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Ŕ periodica polytechnica
Civil Engineering
56/2 (2012) 167–173
doi: 10.3311/pp.ci.2012-2.03
web: http://www.pp.bme.hu/ci
c
Periodica Polytechnica 2012
RESEARCH ARTICLE
The sensitivity of the flutter derivatives
and the flutter speed to the eccentricity
of the cross section
Mátyás Hunyadi /István Heged˝us
Received 2011-10-26, revised 2012-02-23, accepted 2012-10-11
Abstract
The flutter instability analysis of a bridge deck is based on
flutter derivatives determined by wind tunnel tests on a section
model having two degrees of freedom, heave and pitch (hereafter
referred to as the heave–pitch model). The imperfections and
the eccentricity that arise during the forced sinusoidal vibra-
tion of the section yield erroneous derivatives. This paper stud-
ies the relationship between these errors and the imperfections.
Rotational excitations around two eccentric axes (hereafter re-
ferred to as the pitch–pitch model) of the section model show
that the determined derivatives are less error-prone to imper-
fections. Determining the derivatives, like angular speed flutter
derivative A∗
2for the aeroelastic torsion moment, gives a more
accurate value, so the flutter instability analysis yields a more
accurate estimate of the flutter wind speed. Numerical values
are presented for the case of a thin airfoil and a bluffbridge
cross section.
Keywords
Flutter; 2DOF; Pitch-pitch section model
Mátyás Hunyadi
Department of Structural Engineering, BME, M˝uegyetem rkp. 3. Budapest, H-
1111, Hungary
e-mail: hunyadi@vbt.bme.hu
István Heged˝us
Department of Structural Engineering, BME, M˝uegyetem rkp. 3. Budapest, H-
1111, Hungary
1 Introduction
The considerable interaction of aerodynamic forces and struc-
tural motions is called aeroelasticity. Flutter refers to the aeroe-
lastic phenomenon where structural motions become oscillatory
with amplitude monotonically increasing in time, which can
even lead to the collapse of the structure (in the wide signi-
fication of flutter [1]). Flutter instability analysis of a bridge
deck begins with wind tunnel investigation of flutter deriva-
tives. These derivatives are used in both simplified and com-
plex verification methods to determine the structure’s aeroelas-
tic behaviour and the flutter wind speed, thus the accuracy of
the determined derivatives is primordial. This raises the ques-
tion of the accuracy of these terms, especially with regard to
model errors. Nowadays computational fluid dynamics (CFD)
simulations tend to give acceptable results [8] and, in the near
future, it seems to surmount the issues present in wind tunnel
tests and will be the preferred solution to the flutter analysis.
The flutter analysis of a recently built extradosed bridge [4]
raised the need to investigate the assumed errors involuntary
present in the wind tunnel test driven by a robot arm. The anal-
ysis of the effects of some geometric imperfections and eccen-
tricity is the motivation of the present paper.
Some hereafter defined imperfections that can occur in a wind
tunnel test, due to geometric constraints and quality of the used
test equipment, are studied is this paper. These imperfections
have their effects on the determined derivatives, which will thus
be prone to be erroneous. Two kinds of errors are analysed for
the case of a wind tunnel test set-up with vertical and angular
displacement degrees of freedom (hereafter referred to as the
heave–pitch model): the eccentricity of the rotation centre and
the centre of gravity of the section; and the rotation of the sec-
tion during vertical displacement excitation. These errors can be
related to the geometry of the model and/or to mistaken theoreti-
cal assumptions concerning the configuration, hereafter referred
to as the imperfections.
The study of the considered imperfections uses a newly devel-
oped two degrees of freedom (DOFs) description of the section.
This model is used in the analysis of the errors due to the im-
perfections and in the development of a test configuration where
The sensitivity of the flutter derivatives and the flutter speed 1672012 56 2
the section is forced by two distinct rotations around two given
axes (hereafter referred to as the pitch–pitch model). This set-
up yields new derivatives, from which the derivatives defined to
the centre of gravity can be calculated. Imperfections of this
model are also studied and presented, which yield derivatives
less prone to error.
Eccentric flutter models can also be used for power genera-
tion, as is presented in a feasibility study [2].
2 Description of the pitch–pitch section model
Consider a symmetric section of a bridge deck on which cen-
tre of gravity aeroelastic vertical lift force Lhand torsion mo-
ment Mαper unit length act due to the oncoming wind with
speed U. The motion of the section subjected to these aeroelas-
tic forces is described by equations [6]
m¨
h+ch˙
h+khh=Lh
Iα¨α+cα˙α+kαα=Mα(1)
where hand αdenote the vertical deflection (heave) and tor-
sional rotation (pitch) of the centre of gravity (c.g.) of the sec-
tion. The section has mand Iαas the mass and mass moment
of inertia per unit length, respectively, chand cαas the viscous
damping constants according to the two movements, khand kα
as the stiffness coefficients of the heaving and pitching modes,
respectively. Based on analytical theories of Theodorsen [7] and
Klöppel et al [5] the aeroelastic forces can be written in a form
with force coefficients in function of the motion of the section.
These force coefficients have been rewritten in a new form by
Scanlan [6]. This latter formulation is used in this paper.
γ
U
c.g.
δ
ηB/2ηB/2
G
G0
D0
D
B/2B/2
Fig. 1. Section model with two rotational degrees of freedom: γand δ
Consider the same section with two independent rotational
degrees of freedom (fig. 1). Define the points G and D in
the horizon of the centre of gravity (c.g.) spaced at distance
ηB/2 from the latter to windward and leeward side, respectively,
where B/2 denotes the half-width of the section. Denote these
same two points by G0and D0in a displaced position of the sec-
tion. Defining by γthe angle of the displaced line G–D0and by δ
the angle of the line D–G0to the original horizon results in a well
determined description of the displaced position. These two an-
gles will be used as the two degrees of freedom (pitch–pitch
DOFs) of the section. There exists a bijective relation between
the DOFs of the heave–pitch model and those of the pitch–pitch
model supposing small displacements.
The force and moment equilibrium written to the windward
rotation axis (denoted by G) yields a lift force equivalent to the
one acting at c.g. of the section (Lγ=Lα) and a moment (Mγ)
which contains the aeroelastic moment Mαacting at c.g. and the
lift force Lhmultiplied by the eccentricity. Both relations can be
organized to obtain coefficients for the incidences γand δand
angular speeds ˙γand ˙
δ. These coefficients represent physical
phenomena similar to the derivatives of the heave–pitch model,
so they can be treated as derivatives and denoted with second
indexes γand δ, respectively to the DOF considered. Writing
the equilibrium to the leeward axis D gives a relationship for
lift force Lδand moment Mδanalogue to the previously defined
ones. These relationships are
Lγ=Lδ=1
2ρU2B KH∗
2γ
˙γB
U+K2H∗
3γγ+KH∗
2δ
˙
δB
U+K2H∗
3δδ!
Mγ=1
2ρU2B2KA∗
2γ
˙γB
U+K2A∗
3γγ
+K(A∗
2δ+ηH∗
2δ)˙
δB
U+K2(A∗
3δ+ηH∗
3δ)δ!
Mδ=1
2ρU2B2K(A∗
2γ−ηH∗
2γ)˙γB
U+K2(A∗
3γ−ηH∗
3γ)γ
+KA∗
2δ
˙
δB
U+K2A∗
3δδ!(2)
where K=2πf B
Uis the reduced circular frequency of the motion
of the section, fis the frequency of the oscillating motion and
Bis the width of the section, Uis the oncoming wind speed, ρ
is the mass density of the air, and, assuming the linearity of the
aeroelastic forces, the newly defined A∗
ij and H∗
ij (i=2,3; j=
γ, δ) derivatives have a bijective relationship with angular and
heave derivatives A∗
iand H∗
i(i=1..4), respectively, defined to
the c.g. This relationship is of the form
A∗
2γ=H∗
1η2
4+H∗
2η
2+A∗
1η
2+A∗
2
A∗
3γ=H∗
3η
2+H∗
4η2
4+A∗
3+A∗
4η
2
H2γ=H∗
1η
2+H∗
2
H3γ=H∗
3+H∗
4η
2(3)
and
A∗
2δ=H∗
1η2
4−H∗
2η
2−A∗
1η
2+A∗
2
A∗
3δ=−H∗
3η
2+H∗
4η2
4+A∗
3−A∗
4η
2
H2δ=−H∗
1η
2+H∗
2
H3δ=H∗
3−H∗
4η
2(4)
The derivatives are functions of the reduced wind speed
Ured =U
f B .
This model will be used to investigate the previously de-
scribed imperfections on the classic heave–pitch section model.
Per. Pol. Civil Eng.168 Mátyás Hunyadi /István Heged˝us
3 Imperfections and errors resulted in the heave–pitch
section model
Consider a heave–pitch section model under a wind-tunnel
investigation. During the test the model is excited with only one
of its DOFs at a time (a sinusoidal heaving motion hand twist
motion αalternately).
Suppose geometric imperfections of the heave–pitch model in
two ways; in the case of rotational and of heave forced vibration.
In the case of the rotational excitation define the imperfection as
an eccentricity ηB
2of the excitation axis to the c.g. of the section.
This geometric imperfection could also be treated as a wrong
assumption concerning the DOFs of the model. During the wind
tunnel test the assumption that no heave occurs holds (h=0),
while the eccentricity makes the heave motion appear. The exact
value of the aeroelastic forces acting on the section are the ones
defined in the previous section. With the false assumption the
determined derivatives contain errors which are
∆H∗
2=H∗
1η
2
∆H∗
3=H∗
4η
2
∆A∗
2=H∗
1η2
4+H∗
2η
2+A∗
1η
2
∆A∗
3=H∗
3η
2+H∗
4η2
4+A∗
4η
2.(5)
The relative errors of these derivatives are shown in figures 2
for a thin airfoil [3], [6]. The relationships of the errors in the
derivatives to the imperfection are linear for some derivatives
and non-linear for others, the magnitudes of the errors are in
the order of the other derivatives than the one examined, and
are functions of the reduced wind speed. The relative errors
depend on the derivatives themselves, so they have a complex
relationship with the imperfection.
The relative error in the angular speed derivative H∗
2(fig. 2
(a)) for the lift force is linearly dependent on the eccentricity
η, but at the zero point of the derivative the relative error pro-
duces huge values. This implies the difficulty in determining the
real zero point of the derivative. The angular derivative for the
lift force contains a large relative error in the precious small re-
duced velocity domain. Precious in the sense of an aeroelastic
instability investigation.
The relative error in the angular speed derivative A∗
2(fig. 2
(c)) is non-linearly dependent of the imperfection, with higher
order dependency at small reduced velocities. The non-linear
relationship is present at a higher value of the imperfection fac-
tor than those shown in the diagram. Due to the fact that many
simplified instability analyses rely on this sole derivative the ap-
plied analysis’ result is also affected by the imperfection, which
leads to inaccurate flutter wind speed. One of the main purposes
of a new model configuration is to diminish this error and assure
reliable instability analysis results. The derivative A∗
3(fig. 2 (d))
is almost constantly sensitive to the imperfection at all reduced
velocities, except for the small reduced velocity range where rel-
ative error is non-linear and is acceptable for small eccentricity
only.
For the case of the heave excitation of the model the assump-
tion is that no rotation occurs during the test series. As the im-
perfection supposes a small rotation α=ν
B/2hof magnitude lin-
ear to the heave excitation, analysis steps analogue to the afore-
mentioned ones produce errors in the heave flutter derivatives
such as:
∆H∗
1
H∗
1
=H∗
2
H∗
1
2ν
B
∆H∗
4
H∗
4
=H∗
3
H∗
4
2ν
∆A∗
1
A∗
1
=A∗
2
A∗
1
2ν
B
∆A∗
4
A∗
4
=A∗
3
A∗
4
2ν(6)
The errors are linear to the imperfection factor ν, with magni-
tude increasing as a function of the reduced wind speed. The
model geometric size Balso affects the uncertainties in the
derivatives H∗
1and A∗
1in a logical way, the larger the model size
the more accurate the results produced. As the orders of A∗
1and
A∗
2, and H∗
1and H∗
2are the same, the error is easy to handle.
On the other hand the same is not true for the remaining two
derivatives as there is an order of magnitude difference between
the derivative and the one that its error is related to. As there
is a zero point in H∗
4its relative error gets unacceptably high at
reduced velocities above the zero point.
4 Errors in the pitch–pitch section model
Similarly to the heave–pitch section model the determination
of the derivatives of the pitch–pitch section model in a wind-
tunnel investigation is to be performed in two steps according
to the two degrees of freedom. The section model is excited
by sinusoidal rotations in its DOFs alternately. In the case of
an oscillating rotation around the windward rotation point the
assumption of having no rotation around the leeward axis holds
(case of excitation in γand δ=0). The recorded lift force and
torsion moment around the axis in point G give the derivatives
with index γ. The same steps apply for the determination of
derivatives corresponding to the rotation around the axis in point
D and result in derivatives with index δ.
The investigation of the errors in the measured derivatives due
to the imperfection in the pitch–pitch model leads to the follow-
ings. Let the imperfection be defined as an unexpected move-
ment of the fixed point described by a rotation δ=λγ in the
non-excited DOF. The errors of the derivatives shall be:
∆A∗
2γ=λ(A∗
2δ+ηH∗
2δ)∆A∗
3γ=λ(A∗
3δ+ηH∗
3δ)
∆H∗
2γ=λH∗
2δ∆H∗
3γ=λH∗
3δ(7)
Similar relations can be concluded for the other pitch excita-
tion test series, with an independent λfactor for this case. The
errors in the derivatives are given as:
∆A∗
2δ=λ(A∗
2γ−ηH∗
2γ)∆A∗
3δ=λ(A∗
3γ−ηH∗
3γ)
∆H∗
2δ=λH∗
2γ∆H∗
3δ=λH∗
3γ(8)
The sensitivity of the flutter derivatives and the flutter speed 1692012 56 2
(a) Relative error in H∗
2(b) Relative error in H∗
3(c) Relative error in A∗
2(d) Relative error in A∗
3
Fig. 2. Relative errors in ∆H∗
i
H∗
iand ∆A∗
i
A∗
i(i=2,3) due to imperfect eccentricity η(case of an thin airfoil)
(a) Relative error in derivative
A∗
2γ
(b) Relative error in derivative
H∗
2γ
Fig. 3. Relative errors in angular speed derivatives with imperfection factor magnitude of λ=0,0.05,0.1 (light grey, mid-grey, black, resp.)
Without presenting diagrams of the relative errors in all
derivatives the remarks are the following for the case of a thin
airfoil. The relative errors in the derivatives A∗
2i(i=γ, δ) (fig. 3
(a)) are only present in the region of small reduced wind speeds.
These errors decrease with increasing eccentricity ηof the rota-
tion axes. This means that the derivative sensibility to the im-
perfection can be weakened by spacing the two rotation points.
Derivative A∗
3γis highly contaminated by the imperfection fac-
tor λat the precious small reduced velocities, which error is not
present in the derivative A∗
3δ. The same applies to the deriva-
tives H∗
3γand H∗
3δfor the lift force. Angular speed derivatives
contributing to the lift force contain high errors in the lower and
higher reduced velocity ranges for H∗
2δand H∗
2γ(fig. 3 (b)), re-
spectively.
4.1 Determination of the derivatives defined to the centre
of gravity
The derivatives determined on the pitch-pitch section model
are converted into the derivatives defined to the centre of gravity
of the section using the equations (3) and (4). The relative errors
in the latter derivatives were determined for a thin airfoil, us-
ing the theoretically defined derivatives of Theodorsen [7], and
the derivatives of the extradosed bridge of the motorway M43
in Hungary [4]. During the analysis we assumed that the two
phases of the wind tunnel tests and their imperfections are in-
dependent from each other. The derivatives of the bluffsection
were fitted on the data measured in a wind tunnel on a heave-
pitch section model and are treated hereafter as the derivatives
without imperfections, see figure 5.
The figures 4 and 5 present the perfect and imperfect (erro-
neous) derivatives with different supposed imperfection factors
in the wind tunnel test (λ=0,0.05,0.1). Although an imperfec-
tion of 10% is relatively high, it is used as a demonstration in
the present theoretical investigation. It can be observed that the
same motion derivatives corresponding to the torsion moment
and the lift force have similar diagrams (e.g. H∗
1–A∗
1, fig. 4 (a)–4
(b) for the case on the thin airfoil, fig. 5 (a)–5 (b) for the bluff
section). The derivatives H∗
3and A∗
3do not depend on neither
the imperfection factor λnor the eccentricity η. The absolute
errors in the derivatives H∗
2and A∗
2(fig. 4 (c) and 5 (c)) increase
with the supposed imperfection and increase with the applied
eccentricity for the airfoil but decrease with it for the bluffsec-
tion. All errors in the other derivatives show a dependency on
the reduced velocity (e.g. 4 (d) and 5 (d)). These errors can be
compensated by increasing the applied eccentricity η, in other
words by spacing the two rotational axes.
Concerning the relative errors in the derivatives (figures 6 and
7) the followings are concluded. Relative errors in some deriva-
tives show dependency on the reduced wind speed, others show
sensibility to the applied eccentricity depending on the reduced
velocity, and others show improvements with increasing eccen-
tricity. The heave speed derivatives H∗
1and A∗
1(fig. 6 (a) and
7 (b)) include errors of great magnitude at almost all reduced
velocities, which errors can be diminished by increasing the ap-
plied eccentricity. The relative errors in the heave derivatives
show similar behaviour, but the errors stay unacceptable at both
small and high eccentricities. This causes no huge practical is-
sues as these derivatives are in several cases omitted from the
flutter analysis. The angular speed derivatives H∗
2(fig. 7 (a)) and
Per. Pol. Civil Eng.170 Mátyás Hunyadi /István Heged˝us
(a) H∗
1(b) A∗
1(c) A∗
2(d) A∗
4
Fig. 4. Thin airfoil derivatives due to the imperfection factor λ. Light grey: without imperfection (λ=0), mid-grey: λ=0.05, black: λ=0.10
A∗
2(fig. 6 (b) and 7 (c)) are contaminated with errors linearly de-
pendent on the eccentricity, and the determination of their zero
becomes inaccurate. The relative errors included in the angu-
lar derivatives show strange relations. The relative error in the
derivative H∗
3for the lift force decreases exponentially as a func-
tion of the reduced velocity, even if the eccentricity increases
it. This gives acceptable measures of this derivative at all re-
duced velocities but the very small ones. The angular derivative
A∗
3for the torsional moment seems to be practically insensible
to the eccentricity and the supposed imperfections. The heave
derivatives H∗
4and A∗
4show relative errors with magnitude of
great significance, the determination of these derivatives with
the pitch–pitch model is questionable.
The relative errors are functions of the supposed imperfec-
tions and it seems that the model size has no effect on them. But
as the size of the model increases, or as the applied eccentricity
increases the imperfection diminishes, due to the easier geomet-
ric manoeuvrability.
5 Sensitivity of the flutter speed to the model imperfec-
tions
The study investigated also the effect of the imperfections on
the critical wind velocity, referred to as flutter speed UF. The
complex eigenvalue analysis [5] of an undamped two degree-
of-freedom cross section was used in the determination of the
flutter speed. The effect is represented as the ratio of the flut-
ter speed resulted with the erroneous derivatives to the one with
no errors. The most unfavourable sign combination of the ec-
centricities and the imperfections were used to obtain the rep-
resentable values. Hereafter the relative mass µis the ratio of
the mass of the cross section to the mass of the circumscribed
air mass, the relative torsional inertia radii rαis the ratio of the
torsional inertial radius to the half width, and represents the
ratio of the considered torsional to the heave eigenfrequencies.
The erroneous flutter speed on the heave–pitch (hand α)
model is influenced by the width Bof the section. The figure
8 (a) presents the case of the heave–pitch model of a thin air-
foil with the torsional imperfections νand the eccentricity ηof
the rotational axis to the centre of gravity. It is concluded, that
although the relative mass µcompensates the error, the latter
reaches quite important values even at high eigenfrequency ra-
tios .
The figure 8 (b) shows the results of a similar analysis this
time based on the pitch–pitch model with the torsional imper-
fections λδand λγfor the two rotational degrees, resp. The vari-
ation of the flutter speed is represented as a function of the ap-
plied ηeccentricity. The error seems to highly depend on the
eccentricity and has an extremum at a singular value of it. The
similar diagram shown on fig. 9 is resulted for the case of the
bluffsection.
The coupled flutter phenomenon of a bridge cross section ap-
pears as an undamped rotation around an eccentric quasi rota-
tional axis. The phase shift of the vertical and angular harmonic
motion causes the vertical movement of this quasi rotational
axis, that is why we have denote it “quasi” (not to be confused
with the body, nor the space rotational axis). One could assume
that the optimal eccentricity, when the flutter speed error is min-
imum, coincides with the quasi rotation point. This coincidence
could be used in the wind tunnel investigation by preliminarily
estimating the required eccentricity of the set-up based on expe-
riences in hope of minimise the errors treated in this paper. Un-
fortunately the analyses showed that the optimal eccentricities
for the presented configurations were at around 1.3–1.8 times
the eccentricity of the quasi rotational point.
6 Conclusions
The analysis of the effects of the supposed (model or ge-
ometric) imperfections and eccentricity of a heave–pitch sec-
tion model on the derivatives was studied. The discussed errors
increase the uncertainties of determination of the zeros of the
derivatives. The imperfections appear as high relative errors in
several derivatives (H∗
1,H∗
3,A∗
2and A∗
4) in the precious range of
small reduced velocities, or in other ranges for A∗
1and A∗
3.
A wind tunnel test carried out on a pitch–pitch section model
yields newly defined derivatives, from which those defined to
the centre of gravity of the section are calculated. On this model
the impact of the supposed imperfections can be decreased for
some specific derivatives (H∗
2,H∗
4,A∗
1and A∗
4). The derivatives
H∗
3and A∗
3show to be practically insensible to the imperfec-
tions. The application of an adequate eccentricity of the forced
rotational points seems to moderate the effects of the rotational
imperfections on the derivatives H∗
1and A∗
2.
The sensitivity of the flutter derivatives and the flutter speed 1712012 56 2
Ured
0 5 10 15 20 25 30
-10
-8
-6
-4
-2
0
2
(a) H∗
1
Ured
0 5 10 15 20 25 30
0.0
0.5
1.0
1.5
2.0
derivative fitted
measured data
on data
(b) A∗
1
Ured
0 5 10 15 20 25 30
-1.5
-1.0
-0.5
0.0
η=0.5
η=0.75
η=1.0
η=1.5
with eccentricity:
studied curves
(c) A∗
2
Ured
0 5 10 15 20 25 30
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
(d) A∗
4
Fig. 5. M43 derivatives: tick line =derivative fitted on measured data and
referred to as without imperfections, thin lines =studied derivatives supposing
imperfection λ=0.05 on the proposed pitch-pitch model
(a) Relative error in derivative
A∗
1
(b) Relative error in derivative
A∗
2
Fig. 6. ∆A∗
i(λ)
A∗
irelative errors in the thin airfoil derivatives as a function of the
applied eccentricity and the imperfection factor λ. Light grey: without imper-
fection (λ=0), mid-grey: λ=0.05, black: λ=0.10
Ured
0 5 10 15 20 25 30
-0.06
-0.04
-0.02
0.00
0.02
0.04 η=0.5
η=0.75
η=1.0
η=1.5
(a) H∗
2
Ured
0 5 10 15 20 25 30
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
η=0.5
η=0.75
η=1.0
η=1.5
(b) A∗
1
Ured
0 5 10 15 20 25 30
-0.15
-0.10
-0.15
0.00
0.05
0.10
0.15 η=0.5
η=0.75
η=1.0
η=1.5
(c) A∗
2
Fig. 7. ∆A∗
i(λ)
A∗
irelative errors in the M43 derivatives as a function of the applied eccentricity ηand the imperfection factor λ=0.05
Per. Pol. Civil Eng.172 Mátyás Hunyadi /István Heged˝us
1.0 1.5 2.0 2.5 3.0 3.5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
µ=10
µ=30
µ=50
ε=ω0α
ω0h
UF,with error
UF,without error
(a) Heave-pitch (h–α) model, B=1 m,η=0.1,
ν=−0.1
ε=ω0α
ω0h
1.0 1.5 2.0 2.5 3.0 3.5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
η=0,2
η=0,5
η=1,0
η=1,5
UF,with error
UF,without error
(b) Pitch-pitch (γ–δ) model, µ=30,
λγ=−λδ=−0.10
Fig. 8. Effect of the imperfection on the flutter speed. Common parameters:
thin airfoil, rα=0.5
It seems that an optimum of the applied eccentricity for
derivatives A∗
1and A∗
2exists where these derivatives contain the
least error. The heave derivative A∗
4contains an error of great
magnitude, but maybe less at the same value of eccentricity. The
relative errors in the derivatives H∗
1,H∗
3and H∗
4for the aeroelas-
tic lift force tend to diminish by augmenting the eccentricity of
the rotation points. The inverse is concluded for the derivative
H∗
2, where the error diminishes as the rotation points get closer
to each other.
The investigation of the effect of the imperfections on the
changes of the flutter speed showed the existence of an opti-
mal eccentricity of the pitch–pitch model. Unfortunately this
optimal eccentricity did not coincide with the quasi rotational
point at flutter and thus the eccentricity to be applied cannot be
preliminarily estimated by the latter to minimise the error of the
flutter speed.
The applicability of the presented set-up requires the further
investigation of the effects on the errors present in the wind tun-
nel tests.
Acknowledgement
This work is connected to the scientific program of the ”De-
velopment of quality-oriented and harmonized R+D+I strat-
egy and functional model at BME” project. This project is
supported by the New Széchenyi Plan (Project ID: TÁMOP-
4.2.1/B-09/1/KMR-2010-0002).
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ε=ω0α
ω0h
UF,with error
UF,without error
1.0 1.5 2.0 2.5 3.0 3.5
0.0
0.2
0.4
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