized governing equations, which are written in a first-order differential form as(2.9)Ẏ=A(U)Y,
where
(2.10)A(U)=(00100001-Iαkh0d-(k1U2-mWxαbkα0d)-c1-c2mWxαbkh0d-(k2U2+mTkα0d)-c3-c4).
The 4×4 matrix A(U) has a set of four eigenvalues, {λj,j=1,2,…,4}. These eigenvalues determine the stability of the trivial solution of (2.6). If the real parts of all of the λj are negative, the trivial solution is asymptotically stable. On the other hand, if the real part of one or more eigenvalues is positive, the trivial solution is unstable. The flutter speed Uf, for which one 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 the real and imaginary parts of the λj with U, which, respectively, correspond to the damping and frequencies of the plunge and pitch motions. We note that the damping of two modes becomes positive at Uf=9.1242 m/s, which corresponds to the flutter speed at which the aeroelastic system undergoes a Hopf bifurcation.Variations of (a) damping(Real(λj)) and (b) frequencies (Imag(λj)) with the freestream velocity U.
(a)(b)


## 3. Static Feedback Control

To manage the Hopf bifurcation and achieve desirable nonlinear dynamics, we follow Nayfeh and Balachandran [12] and use a static feedback control. To the system given by (2.6), we add a static feedback u(Y), which includes linear, LuY, quadratic Qu(Y,Y), and cubic Cu(Y,Y,Y) components; that is,
(3.1)u(Y)=LuY+Qu(Y,Y)+Cu(Y,Y,Y).
Hence, the controlled system takes the form
(3.2)Ẏ=F(Y,U)+u(Y).


### 3.1. Normal Form of Hopf Bifurcation

To compute the normal form of the Hopf bifurcation of (3.2) near U=Uf, we follow Nayfeh and 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 write U=Uf+ϵ2σUUf and seek a third-order approximate solution of (3.2) in the form
(3.3)Y(t,σU,σα,σh)=ϵY1(T0,T2)+ϵ2Y2(T0,T2)+ϵ3Y3(T0,T2)+⋯,
where the time scales Tm=ϵmt. In terms of these scales, the time derivative d/dt is written as
(3.4)ddt=∂∂T0+ϵ2∂∂T2+⋯=D0+ϵ2D2+⋯.
Scaling Lu as ϵ2Lu, substituting (3.3) and (3.4) into (3.2), and equating coefficients of like powers of ϵ, we obtainOrder (ϵ),
(3.5)D0Y1-A(Uf)Y1=0,Order (ϵ2),
(3.6)D0Y2-A(Uf)Y2=Q(Y1,Y1)+Qu(Y1,Y1),Order (ϵ3),
(3.7)D0Y3-A(Uf)Y3=-D2Y1+σUBY1+LuY1+2[Q(Y1,Y2)+Qu(Y1,Y2)]+C(Y1,Y1,Y1)+Cu(Y1,Y1,Y1),
where
(3.8)B=-2k1Uf2I1-2k2Uf2I2,I1=(0000000001000000),I2=(0000000000000100).
The general solution of (3.5) is the superposition of four linearly independent solutions corresponding to the four eigenvalues: two of these eigenvalues have negative real parts and the other two are purely imaginary (±iω). Because the two solutions corresponding to the two eigenvalues with negative real parts decay as T0→∞, we retain only the nondecaying solutions and express the general solution of the first-order problem as
(3.9)Y1(T0,T2)=η(T2)peiωT0+η¯(T2)p¯e-iωT0,
where η(T2) is determined by imposing the solvability condition at the third-order level and p is the eigenvector of A(Uf) corresponding to the eigenvalue iω; that is,
(3.10)A(Uf)p=iωp.
Substituting (3.9) into (3.6) yields
(3.11)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¯)]η¯2e-2iωT0.
The solution of (3.11) can be written as
(3.12)Y2=(ζ2+ζ2u)η2e2iωT0+2(ζ0+ζ0u)ηη¯+(ζ¯2+ζ¯2u)η¯2e-2iωT0,
where
(3.13)[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¯).
Substituting (3.9) and (3.12) into (3.7), we obtain
(3.14)D0Y3-A(Uf)Y3=-[+3Cu(p,p,p¯)+4Q(p,ζ0u)+2Q(p¯,ζ2u))η2η¯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,
where cc stands for the complex conjugate of the preceding terms and NST stands for terms that do not produce secular terms. We let q be the left eigenvector of A(Uf) corresponding to the eigenvalue iω; that is,
(3.15)A(Uf)Tq=iωq.
We normalize it so that qTp=1. Then, the solvability condition requires that terms proportional to eiωT0 in (3.14) be orthogonal to q. Imposing this condition, we obtain the following normal of the Hopf bifurcation:
(3.16)D2η=β̂η+Λ̂η2η¯,
where
(3.17)β̂=β+βu,
with
(3.18)β=qTσUBp,βu=qTLup,Λ̂=Λ+Λu,
with
(3.19)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).
Letting η=(1/2)aexp(iθ) and separating the real and imaginary parts in (3.16), we obtain the following alternate normal form of the Hopf bifurcation:
(3.20)ȧ=β̂ra+14Λ̂ra3,(3.21)θ̇=β̂i+14Λ̂ia2,
where (·)r and (·)i stand for the real and imaginary parts, respectively, a is the amplitude and θ̇ is the frequency of the oscillatory motion associated with the Hopf bifurcation.We note that, because thea component is independent of θ, the system's stability is reduced to a one-dimensional dynamical system given by (3.20). Assuming that Λ̂r≠0, a admits three steady-state solutions, namely,
(3.22)a=0,a=±-4β̂rΛ̂r.
The trivial fixed point of (3.20) corresponds to the fixed point (0,0) of (3.2), and a nontrivial fixed point (i.e., a≠0) of (3.20) corresponds to a periodic solution of (3.2). The origin is asymptotically stable when β̂r<0, unstable when β̂r>0, unstable when β̂r=0 and Λ̂r>0, and asymptotically stable when β̂r=0 and Λ̂r<0. On the other hand, the nontrivial fixed points exist when -β̂rΛ̂r>0. They are stable when β̂r>0 and Λ̂r<0 (supercritical Hopf bifurcation) and unstable when β̂r<0 and Λ̂r>0 (subcritical Hopf bifurcation). We note that 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 periodic solution of (3.2).Therefore, to delay the occurrence of Hopf bifurcation (i.e., stabilize the aeroelastic system at speeds higher than the flutter speed), one needs to set the real part ofβ̂ to a negative value by appropriately managing the linear control represented by Lu in (3.2). To eliminate subcritical instabilities and limit LCO amplitudes to small values at speeds higher than 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 so that 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; that is, kh1=kh2=cs=0, kα1=9.9967, and kα2=167.685. The hysteretic response as a function of the freestream velocity, obtained through the numerical integration of (2.6) for these parameters, is presented in Figure 3. The onset of flutter takes place at Uf=9.1242 m/s and is characterized by a jump to a large-amplitude LCO when transitioning through the Hopf bifurcation. As the speed is increased beyond the flutter speed, the LCO amplitudes of both of the pitch and plunge motions increase. Furthermore, LCO take place at speeds lower than Uf if the disturbances to the system are sufficiently large. Clearly, this configuration exhibits a subcritical instability (Real(Λ)>0).Hysteretic response of the aeroelastic system (subcritical instability). The steady-state amplitudes are plotted as a function ofU.
(a)
Plunge motion(b)
Pitch motionFor linear control, we consider the matrixLu defined in (3.2) in the form of
(3.23)Lu=(-kl000000000000000),
where kl is the linear feedback control gain. Then, for the specific values of the system parameters given in [9], we obtain
(3.24)β̂=(0.433891σUUf-0.110363kl)+ı(0.323942σUUf-0.303804kl).
To guarantee damped oscillations of the airfoil at speeds higher than the flutter speed, one needs to set kl to a value such that Real(β̂)<0. Using a gain of 10, we plot in Figure 4 the plunge and pitch displacements for a freestream velocity of U=10 m/s with and without linear control. Clearly, linear control damps the LCO of the uncontrolled system. We note that, by increasing the linear feedback control gain kl, the amplitudes of pitch and plunge decay more rapidly.Measured pitch and plunge responses::-, without linear control, *, with linear control.
(a)
Plunge motion(b)
Pitch motionAlthough linear control is capable of delaying the onset of flutter in terms of speed and reducing the LCO amplitude, the system would require higher gains at higher speeds. Furthermore, it maintains its subcritical response. To overcome these difficulties and convert the subcritical instability to a supercritical one, we introduce the following nonlinear feedback control law:(3.25)uT=-(knl1knl2knl3knl4)α̇3,
where the knli are the nonlinear feedback control gains. For the specific airfoil's geometry given in [9], we obtain
(3.26)Λ̂r=0.866899-(132.844knl1+13.3643knl2+2.9858knl3+1.32415knl4).
This equation shows that applying gain to the plunge displacement is more effective than applying it to the pitch displacement or plunge velocity or pitch velocity.The subcritical instability takes place for positive values ofΛ̂r. As such it can be eliminated by forcing Λ̂r to be negative. This can be achieved by using nonlinear control gains knl1=0.02  (knl2=knl3=knl4=0). The results are presented in Figure 5. The subcritical Hopf bifurcation at Uf observed in Figure 3 has been transformed into the supercritical Hopf bifurcation of Figure 5. A comparison of the two figures shows that the unstable limit cycles observed over the freestream speed between U=6.5 m/s and U=9.12 m/s have been eliminated. Furthermore, the exponentially growing oscillations predicted by the linear model are limited to a periodic solution whose amplitude increases slowly with increasing freestream velocity. Moreover, increasing the value of knl1 reduces the amplitude of the limit cycles created due to Hopf bifurcation.LCO amplitudes of plunge and pitch motions (controlled configuration):-, analytical prediction, *, numerical integration.
(a)
Plunge motion(b)
Pitch motionIn order to check the accuracy of the analytical formulation given by the normal form in predicting the amplitude of LCO near the Hopf bifurcation, we consider the first-order solution given by (3.9). The amplitude of plunge and pitch LCO, Ah and Aα, respectively, are given by
(3.27)Ah=ap1r2+p1i2,Aα=ap2r2+p2i2,
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 plunge motions 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.


## 3.1. Normal Form of Hopf Bifurcation

To compute the normal form of the Hopf bifurcation of (3.2) near U=Uf, we follow Nayfeh and 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 write U=Uf+ϵ2σUUf and seek a third-order approximate solution of (3.2) in the form
(3.3)Y(t,σU,σα,σh)=ϵY1(T0,T2)+ϵ2Y2(T0,T2)+ϵ3Y3(T0,T2)+⋯,
where the time scales Tm=ϵmt. In terms of these scales, the time derivative d/dt is written as
(3.4)ddt=∂∂T0+ϵ2∂∂T2+⋯=D0+ϵ2D2+⋯.
Scaling Lu as ϵ2Lu, substituting (3.3) and (3.4) into (3.2), and equating coefficients of like powers of ϵ, we obtainOrder (ϵ),
(3.5)D0Y1-A(Uf)Y1=0,Order (ϵ2),
(3.6)D0Y2-A(Uf)Y2=Q(Y1,Y1)+Qu(Y1,Y1),Order (ϵ3),
(3.7)D0Y3-A(Uf)Y3=-D2Y1+σUBY1+LuY1+2[Q(Y1,Y2)+Qu(Y1,Y2)]+C(Y1,Y1,Y1)+Cu(Y1,Y1,Y1),
where
(3.8)B=-2k1Uf2I1-2k2Uf2I2,I1=(0000000001000000),I2=(0000000000000100).
The general solution of (3.5) is the superposition of four linearly independent solutions corresponding to the four eigenvalues: two of these eigenvalues have negative real parts and the other two are purely imaginary (±iω). Because the two solutions corresponding to the two eigenvalues with negative real parts decay as T0→∞, we retain only the nondecaying solutions and express the general solution of the first-order problem as
(3.9)Y1(T0,T2)=η(T2)peiωT0+η¯(T2)p¯e-iωT0,
where η(T2) is determined by imposing the solvability condition at the third-order level and p is the eigenvector of A(Uf) corresponding to the eigenvalue iω; that is,
(3.10)A(Uf)p=iωp.
Substituting (3.9) into (3.6) yields
(3.11)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¯)]η¯2e-2iωT0.
The solution of (3.11) can be written as
(3.12)Y2=(ζ2+ζ2u)η2e2iωT0+2(ζ0+ζ0u)ηη¯+(ζ¯2+ζ¯2u)η¯2e-2iωT0,
where
(3.13)[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¯).
Substituting (3.9) and (3.12) into (3.7), we obtain
(3.14)D0Y3-A(Uf)Y3=-[+3Cu(p,p,p¯)+4Q(p,ζ0u)+2Q(p¯,ζ2u))η2η¯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,
where cc stands for the complex conjugate of the preceding terms and NST stands for terms that do not produce secular terms. We let q be the left eigenvector of A(Uf) corresponding to the eigenvalue iω; that is,
(3.15)A(Uf)Tq=iωq.
We normalize it so that qTp=1. Then, the solvability condition requires that terms proportional to eiωT0 in (3.14) be orthogonal to q. Imposing this condition, we obtain the following normal of the Hopf bifurcation:
(3.16)D2η=β̂η+Λ̂η2η¯,
where
(3.17)β̂=β+βu,
with
(3.18)β=qTσUBp,βu=qTLup,Λ̂=Λ+Λu,
with
(3.19)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).
Letting η=(1/2)aexp(iθ) and separating the real and imaginary parts in (3.16), we obtain the following alternate normal form of the Hopf bifurcation:
(3.20)ȧ=β̂ra+14Λ̂ra3,(3.21)θ̇=β̂i+14Λ̂ia2,
where (·)r and (·)i stand for the real and imaginary parts, respectively, a is the amplitude and θ̇ is the frequency of the oscillatory motion associated with the Hopf bifurcation.We note that, because thea component is independent of θ, the system's stability is reduced to a one-dimensional dynamical system given by (3.20). Assuming that Λ̂r≠0, a admits three steady-state solutions, namely,
(3.22)a=0,a=±-4β̂rΛ̂r.
The trivial fixed point of (3.20) corresponds to the fixed point (0,0) of (3.2), and a nontrivial fixed point (i.e., a≠0) of (3.20) corresponds to a periodic solution of (3.2). The origin is asymptotically stable when β̂r<0, unstable when β̂r>0, unstable when β̂r=0 and Λ̂r>0, and asymptotically stable when β̂r=0 and Λ̂r<0. On the other hand, the nontrivial fixed points exist when -β̂rΛ̂r>0. They are stable when β̂r>0 and Λ̂r<0 (supercritical Hopf bifurcation) and unstable when β̂r<0 and Λ̂r>0 (subcritical Hopf bifurcation). We note that 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 periodic solution of (3.2).Therefore, to delay the occurrence of Hopf bifurcation (i.e., stabilize the aeroelastic system at speeds higher than the flutter speed), one needs to set the real part ofβ̂ to a negative value by appropriately managing the linear control represented by Lu in (3.2). To eliminate subcritical instabilities and limit LCO amplitudes to small values at speeds higher than 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 so that 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; that is, kh1=kh2=cs=0, kα1=9.9967, and kα2=167.685. The hysteretic response as a function of the freestream velocity, obtained through the numerical integration of (2.6) for these parameters, is presented in Figure 3. The onset of flutter takes place at Uf=9.1242 m/s and is characterized by a jump to a large-amplitude LCO when transitioning through the Hopf bifurcation. As the speed is increased beyond the flutter speed, the LCO amplitudes of both of the pitch and plunge motions increase. Furthermore, LCO take place at speeds lower than Uf if the disturbances to the system are sufficiently large. Clearly, this configuration exhibits a subcritical instability (Real(Λ)>0).Hysteretic response of the aeroelastic system (subcritical instability). The steady-state amplitudes are plotted as a function ofU.
(a)
Plunge motion(b)
Pitch motionFor linear control, we consider the matrixLu defined in (3.2) in the form of
(3.23)Lu=(-kl000000000000000),
where kl is the linear feedback control gain. Then, for the specific values of the system parameters given in [9], we obtain
(3.24)β̂=(0.433891σUUf-0.110363kl)+ı(0.323942σUUf-0.303804kl).
To guarantee damped oscillations of the airfoil at speeds higher than the flutter speed, one needs to set kl to a value such that Real(β̂)<0. Using a gain of 10, we plot in Figure 4 the plunge and pitch displacements for a freestream velocity of U=10 m/s with and without linear control. Clearly, linear control damps the LCO of the uncontrolled system. We note that, by increasing the linear feedback control gain kl, the amplitudes of pitch and plunge decay more rapidly.Measured pitch and plunge responses::-, without linear control, *, with linear control.
(a)
Plunge motion(b)
Pitch motionAlthough linear control is capable of delaying the onset of flutter in terms of speed and reducing the LCO amplitude, the system would require higher gains at higher speeds. Furthermore, it maintains its subcritical response. To overcome these difficulties and convert the subcritical instability to a supercritical one, we introduce the following nonlinear feedback control law:(3.25)uT=-(knl1knl2knl3knl4)α̇3,
where the knli are the nonlinear feedback control gains. For the specific airfoil's geometry given in [9], we obtain
(3.26)Λ̂r=0.866899-(132.844knl1+13.3643knl2+2.9858knl3+1.32415knl4).
This equation shows that applying gain to the plunge displacement is more effective than applying it to the pitch displacement or plunge velocity or pitch velocity.The subcritical instability takes place for positive values ofΛ̂r. As such it can be eliminated by forcing Λ̂r to be negative. This can be achieved by using nonlinear control gains knl1=0.02  (knl2=knl3=knl4=0). The results are presented in Figure 5. The subcritical Hopf bifurcation at Uf observed in Figure 3 has been transformed into the supercritical Hopf bifurcation of Figure 5. A comparison of the two figures shows that the unstable limit cycles observed over the freestream speed between U=6.5 m/s and U=9.12 m/s have been eliminated. Furthermore, the exponentially growing oscillations predicted by the linear model are limited to a periodic solution whose amplitude increases slowly with increasing freestream velocity. Moreover, increasing the value of knl1 reduces the amplitude of the limit cycles created due to Hopf bifurcation.LCO amplitudes of plunge and pitch motions (controlled configuration):-, analytical prediction, *, numerical integration.
(a)
Plunge motion(b)
Pitch motionIn order to check the accuracy of the analytical formulation given by the normal form in predicting the amplitude of LCO near the Hopf bifurcation, we consider the first-order solution given by (3.9). The amplitude of plunge and pitch LCO, Ah and Aα, respectively, are given by
(3.27)Ah=ap1r2+p1i2,Aα=ap2r2+p2i2,
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 plunge motions 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 plunge motions. The method of multiple scales is applied to the governing system of equations to derive the normal form of the Hopf bifurcation near the flutter onset. The linear and nonlinear parameters of the normal form are used to determine the stability characteristics of the bifurcation and efficiency of the linear and nonlinear control components. The results show that linear control can be used to delay the flutter onset and dampen LCO. Yet, its required gains remain a function of the speed. On the other hand, nonlinear control can be efficiently implemented to convert subcritical to supercritical Hopf bifurcations and to significantly reduce LCO amplitudes.



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*Source: 782457-2010-02-01.xml*