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SUPERCRITICAL AND SUBCRITICAL BIFURCATION IN TYPICAL
AEROELASTIC SECTION WITH STRUCTURAL NONLINEARITIES
D. A. Pereira
F. D. Marques
Engineering School of São Carlos – University of São Paulo
daniel.almeida.pereira@usp.br, fmarques@sc.usp.br
Abstract. Aeroelasticity studies the mutual interaction between aerodynamic and structural dynamics loading. This
physical relationship often behaves nonlinearly, causing various problems such as instabilities, limit cycle oscillations,
bifurcations, and chaos that may affect aircraft operations. The objective of this work is to analyze and identify the
dynamic response of a three degree of freedom airfoil section with hardening nonlinearity in the pitching stiffness and free-
play nonlinearity in the control surface stiffness using the bifurcation analysis. The effect of hardening corresponds to a
smooth nonlinear increasing in restoring structural loading for a given displacement. This effect is modeled here by means
of the rational polynomial function while the free-play is represented by hyperbolic functions combination. Aeroelastic
responses are analyzed from computational and experimental results. The numerical model is based on the classical
theory for the linear unsteady aerodynamics with corrections for arbitrary motions coupled to a three degrees of freedom
typical aeroelastic section. The study of the phenomena associated with the hardening, free-play, and their intensity
variation effects provide ways to mitigate or circumvent any undesired responses to those behaviors. This contributes to
determining safety margins for aircraft even with these nonlinearities, as fully linear systems represent major challenges
to achieving, and results in very high costs.
Keywords: Aeroelasticity, subcritical bifurcation, supercritical bifurcation, hardening, free-play
1. INTRODUCTION
The aeroelasticity is a multidisciplinary field of the aerospace engineering that deals with the mutual interaction be-
tween structural dynamics and non-stationary aerodynamic flow (Bisphinghoff et al., 1996). Aeroelastic systems may
behave nonlinearly, therefore, subject to behavior such as bifurcations, limit cycle oscillations (LCO), and chaos (Sheta
et al., 2002). These phenomena, whose origins are both from structural dynamics and/or from the unsteady aerodynamic
loading, may be difficult to predict. The concentrated structural nonlinear effects can be incorporated into numerical mod-
els through the elastic restoring forces or moments representations. The most common types of concentrated nonlinearities
representations can be given by polynomial functions, nonlinear damping effects, free-play, hysteresis, etc.
The literature is vast and many authors have been studying nonlinear aeroelastic problems in the aviation, for example,
the limit cycle oscillations have caused persistent problems in many aircraft designs, such as the F-16, where Denegri and
Cutchins (1997) and Chen et al. (1998) have observed the existence of hardening nonlinearity in wings pitching moment
stiffness. O’Neil and Strganac (1998) have developed an experimental model that provides direct measurements from the
typical aeroelastic section with cubic nonlinearity in the pitch and plunge motion where they examined the sensitivity of
the response to system parameters. Recently, Vasconcellos et al. (2012, 2014) have shown that the hyperbolic tangent
function combination approach for modeling discontinuous nonlinearities is appropriate for detecting different nonlinear
behaviors, including the experimentally observed LCO, chaos and transitions.
It is known that the system behavior is directly related to the nonlinearities involved, for example, the system under
free-play shows subcritical behavior. Figure 1 represent the classical bifurcation diagram for the supercritical and subcrit-
ical behavior where the limit cycle amplitude is plotted versus some system parameter, for example, the flight speed. In
general, if the system depends on the initial condition and has different solution when the airspeed velocity is increased
and decreased near the nonlinear critical velocity, the bifurcation is called subcritical, i.e. the LCOs may also exist below
the flutter boundary (unstable LCO, dashed line). However, if the system are independent on the initial condition and
the system stability changes only after the critical flutter velocity, the bifurcation is called supercritical and the system
has only stable LCO (solid line) which sometimes is welcome because without it the LCO would instead be replaced by
catastrophic flutter leading to loss of the flight vehicle (Dowell and Tang, 2002; Dowell et al., 2003; Ghommem et al.,
2010).
Bifurcation analysis is used to indicate a quantitative and qualitative changes in the features of a system, such as the
number and type of solutions, under the variation of one or more parameters on which the considered system depends
on (Nayfeh and Balachandran, 1995). Abdelkefi et al. (2012) have analyzed changes in the system behavior of a typical
D. A. Pereira, F. D. Marques
Supercritical and Subcritical Bifurcation Analysis
LCO amplitude
LCO amplitude
bifurcation
onset bifurcation
onset
airspeed airspeed
SUPERCRITICAL SUBCRITICAL
Figure 1. Supercritical bifurcation diagram leading to only stable LCO (solid line) and subcritical bifurcation diagram
leading to both stable and unstable LCO (dotted line).
section presenting cubic structural nonlinearity in three degrees of freedom using bifurcation analyzes, they showed that
the stability of the system are related with the influence of the different nonlinear couplings.
This work presents an investigation on the influence of the control surface free-play on the bifurcation behavior of
the dynamics system when it presents a LCO induced by different hardening nonlinearity present in the pitch moment on
typical aeroelastic section. The experimental and computational analyses were performed for the three degree-of-freedom
airfoil and the numerical model is based on the classical theory for the unsteady aerodynamic with corrections for arbitrary
motion.
2. METHODOLOGY
2.1 Mathematical Model
The mathematical model for the typical section with three degrees of freedom and semi-chord b(cf. Fig. 2) is derived
Bisphinghoff et al. (1996); Theodorsen (1935), that is,
µexαxβ
xαrα2[rβ2+ (c−a)xβ]
xβ[rβ2+ (c−a)xβ]rβ2
¨
ξ(t)
¨α(t)
¨
β(t)
+
d1,1d1,2d1,3
d2,1d2,2d2,3
d3,1d3,2d3,3
˙
ξ(t)
˙α(t)
˙
β(t)
+
ωh20 0
0rα2ωα2F(α)
α(t)0
0 0 rβ2ωβ2F(β)
β(t)
ξ(t)
α(t)
β(t)
=
−L(t)/mWb
Mα(t)/mWb2
Mβ(t)/mWb2
,(1)
where the plunge, pitch and control surface displacements are given by h,α, and β, respectively, aand care the dimen-
sionless distances from the elastic axis to the middle and to the surface control, Uis the airspeed, xαand xβare the
dimensionless distances from elastic axis to the center of gravity (CG) and the distance between the control surface and its
CG, respectively, rαand rβare the dimensionless rotational inertia term about elastic axis and surface control hinge line,
respectively, kh,kα, and kβare the plunge, pitch and control surface stiffness, respectively, ξ(t) = h(t)
b,µe=mT
mW,di,j
are the damping factors with respect to each airfoil motion and their influences (Rayleigh approach), L(t)is the lift force,
Mα(t)and Mβ(t)are the aerodynamic pitch and hinge moments, respectively, F(α)and F(β)are functions related to
nonlinearities applied to the stiffness.
k
α"
k
β"
elastic axis
(nonlinear)
(nonlinear)
x
β
b
L(t)
"
α(
t
)"
h(t)
+"
β(
t
)"
+b
"
-b
"
M
α
(t)
M
β
(t)
U
Figure 2. Typical aeroelastic section.
Aerodynamics modeling is based on the formulation by Theodorsen (1935) that presents an analytical solution to the
pressure distribution around a typical airfoil section using 2-D irrotational, incompressible and potential flow and written
23rd ABCM International Congress of Mechanical Engineering (COBEM 2015)
December 6-11, 2015, Rio de Janeiro, RJ, Brazil
respectively as,
L(t) = LNC +LC,(2)
Mα(t) = MNC
α+MC
α,(3)
Mβ(t) = MNC
β+MC
β,(4)
where the non-circulatory terms (superscripts NC) are,
LNC =πρb2¨
h+U˙α−ba ¨α−U
πT4˙
β−b
πT1¨
β,(5)
MNC
α=πρb2ba¨
h−U b 1
2−a˙α−b21
8+a2¨α−U2
π(T4+T10)β+
Ub
π−T1+T8+ (c−a)T4−1
2T11˙
β+b2
π[T7+ (c−a)T1]¨
β,(6)
MNC
β=πρb2b
πT1¨
h+Ub
π2T9+T1−a−1
2T4˙α−2b2
πT13 ¨α−
U2
π2(T5−T4T10)β+U bT4T11
2π2˙
β+b2T3
π2¨
β,(7)
and the circulatory terms (superscripts C) are,
LC= 2πρU bC(k)f(t),(8)
MC
α= 2πρb2(a+1
2)C(k)f(t),(9)
MC
β=−ρUb2T12C(k)f(t),(10)
where C(k)is the so-called Theodorsen function and,
f(t) = Uα +˙
h+ ˙αb 1
2−a+U
πT10β+b
2πT11 ˙
β , (11)
is the quasi-steady term and the T-functions (Ti, for i= 1,2, . . .) are as defined in Theodorsen (1935).
The aerodynamic loading, as given from Eqs. (2) to (4), depends on the Theodorsen function C(k), where k=ωb
Uis
the reduced frequency of harmonic oscillations. However, to account for arbitrary motions an alternative is to manipulate
the Theodorsen function by convolution based on Duhamel formulation in the time domain (Li et al., 2010). If the Wagner
function is also considered, it implies that, for instance, circulatory lift term can be written as,
C(k)f(t) = f(0)φ(τ) + Zτ
0
∂f (σ)
∂σ φ(τ−σ)dσ , (12)
where σis a dummy integration variable, and φ(τ)is the Wagner function in the non-dimensional time τ=tu
b, approxi-
mated by Sear’s approach (Sears, 1940),
φ(τ)≈c0−c1e−c2τ−c3e−c4τ,(13)
where c0= 1.0,c1= 0.165,c2= 0.0455,c3= 0.335 and c4= 0.3.
Using integration by parts and following the state space method proposed by Lee et al. (1997, 2005); Li et al. (2010),
Eq. (12) leads to a circulatory term as,
C(k)f(t)=(c0−c1−c3)f(t) + c2c4(c1+c3)x+ (c1c2+c3c4)˙
x , (14)
where xand ˙
xare augmented aerodynamic states. Substituting the Eq. (14) in Eqs. (8), (9) and (10) we obtain the
circulatory aerodynamic loads for the three degrees-of-freedom.
Functions F(α)and F(β)in Eq. (1) are used to feed that aeroelastic set with the respective terms to account for
nonlinearities in pitch and control surface responses. Here the hardening and free-play effects are combined for that
D. A. Pereira, F. D. Marques
Supercritical and Subcritical Bifurcation Analysis
motions (cf. Fig. 3), being: (i) for pitching motion only hardening nonlinearity is considered, while, (ii) free-play is only
affecting the control surface motion.
The hardening nonlinearity in pitching depicted in Fig. 3(a) has been obtained using the rational polynomials appro-
ximation (RP) (Li et al., 2012) from the curve experimentally measured which is given by,
F(α) = a3α3+a2α2+a1α+a0
b2α2+b1α+b0
,(15)
where a0to a3, and b0to b2are real-valued coefficients obtained numerically from measured experimental data.
F(
α
)!
α
!
linear
hardening
representation
(polynomial)
(a) Hardening representation
F(
β
)!
β
!
+δ
!
δ
!
increasing
ε$
increasing
ε$
linear
(b) Free-play representation
Figure 3. Representations for hardening in pitching (rational polynomial approximation) and free-play in control surface
deflections (hyperbolic tangent functions combination).
The free-play nonlinearity combinations of hyperbolic tangent function have been used as proposed and validated by
Vasconcellos et al. (2014) to represent the restoring torque in the control surface. The mathematical formula for this
combination is given by:
F(β) = 1
2[1 −tanh(ε(β−βl))] (β−βl) + 1
2[1 −tanh(ε(β−βu))] (β−βu),(16)
where βland βudenotes the lower and the upper free-play boundary region (length of 2δ), and εis a variable which
affects the smoothness of the function, thereby determining the accuracy of the approximation (cf. Fig. 3(b)).
2.2 Experimental Model
The experimental apparatus comprises a rigid wing mounted along its span and it is supposed to correlate the bi-
dimensional behavior of typical aeroelastic sections. The system allows three degrees of freedom (plunge, pitch, and
control surface). The plunge motion is restrained by four elastic steel beams and the pitch stiffness is given by two tension
springs connected by a nonlinear cam (cf. Fig. 5), which is responsible for the hardening nonlinearity effect. Control
surface stiffness is provided by a piano wire and free-play controlled by a pair of adjustable bolts depicted in the Fig. 6.
The measurements of the three degrees of freedom are done using encoders (angular and linear ones). The DSpace R
system together with Simulink R
are used to signals acquisition and processing. The experimental apparatus is mounted
in an open-section wind tunnel (500 ×500mm) (cf. Fig. 4).
Figure 4. Experimental apparatus under preparation for wind tunnel tests.
23rd ABCM International Congress of Mechanical Engineering (COBEM 2015)
December 6-11, 2015, Rio de Janeiro, RJ, Brazil
(a) Nonlinear Cam – front view (b) Nonlinear Cam – side view
Figure 5. Details of the experimental apparatus responsible for the hardening nonlinearity.
(a) Surface control set-up (b) Free-play adjustment device
Figure 6. Details of the experimental apparatus responsible for the free-play nonlinearity.
3. RESULTS
The aeroelastic system (cf. Eq. (1)) is numerically integrated using the Runge-Kutta method. The Table 1 presents the
value under consideration in this work.
Table 1. Experimental values used for the numerical model.
VARIABLES DESCRIPTION VAL UES
b Mid-chord (m)0.125
a Distance from semichord to elastic axis (nondimensional) -0.5
c Hinge line location measured from mid-chord (nondimensional) 0.5
ρAir density (kg/m3)1.078
mWWing mass (kg)1.5
mTTotal mass (kg)4.3723
ωhDecoupled plunge natural frequency (rad/s)27.3268
ωαDecoupled pitch natural frequency (rad/s)12.11
ωβDecoupled control surface natural frequency (rad/s)50.2761
xαNondimensional distance between elastic axis and CG of wing 0.66
xβNondimensional distance between hinge line and CG of flap 0.0028
rαNondimensional rotational inertia term about elastic axis 0.7303
rβNondimensional rotational inertia term about hinge line 0.0742
µNondimensional mass ratio 28.3467
ζhPlunge modal damping ratio 0.1275
ζαPitch modal damping ratio 0.3697
ζβFlap modal damping ratio 0.0106
UfLinear flutter velocity (Numeric) (m/s)11.465
U∗
fCritical flutter velocity (Experiment) (m/s)12.0 ≤U∗
f≤12.20
Hopf bifurcation is a typical phenomenon in nonlinear aeroelastic systems, where limit cycle oscillations (LCOs)
manifest at a particular airflow velocity. This bifurcation onset can be either observed experimentally or simulated nume-
rically. Here, numerically predicted flutter velocity for linear structural behavior is taken as the reference to compare with
experimental and numerical results. The linear flutter velocity, Uf, is 11.465m/s and the experimental flutter velocity,
D. A. Pereira, F. D. Marques
Supercritical and Subcritical Bifurcation Analysis
U∗
f, is observed in the interval 12.0 < U∗
f<12.2m/s, thereby representing an error in between 4.45 to 6.0%. The results
are given in terms of normalized numerical and experimental velocities, which are obtained dividing these velocities by
Uf= 11.465m/s and U∗
f≈12.10m/s, respectively, but represented as a reference to critical velocity, Uc, which is
useful to compare both numerical and experimental results in the same plottings.
3.1 Hardening nonlinearity investigation
Simulations and experiments were carried out for three different intensities of the hardening nonlinearity in pitching
motion as illustrated in Fig. 7(a). The hardening effects vary from large, medium, and lower increasing in restoring struc-
tural loading for the same displacement represented by the rational approximation (cf. Eq. (15)), where the coefficients
are presented in the Tab. 2 .
Table 2. Rational polynomial coefficients for three different intensities of hardening nonlinearities.
Hardening a3a2a1a0b2b1b0
16.403 −4.76 ×10−31.26 ×10−1−3.03 ×10−41.0 2.61 ×10−76.37 ×10−2
26.313 −4.58 ×10−23.64 ×10−2−2.48 ×10−41.0 −1.06 ×10−22.54 ×10−2
37.281 3.01 ×10−21.33 ×10−2−1.44 ×10−41.0 6.39 ×10−31.91 ×10−2
−0.2 −0.1 0 0.1 0.2 0.3
−4
−3
−2
−1
0
1
2
3
4
Moment [N.m]
α [degrees]
Hardening 3 − Predicted
Hardening 2
Hardening 1
Hardening 3 − Measured
Hardening 2
Hardening 1
(a) Pitching moment stiffness
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
0
2
4
6
8
10
12
14
16
18
20
U/Uc
α amplitude [degrees]
Hardening 3 − Predicted
Hardening 2
Hardening 1
Hardening 3 − Measured
Hardening 2
Hardening 1
(b) Hopf bifurcation diagram
Figure 7. Numerical and experimental results for the aeroelastic system with three different intensities of hardening
nonlinearity in the pitch motion.
Figure 7(b) depict the computational and experimental results of pitching motion with no free-play in the control
surface for the three different hardening effects considering increasing wind tunnel airflow velocity. The results reveal a
smooth subcritical behavior, therefore, the system jumps from his fixed point stability to start LCO with high amplitude
before the air velocity reaches the critical flutter speed (U=Uc). In the numerical results, the LCO amplitude starts with
3.2◦,3.6◦,3.9◦for hardenings 3, 2, and 1, respectively, when U= 0.945Uc. For the experimental results, the bifurcation
onset is close to 0.970Uc, since the accuracy of the tunnel does not allow capturing the precise instability onset point.
Although the hardening nonlinearity has been responsible for the appearance of LCOs, it has not been responsible to
changes in supercritical to subcritical behavior, or vice-versa. Moreover, the linear flutter velocity of the system for all
LCOs starts and finishes at the same speed. The numerical results have shown good agreement with the experimental
counterparts in which is possible to observe the stronger nonlinearity giving smaller LCO amplitude when the airflow
velocity is increased and decreased.
3.2 Free-play nonlinearity investigation
Admitting hardening 3 nonlinearity in pitching, free-play is included to the control surface hinge. For the analysis
different free-play sizes (2δ) were considered, that is, 2◦and 4◦(cf. Fig. 3(b)). The Figure 8 shows the computational and
experimental results of the surface control LCO amplitudes to increasing of the airspeed. Subcritical bifurcations can be
observed in all cases, but different levels of LCO amplitudes are revealed as the free-play gaps changes.
Here, it is observed that the LCOs for the computational analysis starts in the U= 0.945Uc,U= 0.927Uc, and U=
0.917Uc, respectively, which demonstrates that the subcritical bifurcation also increase as free-play gap size increases.
The numerical results are in close agreement with the experimental data, whose results are U= 0.977Uc,U= 0.958Uc,
23rd ABCM International Congress of Mechanical Engineering (COBEM 2015)
December 6-11, 2015, Rio de Janeiro, RJ, Brazil
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
0
1
2
3
4
5
6
7
U/Uc
β amplitude [degrees]
2δ = 0o − Predicted
2δ = 2o
2δ = 4o
2δ = 0o − Measured
2δ = 2o
2δ = 4o
Figure 8. Hopf bifurcation diagram for the surface control displacement with hardening and free-play nonlinearity.
and U= 0.948Uc, respectively. Major discrepancies are observed from control surface deflections amplitudes in LCOs,
with larger differences in bifurcation progress after the jump effect. Although numerical simulations indicate the need for
adjustments on, perhaps, adding extra effects on free-play modeling, the results are considered satisfactory under those
conditions.
3.3 Supercritical and Subcritical bifurcations analysis
Although the changes in the aeroelastic system dynamics due to structural nonlinearities, it has been observed that the
linear behavior are responsible for determining whether subcritical or supercritical bifurcation occurs. To demonstrate
that phenomena, the pitch moment spring stiffness has its value altered arbitrarily by increasing leading to a new ωα=
17.1261rad/s (this corresponds to an increase of 100% in pitch stiffness). All other parameters have been left the same
as in Tab. 1. The Figure 9(a) presents the bifurcation diagram for the new aeroelastic system (solid line), which results
in a new critical flutter velocity of Uf= 10.3m/s. One can also observe a supercritical behavior, compared with the old
one (dashed line), where ωα= 12.11rad/s, for the hardening 3 and 1 present a subcritical behavior. The Figure 9(b)
shows the aeroelastic system (ωα= 17.1261rad/s) when the subcritical behavior is induced by the free-play nonlinearity
(2δ= 2◦), one can recognize that the free-play here induces a supercritical system to be a subcritical behavior as expected.
0.9 0.95 1 1.05 1.1
0
2
4
6
8
10
12
U/Uc
α amplitude [degrees]
Bifurcation onset
3
1
3
1
ωα = 17.1261 rad/s
ωα = 12.11 rad/s
(a) Subcritical to supercritical bifurcation
0.98 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
U/Uc
α amplitude [degrees]
Bifurcation onset
3
1
3
1
No free−play
Free−play (2δ= 2o)
(b) Supercritical to subcritical bifurcation
Figure 9. Computational results for the aeroelastic system when the new pitch stiffness is responsible to change the Hopf
bifurcation behavior.
4. CONCLUDING REMARKS
This paper has presented a study on the influence of structural nonlinearities in a three-degrees-of-freedom typical
aeroelastic section using the bifurcation analysis. The computational and experimental results have focused on examining
D. A. Pereira, F. D. Marques
Supercritical and Subcritical Bifurcation Analysis
the effect of hardening in pitching motion and free-play in the control surface hinge. As observed, the hardening nonlin-
earity does not change the subcritical or supercritical behavior of the system, but it handles the appearance and amplitude
control of LCOs. Otherwise, the free-play leads to the appearance of subcritical behavior and increase the region of dan-
gerous LCO. In addition to that, it has also been observed the influence of the linear parameters on airfoil dynamics that
determine the supercritical/subcritical behavior when the system presents only hardening nonlinearity in pitching.
Further analysis are planned to study the influence of the parameters of the system in its behavior, for instance, by
changing the position of the elastic axis, CG, the moment of inertia, etc. The idea is to try to understand how they are
related and how it is possible to avoid the subcritical behavior.
5. ACKNOWLEDGMENTS
The authors acknowledge the financial support of the CNPq (grant 305700/2013-8) and the Coordination for the Im-
provement of Higher Education Personnel (CAPES, grant 0011/07-0). The authors are also thankful to the Brazilian
Research Agencies, CNPq and FAPEMIG, for funding this present research work through the INCT-EIE (National Insti-
tute of Science and Technology in Smart Structures in Engineering).
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7. RESPONSIBILITY NOTICE
The authors are the only responsible for the printed material included in this paper.
