Characterization of Grazing Bifurcation in Airfoils with ControlSurface Freeplay Nonlinearity

Rui M. G. Vasconcellos1 , Abdessattar Abdelkefi2 , Flavio D. Marques3 and Muhammad R. Hajj41 Sao Paulo State University (UNESP), Sao Joao da Boa Vista, SP, Brazil

2,4 Department of Engineering Science and Mechanics, MC 0219, Virginia Tech, Blacksburg, Virginia 24061, USA.3 Engineering School of Sao Carlos, University of Sao Paulo, Sao Carlos, SP, Brazil

ABSTRACT: A variety of nonlinear features is obtained from aeroelastic systems with discontinuous nonlinearity mo-

tivates investigations that may support future applications in controls design, flutter prediction problems, and energy

harvesting exploration. The freeplay nonlinearity leads to bifurcations and abrupt response changes which can result

in undesirable or catastrophic responses. Grazing bifurcations of limit cycles are one of the most commonly found

discontinuity-induced bifurcations (DIBs) and are caused by a limit cycle that becomes tangent to the discontinuity bound-

ary of the available piecewise-smooth function. The abrupt transition from periodic to aperiodic is directly related with the

discontinuous nature of freeplay nonlinearity. In fact, recent studies in different areas discussed the presence of grazing

bifurcations and the associated behavior changes. These abrupt transitions caused by grazing bifurcations are different

from the well-known routes to chaos. In this work, a nonlinear analysis based on modern methods of nonlinear dynamics,

such as power spectra and phase portraits is performed to characterize the sudden transitions in a three-degree of freedom

aeroelastic system with freeplay nonlinearity in the flap degree of freedom. The results show that the main transition is

due to a grazing bifurcation.

KEY WORDS: Aeroelasticity, freeplay, grazing bifurcation, nonlinear dynamics, piecewise-smooth systems.

1 Introduction

Discontinuous nonlinearities, such as freeplay, bilinear,

multi-segmented stiffness, and impacts are the most dan-

gerous types of nonlinear problems that aeroelastic sys-

tems can face. Its discontinuous characteristics can lead to

the occurrence of sudden transitions as a consequence of

DIB’s (Discontinuity-Induced Bifurcations) [10, 14, 9, 8],

causing aeroelastic systems to face transitions from Limit

Cycle Oscillations (LCOs) to chaos or from chaos to or-

der in a frequently unpredictable way, sometimes with

high amplitude variations, leading to possible catastrophic

structural damages.

Characterizing and understanding these undesirable

behaviors has been the topic of many investigations. Vir-

gin et al. [29], Conner et al. [5], Trickey et al. [25],

and Vasconcellos et al. [27] have evaluated numerically

and experimentally the effects of a freeplay nonlinearity

in the flap degree of freedom on the response of an aeroe-

lastic system. They showed that transitions from damped

to periodic LCOs to quasi-periodic responses, and then, to

chaotic motions can occur. In these works, the behavior

and evolution were investigated, but the mechanism lead-

ing to the observed abrupt transitions was not discussed in

details.

Grazing bifurcations of limit cycles are one of the most

common discontinuity-induced bifurcations (DIBs)[10,

14, 9, 8]. This type of bifurcations is caused when a

periodic orbit reaches a boundary tangentially and, as

such, can occur only in discontinuous systems. Many re-

searchers identified grazing bifurcations in different elas-

tic structures undergoing impacts [19, 22, 21, 30, 20, 4].

For this bifurcation, a special phenomenon arises during

zero-velocity incidence which is refereed to ”grazing con-

tacts”. Grazing bifurcations have also been found in struc-

tural systems, such as spring-mass systems [22, 20, 23, 4,

28, 18, 6] and cantilever beams [19, 21, 7, 13, 11, 3]. Luo

and Brandon [16, 17, 15] presented an extensive investiga-

tion on sliding and grazing bifurcations in forced oscilla-

tors with dry friction. In these and other studies, the role of

different parameters and excitations sources, such as low-

velocity impacts [23], friction and hard impacts [4], har-

monic and aharmonic impacts [2], and off-resonance exci-

tations [11] in the generation of grazing bifurcations were

investigated. Recently, Vasconcellos et al. [26] investi-

gated the effects of a freeplay nonlinearity on the response

of a two-degree of freedom aeroelastic system. The nons-

mooth freeplay is associated with the pitch degree of free-

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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

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dom. They reported that there are two sudden jumps in the

aerelastic response when varying the freestream velocity.

They demonstrated that these transitions are accompanied

with the appearance and disappearance of quadratic non-

linearity induced by discontinuity. They also showed that

these sudden transitions are associated with a tangential

contact between the trajectory and the freeplay disconti-

nuity boundary which is a characteristic of a grazing bi-

furcation.

In this work, a nonlinear analysis is performed to

deeply investigate the behavior of a three-degree of free-

dom aeroelastic system with a freeplay nonlinearity in the

flap degree of freedom. The governing equations of the

considered aeroelastic system are presented in Section 2

and the nonsmooth function representing the freeplay non-

linearity in the flap degree of freedom are presented in Sec-

tion 3. In Section 4, the aerodynamic loads are modeled

based on the unsteady formulation. Nonlinear analyses re-

sults are presented in Section 5 . Summary and conclu-

sions are presented in Section 6.

2 Governing Equations

The aeroelastic system considered here and shown in Fig-

ure 1 consists of a two-dimensional airfoil that has three

degrees of freedom including pitch, plunge and control

surface motions. The plunge and pitch motions are de-

noted by w and α, respectively, and the control surface

motion is denoted by β. The plunge and the pitch are mea-

sured at the elastic axis and the β angle of control surface

is measured at the hinge line. The distance from the elastic

axis to midchord is represented by ab where a is a constant

and b is the semichord length of the entire airfoil section.

The distance between the elastic axis and the hinge line of

control surface is represented by c. The mass center of the

entire airfoil is located at a distance xα from the elastic

axis and the mass center of the control surface is located

at a distance xβ from the hinge line, kw and kα are used to

represent the plunge and pitch stiffnesses, respectively and

kβ is used to represent the stiffness of the control surface

hinge. Finally, U is used to denote the freestream velocity.

Figure 1: Structural representation of the aeroelastic model

Using Lagrange’s equation, the equations of motion of

the typical airfoil section as considered above are written

as [27].

MsX+BsX+KsX = Ae (1)

where

Ms =

⎡⎣

r2α r2β + (c− a)xβ xα

r2β + (c− a)xβ r2β xβ

xα xβ MT /mW

⎤⎦

Bs = (ΛT)−1

⎡⎣

2mαωαξα 0 00 2mβωβξβ 00 0 2mwωwξw

⎤⎦Λ−1

Ks =

⎡⎣

r2αω2α 0 0

0 r2βω2βF (β)/β 0

0 0 ω2w

⎤⎦

X =

⎡⎣

αβw

⎤⎦

and

Ae =

⎡⎣

Mα

Mβ

L

⎤⎦

where

xα = Sα

mW b ; xβ =Sβ

mW b ; r2α = IαmW b2 ; r2β =

IβmW b2 ;

and mW is the mass of the wing, mT is the mass of the

entire system (wing + support blocks), dh, dα and dβ are

damping coefficients for the plunge, pitch, flap motions,

respectively, Iα is the airfoil mass moment of inertia about

the elastic axis, Iβ is the control surface mass moment of

inertia about the elastic axis, L and Mα are the aerody-

namic lift and moment measured about the elastic axis,

respectively, Mβ is the aerodynamic moment on the flap

about the flap hinge, Sα and Sβ are the static moments of

the wing mass, and, finally, F (β) is a function used to rep-

resent the control surface freeplay nonlinearity. Detailed

formulation can be found in [27].

3 Control Surface Freeplay Repre-sentations

The freeplay nonlinearity is considered in the the discon-tinuous representation, F (β), presented in Figure 2 andgiven by:

F (β) =

⎧⎨⎩

β + δ , if β < −δ ,0 , if | β |≤ δ ,β − δ , if β > δ .

(2)

To integrate the equations of motion with discontinuous

freeplay representation, the method described by Henon

[12] is used to locate and integrate at the discontinuity.

This method is usually known as the technique of inverse

interpolation and is well described by Conner et al.[5].

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−4 −3 −2 −1 0 1 2 3 4−2

−1

0

1

2

β (deg)

F(β

) (N

.m)

2δ

Figure 2: Flap displacement β versus torque obtained with Eq. (2)

4 Aerodynamic LoadsThe Theodorsen approach is used to model the aerody-namic lift and moments L, Mα and Mβ . In this approach,the unsteady aerodynamic forces and moments are calcu-lated using the linearized thin airfoil theory and written as[24]

L = −πρb2

[w + Uα − baα − U

πT4β − b

πT1β

]−2πρUbQC(k) (3)

Mα = πρb2

[baw − Ub

(1

2− a

)α − b

2

(1

8+ a

2

)α

−U2

π(T4 + T10)β+

Ub

π

{−T1 + T8 + (c − a)T4 − 1

2T11}β +

b2

π{T7 + (c − a)T1

}β

]

+2πρb2(a +

1

2)QC(k)

(4)

Mβ = πρb2

[b

πT1w +

Ub

π

{2T9 + T1 − (a − 1

2)T4

}α

− 2b2

πT13α −

(U

π

)2

(T5 − T4T10)β

+Ub

2π2T4T11β +

(b

π

)2

T3β

]− ρUb

2T12QC(k)

(5)

where

Q = Uα + w + αb

(1

2− a

)+

U

πT10β +

b

2πT11β (6)

and the T functions can be found in Vasconcellos et al.

[27].

The aerodynamic loads are dependent on

Theodorsen’s function C(k), where k is the reduced fre-

quency of harmonic oscillation. In the quasi-steady ap-

proximation, C(k) is set equal to one. To simulate the

arbitrary motion of the system, the loads associated with

Theodorsen’s function are replaced by the Duhamel for-

mulation in the time domain. More details for the deriva-

tion of the aerodynamic loads based on the Duhamel for-

mulation can be found in Abdelkefi et al. [1].

5 Results and DiscussionsThe values of the structural parameters of the aeroelastic

system considered in this work are similar to the used ones

in the experiments of Trickey et al. [25] and presented in

Table 1.

Table 1: Concentrated typical section parameters of the aeroelastic wing.

Wing span (m) 0.5

b(m) 0.125

a -0.5

c 0.5

ρp(kg/m3) 1.1

mw(kg) 1.716

mT (kg) 3.53

r2α(kgm2) 0.684

r2β (kgm2) 0.01795

ωα(rad/s) 48.7963

ωβ (rad/s) 233.0452

ωh(rad/s) 41.5211

xα 0.3294

xβ 0.01795

The bifurcation analysis is performed when the

freeplay is set to δ = ±2deg for a decreasing freestream

velocity U , from near flutter (20m/s) to near damped mo-

tion (4m/s) at a small step of -0.1m/s at each 6 seconds

to reach stationary state, in a region where LCO, chaos and

significant transitions happens. As the freeplay nonlinear-

ity is in the flap degree of freedom of the considered wing,

the focus of all analyses in this paper will be on the motion

of the flap component.

In Figure 3, a bifurcation diagram is presented for

the flap motion (β) time series. It follows from this plot

that there is a sudden transition from periodic LCO to

aperiodic one at a freestream velocity almost equal to

10m/s and other transitions between aperiodic/order and

order/aperiodic happens at lower freestream velocities. As

the system is structurally and aerodynamically coupled,

this behavior will contaminate the entire wing behavior,

as observed in the pitch and plunge bifurcation diagrams

in Figures 4 and 5, respectively.

20 15 10 5−3

−2

−1

0

1

2

3

U(m/s)

β(deg)

Figure 3: Bifurcation diagram of Flap displacement β for decreasing velocity.

20 15 10 5−2

−1

0

1

2

U(m/s)

α(deg)

Figure 4: Bifurcation diagram of Pitch displacement α for decreasing velocity.

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20 15 10 5−0.03

−0.02

−0.01

0

0.01

0.02

0.03

U(m/s)

w/b

(m)

Figure 5: Bifurcation diagram of Plunge displacement w for decreasing velocity.

In order to characterize the behavior and search for

the phenomena inducing the observed sudden transitions,

a grazing contact detection is performed over the entire

flap time series with a velocity variation. As the grazing

contact is a tangential contact on the discontinuity bound-

aries, the condition (β = 0;β = ±δ) is sought in the

decreasing-velocity β time series. The resulting bifurca-

tion diagram is then presented in Figure 6. It follows from

this plot that, as the freestream velocity decreases, the be-

ginning of grazing contacts (black squares) corresponds to

the major transition from LCO to aperiodic motion. Fur-

thermore, the control surface behavior remains aperiodic

while grazing contacts are occurring.

20 15 10 5−3

−2

−1

0

1

2

3

U(m/s)

β(deg)

Figure 6: Grazing contacts for decreasing velocity (black squares).

To better characterize these associated phenomena, we

present, respectively, in Figures 7, 8, and 9 the time his-

tory, power spectrum, and phase portrait of the flap de-

gree of freedom when the freestream velocity is set equal

to 10m/s (before the transition). As observed in these

plotted curves, the flap behaves periodically. In fact, the

power spectrum shows the main frequency and its cubic

harmonics and the trajectory is crossing the discontinuity

boundaries (dashed lines in Figure 9). As expected and

observed in Figure 6, no grazing contacts are detected at

this freestream velocity.

0 1 2 3 4 5 6−3

−2

−1

0

1

2

3

time(s)

β(deg)

Figure 7: Time series of Flap displacement β at 10m/s.

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

βMagnitude

frequency(Hz)

Figure 8: Power spectrum of Flap time series at 10m/s.

−3 −2 −1 0 1 2 3−150

−100

−50

0

50

100

150

β(deg)

β(deg/s)

Figure 9: Phase portrait of Flap at 10m/s.

At a lower freestream velocity, just after the main tran-

sition (U=9.5m/s), same analysis is performed, as shown

in Figures 10, 11, and 12. At this velocity, in contrast

with the previous case (10m/s), grazing contacts are de-

tected. Inspecting Figures 10, 11, and 12, we can con-

clude that the motion is aperiodic as observed in the broad-

band power spectrum in Figure 11. The abrupt transi-

tion to this new behavior coincides with the appearance

of grazing contacts in the right side of phase portrait, as

shown in Figure 12. In other words, the observed aperi-

odic behavior at freestream velocities lower than 10m/s is

a consequence of a grazing bifurcation, since the flap has

not sufficient energy to cross the discontinuity boundaries

all times.

0 1 2 3 4 5 6−3

−2

−1

0

1

2

3

time(s)

β(deg)

Figure 10: Time series of Flap displacement β at 9.5m/s.

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

βMagnitude

frequency(Hz)

Figure 11: Power spectrum of Flap time series at 9.5m/s.

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3164

−3 −2 −1 0 1 2 3−150

−100

−50

0

50

100

150

β(deg)

β(deg/s)

Figure 12: Phase portrait of Flap at 9.5m/s with grazing contact (black square).

As the freestream velocity is decreased to lower val-

ues, different aperiodic behaviors occur and the motion

of flap seems to become more complicated, as shown in

Figure 6. These aperiodic-aperiodic transitions are proba-

bly related with the decrease of available energy to cross

the discontinuity boundaries, due to the decrease in the

freestream velocity.

We plot in Figures 13, 14 and 15 the time history,

power spectrum, and phase portrait when the freestream

velocity is set equal to 8.0m/s. It follows from these plots

that the flap motion remains aperiodic. It is noted that

grazing contacts are also detected in the left side of the

phase portrait, as shown in Figure 15. The results con-

firms that the main transition and the subsequent behavior

at lower freestream velocities are a consequence of grazing

contacts of control surface.

0 1 2 3 4 5 6−3

−2

−1

0

1

2

3

time(s)

β(deg)

Figure 13: Time series of Flap displacement β at 8.0m/s.

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

βMagnitude

frequency(Hz)

Figure 14: Power spectrum of Flap time series at 8.0m/s.

−3 −2 −1 0 1 2 3−150

−100

−50

0

50

100

150

β(deg)

β(deg/s)

Figure 15: Phase portrait of Flap at 8.0m/s with grazing contact (black square).

6 ConclusionsA numerical analysis of nonlinear responses of a three-

degree of freedom aeroelastic typical section with freeplay

nonlinearity in the flap degree of freedom has been per-

formed in order to identify the mechanisms leading to the

observed abrupt transitions. A decreasing freestream ve-

locity sweep from near flutter to near damped motions was

performed and a grazing bifurcation detection condition

was tested over the velocity range.

The results show that the main abrupt transition be-

tween 10m/s and 9.5m/s matches with the beginning of

grazing contacts of control surface angle with the bound-

aries of freeplay discontinuity. The observed behavior af-

ter the main transition, at lower velocities, are dramatically

influenced by grazing contacts, being aperiodic, with an

increasing in the complexity as the freestream velocity is

decreased.

7 AcknowledgmentThe authors acknowledge the financial support of CNPq

(grant 303314/2010-9) and FAPESP (grants 2012/00325-

4 and 2012/08459-1).

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