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Sarkar2006.PDF
European Conference on Computational Fluid Dynamics
ECCOMAS CFD 2006
P. Wesseling, E. Oñate and J. Périaux (Eds)
c TU Delft, The Netherlands, 2006
°
NONLINEAR AEROELASTIC STUDY OF STALL INDUCED
OSCILLATION IN A SYMMETRIC AIRFOIL(ECCOMAS
CFD 2006)
Sunetra Sarkar∗ and Hester Bijl†
∗ Delft
University of Technology, Aerospace Engineering
Kluyverweg 2, 2629 HT, The Netherlands
e-mail: [email protected]
web page: http://www.lr.tudelft.nl/aerodynamics
† Delft University of Technology, Faculty L&R,
Kluyverweg 1, 2629 HS Delft, The Netherlands
e-mail: [email protected]
Key words: dynamic stall, aeroelastic stability, nonlinear behavior
Abstract. In this paper the aeroelastic stability of a wind turbine rotor in the dynamic
stall regime is investigated. Increased flexibility of modern turbine blades makes them more
susceptible to aeroelastic instabilities. Complex oscillation modes like flap/lead-lag are of
particular concern, which give way to potential structural damage. We study the stall induced oscillations in pitching direction and in combined flapwise-leadlag wise directions.
The aerodynamic loads acting on the rotor body in the stall regime are nonlinear. We consider a wide ranging parametric variation and investigate their effect on the aeroelastic instability and overall nonlinear dynamical response of the system. An engineering dynamic
stall model (Onera) has been used to calculate the aerodynamic loads. The aerodynamic
loads are given in terms of differential equations which are combined with the governing
equations of the aeroelastic system; the resulting system of equations are solved by a 4th
order Runge-Kutta method. In the pitching oscillation study we consider the following
parameters: nondimensional airspeed, mean angle of attack, initial condition, structural
nonlinearity and reduced frequency and amplitude of external forcing. Quasi-periodic and
chaotic response have been observed. The second case of flap/edgewise oscillation in the
stall regime identifies nondimensional rotational speed of the rotor along with structural
stiffnesses and nonlinearity as most important parameters of the self excited system. However, no chaotic response has been obtained. External forcing shows presence of higher
harmonics and quasi-harmonics in the response.
1
INTRODUCTION
Aeroelastic stability remains an important concern for the design of wind turbine
blades, more so with the use of increasingly flexible blades. This also brings complex
1
Sunetra Sarkar and Hester Bijl
oscillation modes like edgewise vibration in the limelight. Chaviaropoulos 1 reported recent cases of structural damage to modern wind turbine blades, in particular, occurrence
of longitudinal cracks which resulted from severe edgewise vibration. This has typically
occurred in stall regulated blades of large size. Wind turbine rotors often have to operate at large angles of attack, in the dynamic stall regime. The resulting flow is largely
separated and viscous effects are important. The physical process involves growth and
evolution of leading edge vortex structures and their subsequent shedding from the body
into the near wake. This largely controls the aerodynamic loads on the airfoil. The flow
field involves flow transition and large turbulent regions as well. All these effects make
the aerodynamic load prediction during a stall flutter problem much more involved than
its traditional bending-torsion counterpart.
The main objective of any computational aeroelastic analysis is to define the instability
boundary and identifying the system parameters affecting it. Nonlinearity could play an
important part, for it not only could influence the stability but also lead to bifurcations
in the dynamical response and chaos. Dunn and Dugundji 5 have presented an analysis and experimental validation of aeroelastic instabilities at the nonlinear dynamic stall
regime for a cantilever plate-like wing structure. Tang and Dowell 3;4 have studied flutter and forced response of a helicopter rotor in bending-torsion mode, using a nonlinear
aerodynamical model and considering structural nonlinearities. The dynamic stall model
used in the analysis is based on a semi-empirical technique called the Onera dynamic
stall model 10;11 . Lee et al. 6 have presented a review of nonlinear aeroelastic studies focusing on the pitch-plunge oscillation of an airfoil. They have discussed problems with
both structural and aerodynamic nonlinearities. Price and Fragiskatos 9 have presented
a nonlinear stall flutter analysis of a symmetric airfoil, using a nonlinear dynamic stall
model by Beddoes-Leishman. In their study, the effect of structural nonlinearity was not
considered. Chaviaropoulos et al. 1;2 have presented a study on stall induced flap and edgewise oscillation in a stall regulated rotor. Nonlinear aerodynamic loads in the dynamic
stall regime have been calculated by a quasi-steady Onera model as well as Navier-Stokes
solvers. Effects of structural nonlinearity have not been studied.
The present study is complementary to the earlier works discussed above. It considers
the problems of stall flutter in two different structural models; a single degree-of-freedom
problem of a pitching airfoil and a two-degree-of-freedom problem of flap-edgewise oscillation in the stall regime. The first model is the simplest stall flutter case representing
a predominant torsional oscillation. The second oscillation problem is encountered in
stall regulated rotors operating at large angles of attack. The present work investigates
the effect of structural nonlinearity and also initial conditions on the aeroelastic stability
of the above systems. Effects of structural nonlinearity on the stability of the response
have been found to be significant. Changing the initial conditions in the self excited system reveals different quasi-periodic routes. Influence of other system parameters has also
been included which helps in a systematic understanding of the stall flutter phenomenon
in the above aeroelastic models. The aerodynamic loads have been calculated by the
2
Sunetra Sarkar and Hester Bijl
semi-empirical Onera dynamic stall model, also been used in some earlier works. This
dynamic stall model compares well with the experimental results and is computationally
much cheaper than a Navier-Stokes solver.
2
EQUATIONS OF MOTION
To derive the equations of motion of a two dimensional blade section, we consider a
strip of unit span of a symmetric airfoil. A NACA 0012 profile has been selected. Two
oscillation cases have been considered: one is an airfoil oscillating in pitch degree-offreedom, the second is an airfoil oscillating in the directions of its flap and edge. The
first case represents the classical stall flutter case of an airfoil which is predominantly a
single degree-of-freedom problem where the torsional motion of the blade prevails. The
second case is obtained in stall regulated rotors showing sustained oscillation in the flap
and edgewise directions.
The equation of motion for the single degree-of-freedom pitching oscillation is given in
nondimensional form as follows 7;8 ,
α00 + α/(U 2 ) + K̄nl = 2Cm /(πµrα2 ) + F0 sin(k1 τ ).
(1)
Here, (0 ) is denoted as derivative with respect to nondimensional time τ = tV̄ /b; b is the
semi-span; V̄ is the relative wind velocity; Cm is the moment coefficient which is calculated
using the Onera dynamic stall model; U is the nondimensional airspeed defined as U =
V̄ /bωα ; µ and rα are nondimensional structural parameters; mass ratio µ = m/(πρb2 ); radius of gyration rα = Iα /(mb2 ); F0 and K̄nl are nondimensional forcing moment amplitude
and structural nonlinear stiffness respectively; k1 is the reduced frequency of oscillation.
The equations of motion for the combined flap/lead-lag flutter case for a stall regulated
rotor have been derived in the nondimensional form as follows 1 :
"
1 0
0 1
#(
ȳ 00
z̄ 00
)
"
+ k2
ω̄y2
0
0 1 + ω̄z2
#(
ȳ
z̄
)
=
1
2πµ
(
+
(
CD cos α − CL sin α
CD sin α + CL cos α
0
F̄0 sin(k1 τ )
)
)
(2)
Here, ȳ and z̄ are nondimensional edge and flapwise displacements respectively; k is
the nondimensional rotor speed Ωb/V̄ ; r is the radial distance of the blade section from
rotor root; b is the semi-span of the airfoil; nondimensional stiffness terms are defined as,
ω̄y = ωy /Ω and ω̄z = ωz /Ω.
3
ONERA DYNAMIC STALL MODEL
The aerodynamic loads are computed by the Onera dynamic stall model for a Reynolds
number Re > 106 . The elastic axis is at the quarter chord point. For both flutter
cases, structural damping has been neglected. Previous work suggests that the influence
3
Sunetra Sarkar and Hester Bijl
of structural damping is insignificant compared to aerodynamic damping below flutter
boundary 8 .
z(flap)
pitch
11111111111111
00000000000000
00000000000000
11111111111111
00000000000000
11111111111111
α
V
y(lead−lag )
U
Ωr
Figure 1: Airfoil coordinate system and oscillation degrees-of-freedom.
The Onera model is constructed in the form of differential equations to model the dynamic stall process. The physical process of dynamic stall involves leading and trailing
edge vortex development, their separation and shedding into the wake. Vortex growth
increases the aerodynamic loads beyond their stall boundary and separation causes a decline in the loads. The loading behavior is dependent on the frequency or rate of airfoil
movement. The aerodynamic loads by the Onera model are divided into two equations;
equation for the inviscid part, modeled with a single phase lag term to capture the aerodynamic lag due to the formation of vortex structure; the other is for the viscous part
which becomes important above the stall angle. The details of the technique are discussed elsewhere 5;10;11 . The coefficients of the inviscid part equation are obtained from
steady state experimental results and are dependent on the airfoil profile. However, the
viscous part of the equation uses parameters from unsteady experiments; the coefficient
values, therefore, are dependent on dynamic parameters like frequency. The coefficients
associated with the appropriate force coefficients, determined empirically by parameter
identification. The coefficients used in this study have been taken from the earlier works
of Dunn & Dugundji 5 for a NACA 0012 airfoil.
The differential equations of the Onera model are combined with the governing equations of the structural system. The resulting system of equations are solved in the time
domain using a numerical integration scheme of 4th order Runge-Kutta technique. The
nondimensional time step size used in the study is 0.01 and chosen based on the frequency
of oscillation. The time step size has been finalized after checking for convergence.
4
Sunetra Sarkar and Hester Bijl
4
PITCHING OSCILLATION
We assume a symmetric NACA 0012 section to study both flutter cases. The influence
of the relevant parameters for individual cases are discussed in the following subsections.
The aerodynamic loads are computed by the Onera dynamic stall model. The elastic axis
is chosen at the quarter chord point. For both flutter cases, structural damping has been
neglected. Previous work suggests that the influence of structural damping is insignificant
compared to aerodynamic damping below flutter boundary 8 .
The equation of motion for this case has been given in Eq. (1). Previous studies by
Price & Fragiskatos 8 have not addressed parameters like initial conditions and structural
nonlinearities which could play significant role in the nonlinear aeroelastic system. These
issues are studied in the present work. Multiple quasi-periodic routes have been found for
the self excited system with different initial conditions. Geometric structural nonlinearity
has also been found to modify the bifurcation pattern of the system response.
4.1
Self Excited System: Effect of Mean Angle Of Attack, Geometric Nonlinearity
In the self excited system, U has been considered as the main bifurcation parameter.
F0 is put to zero. First, effect of different mean angles of attack are considered, with
structural nonlinearity zero. Initial perturbations are given around the mean angle. The
resulting response for a mean angle of 4o is plotted in Fig. 2 for different U . The y-axis
of the plot gives the pitch response at the maximum and minimum points, that is, where
the derivative of the response is zero. The initial perturbation for α is 10o . The response
goes to a limit cycle oscillation (LCO) around U = 16. Below this U , the response is
damped. Therefore, from a fixed point, the response becomes a fixed orbit, indicating a
supercritical Hopf bifurcation. Next we consider a different mean angle of attack of 2o .
Critical U at which LCO is obtained is 22 as shown in Fig. 3. Comparing this with the
case of 4o mean angle of attack, we surmise that a smaller initial αm takes a higher U to
reach the LCO. It is also apparent that, as U increases, the mean angle of incidence αm
about which the oscillation occurs or dies down, increases. LCO occurs only when αm
reaches past the stall angle, at a critical value of U (Ucr ). Next, a geometric structural
nonlinearity is added to the system in order to see the effect on the dynamic response.
A cubic nonlinear term of 0.05α3 is introduced into Eq. 1 to this effect. In this case, the
LCO occurs even later at U = 28, much beyond the previous two cases. This is shown in
Fig. 4. Thus, cubic structural nonlinearity has made the Ucr value much higher than its
linear counterpart. This could be attributed to the extra stiffness added to the system.
4.2
Self Excited System: Influence Of Initial Condition
Next we present some more bifurcation routes of the self excited system. A mean angle
of incidence around 0.5o is chosen. Initial angular perturbation is 20◦ , much larger than
the earlier cases. U is varied as the bifurcation parameter. A single periodic response goes
5
Sunetra Sarkar and Hester Bijl
16
14
αo at α’ = 0
12
10
8
6
4
2
5
10
15
20
25
30
U
Figure 2: Bifurcation plot,αm = 4o ,αinit = 10◦ .
14
12
αo at α’ = 0
10
8
6
4
2
0
5
10
15
20
25
30
U
Figure 3: Bifurcation plot for αm = 2o ,αinit = 10◦ .
to period-2 at around U = 6 which leads to aperiodic response at U = 9.7, after following
a series of period doubling bifurcations. Fig. 5 shows the aperiodic pattern obtained at
U = 9.7. In this plot, the attractor in the phase plane looks similar to a quasi-periodic
one; the frequency content shown in Fig. 6 reveals incommensurate behavior. When a
Poincare section is taken, the Poincare plot shows a closed loop. Also, simulating with a
slightly different initial condition does not make the solution diverge. Hence, we conclude
that the response is indeed quasi-periodic and not chaotic.
Next, we consider the mean angle of attack to be 5.5o , again a similar quasi-periodic
route is observed. However, the quasi-periodic response is obtained at a much lower U
6
Sunetra Sarkar and Hester Bijl
14
12
αo at α’ = 0
10
8
6
4
2
0
5
10
15
20
U
25
30
35
Figure 4: Bifurcation plot, αm = 4o and K̄nl = 0.05α3 ,αinit = 10◦ .
6
α ’(degrees/nd time
4
2
0
−2
−4
−6
−40
−30
−20
−10
αo
0
10
20
30
Figure 5: αm = 0.5 and U = 9.7,αinit = 20◦ , phase plot.
value and with a larger angular perturbation of 25o . For this case, LCO appears around
U = 3; period doubling is observed at U = 4 and quasi-periodic response is seen at
U = 5.1. Though we obtain similar bifurcation plots for both the mean angles of attack
cases discussed above, the difference between them is that they occur at different ranges
of U . The bifurcation pattern and the quasi-periodic route for αm = 0.5◦ is presented
in Fig. 7. Now we add a geometric nonlinearity and consider αm = 0.5o case again. A
cubic spring stiffness of 0.01α3 defers the quasi-periodic response slightly to around U =
10.1 from 9.7 of the first case of αm = 0.5o . Otherwise, the bifurcation pattern remains
the same. However, a larger cubic stiffness of 0.1α3 significantly modifies the response,
7
Sunetra Sarkar and Hester Bijl
20
Psd
15
10
5
0
0.05
0.1
0.15
0.2
0.25
0.3
nondimensional frequency
0.35
Figure 6: αm = 0.5 and U = 9.7,αinit = 20◦ ,frequency content.
30
20
o
α , when α′ =0
10
0
−10
−20
−30
−40
2
3
4
5
6
7
8
9
10
U
Figure 7: Bifurcation plot, αm = 0.5o ,αinit = 20◦ ,K̄nl =0.
as shown in Fig. 8. For this case, period-2 response appears at U value around 9 while
no further period doubling is obtained within the range of U up to 30. No chaotic or
quasi-periodic response is found within this range.
4.3
Forced Oscillation
A harmonic forcing couple of the form F0 sin(k1 τ ) is considered. First, we choose,
F0 = 0.002, U = 15, k1 = 0.2,Knl = 0. A chaotic response is observed. We follow the
chaotic path backward by decreasing F0 and the resulting bifurcation plot is presented in
Fig. 9. The system shows period doubling bifurcation routes to chaos. Like the previous
bifurcation diagrams, this one also presents the α values at the extrema points, that is
8
Sunetra Sarkar and Hester Bijl
30
20
αo, when α′ =0
10
0
−10
−20
−30
−40
5
10
15
20
25
30
U
Figure 8: Bifurcation plot,αm = 0.5o ,αinit = 20◦ ,K̄nl = 0.1α3 .
when α is maximum and minimum during one oscillation cycle. The movement of both
the extrema points are captured in the bifurcation plane. Next, we vary the oscillation
frequency k1 for F0 = 0.002 and U = 15. The bifurcation diagram is presented in Fig. 10.
This case shows an interesting behavior, a period three response. We have varied k1
between 0.2 and 0.4 with an interval of 0.001. The response starts with chaos and remains
so at the lower values of k1 . As k1 increases to 0.22, a period two response is clearly visible.
These again becomes chaotic through a period doubling cascade at k1 = 0.286. At k1 =
0.295, the chaotic response gives way to a period three response. The response remains
so for the rest of the k1 range.
20
15
αo, when α′ =0
10
5
0
−5
−10
−15
−20
0
0.5
1
F0
1.5
2
−3
x 10
Figure 9: F0 as a bifurcation parameter, U = 15, k1 = 0.2, αm = 5.5o .
9
Sunetra Sarkar and Hester Bijl
20
o
α , max amplitude
15
10
5
0
−5
−10
0.2
0.25
0.3
k1
0.35
0.4
Figure 10: k1 as a bifurcation parameter, U = 15, F0 = 0.002, αm = 5.5o .
5
FLAP-EDGEWISE OSCILLATION
The combined flap and edgewise oscillation case in stall regulated rotors has not received much attention in the nonlinear aeroelastic community. Nevertheless, such oscillations could potentially lead to structural damage 1 . Chaviaropoulos 1;2 has presented a
linear stability analysis for such a system, along with time domain analysis results for a
self excited system with different nondimensional rotor speeds. No other parameters have
been investigated. The present study considers the influence of the following parameters:
nondimensional rotor speed k, linear structural stiffnesses and their ratio; effects of initial
conditions; geometric structural nonlinearity. Finally a forced response study is presented.
The system shows interesting dynamics like super-harmonic and quasi-harmonic response
at different forcing frequencies. However, no chaotic route has been found.
1
The equation of motion is given in the nondimensional form in Eq. (2). We assume,
πµ
= 0.02 and pitch angle θ = 18◦ , which is beyond the stall angle of attack. These values are
kept constant throughout the simulations. The total angle of attack α is obtained at each
time instant by modifying θ for the instantaneous flapping movement. The structural
stiffnesses are, ω̄y = 7, ω̄z = 4 and Reynolds number is assumed to be Re > 106 , with
incompressible flow conditions.
It has been shown by Chaviaropoulos et al. 2 that aeroelastic instability occurs at low
nondimensional rotor speeds (k = Ωb/V̄ ). The results were predicted using an eigenvalue
based linear stability model as well as time domain viscous solvers. We consider two test
values of k to compare with their results. We consider, k = 0.05 and 0.1 with zero initial
conditions. For k = 0.05, an unstable response has been predicted by our time domain
Onera model, as was also reported by Chaviaropoulos et al. 2 . The time response for both
flap and edgewise oscillations are shown in Fig. 11. For k = 0.1, a stable damped response
is seen, as was also seen by 2 . We will refer to this latter case again when we study the
10
Sunetra Sarkar and Hester Bijl
effect of structural stiffness.
3
0.5
Edgewise oscillation
Flapwise oscillation
2
1
0
0
−1
−2
0
50
100
τ
150
200
250
(a)
−0.5
0
50
100
τ
150
200
250
(b)
Figure 11: Unstable response shown for combined flap/lead-lag oscillation case for k = 0.05,
ω̄y = 7, ω̄z = 4.
5.1
1
= 0.02,
πµ
Self Excited System: Effect Of Initial Conditions & Structural Stiffness
1
Continuing with the above mentioned values for θ, ω̄y , ω̄z and
, we investigate the
πµ
effect of varying k between 0.05 and 0.2 in steps of 0.001. The initial perturbation is zero.
The system response is unstable at k = 0.05, as mentioned earlier but at all other values
above it, only stable period-1 solutions are obtained. However, we observe that adding
a nonzero initial condition could change the type of response. This has been presented
in Fig. 12 for k = 0.06. With an initial perturbation of 0.5 in the flapping direction the
response has become unstable. We also study the influence of linear structural stiffnesses
in the flap and edgewise directions, both the effect of their ratio as well as individual values.
We increase both the stiffnesses in the edge and flapwise direction in such a way that their
ω̄z
remains the same. The response in both the directions are seen to decrease,
ratio
ω̄y
but no bifurcation occurs. We present the response in the flapwise direction in Fig. 13
with the same stiffness ratio but individual values being doubled. Next, we introduce a
cubic nonlinearity in the flapwise direction. This is because, oscillation amplitude in the
flapwise direction is much larger than that in the edgewise direction, thus it is expected
that the effect of geometric nonlinearity will be more pronounced. The effect of adding
a cubic nonlinear stiffness term has been similar to adding extra stiffness to the system,
the resulting amplitude being smaller.
11
Sunetra Sarkar and Hester Bijl
0.4
4
0.3
Flapwise oscillation
Flapwise oscillation
0.35
0.25
0.2
0.15
0
−2
−4
0.1
0.05
0
100
2
−6
200
300
τ
400
500
100
600
200
(a)
300
τ
400
(a) zero initial perturbation , (b) initial perturbation of 0.5 in flap.
0.12
0.04
Flapwise oscillation
Flapwise oscillation
1
= 0.02, ω̄y = 7, ω̄z = 4;
πµ
0.05
0.1
0.08
0.06
0.04
0.03
0.02
0.01
0.02
20
40
τ
60
80
100
(a)
0
0
20
40
τ
60
80
100
(b)
Figure 13: Effect of structural stiffness, flapwise oscillation shown for k = 0.1,
= 4, (b)ω̄y = 7 × 2, ω̄z = 4 × 2.
5.2
600
(b)
Figure 12: Effect of initial condition, flapwise response shown for k = 0.06,
0
0
500
1
= 0.02; (a) ω̄y = 7, ω̄z
πµ
Forced Oscillation
The unforced system shows instability at low k; no significant nonlinear dynamical
behavior has been observed. To investigate the system dynamics further, we now consider
an external forcing. A sinusoidal forcing term is added to the system. In the previous
12
Sunetra Sarkar and Hester Bijl
case, we saw chaotic responses appear in the otherwise damped system response after
adding forcing term to it. Like in the previous case, we once again vary the forcing
amplitude (F̄0 ) and the nondimensional frequency of the forcing (k1 ). Other parameters
have been kept at ω̄y = 7, ω̄z = 4, k = 0.1. Variation of nondimensional frequency k1 of
the sinusoidal forcing shows interesting dynamical behavior. The system shows presence
of two frequencies, one is the forcing frequency of oscillation and the other is the frequency
of the unforced system. The nondimensional frequency of the unforced system is found
to be 0.7. The influence of this frequency is more predominant at lower values of k1 .
We present the time history and frequency content plots of the response for k1 = 0.05 in
Fig. 14. The response shows two main frequencies, 0.05 and 0.7, that is, a higher harmonic
of fourteenth order is present. The time history and frequency content plot for a quasiperiodic response for k1 = 0.15, have been presented in Fig. 15. The frequencies present
are 0.15 and 0.7 which are incommensurate, hence a quasi-periodic behavior. As the
reduced forcing frequency k1 is increased further, the response gradually shows stronger
influence of k1 only. We present the time history for k1 = 0.2 in Fig. 16(a), where the
presence of higher harmonics is not visible. The frequency content of this response, shown
in Fig. 16(b), shows a strong peak around 0.2. Though an extremely weak presence of
frequency 0.7 is also observed, this is not reflected on the time history of the response.
−3
−1.85
−3
x 10
1
x 10
−1.9
0.8
−2
0.6
−2.05
Psd
Edgewise oscillation
−1.95
−2.1
0.4
−2.15
−2.2
0.2
−2.25
−2.3
4500
4600
4700
τ
4800
4900
5000
(a)
0
0.2
0.4
0.6
0.8
nondimensional frequency
1
(b)
Figure 14: Higher harmonic response, k = 0.1, k1 = 0.05, F0 = 0.001, ω̄y = 7, ω̄z = 4.(a)edgewise
oscillation time history , (b) edgewise oscillation frequency content.
6
CONCLUSIONS
In this study the nonlinear aeroelastic behavior of two dimensional rotor blades during stall induced oscillation is analyzed. The motivation has been to investigate the
13
Sunetra Sarkar and Hester Bijl
−3
−3
x 10
1
x 10
−1.8
0.6
−2
Psd
Edgewise oscillation
0.8
−1.9
−2.1
0.4
−2.2
0.2
−2.3
4500
4600
4700
τ
4800
4900
0
5000
0.2
(a)
0.4
0.6
0.8
nondimensional frequency
1
(b)
Figure 15: Quasi-periodic response, k = 0.1, k1 = 0.15, F0 = 0.001, ω̄y = 7, ω̄z = 4.(a)edgewise oscillation
time history , (b) edgewise oscillation frequency content.
−3
−4
x 10
x 10
−1.9
2
−2
Psd
Edgewise oscillation
−1.8
−2.1
1
−2.2
−2.3
4650
4700
4750
τ
4800
0
4850
(a)
0.2
0.4
0.6
0.8
nondimensional frequency
1
(b)
Figure 16: Predominantly single periodic response with k = 0.1, k1 = 0.2, F0 = 0.001, ω̄y = 7, ω̄z =
4.(a)edgewise oscillation time history , (b) edgewise oscillation frequency content.
influence of various systems parameters on the aeroelastic stability and nonlinear dynamical response. Emphasis is given on parameters like structural nonlinearity and initial
conditions. Two different stall flutter oscillation cases with pitching oscillation and flapedgewise oscillation are considered. For both systems, self excited oscillation as well as
forced oscillation are studied. An engineering stall model by Onera is used to calculate
the aerodynamic loads in the dynamic stall regime. This model is well established and is
14
Sunetra Sarkar and Hester Bijl
based on experimental results. Comparison of Onera model with experiments are widely
reported in the literature.
A direct numerical integration technique of 4th order Runge-Kutta method has been
used. At each time level the technique uses internal time step sizes smaller than the user
prescribed value till convergence is obtained. Nevertheless, we have checked the time step
size for convergence by comparing the results with those obtained by smaller time steps.
A nonlinear aeroelastic analysis has performed with wide ranging parametric variation and interesting bifurcation behavior has been observed. The first system of pitching
oscillation has shown the presence of strong nonlinear behavior even without structural
nonlinearity and external forcing. For example, quasi-periodic orbits have been found in
the self excited system without structural nonlinearity. The bifurcation behavior of the
overall nonlinear system is influenced by many parameters: the mean angle of attack,
initial conditions and structural nonlinearity. For a forced system, the bifurcation parameters are the forcing frequency and amplitude. Forcing induces chaotic response in the
otherwise damped system. Forcing amplitude as a bifurcation parameter shows chaotic
routes through a series of period doubling bifurcations. Forcing frequency shows presence
of period-3 oscillation, culminating into chaos.
For the second system, flap-edgewise oscillation, not many previous studies have been
available. Therefore, we have taken some test cases from the literature and verified our
results with those. We have varied system parameters like structural stiffnesses and
their ratio, initial conditions and structural nonlinearity. Initial conditions influence the
stability behavior significantly. However, varying structural stiffnesses and their ratio
in the flap and edgewise directions do not show any nonlinear dynamical behavior of the
system. For the forced system, there are some interesting patterns as the forcing frequency
is varied. Super-harmonic and quasi-periodic response are observed at lower values of
forcing frequency. However, at higher values the response becomes predominantly single
periodic.
It should be noted that, direct integration methods are capable of showing only stable
fixed orbits of a nonlinear system. In order to capture the unstable orbits of the system, some frequency domain techniques should be employed. A nonlinear system with
geometric structural nonlinearity often shows fold or jump bifurcations which have not
been found in the present analysis. It would be interesting to investigate further for the
presence of other bifurcation routes by considering a wider variety of initial conditions. It
would also be interesting to see the effect of other classical structural nonlinearities like
free-play and hysteresis.
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Sunetra Sarkar and Hester Bijl
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