Influence of structural nonlinearities in stall-induced aeroelastic response of pitching airfoils Daniel A. Pereira 1 , Rui M. G. Vasconcellos 2 , and Fl´ avio D. Marques 1 1 Engineering School of S˜ ao Carlos, University of S˜ ao Paulo, S˜ ao Carlos, SP, Brazil 2 S˜ ao Paulo State University (UNESP), S˜ ao Jo˜ ao da Boa Vista, SP, Brazil ABSTRACT: Stall-induced vibrations are a relevant aeroelastic problem for very flexible aero-structures. Helicopter blades, wind turbines, or other rotating components are severely inflicted to vibrate in stall condition during each revolu- tion of its rotor. Despite a significant effort to model the aerodynamics associated to the stall phenomena, non-linear aeroe- lastic behavior prediction and analysis in such flow regime remain formidable challenges. Another source of nonlinearity with influence to aeroelastic response may be associated to structural dynamics. The combination of both separated flow aerodynamic and structural nonlinearities lead to complex dynamics, for instance, bifurcations and chaos. The purpose of this work is to present the analysis of stall-induced vibrations of an airfoil in pitching when concentrated nonlinearities are associated to its structural dynamics. Limit cycles oscillations at higher angles of attach and complex non-linear features are analyzed for different nonlinear models for concentrated restoring pitching moment. The pitching-only typical sec- tion dynamics is coupled with an unsteady aerodynamic model based on Beddoes-Leishmann semi-empirical approach to produce the proper framework for gathering time series of aeroelastic responses. The analyses are performed by checking the content of the aeroelastic responses prior and after limit cycle oscillations occur. Evolutions on limit cycles ampli- tudes are used to reveal bifurcation points, thereby providing important information to assess, characterize, and qualify the nonlinear behavior associated with combinations of different forms to represent concentrated pitching spring of the typical section. KEY WORDS: Aeroelasticity, stall-induced vibrations, dynamic stall, nonlinear dynamics, nonlinear vibrations. 1 INTRODUCTION Aeroelastic problems related to stall-induced vibra- tions represent great challenge in modeling and analysis. These problems may lead to highly non-linear phenom- ena, when the unsteady aerodynamics gives a major con- tribution to the aeroelastic system complexity. Helicopter industry is always aware of the complex effects of stall- induced vibrations, since the helicopter blades are con- stantly subjected to the effect of dynamic stall per rotor revolution, particularly in forward flight [1, 2, 3]. In wing energy industry, modern blade design (slender shapes) and pitching control approaches may induce severe blade reac- tions at dynamic stall regime [4, 5]. Non-linear effects are difficult to predict or model, whatever the dynamic system in question. Aeroelastic sys- tems are influenced by non-linear behavior from structural dynamics and/or aerodynamics loading. Structural non- linearities may be related to the effect of aging, loose at- tachments, certain material features, and large motions or deformations. They can be subdivided into distributed and concentrated ones. Distributed nonlinearities are spread over the entire structure representing the characteristic of materials and large motions, for example [6]. Concen- trated nonlinearities act locally, representing loose of at- tachments, worn hinges of control surfaces, aging, and presence of external stores [7]. The concentrated nonlin- earities can usually be approximated by one of the classi- cal structural nonlinearities, namely, cubic, free-play and hysteresis, or by a combination of these. For unsteady aerodynamic modeling, the non-linear flow effects of interest are mostly due to separated flows and compressibility effects leading to the appearance and dynamic excursion of shock waves. Their modeling is particularly difficult because of the lack of complete un- derstanding on some physical aspects of unsteady flows; for example, separation and turbulence mechanisms. For aeroelastic applications, the ideal and, perhaps, most gen- eral aero-structural model would be based on solutions of the non-linear fluid mechanics equations, which considers unsteady, compressibility and viscous effects, simultane- ously with the solution of the equations of motion. The in- stantaneous states, which are generated by each of the cor- responding equations, would be exchanged and the global Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Porto, Portugal, 30 June - 2 July 2014 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 2311-9020; ISBN: 978-972-752-165-4 3153
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Influence of structural nonlinearities in stall-induced aeroelasticresponse of pitching airfoils
Daniel A. Pereira1 , Rui M. G. Vasconcellos2 , and Flavio D. Marques11 Engineering School of Sao Carlos, University of Sao Paulo, Sao Carlos, SP, Brazil
2 Sao Paulo State University (UNESP), Sao Joao da Boa Vista, SP, Brazil
ABSTRACT: Stall-induced vibrations are a relevant aeroelastic problem for very flexible aero-structures. Helicopter
blades, wind turbines, or other rotating components are severely inflicted to vibrate in stall condition during each revolu-
tion of its rotor. Despite a significant effort to model the aerodynamics associated to the stall phenomena, non-linear aeroe-
lastic behavior prediction and analysis in such flow regime remain formidable challenges. Another source of nonlinearity
with influence to aeroelastic response may be associated to structural dynamics. The combination of both separated flow
aerodynamic and structural nonlinearities lead to complex dynamics, for instance, bifurcations and chaos. The purpose of
this work is to present the analysis of stall-induced vibrations of an airfoil in pitching when concentrated nonlinearities are
associated to its structural dynamics. Limit cycles oscillations at higher angles of attach and complex non-linear features
are analyzed for different nonlinear models for concentrated restoring pitching moment. The pitching-only typical sec-
tion dynamics is coupled with an unsteady aerodynamic model based on Beddoes-Leishmann semi-empirical approach to
produce the proper framework for gathering time series of aeroelastic responses. The analyses are performed by checking
the content of the aeroelastic responses prior and after limit cycle oscillations occur. Evolutions on limit cycles ampli-
tudes are used to reveal bifurcation points, thereby providing important information to assess, characterize, and qualify
the nonlinear behavior associated with combinations of different forms to represent concentrated pitching spring of the
34.0, 51.1, and 68.1 m/s, respectively. As transient re-
sponses are neglected, thus each time window is related
to steady state dynamic responses. Figure 4 also shows a
spectrogram [30] closely related to the time windows per
airspeeds. For each time window the frequency content
can be observed in the spectrogram. It is clear to ob-
serve the harmonic coupling for the LCO conditions af-
ter bifurcation. Moreover, harmonic frequencies are also
affected by increasing airspeed, when there is a trend of
increasing the fundamental LCO frequency. This results
demonstrates the complex feature of LCO responses of
stall-induced aeroelastic vibrations.
0 5 10 15 20 25 30 35 40
0
5
10
15
Time (s)
α(t)
(de
gree
)
Time
Fre
quen
cy (
Hz)
5 10 15 20 25 30 350
0.5
1
1.5
2
2.5
3
Figure 4: Concatenated time windows of increasing airspeeds and
spectrogram (linear structure).
Results have shown that stall-induced responses lead
to bifurcation and LCO phenomena. To survey on the
features of those responses, Figs. 5 and 6 present time-
histories and phase portraits for the case of aeroelastic sys-
tems with smooth structural nonlinearities in comparison
with linear structure.
Figure 5 depicts LCO responses for the airfoil with
linear structure and for intermediary ε per δ values. For
linear structure, the aeroelastic response in LCO can be
seen, where at V = 34.05m/s bifurcation phenomenon
onset is associate to LCO amplitude of ≈ 10.2◦. As air-
speed increases, LCO amplitude changes to ≈ 4.7◦ at
V = 40.85m/s. Remaining plots are related to LCO at bi-
furcation onset and for higher airspeed related to smooth
structural nonlinearities respectively for δ equals to 1.0,
3.0, and 5.0◦ (ε = 28.125, 9.375, and 5.625). All these
LCO responses have a corresponding point in Fig. 3; the
first at bifurcation point and another at the subsequent air-
speed.
Variation in the range of smoothness of structural non-
linearity clearly affects the aeroelastic dynamics in stall-
induced LCO. At bifurcation onset, it was observed that
δ value is responsible to anticipate this phenomenon (cf.Fig. 3) when it equals 3.0 and 5.0◦. However, even for
δ = 1.0◦ it is observed that the aeroelastic system fre-
quency content is already changed to a higher value, de-
spite the bifurcation onset was kept basically the same as
for linear structure counterpart. Figure 5 also illustrates
how aeroelastic time histories become more complex as δincreases, for instance, for δ = 5.0◦ one can observe a
secondary frequency coupling at stall condition (LCO am-
plitude is ≈ 9.0◦). For all cases, airspeeds higher than
that of bifurcation onset reveals the same LCO aspect and
amplitudes of ≈ 5◦.
0 2 4 65
10
15
20V = 34.05m/s; linear structure
α (
deg)
0 0.5 1 1.5 25
10
15
20V = 40.85m/s; linear structure
0 2 4 65
10
15
20
α (
deg)
V = 34.05m/s; nonlinear (δ = 1.0 deg)
0 0.5 1 1.5 25
10
15
20V = 40.85m/s; nonlinear (δ = 1.0 deg)
0 2 4 65
10
15
20
α (
deg)
V = 23.83m/s; nonlinear (δ = 3.0 deg)
0 0.5 1 1.5 25
10
15
20V = 34.05m/s; nonlinear (δ = 3.0 deg)
0 2 4 65
10
15
20
time (s)
α (
deg)
V = 23.83m/s; nonlinear (δ = 5.0 deg)
0 0.5 1 1.5 25
10
15
20
time (s)
V = 34.05m/s; nonlinear (δ = 5.0 deg)
Figure 5: Typical LCO responses for the airfoil with linear stiffness
and smooth nonlinearities at bifurcation airspeed and higher
(intermediary ε).
5 10 15 20−150
−100
−50
0
50
dα /
dt (
deg/
s)
Linear structure
5 10 15 20−150
−100
−50
0
50Smooth nonlinear, δ = 1.0 deg
5 10 15 20−80
−60
−40
−20
0
20
40
α (deg)
dα /
dt (
deg/
s)
Smooth nonlinear, δ = 3.0 deg
5 10 15 20−100
−50
0
50
α (deg)
Smooth nonlinear, δ = 5.0 deg
Figure 6: Portraits for the airfoil with linear stiffness and smooth
nonlinearities at bifurcation airspeed and higher (intermediary ε):
black – at bifurcation onset; red – at higher airspeed.
Figure 6 relates the time histories in Fig. 5 to their re-
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
3158
spective phase portraits. These plots help to verify how
complex are the aeroelastic dynamics behind changes in
nonlinearity representations. An interesting dynamics can
be observed for smooth nonlinearity with δ = 5.0◦ at the
bifurcation onset (V = 23.83m/s). The portrait for this
case clearly demonstrates the complex frequency content.
For the case of aeroelastic responses with freeplay
nonlinearity, comparing with linear structure case, Fig. 7
shows the respective phase portraits. In these cases at
the bifurcation speed and higher values, it is observed
that freeplay gap does not represent substantial influence
on changing the portrait orbits shape. Time history plots
have been suppressed from this paper because they present
mostly the same aspect as the linear structure one. The
only observation is regarding the case when freeplay gap
is δ = 3.0◦, in which bifurcation have been assessed at
V = 34.05m/s and LCO amplitudes are ≈ 5◦. It is possi-
ble that the bifurcation onset for this case occur before the
aforementioned airspeed, with the same oscillatory motion
as for the other conditions. To verify this condition, more
simulations can be carried out at airspeeds within the range
in consideration (around 20 to 35m/s).
5 10 15 20−150
−100
−50
0
50
dα /
dt (
deg/
s)
Linear structure
5 10 15 20−150
−100
−50
0
50
δ = 1.0 deg
8 10 12 14 16−100
−50
0
50
100
α (deg)
dα /
dt (
deg/
s)
δ = 3.0 deg
5 10 15 20−80
−60
−40
−20
0
20
40
60
α (deg)
δ = 5.0 deg
Figure 7: Portraits for the airfoil with linear stiffness and freeplay
nonlinearity at bifurcation airspeed and higher:
black – at bifurcation onset; red – at higher airspeed.
Bifurcation phenomenon and LCO occurrence due to
stall-induced vibrations have been investigated for fixed
typical section structural parameters. The study so far has
been based in changing the characteristics of the structural
nonlinearity in pitching. This has been sufficient to show
that bifurcation onset is influenced, as well as LCO fea-
tures. A further investigation has also been carried out, be-
ing now presented the influence of varying the airfoil ref-
erence pitch frequency ωα (cf. Eq. 2). Admitting the lin-
ear structure as reference, LCO amplitude variation with
respect to airspeed is evaluated for different range of ωα.
Pitching frequency determines the system stiffness, there-
fore, the higher is that value the higher must be the aerody-
namic energy involved to maintain stall-induced LCO. In
fact, it is reasonable to infer that there is a ωα value where
no LCO occurs at airspeeds assumed in this paper (i.e., 0.0
to 70.0m/s). Based on this consideration, aeroelastic case
where smooth structural nonlinearity defined by δ = 1.0◦
and ε = 35.156 is assumed to explore changes in ωα.
10 20 30 40 50 60 700
2
4
6
8
10
12
LCO
am
plitu
de (
deg)
Linear structure
1.0 Hz2.0 Hz3.5 Hz4.6 Hz
10 20 30 40 50 60 700
2
4
6
8
10
12
Aispeed (m/s)
LCO
am
plitu
de (
deg)
Smooth nonlinear structure (δ = 1.0 deg)
1.0 Hz2.0 Hz3.5 Hz4.0 Hz4.93 Hz
Figure 8: Bifurcation diagrams for variations of airfoil reference pitch
frequency.
Figure 8 depicts LCO evolutions for a range of airfoil
reference pitching frequencies. From the case of a linear
structure it is clear that as ωα increases, bifurcation mani-
fests later in terms of airspeeds. Simulations demonstrate
that for ωα > 4.6Hz LCO are suppressed, which is phys-
ically comprehensive. The case when nonlinearities are
admitted to the pitch stiffness, bifurcation is anticipated
for ωα higher than 2.0Hz. Moreover, the LCO suppres-
sion occurs at higher ωα comparing to the linear structure
case.
4 CONCLUSIONS
An analysis of non-linear aeroelastic responses for
stall-induced oscillations of an airfoil in pitching moment
considering structural nonlinearities is presented. The
aeroelastic model is based on linear 1–dof structural dy-
namics in pitching coupled with a dynamic stall aerody-
namic model, thereby allowing realistic higher angles of
attack motions in low speed flow fields. The dynamic stall
model is given by Beddoes-Leishman semi-empirical ap-
proach [21] and NACA0012 parameters. Nonlinear struc-
tural behavior is accounted with smooth (polynomial-type)
representation based on combination of hyperbolic tangent
functions. The approach also permits freeplay representa-
tion with the same function representation.
The modeling has been effective to capture the stall-
induced loading fluctuations responsible to lead the aeroe-
lastic system to high angle of attack LCO. LCO has mani-
fested itself after a system bifurcation from stable equilib-
rium condition when linear structure is considered at ap-
proximately 30m/s. When structural nonlinearity is con-
sidered, bifurcation occurrence is clearly anticipated for
higher range of smoothing effect associated to the nonlin-
earity. The same is valid for the freeplay case, that is, the
higher is the gap, bifurcation happens in lower airspeeds.
LCO responses inspection also reveals complex frequency
couplings as the smoothness of the nonlinearity is higher.
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
3159
For freeplay case, such complexity seems to be more dis-
crete. All LCO range in the aeroelastic system responses
have also demonstrated to be stable ones. When inspected
in terms of varying pitch frequency it has been observed
that bifurcation onset is sensitive. For increasing ωα, bi-
furcation occurs at higher airspeeds, since the airfoil sus-
pension becomes stiffer. Nonlinear structure also leads to
higher ωα prior to LCO suppression.
Further investigation will consider expanded param-
eter analysis of aero-structural parameters, as well as to
modify the typical section model for traditional pitch and
plunge motion.
AcknowledgementsThe authors acknowledge the financial support of
CNPq (grant 303314/2010-9) and FAPESP (grants
2012/00325-4 and 2012/08459-1). They are also thank-
ful to CAPES, CNPq, and FAPEMIG for funding this
present research work through the INCT–EIE (CNPq
574001/2008-5).
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