ask 21 flight sim
Post on 26-Dec-2015
65 Views
Preview:
DESCRIPTION
TRANSCRIPT
Aeroelastic Simulations of a Sailplane
David Eller and Ulf RingertzAeronautical and Vehicle Engineering
Royal Institute of TechnologySE - 100 44 Stockholm, Sweden
December 2005
Abstract
Using the ASK 21 training and aerobatics sailplane as a test case,different structural and aerodynamic modeling approaches are compared.For the structure, a simplified model consisting of a beam framework isassembled and compared with a more complex shell finite element model.Frequency-domain lifting-surface aerodynamics is employed for flutteranalysis and compared with results obtained from three-dimensional time-domain potential flow aerodynamics. Finally, a 4g pull-up maneuver isperformed in flight testing and numerical aeroelastic simulations of thesame maneuver are evaluated.
Introduction
The purpose of the present study is to compare and evaluate different levels
of modeling detail when trying to assess the aeroelastic behavior of a complete
aircraft configuration. Many different modeling approaches are possible depend-
ing on the application at hand. If the analysis is part of a preliminary design
process, a computational approach is the only option and standard procedures
involve a finite element model to assess the structural dynamics properties and
a linear potential flow model to obtain the unsteady aerodynamic forces.
Combining the structural model with the model of unsteady aerodynamics,it is possible to pose the aeroelastic stability problem as a nonlinear eigenvalue
problem for which there are efficient methods available [1]. Essentially the same
computational model can be used to analyze other aeroelastic phenomena such
as control surface efficiency and gust response [2].
Different levels of detail can be used for both the structural and aerody-
namic model. The simplest structural model is usually obtained in the form
1
E 2 D. Eller and U. Ringertz
of a beam model of the wings, fuselage and tail. Even though the topology ofthe beam model is simple, significant difficulties arise when trying to accurately
estimate the properties of the beams and their interaction at connections such
as the wing-fuselage interface. A more detailed structural model, using for ex-
ample shell elements, may make it easier to define structural properties. The
drawback is that one may have to model much more details in terms of local re-
inforcements in order to obtain accurate global stiffness properties and accurate
representations of resonance frequencies and the associated modal eigenvectors.
If the analysis is concerned with the certification process of an already exist-ing design for which a prototype has been manufactured, one may eliminate the
numerical structural analysis all together. In this case, a detailed ground vibra-
tion test is used to experimentally determine the structural dynamics in terms
of resonance frequencies and modal eigenvectors. This procedure is well de-
scribed in [3]. The measured structural dynamics properties are then combined
with a numerical model of the aerodynamic forces to perform the actual flutter
analysis. This approach is mainly used for smaller general aviation aircraft and
gliders. However, even though general design guides such as those provided
by the DLR [4] are used, costly design changes to the already existing aircraft
prototype may be needed to demonstrate the aeroelastic stability required by the
certification authority as defined in the airworthiness code (CS-22 [5]).
In case the needed engineering manpower and resources are available, a nu-
merical model is used for the structural dynamics even in the early design stages.
The numerical model is then refined during the design process and finally cor-
related to ground vibration test data as it becomes available. A structural op-
timization approach [6] may be used to improve the correlation between the
numerical and the experimental data. Such an approach helps to avoid surprises
late in the development process.
The model of the unsteady aerodynamics can also be of very different detail
even if it is restricted to unsteady potential flow. The least complex model
would be to use a strip model were the flow is assumed to be well represented
by a locally two-dimensional analysis as described by Rodden and Johnson [7].
A more detailed model involves the use of a panel method to approximate all
lifting bodies by their mean surfaces, but still neglecting the fuselage and the
thickness of the wing and tail surfaces, using the well known Doublet-Lattice
method [7] or the similar methods described in Refs. [8, 9]. An even more
detailed model would involve a model of the fuselage and also the actual wingprofile of the lifting surfaces.
In this study, different modeling approaches for the structural and aerody-
namic behavior of the ASK 21 sailplane will be investigated and compared with
respect to flight mechanics and aeroelastic properties. Numerical models are
Aeroelastic Simulations of a Sailplane E 3
compared with data from ground vibration and flight testing in order to assess
the accuracy which can be obtained with the different simulation methods. Fi-
nally, some remarks regarding the relative effort required to create and validate
the different models are provided.
The ASK 21 sailplane
The reason for using the ASK 21 sailplane [10] in this study is the availability
of an instrumented aircraft of this type and also that detailed design data has
been kindly provided to the authors by the manufacturer. The ASK 21, whichis shown in Figure 1, is a two-seat sailplane for pilot training and aerobatics. It
Figure 1: The ASK 21 glider.
features a wing aspect ratio of 16.1, large wing thickness and robust glass fiber
composite construction. In comparison to modern high-performance sailplanes,
the ASK 21 is relatively stiff, and hence less susceptible to aeroelastic instability.It operates at flight speeds between 80 km/h and 280 km/h and, as it is certified
for aerobatics, may be subjected to load factors from −4g to +6.5g. Due to the
high allowed load factor, elastic deformations experienced in service may still be
fairly high and likely exceed those encountered in other general aviation aircraft
considerably.
Structural model
With a relatively large wing thickness of 19.6% at the root, a moderate wing
aspect ratio, and sandwich shell construction throughout, the ASK 21 is rela-
tively light for its size. Wings, fuselage and tail shells are built from glass-fiber
E 4 D. Eller and U. Ringertz
reinforced epoxy resin and polymer foam or tubus core. Most sandwich compo-
nents employ outer face sheets which are thicker than the corresponding inner
faces in order to improve the impact strength of the exposed external layers. In
lightly loaded regions such as the control surfaces, layer thickness is as low as0.09 mm/0.18 mm (inner/outer face) with 4 mm light foam core, while even
the outboard wing shell is not thicker than 0.18 mm/0.6 mm with 9 mm foam
core. In comparison to more recent sailplane designs featuring carbon fiber
sandwich construction and much higher aspect ratio, the ASK 21 is expected
to be relatively stiff in terms of global deformations, but comparatively flexible
with respect to localized loads.
Modal subspace formulation
When considering small elastic deformations, linear equations of motion ac-
cording to
Mx + Kx = fa(t,x, x, . . . ) (1)
are used. Here, M is the mass matrix relating accelerations in the structural
degrees of freedom x to inertial forces, K is the stiffness matrix and fa a vector
of aerodynamic forces. Note that the aerodynamic forces fa also depend on the
time history of motion and loads because these determine the shape and strength
of the wake. In order to drastically reduce the number of degrees of freedom,
the standard modal subspace technique is adopted. Let zj be eigenvectors and
ω2
j eigenvalues solving the generalized eigenvalue problem
(
K − ω2
j M)
zj = 0. (2)
Then, a reduced number of eigenvectors with low eigenfrequencies ωj are used
to define a subspace Z = [z1,z2, . . . ]. Low-frequency eigenvectors are utilizedbecause it is assumed that higher order modal deformations do not contribute
relevant aerodynamic loads. With mass-normalized eigenvectors, the equations
of motion in modal subspace become uncoupled according to
Iq + Ωq = fq(t,q, q, . . . ), (3)
with the modal displacements q such that Zq = x, modal loads are found by
projection of the aerodynamic forces into the subspace f q = ZT fa. For the
matrices holds
ZT MZ = I, and ZT KZ = Ω. (4)
Here, I is the identity matrix and Ω is a diagonal matrix of the squares of
the eigenfrequencies ωj . The modal equation of motion (3) only has a small
number of degrees of freedom, but retains the aeroelastically relevant dynamic
Aeroelastic Simulations of a Sailplane E 5
characteristics of the structure if sufficiently many eigenmodes are included in
the subspace Z.
Finite element models
Two finite element models of different type were assembled for comparison
purposes. The first, much simpler model is based on a framework of beams,
the properties of which were chosen to match ground vibration test data, and
was initially developed by Keller [11] and co-workers [12, 13, 14]. It was later
modified in order to improve the accuracy of the control system modeling. The
final beam finite element model contains approximately 2900 structural degrees
of freedom.
The second model is based on triangular and quadrilateral shell finite ele-
ments, as shown in Figure 2, where different colors are used for different shell
properties. In contrast to the beam model, stiffness properties are not designed
to match test data, but derived from lamination plans which were kindly pro-vided by the manufacturer. The main load-carrying structure accounts for only
roughly 50% of the empty mass of the aircraft, and the distribution of the re-
maining mass is to a large extent unknown. Therefore, mass properties were
used to match the model to data obtained from vibration testing. The model
for which results are discussed below has about 23 000 elements and 70 000
structural degrees of freedom. Both finite element models were constructed for
NASTRAN [15].
Figure 2: Shell FE model including control surfaces.
Testing and validation
The main source of validation data for the structural model was the ground
vibration test (GVT), shown in Figure 3, of the full aircraft performed at KTH
[16], along with additional vibration testing of the fuselage performed as part of
E 6 D. Eller and U. Ringertz
the present investigation. These ground vibration tests were also compared to the
Figure 3: Ground vibration test of the ASK 21 glider.
tests by Niedbal [17] performed during the original certification process. Due
to friction in the control system, most control surfaces do not move when the
aircraft is excited to vibrate, at least not with moderate excitation amplitudes.
Therefore, all flaps were fixed rigidly to the adjacent wing or tail surface in orderto create a well-defined structural configuration.
In Table 1, eigenfrequencies obtained from modal analysis of the beam (fb)
and shell (fs) finite element models are compared with test data (fexp). For
a more compact notation, the abbreviations given in Table 2 are used in the
following presentation.
In general, it should be noted that due to the placement of acceleration
sensors and excitation devices employed in the GVT, some eigenmode shapes
present in the FE models could not be detected experimentally. The rather
distinct mode at 15 Hz involves mainly torsion of the vertical fin, which can
hardly be excited with a laterally acting shaker placed at the rear wheel, i.e. close
to the elastic axis of the fin. Even for the higher wing bending modes above20 Hz, accelerometer data sometimes leaves doubts about the bending mode
shape. Furthermore, very few eigenmodes can be unambiguously identified as
pure fundamental modeshapes. Instead, complex coupling patterns of fuselage,
wing and tail deformations are observed. Comparisons of eigenfrequencies alone
Aeroelastic Simulations of a Sailplane E 7
fexp [Hz] fb [Hz] fs [Hz] Description
2.95 2.90 2.96 sym. wing 1b.
4.09 4.00 4.11 asym. fuselage 1t.6.50 6.64 6.23 asym. wing 1b. + fuselage 1b.
- 7.28 7.07 sym. fuselage 1b. + wing 1ipb.
7.29 7.09 7.34 asym. fuselage 1b. + fin b.
8.25 - - asym. fuselage 1b. + stabilizer rotation
8.74 9.25 8.56 sym. wing 2b. + fuselage 1b. ip.
10.5 10.7 10.3 sym. wing 2b. + fuselage 1b. cp.
12.2 11.6 12.0 asym. fuselage 1b. + fin t.
- 15.5 14.2 asym. fin torsion + fuselage 1b.
16.9 18.8 16.5 asym. wing 2b. + stabilizer b. cp.
17.4 - - asym. wing 2b. + stabilizer b. ip.
20.4 19.8 19.7 sym. wing 2b. + fuselage 2b. + stab. b. cp.
- 27.9 20.9 asym. fuselage b. + stabilizer b22.3 - 22.7 sym. wing 2b. + fuselage 2b. + stab. b. ip.
22.5 20.6 24.3 asym. wing 1ipb.
- 26.2 - sym. wing t. + stabilizer b. ip.
- 27.0 - sym. wing t. + stabilizer b. cp.
- - 28.1 sym. wing 2ipb. + stabilizer b.
26.9 - 28.7 sym. stabilizer 1b.
- - 28.8 asym. wing t. + wing 3b.
29.3 - 29.5 sym. wing t.
- 26.1 30.0 asym. wing t.
32.7 36.2 32.4 asym. wing 3b.
Table 1: Comparison of measured and computed eigenfrequencies.
are hence not necessarily useful, as modes with the same description in Table 1
may combine different magnitudes of fuselage and wing deformations, although
the modeshape appears visually similar.
For the lowest-frequency eigenmode shapes up to about 15 Hz, both finite
element models compare reasonably well with GVT data. There is one low-
frequency modeshape at 8.25 Hz which remarkably is not present in either nu-
merical model. A possible interpretation of the accelerometer data for this modeis that the stabilizer which is attached to the vertical tail using two detachable
centerline connections, rotates about the line through the attachment points as
in a joint. The motion may even be combined with some degree of local fin
bending and rear fuselage torsion. Both finite element models fail to reproduce
E 8 D. Eller and U. Ringertz
b. modeshape dominated by bending
t. modeshape dominated by torsion
2b. second fundamental bending mode
ipb. in-plane bending (xy-plane)
sym. symmetrical
asym. antisymmetrical
ip. wing & stabilizer tips moving in phase
cp. wing & stabilizer tips moving counter-phase
Table 2: Abbreviations used in Table 1
this particular type of motion accurately, although elastic connections between
fin and stabilizer bodies are used. The modeled attachment stiffness has a strong
impact on the frequency of the mode at 7.3 Hz, but does not appear to allow a
motion such as that observed in the GVT at 8.25 Hz.
For higher frequencies above 20 Hz, differences become more pronounced.
Evidently, the shell finite element model produces a number of eigenmode
shapes not observed in vibration testing. As noted previously, this may partly
be due to the location of excitation devices and accelerometers. Moreover, com-
puted fuselage deformations for modes with frequencies exceeding 22 Hz and
wing deformations for all eigenmodes involving torsion show significant local
shell bending deformations. It is considered likely that the type of local fuselage
deformations observed in the shell finite element model would not occur in
reality since inertial loads are introduced by means of local reinforcements, not
all of which are included in the finite element model. Computed shell defor-
mations for the wings, however, are considered realistic as the stiffness of thewing shell is well defined and the mass fairly evenly distributed. In contrast to
the forward fuselage, no large inertial loads, except the mass forces of the wing
itself, act normal to the surface of the wing. Local shell deformations of this
kind would be hard to detect in vibration testing of the full aircraft unless a
very large number of accelerometers were used.
Five eigenmodes of the shell finite element model have eigenfrequencies be-
tween 28 Hz and 30 Hz, while only one symmetric wing torsion mode with
possibly some bending component was experimentally observed in this interval.
Not a single eigenmode of the beam model is located in the same frequency in-terval, which, instead, has four eigenmodes with frequencies between 26 Hz and
28 Hz, none of which appeared in vibration testing with a similarly coupled
deformation pattern. Considering the substantial differences between experi-
mentally identified and computed eigenmodes, results obtained with the above
Aeroelastic Simulations of a Sailplane E 9
structural models should be regarded as approximations. At this point, the
beam model appears to be globally somewhat less accurate than the shell finite
element model for moderate to high eigenfrequencies.
Fuselage vibration testing
The mass distribution in the fuselage is affected by appreciable uncertainties.
A number of heavy local reinforcements, composite landing gear frames and
cockpit installations were found difficult to model. Most of these items can
probably not be represented well by means of shell elements, and accurate mass
distributions were unknown. Therefore, the isolated fuselage was subjected to
vibration testing, as shown in Figure 4 in order to collect more detailed eigen-
mode shapes. Following the testing, the shell finite element model was updated
Figure 4: Ground vibration test of the fuselage.
with three concentrated masses which in total account for the difference between
measured weight and the structural mass of the finite element model.
Using a trial-and-error approach, the concentrated masses were placed in theforward and central fuselage and connected to the shell structure using inter-
polation elements (Nastran element type RBE3). These elements compute the
displacement of the connected concentrated point mass as a weighted average
of a supplied set of mesh nodes. In the current case, all three masses were
E 10 D. Eller and U. Ringertz
connected to all mesh nodes in the lower half of the forward fuselage. In this
manner, the resulting inertial forces were effectively distributed among a large
number of shell elements, thus avoiding locally concentrated forces. Neverthe-
less, it eventually turned out that even the load-carrying pilot seat shells andsome minor local reinforcements in the lower cockpit shell needed to be mod-
eled in order to avoid that local shell deformation modes (’breathing modes’)
appear at unrealistically low frequencies.
Figures 5 and 6 show the front view of the undeformed mesh and the same
view of the deformed surface for the 14th mode. The 14th mode is dominated
by antisymmetric in-plane bending of the wing but contains a significant com-
ponent of lateral fuselage bending. At the relatively high frequency of 24.3 Hz,
this fuselage bending combines with significant local shell deformations.
X
Z
X
Z
Figure 5: Undeformed mesh.
X
Z
X
Z
Figure 6: ’Breathing’ mode.
A comparison of the first two longitudinal (vertical) bending eigenmode
shapes of the fuselage is shown in Figure 7. Note that here, the classification
of modes refers to the isolated fuselage alone and that these modeshapes are
not directly related to the full aircraft eigenmodes discussed above. To make
measurements and numerical results directly comparable, the computed shellmodel modeshape was interpolated to the known locations of the accelerometers.
Then, both the measured and computed vector of displacement magnitudes were
normalized to unit length. The choice of normalization is admittedly rather
arbitrary.
Aeroelastic Simulations of a Sailplane E 11
0 1 2 3 4 5 6 7 8
−0.4
−0.2
0
0.2
0.4
Position [m]
Ver
tical
dis
plac
emen
t [−
]
1st longitudinal bending
0 1 2 3 4 5 6 7 8−0.6
−0.4
−0.2
0
0.2
Position [m]V
ertic
al d
ispl
acem
ent [
−]
2nd longitudinal bending
ExperimentShell FE model
Figure 7: Modeshape comparison for two longitudinal fuselage modes.
The normalized eigenmode shapes for the first two longitudinal modes
match fairly well. In both cases, there is a small parallel shift between ex-
perimental and computed modeshapes, which indicates that the longitudinal
mass distribution is not accurate, while the fuselage stiffness is matched well.
In the left part of Figure 7, the location of the forward and rear node of thefirst longitudinal bending mode is shown to lie slightly farther apart in the
experiment than the modal finite element solution predicts. When considering
that both the fuselage mass and the center of gravity match closely between the
experimental setup and the corresponding model, this indicates that the mass
in the real aircraft is less concentrated near the center than in the finite element
model.
Figure 8 shows the first lateral bending and torsion modes of the fuselage.
In this case, there is a small irregularity at the 3 m position, just behind the
cockpit, where the measured acceleration data shows a localized peak. For the
torsion mode, the irregularity may be related to measurement noise, since at
this position, there are only small displacements from which the torsion angle is
computed by finite differences. Even the inconsistency between two experiments
performed for the same excitation hints at this explanation.
The irregularity in the first lateral bending mode occurs at the same posi-
tion. In this case, a repetition of the experiment yields practically the same data.Since the fuselage shell in this region is locally reinforced with internal frames,
it is considered unlikely that the accelerometer senses local shell deformation.
It rather appears as if the effect of the said reinforcements is not represented
sufficiently accurately in the FE model. Similar to the longitudinal case, the
E 12 D. Eller and U. Ringertz
0 1 2 3 4 5 6 7
−0.4
−0.2
0
0.2
0.4
Position [m]
Late
ral d
ispl
acem
ent [
−]
1st lateral bending
0 1 2 3 4 5 6 7
−0.4
−0.2
0
0.2
0.4
Position [m]T
orsi
on [−
]
1st torsion
ExperimentShell FE model
Experiment 1Experiment 2Shell FE model
Figure 8: Modeshape comparison for lateral fuselage modes.
overall match of the lateral eigenmode shapes is fairly good.
Looking at the eigenfrequencies for the isolated fuselage shown in Table 3,
it appears that the shell model of the fuselage yields slightly higher frequencies
for all three bending modes. This difference could be related to the placement
of concentrated masses near the centerline. As a result, the inertia in bending is
too low for the same total mass. The torsion frequency, on the other hand, is
matched with better accuracy.
fexp [Hz] fs [Hz] Description
10.8 11.4 1st lateral bending
13.2 13.7 1st longitudinal bending
19.8 19.6 fuselage torsion
32.5 32.9 2nd longitudinal bending
Table 3: Measured and computed eigenfrequencies for the fuselage.
Control system model
For the beam finite element model, control surfaces are modeled as rigid el-ements connected to the beam by means of rigid bar elements, which allow
control surface rotation as the only degree of freedom. Aileron rotation is as-
sumed to occur about the y-axis instead of the actual aileron hinge axis, which
makes an angle of only 7.2 with the y-axis. The rotational degree of freedom
Aeroelastic Simulations of a Sailplane E 13
is connected to a concentrated mass and rotational inertia corresponding to val-
ues measured by the manufacturer. Since the internal kinematics of the control
system introduces a small stiffness, rotational springs are added to constrain the
control surface motion accordingly.
The shell finite element model shown in Figure 2 includes shell models
of the control surfaces. In this case, all surfaces are attached by means ofmultiple hinges positioned according to the drawings. Hinge nodes on the
wing, fin and stabilizer are locally reinforced with small beam elements in order
to better distribute inertial loads into the rather flexible sandwich shell.The
rather detailed modeling has the advantage of including the effect of elastic
control surface motion. For this particular aircraft, the first elastic eigenmodes
involving aileron torsion and bending appear around 25 Hz, close to the wing’s
fundamental torsion modes, and may hence be relevant from an aeroelastic
point of view.
Unfortunately, the detailed structural model caused difficulties when at-
tempting to interpolate eigenmode shapes to the aerodynamic surface mesh.
The elastic motion of the hinged control surfaces which is part of most relevant
modeshapes could not be represented as a smoothly varying deformation of the
closed aerodynamic surface. Even comparatively small deformations involvingcontrol surface rotation or twist lead to geometric inconsistencies such as locally
self-intersecting surface elements or much too large variation of the local surface
normal. Even when the elastic surfaces were constrained to the wing, stabilizer
and fin in order to approximate a fixed-stick configuration, the pronounced flex-
ibility of the very light control surface shells caused discontinuous deformations
e.g. at the outboard end of the aileron.
As an alternative, the control surface motion was modeled separately. The
aerodynamic surface mesh was deformed according to the flap deflection using a
previously developed procedure which guarantees smooth surface deformations
up to moderate deflection angles of at least ±15. The structural model, on the
other hand, was simplified by removing the explicit control surface components
and replacing them with a continuation of the wing, stabilizer and fin shell.
In this manner, fully continuous modeshapes could be obtained for which suf-ficiently smooth interpolated deformations of the aerodynamic mesh could be
computed.
In order to account for the mass coupling effect between modal displace-
ments and control surface deflections, a point-mass model was used. The equa-
tions of motion in modal form (3) are augmented with the control surface
deflections δi according to
E 14 D. Eller and U. Ringertz
[
I CT
C diag(Jδ,i)
] [
q
δ
]
+
[
Ω 00 diag(kδ,i)
] [
q
δ
]
=
[
f q
Mh
]
. (5)
Here, Jδ,i are the inertia terms associated with control surface rotation, in-
cluding the rotational inertia of the control surface itself and components of
the control system mechanics which are coupled to the flap deflection. The cor-
responding stiffness terms kδ,i constitute the lower diagonal of the new stiffness
matrix. On the right hand side, the excitation force vector is augmented with
the aerodynamic hinge moments Mh.
Since there are significant coupling effects between the structural motion
as described by the modal coordinates q and the control surface rotation, off-
diagonal terms C are added to the mass matrix. The coupling term Ci,j is the
hinge moment experienced by the control surface i due to an acceleration inmode j. It is computed by considering a point mass mi located at the center of
mass of the control surface. Taking ~ri as the vector from the center of mass to its
projection on the hinge axis and ~aj as the translational acceleration experienced
by the center of mass due to an acceleration qj in mode j, the coupling term is
found to be
Ci,j = ~hi · (mi~aj × ~ri), (6)
where ~hi is the normalized direction vector of the hinge axis.
In practice, there are also kinematic coupling effects, i.e. a steady modal
deformation would lead to certain flap deflections. Such effects are caused by
the kinematics of the complex internal actuation mechanism, which is not only
difficult to include in a structural model but also nonlinear, at least for some
control surface positions. Kinematic couplings are therefore not present in the
current structural models.
Remarks on structural modeling effort
Creating the structural models required a significant amount of time. The beam
model is a strongly reduced representation of the structure, so that the deriva-
tion of beam properties involves a certain degree of heuristic approximations
and fairly many iterations until a reasonable match with experimental data can
be obtained. Without GVT data, creating such a model would be even more
difficult. Furthermore, the credibility of simulations performed with a beam
model not validated with experimental data would be poor due to the largeamount of approximations involved.
The process of creating a much more complex shell model revealed that,
with drawings and lamination plans available, a finite element model with rea-
sonably accurate global stiffness properties can be created comparatively easily.
Aeroelastic Simulations of a Sailplane E 15
The by far most demanding problem in this context is the realistic modeling
of component connections such as the interface of the fuselage with the detach-
able wing. These points are usually designed with complex local reinforcements
which are not easily included in a shell model and must hence be approximated.A finite element model with accurate stiffness behavior is obviously not suffi-
cient for dynamic analyses, and mass properties are most likely less well defined
than laminate properties. Unless the mass distribution can be established from
accurate weight and balance calculations, it is unlikely that a credible shell model
can be created without access to vibration testing data.
Aerodynamic model
Two different aerodynamic models based on subsonic potential flow theory are
used in this study. The first model is based on the ZONA6 method implemented
in the commercial code ZAERO [18], which is a frequency domain formulation
similar to the Doublet-Lattice Method (DLM). In Figure 9, the discretization of
the lifting surfaces used with this method is shown.
X Y
Z
X Y
Z
Figure 9: Panel mesh used with ZAERO.
The second model uses a time-domain formulation based on linear potentialflow. In contrast to lifting-surface methods, the actual aircraft geometry is
discretized, as illustrated in Figure 10. The mesh is shown in the deformed state
for a positive loadfactor of 4. The computational method is described in detail
in [19].
Comparison of aerodynamic coefficients for rigid-body motion
The aerodynamic response of the aircraft to prescribed rigid-body motion is
computed in order to investigate if the numerical methods yield similar results
E 16 D. Eller and U. Ringertz
Figure 10: Surface mesh for time-domain aerodynamic method.
despite the considerably different discrete surface representations. Since ZONA6
is formulated in the frequency domain, the results are complex pressure coeffi-
cient differences ∆Cp between upper and lower side of the lifting surfaces as a
function of frequency. In this context, it is common to use the reduced formof the angular frequency ω according to
k =ωc
2u∞
, (7)
which is made nondimensional with the reference chord c and the airspeed
u∞. Integration of the pressure coefficient over the aerodynamic surface yieldsintegral aerodynamic coefficients, which depend on type and frequency of the
unsteady rigid-body motion. Harmonic heave and pitch motion about the center
of gravity of the aircraft were used as prescribed motion.
Frequency-domain coefficients cannot be computed directly with a time-
domain method, so that an indirect approach must be taken. Here, time-
marching simulations were performed for a prescribed harmonic rigid-body
motion with reduced frequency k. The coefficients are then evaluated by di-
viding the discrete Fourier transform of the output lift coefficient time series
by the (known) Fourier transform of the prescribed motion. In this manner,
coefficients can be determined at discrete frequencies, which is a quite costly
procedure since a new simulation is necessary for each desired frequency. Fig-
ure 11 shows the results of a number of harmonic excitation simulations next
to ZONA6 results for the same four coefficients, namely lift and moment coef-ficient as a function of heave and pitch motion frequency. These time-domain
simulations were performed using timesteps of 2.5 ms, and repeated with a
timestep of 1 ms with the same mesh for comparison. Despite the considerably
higher computational cost, no significant differences could be found.
Aeroelastic Simulations of a Sailplane E 17
0 0.5 1 1.5−20
−15
−10
−5
0
5
10
15
Reduced frequency
CLh
(k)
DLM, real partDLM, imag. parttime−domain, realtime−domain, imag.
0 0.5 1 1.5−5
0
5
10
15
20
Reduced frequency
CL
α(k)
0 0.5 1 1.5−8
−6
−4
−2
0
2
4
6
Reduced frequency
CM
h(k)
0 0.5 1 1.5−40
−30
−20
−10
0
10
20
Reduced frequency
CM
α(k
)
Figure 11: Dependence of integral aerodynamic coefficients on reduced fre-
quency.
At low to moderate reduced frequencies, both methods yield similar results.
Even the characteristic ’wavy’ behavior for the moment coefficient response to
harmonic heave motion, CMh(k) is represented well. For k in excess of about
0.8, more differences appear. A possible reason may be the added mass effect
of the displacement bodies, i.e. wing thickness and fuselage, which is neglected
completely in the ZONA6 model. This effect would normally be assumed to
be very small for low reduced frequencies, it increases however linearly with
frequency and may play a certain role at high reduced frequencies.
Additionally, it should be noted that the time-domain method requires the
explicit enforcement of the Kutta condition due to the location of collocation
points exactly at the trailing edge, which is not the case in the same way forZONA6. It is not clear to what extent the kinematic (velocity) formulation
chosen here for the Kutta condition yields accurate results for high reduced
frequencies [20, 21].
For better computational efficiency, time-domain simulations can be per-
formed with a non-harmonic input signal, which contains more than a single
frequency. Good results were obtained with a frequency ramp (’chirp’) signal,
E 18 D. Eller and U. Ringertz
for which the frequency increases linearly with time. Using the output of a simu-
lation with a ramped frequency input, the complex coefficients can be computed
at a number of frequencies from the ratio of the Fourier transform of the coef-
ficient time series to the Fourier transform of the input signal. Figure 12 showslift and moment coefficient response to a chirp excitation in heave.
0 0.5 1 1.5−20
−15
−10
−5
0
5
10
15
Reduced frequency
CLh
(k)
0 0.5 1 1.5−8
−6
−4
−2
0
2
4
6
Reduced frequency
CM
h(k)
harmonic motion, realharmonic motion, imag.identification, realidentification, imag.
Figure 12: Identification from harmonic input and chirp response.
Coefficients obtained from the identification procedure are overall fairly
close to those found from single harmonic motion simulations, although small-
scale variations occur between adjacent frequencies. Such a wiggling behavior in
frequency is not realistic and must hence be regarded as computational noise.
The thin solid lines in Figure 12 are approximating 5th order polynomialsfitted to the discrete frequency response data. Alternatively, cubic spline fits
with multiple segments can be used to approximate the frequency dependency.
Frequency-domain stability analysis
Substituting the damped harmonic ansatz x = xept into the aeroelastic equa-
tion of motion (1) yields(
Mp2 + K)
xept = fa(t, p, x, . . . ) (8)
for p = σ + iω, where σ is the damping constant and ω the angular frequency.
Using a modal subspace approach according to (3), the above becomes(
Ip2 + Ω)
qept = ZT fa, (9)
where the deformations are now expressed in terms of the structural eigenmode
shapes Z obtained from the solution of the free vibration eigenproblem (2).
Aeroelastic Simulations of a Sailplane E 19
Hence, q is a vector of complex modal (or generalized) displacements. For
small displacements, the modal aerodynamic loads (or generalized aerodynamic
forces) ZT fa are assumed to depend linearly upon the magnitude of the modal
displacements q, so that (9) can be written as a nonlinear eigenvalue problemaccording to
(
(
c
2u∞
)2
Ip2 + Ω − q∞A(p)
)
q = 0, (10)
where p =σc
2u∞
+ ik. (11)
Here, q∞ is the dynamic pressure and A(p) relates the modal aerodynamic
forces to modal displacements. The real part of the nonlinear eigenvalue p is
the nondimensional damping value and its imaginary part corresponds to the
reduced frequency. In terms of actual deformations x, the solution can be
expressed as
x(t) = Z Re(
qept)
, (12)
where Re(·) extracts the real part. In order to simplify the computation of
frequency-domain aerodynamic loads, it is often assumed that A depends on
the reduced frequency k alone, and not on the damping. This approximation is
exactly fulfilled for a purely imaginary solution pcrit = ik of (10), corresponding
to neutral stability.
Results for the shell finite element model
The nonlinear eigenvalue problem (10) is solved using the robust pk−algorithm
by Back and Ringertz [22], which yields eigenvalues and eigenvectors solving
(10) for a particular dynamic pressure. Figures 13 to 14 show the damping g
and frequency of solutions for airspeeds up to 160 m/s, where the damping isdefined as
g = 2Re(p)
Im(p). (13)
This type of diagram is usually called vg-plot. As the nonlinear eigenvalue
solver does not track modes but instead employs a more robust scheme based
on sorting eigenvalues by reduced frequency, the line colors in the graphs below
identify the position of a solution in the frequency order.
In Figure 13, solutions to the eigenvalue problem are shown for the shellFE model in combination with aerodynamic loads A(k) computed using the
ZONA6 method implemented in ZAERO 6.2 [18]. Although the load matrices
are computed using this software, the pk−solution is performed externally us-
ing a Matlab implementation of the algorithm by Back and Ringertz [22]. The
E 20 D. Eller and U. Ringertz
50 100 150−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
Airspeed [m/s]
Dam
ping
[−]
50 100 1500
5
10
15
20
25
30
Airspeed [m/s]
Fre
quen
cy [H
z]Figure 13: Damping and frequency for shell model and ZONA6 aerodynamics.
shell finite element model used is the simplified version which does not con-
tain structurally modelled control surface mechanisms. Aerodynamic effects of
control surface motions are not modelled either.
Figure 14 shows results computed using aerodynamic loads obtained fromthe identification process described previously, applied to modal displacements
instead of rigid body motion. Time-domain simulations are performed for
50 100 150−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
Airspeed [m/s]
Dam
ping
[−]
50 100 1500
5
10
15
20
25
30
Airspeed [m/s]
Fre
quen
cy [H
z]
Figure 14: Damping and frequency for identified aerodynamics matrices.
prescribed motions in one or two modal coordinates, and the time history of themodal aerodynamic forces is recorded. The input signal is a sine which linearly
increases in frequency up to 40 Hz, corresponding to a reduced frequency of
1.47. One symmetric and one antisymmetric mode are excited simultaneously
in each simulation, and the resulting modal forces are partitioned according
Aeroelastic Simulations of a Sailplane E 21
to symmetry, relying on the assumption that symmetric deformations do not
cause antisymmetric loads and vice versa. After the simulations are completed,
complex frequency-domain loads are computed from the ratio of the Fourier
transforms of output (modal force) and input (modal displacement) signals.
Figures 13 and 14 indicate that both approaches to compute frequency-
domain aerodynamic loads give similar flutter results. In Table 4, the airspeeds
for the first two critical modes are listed with their (absolute and reduced) fre-
quencies at the flutter boundary (g = 0). Note that the computed instabilities
occur far beyond the flight envelope of the ASK 21, which is certified for amaximum permitted flight speed of 78 m/s.
Mode ZONA6 Identification Difference
1 airspeed 126 m/s 126 m/s <1%
frequency 9.65 Hz 9.52 Hz -1.3%k 0.254 0.251
2 airspeed 150 m/s 150 m/s <1%
frequency 16.1 Hz 15.8 Hz -1.9%k 0.356 0.340
Table 4: Critical aeroelastic modes
While there are small differences in the frequencies of the critical modes,
the overall behavior is quite similar. Some of the aeroelastic modes show slightly
more significant differences. As an example, consider the mode which becomes
unstable at 150 m/s. While the aeroelastic system using identified aerody-
namic frequency-domain loads predicts that this particular mode will achieve
its strongest damping of g = −0.18 at about 130 m/s, the model employing
ZONA6 aerodynamics predicts only g = −0.13 at the same airspeed. For air-
speeds slightly larger than 130 m/s, the same mode quickly becomes less strongly
damped.
Figure 15 to 18 show the first critical flutter modeshape for each of the two
aerodynamic methods. Since aeroelastic modes q are complex-valued, real and
imaginary part of the mode are plotted separately. The real part can be under-
stood as the deformation shape at the beginning of the neutrally stable flutter
oscillation, while the imaginary part would be the shape after one quarter of a
period, according to (12). Comparing the modeshapes obtained using differentaerodynamic methods, considerable similarities in the deformation pattern can
be found. Symmetric wing bending is visible in both flutter modes. The bending
pattern is a combination of first and second fundamental wing bending in such
a way that mainly the outboard third of the wing appears to move. Vertical
E 22 D. Eller and U. Ringertz
Figure 15: Real part of the first critical
flutter mode (ZONA6).Figure 16: Imaginary part of the first
critical flutter mode (ZONA6).
Figure 17: Real part of the first critical
flutter mode (identified loads).
Figure 18: Imaginary part of the first
critical flutter mode (identified loads).
fuselage bending and a minor component of symmetric stabilizer bending are
also present in both modeshapes. Even the ratio of fuselage to wing bending
appears to be similar for the two methods. However, the phase between fuselage
bending and wing tip motion is slightly different, as the comparison of the
imaginary part of the modes shows. Since, in this case, the same structural
model is used, the differences must either be caused by different interpolation
of structural eigenmode shapes to the aerodynamic mesh, or by differences in
the computation of modal aerodynamic forces.
The interpolation procedure employed by ZAERO is based on the fitting
of a plate spline to a subset of structural mesh nodes. Visualization of the
ZONA6 mesh, deformed according to structural eigenmodes reveals that the
quality of the spline interpolation procedure depends on the chosen node subset.
Subsets with too many structural nodes lead to ’wavy’ interpolated deformation
shapes, while too small node sets do not define the structural deformationwith sufficient accuracy. For the results shown above, only a small number of
structural, i.e. only 60 of 3800 finite element nodes per half-wing were used, in
order to suppress waviness in the interpolated modeshapes. The selected nodes
cover most of the structural surface, but are not regularly spaced in any way,
which may decrease interpolation accuracy.
For the time-domain aerodynamics used in the identification process, the
deformation of the surface mesh is computed from the shell FE model by as-
Aeroelastic Simulations of a Sailplane E 23
suming that surface mesh points are connected rigidly to the nearest structural
element. Since the distance is very small (typically below 3 mm), the surface
mesh follows the shell model surface very closely except where the structural
model is not defined. Such small ’gaps’ in the FE model exist e.g. between wingroot rib and fuselage. Due to the different interpolation method, the deformed
shape can differ significantly from the deformation of the ZONA6 mesh for the
same modeshape. Such differences are more likely to occur for higher frequency
modes, where local shell deformations become important.
The first and fourth structural eigenmode contain deformations which ap-
pear prominently in both flutter modeshapes, namely first wing bending and
first vertical fuselage bending. In Figure 19, frequency-domain modal aerody-
namic loads computed using the two methods are compared for reduced fre-
quencies up to 1.5.
0 0.5 1 1.5−2
−1.5
−1
−0.5
0
0.5
1
Reduced frequency
A1,
1
DLM, real partDLM, imag. partResponse, realResponse, imag.
0 0.5 1 1.5−0.04
−0.02
0
0.02
0.04
0.06
0.08
Reduced frequency
A1,
4
0 0.5 1 1.5−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
Reduced frequency
A4,
1
0 0.5 1 1.5−0.15
−0.1
−0.05
0
0.05
0.1
Reduced frequency
A4,
4
Figure 19: Four elements of A(k) compared.
Both methods generate similar results for the elements A1,1(k) and A4,4(k),i.e. the aerodynamic loads in modes one and four due to harmonic excitation
of the mode itself, for reduced frequencies up to 1.2. Much less similarity can
be found in the off-diagonal terms A1,4(k) and A4,1(k). The imaginary part
E 24 D. Eller and U. Ringertz
of the former shows quite significant deviations, which appear to be systematic
differences in the aerodynamic model, as the values obtained from the discrete
Fourier transform are not too far from some smooth function of frequency. In
the latter case, however, the identification process using the FFT failed. Realand imaginary part of the response are found to vary wildly with frequency
and deviate strongly from values computed with the ZONA6. Such an erratic
behavior indicates that the modal force response to chirp excitation for the
corresponding time-domain simulation is either noisy or does not resemble the
input signal sufficiently.
Figure 20 shows the time history of aerodynamic forces in mode 1 and 4
due to a chirp excitation of the first structural eigenmode. While the response
0 0.2 0.4 0.6 0.8 1−3000
−2000
−1000
0
1000
2000
3000
Time [s]
Fq,
1
0 0.2 0.4 0.6 0.8 1220
240
260
280
300
320
340
Time [s]
Fq,
4
Figure 20: Time-domain response to excitation of the first mode.
of the first mode itself is smooth and directly correlated to the modal veloc-
ity of the excitation signal, the force in the fourth mode is of much smaller
amplitude and more noisy. Furthermore, the time history looks more like a
response to two superimposed excitation signals. Since such a signal clearly is
not properly correlated to a single input, the simple identification procedure
fails to yield reasonable results. In this particular case, the level of noise may
be related to the extreme difference in magnitude between modal force compo-
nents. The large values of modal force in the first mode indicate that the surface
pressure distribution must differ substantially from the equilibrium state. Sincethe modal force amplitude in the fourth mode is about 40 times smaller, the
apparent superposition in this mode can be related to a numerical integration
error. This problem is also encountered when computing small integrated force
coefficients such as induced drag in the presence of large pressure differences.
Aeroelastic Simulations of a Sailplane E 25
Results for the beam finite element model
Finally, the same flutter analysis as above has been performed with the shell
finite element model replaced by the beam model and ZONA6 aerodynamics.
Damping and frequency over velocity are shown in Figure 21. Interestingly,there is no instability in the speed range of interest, and even the development
of the eigenfrequencies of the aeroelastic modes is quite different. In contrast
to the flutter solution based on the shell model, almost all eigenfrequencies
below 30 Hz rise with airspeed. With the shell FE model, several of the higher
frequency eigenmodes show a distinct drop in frequency above 100 m/s, with
possible divergence in the vicinity of 160 m/s.
50 100 150−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
Airspeed [m/s]
Dam
ping
[−]
50 100 1500
5
10
15
20
25
30
Airspeed [m/s]
Fre
quen
cy [H
z]
Figure 21: Damping and frequency for beam FE model.
The very significant differences between flutter solutions using the beamand the shell models emphasize the utmost importance of an accurate finite
element model for aeroelastic analysis. That does not necessarily mean that a
high-fidelity shell model must be used in every case. As seen in Table 1, the
differences in modal properties of beam and shell model cannot be regarded as
drastic, but still the flutter solutions show very few similarities. This indicates
that an aeroelastic analysis may not be meaningful unless the structural model
reproduces the actual flying structure very accurately.
Maneuver simulation
Due to the difficulties and risks associated with flutter flight testing, a more
benign maneuver flight test serves to compare aeroelastic simulation results with
experimental data. The flight maneuver is a pull-up with a maximum load
factor of about 4, close to the saturation limit of the installed attitude and
E 26 D. Eller and U. Ringertz
heading reference system (AHRS). The test aircraft is the ASK 21 shown in
Figure 1 which is equipped with position sensors at the control surfaces, solid-
state AHRS and fitted with an air data boom with vanes to measure angle of
attack and sideslip. All of these sensors are currently sampled with 10 Hz usinga digital on-board data acquisition computer. Furthermore, 16 piezoelectric
accelerometers are installed in the aircraft, which are sampled with 1 kHz.
The recorded elevator deflection is used as input for the numerical sim-
ulations, where its value is shifted by a constant in order to achieve initial
equilibrium conditions. The first simulation is performed with a flexible air-
craft model using the shell finite element model with eigenmodes up to 40 Hz.
A further numerical simulation is performed with exactly the same simulation
code, but without accounting for flexibility. The only difference in the input
signal between the two simulations is in the definition of the initial state, as
both computations must be started from an equilibrium state. This requires a
different initial elevator deflection (shifted by +2.3) for the rigid model, and
a slightly smaller angle of attack (reduced by 0.8). Moreover, both numeri-
cal simulations employ a simple model for the parasite drag which is addedto the induced drag obtained from surface pressure integration. Drag data was
extracted from the measured speed polar diagram in the aircraft datasheet [10]
and extrapolated for airspeeds in excess of 200 km/h. Since the maneuver is
performed in the vertical plane only, the numerical simulation of the flight me-
chanics was limited to longitudinal states, and lateral velocities and angles were
held fixed.
The time history of the pull-up in terms of altitude, calibrated airspeed and
load factor is shown in Figures 22 – 24. Measured airspeed and altitude are de-
0 2 4 6−8
−6
−4
−2
0
2
Time [s]
Ele
vato
r de
flect
ion
[deg
]
0 2 4 6
−20
−10
0
10
20
30
40
50
Time [s]
Flig
ht p
ath
angl
e [d
eg]
Flight testSimulation, flexibleSimulation, rigid
Figure 22: Measured elevator deflection and processed flightpath angle.
Aeroelastic Simulations of a Sailplane E 27
rived from pressure measurements using standard atmosphere conditions, while
the flight path angle in Figure 22 is computed from low-pass filtered altitude
and velocity data. The latter should therefore be understood as approximate be-
cause the angle is not measured directly. However, the initial short drop by 5 aswell as the steep change at t = 5.5 s hint at the presence of thermal drafts, since
comparison with the elevator position in the same figure indicate that these
rapid changes are not related to the commanded maneuver. Figure 23 shows
pressure altitude and airspeed, which are both reasonably well reproduced by
the numerical simulations. After six seconds simulated flight, the accumulated
0 2 4 6140
160
180
200
220
240
Time [s]
Airs
peed
[km
/h]
Flight testSimulation, flexibleSimulation, rigid
0 2 4 6
420
440
460
480
500
520
540
Time [s]
Alti
tude
[m]
Figure 23: Time history for pressure altitude and calibrated airspeed.
error reaches about 10 km/h and 15 m in altitude. The measured altitude data
contains a sudden jump of about 5 m at t ≈ 0.8 s, which may be caused by pres-
sure measurement noise. Accounting for the use of an approximate drag model,
the accuracy of the numerical simulation in terms of velocity and altitude is
considered reasonable.
Figure 24 shows load factor and pitch rate as obtained from flight test, along
with values computed by numerical simulation with rigid and flexible aircraft.
The global development of load factor and pitch rate is matched quite accurately
by the numerical time-domain simulation. In particular, the accumulated error
after completion of the maneuver is surprisingly small. It was expected that
differences between computed and actual control surface effectiveness would
lead to more significant error accumulation over the simulated 3000 timesteps.
The magnitude of vertical accelerations caused by thermal drafts can be esti-
mated by considering the measured load factor at t ≈ 5.5 s, just before the end
of the time window. Here, the load factor drops by about 0.2 for less than 0.5 s,
while the recorded elevator deflection is essentially flat. Consequently, neither
E 28 D. Eller and U. Ringertz
of the simulations contains this load excursion. Contrary to expectations, the
0 2 4 60
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time [s]
Load
fact
or [−
]
0 2 4 6−10
−5
0
5
10
15
20
25
30
Time [s]P
itch
rate
[deg
/s]
Flight testSimulation, flexibleSimulation, rigid
Figure 24: Load factor and pitch rate.
simulation of the elastic aircraft did not show large differences from the rigid
aircraft flight simulation. Only during the second half of the maneuver, an
error in pitch rate appears to accumulate for the rigid case. At the end, the flexi-
ble aircraft simulation still matches the experimentally obtained pitch rate fairly
well, while the rigid aircraft simulation is off by -3/s. Similarly, the flight path
angles of the two simulations diverge in the second half of the maneuver, to
reach a significant difference of about 10 at the end of the time window. How-
ever, since direct measurements for this angle are not available, it is difficult to
conclude if the elastic aircraft simulation improved accuracy.
The test aircraft is fitted with piezoelectric acceleration sensors which are
primarily intended for vibration testing. Each sensor contains a capacitance-
resistance circuit acting as a high-pass filter with a time constant of one second.
In order to enable interpretation of the measured data over the relatively long
time range considered, the raw accelerometer data was processed by filtering
with the inverted high-pass filter. This process introduces a linear drift sincethe raw electrical accelerometer data also contains a constant feed voltage signal.
Therefore, the corresponding linear term was subtracted from the acceleration
signal.
Figure 25 shows a comparison of the time history of measured wing tip
acceleration, when the cockpit acceleration signal as provided by the AHRS issubtracted. This difference can be regarded as an approximation of the tip mo-
tion due to elastic deflection. Furthermore, acceleration data for the simulation
is shown. It is obtained by finite-differencing the velocity of a mesh node at the
accelerometer position.
Aeroelastic Simulations of a Sailplane E 29
0 2 4 6−10
−5
0
5
10
Time [s]
Acc
eler
atio
n [m
/s2 ]
Flight testSimulation
0 10 20 30 40−40
−35
−30
−25
−20
−15
−10
−5
0
5
Frequency [Hz]
Pow
er s
pect
ral d
ensi
ty [d
B/H
z]Figure 25: Time history and PSD estimate for wing tip acceleration.
The measured time history contains considerably larger acceleration ampli-
tudes than simulation data. Moreover, the strong peaks do not correlate to the
known flight mechanic motion in an obvious manner. The acceleration data
from the simulated maneuver shows a considerably smaller variation and even
allows to recognize two points with particularly strong signals, i.e. pronounced
wing bending and torsion motion, at t = 1.5 s and t = 3.5 s. These points
correspond to the beginning and the end of the pull-up maneuver as seen in
Figure 24, where the load factor varies rapidly. There may be a similar patternin the measured data, but that is somewhat masked by the much stronger and
apparently noise-related peaks.
The right part of Figure 25 shows the Yule-Walker power spectral density
(PSD) estimate of the time series shown to the left. This estimate is the PSD of
a discrete autoregressive time-domain model fitted to the time series data and
yields spectrum information which contains much less variation in frequency
than the raw FFT. It is thus easier to identify the frequency content of thecorresponding signals, although the absolute value of the PSD would not be
directly useful, as it depends somewhat on the parameters used in the method.
Here, the autoregressive model was chosen to span 320 milliseconds, that is
order 320 for the measured data and order 160 for the accelerations obtained
from simulation.
First of all, the measured data comprises significantly higher magnitudesthroughout most of the frequency range considered, which is expected from the
time history. In addition, its PSD is considerably less varying, which indicates
that random processes (such as atmospheric turbulence) could be responsible
for some of the large magnitudes. The PSD estimate of the simulated accelera-
E 30 D. Eller and U. Ringertz
tions, on the other hand, contains several articulated peaks at 10 Hz, 20.5 Hz,
23.5 Hz, 28 Hz and 30 Hz. Most of these frequencies can be identified with
structural eigenfrequencies associated with symmetric modes (see Table 1), which
is expected since the representation of the structural behavior underlying the nu-merical simulation is based on the same modal data.
Computational effort
Structural modal analysis of the NASTRAN shell model shown in Figure 2 con-
tributed only a very minor amount of computational work; the extraction of the
first 30 eigenmodes was usually computed within 5 minutes. Evaluation of the
aerodynamic influence coefficient matrices required for the flutter analysis usingZAERO took approximately 2 hours on the same machine. This computation
need only be performed when the geometry or discretization of the panel mesh
has changed.
Time-domain simulations for the evaluation of frequency-domain load ma-
trices using the chirp response approach were far more computationally inten-
sive. The results presented here required approximately 11 hours computation
time on a dual-processor 2.4 GHz Opteron computer. On the same machine,
the maneuver simulation of the flexible aircraft with 3000 timesteps ran for 26hours. The current mesh-deformation procedure is adapted for arbitrary struc-
tural deformations and therefore performs a number of computations which
may not be necessary if deflections remain small. For that particular case, fur-
ther improvements of computational efficiency are possible.
Conclusions
When comparing aeroelastic stability analysis performed with a beam and ashell finite element model of the same aircraft, considerable differences in aeroe-
lastic stability are found although the structural models match fairly well in
the eigenfrequencies up to 15 Hz. Even though the first flutter mode detected
in the analyses involving the shell models is close to 10 Hz, the aeroelastic
system based on the beam model is stable throughout the whole speed range
investigated. This behavior leads to the conclusion that aeroelastic stability is
highly sensitive to differences in the structural model, even if these differences
are limited to eigenmode shapes with moderate and high frequencies.
A time-domain boundary element method was used to identify frequency-
domain aerodynamic loads from ramped frequency response simulations. In
most cases, the aerodynamic behavior thus obtained was found to differ lit-
tle from the aerodynamic loads computed by the frequency domain method
Aeroelastic Simulations of a Sailplane E 31
ZONA6. Flutter computations performed using the pk−method and with both
types of aerodynamic loads showed closely matching flutter speeds and similar
flutter modeshapes. However, the accuracy of the identification method degrades
significantly for modal aerodynamic loads of small relative magnitude and thecomputational cost of the identification process is comparatively high.
Finally, a 4g pull-up maneuver was performed with an instrumented flight
test aircraft. Numerical simulations which used the measured time history of
the elevator deflection as input demonstrated that the flight mechanic behav-
ior of the sailplane was represented fairly well by the simulation. Comparingsimulations including elastic deformations with those for the rigid aircraft, dif-
ferences were found to be present, but smaller than anticipated from the large
load factor.
Since control system modeling is known to have a strong effect on aeroelas-tic stability, in particular if the control surfaces are not mass balanced, future
efforts in this project will be concerned with the accuracy of control surface
aerodynamics and the mechanical properties of the control system. It is not
unusual that mild instabilities are tolerated in the certification analysis if the
damping required to stabilize is very small. A positive (unstable) value of the
damping parameter g less than 0.03 is sometimes accepted to account for un-
modeled damping in the structure and friction in the mechanical control system.
However, the damping defined by (13) is physically very different in comparison
to the stick-slip type of Coulomb friction that typical control surfaces experi-
ence. Accurate modeling of Coulomb friction requires a nonlinear time-domain
analysis for which the developed aerodynamic model is very suitable.
Acknowledgments
The authors would like to thank Michael Greiner and Gerhard Waibel of Alexan-der Schleicher Flugzeugbau for their valuable support in providing design draw-
ings, lamination plans and even rotational inertia data for control system com-
ponents. Sebastian Heinze implemented and tested the interpolation spline data
required to use the shell finite element model with ZAERO.
References
[1] E. H. Dowell, H. C. Curtiss, R. H. Scanlan, and F. Sisto. A modern course
in aeroelasticity. Kluwer, Dordrecht, 1989.
[2] R. L. Bisplinghoff, H. Ashley, and R. L. Halfman. Aeroelasticity. Dover
Publications, 1955.
E 32 D. Eller and U. Ringertz
[3] W. Stender and F. Kiessling. Aeroelastic flutter prevention in gliders and
small aircraft. Technical report, DFVLR e.V., Gottingen, 1990.
[4] W. Stender. Praxisnahe Abschatzungs- und Vorbeugungsmoglichkeiten
gegen die Flattergefahrdung von Segelflugzeugen und kleinen Motor-
flugzeugen. Technical Report IB 151-74/6, 151-74/20 and 151-74/22,
DFVLR e.V., Braunschweig, October 1974. Part I-III.
[5] European Aviation Safety Agency. CS-22, Certification specifications
for sailplanes and powered sailplanes, November 2003. Available at
www.easa.eu.int.
[6] T. Brama. The structural optimization system OPTSYS. International Series
of Numerical Mathematics, 110:187–206, 1993.
[7] W. P. Rodden and E. H. Johnson. Msc.Nastran Aeroelastic Analysis User’s
Guide. MacNeal-Schwendler Corp., Los Angeles, 1994.
[8] V. J. Stark. The AEREL flutter prediction system. In ICAS-90-1.2.3, 1990.
[9] ZONA Technology. ZAERO Version 6.2 Theoretical Manual, 17th edition,
October 2002. Available at www.zonatech.com.
[10] Alexander Schleicher Flugzeugbau, Poppenhausen. Datasheet for the ASK 21,
July 2003. Available at www.alexander-schleicher.de.
[11] A. Keller. Aeroelastic model development for the ASK 21 glider aircraft.
Master’s thesis, AVE, Kungliga Tekniska Hogskolan, Stockholm, October
2004.
[12] A. Westergren. Structural dynamics of the ASK 21 glider wing. Master’s
thesis, AVE, Kungliga Tekniska Hogskolan, Stockholm, June 2003.
[13] O. Bergogne. A simulation model of the ASK 21. Technical report, AVE,
Kungliga Tekniska Hogskolan, Stockholm, August 2003.
[14] R. Lardet. Flight dynamics model of the ASK 21 sailplane. Master’s thesis,
AVE, Kungliga Tekniska Hogskolan, Stockholm, May 2003.
[15] MacNeal-Schwendler Corp. Nastran Reference Manual, 2004. Msc.Nastran
2004.
[16] U. Nilsson, M. Norsell, and U. Carlsson. Ground vibration testing of
the ASK 21 glider aircraft. Technical Report TRITA/AVE 2003:32, AVE,
Kungliga Tekniska Hogskolan, Stockholm, May 2003.
Aeroelastic Simulations of a Sailplane E 33
[17] N. Niedbal and A. Bertram. Flatteruntersuchung ASK 21. Technical report,
DFVLR e.V., Gottingen, 1979.
[18] P. C. Chen, H. W. Lee, and D. D. Liu. Unsteady subsonic aerodynamics
for bodies and wings with external stores including wake effort. Journal of
Aircraft, 30(5):618–628, September 1993.
[19] D. Eller. An efficient boundary element method for unsteady low-speed
aerodynamics in the time domain. Technical Report TRITA/AVE 2005:40,
AVE, Kungliga Tekniska Hogskolan, Stockholm, December 2005.
[20] J. Katz and D. Weihs. Wake rollup and the Kutta condition for airfoils
oscillating at high frequency. AIAA Journal, 19(12):1604–1606, December
1981.
[21] J. Katz and A. Plotkin. Low Speed Aerodynamics. Cambridge University
Press, second edition, 2001.
[22] P. Back and U. T. Ringertz. Convergence for methods for nonlinear eigen-
value problems. AIAA Journal, 35(6):1084–1087, June 1997.
top related