characterization of grazing bifurcation in airfoils with control...
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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
3161
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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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
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).
References[1] A. Abdelkefi, R. Vasconcellos, A. H. Nayfeh, and M. R.
Hajj. An analytical and experimental investigation into
limit-cycle oscillations of an aeroelastic system. NonlinearDynamics, 71:159–173, 2012.
[2] B. Balachandran. Dynamics of an elastic structure excited
by harmonic and aharmonic impactor motions. Journal ofVibration and Control, 9:265–279, 2003.
[3] I. Chakraborty and B. Balachandran. Near-grazing dynam-
ics of base excited cantilevers with nonlinear tip interac-
tions. Nonlinear Dynamics, 70:1297–1310, 2012.
[4] W. Chin, E. Ott, H. E. Nusse, and C. Grebogi. Graz-
ing bifurcations in impact oscillators. Physical Review E,
50:4427–4444, 1994.
[5] M. D. Conner, D. M. Tang, E. H. Dowell, and L. N. Virgin.
Nonlinear behavior of a typical airfoil section with control
surface freeplay. Journal of Fluids and Structures, (11):89–
109, 1996.
[6] H. Dankowicz, X. Zhao, and S. Misra. Near-grazing in
tapping-mode atomic force microscopy. Int. J. Non-LinearMechanics, 42:697–709, 2007.
[7] J. de Weger, D. Binks, J. Molenaar, and W. de Water.
Generic behavior of gazing impact oscillators. Physical Re-view Letters, 76:3951–3954, 1996.
[8] M. di Bernardo, C. J. Budd, A. R. Champneys, and
P. Kowalczyk. Piecewise-smooth dynamical systems: the-
ory and applications. London: Springer-Verlag, London.(Applied mathematical science; 163), 2008.
[9] M. di Bernardo, C. J. Budd, A. R. Champneys, P. Kowal-
czyk, A. B. Nordmark, G. O. Tost, and P. T. Piiroinen. Bi-
furcations in nonsmooth dynamical systems. SIAM Review,
50:629–701, 2008.
5
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
3165
[10] M. di Bernardo, A. R. Champneys, E. M. Izhikevich,
B. Bronner, N. Orbeck, J. Bao, and Andrey Shilnikov. Two-
parameter discontinuity-induced bifurcations of limit cy-
cles: Classification and open problems. International Jour-nal of Bifurcation and Chaos, 16:601–629, 2006.
[11] A. J. Dick, B. Balachandran, H. Yabuno, K. Numatsu,
K. Hayashi, M. Kuroda, and K. Ashida. Utilizing nonlinear
phenomena to locate grazing in the constrained motion of a
cantilever beam. Nonlinear Dynamics, 57:335–349, 2009.
[12] M. Henon. On the numerical computation of poincare
maps. Physica D, (5):412–414, 1982.
[13] X. H. Long, G. Lin, and B. Balachandran. Grazing bifur-
cation in elastic structures excited by harmonic impactor
motions. Physica D, 237:1129–1138, 2008.
[14] Albert C. J. Luo. Discrete and Switching Dynamical Sys-tems. L& H Scientific Publishing, LLC, Glen Carbon,
USA, 1st edition, 2012.
[15] Albert C. J. Luo and Brandon C. Gegg. Dynamics
of a harmonically excited oscillator with dry-friction on
a sinusoidally time-varying, traveling surface. Interna-tional Journal of Bifurcation and Chaos, 16(12):3539–
3566, 2006.
[16] Albert C.J. Luo and Brandon C. Gegg. On the mechanism
of stick and nonstick, periodic motions in a periodically
forced, linear oscillator with dry friction. Journal of Vi-bration and Acoustics, 128(1):97 – 105, 2005.
[17] Albert C.J. Luo and Brandon C. Gegg. Stick and non-stick
periodic motions in periodically forced oscillators with dry
friction. Journal of Sound and Vibration, 291(1-2):132 –
168, 2006.
[18] J. Molenaar, J. G. de Weger, and W. de Water. Mappings
of grazing impact oscillators. Nonlinearity, 14:301–321,
2001.
[19] F. C. Moon and S. W. Shaw. Chaotic vibrations of a beam
with non-linear boundary condidtions. Int. J. Non-LinearMechanics, 18:465–477, 1983.
[20] A. B. Nordmark. Non-periodic motion caused by grazing
incidence in an impact oscillator. Journal of Sound andVibration, 145:279–297, 1991.
[21] S. W. Shaw. Forced vibrations of a beam with one-sided
amplitude constraint: Theory and experiment. Journal ofSound and Vibration, 99:199–212, 1985.
[22] S. W. Shaw and P. J. Holmes. A periodically forced piece-
wise linear oscillator. Journal of Sound and Vibration,
90:129–155, 1983.
[23] A. Stensson and A. B. Nordmark. Experimental investiga-
tion of some consequences of low velocity impacts in the
chaotic dynamics of a mechanical system. Philos. Trans.R. Soc. A, 347:439–448, 1994.
[24] T. Theodorsen. General theory of aerodynamic instabil-
ity and the mechanism of flutter. Technical Report 496,
NACA, 1935.
[25] T. Trickey, L. N. Virgin, and H. Dowell. The stability
of limit-cycle oscillations in an nonlinear aeroelastic sys-
tem. Proceedings: Mathematical, Physical and Engineer-ing Sciences, 458(2025):2203–2226, Sep. 2002.
[26] R. Vasconcellos, A. Abdelkefi, M.R. Hajj, and F.D. Mar-
ques. Grazing bifurcation in aeroelastic systems with
freeplay nonlinearity. Communications in Nonlinear Sci-ence and Numerical Simulation, 19(5):1611 – 1625, 2014.
[27] R. Vasconcellos, A Abdelkefi, F. D. Marques, and M. R.
Hajj. Representation and analysis of control surface
freeplay nonlinearity. Journal of Fluids and Structures,
31:79–91, 2012.
[28] L. N. Virgin and C. G. Begley. Grazing bifurcations and
basins of attraction in an impact-friction. Physica D,
130:43–57, 1999.
[29] L. N. Virgin, E. H. Dowell, and M. D. Conner. On the evo-
lution of deterministic non-periodic behavior of an airfoil.
Int. J. Nonlin. Mech., 34:499–514, 1999.
[30] G. S. Whiston. Global dynamics of a vibro-impacting linear
oscillator. Journal of Sound and Vibration, 118:395–424,
1987.
6
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
3166