AIAA 2015-0177
AIAA SciTech
5-9 January 2015, Kissimmee, Florida
56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
Reduced-order modeling of loads and deformation of a
flexible flapping wing
Jason Tran∗ and Jayant Sirohi†
The University of Texas at Austin, Austin, TX, 78712, USA
Haotian Gao‡ and Mingjun Wei§
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New Mexico State University, Las Cruces, NM, 88003, USA
This paper describes a methodology to predict the loads and deformation of a flexible
flapping wing. An experimental setup capable of imparting a single degree of freedom
flapping motion while measuring loads and deformation of a flapping wing was developed.
Experimental results were obtained for a simple rectangular plate flapping in air at hover
conditions. The deformation results show agreement with previously published data. The
experimental setup is developed in conjunction with a reduced-order model for loads and
deformation. A POD-Galerkin projection method is used to develop the model. The analytical results show a good correlation between direct numerical simulation and projection.
The focus of the present work is hovering flight with simple kinematics and wing properties.
I.
Introduction
ecently, there has been considerable interest in bio-inspired, flapping wing micro aerial vehicles (MAVs)
R
of a size comparable to that of small birds and large insects. These MAVs have the potential to perform
stealthy surveillance and reconnaissance missions by hiding in plain sight. In addition, it may be possible to
improve the performance and maneuverability of MAVs by incorporating bio-inspired design concepts. For
example, Keennon et al.1 developed a mechanical hummingbird with a wing span of 16.5 cm and a total
mass of 19 g. deCroon et al.2 developed a family of flapping wing MAVs with total mass ranging from 21 g
to 3 g. In spite of successful flight demonstrations, these MAVs are heavier, have lower endurance and lower
maneuverability compared to similarly sized natural flyers.
Development of practical, bio-inspired, flapping wing MAVs requires optimization of the aerodynamic
efficiency of flapping wings, accurate modeling of the vehicle flight dynamics and incorporation of a robust
control system. Each of these tasks relies upon the calculation of the loads generated by flapping wings
across a range of operating conditions, such as flapping frequency, stroke kinematics and flight speed. Several researchers have developed analytical and numerical models of the forces generated by flapping wings.
Examples of structural analyses include: prescribed elastic deformation,3 finite element structural models4, 5
and multibody dynamics.6, 7 Aerodynamic analyses include strip theory,8 panel methods,9 unsteady vortex
lattice methods,10 indicial functions5 and computational fluid dynamics using a Reynolds-Averaged Navier
Stokes solver in conjunction with deformable, body conforming grids.11 The calculation of the coupled
aeroelastic response of a flapping wing becomes increasingly challenging as the flexibility of the wing and
the flapping frequency increase. Accurate prediction of the unsteady aerodynamics and large structural
deformations, especially at high flapping frequencies, requires refined coupled computational fluid dynamics
and computational structural dynamics tools. These can yield accurate results but are computationally
expensive and are not amenable to implementation of real-time control.
Spurred by the need for real-time implementation, researchers are developing reduced-order models of
flight dynamics12 and performing system identification of flapping wing MAVs in flight.13 Implementation
∗ Graduate
Research Assistant, Department of Aerospace Engineering and Engineering Mechanics.
Professor, Department of Aerospace Engineering and Engineering Mechanics, AIAA Associate Fellow.
‡ Graduate Research Assistant, Department of Mechanical and Aerospace Engineering.
§ Associate Professor, Department of Mechanical and Aerospace Engineering, AIAA Associate Fellow.
† Associate
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Copyright © 2015 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
of these models in practical flapping wing MAVs requires not only real-time calculation of the vehicle flight
dynamics but also the measurement of loads generated by the flapping wings. In addition, optimization of
the wing structure, in terms of planform as well as flexibility and mass distribution, requires parametric
trade studies of many variables.
This paper describes the reduced-order modeling of a flexible flapping wing, focused on the prediction
of loads and wing deformation. Experiments were performed on a 100 mm span flat plate wing, undergoing
a sinusoidal flapping motion in quiescent air. The loads and deformation of the wing are measured as a
function of flap angle. The wing deformation was measured using Digital Image Correlation(DIC) and the
loads were measured using a six-component load cell. The experiments are repeated with the wing clamped at
its mid-chord and quarter-chord. The measured loads and deformation are used to validate a reduced-order
model developed using a POD-Galerkin projection.
II.
Experimental Setup
A picture of a rectangular wing mounted in the test setup is shown in figure 2. The whole assembly was
mounted approximately 1 m above the ground. The rectangular flat plate composite wings were constructed
from two plies of carbon fiber prepreg (AS4/3501) in a hot compression mold. The wing was clamped at
the root 14 mm from the motor shaft axis. The parameters of the wings are listed in table 1. A single-wing
configuration clamped at either mid or quarter-chord was tested. The mid-chord clamped wing will induce
only bending deformation, and the quarter-chord clamped wing will induce bending as well as twisting
deformation.
Table 1: Wing parameters
Span, mm (in)
Chord, mm (in)
Thickness, mm (in)
Root cut-out, mm (in)
Aspect ratio, Mass, g (oz)
Young’s modulus, GPa
Shear modulus, GPa
Figure 1: Rectangular wing with speckled pattern
L
c
t
x0
AR
m
E
G
102 (4.01)
25 (0.98)
0.43 (0.017)
9 (0.35)
4.08
1.5 (0.053)
77
6.5
40
Z
Y
X
Flap Angle (deg)
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Wing Properties
20
0
−20
−40
0
Figure 2: Experimental setup
Target profile
PVT points
Measured response
0.0667
0.1333
0.2
Time (sec)
Figure 3: Target flapping motion: 15 Hz, 80◦ stroke
amplitude sinusoid
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The wings were tested on an experimental setup capable of imparting an arbitrary flapping motion.
The majority of previous work in flapping wings has considered only purely sinusoidal flapping. Biological
flyers do not limit themselves to a symmetric flapping motion, for example certain insects are seen to use
asymmetric periodic flapping motions.14 This study only uses a sinusoidal flapping profile, but future work
will include arbitrary motion profiles.
The test setup consists of a 60 W (peak) brushless DC servomotor (Maxon EC 16) output attached to a
gearhead with a reduction ratio of 4.4:1 (Maxon GP 22 C). The wing is rigidly attached to the shaft such
that rotation of the shaft results in flapping of the wing. The DC motor incorporates a 3-channel encoder
(Maxon Encoder MR type M) with a resolution of 512 pulse/turn which gives a total of 2048 quadrature
counts/turn for an angular resolution of 0.176◦ .
Shaft angular position feedback from this encoder, in conjunction with a motor controller (Maxon EPOS2
24/5), is used to impose arbitrary flapping motion on the wing. The motor controller uses a PID control
mechanism combined with acceleration and velocity feed forwards to control the flapping wing. The desired
arbitrary motion is prescribed to the motor controller using the Interpolated Position Mode described by
Maxon.15 The motion profile is sent to the controller in the form of discrete points consisting of position,
velocity, and timestep (PVT) information that is used for a cubic interpolation. The current setup limits
the minimum timestep of data that can be used to generate a motion profile to 3 ms. Therefore, a 15 Hz
sinusoidal motion profile is interpolated using about 22 PVT points per period.
Flapping a wing directly attached to the motor shaft results in a high inertia load on the shaft. Because of
the wing inertia, the prescribed motion is not exactly followed by the motor. This is remedied by modifying
the input motion profile so that the desired motion profile is recorded (i.e. a 30 degree amplitude, 15 Hz
sinusoidal input gives a 40 degree amplitude, 15 Hz sinusoidal measured response). An example of target
motion profile, input PVT points, and measured response are shown in figure 3.
Wing Loads Measurement
The entire motor-wing assembly is mounted on a six-component, strain-gage based load cell (ATI Nano
43) to measure forces and moments produced by the flapping wing. The full scale range of the load cell is 9
N and 125 N-mm with resolutions of 0.002 N and 0.025 N-mm. A LabVIEW virtual instrument is used to
control both the load cell and the motor controller. Data from the load cell and motor encoder is collected
simultaneously at a sampling frequency of 7.5 kHz so that the load measurements can be correlated to the
motion profile.
The load measurements include all inertial and aerodynamic loads. Data is taken for at least 1000 cycles
and then synchronously averaged to increase the signal to noise ratio. A fast Fourier transform is applied to
the synchronously averaged data that has been concatenated to itself multiple times to increase resolution
in frequency. The loads introduced by the motor and gearbox are at a higher frequency than the flapping
motion profile and are filtered out using a zero-phase digital Butterworth filter with a cutoff frequency of 90
Hz.
1
2
0.75
Raw data
Phase averaged
Averaged and filtered
1
Fz (N)
Fz magnitude (N)
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Flapping Wing Mechanism
0.5
0
−1
0.25
−2
0
0
100
200
300
400
500
0
Time (sec)
Figure 5: Load data post-processing of Fz
Frequency (Hz)
Figure 4: FFT Results of Fz
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0.0667
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Deformation Measurement
The three-dimensional, whole-field wing deformation is measured using a non-contact optical technique
called Digital Image Correlation (DIC).16, 17 In this technique, a high-contrast, random speckle pattern is
painted on the wing surface (see the wing in figure 1). Digital images of the test specimen captured before
and after application of the loading are cross-correlated to determine how points on the specimen surface
have moved in response to the load. Using photogrammetric principles, the absolute positions of the points
on the surface are calculated. Two digital cameras (Imager ProX 2M) with a resolution of 1600x1200 pixel
and 29.5 Hz operation speed are placed below the wing and are oriented such that the wing span is aligned
with the longer dimension of their CCD sensor. The exposure time of the cameras can be as low as 500 ns
and their aperture can be adjusted from f1.8 to f22.
The motor controller generates a signal at prescribed locations to trigger a strobe light (Shimpo DT311A). The light source is a 10W Xenon bulb and the flash has a duration of 10 to 40 µs. With this method,
high resolution images of the flapping can be captured without motion blur.
For each wing studied, images are taken at several points along the motion profile. This is done by
indexing the mean flapping plane with respect to the imaging plane, so that the 3D wing deformation is
measured at several instances over the flapping cycle. With these discrete measurements, the dynamic wing
deformation over a flapping cycle can be reconstructed. Images are compared to an undeformed reference
image that is taken at zero velocity. It is assumed in this study that deformation due to gravity is negligible
compared to deformation due to inertial and aerodynamic loads.
The post-processing of the images is done in commercial software (StrainMaster 3D)18 to find surface
heights and 3D displacement vectors. An interrogation window size of 31x31 and a step size of 14 pixels was
used. The pixel/mm scale factor of the setup is 10.9. The result of these parameters is a displacement field
with a grid spacing of about 1.3 mm. Theoretical error estimates of DIC measurements can be made using
parameters such as the pixel/mm scaling factor, interrogation window size, and post-processing interpolations
as described in.18
III.
Analytical Methods
POD-Galerkin projection provides a systematic way to build reduced-order models (ROM) for complex
flow systems.19, 20 There have been many studies for the method being applied on Navier-Stokes equations:
1 2
∂u
+ u · ∇u = −∇p +
∇ u,
∂t
Re
where u is velocity vector, p is pressure, and Re is the Reynolds number. The corresponding ROM is:
N
ȧi =
N
(1)
N
XX
1 X
qijk aj ak ,
lij aj +
Re j=0
j=0
i = 1, ...N,
(2)
k=0
where lij = h∇2 φj , φi iΩ and qijk = h∇·(φj φk ), φi iΩ . However, this equation is only valid for flows with fixed
domains. To study flapping wings, a modified approach to apply POD-Galerkin projection in a combined
domain with a modified Navier-Stokes equation for a fluid and moving structures was recently proposed:21–23
∂u
1 2
+ u · ∇u = −∇p +
∇ u + f,
(3)
∂t
Re
where f is the extra bodyforce term added in solid domain Ωs to define the trajectory of solid boundary/structure:

(u · ∇u − 1 ∇2 u)n + 1 (V − un ) in Ω
s
Re
4t
f=
,
(4)
0
otherwise
with V being the prescribed velocity of solid, the pressure term has been neglected. The classical approach
for fixed fluid domain can then be directly applied on the whole fluid-solid domain to get
ȧi =
N
N X
N
X
X
1
0
0
)aj −
(qijk − qijk
)aj ak + ci ,
( lij − lij
Re
j=0
j=0
k=0
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i = 1, ...N,
(5)
which is similar to the traditional form in (2). However, this dynamic equation (5) has some new parameters
defined by inner products with support only in the solid domain:
1 2
1
∇ φj +
φ ), φi iΩs (t) ,
Re
∆t j
0
qijk
= h∇ · (φj φk ), φi iΩs (t) ,
1
c0i = h Vn , φi iΩs (t) ,
∆t
0
lij
= h(
(6)
(7)
(8)
IV.
Results and Discussion
Experimental Results
This section reviews the relations that can be seen from the loading and deformation measurements. The
Reynolds number for the hovering case studied in the experimental setup is approximately 7000. The motion
profile and its relation to the measured forces and deformations is shown in figure 6. The labeled points are
corresponded to its time within a flapping cycle in table 2. Stroke reversal occurs at points E and M. The
coordinate system follows figure 2.
40
E
D
30
Flap Angle (deg)
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Also included are the old parameters that have a similar definition, but are extended to support the
entire domain, combining both fluid and solid. It is worth noting that the corresponding global POD modes
are also defined by inner products in the combined fluid-solid domain without differentiating between fluid
and solid. The results from preliminary numerical study on simple configurations are encouraging.24 In the
present paper, this same approach is applied on the flapping wing used in our experiments to test the ROM
in a more practical manner.
C
20
10
0 A
B
Downstroke
Upstroke
F
Table 2: Time along flapping cycle
G
H
I
J
−10
P
K
−20
O
L
−30
N
M
−40
0
0.5
1
t/T
A
B
C
D
E
F
G
H
t/T
0.000
0.040
0.083
0.135
0.283
0.365
0.417
0.460
I
J
K
L
M
N
O
P
t/T
0.500
0.540
0.583
0.635
0.783
0.865
0.917
0.960
Figure 6: Motion profile with labeled points denoting deformation measurement points
The synchronously averaged and filtered force and moment data are shown in figures 7 and 8. The
chordwise direction force, Fy , should be zero given that the wing is uniformly stiff. The measured Fy can
be seen to be very small when compared to Fx and Fz . The largest frequency component of Fx and Mz is
30 Hz while the other loads follow the 15 Hz flapping frequency. Fx is max just after midstroke and stroke
reversal while Fz is max a few moments after the stroke reversal at points F and N.
Figure 10 shows bending displacements, W, for several points along the motion profile. The displacements
for the quarter-chord clamped wing are greater than those of the mid-chord clamped wing. The same
relationship can be seen in the load measurements. The plot of tip displacement versus point on motion
profile on the 3D line-plot shows an out of phase sinusoidal profile compared to the motion profile. The
shifted phase delays the zero-tip displacement point until points B or J. The surface plots in figure 9 show
the deformed shape of the wing at point F on the motion profile. Twist can clearly be seen in the quarterchord clamped wing while there is not any in the mid-chord clamped wing.
The max tip displacements were found to occur at stroke reversals (points E, M) and the minimum tip
displacements were found to be after the midstroke (points B, J). The max displacement points occur where
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1
Downstroke
Upstroke
0.75
30
20
Mx (N−mm)
0.5
Fx (N)
0.25
0
0
−10
−0.5
−20
−0.75
−30
−1
0.04
75
50
My (N−mm)
Fy (N)
0.02
0
25
0
−25
−0.02
−50
−0.04
−75
1
30
0.75
20
Mz (N−mm)
0.5
0.25
Fz (N)
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−0.25
10
0
−0.25
−0.5
10
0
−10
−20
−0.75
−30
−1
AB C D
E
F GH I J K L
M N OP
AB C D
E
t/T
F GH I J K L
t/T
Figure 7: Loads measured with mid-chord clamped wing
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M
N OP
1
Downstroke
Upstroke
0.75
30
20
Mx (N−mm)
0.5
Fx (N)
0.25
0
−0.25
10
0
−10
−0.5
−20
−0.75
−30
−1
0.04
75
50
My (N−mm)
Fy (N)
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0.02
0
25
0
−25
−0.02
−50
−0.04
−75
1
30
0.75
20
Mz (N−mm)
0.5
Fz (N)
0.25
0
−0.25
−0.5
10
0
−10
−20
−0.75
−30
−1
AB C D
E
F GH I J K L
M N OP
AB C D
E
F GH I J K L
t/T
M
N OP
t/T
Figure 8: Loads measured with quarter-chord clamped wing
(b) Quarter-chord clamped wing
0
25
y (mm)
W (mm)
y (mm)
−1.5
0.0
−7.5
0
0
−15.0
−1.25
0
25
x (mm)
−2.5
0.0
−10.0
0
0
10 20 30 40 50 60 70 80 90 100
0.0
25
U (mm)
y (mm)
y (mm)
−0.75
U (mm)
0.0
25
W (mm)
(a) Mid-chord clamped wing
−20.0
10 20 30 40 50 60 70 80 90 100
x (mm)
Figure 9: Surface plots of displacement vectors at point N of the motion profile. U is in the x-direction and
W is in the z-direction.
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25
25
20
20
E
15
F
G
D
10
H
5
CI
0
J
B
−5
A
K
−10
P
−15
L
O
N
E
F
DG
10
H
C
I
J
B
A
K
P
5
0
−5
−10
LO
N
−15
Bending displacement (mm)
15
M
−20
−20
−25
0
−25
0
M
10
20
30
40
50
60
70
80
90
100
Distance along mid−chord wing span (mm)
10
20
30
40
20
15
10
5
0
−5
−10
−15
−20
−25
0
20
40
g sp
60
an (
60
70
80
(b) Quarter-chord clamped wing
25
Win
50
80
mm
)
90
100
Distance along quarter−chord wing span (mm)
(a) Mid-chord clamped wing
Bending displacement (mm)
Bending displacement (mm)
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both Fx and Fz are near maximum. The max in Fz occurs at points F and N but these points are not where
max deflection occur because Fx rapidly drops off due to its higher frequency. The jump in Fz at points F
and N is a result of aerodynamic loads because it occurs after max acceleration of a sinusoidal motion.
Similar bending displacement measurements have been performed by Wu et al.16 They tested Zimmerman planform wings composed of a carbon fiber skeleton and membrane skin. They found that max tip
displacement occurred at approximately point F and N of the sinusoidal motion profile. The max displacements at stroke reversals found in this paper does not correlate with the findings by Wu et al. This is likely
due to the much higher stiffness and weight of the wing used in this paper. The more flexible and lighter
wing of Wu is more affected by aerodynamic loads while the stiffer and heavier wing of this study is more
affected by inertial loads. However, the vacuum condition tests performed by Wu et al. removes the effects of
aerodynamics so that deformation are due only to inertia. The vacuum tests showed max tip displacements
at stroke reversal which matches what was found in this paper.
Max error estimates for this study can be extracted directly from the displacement vectors calculated.
The wing has been clamped with a rigid bar that is visible in the DIC images, and the displacement vectors
of that bar should be zero. These have been zeroed in figure 10 but are actually calculated to be close to
zero in the post-processing. The max calculated W displacement vector along the rigid clamp ranged from
0.1 to 0.5 mm along the motion profile. This may be reduced if a smaller interrogation window is used.
100 AB C D
L
E F G HI J K
M N OP
n profile
Point on motio
(c) 3D line-plot of quarter-chord clamped wing
Figure 10: Bending Displacements
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Z
Thickness: 0.55 mm
28 mm
X
79 mm
85 mm
Y
Y
Figure 11: Schematic for the (left) shape and (right) kinematics of the flapping plate used in experiment
and numerical simulation
The same geometry has been used to run numerical simulations as shown in figure 11. However, the
Reynolds number in experiment is still too high for direct numerical simulation. In numerical simulation,
we reduced the Reynolds number to Re = 700. Based on the simulation data, we computed the first 8 POD
modes to be used in ROM.
(a) mode 1
(b) mode 2
(c) mode 3
(d) mode 4
Figure 12: The votex structures of the first 4 POD modes.
1
POD
ROM
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.5
0
0.5
POD
ROM
0.4
-0.5
0
Normalized eigenmodes energy
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Analytical Results
0.9
0.8
0.7
0.6
0.5
10
20
eigenmodes
Figure 13: Phase portrait of coefficients (left) a1 and a2 (middle) a1 and a3 , and (right) the accumulated
energy distribution among POD modes.
(a) DNS
(b) POD reconstruction
(c) ROM
Figure 14: Comparison of snapshots (at the same time) of flow fields reconstructed from the DNS, the direct
projection of DNS data on the first 8 POD modes, and the computation of the 8-mode ROM.
Figure 12 shows the first 4 POD modes, which capture 88% of the total energy. We can see most
large vortex structures in these modes. A reduced-order model was built from the first 8 modes which
capture 96.4% of the total energy. The phase portrait shows that the first and second modes have the
fundamental frequency f , while the third and fourth modes have the frequency 2f (figure 13). With only
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8 modes, the coefficients computed by ROM are smaller than the direct projection (i.e. POD) from DNS.
The fundamental dynamics is well kept as it is shown in figure 13 (a) and (b). Figure 13 (c) gives a better
idea of the percentage of energy accumulatively captured by different number of modes. In figure 14, with
8 modes, we can reasonably reconstruct the same vortex structure as the DNS and its projection onto the
first 8 modes (i.e. POD). Consistent with the observation in phase portrait, the ROM result is somehow
dissipated more and shows weaker structures.
0.15
0.001
0.1
0.0005
0.05
0
0
-0.05
-0.0005
DNS
POD
ROM
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-0.1
DNS
POD
ROM
-0.15
-0.001
0
0.2
0.4
0.6
t/T
0.8
1
0
0.2
0.4
(a) Fx
t/T
0.6
0.8
1
(b) Fy
0.15
0.1
0.05
0
-0.05
DNS
POD
ROM
-0.1
-0.15
0
0.2
0.4
t/T
0.6
0.8
1
(c) Fz
Figure 15: Forces along three directions to compare the results from DNS, POD reconstruction, and ROM.
As an initial effort to benchmark with experimental measurements, though Reynolds numbers are different, we computed and plotted the forces from DNS, direct POD projection with 8 modes, and the ROM
computation with 8 modes in figure 15. The overall forces are smaller in simulation at lower Reynolds
number, but the basic features are similar. With the symmetry in upstroke and downstroke for x direction,
the force along x direction shows the period with only half of one flapping cycle in both experiment and
computation; in y direction, the force is much smaller as expected; in z direction, the force response has the
period the same as the flapping cycle, since the upstroke and downstroke have different contribution. Most
important, within the computational results, the POD projection captures the main features from the DNS,
and the ROM captures the main features with less but reasonable accuracy.
V.
Conclusion
The objective of this study was to develop the methodology necessary for the reduced-order modeling
of the loads and deformations of a flapping wing. An experimental setup capable of imparting an arbitrary
flapping motion while measuring loads and deformations of a flapping wing was developed. Tests were
performed on an uniformly stiff rectangular plate flapping in a 15 Hz, 80 degree stroke amplitude sinusoidal
motion. A POD-Galerkin projection applied on a modified Navier-Stokes equation is able to capture the
trend of direct numerical simulation.
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The pattern of max tip displacement at stroke reversal matches the pattern found in inertially dominated
flapping wing deformation experiments performed in previous literature. The loading and deformation data
can be correlated at specified points along the motion profile through the use of the encoder. The loads
and deformations were larger for the quarter-chord clamped wing, and the surface plots showed an expected
visible twisting. These experimental results for a simple rectangular plate will serve as a necessary validation
step for future analytical reduced-order models.
While the current experimental setup does not capture the more complex kinematics and multi-wing
aerodynamic interactions of biological flyers, the setup does provide a starting point from which a simple
reduced-order model may be developed. In addition to the top-down POD-Galerkin approach, further work
into reduced-order modeling will also include a bottom-up physics approach. As the models are refined to
satisfactory agreement with experimental results, the complexity of the experimental setup and models will
be concurrently increased. This step-by-step process has the benefit of developing several models from which
a MAV may choose from depending on its own level of complexity in kinematics and wing properties.
Downloaded by Mingjun Wei on May 13, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-0177
VI.
Acknowledgement
This research was supported by the Army’s MAST CTA for Microsystem Mechanics with Dr. Brett
Piekarski (ARL) and Dr. Chris Kroninger (ARL-VTD) as Technical Monitors.
References
1 Keennon, M., Klingebiel, K., and Won, H., “Development of the Nano Hummingbird: A Tailless Flapping Wing Micro
Air Vehicle,” AIAA 2012-588, 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace
Exposition, Nashville, Tennessee, Jan. 9-12 , 2012.
2 de Croon, G. C. H. E., de Clerq, K. M. E., Ruijsink, R., Remes, B., and de Wagter, C., “Design, Aerodynamics, And
Vision-Based Control Of The Delfly,” International Journal of Micro Air Vehicles, Vol. 1, No. 2, 2009, pp. 71–97.
3 DeLaurier, J. D., “The Development Of An Efficient Ornithopter Wing,” Aeronautical journal, Vol. 97, 1993, pp. 153–153.
4 Larijani, R. F. and DeLaurier, J. D., “A Nonlinear Aeroelastic Model For The Study Of Flapping Wing Flight,” Progress
in Astronautics and Aeronautics, Vol. 195, 2001, pp. 399–428.
5 Singh, B. and Chopra, I., “Insect-Based Hover-Capable Flapping Wings for Micro Air Vehicles: Experiments and Analysis,” AIAA Journal, Vol. 46, No. 9, 2008, pp. 2115–2135.
6 Orlowski, C. and Girard, A., “Modeling and Simulation of Nonlinear Dynamics of Flapping Wing Micro Air Vehicles,”
AIAA Journal, Vol. 49, No. 5, 2011, pp. 969–981.
7 Grauer, J. A. and Hubbard, J. E., “Multibody Model of an Ornithopter,” Journal of Guidance, Control and Dynamics,
Vol. 32, No. 5, 2009, pp. 1675–1679.
8 DeLaurier, J. D., “An Aerodynamic Model For Flapping-Wing Flight,” Aeronautical Journal, Vol. 97, No. 964, 1993,
pp. 125–130.
9 Vest, M. and Katz, J., “Unsteady Aerodynamic Model of Flapping Wings,” AIAA Journal, Vol. 34, No. 7, July 1996,
pp. 1435–1440.
10 Fritz, T. and Long, L., “Object-Oriented Unsteady Vortex Lattice Method for Flapping Flight,” Journal of Aircraft,
Vol. 41, No. 6, 2004, pp. 1275–1290.
11 Roget, B., Sitaraman, J., Harmon, R., Grauer, J., Hubbard, J., and Humbert, S., “Computational Study of Flexible
Wing Ornithopter Flight,” Journal of Aircraft, Vol. 46, No. 6, 2009, pp. 2016–2031.
12 Faruque, I. and Humbert, S. J., “Dipteran insect flightdynamics. Part 1 Longitudinal motion about hover,” Journal of
Theoretical Biology, Vol. 264, No. 2, 2010, pp. 538–552.
13 Grauer, J., Ulrich, E., Hubbard, J., Pines, D., and Humbert, J., “Testing and System Identification of an Ornithopter in
Longitudinal Flight,” Journal of Aircraft, Vol. 48, No. 2, 2011, pp. 660–667.
14 Willmott, A. P. and Ellington, C. P., “The mechanics of flight in the hawkmoth manduca sexta,” The Journal of
Experimental Biology, Vol. 200, 1997, pp. 2723–2745.
15 Maxon Motor, Switzerland, EPOS2 Application Notes Collection, December 2013.
16 Wu, P., Ifju, P., and Stanford, B., “Flapping Wing Structural Deformation and Thrust Correlation Study with Flexible
Membrane Wings,” AIAA Journal, Vol. 48, No. 9, 2010, pp. 2111–2122.
17 Sicard, J. and Sirohi, J., “Measurement of the deformation of an extremely flexible rotor blade using digital image
correlation,” Measurement Science and Technology, Vol. 24, No. 6, 2013, pp. 065203.
18 LaVision, Germany, StrainMaster DaVis 8.1 , December 2012.
19 Holmes, P., Lumley, J. L., and Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry,
Cambridge University Press, Cambridge, 1996.
20 Noack, B. R., Afanasiev, K., Morzynski, M., Tadmor, G., and Thiele, F., “A hierarchy of low-dimensional models for the
transient and post-transient cylinder wake,” Journal of Fluid Mechanics, Vol. 497, No. 1, 2003, pp. 335–363.
21 Zhao, H., Freund, J. B., and Moser, R. D., “A fixed-mesh method for incompressible flow-structure systems with finite
solid deformations,” J. Comp. Phys., Vol. 227, 2008, pp. 3114–3140.
11 of 12
American Institute of Aeronautics and Astronautics
22 Yang,
T., Wei, M., and Zhao, H., “Numerical study of flexible flapping wing propulsion,” AIAA Journal, Vol. 48, No. 12,
2010.
Downloaded by Mingjun Wei on May 13, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-0177
23 Xu, M., Wei, M., Yang, T., Lee, Y. S., and Burton, T. D., “Global model reduction for flows with moving boundary,”
49th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, FL, AIAA paper 2011-376, 2014.
24 Gao, H. and Wei, M., “Nonlinear Structural Response in Flexible Flapping Wings with Different Density Ratio,” AIAA
paper 2014-0222, 2014.
12 of 12
American Institute of Aeronautics and Astronautics