Report1 Divergence

28
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

Transcript of Report1 Divergence

Page 1: Report1 Divergence

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

Page 2: Report1 Divergence

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

Page 3: Report1 Divergence

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

Page 4: Report1 Divergence

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).

Page 5: Report1 Divergence

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

Page 6: Report1 Divergence

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.

Page 7: Report1 Divergence

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.

Page 8: Report1 Divergence

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

Page 9: Report1 Divergence

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

Page 10: Report1 Divergence

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.

Page 11: Report1 Divergence

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.

Page 12: Report1 Divergence

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%.

Page 13: Report1 Divergence

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

linear regression yields 0696.0*4144.11 −

=

q

θθ , with an error of r

2 = 0.9432. The

experimental divergence dynamic pressure is qD = 11.4144 ± 2.2872 psf (20.04% uncertainty),

and so the divergence speed for the second test is VD = 98.002 ± 9.37 ft/s (9.56%). The

uncertainty in this experimental qD value is greater than that the other test because the data is

slightly less linear. It should also be noted that neither of the y-intercepts of the best-fit lines

exactly correspond to αz as predicted by theory. This is because they are not perfect fits to the

data, they are approximate.

When compared to the theoretical values, the experimental values of qD for test 1 and test

2 are smaller by about 53% and 44%, respectively. Since there is a 9% difference in the percent

Page 14: Report1 Divergence

differences, it cannot be said necessarily that qD varies linearly with kα as predicted by the theory.

qD decreased less rapidly than kα from test 1 to test 2, so the relationship between them could be

more complex than a simple linear relationship. The 9% difference could be due to experimental

errors, such as not enough data points or too narrow (or wide) a range, so it is inconclusive

whether or not the relationship is purely linear.

The reason that the second test produced a result closer to the theory is not clear. One

possible explanation is that when kα was decreased by increasing the length of the beam, the

airfoil had to take a larger portion of the torsional load, therefore acting less rigid and twisting

more like a real wing would. Another contributing factor could be how much the airfoil rotated

overall. In the second test, the model began at a higher angle attack, so each change in lift (by

increasing the flow speed) produced a greater angle change than in test 1. This reduced the

number of data points but also reduced the relative percent error of each point, since, as

discussed before, 0.05/0.1 is much greater than 0.05/0.5.

Table 4 summarizes the results of the tests to determine qD. The table shows that qD

decreases as kα decreases. This result agrees with the theory and makes sense physically. If kα for

a given wing is low, then the wing resists torsion poorly, and so it would fail under a relatively

weak applied load, i.e. at a lower dynamic pressure. The higher the kα, the better the wing resists

twisting motions, so it will be able to fly at higher speeds without experiencing divergence.

Table 4. Summary of results for determining qD.

Test L (in) αz (deg) kα,theory (lb*in) qD,theory (psf) qD,experiment (psf)

1 18.25 4.8 458.7 27.266 12.7491

2 24.5 6.9 341.7 20.311 11.4144

Page 15: Report1 Divergence

Table 4 also gives insight into how to best shape wings to resist torsion. It shows that as L

increases, kα decreases, which means qD also decreases. For a wing in flight, the L dimension is

chord-wise. This means that wide wings, will be weaker in torsion, and, therefore, have lower

values of qD than narrow, high aspect ratio wings. For a 2D airfoil, the aspect ratio is defined as

the ratio of the wing span to the chord length. So in order to best resist torsion, long, narrow

wings with high aspect ratios are desirable.

A source of error not yet discussed comes from the deformation of the flexible beam. Each

time the beam is loaded, it deforms by bending, which shrinks the effective length of the beam.

The bending also temporarily weakens the beam; a loaded, bent beam is weaker than the same

straight, unloaded beam. Once the load is removed, the beam’s length should return to its

original value, however this was not observed. The beam retained some deformation from the

loading which evidenced itself in the form of an αz change. After the torsional stiffness test and

the first wind-tunnel test, the airfoil returned to its original αz, although the beam appeared to be

slightly bent. After the second wind-tunnel test, however, αz was 0.1° higher than its initial value,

indicating that the beam was indeed still bent. It seems that this source of error increased as the

experiment progressed, but overall it still contributed negligibly to the results since the length

contraction was small compared to the total length of the beam.

4. CONCLUSIONS

The preceding experiment was performed in order to determine the torsional stiffness and the

divergence dynamic pressure of a GAW-1 airfoil so as to predict and avoid torsional divergence

during flight. The wooden airfoil is connected to a beam that flexes instead of the rigid model

when the loads are applied. By carefully applying a moment to the airfoil and observing the

Page 16: Report1 Divergence

resulting change in angle of attack, the torsional stiffness of the airfoil-beam system can be

experimentally determined and compared to its theoretical value (based on statics). After

determining the torsional stiffness, the open-jet wind tunnel was used to create an incompressible

flow over the airfoil. The flow speed was incrementally increased and the dynamic pressures for

each speed were recorded. The change in lift force on the model acted at the quarter-chord and

caused it to rotate. By measuring the angle change between speed increments and creating a

Divergence Southwell Plot to display the data, an experimental value for the divergence dynamic

pressure can be deduced. This pressure was determined for two different beam configurations,

and so, two different torsional stiffness values. The following conclusions can be made:

1) The experimentally determined value for the torsional stiffness of the beam-airfoil

configuration is kα,exp = 309.9052 ± 1.3025 lb*in., which is about 33% less than the

theoretical value of kα,theory = 458.7 ± 91.5 lb*in.

2) For the first beam-airfoil arrangement, the divergence dynamic pressure was calculated to

be qD,exp = 12.7491 ± 2.3752 psf. This value is about 53% less than the theoretical value

of qD,theory = 27.266 ± 5.440 psf.

3) For the second beam-airfoil arrangement, L of the beam was increased and the divergence

dynamic pressure was calculated to be qD = 11.4144 ± 2.2872 psf. This value is about

44% less than the theoretical value of qD,theory = 20.311 ± 4.052 psf.

4) As kα decreases, qD also decreases, but not necessarily linearly.

5) Narrow wings with high aspect ratios resist torsion better and can achieve a higher qD

than wider, low aspect ratios wings.

The results indicate that the experiment should be performed with much greater attention to

detail in order to get results that more closely match the theoretical results. Every experimentally

Page 17: Report1 Divergence

determined value was significantly less than its corresponding theoretical value. This could mean

either a consistent error is present in all calculations or the experiment was poorly executed.

Since the results are so different from the theoretical results, they should not be considered

accurate. Therefore, little can be inferred from the results. However, conclusions 4 and 5 above

can be considered accurate since they are based on relative changes in the results. These 2

conclusions can be coupled to aid in airplane design; to resist torsion and increase qD, it would be

advantageous for the wings to have a high kα and a high aspect ratio.

Other than making measurements more carefully, this experiment could have been improved.

One improvement would be to use a rotational spring instead of a beam to resist twisting motion.

Another change that could be made would be to use a more flexible airfoil that can support a

torsional load. This would also require more advanced measurement techniques to measure the

deflection of the leading edge relative to other chord-wise locations of the airfoil. A final

improvement would be to perform the test using a setup more like an actual wing-fuselage

configuration. The wing would be much more sensitive to torsional loads if it were free at one

end rather than fixed at both ends. Again, this would complicate the problem as the flow over the

wing would be three-dimensional.

Page 18: Report1 Divergence

REFERENCES

Borgoltz, Aurelien. "Experimental Error." AOE 3054 Class Lecture. Virginia Tech. Blacksburg.

01Feb2011.

Gordon, J.E. Structures: Or Why Things Don’t Fall Down. Penguin Books Ltd, 1978. 260-271.

eBook.

"Linear Regression." University of Adelaide. Australia, Web. 5 Sep. 2011.

<http://www.chemistry.adelaide.edu.au/external/soc-rel/content/lin-regr.htm>.

APPENDIX A: UNCERTAINTY CALCULATIONS

For a result R that depends on primary measurements a, b, c, …, the equation to calculate

derived uncertainties is given as:

...)()()()(

222

+

∂+

∂+

∂= c

c

Rb

b

Ra

a

RR δδδδ (8)

where the partial derivatives are the primary uncertainties which have been given numerical

estimates (Borgoltz, 2011). This equation can be applied analytically for a known function or

numerically with a spreadsheet for discrete data. For example, to determine the uncertainty in the

divergence speed VD, express it explicitly as a function of known variables, so ρ

D

D

qV

2= .

Since ρ is assumed to be exact, δ(ρ) = 0, so δ(VD) becomes simply )()( D

D

D

D qq

VV δδ

∂= . For test

1, δ(qD)=2.375 psf so for ρ equal to its sea-level value and qD equal to that determined in test 1,

δ(VD) = 9.24 ft/s. VD = 103.57 ft/s, so δ(VD) can also be expressed as δ(VD) = 8.92%. This

process was used to find uncertainties for all other known functions.

Page 19: Report1 Divergence

If uncertainty is needed for discrete data, spreadsheets, like those shown in Tables 5 and

6, must be used to calculate the uncertainty. Table 5 is for the torsional stiffness test and the first

wind-tunnel test with L = 18.25 in.; Table 6 is for the second wind-tunnel test with L = 24.5 in.

Both tables give an uncertainty in the theoretical kα of about 19.95%, which is quite high. This

large uncertainty is due to the uncertainty in h which appears to the third power in the calculation

of kα. h also has a larger individual percent error (about 6.25%) than all of the other variables

because its length is comparable to the uncertainty in the measuring tool (calipers). Also, since e,

S, and a0 are assumed to be exact values, the uncertainty in the theoretical qD from Equation 4 is

exactly the same as that of the theoretical kα.

Page 20: Report1 Divergence

Table 5. Table for calculating uncertainty in M0 and kα for L=18.25 in.

Page 21: Report1 Divergence

Table 6. Table for calculating uncertainty in kα for L=24.5 in.

Page 22: Report1 Divergence

APPENDIX B: DIVERGENCE DYNAMIC PRESSURE EQUATION

To derive Equation 7, the configuration of Figure 8 must be used. For a zero airspeed

angle of attack, αz, and current angle of attack α, let a positive change in angle of attack be

defined counter-clockwise. The sum of the moments about the hinge must be zero for static

equilibrium, so

0=+−−=Σ ααkWdeLM w (B.1)

Substituting α0qSaqSCL lw == into equation B.1 yields

WdeqSakkWdeqSa −−=+−−= ααα αα )()(0 00 (B.2)

B.2 can be solved explicitly for α,

eqSak

Wd

0−=

α

α (B.3)

It is clear from this equation that αα kWdz = since q=0 for zero airspeed. Next define the

theoretical divergence dynamic pressure as 0eSa

kqD

α≡ so that B.3 can be rewritten as

D

z

qq−

=1

αα (B.4)

Equation B.4 diverges as q approaches qD but the airfoil would break before qD could possibly be

reached. In order to predict the value of qD without breaking the test model, Equation B.4 must

be transformed from hyberbolic to linear. This is accomplished through the use of θ, the change

in angle of attack relative to the zero airspeed position and some algebra. zααθ −≡ , so

Page 23: Report1 Divergence

D

z

z

qq−

=+=1

ααθα (B.5)

D

D

z

D

D

zz

q

q

q

q

q

q

q

q

=

−−

=

11

1 ααα

θ (B.6)

D

z

D q

q

q

qαθθ =

− (B.7)

zDq

q αθθ

=−

(B.8)

zDq

q αθ

θ −

= (B.9)

The final equation here shows θ varying linearly with the quantity θ/q. The slope of the line is

the divergence dynamic pressure, qD, and the y-intercept is –αz.

Page 24: Report1 Divergence

FIGURES

Figure 1. GAW-1 airfoil mounted in the open-jet wind tunnel. The incoming flow from the

tunnel is almost entirely uniform and is moving in the +x-direction. The origin of the x-y-z

coordinate system is in the center of the plane of the tunnel exit (not shown in this picture).

Figure 2. Side view of the airfoil mounted in the open-jet wind tunnel. The digital protractor is

used to measure angles and angle changes. The boundary layer fences (only one seen), help

x

y

z

U∞

Digital

Protractor

Boundary

Layer Fence

Span

b = 30 in.

Chord

c = 10 in.

Weights

Hang Here

Axis of

Rotation

Page 25: Report1 Divergence

prevent the flow from leaking around the sides of the airfoil so that the flow can remain 2D. The

weights that hang from the end create a moment about the axis of rotation (mid-chord).

Figure 3. Pitot probe. The Pitot probe has ports used to measure the static and stagnation

pressures, p and p0 respectively. The dynamic pressure is the difference between p0 and p.

Figure 4. Flexible beam view. Since this wooden airfoil does not bend very much like a real

wing would, the flexible beam shown is used to determine the torsional stiffness, kα, of the

otherwise rigid airfoil. The length of the beam, L, is inversely proportional to kα.

Flexible Beam

Pitot Probe

Axis of

Rotation

Height, h

Base, b

Page 26: Report1 Divergence

Figure 5. Simply-supported beam. The angle θ is the change in the angle of attack of the airfoil

due the moment caused by a hanging weight or a change in lift on the airfoil.

Figure 6. GAW-1 airfoil dimensions. All values shown are assumed to be exact.

Page 27: Report1 Divergence

Figure 7. Linear regression of applied moment vs. angle change data. The line shown is the best-

fit line through the four data points shown and its equation is 0153.0*9052.3090 −= θM .

Figure 8. Theoretical experiment setup to determine kα. This differs from the actual experiment

setup in that the object that resists twisting motion is a spring here instead of a simply-supported

beam, which was used in the experiment. θ is positive counter-clockwise.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045-2

0

2

4

6

8

10

12

14

Angle Change, θ (rad)

Applie

d M

om

ent, M

0 (

lb*i

n)

Page 28: Report1 Divergence

Figure 9. Divergence Southwell Plot for αz=4.8° and L=18.25 in. For the first two data points,

there was no rotation, so the two points at (0,0) were not used in the linear regression. The

equation of the best-fit line shown is 0186.0*7491.12 −

=

q

θθ .

Figure 10. Divergence Southwell Plot for αz=6.9° and L=24.5 in. The equation of the best-fit

line shown is 0696.0*4144.11 −

=

q

θθ .

0 0.5 1 1.5 2 2.5 3 3.5

x 10-3

-0.005

0

0.005

0.01

0.015

0.02

0.025

θ (

rad)

θ/q (rad/psf)

6 7 8 9 10 11 12

x 10-3

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

θ (

rad)

θ/q (rad/psf)