Experiment No. 1 AN EXPERIMENTAL STUDY OF AEROELASTICITY: TORSIONAL DIVERGENCE OF A FLEXIBLE WING Submitted by: STEPHEN LUTZ AEROSPACE AND OCEAN ENGINEERING DEPARTMENT VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY BLACKSBURG, VIRGINIA 7 SEPTEMBER 2011 EXPERIMENT PERFORMED 31 AUGUST 2011 AOE 4154 – AERO ENGR LAB – WEDNESDAY, 10:30 A.M. LAB INSTRUCTOR: ARTUR WOLEK LAB COORDINATOR: DR. ROGER L. SIMPSON Honor Pledge: By electronically submitting this report I pledge that I have neither given nor received unauthorized assistance on this assignment. 905343280 __ 9/7/2011 Student ID Number Date
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Experiment No. 1
AN EXPERIMENTAL STUDY OF AEROELASTICITY:
TORSIONAL DIVERGENCE OF A FLEXIBLE WING
Submitted by:
STEPHEN LUTZ
AEROSPACE AND OCEAN ENGINEERING DEPARTMENT
VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY
BLACKSBURG, VIRGINIA
7 SEPTEMBER 2011
EXPERIMENT PERFORMED 31 AUGUST 2011
AOE 4154 – AERO ENGR LAB – WEDNESDAY, 10:30 A.M.
LAB INSTRUCTOR: ARTUR WOLEK
LAB COORDINATOR: DR. ROGER L. SIMPSON
Honor Pledge:
By electronically submitting this report I pledge that I have neither given nor received
unauthorized assistance on this assignment.
905343280 __ 9/7/2011
Student ID Number Date
The goal of this experiment was to predict the torsional divergence of a wing by determining its
torsional stiffness and divergence dynamic pressure. The wooden GAW-1 airfoil was too rigid to
twist without breaking, so a beam of known material and dimensions was connected to the
airfoil’s axis of rotation to simulate a rotational spring connected to the airfoil. The torsional
stiffness of the airfoil-beam arrangement was determined by applying moments to the airfoil and
observing the resulting angle changes. This stiffness value turned out to be significantly less than
the theoretical value based on the end rotation of a beam. The divergence dynamic pressure, the
pressure at which the wing fails due to excessive twisting, was then investigated for two different
values of torsional stiffness. By placing the airfoil in an incompressible flow, increasing the flow
speed by a known amount, and measuring the change in angle of attack each time the speed was
increased, the divergence dynamic pressure was deduced. Again, these experimental values were
significantly less than the theoretical values found from free-body-diagram equilibrium. It was
discovered that wings with high torsional stiffness and aspect ratios can carry torsional loads
much better than low stiffness, low aspect ratio wings.
1. INTRODUCTION
The goals of this experiment are to:
1. Experimentally determine the value of the torsional stiffness, kα, of a GAW-1 airfoil and
compare it to the theoretical value for the torsional stiffness of a rotation spring.
2. Subcritically determine the divergence dynamic pressure, qD, for the airfoil and compare
it to the theoretical qD for two different theoretical values of kα.
To accomplish these goals, a GAW-1 airfoil was mounted in the test section of an open-jet wind
tunnel, as seen in Figure 1. For the first goal, various weights were attached to the rear of the
airfoil at the location shown in Figure 2. The resulting force of the weight at a measured
distance, b, from the axis of rotation creates a moment, M0, on the airfoil that causes the airfoil to
rotate by an angle α*. The experimental value of the torsional stiffness of the airfoil, kα, can be
determined from a plot of M0 vs. α*. The second goal involves subjecting the airfoil to a steady
aerodynamic force rather than a static force in order to determine the divergence dynamic
pressure, qD. By incrementally increasing the wind speed (and the dynamic pressure, q) and
measuring the resulting change in angle of attack, qD can be inferred from a Divergence
Southwell plot. The dynamic pressure is measured by a Pitot probe, shown in Figure 3.
Aeroelsaticity is the study of aerodynamic, elastic (structural), and inertial forces and
their mutual interaction on a flexible body in a fluid. It can be broken down into static and
dynamic aeroelasticity problems. Dynamic problems involve all three of aerodynamic, elastic,
and inertial forces. They are also time-dependent, so they are generally much more difficult to
solve. Some common examples of dynamic aeroelastic phenomena are flutter (wing vibrations),
buffeting (from shed vorticies), and transient response of control surfaces. Static aeroelasticity
problems involve only aerodynamic and elastic forces and are time-independent, so the difficulty
of the problems is significantly reduced from dynamic ones. Static aeroelastic phenomena
include load distributions on the structure, divergence, and control system reversal.
This experiment deals with a static aeroelasticity problem: the torsional divergence of a
flexible wing. Since the wooden airfoil is rigid, the twisting of the wing is simulated by using the
flexible beam setup seen in Figure 4. The beam flexes instead of the airfoil, so the kα for the
beam-wing setup can be thought of as the kα for the wing. The theoretical kα for a simply-
supported beam can be easily determined using the configuration in Figure 5. Based on Figure 5,
the rotation at the end of the beam, θ, due to an applied moment, M0, can be computed as
EI
LM 0
3
1=θ (1)
where E is Young’s Modulus of the material, L is the length of the beam, and I is the second
moment of inertia defined by
3
12
1bhI = (2)
for a beam of given width, b, and thickness, h. kα is related to θ and the M0 via the equation
θαkM =0 . Substituting this equation into Equation 1 and rearranging yields
L
EIk
3=α (3)
Equation 3 is a useful formula that can be used to calculate the theoretical value of kα, which
depends on both the dimensions of the object and the material being used.
When a wing experiences its divergence dynamic pressure, qD, it can fail completely due
to excessive twisting. A classic example of torsional divergence is the Fokker D8 monoplanes of
WWI which experienced this static instability during combat conditions. The torsional stiffness
in the wings was because of poorly placed spars. The center of flexure (rotation axis) between
the two spars of the D8 was too far aft from the center of pressure of the wing, so the front spar
deflected upward more than the rear spar under a steady, level flight lift load. When pulling out
of dives dogfights, the wings completely detached from the aircraft, resulting in the loss of many
pilots. It is essential to know qD for a given aircraft so that catastrophic failure can be predicted
and avoided (Gordon).
The most intuitive way to combat torsional divergence, or increase qD, is to increase the
torsional stiffness, kα. Based on Equation 3, the only ways to increase kα are to increase the cross-
sectional area of the beam, which adds more weight, or to use a material with a higher Young’s
Modulus, which could cost and weigh more. Other ways to increase qD are to decrease the
planform area or the wing, S, or decrease the distance between the rotation axis and the quarter-
chord of the airfoil, e. This is clearly seen in the following equation for theoretical dynamic
divergence pressure, where a0 is the lift curve slope for the airfoil:
0eSa
kqD
α= (4)
As evident in Equation 4, decreasing a0 would also increase qD. Compared to its 2D
counterpart, a 3D wing has a decreased lift-curve slope; this is true for any wing. This means that
if the GAW-1 airfoil were in a real flight with a 3D flow, qD would be higher than that calculated
for the 2D model in this experiment due to 3D effects, such as downwash.
2. DESCRIPTION OF EXPERIMENT
2.1 Apparatus
2.1.1 Open-Jet Wind Tunnel
The open-jet wind tunnel used in this experiment, shown in Figures 1 and 2, is used to
study incompressible flow. The main difference between and open-jet wind tunnel and a
conventional wind tunnel is that the test model is subject to atmospheric conditions, rather than
controlled values of pressure and temperature. Its square cross section has a side length of about
2.5 feet and can exhaust air at a wide range of speeds from less than 0.5 m/s to over 25 m/s. Flow
straighteners within the tunnel help make the exiting flow almost entirely uniform, so for the
purposes of this experiment, the flow can be assumed to be perfectly uniform across the span (y-
direction) and thickness (z-direction) of the airfoil.
Rather than choosing a specific value for the free-stream velocity, the speed of the open-
jet wind tunnel is controlled by setting the RPM. The speed is monitored by multiple pressure
probes, including the Pitot probe (Figure 4) used in this experiment, which measures the static
and stagnation pressures, p and p0 respectively. As per Bernoulli’s equation, the difference
between p0 and p is the dynamic pressure, q, which is related to the speed, U∞, of the fluid with
density ρ by the following equation
∞= Uq ρ2
1 (5)
The Pitot probe was mounted on a traverse that was movable in all three directions: x, y, and z.
For this experiment, it was sufficient to keep the probe fixed in one location.
2.1.2 GAW-1 Airfoil Test Model
The GAW-1 airfoil is a wooden airfoil shown schematically in Figure 6. Its span, b, is 30
in., its chord, c, is 10 in., and the value of the lift curve slope for this airfoil is a0 = 3.23 rad-1
.
These values were given in the lab manual, so they are assumed to be exact. The axis of rotation
is mid-chord and since the lift is assumed to act at the quarter chord, the distance between the
two, e, is exactly 2.5 in. A clip attached to the airfoil used for hanging weights, shown in Figure
2, does not interfere with the experiment since it is so far aft and out of the way of the incoming
flow. Also shown in Figure 2 are the boundary layer fences. Boundary layer fences are large flat
plates attached to the ends of the airfoil to keep the flow 2D. Without the fences, the flow could
leak around the sides of the airfoil which would make the problem 3D and much more difficult.
2.1.3 Flexible Beam and Test Rig
A rod goes directly through the center of the airfoil and acts as the main spar. The center
of this rod is also the axis of rotation for the airfoil, which is clearly seen in Figures 2 and 4.
When the loads are applied to the rigid airfoil, it rotates, rather than flexing like a real wing
would. To deal with this issue, a beam attached to the rod bends when the airfoil rotates, as
shown in Figure 4. When the airfoil rotates counter-clockwise, the rod deflects downward in the
middle based on its dimensions and material properties; the thinner and weaker the beam is, the
more it will deflect, so its kα would be low. The beam used in this experiment is made of a
material with a Young’s modulus of E = 30×106 psi and dimensions b = 2.18 ± 0.005 in., h =
0.08 ± 0.005 in., and L = 18.25, 24.5 ± 0.0625 in. (L varies in the second part of the experiment).
2.1.4 Other Items
The rotation of the airfoil was monitored by a digital protractor mounted on the outer
surface of one of the boundary layer fences, as shown in Figure 2. The protractor displayed the
current angle of the airfoil relative to horizontal and also whether the measured angle was nose
up or nose down. The uncertainty associated with the protractor’s reading is 0.05 deg.
Three different weights were used apply a moment to the airfoil. Their weights are 0.660
lb, 0.996 lb, and 1.996 lb. The digital scale used to obtain these values has an uncertainty of
0.005 lb. Each weight was applied individually, and hung from the rear of the airfoil by a string
of negligible weight.
2.2 Procedure
There were two main parts to this experiment. The first part consisted of hanging weights
to the end of the airfoil to obtain an experimental value of kα for the airfoil. The second part
made use of the open-jet wind tunnel to apply an aerodynamic load to the airfoil.
To determine kα, weights were hung at a measured distance of 6.5 ± 0.0625 inches from
the axis of rotation of the airfoil. This distance is called the moment arm of the applied load.
Each weight was carefully weighed on a digital scale and then attached to the rear of the airfoil
to create a moment. The moment, M0, causes the airfoil to rotate positively and the beam to flex.
Assuming the airfoil was at an initial angle of attack αz and is now at a new angle of attack due
to the applied moment, α, then the change in angle of attack, θ, is defined as
zααθ −= (6)
θ was obtained from protractor readings and M0 is found from multiplying the moment arm by
the weight. M0 in lb*in was then plotted vs. θ in rad. The result was a straight line with a slope
that is equal to kα (lb/in) since θαkM =0 . This was done for 3 different weights, so including
(0,0), there were four total data points with which to obtain kα through a linear regression.
Once kα had been determined, the airfoil was subjected to a uniform incompressible flow
in order to find the dynamic divergence pressure, qD. The low-speed flow was generated and
straightened by the open-jet wind tunnel and was steadily increased in speed in increments of
100 RPM. Increasing the speed increased q, which was displayed on the side of the tunnel in
inches of water (± 0.01 in. of water) and recorded for each speed. q is directly related to the lift
force on the airfoil, and the change in lift acted at the quarter chord of the airfoil, so it created a
moment that rotated the airfoil counter-clockwise about its axis. The change in angle of attack
due to the aerodynamic load, θ, was measured and recorded, as well as q. The speed was
increased until the airfoil began to shake. This test was performed for two different values of L
for the beam.
3. RESULTS OF EXPERIMENT
3.1 Determining kα
Based on Equation 3, for L = 18.25 in., the theoretical value for the torsional spring
stiffness is kα = 458.7 ± 91.5 lb*in., (19.95% uncertainty). The large error here indicates that a
measurement could have been made better. In this case, it seems that the uncertainty in the height
measurement was the biggest contributor to the overall uncertainty. This is likely due to the fact
that there is an h3 term in the equation and all other variables used to calculate kα only appear to
the 1st power. Also, h has the largest individual percent error since it is the smallest dimension
being measured.
To experimentally determine kα, a linear regression was performed on the data in Table 1
for an αz of -2.2° (nose down). A graph of the data is seen in Figure 7, and the equation obtained
from the regression is 0153.0*9052.3090 −= θM with an error term of r2=0.999991 (r
2=1 being
perfectly linear). The slope of this line represents the experimental value of kα and is 309.9052 ±
1.3025 lb*in, which is about 1/3 less than the theoretical value. The error in the experimental
value of kα is low because the data falls on a straight line so well, as predicted by the theory.
Qualitatively, the data agrees with the theory since it falls on a straight line almost perfectly, but
quantitatively is does not agree with the value for kα. Given the large uncertainty in the
theoretical value, however, the two values for kα can vary by as little as 15% or as much as 44%.
This significant variation was not expected.
Table 1. Data for determining kα experimentally.
Test Weight (lb) Moment, M0 (lb*in) Angle Change, θ (rad)
1 0 0 0
2 0.660 4.290 0.0140
3 0.996 6.474 0.0209
4 1.996 12.9740 0.0419
The experimental value for kα could be significantly less than the theoretical value for a
few reasons. One reason is that the object resisting the twisting motion is a beam instead of a
torsional spring. They both serve the same purpose for this experiment but they are
fundamentally different objects. The experimentally determined kα was found assuming that a
torsional spring was rotated through an angle θ, like in the setup shown in Figure 8. On the other
hand, the theoretical value for kα was derived using statics to find the end rotation of a beam with
the setup in Figure 5. In addition, the hinge shown in Figure 8 is assumed to be frictionless, so
actual energy losses due to friction could affect the results.
Another plausible reason is that the airfoil is not perfectly rigid like it is assumed to be.
The wooden airfoil is much more rigid than its full-sized counterpart, but it still deforms slightly.
Because of this, the airfoil could be taking part of the torsional load and twist slightly and leave
the rest of the rotation to the beam, which could affect the experimental kα. The actual bending
moment that the beam experiences is less than the calculated moment (force times moment arm)
because the airfoil “absorbs” some of the moment by twisting.
3.2 Deducing qD
The goal of this experiment was to determine qD subcritically, which means find the
value of qD without actually subjecting the airfoil to that pressure, which would destroy the
model. In order to do this easily, some equations needed to be developed to allow data to be
presented linearly in qD. The resulting equation needed to determine qD subcritically is
zDq
q αθ
θ −
= (7)
A graph with θ on the vertical axis and θ/q on the horizontal axis (known as a Divergence
Southwell Plot), shows the data as a straight line with slope qD and y-intercept αz. For the full
development of this equation see Appendix B.
The first test began at αz = 4.8° with the beam at length L = 18.25 in. The air density, ρ,
for this experiment is assumed to be its sea-level value of 0.0023769 slug/ft3. Equation 5 can
then be used to compute the velocity, U∞, given the dynamic pressure, q. The data for the first
wind tunnel test is summarized in Table 2 below. For a theoretical kα of 458.7 lb*in, the
theoretical value for qD, as given by Equation 4, is qD = 27.266 ± 5.440 psf. The uncertainty in
the theoretical qD is the same as for the theoretical kα (19.95%) because e, a0, and S are assumed
to be exact values.
Table 2. Data from first wind tunnel test to determine qD.
RPM q (in. H2O)* U∞ (ft/s) α (deg) θ (deg)
300 0.09 19.84 4.8 0
400 0.15 25.62 4.8 0
500 0.25 33.07 4.9 0.1
600 0.36 39.69 5 0.2
700 0.5 46.77 5.1 0.3
800 0.66 53.74 5.2 0.4
900 0.84 60.62 5.4 0.6
1000 1.05 67.78 5.6 0.8
1100 1.26 74.25 5.9 1.1
1200 1.45 79.65 6.2 1.4
*1 in. H2O = 5.2 lb/ft2
A Divergence Southwell Plot of the data in Table 2 is shown in Figure 9. Assuming all of
the data falls on a straight line, a linear regression can be performed to determine qD, which is the
slope of the best-fit line. For the first two airspeeds, no rotation occurred, i.e. θ=0°. Therefore,
neither of the first two points (both at (0,0)) line up with the rest of the data, and so they were not
used in the linear regression. The equation of the best-fit line through the other 8 points is
0186.0*7491.12 −
=
q
θθ , with an error term of r
2 = 0.9505. This means that qD is 12.7491
lb/ft2. For this value of qD at sea-level, the divergence speed is VD = 103.57ft/s.
The uncertainties in θ and θ/q vary from point to point. For example, for θ=0.1°, the
uncertainty of 0.05° is 50%, but when θ=1.4°, the uncertainty of 0.05° is less than 4%. The same
goes for the values of θ/q. Since this is the case, the uncertainty in qD is determined by assuming
that its uncertainty is twice its standard deviation. The standard deviation of a slope of a linear
regression of data sets x (θ/q) and y (θ) can be computed with various techniques and is 2.3752
psf, which is about 18.6% uncertainty (“Linear Regression”). Using the methods described in
Appendix A, the uncertainty in VD is then 9.24 ft/s, or about 8.92%.
For the second test, L was extended to 24.5 in. and αz was set to 6.9°. The change in
length changed the theoretical spring stiffness to kα = 341.7 ± 68.166 lb*in. Therefore, using
Equation 4 again, the theoretical divergence dynamic pressure for the second test is qD = 20.311
± 4.052 psf (± 19.95%). The data for the second test is shown below in Table 3.
Table 3. Data from second wind tunnel test to determine qD.
RPM q (in. H2O) U∞ (ft/s) α (deg) θ (deg)
300 0.09 19.8441 7.1 0.2
400 0.16 26.4589 7.2 0.3
500 0.26 33.7286 7.4 0.5
600 0.38 40.7758 7.7 0.8
700 0.52 47.6994 8.2 1.3
800 0.7 55.3427 8.7 1.8
900 0.89 62.4031 9.6 2.7
1000 1.1 69.3757 10.8 3.9
Figure 10 is a Divergence Southwell Plot for the data in Table 3. All of the data seemed
to be ‘good data’ (no outliers), so all eight points above were used in the analysis. Performing a