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

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

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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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simultaneous solution would produce both aerodynamic

response and structural motion histories, which depend on

the given initial conditions [8].

The problem in applying the general aeroelastic model

is mainly related to the unsteady aerodynamic model in

use. Solutions to the non-linear fluid mechanics equations

have been the focus of a great amount of research effort.

For practical applications, however, solutions of the gen-

eral fluid mechanics equations can usually be attained only

by means of numerical techniques, or computational fluid

dynamics (CFD) methods [9, 10], that normally demand

extensive computations. These methods encompass any

numerical technique for specific fluid mechanics applica-

tions. For instance, finite-difference, finite volume, and

finite element techniques are frequently used in numerical

solutions in fluid mechanics applications. Limitations of

CFD methods are basically the ones concerning the great

amount of computations required.

Alternative models of non-linear unsteady aerodynam-

ics for aeroelastic applications have been achieved on the

basis of some essential assumptions. Primarily, in aeroe-

lastic models, the decoupling between the fluid mechan-

ics equations and the equations of motion is an assump-

tion that eliminates the need for simultaneous solution of

the combined aero-structural set of equations. Therefore,

by this premise the unsteady aerodynamic model is deter-

mined in isolation of the physical laws governing the struc-

tural motion. An intrinsic element of this decoupling pro-

cess is that any alternative unsteady aerodynamic response

model should account for the spatio-temporal behavior of

the internal aerodynamic states.

Formal mathematical approaches to determine the

functional relationship of the hereditary behavior of un-

steady aerodynamic responses, are originally due to the

use of the superposition principle over transient responses

to step changes, namely, indicial responses [11, 12, 13].

This approach, which provides exact representation of the

linear unsteady aerodynamic behavior, is categorized as a

functional due to its dependence on the complete (or par-

tial) motion histories. However, practical use of the re-

sulting complex integral equations is only permitted by

simplifications, for example, by replacing the mathemat-

ical description of the motion history by its Taylor series

expansion, or by assuming a limited dependence on the

motion past values. Other functional forms; for instance,

the Volterra series [14, 15] also provide appropriate frame-

works to the production of non-linear unsteady aerody-

namic functionals.

Semi-empirical methods, or phenomenological mod-

els, comprise a class of aerodynamic models based on the

premise of modeling unsteady flow response by consid-

ering its functional relationship with respect to the mo-

tion histories. Indeed, most of the knowledge on unsteady

flow behavior is due to experimental work, and basically,

semi-empirical models use the information from these ex-

periments to establish a mathematical and logic formu-

lation of the events that determine the unsteady aerody-

namic response over a range of incidence motions and flow

regimes. The works by [16, 17, 18, 19, 20, 21, 22], are ex-

amples of contributions to semi-empirical modeling. The

nature of semi-empirical methods facilitates their incor-

poration into aeroelastic stability and control design. In

addition, semi-empirical methods have the advantage of

being computationally fast. Nevertheless, semi-empirical

models need extensive, specific and precise experimental

data. There is also the problem of correlating this data with

mathematical and logic formulations.

The Beddoes-Leishman model [17, 18, 19, 21] is a

very popular choice of dynamic stall model for helicopter

industry, as well as for most recent analyses in wind en-

ergy applications. Nonetheless, the Beddoes-Leishman

approach is also more complex compared to other semi-

empirical models. It is based originally based on con-

volution of indicial response functions because of more

effective computational formulation. To account for the

flow physics involved in the dynamic stall phenomenon,

this approach includes contributions to each stage when

important separated flow events occur. Original Beddoes-

Leishman model considers the effects of trailing edge sep-

aration, vortex flow (dynamic stall effect), and flow reat-

tachment when rebuilding lower angles of attack. Com-

pressibility effects are also incorporated to this approach

as it is significant to the helicopter blade unsteady loading,

however, for the most recent applications to wind turbine

blades, novel formulations have been developed neglect-

ing the Mach number influence [23, 24].

The purpose of this paper is to present an investigation

on the influence of structural nonlinearities to the aeroe-

lastic behavior due to stall-induced unsteady aerodynamic

loading. The investigation has been carried out using 1-doftypical section in pitching motion and dynamic stall model

based on Beddoes-Leishman approach [17, 18, 19, 21].

Concentrated nonlinearities in pitching stiffness were ad-

mitted from different smooth (polynomial-type) represen-

tations of various intensities and the freeplay. Nonlin-

ear analyses are performed by inspecting the time histo-

ries from simulated aeroelastic responses and the result-

ing LCOs for a range of nonlinearity diversities, which

includes the sensitivity to pitching frequency. Time and

frequency related analysis are performed comparing with

the linear structure cases.

2 AEROELASTIC MODEL

The aeroelastic model is based on typical section for

pitching motion only, admitting large angles of attack.

To take into account the effects of separated flow around

the airfoil the Beddoes-Leishman dynamic stall model is

used [17, 18, 19, 21]. Structural behavior are admitted as

non-linear with variables that ensure proper aeroelastic dy-

namics with realistic-valued parameters. Pitching stiffness

nonlinearities are taken in a range for cubic polynomials

of various intensities towards the freeplay case. More de-

tails on the aeroelastic model are presented in the follow-

ing sections.

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2.1 Aeroelastic equation for pitching airfoil

Pitching-only typical aeroelastic section of chord c is

adopted. Linear 1–dof structural dynamics depend on the

pitching stiffness and inertia moment around the elastic

axis. The Beddoes-Leishman model is used to calculate

the unsteady aerodynamic loading of the airfoil under sep-

arated flow effects at higher angles of attack.

Aeroelastic 1–dof equation of motion is given by,

Iαα+ kαF (α) = Mea , (1)

where α is the pitching angle (angle of attack), Iα is the in-

ertial moment, kα is the pitching spring stiffness, F (α) is

a function that represents the non-linear effect in pitching,

and Mea is the aerodynamic pitching moment (positive for

airfoil leading edge up).

A convenient way to represent the equation of mo-

tion can be with non-dimensional form. Therefore, for the

pitching natural frequency, ωα = (kα/Iα)12 , the radius of

gyration, r2α = 4Iαmc2 , and the mass ratio, μ = 4m

πρc2 , where

m is the airfoil mass for unit length, ρ is the air density, the

non-dimensional form of the aeroelastic equation results,

α+ ω2αF (α) =

4V 2

r2αμπc2Cmea

, (2)

where V is the airspeed and Cmeais the pitching moment

coefficient with respect to the elastic axis.

For the purpose of aeroelastic simulations, Eq. (2)

can be integrated in time with any conventional numeri-

cal method, but one must be careful for the case of F (α)as a piecewise function. The function F (α) in this work is

given by a particular way that enables to keep the Runge-

Kutta method for ODE integration. Details are presented

in Section 2.3.

2.2 Nonlinear aerodynamic model

This paper considers the original formulation of the

Beddoes-Leishman model [17, 18, 19, 21]. This model

deals with the prediction of dynamic stall, thereby ac-

counting for non-linear unsteady aerodynamic effects due

to separated flow fields surrounding airfoils. Original

Beddoes-Leishman model considers leading edge sepa-

ration and impulse forces for compressibility effects (al-

though the airspeed range under consideration for this

work will not reach Mach numbers higher than 0.4). The

Beddoes-Leishman model admits three contributions to

the total loading, namely: (i) unsteady attached flow load-

ing; (ii) trailing edge separation effect; and (iii) dynamic

stall or vortex flow effect. General aspects of each con-

tribution are presented in the following paragraphs. For

more detailed information on Beddoes-Leishman model

the reader must refer to [17, 18, 19, 21].

The unsteady attached loading is computed using

changes in aerodynamic forces with respect to a step

change in airfoil pitch angle or pitch rate, the so-called in-dicial aerodynamic responses. These indicial lift functions

can be expressed in terms of exponential functions in non-

dimensional time likewise Wagner function classical ap-

proximations [25, 26]. Therefore, lift transfer function can

be derived straight from the indicial function, which fa-

cilitates the numerical solution for arbitrary loading terms

using superposition assumption of a step inputs set. This

means that Duhamel’s integral formulation can be applied

and the indicial lift coefficient response is given by:

CL(s) = CLα(s)

[α(0)φ(s) +

∫ s

0

dα(σ)

dsφ (s− σ) dσ

],

(3)

where s = (2V∞t)/c represents the non-dimensional time

(relative distance travelled by the airfoil in terms of semi-

chords), α(0) is the angle of attack initial condition, and

φ(s) is the indicial response function.

The indicial response function can be written in a con-

venient way so that compressible and time-delay effects

are accounted into the model [18]. A simplified represen-

tation of this assumption is,

φ(s′) = φc(s′) + φI(s

′) + φq(s′) , (4)

where s′ = s(1 − M2) is the non-dimensional time

parametrized by the Mach number (M ), φc(s′) =

1 − A1e−b1s

′ − A2e−b2s

′is the circulatory compo-

nent (A1,2 and b1,2 are real-valued constants), φI(s′) =

(4/M)e−s′/TI is the impulsive component (TI is a time

constant – first-order system delay), and φq(s′) =

(−1/M)e−s′/Tq is the impulsive component of the indi-

cial lift response to pitch rate about 3/4 chord (Tq is re-

spective time constant).

Trailing edge separation effects are assessed with the

Kirchhoff theory [27]. This approach can be used to quan-

tify the associated loss of circulation with the progressive

trailing edge separated flow. Circulatory component of the

lift coefficient can be corrected with,

CL = 0.25CLαα(1 +

√f)2

, (5)

where f is the separation point with respect to the airfoil

chord, as function of airfoil incidence and empirical pa-

rameters.

The final contribution to the unsteady aerodynamic

loading is related to the effects of deep stall regime [28].

This highly non-linear dynamics is related to vortex shed-

ding intensified by trailing edge separation progress up to

the release of leading edge vortices. The physical events

associated to this flow regime are complex and an ac-

ceptable modeling requires precise understanding of them.

Several stages in the physical events related to dynamic

stall can be observed. Admitting progressive increase in

airfoil incidence and after reaching the static stall in the

lift curve (beyond the limits imposed by Kirchhoff theory),

separation delay effects start to dominate. The separation

delay leads to the continuation of linear increase in lift,

which goes beyond the static stall angle. Moreover, dur-

ing lift increment beyond static stall the unsteadiness of

the shed circulation at high angle of attack results in lift

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reduction and adverse pressure gradients. Because of ad-

verse pressure gradients, additional unsteady effects rise in

the form of attached flow reversal. All these events result

in a lag effect in the boundary layer that ultimately provide

conditions to the onset of leading edge separation.

Leading edge separation starts a vortex departing and

shedding toward trailing edge. Pressure gradient by this

convected vortex leads to lift overshoot, sometimes as sig-

nificant as 50 to 100% of the static maximum lift, and this

generated lift is also called stall or vortex lift. This large

contribution from the upstream moving vortex also gives

rise to nose-down pitching moment. Then, when the vor-

tex reaches the trailing edge the entire upper surface of the

airfoil is under separated flow.

As pitching moment reduces significantly at stall lift

condition the airfoil motion will be experience angle of

attach reduction (it may also be prescribe as in pitching

adjustments per revolution like in helicopter blades in for-

ward flight). When the angle of attack is low enough, flow

reattachment is possible. The reattachment occurs within

a large amount of dynamic lagging effects. The reasons

for that are due to flow reorganization from the fully sep-

arated regime, and by a reverse kinematic induced camber

effect on the leading-edge pressure gradient by the nega-

tive pitch rate. As result fully reattached flow around the

airfoil is only reached below the static stall angle, intro-

ducing a large amount of hysteresis to the unsteady loads.

The Beddoes-Leishman model uses a practical ap-

proach to account to the vortex shedding process. The

stall or vortex lift is modeled assuming the increment in

lift during vortex shedding in terms of the difference be-

tween instantaneous linear value of circulatory lift (CcL)

and the corresponding lift as given by the Kirchhoff the-

ory, that is,

CV = CcL

[1− (1 +

√f)2

4

], (6)

where f as in Eq. (5).

Simultaneously, the total vortex lift is allowed to decay

exponentially with time, but incremented in value. This is

considered so that when lower lift rate of change occurs

the vortex lift is rapidly dissipated. The lift exponential

decay is given as,

CvL(t) = Cc

L(t− 1)Ev + [CV (t)− CV (t− 1)]√

Ev ,(7)

where Ev = e(−c Δt

2TvV∞ ), for Tv denoting a time constant.

2.3 Nonlinear structural model

To represent nonlinear behavior in pitching degree of

freedom, concentrated stiffness is affected by a function

F (α) (cf. Eqs. (1) and (2)). Commonly, smooth or piece-

wise representations of structural restoring loadings versusdisplacements have been adopted to account for nonlinear

responses. Here F (α) is assumed to allow a variety of

possible behaviors. It is desirable to represent nonlineari-

ties having smooth changes and deviations from the linear

stiffness, typically taken with polynomial fitting. By vary-

ing the polynomial parameters one may reach different

intensities of nonlinear restoring loading, permitting rep-

resentations where dramatic loss in stiffness can happen.

The extreme case is the one when the restoring loading is

null for a range of displacement, the so-called, freeplay.

An alternative function representation to account for

concentrated nonlinearities from smooth to piecewise, al-

lowing freeplay, is used in this work. This is the case of

using combinations of hyperbolic tangent function to rep-

resent the restoring pitching moment due to α deflections

[29]. The mathematical formulation for this combination

is given by:

F (α) =1

2[1− tanh(ε(α− δ))] (α− δ) +

1

2[1 + tanh(ε(α− δ))] (α− δ) , (8)

where δ denotes the main influenced range for the smooth-

ing effect associated to the nonlinearity (the extreme case

is the freeplay, therefore δ would represent the boundary

edges), and ε is a real-valued parameter which affects the

smoothness of the function, thereby determining the accu-

racy of the approximation.

In Eq.(8), as ε value increases, the hyperbolic tangent

function combination becomes more representative of the

real freeplay effect, while goes from a range of typical

polynomial-like representations. This feature can be seen

in Figure 1 as obtained by using Eq. (8), and for examples

of hyperbolic tangent function combination for increasing

ε values. Clearly, as ε goes to infinity, the representation

for F (α) leads to the real freeplay discontinuous effect.

� � � � � � �����

����

����

����

����

�

����

����

����

����

����

F(α)�

α�+δ�

δ�

increasing ε�

increasing ε�

Figure 1: Pitch deflection versus restoring moment described in

Eq. (8), ε increasing from 0 to 1000 (solid line would be an acceptable

freeplay representation).

The advantage of Eq. (8) for this investigation is in ex-

ploring smooth nonlinearities up to freeplay with only one

formulation, by only changing few parameters of the func-

tion F (α), i.e. the value of ε and the range δ.

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3 RESULTS

The aeroelastic model admitted in this study presents

the following parameters: chord length, c = 0.3m; air

density, ρ = 1.225kg/m3; elastic axis position at 30%

of c; mass ratio, μ = 11.5486; reference pitching fre-

quency, ωα = 2Hz; and radius of gyration, rα =√0.5.

Moreover, NACA0012 airfoil is considered to adjust the

unsteady aerodynamic model.

For airspeed range of 0.0 to 70.0 m/s, simulations

have been performed to assess non-linear response since

limit cycle oscillations (LCO) are expected. To account

for nonlinear structural effects, F (α) given by Eq. (8) is

considered in the following conditions:

(i) linear structure, when F (α) = 1.0;

(ii) smooth structural nonlinearities values for δ and ε are

taken respectively as:

δ = 1.0◦ ⇒ ε = [22.5 28.125 35.156]T ,δ = 3.0◦ ⇒ ε = [7.5 9.375 11.719]T ,δ = 5.0◦ ⇒ ε = [4.5 5.625 7.0312]T ;(iii) freeplay nonlinearity for δ = [1.0 3.0 5.0]T (in de-

grees) and ε = 104.

Simulations are carried out adopting an initial condi-

tion in angle of attack (α0) followed by leaving the aeroe-

lastic system to react. An evolution with respect to air-

speed reveals the condition in which limit cycle oscilla-

tion occurs. As nonlinear behavior has been reached, it

is reasonable to check LCO in terms of its stability. A

straightforward way to verify stability can be by simulat-

ing the system with different initial conditions and how the

responses lead to LCOs. For example, this procedure has

been performed for a fixed airspeed (V = 51.1ms ), lin-

ear structure, and the resulting LCOs for α0 equals to 5,

20 and 45◦ are depicted in Fig. 2. The phase portrait of

pitching motion indicates that all α0 initial values results

in the same LCOs. Stability is guaranteed since LCO can

be reached from both sides (internally and externally). The

same procedure has been performed for other airspeeds

and admitting structural nonlinearities within the range of

LCO existence, and for all cases stability is observed.

−30 −20 −10 0 10 20 30 40 50−1500

−1000

−500

0

500

1000

1500

AoA (degree)

AoA

rat

io (

deg/

s)

α(0)=5degα(0)=20degα(0)=45degportrait for α(0)=5degportrait for α(0)=20degportrait for α(0)=45deg

Figure 2: Phase portraits for stable LCOs (three different α0, linear

structure, and V = 51.1ms

).

Figure 3 illustrates the LCO amplitude progress, i.e.bifurcation diagrams, with respect to airspeed. The plot-

tings encompass linear and nonlinear structural cases re-

vealing that the most important change to the aeroelastic

responses are related to the size of δ, or the size in inci-

dence angle where one gets lower pitch stiffness. These

LCOs after bifurcation point are all due to stall-induced

excitation, which restricts the motion at higher angles of

attack. An explanation for this condition can be related to

loading degeneration due to dynamic stall at higher angles

of attack and subsequent unsteady loading recovering due

to reattached flow effects when traveling in lower airfoil

incidence angles. If the range is small, as in δ = 1.0◦,

it is observed there is no significant change in bifurcation

and LCO amplitudes from linear to any other nonlinear

structural cases. Moreover, it can observed that bifurca-

tion occurs within the range of airspeeds around 30.0m/s,

and the first jump in LCO amplitude is significantly higher

than the following ones. From all these cases one can infer

that unsteady aerodynamic effects rise as a major influence

to the aeroelastic dynamics. Therefore, the influence of a

small range in nonlinear structural effect is not enough to

affect the major stall-induced LCO at higher incidence an-

gles.

For δ equals to 3.0◦ and 5.0◦, that is for larger region

of angle of attack where smooth or freeplay nonlinearities

dominates, it is clear that bifurcation phenomenon is antic-

ipated for airspeed around 20.0m/s. The same larger jump

in LCO amplitude at bifurcation airspeed is also observed.

For increasing airspeeds LCO amplitudes are higher than

the linear structure reference case. This demonstrates that

nonlinear effects are now influencing the airfoil aeroelastic

dynamics.

A peculiar case is observed in the δ = 3.0◦ plotting of

Fig. 3. It is the case of freeplay nonlinearity, where the bi-

furcation is observed around 30.0m/s without higher ini-

tial LCO amplitude. For this condition it is reasonable to

consider that smooth nonlinearities is a better environment

to promote bifurcation anticipation rather than the severe

case of discontinuous behavior of a freeplay.

10 20 30 40 50 60 700

5

10

15

LCO

am

plitu

de (

deg)

δ = 1.0 deglinearε1

ε2

ε3

freeplay

10 20 30 40 50 60 700

5

10

15

LCO

am

plitu

de (

deg)

δ = 3.0 deg

10 20 30 40 50 60 700

5

10

15

Aispeed (m/s)

LCO

am

plitu

de (

deg)

δ = 5.0 deg

Figure 3: Bifurcation diagrams: LCO amplitudes evolution to airspeed.

The existence of bifurcation leading to LCO response

allows inferring that exists a fundamental LCO frequency

with its respective harmonics. This complex frequency

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3157

distribution can be computed and observed from Fourier

analysis of the respective time series. As non-linear aero-

dynamics plays an important role on the frequency content

on LCO condition, it is reasonable that variations with re-

spect to airspeed can occur. To help understanding how

airspeed contributed in the frequency content and harmon-

ics distribution after bifurcation, time-frequency analysis

can be used. Figure 4 shows concatenated time history

windows of same size at different increasing airspeeds for

the reference case of airfoil with linear structure. Each

distinct time window in Fig. 4 has been taken with the fol-

lowing airspeeds: 3.4, 10.2, 17.0, 20.4, 23.8, 27.2, 30.6,

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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