Stability Analysis of a Segmented Free-wing Concept for UAS Gust
Alleviation in Adverse Environments
Except where reference is made to the work of others, the work described in this
thesis is my own or was done in collaboration with my advisory committee. This
thesis does not include proprietary or classi ed information.
Jason Welstead
Certi cate of Approval:
Brian S. Thurow
Assistant Professor
Aerospace Engineering
Gilbert L. Crouse Jr., Chair
Associate Professor
Aerospace Engineering
John E. Cochran Jr.
Professor and Head
Aerospace Engineering
George T. Flowers
Dean
Graduate School
Stability Analysis of a Segmented Free-wing Concept for UAS Gust
Alleviation in Adverse Environments
Jason Welstead
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Ful llment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
August 10, 2009
Stability Analysis of a Segmented Free-wing Concept for UAS Gust
Alleviation in Adverse Environments
Jason Welstead
Permission is granted to Auburn University to make copies of this thesis at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Jason Robert Welstead, son of Robert "Bob" L. Welstead and Teresa "Tess" A.
Welstead, was born on September 25, 1984 in Sioux City, Iowa. He attended school
in Elkhorn, Nebraska where he participated in choir, soccer, and OPPD Powerdrive.
In 2003, he graduated with honors from Elkhorn High School and the following fall
enrolled to Wartburg College in Waverly, Iowa. Balancing collegiate athletics and
academics, he graduated summa cum laude with a B.A. in Engineering Science and
Mathematics in 2007. In the fall of 2007, he began his graduate studies at Auburn
University. At Auburn University, he was the team leader of the American Insti-
tute of Aeronautics and Astronautics (AIAA) Design/Build/Fly student competition
where the team successfully completed all three missions. Only 9 teams completed all
missions out of a total of sixty. The following year, he was an advisor to the team that
had a successful prototype four months before the competition. Directing numerous
improvements, the team went to competition and successfully met the goal of having
the lightest aircraft at the competition. In April of 2009, he presented and published
a technical paper at the AIAA Unmanned...Unlimited conference which formed the
foundation for this thesis.
iv
Thesis Abstract
Stability Analysis of a Segmented Free-wing Concept for UAS Gust
Alleviation in Adverse Environments
Jason Welstead
Master of Science, August 10, 2009
(B.A., Wartburg College, 2007)
112 Typed Pages
Directed by Gilbert L. Crouse Jr.
High altitude, long endurance (HALE) aircraft feature large wing spans and
have very low wing loadings resulting in sensitivity to turbulence. While turbulence
is usually quite low in the stratosphere where HALE aircraft typically operate, even
high altitude aircraft must transition through the lower atmosphere during takeo
and landing operations. Sensitivity to turbulence may restrict the weather conditions
under which HALE aircraft can be launched or retrieved. A compounding considera-
tion for HALE aircraft is that because of their large wing spans, their wings may be
longer than the length scale of the turbulence they encounter. This means that di er-
ent portions of the aircraft?s wings will see di erent aerodynamic conditions and will
result in signi cant additional structural loads on the wing structure. Alleviating the
aircraft?s response to time-varying gust elds as well as spatially-varying gust elds
is thus important for HALE aircraft. One promising technology for gust alleviation
is the \free wing". A free-wing design allows the wing to adjust itself in pitch about
a spanwise axis in response to aerodynamic loads rather than being rigidly attached
v
to the aircraft fuselage. Free wings historically have shown the ability to reduce an
airplane?s response to turbulence. An extension of the concept proposed here is called
a \segmented free wing". A segmented free wing di ers from the conventional free
wing by sectioning the wings into multiple, independent segments. This design pro-
vides a greater reduction in turbulence response than both the standard free wing
and the xed wing as demonstrated in initial wind tunnel tests. A conceptual design
of such a planform along with a study of its stability characteristics was examined.
Initial results from a wind tunnel model showed a reduced rolling moment coe cient
when compared to a traditional free-wing design. Experimental tests of the larger
model showed a divergent oscillatory mode that appears with increasing velocity. An
analytical model of the experimental test was developed and successfully predicts the
instability. Comparison of the analytical model versus the experimental results shows
an over-prediction of the stability of the system by the analytical model and causes
for the over-estimation were investigated. The e ects of unsteady aerodynamics, ap-
parent mass terms, and wake e ects on the analytical model were studied and all were
determined to signi cant in the aerodynamic model. The analytical model was used
to predict the crossover velocity of a wind tunnel model but the wind tunnel model
failed to become unstable due to the stabilizing friction force in the bearing surfaces.
vi
Acknowledgments
The author would like to thank Dr. Gilbert L. Crouse Jr. for giving him the op-
portunity to work on this research. Dr. Crouse has been invaluable in his support on
the research project. The author would also like to acknowledge Dr. Robert S. Gross
for allowing him to use his vehicle to obtain experimental data. The author would
like to thank Brian C. Reitz for his help with the electronics for the experimental
model and working with the RC transmitter. Kevin Albarado?s help with the wind
tunnel testing was extremely appreciated. Finally, the author would like to thank
Neal A. Allgood for his help in the construction of the experimental model.
vii
Style manual or journal used American Institute of Aeronautics and Astronautics
(AIAA)
Computer software used Matlab 7, Microsoft Word 2007, Microsoft Excel 2007,
WinEdt, Latex, Microsoft Paint, QuickTime, Adobe Photoshop
viii
Table of Contents
List of Figures xi
List of Tables xiv
1 Introduction 1
2 Review of Literature 5
2.1 Active Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Passive Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Conceptual Design 13
3.1 Sizing and Con guration . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Airfoil Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Wind Tunnel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Analytical Model 21
4.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2.1 Validation of Aerodynamic Model . . . . . . . . . . . . . . . . 29
4.3 Wake E ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.5 Wind Tunnel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Results 40
5.1 Initial Wind Tunnel Test . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2 Experimental Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3 Analytical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3.1 System Properties . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3.2 Analytical Model Without Fixed Segments . . . . . . . . . . . 52
5.3.3 Analytical Model With Fixed Segments . . . . . . . . . . . . . 57
5.3.4 Analytical Model Compared to Experimental Results . . . . . 62
5.3.5 Full Aerodynamic Model Sensitivities . . . . . . . . . . . . . . 64
5.3.6 Variation of Input Parameters Based Upon Error Estimates . 70
5.3.7 Modeling Options . . . . . . . . . . . . . . . . . . . . . . . . . 74
ix
5.4 Wind Tunnel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4.1 Wind Tunnel Test Results . . . . . . . . . . . . . . . . . . . . 82
5.4.2 System Properties . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4.3 Analysis of Results . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4.4 Friction Discussion . . . . . . . . . . . . . . . . . . . . . . . . 88
6 Conclusions 92
7 Future Work 95
Bibliography 96
x
List of Figures
1.1 Cross-sectional illustration of the free-wing design . . . . . . . . . . . 2
3.1 Segmented free wing viewed from the top . . . . . . . . . . . . . . . . 13
3.2 Free-wing airfoil design with control tab . . . . . . . . . . . . . . . . 14
3.3 Lift coe cient versus angle of attack . . . . . . . . . . . . . . . . . . 15
3.4 Pitching moment versus lift coe cient . . . . . . . . . . . . . . . . . 16
3.5 Diagram of the experimental setup . . . . . . . . . . . . . . . . . . . 17
3.6 Segmented free wing mounted to beam on truck looking through the
windshield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.7 Wind tunnel test schematic . . . . . . . . . . . . . . . . . . . . . . . 19
3.8 Segmented free-wing model in AU wind tunnel . . . . . . . . . . . . . 20
4.1 Axis con guration for analytical model . . . . . . . . . . . . . . . . . 21
4.2 Free-body diagram of the segmented free-wing system in roll, looking
from rear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3 Free-body diagram of a wing segment . . . . . . . . . . . . . . . . . . 23
4.4 The real and imaginary parts of Theodorsen?s function, F(k) and G(k)
respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.5 Comparison of the analytical model circulatory lift to Theodorsen?s
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.6 Comparison of the analytical model total lift response with pitching
data from Rainey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.7 Setup of the Vortex Lattice Method on segmented free wing . . . . . 33
xi
4.8 \Nomenclature for calculating the velocity induced by a nite length
vortex segment" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1 Segmented free wing with xed segments attached to the tip ying
with a visible dihedral . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Diagram of the bi lar pendulum setup . . . . . . . . . . . . . . . . . 47
5.3 Experimental model mounted on the bi lar pendulum in Auburn Uni-
versity?s Adaptive Aerostructures Lab . . . . . . . . . . . . . . . . . . 50
5.4 Experimental segment mounted on the bi lar pendulum in Auburn
University?s Adaptive Aerostructures Lab . . . . . . . . . . . . . . . . 51
5.5 Axis orientation for modulus calculations . . . . . . . . . . . . . . . . 52
5.6 Root locus plot of the full aerodynamic model for the mechanically
restrained model without xed segments . . . . . . . . . . . . . . . . 53
5.7 Root locus plot of the full aerodynamic model for the mechanically
restrained model without xed segments from a zoomed in perspective 54
5.8 Root locus plot for the full aerodynamic model for the system mechan-
ically free in roll without xed segments . . . . . . . . . . . . . . . . 55
5.9 Root locus plot for the full aerodynamic model for the system mechan-
ically free in roll without xed segments zoomed in about the origin . 56
5.10 Root locus plot for the full aerodynamic model for the system mechan-
ically restrained in roll with xed segments . . . . . . . . . . . . . . . 58
5.11 Root locus plot for the full aerodynamic model for the system mechan-
ically restrained in roll with xed segments attached, zoomed about
the origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.12 Root locus plot for the full aerodynamic model for the system mechan-
ically free in roll with xed segments attached . . . . . . . . . . . . . 60
5.13 Root locus plot for the full aerodynamic model for the system mechan-
ically free in roll with xed segments attached, zoomed in about the
origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.14 Full aerodynamic model sensitivity to changes in pitching mass mo-
ment of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
xii
5.15 Full aerodynamic model sensitivity to changes in rolling mass moment
of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.16 Full aerodynamic model sensitivity to changes in the aerodynamic center 67
5.17 Full aerodynamic model sensitivity to changes in the center of gravity
position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.18 Full aerodynamic model sensitivity to changes in the spring constant K 69
5.19 Root locus plot of the full aerodynamic model with modi ed input
parameters but without xed segments . . . . . . . . . . . . . . . . . 72
5.20 Root locus plot of the full aerodynamic model with modi ed input
parameters but without xed segments zoomed in to reveal unstable
roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.21 Root locus plot of the basic model . . . . . . . . . . . . . . . . . . . . 75
5.22 Root locus plot of the basic model with wake e ects . . . . . . . . . . 77
5.23 Root locus plot of the basic model with unsteady aerodynamics . . . 79
5.24 Root locus plot of the basic model with apparent mass terms . . . . . 80
5.25 Root locus plot of the full aerodynamic model . . . . . . . . . . . . . 81
5.26 Root locus plot for the wind tunnel model . . . . . . . . . . . . . . . 85
5.27 Friction coe cients for maintained stability with unsteady aerodynamics 86
5.28 Friction coe cients for maintained stability without unsteady aerody-
namics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
xiii
List of Tables
5.1 Rolling moments of three con gurations . . . . . . . . . . . . . . . . 41
5.2 Results of the experimental testing on the segmented free wing . . . . 43
5.3 Lodgepole Pine elastic properties . . . . . . . . . . . . . . . . . . . . 50
5.4 Data from spring constant test on mounting beam . . . . . . . . . . . 52
5.5 Results of the di erent modeling features . . . . . . . . . . . . . . . . 76
5.6 Selected values of coe cients of friction . . . . . . . . . . . . . . . . . 89
xiv
Chapter 1
Introduction
Turbulence in the troposphere has long been an issue for aircraft of all sizes.
However, aircraft with lower wing loadings are much more sensitive to turbulence than
aircraft with high wing loadings. The high sensitivity to turbulence as seen in general
aviation aircraft and unmanned aerial systems (UAS) causes passenger discomfort,
sensor malfunction, or potentially fatal loss of control. High altitude, long endurance
(HALE) aircraft are even more sensitive to turbulence than the standard UAS or small
aircraft because of their very low wing loading and light construction. An example of
this type of problem is demonstrated in NASA?s Helios project, a HALE aircraft with
a wing span of 247 feet. The Helios aircraft took o from Kauai, Hawaii on June 26,
2003 and ew approximately 30 minutes until a divergent pitch oscillation occurred
\in which the airspeed excursions from the nominal ight speed about doubled every
cycle of the oscillation." The design speed of the aircraft was subsequently exceeded
causing dynamic pressures to increase beyond the designed limits of the structure.
Under the intense load, the vehicle began to break up and it eventually crashed into
the ocean. The nal determination of the crash investigation was that the fatal pitch
oscillation was caused by atmosphere disturbances, i.e. turbulence, which resulted in
a structural failure of the system [1].
A possible solution to this sensitivity to turbulence is to increase the wing loading
to be comparable to that of airliners; however, this is unpractical in small aircraft [2]
and UAS?s, especially HALE aircraft. Reducing wing area of a HALE aircraft would
1
reduce wingspan and thus increase induced drag thereby greatly reducing its max-
imum altitude and endurance capabilities. Reducing wing area would also increase
stall speed which would negatively a ect safety of operation. Another solution is to
increase the structural strength of the HALE aircraft to handle the dynamic loading
under oscillatory conditions. However, this would increase the system weight which
would decrease performance taking away from the HALE mission.
Alternatively, an approach to decrease the sensitivity to turbulence is a free-wing
design as depicted in Fig. 1.1. A free wing di ers from a xed wing in that the wing is
Figure 1.1: Cross-sectional illustration of the free-wing design [3]
free to pivot about a spanwise axis. Another di erence is that the aerodynamic forces
control the pitch of the wing, whereas, with a xed wing the fuselage controls the pitch
of the wing. A control tab on the trailing-edge of the wing is the only control device
2
used to increase or decrease lift. NASA conducted several studies on the free-wing
planform throughout the 1970?s and early 1980?s dealing with aircraft ranging from
3,000 to 50,000 pounds [3]. It was determined in Ref. [3] that the free wing drastically
reduced the load-factor response, as high as 4 to 1 in all con gurations. The major
conclusion from the study was that the free wing was extremely e ective at reducing
the longitudinal response to disturbances for aircraft of common wingspans.
A limitation of the free-wing planform is that it can only adjust to time-varying
turbulence. A feature of HALE aircraft is their large wingspans which makes them
susceptible to both time-varying and span-varying turbulence. The extension to the
free-wing concept proposed here is called a segmented free wing and breaks the wing
into multiple independent free-wing segments. Each segment is allowed to move and
to be controlled independently from all of the other segments. The segmented free-
wing design shows promise for application to HALE aircraft due to its ability to adjust
to both time-varying and span-varying turbulence. Being a passive system, just like
the free wing, no complex control systems need to be designed and implemented into
the system; thus saving system complexity and weight. Each segment is free to pivot
about its span-wise axis allowing the entire wing to maintain a constant angle of
attack even as the velocity eld is changing. Additionally, the wing can alter its local
pitch as disturbances vary across the span of the wing. Regardless of the turbulence
pro le across the planform, the segmented free wing can adjust to maintain constant
lift across the span and alleviate much of the lateral response of the aircraft. With this
ability to deal with both time-varying and span-varying turbulence, the segmented
free wing is ideal for use in HALE aircraft design.
The initial thrust of this research was to design and fabricate a conceptual model
and perform preliminary testing on the model. With the preliminary testing, the
study of the dynamic stability of the segmented free wing became the focus of this
3
research. An experimental model was constructed and tested to determine the sta-
bility of the system. An analytical model was also developed to model the behavior
of the segmented free wing and to predict the stability of the system. A wind tunnel
model was constructed to give preliminary results of the spanwise varying turbulence
response, along with testing the validity of the analytical model.
4
Chapter 2
Review of Literature
Aircraft gust response is a topic that has been studied extensively since the early
1900?s when the Wright Brothers were ushering in the new age of ight. It is an area
of intense interest due to its a ect on passengers and the increased requirements it
places on the strength of the aircraft structure. Additionally, the aeroelastic e ects of
the aircraft?s gust response can potentially lead to negative damping of a structural or
ight mechanics mode causing extremely large, potentially fatal vibrations or oscilla-
tions. Not only can the gust response of an aircraft cause structural problems, it also
creates issues with passenger comfort and scienti c equipment. Air travelers often
avoid smaller aircraft due to their magni ed gust response when compared to heavier
aircraft. Additionally, aircraft ying scienti c missions in extremely adverse environ-
ments, such as hurricanes, may require sensitive instruments that could be damaged or
their e ectiveness degraded by the gust response of the aircraft. Aircraft ying High
Altitude, Long Endurance (HALE) missions typically have extremely high aspect ra-
tios with low wing loadings and thus are more sensitive to adverse gust loadings. The
low fuel fractions and light structure of HALE aircraft result in an aircraft that is
both exible and sensitive to adverse gust conditions. Flight dynamics of very exi-
ble aircraft were studied by Shearer and Cesnik [4] where they implemented 6-DOF
vehicle dynamics with a modi ed version of the non-linear strain-based structural for-
mulation originally developed by Cesnik and Brown [5]. In their heavy weight cases,
rigid-body solutions were not su cient to accurately capture the ight dynamics of
5
exible aircraft. Shearer and Cesnik also determined that linear analysis was su -
cient for modeling simple symmetric maneuvers but nonlinear analysis was required
to accurately model the vehicle response due to asymmetric maneuvering. An under-
standing of the nonlinear ight dynamics of exible aircraft is necessary as shown by
Cesnik and Brown, but solutions for dealing with the adverse gust conditions causing
the nonlinear ight dynamics are needed.
Two broad categories of methods for alleviating the gust response have been
researched. One is active control systems where the aircraft actively responds based
on measurements of its ight conditions. This has been a topic of more recent research
with the advancement of new materials and computing capabilities for active control
laws. The second is passive systems whereby the aircraft automatically responds
to the ight conditions without any active control inputs or measurements. This is
extremely bene cial in that it maintains simplicity of the design. The second method
is the focus of this research.
2.1 Active Control Systems
Active control systems and aeroservoelasticity have been a popular area of re-
search in addressing gust alleviation. A di cult problem with using active control
systems has been accurately modeling the aerodynamic behavior of the system so
control laws could be derived. The Laplace domain has been used to calculate the un-
steady aerodynamic loads and aerodynamic coe cients which tend to be nonrational
and mathematically intensive [6], [7]. A nite-order, state-form, matrix equation for
stability analysis is desired which requires the aerodynamic in uence coe cients to
be approximated [6]. Because of the complexity of these calculations and the non-
rationality of the coe cients, a rational approximation of the unsteady loads on a
6
typical section in incompressible ow was formulated by R. T. Jones [8]. Approxi-
mating the nonrational coe cients caused a degradation in the accuracy of the model
but is necessary to be implemented into the control systems [9]. Karpel [6] developed
a new minimum-state method for e ciently approximating rational unsteady aerody-
namic loads. With an acceptable approximation of the unsteady aerodynamic loads, a
state-space set of matrix equations could be formulated and control laws implemented
for the active control system [6].
Within active control systems, two methods that were studied provide increased
system performance by reducing the gust response and delaying/eliminating the on-
set of utter. The rst method used aerodynamic control surfaces such as ailerons
and the second used aeroservoelasticity to alleviate the gust response. Block and
Gilliat [10] successfully doubled the stability of a simple wing with the use of a single
control surface that was 20% of the wing and spanned the entire trailing edge. Cruise
aps, similar to the previous study, were shown to successfully reduce disturbances
in aircraft lift and vertical acceleration at the aircraft center of gravity by 65%. Mea-
surements of the pressure di erentials allowed the cruise ap to adjust and maintain
the ideal stagnation point in adverse gust conditions [11]. Additionally, Rennie and
Jumper [7] used a trailing-edge ap on a NACA 0009 airfoil that was 27% of the
chord. Measurements of the lift perturbations and the pressure distributions were
used to control the ap and successfully reduce the gust response. Several problems
exist with these methods of gust alleviation. First, the control surface must be signi -
cant in size as characterized by all three studies where the control surface was at least
20% of the chord. Additionally, as determined by the trailing-edge ap study [7],
the control rates for the ap were high and the lift and pressure measurements to
determine the aeroelastic properties were extensive adding system complexity. Also,
7
Ref. [11] was limited in its results because it did not include the aeroelastic e ects of
the unsteady aerodynamics resulting from the rapid ap motion.
Research has been conducted to see if an optimization algorithm can be used to
optimize the planform design in order to have the greatest aeroelastic control for a
given design using control surfaces [12], [13]. Nam and Chattopadhyay [13] varied
aspect ratio, sweep angle, control surface location, and control surface size whereas
Pettit and Grandhi [12] used aileron e ectiveness and gust response as the driver of
the design. In both cases, the gust response of the designed aircraft was reduced
when compared to a base model.
An approach di erent to using standard control surfaces for gust alleviation is to
use several special control surfaces. Instead of a single trailing-edge control surface,
Moulin and Karpel [9] used three control surfaces: ailerons, special underwing forward
positioned control surfaces, and special wing-tip forward-positioned control surfaces.
At intermediate velocities, the movement of all control surfaces successfully reduced
the gust loads and the wing tip accelerations in the wind tunnel tests. However, as
the velocity was increased the e ectiveness of the ailerons was reduced but the special
surfaces became more e cient with the wing-tip surfaces being the most e cient.
Another active control system involved manipulating the aeroelastic properties
of the system in order to produce bene cial results, otherwise known as aeroservoe-
lasticity. The use of piezoelectric materials was studied on a cantilevered composite
plate to determine if aeroservoelastic control was capable of reducing the gust re-
sponse. Using four actuator units, \tip displacement, twist, and acceleration [were]
reduced by 36%, 72%, and 54% respectively" [14]. Additionally, Lazarus, Crawley,
and Lin [15] used actuators to control both the shape and the strain forces of the lift-
ing surface. By controlling those two parameters, the dynamic stability of the lifting
plate was increased. Use of the strain-actuated lifting surface reduced the rms gust
8
response by 60% in addition to the utter speed being increased by 11%. Several
problems with aerosevoelasticity consist of the power required by the actuators to
deform the surface appropriately. This especially becomes a problem as the thickness
ratio of the lifting surface is increased beyond that of a thin plate [15]. Along with the
power requirement, the actuator size increases as the thickness ratio becomes larger.
Finally, the density of the piezoceramics used in aeroservoelastic control is more than
double that of aluminum, which makes it an extremely unfriendly aircraft material.
Overall, active control systems show promise in being able to successfully allevi-
ate or reduce the gust response of an aircraft if properly designed. The drawbacks of
using active control systems for gust alleviation are the increase in system complexity,
computing power, weight, sizing, and maintenance. In order to properly model the
aeroelastic properties of the wing, additional sensors need added and implemented
into the controller which is nontrivial. The additional weight of the sensors, wires,
actuators, and control surfaces can also add up quickly. Finally, a signi cant nancial
cost will be associated with the maintenance of these systems.
2.2 Passive Control Systems
A passive control system for gust alleviation in aircraft is the torsionally free wing,
otherwise known as the \free wing". The free-wing design allows the wing to freely
rotate about a spanwise axis, usually the structural spar, where the aerodynamic
forces control the pitch of the wing. Extensive research was conducted on the free
wing by NASA throughout the late sixties through the early 1980?s. The rst major
study of the free wing was conducted by Porter and Brown in 1970 where the stability
and gust alleviation properties were studied analytically [3]. Three separate aircraft
classes were considered to analyze the capabilities of the free wing. A general aviation
9
(Cessna style) aircraft, a utility aircraft, and a transport aircraft were all analyzed
using the free wing. In all ight regimes, the gust responses of all three aircraft classes
were signi cantly decreased. Along with the excellent reductions in gust response,
results showed a 25% reduction in roll disturbances as well [3].
Additional studies on the aerodynamics and the potential for dynamic stall and
utter of the free wing were studied by Ormiston in 1972 [16]. Dynamic stall and
utter characteristics were achievable with the free wing but could be easily eliminated
by the reduction in the size of the trailing-edge control tab. With a properly designed
control tab, the stability characteristics of a free-wing wing section were excellent and
maintained the extreme di culty in stalling the wing.
As a followup to Porter and Brown [3], results of a study of the applications
of the free wing for light, general aviation aircraft were reported in Ref. [17]. It
was determined that the free wing could be applied to light, general aviation aircraft
because of the excellent gust alleviation characteristics of the free wing. A 54% reduc-
tion in the rms load factor was found when the aircraft was subjected to continuous
turbulence. A \free-wing/free-trimmer" design was studied by NASA in 1978 where
the free-trimmer was a free-wing design that controlled the pitch angle of the main
free wing [18]. The same bene cial gust alleviation was experienced but the free-
trimmer increased the maximum lift capability when compared to a pure free-wing
con guration.
The free-wing design was most e ective in aircraft that have low wing loadings,
such as light, general aviation aircraft and many unmanned aerial vehicles. The
greater the wing loading, the less the turbulence response. However, having a low
wing loading free-wing aircraft was like having a xed wing aircraft with a large wing
loading. This was shown in Ref. [2] where the free-wing design had a similar peak
vertical acceleration from the vertical gust pro le as a high wing loading xed wing.
10
A xed wing aircraft with a wing loading similar to that of the free-wing aircraft had
a vertical response ve times greater.
Free wings have been considered for the use in unmanned aerial vehicles. Un-
manned aerial vehicles have become increasingly popular for use in long endurance
missions, experimentation, and surveillance. Kraeger [19] studied the use of a free-
wing UAV for a microgravity facility where the UAV could y exact mission pro les
at a lower cost compared to larger microgravity facilities operated by manned air-
craft. A xed wing UAV microgravity facility was more sensitive to gusts compared
to the much larger manned microgravity facility. The gust response could cause the
UAV to deviate from the ideal ight path and be detrimental to the microgravity
performance. The free-wing UAV microgravity facility was capable of handling these
disturbances and maintain the desired microgravity ight performance.
The free wing is a passive control system that is very capable of alleviating gust
and rolling disturbances on aircraft of all sizes. A great bene t of the free wing is
that it requires no wiring, no computer power, no power to work any actuators other
than the trailing edge control tab which is no di erent than an aileron, and maintains
simplicity so maintenance and maintenance costs would be considerably less than
an active control system. A drawback of the free wing is the loss of maximum lift
when compared to a xed wing aircraft. Because the trailing edge of the free wing
is used to control the pitch of the wing, the maximum lift produced by the wing
was signi cantly reduced and leading edge aps were highly suggested to make up
for the lost lift [17]. Additionally, a weight penalty is usually incurred using the free
wing due to the ballast required to balance the free wing on the spanwise hinge axis.
The loss can be as much as 1.5% to 7% of the aircraft gross weight depending on the
location of the hinge axis [17]. Finally, when the right and left wings are free to rotate
11
with respect to each other, an unstable spiral mode exists that must be stabilized,
especially in the lower ight speeds [3].
12
Chapter 3
Conceptual Design
3.1 Sizing and Con guration
A conceptual design was created in order to study the stability characteristics
of the segmented free-wing system and to test its gust response for use on a HALE
aircraft. The segmented free wing was designed to t onto an existing, modular
testbed aircraft by replacing the existing xed wings of the aircraft. The overall
wingspan of the system was approximately 13.3 feet and had a chord of 16 inches.
The planform had no taper or sweep. Each half of the wing wing was divided into
5 equal segments approximately 14 inches wide with a 1/16 inch gap between each
segment to prevent any interference between segments. A top view of the segmented
free wing is shown in Fig. 3.1. Each individual segment was balanced about its hinge
Figure 3.1: Segmented free wing viewed from the top
axis by adding counterweights extending in front of the segment. The total weight of
the wing was 8.5 pounds including the counterweights.
13
Figure 3.2: Free-wing airfoil design with control tab
3.2 Airfoil Design
To maintain static pitch stability of each wing segment the pivot axis must be
ahead of the aerodynamic center of the segment, which is most often near the quarter-
chord [3]. The hinge margin is the distance the pivot axis is ahead of the quarter-chord
line and is represented in percentage of the chord. Hinge margins ranging from 0.05c
to 0.20c were examined in Ref. [3]. Additionally, the e ect of the hinge margin on
the control tab sizing at the trailing edge of the wing was examined [3]. In this study,
a hinge margin of 0.05c was chosen in order to reduce the amount of ballast required
to balance the wing segments. With this hinge margin the pivot axis, which was the
main wing spar, was located at 20% of the chord. Vertically, the spar was centered
on the chord line.
A control tab was located at the trailing edge and was the only control device on
the segmented free wing. The control tab could de ect +/- 20 degrees, had a chord
length of 25% of the wing chord, and spanned the entire trailing edge of the segment.
Each segment?s control tab was actuated using an electric servo motor embedded in
the airfoil and covered using a plastic skinning material.
14
The airfoil shape selected was an airfoil designed by John Roncz for Freewing
Aerial Robotics and used on their Scorpion UAV. This airfoil shape is shown in Fig-
ure 3.2. The wing was constructed out of extruded polystyrene foam. Several sections
of the inner wing were cut out to allow the passing of wires and to reduce weight. A
counterweight was attached to the bottom side of each segment and extended forward
of the leading edge with approximately a 1 oz weight attached. The counterweight
moved the center of gravity forward to balance the segment on the spanwise pivot
axis.
The airfoil was analyzed using Xfoil [20], [21] to determine its aerodynamic center
and lift curve slope. A Reynolds number was calculated to be around 200,000 based
upon a 22 ft=s velocity and a 16 inch chord length. This was used for the viscous
boundary layer analysis. As shown in Fig. 3.3, the lift curve slope, Cl , was calculated
to be 7.66 per radian (0.132 per degree). Figure 3.4 shows the pitching moment of
Figure 3.3: Lift coe cient versus angle of attack
15
Figure 3.4: Pitching moment versus lift coe cient
the airfoil about its 1/4 chord point as a function of lift coe cient. A linear trend
line from regression analysis has been drawn on the graph along with the trend line
equation. The aerodynamic center of the airfoil was found by adding 1/4 to the
inverse slope of the trend line. At 29% of the chord, the aerodynamic center for this
airfoil is considerably further aft than the typical thin airfoil value of 25% of the
chord.
3.3 Experimental Setup
The segmented free-wing model was tested experimentally by mounting the
model on a truck. The wing was mounted to a 3" x 3 1/2" wood beam (constructed
from a pair of 2x4?s) that extended 7 feet in front of the truck as shown in Fig. 3.5.
16
Figure 3.5: Diagram of the experimental setup
Mounting the wing a considerable distance in front of the truck reduced the aerody-
namic in uence of the truck on the wing. The wing was rigidly mounted to the beam
so that roll was mechanically restrained. The mount was later modi ed to allow the
wing to freely roll. The control tabs on each of the wing segments were controlled
via a radio transmitter so the angle of attack of the segments could be varied from
within the truck cab during testing.
3.4 Wind Tunnel Model
A wind tunnel model of the wing was designed and fabricated for use in the
Auburn University 3x4 foot subsonic wind tunnel. The model had a wing span of
three feet divided into six segments of equal size (approximately 6 inch span). Each
segment had an eight inch chord with the axis of rotation placed at fteen percent of
the chord. Each segment had a negative ve degree control surface de ection. Unlike
the larger model discussed in Section 3.1, the wind tunnel model had no internal
control wires. The wing segments were balanced about the pivot axis using lead
shot so the only net forces acting on the wing segments were the aerodynamic forces.
The spar was made of 12 inch diameter steel round stock to maintain the sti ness
in the model. A 116 inch spacer was placed between each segment to prevent the
17
Figure 3.6: Segmented free wing mounted to beam on truck looking through the
windshield
18
segments from interfering with each other. To create a spanwise ow variation in the
wind tunnel, an oscillating wing was placed vertically in the wind tunnel generating
a trailed vortex across the segmented free-wing wind tunnel model. A schematic of
the wind tunnel setup can be seen in Fig. 3.7.
Figure 3.7: Wind tunnel test schematic
The wind tunnel model was designed so that each individual segment could be
connected to its neighbors. Thus, it could simulate a single free wing, a left and right
free wing, or a segmented free wing.
The variation in angle of incidence across the six segments is easily seen in Fig. 3.8
as each individual segment adjusts to maintain constant aerodynamic angle of attack
across the span in the presence of the trailed vortex above the wing.
19
Figure 3.8: Segmented free-wing model in AU wind tunnel
20
Chapter 4
Analytical Model
A mathematical model of the segmented free wing was developed in order to
accurately predict the behavior of the system. MATLAB 7.0 was used to run the
calculations and to plot the results of the stability analysis. A linear model of the
system dynamics was developed and eigen analysis was used to evaluate the behavior
of the segmented free-wing system.
The origin of the system was located at the intersection of the axis of symmetry
and the centerline of the spar. The x-direction points forward from the leading edge
of the wings and y-direction extends out the right wing as shown in Fig. 4.1. The
z-direction points downward to complete the triad. The system was modeled with
11 degrees of freedom: ten di erent pitch angles, one for each segment, and a roll
angle. The system was mechanically restrained to the support beam resulting in no
translational degrees of freedom. The translation of the truck was assumed to be a
constant velocity in the x-direction.
Figure 4.1: Axis con guration for analytical model
21
4.1 Governing Equations
Free-body diagrams of the segmented free-wing system and a wing segment are
shown in Figs. 4.2 and 4.3. From the free-body diagrams, equations modeling the
segmented free-wing system could be formulated.
Figure 4.2: Free-body diagram of the segmented free-wing system in roll, looking
from rear
The equation modeling rolling motion of the system is
Mx = Ixx
Mx =
X
i
Liyi K C _ +
X
i
myi xi i
(4.1)
where is the rolling acceleration in rad=s2, K is the spring force of the mounting
beam, and C is the damping term in the wood. The nal term in Eqn. 4.1 takes into
22
Figure 4.3: Free-body diagram of a wing segment
account the possible o set of the center of gravity of the wing segments causing a
slight imbalance in each segment.
The equation for the pitching motion of each segment is
Myi = Iyy i
Myi = MAEROi +myi xi
(4.2)
where is the pitching acceleration in rad=s2, MAERO is the aerodynamic moment,
and the nal term is the center of gravity o set.
4.2 Aerodynamics
The aerodynamic lift and moment, Li and MAEROi, in Eqns. 4.1 and 4.2 are
the most complex terms in the analytical model. Unsteady aerodynamic calcula-
tions were used to accurately model the aerodynamic forces and moments on the
segmented free wing. To determine whether unsteady aerodynamics are signi cant
for this problem or whether a quasi-steady approximation could be justi ed, a brief
analysis using Theodorsen?s function was completed. Theodorsen?s function is one
23
method for modeling unsteady aerodynamics in the frequency domain. Theodorsen?s
function predicts the lift response, both in-phase and out-of-phase, to a sinusoidal
oscillation of an airfoil as a function of the reduced frequency. Figure 4.4 presents
the real (in-phase) and imaginary (out-of-phase) parts of Theodorsen?s function for a
range of reduced frequency. The reduced frequency can be calculated by
k = !bU (4.3)
where ! is the oscillation frequency in rad=s, b is half the chord length, and U is the
velocity in feet per second [22]. During experimental testing of the model described
in Section 3.1, an oscillation at a frequency of 1.2 Hz was observed at an airspeed
of 22 fps. This corresponds to a reduced frequency of k = 0:23. Using Fig. 4.4, the
magnitude of the lift response was 0.72 and the phase shift is 0.2584 rad or 14.80
degrees. This indicates that a quasi-steady assumption would over predict the lift
response by over 20% and clearly unsteady e ects are signi cant.
In unsteady aerodynamics the lift response to a step change in angle of attack,
known as the indicial response, can be modeled as [23]
CL(s)
= CL (s) (4.4)
where s represents the distance traveled in half chords
s = Utb (4.5)
24
Figure 4.4: The real and imaginary parts of Theodorsen?s function, F(k) and G(k)
respectively [22]
25
and (s) is the Wagner lift de ciency function. One approximation to the Wagner
function is by Jones and is highly accurate for intermediate values of s [23]
(s) = 1 A1exp( b1s) A2exp( b2s) (4.6)
where A1 = 0:165, A2 = 0:335, b1 = 0:0455, and b2 = 0:3. Given the indicial
response, with Jones?s approximation to the Wagner function, Duhamel?s convolution
integral [23] can be used to get the overall lift response to an arbitrary forcing function,
(t).
CL(t) = CL
(0) +
Z t
0
d
d ( )
U
b (t )
d
(4.7)
For the analytical model, it is desirable to have the unsteady terms in a state-space
form as presented in Eqn. 4.8.
8>
<
>:
_X1
_X2
9>
=
>; = A
8>
<
>:
X1
X2
9>
=
>;+
2
64 1
1
3
75 (t)
CL = C
8
><
>:
X1
X2
9
>=
>;+ D (t)
(4.8)
To nd this state-space representation, the Laplace transforms of Eqns. 4.7 and 4.8
were taken and are shown in Eqns. 4.9 and 4.10 where p is the Laplace variable.
CL (p)
CL = (p)
1
2 +
A1b1
p+b1 +
A2b2
p+b2
(4.9)
26
pX (p) = AX (p) +
0
B@ 1
1
1
CA (p)
CL (p) = CX (p) + D (p)
(4.10)
These equations can then be equated to solve for the elements of the matrices of A,
C, and D in the equations for the unsteady circulatory lift per unit span:
8
><
>:
_X1
_X2
9
>=
>; =
2
64 b1Ub 0
0 b2Ub
3
75
8
><
>:
X1
X2
9
>=
>;+
2
64 1
1
3
75 (t)
Lc0 = U2bCL
0
B@U
b
A1b1 A2b2
8
><
>:
X1
X2
9
>=
>;
1
CA+
2
64 1
1
3
75 (t)
(4.11)
The circulatory lift on the airfoil is a ected by both the angle of attack and also
by the pitch rate (rate of change of angle of attack). From thin-airfoil theory, it is
known that these two can be combined by using the angle of attack at the 3/4 chord
point [23]. Combining the pitching motion and the rolling motion of the wing, the
angle of attack at the 3/4 chord point can found from
3
4c
= + 1U
y _ + _ b
1
2 a
(4.12)
which replaces the angle of attack term of the circulatory lift in Eqn. 4.11.
In addition to the circulatory lift on the wing, there are also aerodynamic forces
on the wing due to the mass of air surrounding the wing that must be pushed out
27
of the way when the wing pitches or plunges up or down. These forces are called
apparent mass or non-circulatory forces, and for lift can be written [24]
L0NC = b2
h+U _ ba
(4.13)
where
_h = y _ (4.14)
and
h = y (4.15)
Including both the circulatory and non-circulatory terms, the total lift per unit
span can be written as
L0 = U2bCL
0
B@U
b
A1b1 A2b2
8
><
>:
X1
X2
9
>=
>;+
1
2 34c(t)
1
CA
+ b2
h+U _ ba
(4.16)
The aerodynamic pitching moment includes both circulatory and non-circulatory
terms just as the lift force does. The circulatory moment is due to a constant term
plus the circulatory lift force acting at the aerodynamic center of the wing segments.
This can be written as
MC0 = 12 U2b2CM0 +b
1
2 +a
LC0 (4.17)
28
The non-circulatory terms for pitching moment are [24]
M0NC = b2
ba h+Ub4
a 12
_ b2
16
(4.18)
which yields a total pitching moment per unit span of
M0 = 12 U2b2CM0 +b
1
2 +a
L0C
+ b2
ba h+Ub4
a 12
_ b2
16
a2 + 18
(4.19)
4.2.1 Validation of Aerodynamic Model
The analytical aerodynamic model was validated by comparing the circulatory lift
response to Theodorsen?s function and the total lift response to experimental data by
Rainey [25] of a purely pitching wing. All roll terms in the lift and unsteady equations
were ignored to allow for a purely pitching system. To compare the circulatory lift to
Theodorsen?s function, the Laplace transform of Eqn. 4.11 was taken which gives
CLc (s) = CL
1
2 (s) +
U
bA1b1X1 (s) +
U
bA2b2X2 (s)
(4.20)
and
sX1 (s) = b1UbX1 (s) + (s)
sX2 (s) = b2UbX2 (s) + (s)
(4.21)
Substituting the unsteady terms, X1 and X2, into the lift equation and dividing by
CLc (s)
(s) = CL
1
2 +A1b1
U
bs+b1U +A2b2
U
bs+b2U
(4.22)
29
The Laplace transform of the non-circulatory lift, Eqn. 4.13, divided by is
shown by
CLNC (s)
(s) = b
s
U
ba
U2s
2
(4.23)
Substituting i! for s and gathering terms to put Eqns. 4.22 and 4.23 in terms of the
reduced frequency, k, the transfer function from angle of attack to lift response in
terms of the reduced frequency is written as
CL (i!)
(i!) = CL
1
2 +A1b1
1
ik +b1 +A2b2
1
ik +b2
+ ik +ak2 (4.24)
As indicated by Fung [22], the circulatory lift portion of this function should
be equivalent to Theodorsen?s function when normalized by the lift curve slope.
Theodorsen?s function can be written as
C (k) = K1 (ik)K
0 (ik) +K1 (ik)
(4.25)
where K0 and K1 are the modi ed Bessel functions.
In Fig. 4.5, the circulatory lift portion of Eqn. 4.24 is compared to Theodorsen?s
function, Eqn. 4.25. The real portion matches within 2% and the imaginary within
10%. The relatively small discrepancy can be attributed to the approximation used
for the Wagner function.
Experimental data on a pitching wing by Rainey [25] was used for comparison
of the total lift response. The total lift response as a function of reduced frequency is
plotted in Fig. 4.6 with Rainey?s data being marked by an \o". The analytical model
follows the experimental data trend quite well in both magnitude and phase. As the
reduced frequency increases to greater than 0.4, a slight deviation occurs between the
experimental data and the total lift response. However, for the experiments presented
30
Figure 4.5: Comparison of the analytical model circulatory lift to Theodorsen?s func-
tion
31
Figure 4.6: Comparison of the analytical model total lift response with pitching data
from Rainey [25]
32
later in this thesis, the reduced frequency remained within the 0.2 to 0.3 range which
matches the Rainey data nicely.
4.3 Wake E ects
Because of the trailed vortex wake system behind any lifting body, the aero-
dynamic lift on one airfoil segment a ects all of the other nine. Consequently, to
accurately predict the behavior of the segmented free wing, it was necessary to add
downwash e ects to the system so the e ective angle of attack on each segment, in-
cluding the in uence of all the others, could be accurately modeled. To model the
downwash, the Vortex Lattice Method (VLM) [26] was used to calculate an e ective
angle of attack on each segment which could then be used to determine the overall
lift.
Figure 4.7: Setup of the Vortex Lattice Method on segmented free wing
33
Each wing segment was modeled using a set of horseshoe vortices that consisted
of a bound vortex along the quarter-chord of the wing segment and two semi-in nite
trailed vortices extending from the ends of the bound segment to in nity aft of the
wing. Each segment had a control point placed at the three-quarter-chord centered
between the two trailed vortices.
The velocity induced by a general horseshoe vortex of strength n with a length
of dl is determined by use of the law of Biot and Savart as shown in Eqn. 4.26
!dV = n
!
dl !r
4 r3 (4.26)
Referring to Fig. 4.8, the magnitude of the induced velocity induced by a vortex
lament is:
dV = n sin dl4 r2 (4.27)
Equation 4.27 can be integrated to obtain the total induced velocity produced by a
straight vortex lament:
V = n4 r
p
Z 2
1
sin d
= n4 r
p
(cos 1 cos 2)
(4.28)
A boundary condition must be applied in order to compute the strength of the
vortices where the \surface of the wing is a streamline" [26]. Since the surface behaves
as a streamline, the ow does not pass through the surface of the wing but remains
34
Figure 4.8: \Nomenclature for calculating the velocity induced by a nite length
vortex segment" [26]
tangent to the surface. For wings with a modest slope of the mean camber line, the
tangency condition can be approximated by [26]:
wm vm tan +U1
dz
dx
m
= 0 (4.29)
For airfoils of modest thickness, a 2-dimensional, planar approximation can be used
to simplify the tangency condition to
wm +U1sin = 0 (4.30)
or for small angles of attack:
wm = U1 (4.31)
35
The downwash at control point i from segment j can be calculated from Eqn. 4.28
and then summed to get downwash, wmi
wmi = 14
X
j
Cij j (4.32)
as derived in Bertin [26]. Equating this to 4.31 gives a set of equations for the vortex
strengths:
1
4
X
j
Cij j = U1 (4.33)
From the Kutta-Joukowski theorem [27] and the de nition of the lift coe cient
L0 = U = 12 U22bCL (4.34)
Solving for CL and substituting for
CL = 4 b C 1 = Cl e (4.35)
Finally, solving for e
e = 4 bC
l
C 1 (4.36)
This matrix equation provides a relationship between the angle of incidence, , of
each segment, and the aerodynamic angle of attack of the segments including the
mutual in uences of each segment on all the others. This e ective angle of attack
could then be substituted into the aerodynamic equations 4.16 and 4.19 to arrive at
the total lift and moment per unit span on each segment.
36
4.4 Equations of Motion
By combining all the derivations from the previous sections the equations of
motion for the full system were formulated. The system was set up in the following
format:
C _~x = A~x (4.37)
The state vector ~x was de ned as
~x =
_ i i X1;i X2;i _
T
, for i = 1..10 (4.38)
The superscript \T" in Eqn. 4.38 denotes the transpose of the matrix. Solving for _~x,
the stability of the system could be analyzed by determining the eigenvalues of the
matrix C 1A. This was performed using the MATLAB program and results will be
presented in the form of root-locus plots. The eigenvectors of the system matrix were
used to determine the di erent modes of oscillation of the system.
37
The equations for pitch, roll, X1, and X2 are shown in order in Eqs. (26-28).
Pitch:
b4xa2 + 18 b4x+Iyy
i + b3xayi 2mb x
iyi
=
b3xU
1
2 +a
+ 18 Ub3xCl Cij 1 4a2
_ i
+
1
4 U
2b2xCl
Cij (1 + 2a)
i
+
1
2 U
3bxCl
A1b1 (1 + 2a)
X1;i
+
1
2 U
3bxCl
A2b2 (1 + 2a)
X2;i
+
1
4 Ub
2xCl
yjCij (1 + 2a)
_
(4.39)
Roll:
xy
i b3a+ 2myi xib
i +
xy2
i b
2 Ixx =
xyi b2U + 14 xyiUCl b2Cij (1 2a)
_ i
+
1
2 xyibU
2Cl
Cij
i
+ xyiU3Cl A1b1 X1;i
+ xyiU3Cl A2b2 X2;i
+
C + 12 xyiyjUCl bCij
_
+K
(4.40)
38
Unsteady Aerodynamic states:
2
64 _X1;i
_X2;i
3
75 =
2
64 Ub b1 0
0 Ub b2
3
75
+
0
B@ 1
1
1
CA b
2UCij (1 2a)
_ i +Cij i + yj
UCij
_
(4.41)
Combining all of these equations produces a set of 42 coupled rst-order di er-
ential equations, 10 for , 10 for _ , 10 for _X1, 10 for _X2, one for , and one for _ .
The equations for _ and _ are trivial and of the form _ = _ and _ = _ .
4.5 Wind Tunnel Model
The center of gravity position was located on the spanwise axis of rotation. The
analytical model used for the wind tunnel model was substantially the same as used
for the truck model. The input parameters were changed to match the geometry of
the smaller model. In addition, two new friction terms were added to the model to
account for friction in the pitch and roll bearings. These friction terms were added
to the equations of motion (Eqns. 4.1 and 4.2) as presented in Eqn. 4.42.
My = MAERO +myi xi kpitch _
Mx =
X
i
Liyi K C _ +
X
i
myi xi i kroll _
(4.42)
39
Chapter 5
Results
5.1 Initial Wind Tunnel Test
An initial wind tunnel model was tested as both a proof of concept and then as
a veri cation of the computer model. The rst experimental test on the wind tunnel
model was to determine the rolling moment generated by the model as a result of
spanwise ow variations created by an upstream vortex-generating oscillating wing.
However, during the rst tests, the wind tunnel model began to have a destructive
oscillation about the rolling axis. The test was immediately terminated. It was
assumed that play in the mount and imbalance of the wing segments were responsible
for the oscillation.
After a new mount was constructed for the model and the segments were bal-
anced about their hinge axis, the rolling moment generated by the spanwise ow
variation over the model was measured. For each of the three con gurations, the
rolling moment was measured with the vortex-generating wing at full positive and
full negative de ection. The di erence between the two rolling moments gives an in-
dication of the amount of aerodynamic load that would be transmitted to the aircraft
structure. This simulates the aircraft encountering a gust that varies across the span
of the wing. The results of the wind tunnel tests are shown in Table 5.1.
As seen in Table 5.1, the single free-wing transfers a load of 1.2 ft-lbs to the fuse-
lage of the model. However, when each wing half is allowed to move freely from the
40
Table 5.1: Rolling moments of three con gurations
other, nearly a 30% reduction in the transferred load occurs. This is even further en-
hanced by the segmented free wing which reduces the transmitted load from the wings
to the fuselage by 64%. These results, although preliminary, show strong evidence
that the segmented free-wing design would successfully reduce the gust response of
HALE aircraft as they pass through the troposphere.
5.2 Experimental Model
Following the success of the initial wind tunnel model, a larger wing was con-
structed as described in Section 3.1. The experimental testing of this model was
performed by attaching the rig to a full size pickup truck and slowly increasing the
speed of the system while observing the results. Digital videos of the tests were
acquired and used to estimate oscillation frequencies and onset velocities.
At a velocity of approximately 22 ft/s an unstable oscillatory rolling motion
occurred during experimental testing. As speed was increased the frequency stayed
the same but the amplitude increased dramatically. The system had an oscillation
frequency of approximately 1.2 Hz with a maximum rolling angle of about 10 degrees
before the speed was reduced. Reference [3] found a divergent spiral mode in their
computational analysis of a free-wing aircraft with independent left and right wing
panels. It is conjectured that this spiral mode is appearing as an unstable oscillatory
41
roll mode in this experimental apparatus due to the mechanical restraints of the
rolling motion.
Several possible solutions to the unstable motion were examined including in-
creasing the rolling moment of inertia, changing the center of gravity of the wing
segments, and changing the number of segments with results shown in Table 5.2. To
increase the rolling moment of inertia, aluminum pipes were attached to the wingtips
extending beyond the wingtips by two feet. This method provided very little damping
to the system and was deemed unsuccessful. The second test consisted of changing
the center of gravity of the segments by adding weight to the counterweight extending
out beyond the leading edge of each segment. This also proved unsuccessful. Reduc-
ing the number of segments increased the speed at which the unstable mode occurred,
but did not eliminate it.
As an additional test, the mounting system for the wing was modi ed to allow
the wing to roll freely as it would on a ying aircraft. Once the mechanical roll
restraint was removed the oscillation was gone, however as anticipated, a divergent
mode took its place. The system would consistently roll at very low speeds, which
would be expected to be the unstable spiral mode. The system was very hard to
control and could never be trimmed.
With the initial simple modi cations to the experimental system proving unsuc-
cessful, the design was modi ed to allow for the attachment of xed segments on each
wingtip. This design change allowed for various sizes of xed segments to be attached
ranging from as six inches wide to two feet wide. It was hoped that the xed segments
would provide adequate damping to stabilize the system.
The xed segments were a di erent design than the free-wing segments. The
chord length of the xed segments was the same as the free-wing segments, however,
the span of the segment varied. Fixed segments of four di erent sizes were made;
42
Table 5.2: Results of the experimental testing on the segmented free wing
43
24, 18, 12, and 6 inches in span. The airfoil for the xed segments was a symmetric
NACA 0012 airfoil so the only lifting force would be from the free-wing segments.
Additionally, the spar was place at the quarter-chord to minimize the moment of the
xed segments. The segments were placed at zero angle of attack.
The experimental platform with xed-segments was tested rst in the mechan-
ically restrained roll setup. The 24 inch segment was the rst xed segment tested
during the experiment. With the xed segments attached to the platform, the system
demonstrated a drastic increase in stability. The system no longer had the catas-
trophic divergent rolling oscillation that was seen in previous experimental attempts.
The platform remained straight and level throughout a large range of velocities with
a maximum sustained velocity of 66 ft=s. A dihedral formed in the wings due to the
lift being produced by the free-wing segments. Very little movement occurred in the
segments once equilibrium of the system was reached.
Testing continued with each of the other xed segment sizes. Mechanically re-
strained in roll, the experimental platform was able to demonstrate satisfactory sta-
bility using all sizes xed segments, including the 6 inch segment. All segment sizes
tested appeared to be su cient to provide dynamic stability to the experimental
platform.
In the next series of tests, the wing was allowed to roll freely with the xed
segments attached to the tips. Using the 24 inch segments, the wing did not behave
as it had without the xed segments. The free wing did not roll to one side or the
other as had been seen before. In order to keep the free wing level, some mechanical
control was required by the transmitter from the cab. The system did not damp
itself out, but the instability did not grow either. As the size of the xed segments
decreased, the sensitivity of the segmented free wing to a disturbance grew and the
pilot controlling the wings had more trouble holding the wings straight and level.
44
Figure 5.1: Segmented free wing with xed segments attached to the tip ying with
a visible dihedral
45
Although di cult, it was possible to trim the wing with the 6 inch segments. It was
much easier to maintain trimmed ight of the segmented free wing with the 24 inch
segments as they were not as sensitive to disturbances.
When the system was own without the xed segments, the segmented free wing
was dynamically unstable in both the mechanically restrained and mechanically free
states. However, when the xed segments were attached to the tips of the experimen-
tal platform the system demonstrated strong dynamic stability in the mechanically
restrained state. Any perturbations were quickly damped out and steady ight was
quickly achieved for all xed segment sizes. In the mechanically free state, the seg-
mented free wing appeared to be neutrally stable. There appeared to be no damping
or divergence in the system resulting from any perturbations.
5.3 Analytical Modeling
An eigen analysis was performed to study the dynamic stability properties of
the segmented free-wing equations of motion. The goals of this analysis were to
understand the nature of the instability, to facilitate predicting its onset, and to nd
methods for eliminating it. The analytical model was able to successfully predict the
oscillation in the mechanically restrained state and the instability in the mechanically
free state, both without the xed segments. With the addition of the xed segments in
the computer model the mechanically restrained system became dynamically stable
as was seen in the experimentation. The analytical model also predicts neutrally
stable dynamic system in the mechanically free state.
46
5.3.1 System Properties
In order to analyze the system, the mass properties of the wing and elastic
properties of the mount were needed. The mass moments of inertia, Ixx and Iyy, were
initially estimated based on approximations of the geometry. After the mass moments
of inertia were estimated, they were tested experimentally using a bi lar pendulum
as pictured in Fig 5.2. To obtain the estimated rolling moment of inertia, the system
was modeled as a rectangular parallelepiped with dimensions of 13.3 feet in span,
16 inch chord, and approximately 3 inches thick. The entire system was weighed to
obtain its mass in slugs and the rolling moment of inertia, Ixx, was calculated using
Eqn. 5.1 [28].
Figure 5.2: Diagram of the bi lar pendulum setup
47
Ixx = 112m(span2 +thickness2) (5.1)
Using AutoCAD 2004, the airfoil shape was drawn and the polar area moment
of inertia was calculated. Knowing the density and the span of the wing segment,
the estimated pitching mass moment of inertia was calculated. In addition to the
wing segment, the counterweight extending beyond the leading edge of the wing was
treated as a point mass and its moments of inertia added accordingly.
Iyy = Jyx seg +l2mcw (5.2)
The bi lar pendulum was then used to experimentally obtain the mass moments
of inertia. The experimental values were compared to the estimated values to deter-
mine validity of the experimental value. To maintain simplicity of the experimental
model, nonlinear aerodynamic damping was ignored and the moment of inertia was
calculated using Eqns. 5.3 and 5.4 [29].
k2 = gT
2
4 2
b
1b2
Y
= gT
2
4 2
b1b2
S2 (b
2 b1)
2 1=2 (5.3)
I = Mk2 = MgT
2
4 2
b1b2
S2 (b
2 b1)
2 1=2 (5.4)
The length of the two strings are denoted by S and are at equal angles from the
vertical. The distance between the two points of suspension and the distance between
the mounting points are denoted by 2b1 and 2b2 respectively. The perpendicular
distance between the suspension point and the mounting point is represented by Y.
T is the period of oscillation and M is the mass of the system. Finally, I is the
moment of inertia about the vertical axis and k is the radius of gyration. Figures 5.3
48
and 5.4 show the experimental model and the single wing segment mounted on the
bi lar pendulum and Auburn University?s Adaptive Aerostructures Lab.
The spring force in the experimental mounting beam was calculated using the
angle of twist equation and solving for the torque [30].
T = JGL (5.5)
where
K = JGL (5.6)
The mounting beam was fabricated from standard dimensional lumber by lami-
nating two 20? long nominally 2" x 4" (1 1/2" x 3 1/2" actual dimension) pieces of
wood stock. The spring force is highly dependent on the species of wood used to make
the mounting beam. The spring force varies 400 ft lbs=rad within just a species
family. Unfortunately, the exact species of wood was unknown, but it was assumed
to be from the Pine family. Using the material properties of Lodgepole Pine shown
in Table 5.3 [31], a modulus of rigidity could be calculated for the radial tangential
plane of the lumber. A diagram of the axis con guration is shown in Fig. 5.5. The
modulus of rigidity, G, was calculated to be 964,800 psf. The estimated modulus of
rigidity was used in the spring constant equation and an estimated value of K was
calculated to be 123.6 ft lbs=rad. No simple analysis for the damping coe cient,
C, was available, so for this initial study the value was assumed to be negligible
compared to the aerodynamic forces on the wing system.
To obtain a more accurate value of K, the spring force of the beam was exper-
imentally determined by applying a torsional loading to the mounting beam about
49
Figure 5.3: Experimental model mounted on the bi lar pendulum in Auburn Univer-
sity?s Adaptive Aerostructures Lab
Table 5.3: Lodgepole Pine elastic properties
50
Figure 5.4: Experimental segment mounted on the bi lar pendulum in Auburn Uni-
versity?s Adaptive Aerostructures Lab
51
Figure 5.5: Axis orientation for modulus calculations [31]
the longitudinal axis. The spring force was measured by twisting the beam to a pre-
determined angle and taking a measurement of the applied torque required to achieve
the angle. The beam was twisted to 5, 10 and 15 degrees with several torque read-
ings at each angle to obtain an adequate average. From the torque measurements,
K was calculated by dividing the measurement by the respective angle of twist. The
experimental results yielded an average K value of 482.5 ft lbs=rad and are shown
in Table 5.4.
Table 5.4: Data from spring constant test on mounting beam
5.3.2 Analytical Model Without Fixed Segments
The root locus plot in Fig. 5.6 shows the di erent modes of the restrained seg-
mented free wing. The rst eigenvalue is denoted by a circle and then the subsequent
eigenvalues are denoted by a plus sign. Ten stable pitching modes can easily be seen
labeled in Fig. 5.6 forming a fan shape originating at zero. The scatter in the pitching
52
mode is due to the wake e ects of each wing segment upon the other. There are two
rolling modes associated with the mechanically restrained system. One of the modes
is stable and is labeled in Fig. 5.6. The second mode is the unstable rolling mode
that was seen in the experimental testing of the segmented free wing.
Figure 5.6: Root locus plot of the full aerodynamic model for the mechanically re-
strained model without xed segments
The analytical model velocity was steadily increased from one foot per second
to 150 ft=s by 0.1 ft=s intervals. With a velocity of 150 ft=s the analytical model
predicts the segmented free wing will be unstable with an oscillation frequency of
1.31 Hz. This frequency is within 10% of the experimental frequency of 1.2 Hz. The
53
analytical model shows that as the velocity is increased, the frequency remains the
same but the real part of the unstable root becomes increasingly positive. Figure 5.7
shows a zoomed view of the root locus plot for the full aerodynamic model that is
mechanically restrained in roll without the xed segments attached. In Fig. 5.7 it is
easy to see the small changes in the unstable mode as the velocity is increased.
Figure 5.7: Root locus plot of the full aerodynamic model for the mechanically re-
strained model without xed segments from a zoomed in perspective
Mechanically freeing the roll in the analytical model produces a result that is
both expected and agrees with previous studies conducted by Porter and Brown [3].
The analytical model predicts an unstable roll mode that grows in magnitude with
54
the increase in velocity. The positive real root can be seen in Fig. 5.8. All the same
modes occur as in the mechanically restrained system except the unstable oscillating
rolling mode has now become an unstable rolling mode. This unstable rolling mode
is analogous to the unstable spiral mode reported in Ref. [3]. The unstable roll mode
is even more visible when looking at the locus plot zoomed in about the origin as in
Fig. 5.9
Figure 5.8: Root locus plot for the full aerodynamic model for the system mechanically
free in roll without xed segments
55
Figure 5.9: Root locus plot for the full aerodynamic model for the system mechanically
free in roll without xed segments zoomed in about the origin
56
5.3.3 Analytical Model With Fixed Segments
Adding xed segments to the analytical model produces a dynamic stability
that matches the experimental results. When adding xed segments to the analytical
model placed at the wingtips of the segmented free wing, the analytical model predicts
a dynamically stable system. The root locus plot of the full aerodynamic model,
restrained in roll, and with xed segments attached, is presented in Fig. 5.10 and
shows the newly stable rolling mode becoming increasing stable as velocity increases.
All the modes from the previous root locus plot of the same system setup are present
and exhibit the same behavior. The previously stable rolling mode has the same
shape, however, it is more heavily damped with the xed segments attached to the
segmented free wing. Zooming into the origin, the behavior of the newly stable rolling
mode can easily be seen. Figure 5.11 clearly shows the stable rolling mode becoming
increasingly damped as the velocity is increased. Also, as the velocity is increased
the oscillation frequency of the stable rolling mode remains almost constant.
The free-to-roll system contains an unstable roll mode in the analytical model
when the xed segments are not attached to the wingtips of the segmented free wing.
Adding the xed segments to the analytical model for the free-to-roll system produces
a system that is no longer unstable. However, with the addition of the xed segments,
several zero eigenvalues appear. Instead of being dynamically unstable, the analytical
model predicts a neutrally stable system. This would agree with experimental results
where the segmented free wing could be trimmed, but if a perturbation entered the
system there was no damping. Figure 5.12 shows the root locus plot for the roll-free
system with the xed segments attached. Figure 5.13 is the same plot but zoomed in
about the origin.
57
Figure 5.10: Root locus plot for the full aerodynamic model for the system mechani-
cally restrained in roll with xed segments
58
Figure 5.11: Root locus plot for the full aerodynamic model for the system mechani-
cally restrained in roll with xed segments attached, zoomed about the origin
59
Figure 5.12: Root locus plot for the full aerodynamic model for the system mechani-
cally free in roll with xed segments attached
60
Figure 5.13: Root locus plot for the full aerodynamic model for the system mechani-
cally free in roll with xed segments attached, zoomed in about the origin
61
5.3.4 Analytical Model Compared to Experimental Results
The analytical model predicts a crossover velocity, the velocity at which the sys-
tem becomes dynamically unstable, of 49.5 ft=s (33.8 mph). In the experimental
results, the velocity at which the segmented free wing became unstable was between
22 and 30 ft=s. The analytical model predicts a signi cantly more stable system.
This discrepancy between the analytical model and experimental results possibly oc-
curs from characteristics not modeled in the analysis. An example of a characteristic
that was not modeled is the small oscillation of the forward mounted counterweights
along with their possible e ects on the aerodynamic properties of the airfoil. During
experimental testing, small oscillations of the 1 ounce counterweight were visible in
both the horizontal and vertical directions. Also, the segmented free-wing mounting
beam was modeled having no movement. However, in the experimental tests, the
beam had a slight bending de ection that would allow the mount to have small, -
nite oscillations. This oscillation could have been caused by the movement of the
segmented free wing or any movement in the suspension of the truck. This was not
believed to a ect the system as it would appear to the wings only as turbulence,
however, it was not modeled so the e ects are unknown. Finally, the modeling pa-
rameters used in the analytical model may have a certain amount of error possibly
causing the analytical model to predict a more stable dynamic system. Examples of
the parameters that may have error are the spring constant, mass moments of iner-
tia, center of gravity position of the airfoil, and nally the aerodynamic center of the
airfoil.
The uncertainty in the spring constant of the mounting beam comes from a
limitation in the testing. A torque wrench was attached to the front of the supporting
beam. The beam was then torqued until a particular angle of twist was achieved,
62
followed by recording the value from the torque wrench. A problem with this method
of measuring the spring constant arises from the downward bending of the beam
from the way the torque was applied. The e ect of the bending of the beam on the
measured torque is unknown.
The pitching and rolling mass moments of inertia were measured via a bil lar
pendulum. Potential error in the pitching moment of inertia comes for the varying
mass in each wing segment. The pitching mass moment of inertia for one segment
was measured and assumed to apply for all of the segments of the segmented free
wing. The quantity of internal wires and their position, the counterweight mass and
position, and variable segment mass all contribute to an uncertainty in the pitching
mass moment of inertia. For both the rolling and pitching mass moments of inertia,
the damping of the displaced volume of air during the bi lar pendulum test was not
modeled and ignored in all calculations of mass moments of inertia. These damping
e ects were ignored because there is no simple means of estimating or measuring
them.
The location of the center of gravity of each individual segment varies in each
segment depending on the location of the control surface wires that run laterally
through the internal spaces of the airfoil aft of the spar. The segments were balanced
as accurately as possible, but during ight the wires have freedom to move and can
shift the center of gravity of the segment a nite amount, depending on the number
of control wires running through the individual wing segment.
Variations in the surface of each wing segment can a ect the location of the
aerodynamic center. As determined by Xfoil, the aerodynamic center of the airfoil
is shifted aft due to a small separation bubble that occurs just aft of the maximum
thickness of the airfoil. However, small imperfections in the surface could cause a
change in the boundary layer. Each free-wing segment was covered with a plastic
63
laminate. Small wrinkles and ridges were present on the surface from the laminate
and could a ect the boundary layer conditions of each segment. Changes in the
boundary layer conditions on each segment could a ect the separation bubble that
occurs and therefore a ect the aerodynamic center position on each free-wing segment.
5.3.5 Full Aerodynamic Model Sensitivities
The potential error in the input parameters of the system could have a large
e ect on the analytical model?s prediction of when the system will become unstable,
measured by the crossover velocity. To better understand the analytical model?s
sensitivities to these parameters a study was performed by slightly changing the input
values and observing the change in the crossover velocity. The parameters examined
were the rolling moment of inertia, pitching moment of inertia, aerodynamic center,
center of gravity position, and the spring constant. Each value was varied to show
the behavior in the general vicinity of the predicted value. The spring force and
mass moments of inertia were varied by plus and minus 25% and 50% whereas the
aerodynamic center was varied between 27% and 31% of the chord. The center of
gravity position was varied from 5% aft of the spar to 3% forward of the spar.
Increasing the pitching mass moment of inertia decreases both the crossover
velocity and the crossover frequency as shown in Fig. 5.14. The change in frequency
is quite small when compared to the change in velocity associated with the increase
in pitching mass moment of inertia. However, as you continually increase the inertia
the frequency fall o rate increases. The change in crossover velocity is more extreme
in the lower regions of the pitching mass moment of inertia and begins to level o as
it increases to 0.008 slugs ft2.
Changes in the rolling mass moment of inertia are similar to that of the pitching
mass moment of inertia. Unlike the crossover frequency associated with the pitching
64
Figure 5.14: Full aerodynamic model sensitivity to changes in pitching mass moment
of inertia
65
mass moment of inertia, the rate of change of the crossover frequency decreases as
the rolling moment of inertia is increased. The crossover frequency for the changing
rolling mass moment of inertia follows the same behavior as the crossover velocity.
As the rolling mass moment of inertia is increased, the system becomes increasingly
unstable and the crossover velocity is decreased as shown in Fig. 5.15.
Figure 5.15: Full aerodynamic model sensitivity to changes in rolling mass moment
of inertia
Shifting the aerodynamic center of the airfoil produces a similar result for the
crossover velocity when compared to changes in the mass moments of inertia. The
crossover frequency, however, behaves in a much di erent manner. As the aerody-
namic center is moved aft, the crossover frequency increases almost linearly. As men-
tioned previously, the crossover velocity follows the same trend as the mass moments
66
of inertia where the changes in velocity are greater when the aerodynamic center is
forward.
Figure 5.16: Full aerodynamic model sensitivity to changes in the aerodynamic center
Changes in the center of gravity position of each wing segment from the hinge
location causes a linear change in both the crossover velocity and frequency. The
crossover frequency decreases linearly as the hinge location is moved forward from
behind the hinge location to ahead of the hinge location. In contrast to the crossover
frequency, the crossover velocity decreases as the center of gravity position moves
aft. It is believed from experience in the experimental trials that the center of gravity
position is aft of the hinge location due to a shift in the position of the internal control
wires during ight. The wires were housed in a chamber that allowed the wires to
vary in position from 4.125 to 6.25 inches aft of the leading edge of the segment. If the
67
wing segments were perfectly balanced with the wires in the most forward position,
the center of gravity would shift aft during ight as the wires shifted aft. This would
produce a segment imbalance and a center of gravity o set of approximately 2%. If
the center of gravity position was shifted 2% aft of the hinge location the predicted
crossover velocity changes to 44.1 ft=s.
Figure 5.17: Full aerodynamic model sensitivity to changes in the center of gravity
position
The spring constant was also varied to show the full aerodynamic model?s sen-
sitivity. Figure 5.18 shows how the crossover velocity and frequency changes as the
spring constant is both increased and decreased. Both the crossover velocity and
frequency follow the same trend and increase as K is increased.
68
Figure 5.18: Full aerodynamic model sensitivity to changes in the spring constant K
69
5.3.6 Variation of Input Parameters Based Upon Error Estimates
Model parameters strongly a ect the predicted crossover velocity and frequency
produced by the analytical model. As discussed in Section 5.3.4, there is some uncer-
tainty in the values that were used in the analytical model. Based upon estimations of
the uncertainty, the model parameters can be modi ed to produce results that are in
closer agreement with the experimental results. The modi cations to the parameters
are simply estimations of the approximate error.
The mass moments of inertia are assumed to be an overestimate due to the
ignored damping of the displaced air from the spinning of the bi lar pendulum. It
is estimated that the damping e ect is greater in the rolling moment of inertia than
the pitching moment of inertia due to the larger surface area perpendicular to the
movement. The rolling mass moment of inertia has been reduced by 20% and the
pitching mass moment of inertia has been decreased by 10%.
Balancing each individual wing segment was always extremely di cult due to
the variability in the internal wires position. Because of the di culty in balancing
the segments, and the wires ability to move inside the segments, the center of gravity
of each individual segment was most likely not centered on the hinge. The tendency
of the wires to move aft in the wing segments during ight would cause the center
of gravity to shift aft. The shift of the CG position aft causes an increase in the
instability of the segment free-wing system. For the analytical model, the center of
gravity position was shifted aft to 4% of the chord behind the hinge location.
Xfoil was used to determine the aerodynamic center of the airfoil. The value
calculated by Xfoil was between 29% and 30% of the chord. However, due to surface
imperfections, the boundary layer may be forced to transition earlier than it would
if it were to freely transition. This causes the aerodynamic center to continue to
70
shift aft to approximately 30%. However, the forced transition only occurs at one
location whereas imperfections that could cause a forced transitions are numerous
in the plastic surface coating. Based upon the forces transition and the numerous
imperfections on the surface of the airfoil, the aerodynamic center has been shifted to
31% of the chord in the analytical model to study the a ect on the predicted stability.
Finally, a moderate beam de ection resulted from the applied torque during
the spring constant veri cation and may have introduced error into the estimated
spring constant, K. Because of the beam de ection, the measured torque was not
as a result of a true twisting of the mounting beam and therefore is considered high.
The initial analytical estimate of the spring constant, based upon published wood
properties [31], was approximately 125 ft lbs=rad, but since the exact species of the
wood is unknown, this is just a ballpark gure. The measured value, including any
de ection force of the beam, was 482.5 ft lbs=rad. The large discrepancy between
the two values may be an indication of the overestimation of the measured spring
constant. For the analytical model, the spring constant K was reduced by 30% as an
estimation of the possible error brought into the system by the beam de ection.
When the analytical model is run with these new estimates of the parameters,
the results are quite promising. The root locus plot for the full aerodynamic model
is similar to previous results for both the mechanically retrained and mechanically
free systems. The crossover velocity and frequency for the mechanically restrained
system has decreased to 29.9 ft=s and the crossover frequency is 1.23 Hz. These
results match the experimental results closely. In Fig. 5.20, the unstable modes are
quite visible and are increasing in instability as velocity is increased. The frequency,
however, is remaining almost constant as the velocity is increased.
71
Figure 5.19: Root locus plot of the full aerodynamic model with modi ed input
parameters but without xed segments
72
Figure 5.20: Root locus plot of the full aerodynamic model with modi ed input
parameters but without xed segments zoomed in to reveal unstable roots
73
The free-to-roll system with the modi ed input parameters produces the same
result as the previous model. When the system is not mechanically restrained, an
unstable roll mode exists immediately.
Adding the xed segments to the modi ed system produces the same stabilizing
results as discussed in Section 5.3.3. The unstable rolling mode in the mechanically
restrained system has been stabilized by the xed segments as it was in the analytical
model without the modi ed inputs. Additionally, the free-to-roll system is neutrally
stable just the same as the analytical model without the modi ed input parameters.
The modi ed input parameters do not change the qualitative behavior of the system,
they just change the velocity and frequency at which they occur. The modi ed input
parameters in the analytical model, based on rough error estimates, produces a result
very similar to the experimental results.
5.3.7 Modeling Options
The analytical model was designed so di erent modeling features could be turned
on and o . The basic model consisted of only circulatory lift and the following features
could be added: apparent mass, unsteady aerodynamics, and wake e ects.
Figure 5.21 shows a root locus plot for the system using the basic aerodynamic
model. In the basic model, only steady state aerodynamic calculations were included
without the unsteady or apparent mass terms. Moreover, the coupling e ects of the
vortex wake were also neglected. The plot clearly shows the instability of the system
with the cross-over from a stable to unstable system. The oscillation frequency at
the velocity where the root crosses over from stable to unstable is 0.90 Hz. This is
low compared to the 1.2 Hz estimated frequency from the experimental analysis. The
analytical model with only the basic aerodynamic model predicts the instability will
occur around 11.5 ft=s which is 48% lower than the experimental case.
74
Figure 5.21: Root locus plot of the basic model
75
Table 5.5: Results of the di erent modeling features
Figure 5.22 shows the root locus plot including the e ects of the vortex wake
system. With this option enabled, the analysis calculated the mutual in uence of
each of the wing segments on each other. The major di erence between this gure
and the results from the basic model is in the pitching modes. The unstable rolling
mode has a di erent shape but the general behavior is the same. In the basic model,
the pitching modes for each of the wing segments have identical eigenvalues. When
wake e ects are enabled however, the pitching modes of the various segments are
slightly modi ed. As can be seen in Fig. 5.22, the pitch modes spread out as velocity
is increased. The spreading of the pitching mode results from di erent segments
having di erent e ects on each other. Hence, their coupled oscillations are slightly
di erent. These interactions also in uence the pitch/roll instability. With the wake
analysis included, the crossover velocity increased from 11.5 ft=s to 14.0 ft=s, an
increase of 22%. The 14.0 ft=s value is closer to the experimental value of around 22
ft=s. The frequency of the oscillation remained almost the same at 0.91 Hz.
76
Figure 5.22: Root locus plot of the basic model with wake e ects
77
Adding unsteady aerodynamic modeling has a larger e ect as shown in Fig-
ure 5.23. In this gure, the circulatory unsteady aerodynamic terms have been in-
cluded but not the apparent mass or non-circulatory terms. The same modes are
present but the shape has changed. The pitching modes maintain the same frequency
but the damping has decreased. The frequency of the stable roll increased compared
to the basic model. The crossover velocity for the unsteady model has decreased
signi cantly to 4.8 ft=s, a 58% drop. In addition to the changing crossover velocity,
the frequency of oscillation at crossover has decreased to 0.32 Hz which is a drop of
73% from the experimental results. The addition of the unsteady model, by itself,
does not closely match the experimental results.
Figure 5.24 adds the apparent mass terms to the aerodynamic model and dramat-
ically alters the mode shapes. The apparent mass terms provide signi cant damping
to the system which causes the crossover velocity to increase to 26.2 ft=s (a 128%
increase). The frequency of the unstable rolling oscillation has also increased by 39%
to 1.25 Hz. The pitching mode is much more heavily damped with the apparent mass
model than the basic, wake, or unsteady models.
For the nal case shown in Figure 5.25, all the modeling options were enabled.
The apparent mass terms and the wake e ects dominate the overall shape of the root
locus plot. The e ects of the wake on each segment can be seen in the pitching mode
with the varying frequencies of oscillation. The unstable rolling mode has a similar
appearance to the apparent mass model however the crossover velocity is much higher
at 49.6 ft=s. Also, the frequency is slightly more than the apparent mass model
at 1.29 Hz and is still higher than experimental results showed. Additionally, the
pitching mode is highly damped as it is in the apparent mass model. The addition of
the unsteady model provides additional instability to the system. With just the wake
and apparent mass models the crossover velocity decreases to 44.9 ft=s even though
78
Figure 5.23: Root locus plot of the basic model with unsteady aerodynamics
79
Figure 5.24: Root locus plot of the basic model with apparent mass terms
80
Figure 5.25: Root locus plot of the full aerodynamic model
81
the shape of the root locus plot is very similar to a combination of the wake models
and apparent mass models. The frequency of the crossover point changes very little
as you add the additional models to the wake and apparent mass models. It remains
in the vicinity of 1.29 Hz. The basic model with the apparent mass terms appears to
closely resemble the experimental results but due to the calculated reduced frequency
unsteady aerodynamics were determined to be necessary to accurately model the
aerodynamics of the system.
5.4 Wind Tunnel Model
To provide additional experimental data for validation of the analytical model,
the wind tunnel model was tested again with a modi ed mount to allow it to rotate
like the truck mount.
5.4.1 Wind Tunnel Test Results
Veri cation of the analytical model was performed by comparing the results from
the analytical model to the wind tunnel tests. The input parameters of the analytical
model had to be changed to those of the wind tunnel model. Several springs with
di erent spring constants were obtained to test with the wind tunnel model on the
modi ed mount. The sti ness of the spring was input into the analytical model for
a prediction of the crossover velocity of the wind tunnel model. Su cient rigidity of
the spring was needed to maintain stability until at least the lower velocity bounds
of the wind tunnel could be reached. The 3x4 foot closed-circuit wind tunnel reaches
steady ow between 30 and 40 ft=s. A 0.25 inch diameter steel rod was found to
be su cient for the spring force. The distance between the hard mount in the wind
tunnel and the wind tunnel model was 6 inches, connected together via the steel rod.
82
With a modulus of rigidity of 11,000,000 psi obtained from Raymer [32], an estimate
of the spring constant K was calculated to be 58.2 ft lbs=rad. With this spring
constant, the estimated crossover velocity was 57.6 ft=s. Additionally, two other rods
were made of di erent lengths to adjust the spring constant and crossover velocity.
Using the 4 inch rod, the crossover velocity was estimated at 70.6 ft=s whereas the
8 inch rod produced an estimated crossover velocity of 49.9 ft=s.
The velocity of the wind tunnel was increased to the estimated crossover velocity
of the wind tunnel model using the six inch steel rod. At the estimated crossover
velocity, the wind tunnel model remained steady and stable. As the velocity in-
creased, the model remained stable until the mount began to oscillate vertically at
approximately 70 ft=s. The source of the oscillation was aerodynamic utter due to
the wing aerodynamic center being well in front of the elastic axis of the mounting
system. The six inch steel rod was replaced by the eight inch steel rod with the lower
spring constant. The velocity was again increased slowly up to 70 ft=s where the
model remained steady and stable the entire time. Again, at 70 ft=s the model began
to oscillate vertically.
Several other springs were tested to see if the system would become unstable.
Spring constants with values of 5.3 ft lbs=rad and 26.5 ft lbs=rad were tested.
Both springs, when placed on the model and tested, didn?t allow the system to become
unstable. The wind tunnel model remained stable at velocities exceeding 70 ft=s.
Because of the continued stability of the wind tunnel model, the spring constant was
drastically reduced to 0.0357 ft lbs=rad. The wind tunnel velocity was continually
increased to 120 ft=s without any e ect on the stability of the model. The wind
tunnel model continued to remain steady and stable at the high velocity.
83
5.4.2 System Properties
The number of wing segments was reduced to six and the mass moments of
inertia was adjusted for the physical properties of the wind tunnel model. The bi lar
pendulum described in Section 5.3.1 was used to measure the mass moments of inertia.
The pitching mass moment was changed to 0.0002 slugs ft2 and the rolling mass
moment of inertia was changed to 0.153 slugs ft2.
5.4.3 Analysis of Results
Additional tests of the wind tunnel model were performed as a veri cation of the
analytical model results. The analytical model was modi ed to predict the stability
of the wind tunnel model with di erent spring constants. The initially estimated
crossover velocity using the steel rod described in Section 5.4.1 was approximately
57.6 ft=s with a frequency of 3.0 Hz. The root locus plot for the wind tunnel model
is shown in Fig. 5.26.
Several di erent explanations were considered for the discrepancy between the
predicted behavior and observed behavior of the wind tunnel model. One hypothesis
was that friction in the roll and pitch bearings was a ecting the results. To evaluate
this hypothesis, friction terms were added to the equations of motion as described in
Section 4.5. The friction terms provide increased damping to the system, but the exact
values were unknown. In order to nd approximate values of the friction coe cients
that would reproduce the observed behavior, the values were varied in order to have
the computer model predict stability up to 120 ft=s. The rolling friction was the
larger driver and therefore was varied to determine the value of pitching friction
required to maintain the stability at 120 ft=s. A plot of the combination of friction
coe cients that provided stability at 120 ft=s is shown in Fig. 5.27. The combination
84
Figure 5.26: Root locus plot for the wind tunnel model
85
Figure 5.27: Friction coe cients for maintained stability with unsteady aerodynamics
86
of the coe cient of friction for roll and pitch has to be greater than the linear trend
of 35:686 k;roll + k;pitch = 2:7628. For all combinations of the friction coe cients
greater than or equal to 2.7628, the model is stable at or above 120 ft=s. Since the
wind tunnel model remained stable to 120 ft=s, it is inferred that the combination
of the coe cients of friction is at or above the trend line.
Figure 5.28: Friction coe cients for maintained stability without unsteady aerody-
namics
In contrast to the larger experimental model, for the wind tunnel model un-
steady aerodynamics were determined to be unnecessary because the reduced fre-
quency ranged from 2.63e-3 to 1.34e-3, depending on the value of the rolling friction
coe cient. When unsteady aerodynamics were removed from the system, the same
linear trend in the friction coe cients occur but the values were slightly changed as
shown in Fig. 5.28. The new linear trend line is 36:923 k;roll + k;pitch = 5:6454. The
87
required value of the rolling friction for stability at 120ft=shas been increased slightly
but the relationship of the pitching friction compared to the rolling friction remains
approximately the same. Just as with the unsteady aerodynamics on, all combina-
tions of pitching and rolling friction coe cients that are greater than or equal to the
linear trend provide stability to the model at speeds greater than or equal to 120
ft=s.
5.4.4 Friction Discussion
The moment produced by the friction could be determined by estimating the
angular velocity of the rolling and pitching motions. The pitching and rolling frictions
were modeled as M = kpitch _ and M = kroll _ respectively with the values of kpitch
and kroll being input parameters in the analytical model as described above. If
the friction coe cient between the steel rod and aluminum block of the wind tunnel
model exceeds the largest rolling coe cient value in Figs. 5.27 and 5.28, then the
model would always be stable up to 120 ft=s.
The angular velocity was modeled by
_ (t) = A!cos!t (5.7)
The maximum angular velocity occurs when cos!t = 1 so the angular velocity sim-
pli es to
_ (t) = A! (5.8)
The frequency, !, and the rolling friction coe cient, kroll, come from the analytical
model and were 0.473 and 0.076 respectively. A maximum roll angle of 10 degrees was
estimated and the total moment induced by the rolling friction was calculated to be
88
0.00629 ft lbs. A dry bearing was used and the friction torque could be calculated
as
M = P d2 (5.9)
where P is the net loading force, d is the shaft diameter, and is the dimensionless
friction coe cient between the two bearing materials.
The net loading force was the di erence between the system weight and the lift
produced by the wind tunnel model. The total system weight for the wind tunnel
model was 5.164 lbs. An estimate of the total lift produced by the wind tunnel model
at 120 ft=s was used to determine the net load on the bearing to be used in Eqn. 5.9.
A CL of 0.2 was used for the lift estimate based upon the 5 degree control surface
de ection. The estimated lift produced by the wind tunnel model at 120 ft=s was
7.43 lbs which gives a net force on the bearing surface of 2.266 lbs. Setting Eqns. 5.8
and 5.9 equal to each other, a value for was calculated to be 0.27. From Table 5.6,
Table 5.6: Selected values of coe cients of friction [33]
89
the coe cient of friction between steel and aluminum is 0.5 which is almost double
the required coe cient of friction to provide stability to the wind tunnel model up to
a velocity of 120 ft=s.
When unsteady aerodynamics were turned o , the rolling coe cient estimated
by the analytical model changed to 0.152 with a slight reduction of the frequency to
0.472 rad=s. Using the same process as described above, an estimated dimensionless
coe cient of friction was calculated as 0.5 which is the same as the coe cient of
friction between aluminum and steel [33]. From the estimates of the angular velocity
and the lift produced by the model, the coe cient of friction between aluminum and
steel is su cient to provide stability to the wind tunnel model up to speeds of 120
ft=s.
The 6 inch steel rod provided a spring constant of 58.2ft lbs=radfrom which the
analytical model predicted a crossover velocity of 57.6 ft=s. The wind tunnel model
remained stable up to 70 ft=s before the system began to oscillate up and down and
the test terminated. The system did not have a rolling bearing and therefore the only
friction present in the system was from the pitch bearing of each wing segment. The
two bearing surfaces consisted of a steel spar and a brass tube inside each segment
which rotated about the spar. For this system, the pitching angular velocity was
estimated using the same method as the rolling angular velocity. The amplitude was
5 degrees with a frequency of 19.046 rad=s. The estimated lift produced by a single
segment was 0.388 lbs. A kpitch of 0.0011 provided a crossover velocity of greater than
70 ft=s in the analytical model. Solving for the dimensionless coe cient of friction,
the value of friction required to stabilize the system was 0.45 which is equal to the
coe cient of friction between brass and steel provided by Muvdi in Table 5.6 [33].
90
Friction forces provide damping to the wind tunnel model and were signi cant
enough to provide stability to the system throughout all tested velocities. The coe -
cients required to maintain stability of the system for both the large spring constant
and the small spring constant were within experimental values provided by Muvdi.
91
Chapter 6
Conclusions
A wind tunnel model of the segmented free wing was constructed and tested in
the Auburn University wind tunnel. Initial results from the wind tunnel tests showed
a 64% reduction in the rolling moment coe cient when compared to a conventional
free wing and a 29% reduction compared to a torsionally free left and right free wing.
These positive results showed that the segmented free wing does have the capability
to respond to both time-varying and span-varying turbulence.
After the initial testing of the wind tunnel model, a conceptual model of a seg-
mented free wing was designed and fabricated at Auburn University. Initial testing
was conducted by mounting the segmented free wing on a truck for aerodynamic test-
ing. A repeatable divergent rolling oscillation emerged at velocities between 22 and 29
ft=s when the segmented free wing was mechanically restrained in roll. When allowed
to roll freely, the segmented free wing would roll over to the left or right but would
not oscillate back. Once the wing rolled over, the wing remained resting against the
stop. From previous results published by Porter and Brown [3], the divergent mode
was presumed to be the divergent spiral mode associated with free wing when the left
and right wings are allowed to be torsionally independent for each other.
Several modi cations to the segmented free-wing properties were made in an
attempt to stabilize the system and included changing the rolling mass moment of
inertia, the center of gravity position, and the number of segments. None of the
modi cations to the properties increased the stability of the segmented free wing.
The crossover velocity at which the wings began to oscillate in the mechanically
92
restrained system was virtually unchanged and a large change in the frequency of the
oscillation was not seen.
The segmented free-wing design was modi ed by adding xed segments to the
wing tips varying in size from six inches to 24 inches in span. All the xed segments
provided dynamic stability to the system for all experimental velocities. When the
system was allowed to freely roll, the xed segments provided greater stability, but
overall, the system was only neutrally stable. With the 24 inch segments, the system
was capable of being trimmed, but no damping occurred if the system was perturbed.
However, the divergent mode that was seen previously did not occur. The segmented
free wing with the six-inch xed segments was capable of being trimmed but was
extremely sensitive to any kind of perturbation.
An analytical model was developed as a result of the experimental testing of
the segmented free wing. The computer model, using eigen analysis, was successfully
able to predict the divergent oscillation in the mechanically restrained system and the
divergent mode in the mechanically free system. The velocity at which the analytical
model predicted the oscillation to occur was higher than what occurred in experimen-
tation. Errors in the model parameters used in the model may have contributed to
this discrepancy. When the parameters were modi ed based upon estimates of the
error, the same divergent modes were predicted but the crossover velocity at which
the oscillation occurred dropped to 29.9 ft=s, which is within the upper bound of
the experimental velocity. When the xed segments were attached in the analytical
model, the results matched the experimental results; the oscillation was damped out
and the system was dynamically stable in the mechanically restrained system and
neutrally stable in the free-to-roll system.
Finally, a wind tunnel model was tested to verify the analytical model with mixed
results. The wind tunnel provided positive preliminary results of the responsiveness of
93
the segmented free-wing design to adverse conditions by having the largest reduction
in the rolling moment when compared to the other con gurations. However, the
wind tunnel model failed to oscillate at the predicted velocity, but due to the smaller
aerodynamic forces, friction was believed to be a larger driver in that system. A
lower bound for the combination of the rolling and pitching friction coe cients was
developed. Friction coe cients above those bounds provide a stable wind tunnel
model up through all tested velocities. Estimates of the friction coe cients required
to stabilize the wind tunnel model were calculated. The calculated friction coe cients
were within experimental values and thus friction in the system was su cient to
stabilize the wind tunnel model at all tested velocities.
Overall, the segmented free-wing conceptual design is intended for use in adverse
environments where span varying conditions are a signi cant problem. This design is
most useful in the High Altitude, Long Endurance (HALE) aircraft design area where
the transition through the troposphere can be extremely hazardous and potentially
fatal to the vehicle. Initial results show promise in the capability in dealing with the
span varying turbulence, but this research focused mainly on the dynamic stability
of the segmented free-wing conceptual design.
94
Chapter 7
Future Work
Additional studies need to be performed to truly understand the abilities of the
segmented free wing to adjust to spanwise varying turbulence. Initial results were
positive but additional tests must be conducted. Aerodynamic properties and control
characteristics such as roll rate need to be determined in order to experimentally y
the segmented free wing. Additionally, the ratio of the area of the xed segments
compared to the wing area of the free-wing segments for su cient dynamic stability
needs to be determined. Finally, the ability to control the segmented free wing needs
to be determined due to the lack of damping from perturbations due to the neutrally
stable free-to-roll system.
95
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