Aeroelastic Studies on a High Aspect Ratio Wing
by
Linda M. Mendenhall
B. S., Aeronautics and Astronautics
B. S., Astronomy and Astrophysics,
The Pennsylvania State University, University Park, Pennsylvania (1997)
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE IN AERONAUTICS AND ASTRONAUTICS
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2003
@ Massachusetts Institute of Technology 2003. All rights reserved.
Author
Department of Aeronautics and Astronautics
January 31, 2003
Certified by
Carlos E. S. Cesnik
Visiting Associate Professor of Aeronautics and Astronautics
Thesis Supervisor
Accepted by
/
Edward M. Greitzer
H.N. Slater Professor of Aeronautics and Astronautics
Chair, Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
AERO
SEP 1 0 2003
LIBRARIES
Aeroelastic Studies on a High Aspect Ratio Wing
by
Linda M. Mendenhall
Submitted to the Department of Aeronautics and Astronautics
on January 31, 2003, in partial fulfillment of the
requirements for the degree of
Master of Science in Aeronautics and Astronautics
Abstract
A high aspect ratio (10.56) composite wing was designed and fabricated in order to verify
a new analysis code developed to predict dynamic behavior of high-aspect-ratio wings. The
chosen wing design was manufactured and tested (both bench-top and wind tunnel testing),
and its dynamic behavior was characterized. Strain gauges and accelerometers were embedded within the wing at various distances along the span to record wing natural frequency
information during bench-top 'tap' testing, and frequency changes as a function of wind
speed and root angle of attack during wind tunnel testing. Bench-top natural frequencies
and mode shapes for the 1 St and
2 nd
Bending, 1 st Chordwise Bending, and 1 st Torsion modes
are in good agreement with the analytical model. Mode shapes with frequencies exceeding
100 Hz were not considered. Within the tunnel, the wing was studies at 10, 20, and 50 root
angles of attack. The 2' root angle of attack configuration was flown to the flutter point by
observing of the wing behavior. This was within 99% of the predicted flutter speed.
Thesis Supervisor: Carlos E. S. Cesnik
Title: Visiting Associate Professor of Aeronautics and Astronautics
2
Acknowledgments
The completion of this research required the assistance of many people and institutions. I
would first like to thank MIT Lincoln Laboratory for allowing me to go back to school to
further my education. In particular, the Lincoln Scholars Commitee for the financial support
and time away from work necessary to complete such an endevor. Also my Group leaders,
Bob Davis and Steve Forman, who were extremely understanding about my absence. A
special thank-you to my Lincoln Tor-mentor (I mean Mentor), John Sultana, who always
provided something to laugh about! Also, thanks to all the people in my group and through
out the laboratory who have helped, especially Dennis Burianek, Vinny Cerrati, Charlie
Nickerson, and Anne Vogel.
A giant thank you goes to my family for putting up with the neglect and all of my
moodiness. Especially my husband, Dr Jeffrey Mendenhall, who took over all the household
and farm duties and still found time to pick me up at the train station. Special thanks to
Rajah, Angie, Jack, Ian, Betty, and Sierra, for providing the devotion and unconditional love
that every graduate student needs. To my parents, Charles and Marion Baker, for instilling
in me the importance of education. Thanks to all my friends who allowed me to drop out of
existance for awhile!
I would also like to thank many people on MIT campus, who provided necessary assitance
along the way. John Kane, Dave Roberts, and Donald Weiner for providing invaluable
assistance during the manufacturing process. Also Dick Perdichizzi for his assistance in the
tunnel. A special thanks to Professor John Dugundji for taking such an interest in my work
and helping me through all of the difficult times. Thanks also to my advisor, Carlos Cesnik,
for providing me with such an interesting topic.
Partial funding for this work was provided by NASA Langley Research Center, under
grant NAG-1-01102, and monitored by Dr. W. Keats Wilkie. Such support does not constitute an endorsement by NASA of the views expressed herein.
This work was also performed for the Air Force under Air Force Contract F19628-00-C002. The opinions, interpretations, conclusions, and recommendations are solely those of the
author and are not necessarily endorsed by the United States Government.
3
" I know where the pieces fit "
-Schism, Tool 2001
4
Contents
Abstract
2
Acknowledgments
3
19
1 Introduction
2
1.1
M otivation . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.2
O bjective
. . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.3
Background . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.3.1
Previous Work
. . . . . . . . . . . . . . . . . . .
20
1.3.2
Present Work . . . . . . . . . . . . . . . . . . . .
21
Design and Analysis of High Aspect Ratio Wing
22
2.1
Design Criteria . . . . . . . . . . . . . . . . . . . . . . .
22
2.2
Two-Layer Design . . . . . . . . . . . . . . . . . . . . . .
23
2.2.1
Basic Characteristics . . . . . . . . . . . . . . . .
23
2.2.2
Wing Non-linear Characteristics . . . . . . . . . .
26
Three-Layer Design . . . . . . . . . . . . . . . . . . . . .
51
2.3.1
Basic Characteristics . . . . . . . . . . . . . . . .
51
2.3.2
Wing Nonlinear Characteristics . . . . . . . . . .
52
. . . . . . . . . . . . . . . . . . . .
78
2.4.1
Case 1 - Maximum Vertical Displacement . . . . .
80
2.4.2
Case 2 - Maximum Pitch Rotation
. . . . . . . .
80
Design Selection . . . . . . . . . . . . . . . . . . . . . . .
84
2.3
2.4
2.5
Nastran Comparison
5
3
88
Experimental Procedure
Wing Manufacture ......
. . . . . . . . .
88
3.1.1
Foam Core ......
. . . . . . . . .
88
3.1.2
Instrumentation ...
. . . . . . . . .
89
3.1.3
Lay-Up
. . . . . . .
. . . . . . . . .
93
3.1.4
Post Cure . . . . . .
. . . . . . . . .
94
3.1.5
Clamp . . . . . . . .
. . . . . . . . .
94
3.2
Data Acquisition Equipment
. . . . . . . . .
97
3.3
Wind Tunnel Load Cell . . .
. . . . . . . . .
97
3.1
99
4 Experimental Studies
Bench-top Tests . . . . . . . . . . . . . . . . .
99
4.1.1
Strain Gauge Calibration . . . . . . . .
99
4.1.2
Tap Tests for Dynamic Properties . . .
104
4.1.3
Shaker Tests for Dynamic Properties .
108
4.2
Re-work of Analytical Model . . . . . . . . . .
109
4.3
Wind Tunnel Tests . . . . . . . . . . . . . . .
136
4.1
5
4.3.1
Wing Calibration in the Wind Tunnel.
136
4.3.2
Wind Tunnel Tests . . . . . . . . . . .
146
180
Concluding Remarks
5.1
Summary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
180
5.2
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
181
5.3
Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . .
181
A Mathematical Formulation
185
. . . . . . . . . . . . . . . . . . . . . .
185
. . . . . . . . . . . . . . . . . . . . . .
185
A.3 Structural Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
186
A.4 Aerodynamic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . .
187
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
188
A.1 Introduction ................
A.2 Geometrical Lay-out
........
A.5 Aeroelastic Formulation
6
B Material Properties
190
C Active Wing Design
192
C .1 A ctive D esign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
C.1.1
Basic Static and Dynamic Properties ..................
193
C.1.2
Wing Non-linear Characteristics ..........................
195
7
List of Figures
2-1
Cross section of two-layer wing (NACA 0012)
2-2
First six normal modes for two-layer wing (in vacuum)
. . . . . . . . . . . .
25
2-3
Static tip deflection for increasing speed at different root angles of attack . .
26
2-4
Elastic tip twist for increasing speed at different root angles of attack . . . .
27
2-5
Frequency change due to root angle of attack change at U
30 m/s . . . . .
28
2-6
Frequency change due to change in speed for 20 root angle of attack . . . . .
30
2-7
Frequency change due to change in speed for 50 root angle of attack . . . . .
31
2-8
Flutter speeds for two-layer design
32
2-9
Root locus plot for 0' root angle of attack
. . . . . . . . . . . . . . . . .
=
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
23
33
2-10 Mode shapes for two-layer design at 0' root angle of attack and its corresponding flutter speed of 55.8 m/s . . . . . . . . . . . . . . . . . . . . . . . .
2-11 Root locus plot for 10 root angle of attack
. . . . . . . . . . . . . . . . . . .
2-12 Magnification of root locus plot for 1 root angle of attack
. . . . . . . . . .
34
35
36
2-13 Mode shapes for two-layer design at 10 root angle of attack and its corresponding flutter speed of 50.2 m/s . . . . . . . . . . . . . . . . . . . . . . . .
37
2-14 Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (10 root angle of attack) for the two-layer design . . . . . . .
38
2-15 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(10 root angle of attack) for the two-layer design . . . . . . . . . . . . . . . .
39
2-16 Nonlinear time simulation of the wing tip displacement motion at 10% above
its flutter speed (1 root angle of attack) for the two-layer design . . . . . . .
8
39
2-17 Maximum ply strain reached at 10% above flutter speed of 10 root angle of
attack for the two-layer design . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2-18 Root locus plot for 20 root angle of attack . . . . . . . . . . . . . . . . . . .
41
2-19 Mode shapes for two-layer design at 2' root angle of attack and at its corresponding flutter speed of 47.5 m/s . . . . . . . . . . . . . . . . . . . . . . . .
42
2-20 Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (2' root angle of attack) for the two-layer design . . . . . . .
43
2-21 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(2' root angle of attack) for the two-layer design . . . . . . . . . . . . . . . .
44
2-22 Nonlinear time simulation of the wing tip motion at 10% above its flutter
speed (2' root angle of attack) for the two-layer design . . . . . . . . . . . .
44
2-23 Maximum ply strain reached at 10% above flutter speed of 20 root angle of
attack for the two-layer design . . . . . . . . . . . . . . . . . . . . . . . . . .
2-24 Root locus plot for 5' root angle of attack
. . . . . . . . . . . . . . . . . . .
45
46
2-25 Mode shapes for two-layer design at 50 root angle of attack and at its corresponding flutter speed of 38.8 m/s . . . . . . . . . . . . . . . . . . . . . . . .
47
2-26 Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (50 root angle of attack) for the two-layer design . . . . . . .
48
2-27 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(50 root angle of attack) for the two-layer design . . . . . . . . . . . . . . . .
49
2-28 Nonlinear time simulation of the wing tip motion at 10% above its flutter
speed (50 root angle of attack) for the two-layer design . . . . . . . . . . . .
49
2-29 Maximum ply strain at reached at 10% above flutter speed of 50 root angle of
attack for the two-layer design . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2-30 Cross section of three-layer wing (NACA 0012) . . . . . . . . . . . . . . . . .
51
2-31 First six modes shapes for the three-layer wing (in vacuum)
. . . . . . . . .
53
2-32 Static tip deflection for increasing speed at different root angles of attack . .
54
2-33 Elastic tip twist for increasing speed at different root angles of attack . . . .
55
2-34 Frequency change due to root angle of attack change at U = 30 m/s .....
57
2-35 Frequency change due to change in speed for 2' root angle of attack . . . . .
58
9
2-36 Frequency change due to change in speed for 50 root angle of attack . . . . .
59
2-37 Flutter speeds for three-layer design . . . . . . . . . . . . . . . . . . . . . . .
60
2-38 Root locus plot for 0' root angle of attack . . . . . . . . . . . . . . . . . . .
61
2-39 Mode shapes for three-layer design at 00 root angle of attack and at its corresponding flutter speed 64.6 m/s . . . . . . . . . . . . . . . . . . . . . . . . .
2-40 Root locus plot for 10 root angle of attack
. . . . . . . . . . . . . . . . . . .
62
63
2-41 Mode shapes for three-layer design at 10 root angle of attack and at its corresponding flutter speed of 62.1 m/s . . . . . . . . . . . . . . . . . . . . . . . .
64
2-42 Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (10 root angle of attack) for the three-layer design . . . . . .
65
2-43 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(10 root angle of attack) for the three-layer design . . . . . . . . . . . . . . .
66
2-44 Nonlinear time simulation of the wing tip motion at 10% above its flutter
speed (10 root angle of attack) for the three-layer design
. . . . . . . . . . .
66
2-45 Maximum ply strain reached at 10% above flutter speed of 10 root angle of
attack for the three-layer design . . . . . . . . . . . . . . . . . . . . . . . . .
2-46 Root locus plot for 20 root angle of attack
. . . . . . . . . . . . . . . . . . .
67
68
2-47 Mode shapes for three-layer design at 20 root angle of attack and at its corresponding flutter speed of 56.9 m/s . . . . . . . . . . . . . . . . . . . . . . . .
69
2-48 Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (20 root angle of attack) for the three-layer design . . . . . .
70
2-49 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(20 root angle of attack) for the three-layer design . . . . . . . . . . . . . . .
71
2-50 Nonlinear time simulation of the wing tip motion at 10% above its flutter
speed (20 root angle of attack) for the three-layer design
. . . . . . . . . . .
71
2-51 Maximum ply strain reached at 10% above flutter speed of 20 root angle of
attack for the three-layer design . . . . . . . . . . . . . . . . . . . . . . . . .
72
2-52 Root locus plot for 50 root angle of attack . . . . . . . . . . . . . . . . . . .
73
2-53 Mode shapes for three-layer design at 5' root angle of attack and at its corresponding flutter speed of 46.4 m/s . . . . . . . . . . . . . . . . . . . . . . . .
10
74
2-54 Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (50 root angle of attack) for the three-layer design . . . . . .
75
2-55 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(50 root angle of attack) for the three-layer design . . . . . . . . . . . . . . .
76
2-56 Nonlinear time simulation of the wing tip motion at 10% above its flutter
speed (50 root angle of attack) for the three-layer design
. . . . . . . . . . .
76
2-57 Maximum ply strain reached at 10% above flutter speed of 50 root angle of
attack .......
..
.......................................
77
2-58 Finite element mesh of the three-layer design used in Nastran (5550 elements,
13617 degrees of freedom)
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
2-59 Nastran normal (chordwise) strain for 5' root angle of attack . . . . . . . . .
81
2-60 Nastran normal (spanwise) strain for 5' root angle of attack . . . . . . . . .
82
2-61 Nastran in-plane shear strain for 5' root angle of attack . . . . . . . . . . . .
83
2-62 Nastran normal (chordwise) strain for 5' root angle of attack . . . . . . . . .
85
2-63 Nastran normal (spanwise) strain for 5' root angle of attack
. . . . . . . . .
86
2-64 Nastran in-plane shear strain for 50 root angle of attack . . . . . . . . . . . .
87
3-1
Four sections of the foam core pieced together in mold
. . . . . . . . . . . .
89
3-2
Location of Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
3-3
General wiring diagram for Wheatstone bridges . . . . . . . . . . . . . . . .
91
3-4
Wiring diagram for Wheatstone bridges in bending
91
3-5
Wiring diagram for Wheatstone bridges in forward/aft bending
. . . . . . .
91
3-6
Wiring diagram for Wheatstone bridges in twist . . . . . . . . . . . . . . . .
91
3-7
Strain gauge at mid-span . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
3-8
Strain gauges wired at the root
. . . . . . . . . . . . . . . . . . . . . . . . .
92
3-9
Top of root clam p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
3-10 Bottom of root clamp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
3-11 Pin diagram for full bridges
97
. . . . . . . . . . . . . . . . . . . . . . . . . . .
3-12 Photo of accelerometer and strain gauge conditioners
4-1
. . . . . . . . . . . . . .
. . . . . . . . . . . . .
98
Experimental set-up for bench-top tests . . . . . . . . . . . . . . . . . . . . . 100
11
4-2
Experimental instrumentation for bench-top tests . . . . . . . . . . . . . . . 100
4-3
Time trace of data from bench-top bending calibration
. . . . . . . . . . . . 101
4-4
Calibration of strain gauge voltage output for bending
. . . . . . . . . . . . 102
4-5
Time trace of data from bench-top twist calibration . . . . . . . . . . . . . . 103
4-6
Calibration of strain gauge voltage output for tip twist . . . . . . . . . . . . 104
4-7
Tap test for bending frequencies-root bending gauge . . . . . . . . . . . . . 105
4-8
Tap test for torsion frequencies-root torsion gauge . . . . . . . . . . . . . . 106
4-9
Tap test for chordwise bending frequencies-forward/aft gauge . . . . . . . . 107
4-10 Shaker at the wing leading edge for the natural frequency bench test
. . . . 108
4-11 Fundamental mode shapes with adjusted weight and length . . . . . . . . . . 110
4-12 Predicted fundamental mode shapes based on added stiffness corrections
. . 112
4-13 Root locus plot for 10 root angle of attack based on modified wing stiffness
properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4-14 Magnified root locus plot for 10 root angle of attack based on modified wing
stiffness properties
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4-15 Root locus plot for 2' root angle of attack based on modified wing stiffness
properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4-16 Magnified root locus plot for 20 root angle of attack based on modified wing
stiffness properties
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4-17 Root locus plot for 50 root angle of attack based on modified wing stiffness
properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4-18 Predicted flutter speeds for various root angles of attack based on modified
wing stiffness properties
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4-19 Predicted mode shapes after final modifications of different wing properties . 120
4-20 Root locus plot for 00 root angle of attack based on final modifications of
different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4-21 Magnified root locus plot for 00 root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . 122
4-22 V-g (frequency part) for 0' root angle of attack based on final modifications
of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
12
4-23 V-g (damping part) for 00 root angle of attack based on final modifications of
different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4-24 Root locus plot for 10 root angle of attack based on final modifications of
different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4-25 Magnified root locus plot for 10 root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . 126
4-26 V-g (frequency part) for 10 root angle of attack based on final modifications
of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4-27 V-g (damping part) for 10 root angle of attack based on final modifications of
different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4-28 Root locus plot for 20 root angle of attack based on final modifications of
different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4-29 Magnified root locus plot for 2' root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . 130
4-30 V-g plot (frequency part) for 20 root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . 131
4-31 V-g plot (damping part) for 20 root angle of Attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . 132
4-32 Root locus plot for 50 root angle of attack based on final modifications of
different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4-33 V-g (frequency part) for 50 root angle of attack based on final modifications
of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4-34 V-g (damping part) for 50 root angle of attack based on final modifications of
different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4-35 Predicted flutter speed for various root angles of attack . . . . . . . . . . . . 137
4-36 Wing set-up in the Wright Brothers wind tunnel . . . . . . . . . . . . . . . . 138
4-37 Set-up for hanging weights to calibrate strain gauges
4-38 Time trace for elastic axis determination
. . . . . . . . . . . . . 139
. . . . . . . . . . . . . . . . . . . . 140
4-39 Time trace for wing loaded at 50% chord in wind tunnel
. . . . . . . . . . . 141
4-40 Time trace for wing loaded at 90% chord in wind tunnel
. . . . . . . . . . . 142
13
4-41 Calibration of root bending gauge from both 50% and 90% chord load cases
143
4-42 Calibration of root torsion gauge from both 50% and 90% chord load cases
144
4-43 Calibration of other bending gauge (located at 35% span) from both 50% and
90% chord load cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4-44 Other bending gauge readings from undisturbed wing mounted in tunnel
. . 147
. . . . 148
4-45 Root bending gauge readings from tap test-wing mounted in tunnel
4-46 Root torsion gauge readings from tap test-wing mounted in tunnel . . . . . 149
4-47 Forward/Aft bending gauge readings from tap test-wing mounted in tunnel
150
4-48 Lift curve for wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4-49 Frequency plot from root bending gauge at 20.1 m/s for 10 root angle of attack152
4-50 Frequency plot from root torsion gauge at 20.1 m/s for 10 root angle of attack 153
4-51 Frequency plot from forward/aft gauge at 20.1 m/s for 10 root angle of attack 154
4-52 Frequency plot from the other bending gauge at 20.1 m/s for 10 root angle of
attack .........
155
.......................................
4-53 Frequency plot from root bending gauge at 36.7 m/s for 10 root angle of attack156
4-54 Frequency plot from root torsion gauge at 36.7 m/s for 1' root angle of attack 157
4-55 Frequency plot from forward/aft gauge at 36.7 m/s for 10 root angle of attack 158
4-56 Frequency plot from the other bending gauge at 36.7 m/s for 10 root angle of
attack . . .. .......
159
.......
...........................
4-57 Time trace of data from root bending gauge at 20.1 m/s for 2' root angle of
attack .........
160
.......................................
4-58 Frequency plot from root bending gauge at 20.1 m/s for 20 root angle of attack161
4-59 Frequency plot from root torsion gauge at 20.1 m/s for 20 root angle of attack 162
4-60 Frequency plot from forward/aft gauge at 20.1 m/s for 2' root angle of attack 163
4-61 Frequency plot from the other bending gauge at 20.1 m/s for 20 root angle of
attack . . .......
..................
..
..........
..... .164
4-62 Time trace of data from root torsion gauge at 52.3 m/s for 2' root angle of
attack . . . ........
.............
166
.......
..............
4-63 Time trace of data from root torsion gauge at 52.3 m/s for 20 root angle of
attack . ..
. .......
.
...................
14
............
..
167
4-64 Frequency plot from root torsion gauge at 52.3 m/s for 20 root angle of attack 168
4-65 Frequency plot from forward/aft gauge at 52.3 m/s for 2' root angle of attack 169
4-66 Frequency plot from the other bending gauge at 52.3 m/s for 2' root angle of
attack .........
170
.......................................
4-67 Frequency plot from root torsion gauge at 20.1 m/s for 50 root angle of attack 171
4-68 Frequency plot from forward/aft gauge at 20.1 m/s for 5' root angle of attack 172
4-69 Frequency plot from the other bending gauge at 20.1 m/s for 5' root angle of
attack ........
....
173
.........
.......................
4-70 Frequency plot from root torsion gauge at 33.5 m/s for 5' root angle of attack 174
4-71 Frequency plot from forward/aft gauge at 33.5 m/s for 5' root angle of attack 175
4-72 Frequency plot from the other bending gauge at 33.5 m/s for 5' root angle of
attack
..........
176
.........
..........................
4-73 Root bending gauge readings from tap test-wing mounted in tunnel
. . . .
177
4-74 Root torsion gauge readings from tap test-wing mounted in tunnel . . . . . 178
4-75 Average tip accelerometer readings from tap test-wing mounted in tunnel . 179
A-i Coordinate systems for the wing . . . . . . . . . . . . . . . . . . . . . . . . . 186
A-2 Variables Defining Airfoil Motion . . . . . . . . . . . . . . . . . . . . . . . . 188
C-1 Cross section of active wing . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
C-2 First six normal modes for active wing design (in vacuum) . . . . . . . . . . 194
C-3 Static tip deflection for increasing speed at different root angles of attack . . 195
C-4 Elastic tip twist for increasing speed at different root angles of attack . . . . 196
C-5 Flutter speeds for active wing design . . . . . . . . . . . . . . . . . . . . . . 197
C-6 Root locus plot for 10 root angle of attack
. . . . . . . . . . . . . . . . . . . 198
C-7 Magnification of root locus plot for 10 root angle of attack
. . . . . . . . . . 199
C-8 Mode Shapes for active wing design at 10 root angle of attack and its corresponding flutter speed of 79.3 m/s . . . . . . . . . . . . . . . . . . . . . . . . 200
C-9 Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (1 root angle of attack) for the active wing design
15
. . . . . 201
C-10 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(10 root angle of attack) for the active wing design . . . . . . . . . . . . . . . 202
C-11 Nonlinear time simulation of the wing tip displacement motion at 10% above
its flutter speed (1' root angle of attack) for the active wing design
C-12 Root locus plot for 20 root angle of attack
. . . . . 202
. . . . . . . . . . . . . . . . . . . 203
C-13 Magnification of root locus plot for 2' root angle of attack . . . . . . . . . . 204
C-14 Mode shapes for active wing design at 2' root angle of attack and its corresponding flutter speed of 77.3 m/s . . . . . . . . . . . . . . . . . . . . . . . . 205
C-15 Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (2' root angle of attack) for the active wing design
. . . . . 206
C-16 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(20 root angle of attack) for the active wing design . . . . . . . . . . . . . . . 207
C-17 Nonlinear time simulation of the wing tip displacement motion at 10% above
its flutter speed (20 root angle of attack) for the active wing design
. . . . . 207
C-18 Root locus plot for 5' root angle of attack . . . . . . . . . . . . . . . . . . . 208
C-19 Mode shapes for active wing design at 50 root angle of attack and its corresponding flutter speed of 68 m/s . . . . . . . . . . . . . . . . . . . . . . . . . 209
C-20 Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (50 root angle of attack) for the active wing design
. . . . . 210
C-21 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(50 root angle of attack) for the active wing design . . . . . . . . . . . . . . . 211
C-22 Nonlinear time simulation of the wing tip displacement motion at 10% above
its flutter speed (50 root angle of attack) for the active wing design
16
. . . . . 211
List of Tables
2.1
Non-zero stiffness matrix terms for two-layer design . . . . . . . . . . . . . .
24
2.2
Non-zero inertial matrix terms for two-layer design
. . . . . . . . . . . . . .
24
2.3
Natural frequencies of the two-layer design in vacuum . . . . . . . . . . . . .
24
2.4
Dynamic properties at U = 30 m/s . . . . . . . . . . . . . . . . . . . . . . .
29
2.5
Dynamic properties at angle of attack = 2 . . . . . . . . . . . . . . . . . . . .
29
2.6
Dynamic properties at angle of attack = 5 . . . . . . . . . . . . . . . . . . .
32
2.7
Non-zero stiffness matrix terms for three-layer design . . . . . . . . . . . . .
51
2.8
Non-zero inertial matrix terms for two-layer design
. . . . . . . . . . . . . .
51
2.9
Natural frequencies of the three-layer design in vacuum . . . . . . . . . . . .
52
2.10 Dynamic properties at U = 30 m/s . . . . . . . . . . . . . . . . . . . . . . .
56
2.11 Dynamic properties at root angle of attack = 20 . . . . . . . . . . . . . . . .
56
2.12 Dynamic properties at root angle of attack = 50 . . . . . . . . . . . . . . . .
60
2.13 Summary of Strain Analysis Results . . . . . . . . . . . . . . . . . . . . . . .
78
2.14 Nastran Case 1: Displacements and Rotations . . . . . . . . . . . . . . . . .
80
2.15 Nastran case 1: summary of results . . . . . . . . . . . . . . . . . . . . . . .
80
2.16 Nastran case 2: displacements and rotations . . . . . . . . . . . . . . . . . .
81
2.17 Nastran case 2: summary of results . . . . . . . . . . . . . . . . . . . . . . .
84
3.1
Accelerometers used in the wing model . . . . . . . . . . . . . . . . . . . . .
93
3.2
Load cell used for the wing tunnel tests . . . . . . . . . . . . . . . . . . . . .
98
4.1
Natural Frequencies From Bench-top Tap Tests
4.2
Natural frequencies with adjusted weight and length . . . . . . . . . . . . . . 109
17
. . . . . . . . . . . . . . . . 106
4.3
Intermediate predicted natural frequencies based on added stiffness corrections 111
4.4
Intermediate predicted flutter speeds based on corrected structural properties 111
4.5
Final values of adjusted parameters . . . . . . . . . . . . . . . . . . . . . . . 119
4.6
Final natural frequencies from wing analysis . . . . . . . . . . . . . . . . . . 120
4.7
Predicted flutter speeds with final model adjustments . . . . . . . . . . . . . 136
4.8
Frequencies detected from strain gauges at 20.1 m/s for 2' root angle of attack165
B.1 Properties of the materials used in this study . . . . . . . . . . . . . . . . . . 191
C. 1 Non-zero stiffness matrix terms for the active wing design . . . . . . . . . . . 193
C.2 Non-zero inertial matrix terms for active wing design . . . . . . . . . . . . . 193
C.3 Natural frequencies of the active wing design in vacuum . . . . . . . . . . . . 194
18
Chapter 1
Introduction
1.1
Motivation
Many industries consider using High Altitude Long Endurance (HALE) aircraft to perform
autonomous sensing (environmental, urban, and reconnaissance), or to assist in civilian or
military communication as a relay [1]. The wings of these vehicles are typically of high
aspect ratio. Long slender wings result in large deflections during flight, especially when
flown at high angles of attack. These large deflections cannot be ignored when analyzing the
wing's flight characteristics. The change in the wing's stiffness, due to nonlinear geometric
behavior, requires new methods of modeling [2]. This change in stiffness results in changes to
the wing's dynamic behavior. There has been considerable research in this area by numerous
groups, resulting in new modeling techniques and codes which need to be validated through
experimentation. One such computational modeling tool is being developed by Cesnik and
Brown [3]. This research is a follow-on of work performed by Cesnik and Ortega-Morales [4].
The ground work for this was set forth by Patil, Hodges, and Cesnik [1], [5], and [6].
1.2
Objective
The primary objective of this research is to design and test an experimental wind tunnel
model wing. This test wing should present nonlinear aeroelastic behavior. The collected
19
data will then support the validation of the MATLAB computer code being developed by
Cesnik and Brown [3]. Originally this would encompass all portions of the code, including
the ability to model active materials embedded within the layers of the composite wing.
However, due to reasons beyond the control of the author, no active structural tests are
included in this study.
1.3
Background
As mentioned above, there has been substantial effort towards developing analytical models
which accurately depict the behavior of high aspect ratio wings undergoing large deflections.
This work is summarized in the next two sections.
1.3.1
Previous Work
In the late 1980s, Minguet and Dugundji studied the effects of deflection on dynamic behavior
of thin composite strips [7]. In their study, it was shown that deflection has significant effect
on the torsion and chordwise bending frequencies. Their beam analysis obtained a set of
linearized equations about a deflected position. The experiments for this research utilized
flat composite beams of various composite orientations and thicknesses. This study showed
that deflections of as little as 3% of the total length affected the torsional and chordwise
bending modes, while the normal bending modes remained relatively uneffected. They also
showed that tip bending deflections between 5% and 10% of the total beam length resulted
in a mix between the 1 st torsion and
1 "t
chordwise bending modes.
Patil, Hodges and Cesnik [8] have worked to improve modeling capabilities for high
aspect ratio wings, which include geometrical non-linearities for structural and aerodynamic
analyses. Their research has shown that accurate modeling of geometric non-linearities is
necessary to fully understand the torsion/bending coupling observed during flutter of wings
with large deflections [8]. Their modeling of nonlinear aeroelasticity is based on a mixed
variational formulation for the dynamics of beams in moving reference frames and finitestate airloads to accommodate deformable airfoil effects [1].
20
Tang and Dowell have studied various ways of modeling high aspect ratio wing aeroelastic
behavior and correlated their work with wind tunnel tests. In the late 1990's, they built a
45-cm wing with an aspect ratio of 8.89
[9).
The wing was flown into the flutter domain to
study limit cycle oscillation behavior. They also used this wing to study the modeling of
gust response [10].
Ortega-Morales and Cesnik created a framework for implementing active aeroelastic tailoring of high aspect ratio wings, as well as gust mitigation [4]. The active tailoring was implemented through embedded piezoelectric actuators within the composite skin of the wing.
This work implemented many of the analysis techniques studied in [1], [8], [2],
[5],
and [6], and
consisted of an asymptotically correct active cross-section formulation, geometrically-exact
mixed formulation for dynamics of moving beams, and finite-state unsteady aerodynamics.
1.3.2
Present Work
Cesnik and Brown are continuing the work done in [4]. This work formulated the MATLAB
code used within this thesis, which employs a strain-based finite element representation of
a nonlinear, large deflection beam model experiencing finite-state unsteady airloads [11].
This program has the capability to model active materials embedded within the composite
lay-up of the wing and to take into account effects from the fuselage and tail [3]. The basic
formulation behind this modeling effort is summarized in Appendix A.
Two different wing designs were generated using the code from [11] and are presented
in Chapter 2 of this thesis. These designs were chosen for their torsional flexibility and
interesting flutter characteristics. Chapter 3 discusses the manufacture and instrumentation
of the wing. The benchtop and wind tunnel experiments, as well as the experimental results
are discussed in Chapter 4. Along with these results is a modified numerical model which
replicates the results of the tests.
Finally, conclusions and suggestions for further work
are presented in Chapter 5. This thesis contains three Appendices. Appendix A provides
a summary of the mathematical formulation for the code, Appendix B contains relevant
material properties, and Appendix C provides the preliminary studies for a future active
wing build-up.
21
Chapter 2
Design and Analysis of High Aspect
Ratio Wing
This chapter describes the design and analysis of the static flexible wing that was built to
validate the model of [3] for subsonic speeds. Some of the design parameters were set by the
wind tunnel limitations and others by ease of fabrication.
2.1
Design Criteria
Since this study targets large deflection aeroelastic response, a high aspect ratio flexible wing
was needed. To reduce the risk of damage to the wind tunnel during the dynamic portion
of testing, the design flutter speed for this wing should be in the range of 40 m/s. This
relatively low flutter speed reduces the amount of momentum the wing would have in case
it becomes unstable and brakes away from the mounting fixture. As a result, a very flexible
wing design, soft in torsion, is expected. Although the wing needed to be as long as possible
to increase the aspect ratio, the span of the wing was also limited by the wind tunnel test
section dimension. The minimum chord of the wing also has limitations. It could not be
too small, as that would significantly increase the complexity of manufacturing. With these
constraints in mind, the length of the wing was set to 1.13 meters (44.5 inches) and the
chord length was set to 0.107 meters (4.22 inches). This yielded an aspect ratio of 10.56.
22
The airfoil shape chosen is the NACA 0012. Safety to the wing tunnel was a major concern,
so the wing needed to possess margins of safety for strain within the E-glass/epoxy of at least
1. For added safety this requirement needed to be met when the wing was analyzed at 10%
above the predicted flutter speed. Also, the wing needed to exhibit limit cycle oscillation at
speeds up to 10% above the predicted flutter values.
After several preliminary layup considerations, two basics ones were selected for detailed
studies: a so-called "two-layer" design and a "three-layer" design. They represent flexible
designs, with flutter speeds within the appropriate range. They also demonstrate interesting
flutter characteristics.
2.2
2.2.1
Two-Layer Design
Basic Characteristics
The two-layer design is composed of two-layers of E-glass/epoxy fabric oriented at 0' around
an inner foam core. A table of the material properties is provided in Appendix B. There
is no spar in this design. This design provides a very flexible wing while still maintaining
closed cell properties. A cross-sectional representation is given in Figure 2-1.
[021
foam
Figure 2-1: Cross section of two-layer wing (NACA 0012)
The cross-sectional properties for the two-layer design are provided below. The stiffness
matrix terms for this cross-section are given in Table 2.1. Detailed definition of the elements
of this matrix are provided in Appendix A.
The inertia matrix for this cross-section is
provided in Table 2.2. The center of gravity for the cross-section is located 0.0182 m aft of
the reference line (located at 30% chord). This places the center of gravity at 47% of the
chord.
23
Table 2.1: Non-zero stiffness matrix terms for two-layer design: 1 = Extension, 2 = Torsion,
3 = Flatwise Bending, 4 = Chordwise Bending
Kn1
K 14 = K 4 1
9.55 * 105 N
-1.68 * 104 N*m
14.24 N*m 2
22.25 N*m 2
1.18 * 103 N*m 2
K2 2
K33
K44
Table 2.2: Non-zero inertial matrix terms for two-layer design
I,
I22
133
0.10 * 10-3 m4
0.22 * 10-5 m 4
0.98 * 10-4 m 4
The dynamic properties of the wing are important to its aeroelastic response. The first
six natural frequencies of the wing are summarized in Table 2.3. The corresponding mode
shapes are provided in Figure 2-2.
Table 2.3: Natural frequencies of the two-layer design in vacuum
Mode
Frequency
Mode Shape
1
2
8 Hz
50 Hz
3
4
5
52 Hz
95 Hz
153 Hz
Bending
1 st Chordwise Bending
2 "d Bending
6
292 Hz
1 st
1" Torsion
3 rd Bending
2 nd
24
Torsion
Speed = 0.00 m/s @ Freq = 7.97 Hz
Speed = 0.00 m/s @ Freq = 49.9 Hz
0.8
Speed = 0.00 m/s @ Freq = 51.5 Hz
Speed = 0.00 m/s @ Freq = 95.1 Hz
0.2
0.4
0.6
0.8
0.1
Speed = 0.00 m/s @ Freq = 153 Hz
Speed = 0.00 m/s @ Freq = 292 Hz
0.2
0.2
0.4
-0.1
-0.05
0.4
0.6
0.6
0.8
0.8
0
1
1
Figure 2-2: First six normal modes for two-layer wing (in vacuum)
25
2.2.2
Wing Non-linear Characteristics
Different root angles of attack of the wing result in different wing tip deflections once the
wing is exposed to airloads. The change in tip deflection is graphed in Figure 2-3 for the
range of angles of attack of interest: 0' to 5'. The tip deflection increases with increased
angle of attack. This is to be expected, since the aerodynamic load is directly proportional
to the angle of attack.
0.6
0.5
0.4
E
0
0.3
a
a)
0
ca.
0.2
0*0
-0.11
0
5
10
15
25
20
30
35
40
45
Speed (m/s)
Figure 2-3: Static tip deflection for increasing speed at different root angles of attack
The wing tip twist also changes with increased root angle of attack. This data is provided
in Figure 2-4. To better illustrate this effect, the initial root angle of attack has been
subtracted from the total tip twist, yielding only the elastic angle change due to gravitational
26
50
and aero-loading.
0.9
0.8
0.7 -.-.-.
AO
0 .
.
.
.... .
-
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...
. .
.
.
. .
.
0.63
. . .. ..
-.
-.
-.
F 0 .4 -.
0.13
.-.-
. .. .
5
5-0
5-0
0
. .. . . . .
30-5
40
5
45
-0
01
Speed (mis)
Figure 2-4: Elastic tip twist for increasing speed at different root angles of attack
The changes in tip deflection and twist result in a change in the dynamic behavior of
the wing, and ultimately a change in the wing stability. This is illustrated by the change in
the mode shapes and their corresponding frequencies for the wing at different root angles of
attack at 30 in/s. Table 2.4 gives the frequencies of the first six modes for 00, 1 , 2~ and
50.
Figure 2-5 contains the first four modes graphically.
It is interesting to note that as the wing is bent upwards due to the force of air pressure,
the natural frequencies of the second mode decrease and the fourth mode increase.
effect will cause some different behaviors in the wing as the different modes interact.
27
This
110
1T
100
90
80
70
60
2B
Cr
50
LL.
1CB
40
30
20
1B
10
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Root Angle of Attack (deg)
Figure 2-5: Frequency change due to root angle of attack change at U
28
=
30 m/s
5
Table 2.4: Dynamic properties at U = 30 m/s
Angle of Attack
Tip Deflection
00
-0.005 m
10
0.03 n
20
0.06 m
50
0.16 n
8
8
8
8
1" Bending
50
49
48
40
2" Bending
1 st Torsional
3 rd Bending
52
95
153
52
96
153
52
98
153
52
107
153
2 "d
292
293
295
303
1"
Mode
Shapes
Chordwise Bending
Torsional
A similar behavior is also seen when the speed increases for a set angle of attack. This
behavior is illustrated in Table 2.5 and Table 2.6. The first four modes for each of these
examples is presented graphically in Figure 2-6 and Figure 2-7, respectively.
Table 2.5: Dynamic properties at angle of attack = 20
Speed
Tip Deflection
Mode
Shapes
30 m/s
0.06 rn
35 m/s
0.09 m
40 m/s
0.12 rn
45 m/s
0.16 rn
1"
Bending
1 st Chordwise Bending
2 " Bending
8
48
52
8
46
52
8
43
52
8
40
52
1' Torsional
3 rd Bending
2 nd Torsional
98
153
295
100
153
297
103
153
300
107
153
303
This change in the dynamic behavior of the wing results in changes in its flutter speed.
This is shown in Figures 2-9, 2-11, 2-18 and 2-24, which present the root locus plots for the
wing at root angles of attack of 0', 10, 20 and 5'. The flutter speeds as function of root angle
of attack determined from these plots are summarized in Figure 2-8.
29
120
1T
80
z
U
C
0)
60
0.
U-
2B
40
1CB
20
1B
0
30
32
34
36
38
40
42
44
46
48
Speed (m/s)
Figure 2-6: Frequency change due to change in speed for 2' root angle of attack
30
50
IT
120
100
80
Cr
0~
60
UL-
2B
40
1CB
20
1B
0
30
32
34
36
38
40
42
44
46
48
Speed (m/s)
Figure 2-7: Frequency change due to change in speed for 50 root angle of attack
31
50
Table 2.6: Dynamic properties at angle of attack = 50
Speed
Tip Deflection
Mode
Shapes
30 m/s
0.16 m
35 m/s
0.22 m
40 m/s
0.30 m
45 m/s
0.38 m
8
40
52
107
8
35
52
114
8
30
51
119
8
25
51
122
153
303
153
300
152
293
151
284
1" Bending
1 st Chordwise Bending
2 " Bending
1 " Torsional
Bending
_ 2 " Torsional
3 rd
60
50
40
E
a>
. 30
C')
a>
i20
10
O'L
0
1
2
3
4
Root Angle of Attack (deg)
5
Figure 2-8: Flutter speeds for two-layer design
32
6
7
Zero-Degree Root Angle of Attack
The flutter speed for zero degree root angle of attack is determined to be 55.8 m/s from
the root locus plot, given in Figure 2-9. The unstable mode is the fourth mode, which is
primarily the first torsion mode. There is some chordwise bending present as well. The
mode shapes for the wing at this speed are given in Figure 2-10.
Root locus for Root AOA =
0*
200
180
- ------ --------- ---------- --
40
30
*
----
140
-
20
----
- --
-
- ------------- - - - --- - - -- --
3B
10
:0
--------------------------- ----------
120
U
100 r
---------- ------------
0~
'I)
LI~
-- - -- T --------- ---
---
- - --- - --- ------------- ----------------------
I
-------------------------------- -
-- --- - - --
80
---- - - -- ----60 -------------------- ----------------
-
-- - - - ---- ------- -2B
~--
1CB
-931
40
-
20
------ - - - - - - - - - - - - - -- - - - - - - - - - - - ---- - ---- - ---- - - ----- - - - - - - - - - ---
- ----- -- -- - - - - - - --- - --- - - - - - - - - - - - - - -
-~
-20
0L
50
40
30
10
0
0
10
Real Part of Root
Figure 2-9: Root locus plot for 00 root angle of attack
33
- -
1B
1e
20
-
20
Speed = 55.80 m/s @ Freq = 7.97 Hz
Speed = 55.80 m/s @ Freq = 49.9 Hz
-0.6
-0.4
Speed = 55.80 m/s @ Freq = 51.5 Hz
Speed = 55.80 m/s @ Freq = 95 Hz
0.2
0.4
0.6
0.8
1
Speed = 55.80 m/s @ Freq = 153 Hz
Speed = 55.80 m/s @ Freq = 292 Hz
0.2
0.2
-0.1
-0.05
0.4
0.4
-
0.6
0.6
0.8
0.8
0
1
1
Figure 2-10: Mode shapes for two-layer design at 0' root angle of attack and its corresponding
flutter speed of 55.8 m/s
34
One-Degree Root Angle of Attack
The flutter speed for one-degree root angle of attack is determined to be 50.2 m/s, shown
in Figure 2-11. The mode of instability is the second mode. A magnification of this mode
is provided in Figure 2-12. This mode is primarily the first chordwise bending. However,
the forces due to air pressure on the tilted wing result in some torsion being present as well.
The first six mode shapes at this speed are given in Figure 2-13.
Root locus for Root AOA =
1*
200
160 3B
ifP]
AfAf
140
-------------
----------100
H- -----------
1T
Cr
!0
4)
LL
-----------
60
2B
14A
lCB
40
20
H- - - - - - - - - - -
------------1B
U
50
50
40
30
10
20
0
10
Real Part of Root
Figure 2-11: Root locus plot for 10 root angle of attack
35
20
Root locus for Root AOA =
1
55
-- -- - -- -- I-- -- I--- I-- -
-
2B
5 0 - --400
8
-- - - -
C 4 5 - - - - ---
-------
1
... . -
. ..-..-- - -.
- .. .- ..--
-
cr2
1GB
LL-
.....
04.
40 -- - - -- - -
10
8
6
I
-- -
4
2
0
2
4
-- - - -
-
6
8
10
Real Part of Root
Figure 2-12: Magnification of root locus plot for 10 root angle of attack
36
-- - - -
Speed = 50.20 m/s @ Freq = 7.97 Hz
Speed = 50.20 m/s @ Freq = 44.7 Hz
-0.05
0.
0.05
0.1
Speed = 50.20 m/s @ Freq = 51.9 Hz
1
Speed = 50.20 m/s @ Freq = 101 Hz
0.2
0.4
0.6
1
Speed = 50.20 m/s @ Freq = 153 Hz
Speed = 50.20 m/s @ Freq = 298 Hz
0.2
0.2
-0.1
-0.05
0.4
0.4
0.6
0.6
0.8
0.8
0
1
1
Figure 2-13: Mode shapes for two-layer design at 1' root angle of attack and its corresponding
flutter speed of 50.2 m/s
37
When the wing is flown at 10% above the flutter speed for a 10 root angle of attack, the
wing enters into a Limit Cycle Oscillation (LCO). The tip deflections and the tip twist are
plotted for this case up to 1 second. These plots are given in Figure 2-14 thru Figure 2-16.
Speed = 55.20 m/s with a Time step = 0.0005 s
0.14
0.12
0.1
E
0)
Ca)
CI)
F0
0.08
0.06
D_
Fr
C.D
0.04
0.02
0
-0.02'
0
0.2
0.4
0.6
0.8
1
Time (s)
Figure 2-14: Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (10 root angle of attack) for the two-layer design
38
Speed = 55.20 m/s with a Time step = 0.0005 s
5
4
3
2
0)
0_0
1
0
1
-2-3 0
0.2
0.4
0.6
0.8
1
Time (s)
Figure 2-15: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(10 root angle of attack) for the two-layer design
Speed = 55.20 m/s with a Time step = 0.0005 s
0.13
0.125
0.12
E
U)
0.1 15 [
0
N
0.11
0. 105 F
0.1 '-0.015
-0.01
-0.005
0
0.005
Y Displacement (m)
0.01
0.015
Figure 2-16: Nonlinear time simulation of the wing tip displacement motion at 10% above
its flutter speed (10 root angle of attack) for the two-layer design
39
A strain analysis was performed on the wing when it was at its maximum deflection.
This occurs during the initial rise shown in Figure 2-14. The maximum tensile strain is
2660 pm/m and the maximum shear strain is -769 pm/m. Both occur at the bottom of
the root of the wing in the outer ply. These results are summarized in Figure 2-17. From
Appendix B, the maximum allowable strain for E-glass/epoxy is 10000 pm/m in tension and
15000 pum/m in shear. This results in a margin of safety (Equation 2.1) of 2.76 for tension
and 18.5 for shear.
Allowable
MS(%) = Actual - 1
Actual
(2.1)
eps1 (longitudinal fiber strain), ply 3
eglass 120: MAX = 0.0026595
eps2 (transverse fiber strain), ply 3
eglass 120: MAX = -0.00039362
eps3 (inplane shear strain), ply 3
eglass 120: MAX = -0.00076936
Figure 2-17: Maximum ply strain reached at 10% above flutter speed of 10 root angle of
attack for the two-layer design
40
Two-Degree Root Angle of Attack
The flutter speed for two-degree root angle of attack is determined to be 47.5 m/s from the
root locus plot shown in Figure 2-18. The instability mode is the same as for 10, except the
amount of torsion present has increased. Also, by the time the angle of attack is 2', the
effects of the large deflections on the wing cause the natural frequency of the fourth mode to
increase with increasing speed instead of decrease as it did in the 00 and 10 cases. The drop
in the second natural frequency due to the increased upward bend also causes the second
mode to turn over faster and amplifies the effects of flow over the wing. The first six modes
at flutter speed are provided in Figure 2-19.
Root locus for Root AOA =
2*
200
180 -- -
---
--- -
------
-
---- --- --- -
160
3B
30
40
2n
1
140
120
V-
---- - - - -- - - -
.
.-
------ - -
N
=
U
C
100 ---
_- -- - ------------------T -
--- -- -
0
~1)
LI~
-
80 -------
60
.
- -44
--
-- - --- - - - -
-
--
--- - -
-
-- - - - - -~- -
- -- ------ - --2B-'
140
1CB
40 -- -
-- -
-
- - - --
- - - - - ---
*~
20
--- - - - -
---
V
B
0
- ------ -_
. ..-..
--.
--
------- ---.- .- -.
-- .-
1B
-20E
50
40
30
10
20
10
Real Part of Root
0)i
0
10
Figure 2-18: Root locus plot for 2' root angle of attack
41
20
Speed = 47.50 m/s @ Freq = 7.98 Hz
Speed = 47.50 m/s @ Freq = 38.2 Hz
Speed = 47.50 m/s @ Freq = 51.8 Hz
1
Speed = 47.50 m/s @ Freq = 109 Hz
Speed = 47.50 m/s @ Freq = 153 Hz
Speed = 47.50 m/s @ Freq = 302 Hz
0.2
0.2
0.4
0.4
0.6
-..
0.6
0.8
0.8
1
Figure 2-19: Mode shapes for two-layer design at 2' root angle of attack and at its corresponding flutter speed of 47.5 m/s
42
For 2' angle of attack, the wing also enters a LCO when flown at 10% above its flutter
speed. The tip deflections and the tip twist are plotted for this case up to 1 second. These
plots are given in Figure 2-20 thru Figure 2-22. Notice from Figure 2-22 that the LCO now
has more than one frequency (no single ellipse on the plot), indicating potentially chaotic
response.
Speed = 52.30 m/s with a Time step = 0.0005 s
0.3
0.25
0.2
E
a)
0
CO
0.15
0.1
0
0.05
0
-0.05-
0
0.2
0.4
0.6
0.8
1
Time (s)
Figure 2-20: Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (20 root angle of attack) for the two-layer design
43
Speed = 52.30 m/s with a Time step = 0.0005 s
15
10
5
0)
U,
V
0
I0.
R
-5
-10 -15 0
0.2
0.4
0.6
1
0.8
Time (s)
Figure 2-21: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(20 root angle of attack) for the two-layer design
Speed = 52.30 m/s with a Time step = 0.0005 s
0.16
0.15
0.14
E
0.13
U>
E
(D
0
0.12
CN
0
N
0.11
0.11-
0.09
0.08'
-0.04
-0.03
-0.02
0.01
0
-0.01
Y Displacement (m)
0.02
0.03
0.04
Figure 2-22: Nonlinear time simulation of the wing tip motion at 10% above its flutter speed
(20 root angle of attack) for the two-layer design
44
A strain analysis was performed on the wing when it was at its maximum deflection.
This occurs during the transient rise shown in Figure 2-20. The maximum tensile strain is
4484 pm/m and the maximum shear strain is -2726 pm/m. Both occur at the bottom of
the root of the wing in the outer ply. These details are summarized in Figure 2-23. These
strains result in a margin of safety of 1.23 and 4.5, respectively.
eps1 (longitudinal fiber strain), ply 3
eglass 120: MAX = 0.0044837
eps2 (transverse fiber strain), ply 3
eglass 120: MAX = -0.00066361
eps3 (inplane shear strain), ply 3
eglass 120: MAX = -0.0027264
Figure 2-23: Maximum ply strain reached at 10% above flutter speed of 2' root angle of
attack for the two-layer design
Five-Degree Root Angle of Attack
The flutter speed for five-degree root angle of attack is determined to be 38.8 m/s from the
root locus plot provided in Figure 2-24. As with the 1 and 2' cases, the mode of instability
is the second mode, which is the 1" chordwise bending with some torsional effects. As was
seen in the 20 case, the changes in the dynamic behavior due to the increased upward bend
result in the second mode rolling over faster to the positive dampening axis and the fourth
45
mode to further curve upward. Although with the fourth mode, we see an interaction with
the fifth mode (3 rd bending) starting to cause the fourth mode to turn towards the instability
line as well. The first six mode shapes for this case are provided in Figure 2-25.
Root locus for Root AOA =
5*
200
180
- - - --
-
-
- -
---
-
-
- - ------
-
-- - - - -- - -----
--
160
3B
140
120
I
Li
100
4
4024
--------------3
20
30
0
--
13
0
--
1T
10
~LI
LL
80
--- - - - -
- -
- ----- - - - - - - - - - - - --- - - - - - -- ---.
60
. . .
. . .
-
. .-
-
--
- --20-
-
- -
-
-- - ---
2B
40 --- ------ --------- ----- ---- - - -----
- - - ---- - .-
- - 0-
--
- 1 - -- -
-
- --
20
-20
0
50
40
30
10
20
10
1B
0
10
Real Part of Root
Figure 2-24: Root locus plot for 50 root angle of attack
46
20
Speed = 38.80 m/s @ Freq = 7.99 Hz
Speed = 38.80 m/s @ Freq = 30.8 Hz
Speed = 38.80 m/s @ Freq = 51.5 Hz
Speed = 38.80 m/s @ Freq = 118 Hz
Speed = 38.80 m/s @ Freq = 152 Hz
Speed = 38.80 m/s @ Freq = 294 Hz
0.1
00
- - - -- -00.
0.8
Figure 2-25: Mode shapes for two-layer design at 5' root angle of attack and at its corresponding flutter speed of 38.8 m/s
47
For 5' root angle of attack, the wing also enters a LCO when flown at 10% above the
flutter speed. The tip deflections and the tip twist are plotted for this case up to 1 second.
These plots are given in Figure 2-26 thru Figure 2-28.
Speed = 42.70 m/s with a Time step = 0.0005 s
0.4
0.35
0.3
E
0
0.25
0.2
CO
CO,
0.15
0.1
0.05
0
-0.05 L
0
0.2
0.4
0.6
0.8
1
Time (s)
Figure 2-26: Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (50 root angle of attack) for the two-layer design
A strain analysis was performed on the wing when it was at its maximum deflection.
This occurs during the initial rise shown in Figure 2-26. The maximum tensile strain is
7405 pm/m and the maximum shear strain is -5018 pm/m. Both occur at the bottom of
the root of the wing in the outer ply. These details are summarized in Figure 2-29. These
strains result in a margin of safety of 0.35 and 1.99, respectively.
48
Speed = 42.70 m/s with a Time step = 0.0005 s
20
15
10
aO
CL
5
P,
0
-5 -
-10 0
0.2
0.4
0.6
0.8
1
Time (s)
Figure 2-27: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(50 root angle of attack) for the two-layer design
Speed = 42.70 m/s with a Time step = 0.0005 s
0.3
0.28-
0.26C
a)
E
C, 0.24C.,
.O
0
N
0 .22 F
0.2 F
0.18'
-0.05
-0.04
-0.03
-0.02
-0.01
Y Displacement (m)
0
0.01
0.02
Figure 2-28: Nonlinear time simulation of the wing tip motion at 10% above its flutter speed
(50 root angle of attack) for the two-layer design
49
eps1 (longitudinal fiber strain), ply 3
eglass 120: MAX = 0.0074051
eps2 (transverse fiber strain), ply 3
eglass 120: MAX = -0.001096
eps3 (in plane shear strain), ply 3
eglass 120: MAX = -0.0050181
.. .
.. . .
I
[IWI, MULL,,, IDIUWI lirnIT
Figure 2-29: Maximum ply strain at reached at 10% above flutter speed of 5' root angle of
attack for the two-layer design
50
2.3
Three-Layer Design
2.3.1
Basic Characteristics
The three-layer design resembles the previous one with the exception of an added layer of
E-glass/epoxy fabric, also oriented at 00 (Figure 2-30.
[03]
foam
Figure 2-30: Cross section of three-layer wing (NACA 0012)
The cross sectional stiffness properties for this design are provided in Table 2.7. A detailed
definition of the terms for this matrix are provided in Appendix A. The inertia matrix for
Table 2.7: Non-zero stiffness matrix terms for three-layer design: 1 = Extension, 2 = Torsion,
3 = Flatwise Bending, 4 = Chordwise Bending
Ku1
K14= K41
1.39 *106 N
-2.35 * 10 4 N*m
K2 2
19.66 N*m
2
K33
33.19 N*m
2
K44
1.63 * 103 N*m
2
this cross-section is provided in Table 2.8. The center of gravity for the cross-section is
Table 2.8: Non-zero inertial matrix terms for two-layer design
Ill
I22
133
0.13 * 10-3 m 4
0.32 * 10- 5 m 4
4
0.13 * 10-3 m
51
located 0.0173 m aft of the reference line (located at 30% chord). This places the center of
gravity at 46% of the chord.
The first six normal modes (in vacuum) and their corresponding frequencies are provided
in Table 2.9. It is interesting that the values for the normal modes have changed very little
from the two layer case. This can be explained by the fact that the increase in stiffness
is comparable to the increase in inertia. The corresponding mode shapes are provided in
Figure 2-31.
Table 2.9: Natural frequencies of the three-layer design in vacuum
2.3.2
Mode
1
Frequency
8 Hz
2
3
4
5
6
51 Hz
54 Hz
97 Hz
161 Hz
299 Hz
Mode Shape
1" Bending
1 st
Chordwise Bending
2 "d Bending
1 st Torsion
3 rd Bending
2 nd Torsion
Wing Nonlinear Characteristics
The effects from changes in root angle of attack are similar to those seen in the two-layer
design. The tip deflection again increases with increase root angle of attack. This is plotted
in Figure 2-32. The tip twist also increases with increased root angle of attack, as shown
in Figure 2-33. As with the two-layer data, the root angle of attack was subtracted from
the tip angle to better illustrate the effects to do gravity and aero-loading only. It should
also be noted that although the trends are the same, the values of deflection and twist are
smaller. This is expected as the stiffness of the wing is increased with the additional layer
of E-glass/epoxy.
Deflection and twist changes versus angle of attack also result in changes in the dynamic
behavior of the wing. The wing's mode shapes and natural frequencies were computed for
a wind speed of 30 m/s. The results are summarized in Table 2.10. This data is presented
52
Speed = 0.00 m/s @ Freq = 8.37 Hz
Speed = 0.00 m/s @ Freq = 50.9 Hz
0.8
Speed = 0.00 m/s @ Freq = 54.1 Hz
Speed = 0.00 m/s @ Freq = 97.1 Hz
0.2
0.4
0.6
0.8
1
Speed = 0.00 m/s @ Freq = 161 Hz
Speed = 0.00 m/s @ Freq = 299 Hz
0.2
0.2
-0.1
-0.05
0.4
0.4
-
0.6
0.6
0.8
0.8
0
1
1
Figure 2-31: First six modes shapes for the three-layer wing (in vacuum)
53
0.35
50
0 .3 -
-..... -.-.-.--
5- -1-
-
0.25
- -.-.-.
203
1
-
- -
5
4-
--
.. . .. .. . .. .. . - - - - - - - - -.-- - - - -.-- -.-. ....... - - - - --..
--..
0 .2
0 .1 5 - --..-.
.-
-.- --..-
.-
-.-.
2*
-0.05
0
5
10
15
20
25
30
35
40
45
Speed (m/s)
Figure 2-32: Static tip deflection for increasing speed at different root angles of attack
54
50
0.8
50
-..- --..
0 .6
-
-
-
0.6
...
.
IQ
. . . . . . . . . .
.
..
..
.
.
. ..
.
. . .
..
.
.
.
.
.
.
. .
.
.
.
_20
4
S 0.3 -
... - - ---- - .... . . . . .. . . . .. . . . ... . . . ..- -. . ... ..-. ... .
-..
-
0.2 -
- - -
-.
-
- -~
0.1 --0*
0
0
5
10
15
25
20
30
35
40
45
Speed (mis)
Figure 2-33: Elastic tip twist for increasing speed at different root angles of attack
55
50
graphically for the first four modes in Figure 2-34.
Table 2.10: Dynamic properties at U
30 m/s
00
-0.005 m
8
10
0.02 m
8
20
0.04 m
8
50
0.10 m
8
1" Chordwise Bending
51
51
50
45
2 "d Bending
1 " Torsional
54
97
54
97
54
98
54
104
Bending
_ 2 " Torsional
161
299
161
299
161
300
161
305
Angle of Attack
Tip Deflection
1 "t Bending
Mode
Shapes
=
3 rd
The three-layer design also shows a change in the chordwise bending and torsional modes
for changes in root angle of attack. As the root angle of attack is increased the frequencies
for the 1I"chordwise bending mode decrease and the frequencies for the 1" torsional mode
increase. The bending modes remain unchanged.
A similar behavior is seen when the speed is varied for a constant root angle of attack.
This is illustrated in Table 2.11 and Table 2.12. This data is also provided graphically for
the first four modes in Figure 2-35 and Figure 2-36.
Table 2.11: Dynamic properties at root angle of attack
Speed
Tip Deflection
Mode
Shapes
30m/s 35m/s 40m/s
0.04 m 0.06 m 0.08 m
=
20
45m/s
0.10 m
1" Bending
8
8
8
8
1" Chordwise Bending
50
49
48
46
2"
Bending
"
Torsional
1
54
98
54
99
54
101
54
103
Bending
2nd Torsional
161
300
161
301
161
302
161
305
3 rd
The flutter speeds for varying angles of attack are given in Figure 2-37. As expected, the
addition of the third layer of E-glass resulted in an increase in the flutter speed.
56
1*10
00
-
1T
90
80
70
60
2B
NU
-~
50--
- ---
LL.
1CB
40
30
20
1B
10
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Root Angle of Attack (deg)
Figure 2-34: Frequency change due to root angle of attack change at U = 30 m/s
57
5
120
100
80
IR
C
6..
60
U-
2B
1CB
40
20
*
II
18
I
30
32
34
36
38
40
42
44
46
48
Speed (m/s)
Figure 2-35: Frequency change due to change in speed for 2' root angle of attack
58
50
120
1T
80
07
60
2B
U-
40
1CB
20
11B
0
30
32
34
36
40
38
42
44
46
48
Speed (m/s)
Figure 2-36: Frequency change due to change in speed for 5' root angle of attack
59
50
Table 2.12: Dynamic properties at root angle of attack = 5'
Speed
30m/s
35m/s
40m/s
45m/s
Tip Deflection
1 " Bending
0.10 m
8
0.15 m
8
0.20 m
8
0.25 m
8
45
54
42
54
38
54
33
54
Torsional
104
108
114
119
Bending
2" Torsional
161
305
161
308
161
308
160
303
Chordwise Bending
2" Bending
1 st
Mode
Shapes
1"
3 rd
70
60
50
E40
U, 0
a
~30
20-
10-
01
0
1
2
4
3
Root Angle of Attack (deg)
5
Figure 2-37: Flutter speeds for three-layer design
60
6
7
Zero-Degree Root Angle of Attack
The flutter speed for 0' root angle of attack is determined to be 64.6 m/s from the root locus
plot provided in Figure 2-38. The unstable mode is the fourth mode, which cooresponds to
the
mode. This is the same as was seen in the two-layer case. The first six
1"t torsional
mode shapes at flutter speed for this wing are shown in Figure 2-39.
Root locus for Root AOA =
00*
200
180
K-----------
160
0
.1.
44
Iflii
4
-2010
!0
140 F-------
120
K-
-
-
-
-
-
-
-
-
-
-
- - - - - -- -
-
I
U
C
LI~
--T
100 |--------
80
---
K-
- 60
2B
-1
-4 -6
--44
4
--------,8
ICB
-- - - --- -
40 - --
---
20
--- ---- 1B
iB
0
50
I
40
30
I
I
10
20
Real Part of Root
I
0
10
Figure 2-38: Root locus plot for 0' root angle of attack
61
20
Speed = 64.60 m/s @ Freq = 8.37 Hz
Speed = 64.60 m/s @ Freq = 50.9 Hz
0
Speed = 64.60 m/s @ Freq = 54.1 Hz
Speed = 64.60 m/s @ Freq = 97 Hz
0.1
0.2
0.4
-0.1
0.6
-0.05
0.8
1
0
Speed = 64.60 m/s @ Freq = 161 Hz
Speed = 64.60 m/s @ Freq = 299 Hz
0.2
0.2
0.4
0.4
-0.1 -0.05
0
0.6
0.6
0.8
0.8
1
1
Figure 2-39: Mode shapes for three-layer design at 0' root angle of attack and at its corresponding flutter speed 64.6 m/s
62
One-Degree Root Angle of Attack
The flutter speed for the 10 root angle of attack is 62.1 m/s. This can be seen on the root
locus plot provided in Figure 2-40. The mode of instability is the second mode. This mode
is primarily the first chordwise bending. However, the forces due to air pressure on the tilted
wing result in some torsion being present as well. The first six mode shapes for this case are
provided in Figure 2-41.
Root locus for Root AOA =
1*
200
180 F--------
160
-
3B- -
------------------------
------
-u
140 |-
U
:3
120
--
10
--
0
U_
80
- ...-
-
60
-4M
44
'4"
-30
-.-
----.-.
-23
1
-
2B
CB
40 F--------
- -- - - -- - - -
20
-
-.-. -.- -.
.
01
50
40
30
20
10
Real Part of Root
--
1B
10 B
0
10
Figure 2-40: Root locus plot for 1 root angle of attack
63
20
Speed = 62.10 m/s @ Freq = 8.38 Hz
Speed = 62.10 m/s @ Freq = 45.1 Hz
-0.05
00
0.05
0.1
1
Speed = 62.10 m/s @ Freq = 54.5 Hz
Speed = 62.10 m/s @ Freq = 104 Hz
0.2
0.4
0.6
1
Speed = 62.10 m/s @ Freq = 161 Hz
Speed = 62.10 m/s @ Freq = 305 Hz
0.2
0.2
-0.1-0.05
0.4
0.4
0.6
0.6
0.8
0.8
0
1
1
Figure 2-41: Mode shapes for three-layer design at 1 root angle of attack and at its corresponding flutter speed of 62.1 m/s
64
When this wing is flown at 10% above the predicted flutter speed, the wing enters into
a LCO. The tip deflections and the tip twist are plotted for this case up to 1 second. These
plots are provided in Figure 2-42 thru Figure 2-44.
Speed = 68.20 m/s with a Time step = 0.0005 s
0.16
0.14
E
0.12
C
CD) 0.1
E
a)
0
0.08
Q_
0.06
CO)
0.04
0.02
0
-0.02'0
0.2
0.4
0.6
0.8
1
Time (s)
Figure 2-42: Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (10 root angle of attack) for the three-layer design
A strain analysis was performed on the wing when it was at its maximum deflection.
This occurs during the initial rise shown in Figure 2-42. The maximum tensile strain is
2697 pm/m and the maximum shear strain is -1765 pm/m. Both occur at the outer ply
of the bottom part of the airfoil at the root of the wing. These results are summarized in
Figure 2-45. These strains result in a margin of safety of 2.71 for tension and 7.50 for torsion.
65
Speed = 68.20 m/s with a Time step = 0.0005 s
10
8
6
CD
4
2
a.
C~ 0
-2-4 -6 0
0.2
0.4
0.6
0.8
1
Time (s)
Figure 2-43: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(10 root angle of attack) for the three-layer design
Speed = 68.20 m/s with a Time step = 0.0005 s
0.13
0.125 F
0. 12 E
0.1 15-
a>
4-
E
CD
0.
CN
0.11
0. 105 I
N
0.1
0.095
0.092
-0.02 -0.015 -0.01
0.005
0
-0.005
Y Displacement (m)
0.01
0.015
0.02
Figure 2-44: Nonlinear time simulation of the wing tip motion at 10% above its flutter speed
(10 root angle of attack) for the three-layer design
66
eps1 (longitudinal fiber strain), ply 4
eglass 120: MAX = 0.0026968
eps2 (transverse fiber strain), ply 4
eglass 120: MAX = -0.00039914
eps3 (in plane shear strain), ply 4
eglass 120: MAX = -0.0017653
.
.
.
.
.
.
.
Figure 2-45: Maximum ply strain reached at 10% above flutter speed of 10 root angle of
attack for the three-layer design
67
Two-Degree Root Angle of Attack
The flutter speed for the 2' root angle of attack is 56.9 m/s. This can be seen on the root
locus plot provided in Figure 2-46. The second mode is again the mode which crosses the
stability axes. The larger tip deflections and tip twists observed for the high angles of attack
cause this mode to turn over and go unstable faster. The first six mode shapes for this case
are provided in Figure 2-47.
Root locus for Root AOA =
20
200
180
160
-------------------- --------- --
-
-I- - - - - - - - - I - - - - - - - - -
- -- ----
----
--- -
-
- - -- - ----------- - -- --- -- - - - - - --
--- 3 B -
- -
--- --- -- - -- -- --
- -- ---------------140 ---------- --------- ---------
120
-
----
- - --- - - --- -
0
U
80- -- -
60
--
q480
4
4
--- - -
3E
2B
2E
- -- - - - -
---
- -
-----
------ --- - -- - -
-- - - -- - - - - - - - - -
--- -- - - -_55 -52
40
- - - - - - - --- -- - - - - - - - - - - - - - ------
- - - -- - -
--
--
8:
- - - -----
---20 -- - - - - - - -- - - ------
*
30
0'
50
40
30
20
20
0
0
10
Real Part of Root
Figure 2-46: Root locus plot for 2' root angle of attack
68
- - - --
1
10
10
----
- ---
-
20
Speed = 56.90 m/s @ Freq = 8.38 Hz
Speed = 56.90 m/s @ Freq = 39.3 Hz
Speed = 56.90 m/s @ Freq = 54.4 Hz
1
Speed = 56.90 m/s @ Freq = 111 Hz
Speed = 56.90 m/s @ Freq = 161 Hz
Speed = 56.90 m/s @ Freq = 309 Hz
0.2
-0.1-0.05
0.2
0.4
0.4
0.6
0.6
0.8
0
0.8
1
1
Figure 2-47: Mode shapes for three-layer design at 2' root angle of attack and at its corresponding flutter speed of 56.9 m/s
69
When this wing is flown at 10% above the predicted flutter speed, the wing enters into
a LCO. The tip deflections and the tip twist are plotted for this case up to 1 second. These
plots are provided in Figure 2-48 thru Figure 2-50.
Speed = 62.60 m/s with a Time step = 0.0005 s
0.3
0.25
E
0.2
E
a>
-Z 0.15
0,,
C
0.1
C.L
F_:
F5
0.05
0 1
-0.05 0
0.2
0.4
0.6
0.8
1
Time (s)
Figure 2-48: Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (20 root angle of attack) for the three-layer design
A strain analysis was performed on the wing when it was at its maximum deflection.
This occurs during the initial rise shown in Figure 2-48. The maximum tensile strain is
4421 ptm/m and the maximum shear strain is -4079 pm/m. Both occur at the outer ply
of the bottom part of the airfoil at the root of the wing. These results are summarized in
Figure 2-51. These strains result in a margin of safety of 1.26 and 2.68, respectively.
70
Speed = 62.60 m/s with a Time step = 0.0005 s
10
5
CD
_0
0
(jO
-5 -
10 I
-15
0.2
0
0.4
0.6
0.8
1
Time (s)
Figure 2-49: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(20 root angle of attack) for the three-layer design
Speed = 62.60 m/s with a Time step = 0.0005 s
0.16
0.15
0.14
E
0.13
CD,
0.
CU,
N
0.12
0.11
0.1
0.09
-0.03
-0.02
-0.01
0
0.01
Y Displacement (m)
0.02
0.03
Figure 2-50: Nonlinear time simulation of the wing tip motion at 10% above its flutter speed
(2' root angle of attack) for the three-layer design
71
eps1 (longitudinal fiber strain), ply 4
eglass 120: MAX = 0.0044209
eps2 (transverse fiber strain), ply 4
eglass 120: MAX = -0.00065431
eps3 (in plane shear strain), ply 4
eglass 120: MAX = -0.0040786
Figure 2-51: Maximum ply strain reached at 10% above flutter speed of 20 root angle of
attack for the three-layer design
72
Five-Degree Root Angle of Attack
The flutter speed for the 50 root angle of attack is 46.4 m/s. This can be seen on the root
locus plot provided in Figure 2-52. This plot shows the mode of instability is again the 1 "
chordwise bending mode (mode 2). Again the increase in angle of attack results in the wing
becoming unstable at lower speeds. The 1 st torsion (mode 4) and 3 rd bending (mode 5) are
also starting to interact with each other, causing the torsion mode to head to the instability
axes. The first six mode shapes for this case are provided in Figure 2-53.
200I
180
I
Root locus for Root AOA =
I
I
H--------
160 F--- -
20
140
120
5*
- ------ 38
10
0
-------
F-
8:
-----
-----
100
:0
U
80
--
60
--
40
--
----
----
---------
--------
---
---------
------- 2B
0.
0 1
------
c B
20 - -- - -- - -- -- -- - -
-B
1B
EU
J~J
50
50
40
30
10
20
Real Part of Root
-u
0
10
Figure 2-52: Root locus plot for 50 root angle of attack
73
20
Speed = 46.40 m/s @ Freq = 8.39 Hz
Speed = 46.40 m/s @ Freq = 31.8 Hz
Speed = 46.40 m/s @ Freq = 54.1 Hz
Speed = 46.40 m/s @ Freq = 120 Hz
-0.1
-0.05
0
0.6
0.8
1
Speed
=
46.40 m/s @ Freq
=
160 Hz
Speed = 46.40 m/s @ Freq = 302 Hz
0.2
0.1
---
0
-
-0.05
-0
..--- - .-0.4
0.6
0.8
0.2
1
Figure 2-53: Mode shapes for three-layer design at 5' root angle of attack and at its corresponding flutter speed of 46.4 m/s
74
When this wing is flown at 10% above the predicted flutter speed, the wing enters into
a LCO. The tip deflections and the tip twist are plotted for this case up to 1 second. These
plots are provided in Figure 2-54 thru Figure 2-56.
Speed = 51.20 m/s with a Time step = 0.0005 s
0.4
0.35
0.3
E
E
0.25
(D
0.
CU)
0.2
0.15
F
0.1
U
0.05
0
-0.05'0
0.2
0.4
0.6
0.8
1
Time (s)
Figure 2-54: Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (5' root angle of attack) for the three-layer design
A strain analysis was performed on the wing when it was at its maximum deflection.
This occurs during the initial rise shown in Figure 2-54. The maximum tensile strain is
7374 tm/m and the maximum shear strain is -6799 pm/m. Both occur at the outer ply
of the bottom part of the airfoil at the root of the wing. These details are summarized in
Figure 2-57. These strains result in a margin of safety of 0.36 and 1.21, respectively.
75
Speed = 51.20 m/s with a Time step = 0.0005 s
20
15
10
a)
F_
5
0
-5-
-10 0
0.2
0.4
0.6
0.8
1
Time (s)
Figure 2-55: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(50 root angle of attack) for the three-layer design
Speed = 51.20 m/s with a Time step = 0.0005 s
0.32
0.31 -
0.30.290.28E
0)
C.~0.27ND
0
N
0.260.250.240.230.22 -0.06
-0.04
-0.02
0
Y Displacement (m)
0.02
0.04
Figure 2-56: Nonlinear time simulation of the wing tip motion at 10% above its flutter speed
(50 root angle of attack) for the three-layer design
76
eps1 (longitudinal fiber strain), ply 4
eglass 120: MAX = 0.0073741
/
eps2 (transverse fiber strain), ply 4
eglass 120: MAX = -0.0010914
eps3 (in plane shear strain), ply 4
eglass 120: MAX = -0.0067985
Figure 2-57: Maximum ply strain reached at 10% above flutter speed of 50 root angle of
attack
77
The following is a table to summarize the strain analysis for the different cases.
Table 2.13: Summary of Strain Analysis Results
Angle of Attack
1 tension
torsion
2' tension
torsion
50
2.4
tension
torsion
Two-Layer Results
2660 p m/m
-769 p m/m
4484 p m/m
-2726 p m/m
7405 y m/m
-5018 p m/m
Three-Layer Results
2697 p m/m
-1765 y m/m
4421 y m/m
-4079 p m/m
7374 p m/m
-6799 p m/m
Nastran Comparison
To ensure that the wing would survive the loadings in the tunnel, a stress analysis was
performed using MSC.Nastran finite element code. Since the MATLAB code from [3] was
not yet validated experimentally, this would provide added assurance that the stress and
strain data predicted by the MATLAB code is correct. Figure 2-58 provides a close-up of
the finite element model mesh. The wing was discretized using 5550 elements. The foam
was modeled as 4-noded solid elements and the composite skin was modeled as 4-noded shell
elements.
The worse stress case scenerio for the three-layer design is the 5' root angle of attack case.
The MATLAB code was used to provide static deflection values for the wing at 51.15 m/s,
which is 10% above the predicted flutter speed. The x,y,z displacements and pitch rotation
for the wing were matched for two different cases: maximum vertical (z) displacement and
maximum pitch rotation. This was achieved by applying varing forces distributed along the
span of the wing at 25% of the chord and then forces to the leading and trailing edges to
establish the desired rotation.
78
Figure 2-58: Finite element mesh of the three-layer design used in Nastran (5550 elements,
13617 degrees of freedom)
79
2.4.1
Case 1 - Maximum Vertical Displacement
This case matched the maximum vertical displacement predicted by the MATLAB code
during the limit cycle oscillation. The original predicted values and the values determined
in Nastran are provided in Table 2.14.
Table 2.14: Nastran Case 1: Displacements and Rotations
chordwise deflection
spanwise deflection
vertical deflection
pitch rotation
MATLAB Prediction
Nastran Result
0.0034 m
0.0603 m
0.3159 m
0.0199 m
0.0539 m
0.3126 m
-9.110
-8.850
The MATLAB and Nastran displacement results match within 11%, except in the chordwise direction. The overall magnitude of the displacements are equivalent and the major
contributors, vertical displacement and pitch rotation, are within 1% and 3%, respectively.
Since the results show such high margins of safety (Equation 2.1), the analysis was considered adequate. The results from this case are summarized in Table 2.15 and fringe plots of
the stress profiles are provided in Figure 2-59 through Figure 2-61.
Table 2.15: Nastran case 1: summary of results
chordwise
spanwise
chord-span shear
2.4.2
Maximum Strain
2570 pm/m
5430 pm/m
-5740 pm/m
Margin of Safety
6.4
2.5
8.0
Case 2 - Maximum Pitch Rotation
This case matched the maximum pitch rotation predicted by the MATLAB code during the
limit cycle oscillation. The original predicted values and the values determined in Nastran
are provided in Table 2.16.
80
MSC.Patran 2001 r3 13-Oct-02 20:38:45
2.57-00
Fringe: lift. PW Linear: 100. % of Load_2: Nonlinear Strains, Strain Tensor-(NON-LAYERED)
2.08-00
Deform: lift, PW Linear: 100. %of Load_2: Displacements. Translational
1.60-003
1.12
6.40-00
aximum value (2570 rn/rn) at the wing root
on the top-side of the wing.
1.58-00
-3.23-00
-8.05-00
4
-1.29-00
-1.77-00
-2.25-00
-2.73-00
-3.21-00
-3.69-00
-4.18-00
-4.66-00
defaultFringe:
Max 2.57-003 @Nd 865
Min -4.66-003 @Nd 461
defaultDeformation:
Max 1.25+001 @Nd 1836
xY
Figure 2-59: Nastran normal (chordwise) strain for 5' root angle of attack
Table 2.16: Nastran case 2: displacements and rotations
chordwise deflection
spanwise deflection
vertical deflection
pitch rotation
MATLAB Prediction
0.0502 m
0.0370 m
0.2391 m
Nastran Result
0.0176 m
0.0348 m
0.2520 m
9.930
10.50
81
MSC.Patran 2001 r3 13-Oct-02 20:38:27
5.43-00
Fringe: lift PW Linear:100. % of Load_2: Nonlinear Strains. Strain Tensor-(NON-LAYERED) (ZZ)
Deform: lift PW Linear: 100. % of Load_2: Displacements. Translational
4.71-00
3.99-00
3.27-00
2.54-003
1.82-00
1.10-003
3.72-004
-3.51-00
-1.07-003
-1.80-00
-2.52-00
-3.24-00
-3.97-00
Maximum value (5430 pim/m) at the wing root,
on the underside of the wing
Y
.Z
X
-4.69-00
-5.41-00
default_- Fringe :
Max 5.43-003 @Nd 461
Min -5.41-003 @Nd 865
defaultDeformation :
Max 1.25+001 @Nd 1836
Figure 2-60: Nastran normal (spanwise) strain for 5' root angle of attack
82
MSC.Patran 2001 r3 13-Oct-02 20:38:55
2.06-00
Fringe: lift, PW Linear: 100. % of Load_2: Nonlinear Strains. Strain Tensor-(NON-LAYERED) (Z>
1.54-003
Deform: lift PW Linear: 100. % of Load-2: Displacements, Translational
1.02-003
5.04-004
Maximum value (2060 gm/m) at the wing root
at the trailing edge
-1.58-005
-5.36-00
-1.06-00
-1.58-00
-2.10-00
-2.62-00
-3.14-003
-3.66-003
-4.18-00
-4.70-003
-5.22-003
Y
-5.74-003
defaultFringe:
Max 2.06-003 @Nd 2
Min -5.74-003 @Nd 1685
defaultDeformation:
Max 1.25+001 @Nd 1836
Figure 2-61: Nastran in-plane shear strain for 50 root angle of attack
83
Similar to the previous case, the MATLAB and Nastran displacements match within
6%. The overall magnitude of displacement is equivalent and the main contributors, vertical
displacement and pitch rotation, are over-predicted. Again, although the Nastran displacements do not match the MATLAB exactly, the margins of safety are high enough to consider
the values sufficient. The results from this case are summarized in Table 2.17 and fringe plots
of the stress profiles are provided in Figure 2-62 through Figure 2-64.
Table 2.17: Nastran case 2: summary of results
chordwise
spanwise
chord-span shear
2.5
Maximum Strain
Margin of Safety
2390 pum/m
4960 pm/m
7950 pm/m
6.9
2.8
5.5
Design Selection
The three-layer design was chosen for the wind tunnel model. This decision was based on
the resulting predicted flutter speeds and on the resulting strains. The flutter speeds for
the two-layer case were on average 16% lower than the three-layer design. However, the
deflections experienced by the two-layer design ranged from 41% to 61% higher than the
three-layer design. This resulted in an increase in the strain levels for the two-layer wing.
To be safe, the increase in deflection and strain was considered more detrimental than an
increase in the flutter speed. Therefore, the three layer design was chosen.
84
MSC Patran 2001 r3 13-Oct-02 19:37:40
2.39-00
Fringe: lift PW Linear: 100. % of Load: Nonlinear Strains, Strain Tensor-(NON-LAYERED)M
1.99-00
Deform: lift, PW Linear: 100. % of Load: Displacements. Translational
1.59-00
1.19-00
7.86-00
Maximum value (2390 pm/m) at the wing root
on the top-side of the wing
3.85-004
-1.51-00
-4.16-00
-8.16-00
-1.22-00
-1.62-003
-2.02-00
-2.42-00
-2.82-003
-3.22-00
Y
-3.62-003
defaultFringe:
Max 2.39-003 @Nd 865
Min -3.62-003 @Nd 206
defaultDeformation:
Max 1.08+001 @Nd 1225
Figure 2-62: Nastran normal (chordwise) strain for 5' root angle of attack
85
MSC.Patran 2001 r3 13-Oct-02 19:38:44
4.96-00
Fringe: lift PW Linear: 100. % of Load: Nonlinear Strains, Strain Tensor-(NON-LAYERED) (ZZ)
4.30-003
Deform: lift PW Linear: 100. % of Load: Displacements, Translational
3.63-00
2.97-00
2.31-00
1.65-00
9.82-00
3.19-004
-3.44-00
-1.01-003
-1.67-00
Maximum value (4960 pm/m) at the wing root
on the under-side of the wing
-2.33-003
-3.00-00
-3.66-00
-4.32-003
y
-4.98-0031
defaultFringe:
Max 4.96-003 @Nd 206
Min -4.98-003 @Nd 865
defaultDeformation:
Max 1.08+001 @Nd 1225
Figure 2-63: Nastran normal (spanwise) strain for 5* root angle of attack
86
7.95-00
MSC.Patran 2001 r3 13-Oct-02 19:38:36
Fringe: lift. PW Linear: 100. % of Load: Nonlinear Strains. Strain Tensor-(NON-LAYERED) (Z7.21-00
Deform: lift PW Linear: 100. % of Load: Displacements. Translational
6.47-003
5.73-003
4.98-003
Maximum value (7950 gm/m) at the wing root
at the trailing edge
4.24-003
3.50-00
2.76-00
2.01-003
1.27-00
5.28-004
-2.14-004
-9.57-00
-1.70-003
-2.44-003
Y
-3.18-003
defaultFringe:
Max 7.95-003 @Nd 1786
Min -3.18-003 @Nd 1071
default Deformation:
Max 1.08+001 @Nd 1225
Figure 2-64: Nastran in-plane shear strain for 50 root angle of attack
87
Chapter 3
Experimental Procedure
This chapter describes the experimental work that took place as part of this project. The
manufacturing methods are presented, as are details for instrumentation location and wiring.
The chapter also contains the description of the experimental tests: bench top and wind
tunnel.
3.1
Wing Manufacture
In this section, different components that make the instrumented wind tunnel model are
discussed.
The foam cores were machined in the Aeronautics and Astronautics machine
shop on the TRAK K3 mill machine with the help of Ricky Watkins and Donald Weiner.
The wing lay-up and cure were performed within the TELAC facilities with the support of
John Kane. The root clamps for holding the wing during the bench top and wind tunnel
characterization were machined at MIT Lincoln Laboratory. The wing mold was provided
by NASA Langley.
3.1.1
Foam Core
The foam core of the wing was manufactured out of Rohacell 31. This material was selected
to ensure survival of the foam at the elevated temperatures of the wing cure process. The
foam for each wing was machined in four seperate pieces, consisting of the top and bottom
88
of the wing from the root to 0.5715 m and the top and bottom of the wing from 0.5715
m to the tip at 1.13 m. The core is slightly oversized by 0.254 mm (10 mil) to ensure
adequate backpressure within the mold during the cure process. After the accelerometers
are embedded within the foam at the tip of the wing, the individual pieces are glued together
with five minute epoxy. The foam pieced together in the mold is shown in Figure 3-1.
Figure 3-1: Four sections of the foam core pieced together in mold
3.1.2
Instrumentation
Seven full strain gauge bridges and two accelerometers were added on the foam core. The
locations of the sensors are provided in Figure 3-2. An additional torsional bridge at the root
was attached after the wing was cured and thus is located on the outer surface of the wing.
The accelerometers were placed at the tip of the wing where maximum deflection happens.
Since the root of the wing experiences the highest strain, one of each of the strain gauge
bridge types was placed there. The locations of the other strain gauges along the span of
the wing correspond to points of high strain for the higher normal modes, i.e.,
bending and 2 "d torsion.
89
2
"d and
3rd
35%
45%
55%
70%
CL
0
oC
80%
+
Leading Edge
Figure 3-2: Location of sensors: bending gauges are represented by squares, forward/aft
gauges by diamonds, torsion gauge by a star, and accelerometers by circles
Strain Gauges
The strain gauges used to make-up the Wheatstone bridges were from the Micro Measurements Division of Measurements Group, Inc. The gauges for the bending bridges were model
EA-06-125AD-120 and the gauges for the torsional and forward/aft bending bridges were
model CEA-06-125UT-120.
Both models consisted of gauges with 120 Q resistance. The
gauges were wired to record spanwise bending and torsion.
The four spanwise bending
bridges consisted of a pair of gauges oriented in the 00 direction on the top and bottom
of the wing. The four torsional bridges consisted of a pair, one in the +45' direction and
one in the -45'
direction on the top and bottom of the wing [12]. A general Wheatstone
bridge wiring diagram is provided in Figure 3-3. The wire diagrams for the specific types
of bridges use the same numbering convention. A map of the proper wiring scheme for a
spanwise bending bridge is presented in Figure 3-4 and Figure 3-5 details the wiring for the
torsional bridges. After some discussion with Professor John Dugundji, MIT, it is believed
that the torsional wiring presented in [12] actually refers to a forward and aft motion, as
opposed to the twisting motion originally thought. A twist bridge was wired to the outer
skin of the wing at the root to measure the twist motion of the wing. The wiring for this
bridge is provided in Figure 3-6.
90
-Vout
Figure 3-3: General wiring diagram for Wheatstone bridges
1,- a
.c
Leading
.b
4 'c
.a
2-
Edge
d
Figure 3-4: Wiring diagram for Wheatstone bridges in bending
<&6
Leading
Edge
tK
Figure 3-5: Wiring diagram for Wheatstone bridges in forward/aft bending
Leading
o'
Edge
&6
Figure 3-6: Wiring diagram for Wheatstone bridges in twist
91
The internal strain gauges where attached to a single square layer (25 mm by 25 mm)
of pre-cured E-glass/epoxy oriented at 00/900. A photo of the internal strain gauges on the
wing is provided in Figure 3-7. The squares were glued down with five minute epoxy to the
foam. The strain gauges were then also afixed with five minute epoxy. Magnet wire was
used to make all of the required electrical connections. The two bridges at the root, with
the wiring of both types of bridges, is shown in Figure 3-8.
Figure 3-7: Strain gauge at mid-span
Figure 3-8: Strain gauges wired at the root
92
Accelerometers
The accelerometers used for this experiment were the Endevco Model 22 piezoelectric accelerometers. These sensors were embedded within the foam core at the wing tip, 14.3 mm
from the leading edge and 39.7 mm from the leading edge. The specifications for the accelerometers are provided in Table 3.1. The wires were run parallel to the leading edge in
channels between the two halves of foam. The pair of accelerometers is used in combination
to calculate the twist and bending frequencies at the wing tip. When the output of one
is subtracted from the other, the tip twist response is obtained. The average of the two
accelerometers produces the bending response.
Table 3.1: Accelerometers used in the wing model
Gauge
CW18-22
CW13-22
3.1.3
Location
14.3 mm from LE
39.7 mm from LE
Charge Sensitivity
0.374 pC/g
0.385 pC/g
II
Lay-Up
In preparation to lay-up the composite on the instrumented foam core, the foam core was
put in the oven at 266'F for 24 hours to remove any residual moisture. Moisture has a
plasticizer effect on the foam, which causes a decrease in its compressibility strength [13]. If
this step is not performed, the foam will collapse under the high temperature and pressure
of the curing process.
The required three layers of E-glass/epoxy were cut to 0.305 m by 1.143 m and stacked
flat on the table. They were then gathered up and placed into the bottom half of the mold
as one piece of cloth. The long edge was placed slightly behind the trailing position and the
short end was aligned with the root. At this point, the additional layers of E-glass/epoxy
around the root section where wrapped around the foam core. These consisted of a 76-mm
strip, a 64-mm strip, a 51-mm strip, and a 38-mm strip, each being two plies thick. These
additional plies were added to the root area to provide additional strength where the bolts
93
from the clamp would be passing through the wing. Once this was done, the foam core was
inserted. The rest of the lay-up was then wrapped around the leading edge of the foam
and pulled tightly toward the trailing edge. Once the lay-up was smooth on both sides of
the wing, a straight edge was placed at the wing trailing edge and the excess was trimmed
off. Care was taken to ensure the wires passing out of the root would not interfere with
the root clamp. The top of the mold was lifted into position and then the two pieces were
clamped together. A thermocouple was attached between the two halves of the mold, as
the temperature of the mold is the one which needs to follow the cure profile. With the
mold assembly completed, the wing was placed into the autoclave and cured at 250'F for 90
minutes. It was then allowed to cool slowly over 48 hours.
3.1.4
Post Cure
The wing was removed from the mold after slowly cooling and only minor cosmetic touch-ups
were required. Some resin had been pulled between the two mold pieces at the leading edge.
This resin was scrapped off and then the leading edge was sanded for a smooth finish. This
resin was also pulled at the trailing edge, but this is desirable as it creates a smooth taper
to the trailing edge, so no modifications were made to the trailing edge. A band saw was
used to remove about 6-mm from the wing tip to provide a smooth edge to that side of the
wing as well. Finally three holes were drilled into the root of the wing, 28.5-mm behind the
leading edge, to accomadate the three bolts from the clamp.
3.1.5
Clamp
The root clamp was made from Aluminum 6061 and was NC machined at MIT Lincoln Laboratory. It consists of two seperate pieces which bolt together through the wing. Drawings
of the top and bottom are provided in Figure 3-9 and Figure 3-10. The clamp is designed
such that a gap between the two pieces exists at the leading and trailing edge. This allows
for some shape mismatch between the final wing and the clamp.
94
Units: Inches
Figure 3-9: Top of root clamp
95
F
r
31/2
NACA 0012 Airfoil
Units: Inches
Figure 3-10: Bottom of root clamp
96
3.2
Data Acquisition Equipment
The strain gauge and accelerometer signals need to be processed through conditioners before
they can be sent to a computer to store. Since all of the bridges were full bridges they needed
to be connected to the conditioner with the pin readout provided in Figure 3-11. The little
letters represent the connections to the strain gauges and the capital letters refer to the pins
on the connector. The conditioner box converts the signal to an amplified voltage which is
B
C
F
a+
-c
b+
-d
A
D
F
Figure 3-11: Pin diagram for full bridges
sent to the computer via a BNC cable. The accelerometer signal was also processed through
a conditioner before the signal voltage was transmitted to the computer. A photo of both
conditioners is provided in Figure 3-12. These are the conditioners used for the bench top
tests. The wind tunnel tests used similar strain gauge conditioners, but in a different box.
3.3
Wind Tunnel Load Cell
The wind tunnel tests required the use of a load cell. The load cell is the interface between
the wing root clamp and the fixed stand in the tunnel. It provides three axis of load data,
as well as the yaw moment. For this wing, a 75-lb Multi-Axes Load Cell from JR3, Inc. was
used. The load settings and ratings are provided in Table 3.2.
To determine the loads, the voltage channel coming from the load cell must be multiplied
by a calibration matrix. The equation for this load cell is:
97
Accelerometer conditioners
Strain Gauge conditioners
Figure 3-12: Photo of accelerometer and strain gauge conditioners
Table 3.2: Load cell used for the wing tunnel tests
Load Cell Channel
F
Fv
Fz
MX
MV
M2
Maximum Load Measurement
75 lb
75 lb
150 lb
0 in-lb
0 in-lb
225 in-lb
Drag (ib)
9.2758
Axial (ib)
0.2977
Lift (ib)
Yaw (in-lb)
-0.1770
9.1166
0.2176 -0.1894
-0.1918
0.0131
98
-0.0589
Sensor Load Rating
75 lb
75 lb
150 lb
225 in-lb
225 in-lb
225 in-lb
-0.1796
(V)
0.0861
0.0407
(V)
18.2251
0.6773
(V)
27.7896
(V)
-0.0766
(3.1)
Chapter 4
Experimental Studies
This chapter discusses the bench top and the wind tunnel test results. There is also a section
on adjustments made to the model to cause the natural frequencies to match those seen in
the bench top tests.
4.1
Bench-top Tests
A series of bench top tests were conducted to determine the characteristics of the manufactured wing and to ensure operation of all the sensors. It was discovered that the forward/aft
strain gauge bridge at the root was electrically shorted during manufacture. It was not possible to be repaired, so it was not monitored during the tests. Aside from simple functionality
tests, only the gauges at the root for bending and torsion were monitored during benchtop tests. These tests were performed in TELAC. A picture of the experimental set-up is
provided in Figure 4-1 and a picture of the monitoring equipment is provided in Figure 4-2.
4.1.1
Strain Gauge Calibration
A static calibration of the strain gauges was performed to determine the deflection-voltage
relationship. This was performed with the strain gauge conditioners within TELAC. So these
results do not correspond with voltages from the wind tunnel tests. This calibration was
performed to establish a process to be used later with the tunnel instrumentation. The wing
99
Figure 4-1: Experimental set-up for bench-top tests
Figure 4-2: Experimental instrumentation for bench-top tests
100
was displaced in the vertical direction by various pre-established tip displacement values
in both directions. This was done by holding the wing tip by hand and aligning it with
pre-drawn tick marks on a reference scale. The wing was then held to these locations for
a few seconds. This provided data on bending deflections versus voltage. A time trace of
this calibration for the root bending strain gauge bridge is provided in Figure 4-3. When
8
7.5 -
6
23
=
7
6.5
M
8=18
6 -
5-5
3.5
0
0
5
4.5
4-
3.5
0
20
40
60
80
100
120
140
160
180
200
Time (s)
Figure 4-3: Time trace of data from bench-top bending calibration
this data is plotted with the known deflections, a curve of voltage versus tip deflection is
generated. This is shown in Figure 4-4.
The wing was also twisted to various amounts at the tip, which provided a calibration for
tip twist. For this, the leading edge of the wing was again deflected by hand to pre-drawn
tick marks, with the trailing edge held at the original position. A time trace of the data
from the root torsion strain gauge bridge is provided in Figure 4-5. This data was plotted
101
8
5c5- 4
604
3-30
-20
-10
0
10
Wing Tip Displacement (mm)
20
30
residuals
0.3
-
Linear: norm of residuals = 0.41744
0.2
0.1
-
---
0
-0.1
-0.2
-0.3
-20
-15
-10
-5
0
5
10
15
20
Figure 4-4: Calibration of strain gauge voltage output for bending
102
25
30
7
6.5
a=2.12 0
6
a =00
0
a =-2.12 0
-1.060
5
a =-3.720
4.51
0
20
40
60
Time (s)
80
100
Figure 4-5: Time trace of data from bench-top twist calibration
103
120
against the known twist values to provide a curve for the voltage output generated through
tip twist. This curve is given in Figure 4-6.
7
> 6.5
()
C
S6
0
0)
0 5.5
t-
4.5'-4
-3
-1
-2
0
1
Wing Tip Twist (deg)
2
3
4
residuals
I
I
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
F
-4
Linear: norm of residuals = 0.28536
I
I
I
-3
-2
-1
I
I
I
0
1
2
1
3
4
Figure 4-6: Calibration of strain gauge voltage output for tip twist
4.1.2
Tap Tests for Dynamic Properties
Various places on the wing were tapped to excite the different normal modes, so that its
natural frequencies could be identified. This data was collected via the strain gauges. The
data from various tap tests is given below in Figure 4-7 through Figure 4-9. The top graph
of each figure is a time trace of the voltage output from the strain gauges and the bottom
is a Fast-Fourier Transform (FFT) of the output. The first tap was to excite the bending
modes. The tap was applied to the tip of the wing around 30% chord line. The frequencies
104
excited were at 6 Hz and at 38 Hz (1st and
2 nd
bending).
The second tap was at the
tip but off the elastic axis, at the trailing edge of the wing tip, thus the torsional modes
were excited. This plot shows dynamics at 70 Hz (1st torsion), as well as the 6-Hz and
38-Hz bending frequencies. The final tap was on the leading edge of the wing tip in the
forward/aft direction. This excited the chordwise bending mode. This plot shows the 1"
chordwise bending frequency at 45 Hz. The 6-Hz 1" bending was also excited.
Volts
5.40-
5.250.00
dBVrns
0 05 0,10 0.15 0.20 0.25 0.30 0.35 0.40
Peak Power 28.14 Vrms^"'2
Peak Frequency:
0,45
0.50
0.00 Hz
14.50.0-20.0-40.0-uA
60.0-Hz
0,0
20.0
40.0
60.0
80.0
100.0
120.0
Figure 4-7: Tap test for bending frequencies-root bending gauge
It can be seen through these graphs that tap tests are not perfect, as exciting purely
one type of mode is extremely difficult. However, the first four modes of vibration can be
determined along with their mode shape. This data is summarized in Table 4.1.
105
Volts
4.30 -
I
I1
4,00
RAA
:3.50
A
Ov
325
AAAK Am;
IN
A)
w v VMWITV
we
5,
0,00
dBVrms
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.00 Hz
Peak Power :12.09 Vrms^2
Peak Frequency :
10.8 0.0-20.0-40.0-
Hz
-57.5 -.
20.0
40.0
60.0
80.0
Figure 4-8: Tap test for torsion frequencies-root torsion gauge
Table 4.1: Natural Frequencies From Bench-top Tap Tests
Mode
Frequency
Mode Shape
1
6.4 Hz
1"
2
38 Hz
2 "d Bending
3
4
45.0 Hz
70.0 Hz
1
Bending
" Chordwise Bending
1 st Torsional
106
100.0
Volts
6.54 -
650 -iv
6.48 0.00
dBVrms
-
-
-
6.52 -
-
1
-
1 1/
1
-i --
V 1v
1
1
I
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
Peak Power 42.31 Vrms^,2
Peak Frequency:
0.00 Hz
16.30.0-
-20.0-40.0-60.00.0
10.0
20.0
30.0
40.0
50.0
60.0
Figure 4-9: Tap test for chordwise bending frequencies-forward/aft gauge
107
4.1.3
Shaker Tests for Dynamic Properties
Finally, a mini shaker type 4810 from Briiel & Kjaer was applied to different spots on the
wing to validate the tap tests and to fully understand the dynamics of the wing. The shaker
was first applied to 30% of the chord at about 0.125-m out from the root clamp. The
shaker was excited with sine waves of frequencies varing from 1 Hz to 110 Hz. This test
was followed by the shaker exciting the leading edge of the wing, as shown in Figure 4-10.
Again the shaker was excited with frequencies varing from 1 Hz to 110 Hz. With the shaker
Figure 4-10: Shaker at the wing leading edge for the natural frequency bench test
at the leading edge, the first torsion mode could be excited. The shaker test confirmed the
frequencies and mode shapes determined through the tap test, as the node line for the wing
could be observed and felt through a light touch. These results support the use of the tap
test in the tunnel, where a shaker test would be difficult.
108
4.2
Re-work of Analytical Model
During the bench top tests the wing was weighed to determine the true cross sectional
weight and the final length of the wing. The wing cross-sectional weight is 0.2176 kg/m.
The length of the wing outside the clamp is 1.038 m. With these adjusted within the model,
the originally calculated natural frequncies for the wing (Table 2.9) changed and the
1 st
bending mode matches that from the bench-top tests. The first six natural frequencies are
given in Table 4.2. A diagram of all the mode shapes is provided in Figure 4-11. The %-error
value is calculated with Equation 4.1.
%error =
Calculated- Measured
Measured
- 100%
(4.1)
Table 4.2: Natural frequencies with adjusted weight and length
Mode
1
2
3
4
5
Calculated Frequency
6.44 Hz
39.1 Hz
41.7 Hz
78.4 Hz
124 Hz
6
241 Hz
Measured Frequency
6.4 Hz
45 Hz
38 Hz
70 Hz
-_-_-
% Error
0%
13%
9.7%
12%
Mode Shape
1 st Bending
st
Chordwise
Bending
1
2"d Bending
1st Torsional
3rd Bending
2 "d
Torsional
Some adjustment of material properties was made to further modify the wing characterists
to match the bench-top results. The shear stiffness, Q66 (defined in Appendix B), of the
E-glass/epoxy was modified from 4.1 GPa to 3.25 GPa. This changed the first torsional
frequency of the modeled wing to 70 Hz without changing the other frequencies. The Young's
Modulus of the foam in the chordwise direction was also adjusted from 36.0 MPa to 558
MPa. This changed the first chordwise bending mode to 45 Hz. The discrepancy between
the true second bending and the predicted second bending is believed due to the non-uniform
distribution of mass along the span of the wing. The cross sectional wing weight takes the
weight of the accelerometers, strain gauges, and epoxy which was used to attach the separate
109
Speed = 0.00 m/s @ Freq = 6.44 Hz
Speed = 0.00 m/s @ Freq = 39.1 Hz
-0.6
-0.2
0
0.8 0.84
0.6
Speed = 0.00 m/s @ Freq = 41.7 Hz
Speed = 0.00 m/s @ Freq =78.4 Hz
0.2
0.4
0.6
0.8
1
1
Speed = 0.00 m/s @ Freq = 124 Hz
Speed = 0.00 m/s @ Freq = 241 Hz
0.2
0.2
0.4
0.4
-0.1
-0.05
0.6
0.6
0
0.8
0.8
1
1
Figure 4-11: Fundamental mode shapes with adjusted weight and length
110
foam pieces and spreads them evenly across the entire wing. This un-even distribution of
mass will cause slight deviations in the second bending, causing the 6.25 times the first
bending frequency rule, value for isotropic homogeneous cantilever beams, to not apply.
With these modifications to the wing properties, the model predictions for the wing are
close to the measured frequencies. The natural frequencies are given in Table 4.3 and the
mode shapes are provided in Figure 4-12.
Table 4.3: Intermediate predicted natural frequencies based on added stiffness corrections
Mode
Calculated Frequency
Measured Frequency
% Error
Mode Shape
1
6.44 Hz
6.4 Hz
0%
2
3
41.5 Hz
45.0 Hz
38 Hz
45 Hz
9.2%
0%
4
5
6
70.0 Hz
123 Hz
216 Hz
70 Hz
0%
-
1"
-
-
2 "d
1 "t
Bending
Bending
1 "t Chordwise Bending
2 nd
Torsional
3 rd Bending
Torsional
These new changes in the model were used to calculate new flutter speeds. The flutter
speeds are given in Table 4.4 along with their corresponding root locus plots in Figure 4-13
through Figure 4-17. A portion of the root locus was magnified for 10 and 20. This data is
provided in Figures 4-14 and 4-16.
Table 4.4: Intermediate predicted flutter speeds based on corrected structural properties
Angle of Attack
10
Flutter Speed
33.0 m/s
20
52.7 m/s
42.75 m/s
50
A more indepth study of flutter speed versus angle of attack has provided a graph depicting some strange behavior of the wing between 1 and 20 angle of attack (Figure 4-18).
Similar behavior of flutter speed versus angle of attack was discussed in [1]. There, a discontinous jump in the flutter speed was seen between 00 and 1 angle of attack. For this
111
Speed = 0.00 m/s @ Freq = 6.44 Hz
Speed = 0.00 m/s @ Freq = 41.5 Hz
0.1
0
-0.1
-01-0.05
0
Speed = 0.00 m/s @ Freq = 45 Hz
Speed = 0.00 m/s @ Freq = 70 Hz
0.01
-0.4
0.2
0.4
\
-0.2
0.6
0..
0.8
1
Speed = 0.00 m/s @ Freq = 123 Hz
Speed = 0.00 m/s @ Freq = 216 Hz
0.2
0.2
0.4
0.4
-0.1
-0.05
0
0.6
0.6
-
0.8
0.8
1
1
Figure 4-12: Predicted fundamental mode shapes based on added stiffness corrections
112
Root locus for Root AOA = 1 *
200
180 |-
140 F --------
120
-
100
-
80
-
U
" 7"
3B
-
1 .- - --0---
-20
I
U
C
0~
'J)
LL
0 1T
60
---- -------
-
2B
~~
40 --
*
7Y0
20 1-
- - - - -- - -- -- -
0
50
-rtr---
-
-
1B
.
-Rn
40
,n
4
.
I
30
1n
a
n
I
I
20
10
0
10
20
Real Part of Root
Figure 4-13: Root locus plot for 10 root angle of attack based on modified wing stiffness
properties
113
Root locus for Root AOA =
1*
50
48
46
- - - - -- -- - -- ------ - - - - - - - - - - - - - - - --
-
-
-- - - --
--
-- -
1CB
44
-
- ---
-
-
-- - -
-- - - - - - -
42
C
LL~
2B
10A
C
-- -- - ---
40
--- - - - -
38
6
-36 - - - ----
34 -- - -
- - -- - - - - ---
---
-- - -- -- - ---- - -
32 -- --- - -
30
10
8
-- -- -- - - -- - -
6
4
2
0
2
4
6
8
10
Real Part of Root
Figure 4-14: Magnified root locus plot for 1' root angle of attack based on modified wing
stiffness properties
114
Root locus for Root AOA =
2*
200
180
-- -
160
--
140
--
---------
3B:
120 -----------
100 F----------
LL
-
80 F- 0
60
F
- -
--- ----- - - -
-
02B
40 F- - - -------
1CB
--------20--
F"
0'
50
0
- - - - - - -- - -- - -
1B:
-Q A
40
30
10
20
Real Part of Root
0
10
20
Figure 4-15: Root locus plot for 2' root angle of attack based on modified wing stiffness
properties
115
Root locus for Root AOA =
2*
50
45
40
Q)
LI-
35
30
25
10
8
6
4
2
2
0
Real Part of Root
4
6
8
10
Figure 4-16: Magnified root locus plot for 2' root angle of attack based on modified wing
stiffness properties
116
Root locus for Root AOA
5*
=
200
-- - - - - - - - -0- 2T
-- -
180
---- -- -----
160
-
-- - - --- -
---- -------
140
---F
-
-
3B
120
-
0
1-
- --
NU
:3
100
-- - ---
U-
-
---- -- ------
80
0
1T:
60 -- 2B
0
0
1CB
20
o1B
"0r
0
50
40
30
10
20
Real Part of Root
0
10
20
Figure 4-17: Root locus plot for 50 root angle of attack based on modified wing stiffness
properties
117
55
50-
45-
E
CD40
a,)
U,
35-
30-
25
-
0
1
2
4
3
Root Angle of Attack (deg)
5
6
7
Figure 4-18: Predicted flutter speeds for various root angles of attack based on modified
wing stiffness properties
118
current wing, however, the behavior of the wing at 10 is not believed to be accurate, as is
proved during the wind tunnel tests. As will be discussed in more detail later, the wing was
flown in the tunnel up to 82 m/s at 1' angle of attack, which is 250% above this predicted
flutter speed.
For this reason the analytical model was again adjusted. A point mass was placed at
42% of the span to simulate the line of epoxy which joined the foam. The size of the point
mass was adjusted until the second bending frequency matched the measured 38 Hz. The
density of the foam was adjusted so that the first bending frequency returned to 6.44 Hz
after the addition of this new mass. Table 4.5 provides a summary of the final values for the
adjusted parameters.
Table 4.5: Final values of adjusted parameters
Parameter
wing span
p foam
Q66 E-glass/epoxy
Et foam
Point mass @ 46% span
Original Value
1m
30 kg/m 3
4.10 GPa
36 MPa
0 kg
Adjusted Value
1.038 m
94.5 kg/m 3
3.265 GPa
558 MPa
0.0286 kg
With this last adjustment, the first four predicted frequencies match those measured.
Table 4.6 shows the frequencies values while Figure 4-19 shows the corresponding mode
shapes.
These new dynamic characteristics resulted in a change in the aeroelastic behaviour of
the wing. The new stability characteristics are shown in the root locus plots provided in
Figure 4-20, Figure 4-24, Figure 4-28, and Figure 4-32. A magnified portion of these graphs
for 0', 10, and 2' root angles of attack is provided in Figures 4-21, 4-25, and 4-29. The
data is also presented in V-g plot format, which is provided in Figure 4-22, Figure 4-23,
Figure 4-26, Figure 4-27, Figure 4-30, Figure 4-31, Figure 4-33, and Figure 4-34.
119
Table 4.6: Final natural frequencies from wing analysis
Mode
Calculated Frequency
Measured Frequency
% Error
Mode Shape
1
6.44 Hz
6.4 Hz
0%
2
3
4
38.0 Hz
45.0 Hz
70.0 Hz
38 Hz
45 Hz
70 Hz
0%
0%
0%
2 "3 Bending
1 " Chordwise Bending
1 st Torsional
5
6
123 Hz
216 Hz
-
-
-
-
Bending
,d
Torsional
2
Speed = 0.00 m/s @ Freq = 6.44 Hz
1 st
Bending
3 rd
Speed = 0.00 m/s @ Freq = 38 Hz
-0.
Speed = 0.00 m/s @ Freq = 45 Hz
Speed = 0.00 m/s @ Freq = 70 Hz
0.2
0.4
0.6
0.8
1
Speed = 0.00 m/s @ Freq = 120 Hz
Speed = 0.00 m/s @ Freq = 208 Hz
0.2
0.2
0.4
0.4
-0.1
-0.05
0.6
0.6
0
0.8
0.8
1
Figure 4-19: Predicted mode shapes after final modifications of different wing properties
120
Root locus for Root AOA =
0*
140
120
48 44 4E)
BE)
2
10
0
H-
100
80 1T
U
Cr
a.)
U-.
60 F-
-
8
1CB
6
450
40
2B
20
K
iR - - 40:
0'
50
40
I
30
-I30
20:
I
20
10
Real Part of Root
13
---------
0
0
10
20
Figure 4-20: Root locus plot for 0' root angle of attack based on final modifications of
different wing properties
121
Root locus for Root AOA =
0*
45.7
45.6
45.5
45.4
-- -
------
1T
45.3
LL
--
-\0
45.1
45
---------
1CB
44.9
F-
44.8 F-------------
44.7
0. 5
I
0
I
I
0.5
1
Real Part of Root
I
1.5
2
Figure 4-21: Magnified root locus plot for 00 root angle of attack based on final modifications
of different wing properties
122
0 Degree Angle of Attack
120
100
80
N
I
C.)
0)
*
*
*
*
*
*
60 H
0~
0)
LL
401
**
20 -
*
*
0
0
*
*
10
20
30
40
Speed (m/s)
50
60
70
Figure 4-22: V-g (frequency part) for 00 root angle of attack based on final modifications of
different wing properties
123
0 Degree Angle of Attack
0.1
-*
0.05
*
*
%I,
-
*
*
*
0.0
-
**
-
-
-
-
-
-
-
-
-
-
*
*
* *
*
*
*
*
*
*
-
-
-
--W
IF
W
*
*
**
*
*
*
*
*
*
*
*
*
*
*
-0.1 F
*
*
*
*
*
-0.15
I
0
10
20
I
I
I
I
30
40
50
60
Speed (m/s)
Figure 4-23: V-g (damping part) for 00 root angle of attack based on final modifications of
different wing properties
124
70
1*
Root locus for Root AOA =
140
120
-------
100
-
- - - - - -- - -- -- 3B --- ------------------
----
-
I E)
0
--------
80 1T:
70
:3
U-
- -- - - -- - - -
60 F-
--
1CB
40 .
213
... .... ...... . .
-0
20
H4-----
-----
--
-
-
-
--
-
-
-
-
-z_
%j
01
50
40
30
20
10
Real Part of Root
--
--
-Iv
-- -
-V
0
10
20
Figure 4-24: Root locus plot for 10 root angle of attack based on final modifications of
different wing properties
125
Root locus for Root AOA =
1*
50
48
-
46
--
--------
1CB
44
--
42 1-- - -
---------C
LL~
-
38
-- - - -- --28
,10
-----
36 1-
34
H
---
---
6
4
-- --
32 F -
30'
10
8
2
0
2
4
6
8
10
Real Part of Root
Figure 4-25: Magnified root locus plot for 10 root angle of attack based on final modifications
of different wing properties
126
1 Degree Angle of Attack
120
100
80
I
C.,
*
*
*
*
*
*
*
*
10
20
601-
U-
40
20 F
0
0
30
40
Speed (m/s)
50
60
70
Figure 4-26: V-g (frequency part) for 10 root angle of attack based on final modifications of
different wing properties
127
1 Degree Angle of Attack
0.1
0Y
*
*
0
**
*
E
*
*
*
*
*
*
-0.1 I*
-0.2
0
10
*
20
30
40
Speed (m/s)
50
*
*
60
Figure 4-27: V-g (damping part) for 10 root angle of attack based on final modifications of
different wing properties
128
70
Root locus for Root AOA =
2*
140
-
--
120 h
-- --- - --- - ---- -3 & ----
-3E)
-3-
-2E)3le20
100
80 l
---- ---
--------
- - - - - - - - - - -- - - -- ------.0
- - - -
Cr
U-
1CB
40 r
20
-0--
1B:
40
0
50
2B
-
40
30-20
30
10
20
10
Real Part of Root
0
0
10
20
Figure 4-28: Root locus plot for 20 root angle of attack based on final modifications of
different wing properties
129
Root locus for Root AOA =
20*
50
40
1CB
''04
45 F-
- - -- - - - - - -
- - --
F-
-- - - - - - --- - -
--
-
--- -
---- -
--0
20
2B
N
V
U)
6
-
35
0~
U)
30 k
-
25 k
20'
10
8
6
--
----------------
4
-
2
0
2
4
6
8
10
Real Part of Root
Figure 4-29: Magnified root locus plot for 2' root angle of attack based on final modifications
of different wing properties
130
2 Degree Angle of Attack
120
100
80
-
-
I
Cr
a)
601
U)
40,
*
*
*
**
201
****
&
0
0
10
20
40
30
50
60
70
Speed (m/s)
Figure 4-30: V-g plot (frequency part) for 2' root angle of attack based on final modifications
of different wing properties
131
2 Degree Angle of Attack
0.2
0.15 -
0.1
*
0
-*
0.05
*
0)
E
0
**
-
*
-0.05
**
-0.1
-0.15
0
10
20
30
40
50
60
Speed (m/s)
Figure 4-31: V-g plot (damping part) for 2' root angle of Attack based on final modifications
of different wing properties
132
70
Root locus for Root AOA =
50*
140
120
F
100
F
--
-
6
*
1T:
C
U-
0
60F
1CB
2B
4
---20 -- - - - - -- --
40
0
50
- - ----------- ---- - -
20:
30
40
30
10
20
Real Part of Root
-
-
-
-- - -
1B
10
0
0
10
20
Figure 4-32: Root locus plot for 5' root angle of attack based on final modifications of
different wing properties
133
5 Degree Angle of Attack
120
100 h
80 I
N
I
*
*
*
*
C.)
a) 60 0)
U-
40
*
***
*
**
**
**
*
20P
0
0
*
*
10
20
30
Speed (m/s)
40
*
50
60
Figure 4-33: V-g (frequency part) for 50 root angle of attack based on final modifications of
different wing properties
134
5 Degree Angle of Attack
0.1
0.05 -
*
*
*
*
0
0
E
::
*
.
-0.05
**
-*
-0.1
-0.15
**
0
10
30
20
40
50
Speed (m/s)
Figure 4-34: V-g (damping part) for 50 root angle of attack based on final modifications of
different wing properties
The strange behaviour observed at 10 root angle of attack in the stiffness adjustments
has shifted and is now occuring at 2'. By plotting the data in V-g plots, it was determined
that the wing is experiencing hump flutter at the lower speed. The mode would not be seen
experimentally, as a very small structural damping within the system would have negated
the weak flutter seen at the first crossing of the instability axis. The second crossing, at
51.8 m/s for 20 angle of attack, is a much stronger flutter and would manifest itself during
experimentation. The predicted flutter speeds for this wing are provided in Table 4.7.
135
60
Table 4.7: Predicted flutter speeds with final model adjustments
Angle of Attack
10
20
Flutter Speed
42.5 m/s
51.8 m/s
50
42.7 m/s
The curve for flutter speed versus root angle of attack is provided in Figure 4-35. The
solid line represents the final predicted flutter speed. The dotted line shows the speed of
the first crossing of the instability axis, or the hump mode speed, and is ignored for reasons
mentioned above. The three individual points represent the maximum wind tunnel tested
speeds.
4.3
Wind Tunnel Tests
The experimental portion of this thesis culminated with the wind tunnel tests. A photo of
the wing in the tunnel is provided in Figure 4-36. Once the set-up was completed, calibration
tests were performed before starting the actual tunnel tests. During the wind tunnel tests,
four strain gauges were monitored-root bending, root torsion, forward/aft (located at 45%
of span) and another bending (located at 35% of span).
4.3.1
Wing Calibration in the Wind Tunnel
As with the bench-top tests, a calibration of the strain gauges with the wing in the tunnel was
performed to determine the deflection-to-voltage ratio of the bridges as different conditioners
were being used. The deflection was produced in a slightly different manner for the tunnel
tests. Weights were suspended from the wing at various locations along the chord at 2/3 the
span of the wing (Figure 4-37).
Three different tests were performed with these weights, which were suspended from the
wing via a hanger (17 gm). The first test used 217 gm (2.13 N) from the wing at points
along the wing chord. This test was to locate the elastic axis. The location of the elastic axis
136
60
1
1
1
predicted flutter speeds
-e- predicted flutter speeds (disregarding hump instability)
* highest wind tunnel tested speed
-0-
55-
5045
E
U)
40-
'0
35-
'0.*
o
30'
0
1
2
0
3
4
Root Angle of Attack (deg)
5
6
Figure 4-35: Predicted flutter speed for various root angles of attack
137
7
Figure 4-36: Wing set-up in the Wright Brothers wind tunnel
138
Figure 4-37: Set-up for hanging weights to calibrate strain gauges
139
was determined to be at 73% of the chord from the leading edge. This result contradicts the
analytical expectation of 30% of the chord. This is due to the location of the strain gauge
used to calculate this result. The root torsion gauge was used and is located about 10 mm
from the clamp. The area near the clamp is subjected to end effects and does not behave
according to St. Venant's principle. If the gauge was located at least a chord-length away
from the clamp, the results would more accurately reflect the analytical value. A plot of the
resulting strain gauge output is provided in Figure 4-38.
0.05
0.04
0.03
a,
0
0.02
0
0
0
0.01
0
0.01
0.02
0
100
200
300
400
500
600
700
Time (s)
Figure 4-38: Time trace for elastic axis determination
The second test was to hang weights(3.109 N)-
117 gm (1.147 N), 217 gm (2.128 N), and 317 gm
at 50% of the chord to calibrate the strain gauge output due to bending. The
locations of the leading and trailing edges at the wing tip were recorded for each weight as
well. This was accomplished by first measuring the height of the unloaded wing tip (leading
140
and trailing edge) and then the height for all subsequent loadings. A time trace of the strain
gauge data is provided in Figure 4-39.
7
6
5
0
324-
2
00
50
100
150
200
250
Time (s)
Figure 4-39: Time trace for wing loaded at 50% chord in wind tunnel
Finally this second test was repeated, except the weights were hung from 90% of the
chord. This provided a second calibration for the gauges. A time trace of this data is shown
in Figure 4-40.
A calibration curve for the root bending gauge as a function of voltage-to-bending moment
at the root is provided in Figure 4-41. A curve for the root torsion gauge depicting the
voltage-to-pitch moment at the root relation is given in Figure 4-42. To get the data from
both loading cases to agree, the moment was found about an axis at 63.81% chord line.
A curve for the other bending gauge provides the voltage-to-bending moment at 35% span
(gauge location) and is shown in Figure 4-43.
141
6
5
0
CC
350
0
50
Time (S)
wing loaded at 90% chord in wind tunnel
Figure 4-40: Time trace for
142
Root Bending Gauge: Weight at 50% and 90% Chord
6
5
y = 2.1035 x + 0.47736
0,
C
-a I-
0
I-
0 2
ME
1 -
0-
0
0.5
1
1.5
Bending Moment at Gauge (N*m)
2
2.5
Figure 4-41: Calibration of root bending gauge from both 50% and 90% chord load cases
143
Root Torsion: Weight at 50% and 90% Chord
-0.015
-0.016-
-0.017-
-0.0180)
y = -0.138 x + -0.022918
Cz
-0.019C
0
-0.02-
0
0
-0.021 -
-0.022-
-0.023-
-0.024'
-0.0 5
I
-0.045
I
I
-0.04
I
-0.035
I
I
I
-0.02
-0.03
-0.025
-0.015
Pitch Moment at Gauge (N*m)
I
I
-0.01
-0.005
Figure 4-42: Calibration of root torsion gauge from both 50% and 90% chord load cases
144
0
Another Bending Gauge: Weight at 50% and 90% Chord
-0.15
-0.16 F
*
-0.17 F
0
C
a>)
-0.18 F
m,
0)
C
>
-0.19 F
y = -0.038802 x + -0.15481
-0.2 F
-0.21
0
0.2
0.4
0.8
0.6
Bending Moment at Gauge (N*m)
1
1.2
Figure 4-43: Calibration of other bending gauge (located at 35% span) from both 50% and
90% chord load cases
145
1.4
In addition to these static tests, the tap tests were also repeated. From these tap tests,
the frequencies of the wing were confirmed with the new mounting set-up to be the same as
in the bench, with the addition of three unexpected frequencies detected. A 15-Hz chorwise
bending mode was introduced, due to flexibility of the load cell mounting. A mode at 20
Hz is present in most of the tunnel data, this is also believed to be associated with the
flexibility of the load cell mounting. Another possibility is that some larger motors produce
a 20-Hz dynamic disturbance. The 60-Hz mode is often found in dynamic data and is due to
electricity noise. Data from the undisturbed system is provided in Figure 4-44. The 100 Hz
dynamics is also believed to be either noise from the tunnel environment, as its magnitude
is quite large, even in the undisturbed wing data. The graphs for the bending, torsion, and
forward/aft taps are provided in Figure 4-45 through Figure 4-47.
The data from the root bending gauge shows the expected 6-Hz 1" bending mode and
the 38-Hz
2 nd
bending mode. It also shows significant dynamics at 15 Hz and 20 Hz.
The data from the root torsion gauge shows the 6-Hz lst bending mode and the 38-Hz
2 nd
bending mode again, but also shows the 70-Hz 1 " torsion mode. Again, an unexpected
dynamics at 15 Hz, 20 Hz, and 100 Hz can be seen from the data.
The data from the forward/aft strain gauge shows the first two bending modes as well,
and also shows the 45 Hz 1 " chordwise bending mode. The dynamics at 15 Hz was again
detected.
4.3.2
Wind Tunnel Tests
With the natural frequencies of the wing confirmed, the wind tunnel tests were conducted.
The first wind tunnel test was to determine the lift curve for the wing. The wing was then
flown at three different root angles of attack (10, 2', and 50). The starting speed for each
of these root angles of attack tests was 20.1 m/s (45 mph) with the upper limits varing
according to the behavior of the wing.
146
10
10
10
N 10
E
CMJ
z
)
10-
10
10-
-
10
0
10
20
30
40
50
60
Frequency (Hz)
70
80
90
100
Figure 4-44: Other bending gauge readings from undisturbed wing mounted in tunnel
147
110
10
102
103
10
I
E_
C\j 10-
z
c.
C)
U)
10
108-
10-9
L -iiI
10-9 -
0
10
20
30
40
60
50
Frequency (Hz)
70
80
90
100
Figure 4-45: Root bending gauge readings from tap test-wing mounted in tunnel
148
110
10-3
10--
10
N
c'J
E
z
ci)
1 -6
10
10-
10-0
10
20
30
40
60
50
Frequency (Hz)
70
80
90
100
Figure 4-46: Root torsion gauge readings from tap test-wing mounted in tunnel
149
110
10-3
10
10--
Cn
10
_
10-
10-
0
10
20
30
40
50
60
Frequency (Hz)
70
80
90
100
Figure 4-47: Forward/Aft bending gauge readings from tap test-wing mounted in tunnel
150
110
Wing Lift Curve
The lift curve was generated by flying the wing at 20 m/s (33.6 mph) and incrementing
the angle of attack from 00 to 12'. The lift generated was monitored through the load cell.
Both the theoretical and experimental lift curves are provided in Figure 4-48. The symbols
represent the experimental data and the solid line is the theoretical curve generated from
[3]. In addition to the calibration matrix coefficient, the load cell data was multiplied by
a constant factor of 10. The exact reason for this multiplier is unknown to the author. A
further calibration of the load cell is necessary to properly identify the correct calibration
term.
30-
y
3*x+1.4
25' 20S15 -theory
10-
experiment
*
linear fit
5--
00
0
1
2
3
4
5
6
Root Angle of Attack (deg)
7
8
9
10
residuals
6-
-
42 -0
U
-2
-4-
Linear: norm of residuals =8.3112
-60
1
2
3
4
5
6
Figure 4-48: Lift curve for wing
151
1
7
1
8
9
1 d10
1' Root Angle of Attack Tests
The 1' test were conducted from 20.1 m/s (45 mph) to 36.7 m/s (82 mph). Since this was
the first round of tests, the flight speed was not driven up near flutter to ensure that data
would be collected for all three angles of attack.
The data from the sensors at 20.1 m/s (45 mph) show many of the natural frequencies
and is provided in Figure 4-49 through Figure 4-52. This data was processed through a
Welsh averaging scheme with a 125 data point window and an overlap of 63 data points. At
a minimum, all flight data was processed through this scheme.
10 1
10
10
10
H
I
I
I
I
30
40
I
I
I
I
I
70
80
90
100
-2
-3
-4
I
C'
E
CMJ
z 10~
CD)
10
107
108
0
10
20
60
50
Frequency (Hz)
110
Figure 4-49: Frequency plot from root bending gauge at 20.1 m/s for 10 root angle of attack
152
103
10
E
z
0
C/)
-6
10-
-
10-
108
0
10
20
30
40
50
60
Frequency (Hz)
70
80
90
100
110
Figure 4-50: Frequency plot from root torsion gauge at 20.1 m/s for 1' root angle of attack
153
10
10--
10-N
N
U 106
10-
10-8
0
10
20
30
40
50
60
Frequency (Hz)
70
80
90
100
110
Figure 4-51: Frequency plot from forward/aft gauge at 20.1 m/s for 1 root angle of attack
154
10-3
10
10
1~
10-
E
10-5-
10-6
10
10-
10100
10
20
30
40
60
50
Frequency (Hz)
70
80
90
100
110
Figure 4-52: Frequency plot from the other bending gauge at 20.1 m/s for 1 root angle of
attack
155
The data from the sensors at 36.7 m/s (82 mph) is noisier and the natural frequencies
are harder to detect. These results are shown in Figure 4-53 through Figure 4-56.
10
10-2
10-3
'N
10
E
z
-5
al 10
-6
10
10~
0
10
20
30
40
50
60
Frequency (Hz)
70
80
90
100
110
Figure 4-53: Frequency plot from root bending gauge at 36.7 m/s for 1 root angle of attack
156
103
10
E
N
'.
C')
10~--
10-
0
10
20
30
40
50
60
Frequency (Hz)
70
80
90
100
110
Figure 4-54: Frequency plot from root torsion gauge at 36.7 m/s for 10 root angle of attack
157
10-3
10
\I
10-5-
10-6
10-
10II|1
0
10
I
20
30
40
I
I
I
60
50
Frequency (Hz)
70
80
Figure 4-55: Frequency plot from forward/aft gauge at 36.7 m/s for 1
158
90
100
110
root angle of attack
10-3
104
-
10-
-6
10
E
z
10
108
10-
I
0
10
I
20
I
30
I
40
I
50
I
60
I
I
70
80
I
90
100
110
Frequency (Hz)
Figure 4-56: Frequency plot from the other bending gauge at 36.7 m/s for 10 root angle of
attack
20 Root Angle of Attack Tests
The 2' tests were conducted from 20.1 m/s (45 mph) to 36.7 m/s (82 mph) and then from
20.1 m/s (45 mph) to 52.3 m/s (117 mph) after the 5' case was completed.
The natural frequencies at 20.1 m/s were fairly straight forward to determine. The added
turbulence from the wind tunnel did not interfere with this 10 s data collect, so this data did
not require extensive processing or filtering. The time trace from the root bending gauge is
provided in Figure 4-57 to show how clean the data looks.
159
-2.5
-3-
E
CMJ
z
CD
m--3.5
CO
C
E
0
2
-4-
C
-4.5-
-5'
0
1
2
3
4
5
Time (s)
6
7
8
9
Figure 4-57: Time trace of data from root bending gauge at 20.1 m/s for 2' root angle of
attack
160
10
The data from the four strain gauges with the aforementioned averaging technique are
provided in Figure 4-58 through Figure 4-61.
10
102
103
104
N
E
N,~
z
0n
10
a-)
106
10
10
0
10
20
30
40
60
50
Frequency (Hz)
70
80
90
100
110
Figure 4-58: Frequency plot from root bending gauge at 20.1 m/s for 2' root angle of attack
161
103
10
10-5
E
z
CMJ
10~
10-
0
10
20
30
40
50
60
Frequency (Hz)
70
80
90
100
110
Figure 4-59: Frequency plot from root torsion gauge at 20.1 m/s for 2' root angle of attack
162
10-3
10-~
10
-7
U)10-6-
10
10-8
0
10
20
30
40
60
50
Frequency (Hz)
70
80
90
100
110
Figure 4-60: Frequency plot from forward/aft gauge at 20.1 m/s for 2' root angle of attack
163
10
10
10
-6
10
E
Cvi
z
0 10
-
108
10-
0
10
20
30
40
60
50
Frequency (Hz)
70
80
90
100
110
Figure 4-61: Frequency plot from the other bending gauge at 20.1 m/s for 20 root angle of
attack
164
All four bridges detected a 20-Hz disturbance as well as one at 37-Hz. The additional
bending gauge only slightly detected the 37 Hz signal. A summary of which gauge detected
which frequencies is provided in Table 4.8.
Table 4.8: Frequencies detected from strain gauges at 20.1 m/s for 2' root angle of attack
Gauge
6 Hz
20 Hz
37 Hz
Root Bending
Root Torsion
Chordwise Bending
Additional Bending
X
X
X
X
X
X
X
X
X
70 Hz
100 Hz
X
X
X
A data set of importance from the 2' root angle of attack tests is at 52.3 m/s. Through
visual observation of the wing behavior, it is believed that the wing was operating just below
the flutter speed. Two root torsion gauge time traces of the wing operating at this speed are
provided in Figure 4-62 and Figure 4-63. The data from the sensors is provided in Figure 4-64
through Figure 4-66.
165
2.2
2-
1.8C'J
E
N'.
z
" 1.6 -
~1.4
E
0
0) 1.2
C,
0.8
0.6'
0
1
2
3
4
5
Time (s)
6
7
8
9
Figure 4-62: Time trace of data from root torsion gauge at 52.3 rn/s for 2' root angle of
attack
166
10
2
1.8
C\J
E
N 1.6
z
Cz
1.4
CD
E
0
~1.2
CL
1
0.8
0
1
2
3
4
5
Time (s)
6
7
8
9
Figure 4-63: Time trace of data from root torsion gauge at 52.3 m/s for 2' root angle of
attack
167
10
10-3
10-4N
z
C,)
IIf
V
I
iiJ
10-
0
10
20
30
40
60
50
Frequency (Hz)
70
80
90
100
110
Figure 4-64: Frequency plot from root torsion gauge at 52.3 m/s for 2' root angle of attack
168
10~3
10
10--
10-6
10
10--
0
10
20
30
40
60
50
Frequency (Hz)
70
80
90
100
110
Figure 4-65: Frequency plot from forward/aft gauge at 52.3 m/s for 2' root angle of attack
169
103
10--
10--
N
-6
10
E
z
10-
10
10--
|
0
10
|
20
I
30
40
I
I
I
I
70
80
90
100
I
1
60
50
Frequency (Hz)
110
Figure 4-66: Frequency plot from the other bending gauge at 52.3 m/s for 2' root angle of
attack
50
Root Angle of Attack Tests
The 50 tests were conducted from 20.1 m/s (45 mph) to 33.5 m/s (75 mph). The wing at
this root angle of attack produced very large tip deflections.
Therefore, the wing was not
flown up to 36.7 m/s (82 mph) as originally desired because of fear that the wing would
break and damage the tunnel.
The data from a 20.1-m/s case is provided in Figure 4-67 through Figure 4-69. The data
from the root bending gauge is not provided as the signal exceeded the recording limits.
170
10-3
10--
10-5
E
z
0
Cl)
10-
0
10
20
30
40
60
50
Frequency (Hz)
70
80
90
100
110
Figure 4-67: Frequency plot from root torsion gauge at 20.1 m/s for 50 root angle of attack
171
10-3
-
10~4
10-
10
10--
10 -
0
10
20
30
40
60
50
Frequency (Hz)
70
80
90
100
110
Figure 4-68: Frequency plot from forward/aft gauge at 20.1 m/s for 50 root angle of attack
172
10
10
10
N
1-6
110
E
10
10-8
10
0
I
I
I
I
10
20
30
40
I
I
50
60
Frequency (Hz)
I
I
70
80
I
90
110
100
Figure 4-69: Frequency plot from the other bending gauge at 20.1 m/s for 50 root angle of
attack
The data from the 33.5-m/s (75-mph) case shows some interesting frequency results.
This data is provided in Figure 4-70 through Figure 4-72. There are three dynamics present
in the root torsion gauge that were not targeted as noise from the tunnel environment:
34
Hz, 42 Hz, and 46 Hz. The forward/aft gauge and the bending gauge at 35% chord have
dynamics at 34 Hz, 38 Hz, and 40 Hz. This range of frequencies is where the root locus plot
(Figure 4-32) predicts both the
2 nd
bending and the 1 st chordwise bending to be interacting.
173
10- 3
10--
E
z
C,)
10-
10-
0
10
20
30
40
50
60
Frequency (Hz)
70
80
90
100
110
Figure 4-70: Frequency plot from root torsion gauge at 33.5 m/s for 5' root angle of attack
174
10-3
10
10-N
\I
U)
10
1
6
10--
108
0
10
20
30
40
50
60
Frequency (Hz)
70
80
90
100
110
Figure 4-71: Frequency plot from forward/aft gauge at 33.5 m/s for 50 root angle of attack
175
10-3
10--
10-
S-6
10
E
z
o
-7
10
10
101-
10
0
20
30
40
50
60
Frequency (Hz)
70
80
90
100
110
Figure 4-72: Frequency plot from the other bending gauge at 33.5 m/s for 5' root angle of
attack
Post-Tunnel Tap Tests
After the tests in the tunnel were completed, a follow-up tap test was conducted to determine
if anything changed during flight. The results from a bending excitation tap are provided in
Figure 4-73. This tap test captured the 6-Hz 1 " bending mode, the 38-Hz
the 45-Hz
1
" chordwise bending mode, and the 70-Hz
1
2 nd
bending mode,
" torsion mode. The torsion mode
was excited through a twisting tap. The data from the torsion strain gage is provided in
Figure 4-74. This graph shows the 60-Hz 18' bending mode and the 70-Hz
1
" torsion mode.
A mode at 100 Hz is also being excited. The forward/aft tap test is not provided as the tap
176
10
10-2
10
10-N
E
z
10
0
10-6
10
10
0
10
20
30
40
50
60
Frequency (Hz)
70
80
90
100
Figure 4-73: Root bending gauge readings from tap test-wing mounted in tunnel
177
110
was not recorded at the correct time. However, the chordwise bending mode was seen in the
bending tap. The chordwise bending mode was excited through a tap on the leading edge.
10-3
E
z
10
100
10
20
30
40
60
50
Frequency (Hz)
70
80
90
100
110
Figure 4-74: Root torsion gauge readings from tap test-wing mounted in tunnel
The data from the forward/aft gauge is shown in Figure 4-75. This data yields the 45 Hz 1 1'
chordwise bending mode. This dynamic is masked quite a bit by the noise from the tunnel,
especially the 15-Hz and 60-Hz signal.
178
10
10-2
10-3
,.0
C')
10~4
10-
0
I
I
10
20
I
30
40
I
iII
I
50
60
Frequency (Hz)
70
80
90
100
Figure 4-75: Average tip accelerometer readings from tap test-wing mounted in tunnel
179
110
Chapter 5
Concluding Remarks
This chapter presents a summary of the findings from the analytical and experimental portions of this thesis. Along with the summary are sections on the conclusions drawn from the
research as well as recommendations for future work. The recommendations section suggests
improvements to the manufacture of the wing and the test set-up.
5.1
Summary
The aeroelastic behavior of high-aspect ratio wings was studied via a new numerical formulation [11] which encompasses a non-linear, large deflection beam model with strain-based
finite element representation and finite-state unsteady airloads. Using this formulation, a
wing was designed, built, and its structural behavior evaluated under a variety of conditions.
Bench-top tests were performed to understand the structural dynamic behavior of the wing.
The first four natural frequencies at zero wind speed measured during bench-top 'tap' testing
were in good agreement with the analytical model. The wing was then tested within the
Wright Brothers Wind Tunnel to further understand the changes in dynamic behavior due
to flight speed and root angle of attack. The wing was flown at speeds ranging from 20 m/s
to 51.8 m/s and at 1*, 2 , and 50 root angles of attack. The tip deflections and natural
frequencies were recorded for these cases.
180
5.2
Conclusions
The results presented herein validate the analytical model of the wing's aeroelastic behavior
under a variety of angle of attacks and wind speeds. Few parameter changes were introduced
to the model to reproduce the actual dynamic behavior. The need for these changes can be
associated with complexities resulting from the manufacturing process. With these changes
implemented, the predicted flutter speed is within 99% of the estimated flutter speed from
observations in the wind tunnel. Experimental validation of the analytical model provides
confidence in the ability of the code to predict the behavior of other systems under similar
conditions. This capability would be useful for predicting the theoretical performance of
high aspect ratio wings currently under consideration for upcoming remote sensing and
reconnaissance vehicle programs.
5.3
Recommendations for Future Work
The manufacture and test of the active wing presented in Appendix C would further validate the capabilities of the analysis code. This would also demonstrate the capabilities of
embedded piezoelectric actuators (in the composite wing skin) to increase the flight envelope
of HALE vehicles. Note that a stress analysis, similar to the one conducted for the passive
wing, will still have to be performed for the active wing design.
Several areas of improvement in the wing manufacturing and the test set-up have been
identified that would facilitate future evaluations. First, although it would be more difficult,
the foam core should be manufactured as a single piece, rather than four seperate ones.
This would provide for more uniform cross-sectional properties.
Second, the use of tip
accelerometers was of limited value and could have been eliminated without a large impact
on the results. The omission of the accelerometers would aleviate one of the reasons for
using four seperate pieces of foam for the core-there would no longer be a need to run wires
through the center of the wing. Omitting the accelerometers also helps with the homogeneity
of the cross section. Finally, the load cell added unwanted noise into the data and had little
practical value in the experiment. If a lift curve is desired it could be attained via a separate
181
set-up. During the dynamic tests of the wing, where mode shapes and frequencies are
important, the flexibility of the load cell mounting fixture added undesirable dynamics to
the system.
182
References
[1] Patil, M. J., D. H. Hodges, and C. E. S. Cesnik, "Nonlinear Aeroelasticity and Flight
Dynamics of High-Altitude Long-Endurance Aircraft," Journal of Aircraft, Vol. 38,
No. 1, 2001, pp. 88-94.
[2] Patil, M. J., D. H. Hodges, and C. E. S. Cesnik, "Characterizing the Effects of Geometrical Nonlinearities on Aeroelastic Behavior of High Aspect-Ratio Wings," in Proceedings
of the InternationalForum on Aeroelasticity and Structural Dynamics, 1999.
[3] Cesnik, C. E. S. and E. L. Brown, "Modeling of High Aspect Ratio Active Flexible
Wings for Roll Control," in Proceedings of AIAA/ASME/ASCE/AHS/ASC
Structures,
Structural Dynamics, and Materials Conference and Exhibit, 43rd, AIAA-2002-1719,
2002.
[4] Ortega-Morales, M. and C. E. S. Cesnik, "Modeling and Control of the Aeroelastic
Responce of Highly Flexible Active Wings," Tech. Rep. AMSL
#
00-2, Massachusetts
Institute of Technology, February 2000.
[5] Patil, M. J., D. H. Hodges, and C. E. S. Cesnik, "Nonlinear Aeroelasticity Analysis of
High Aspect Ratio Wings," in Proceedings of the 39th Structures, Structural Dynamics,
and Materials Conference, 1998.
[6] Cesnik, C. E. S., D. H. Hodges, and M. J. Patil, "Aeroelastic Analysis of Composite Wings," in Proceedings of the 37th Structures, Structural Dynamics, and Materials
Conference, 1996.
183
[7] Minguet, P. and J. Dugundji, "Experiments and Analysis for Composite Blades Under
Large Deflections Part II: Dynamic Behavior," AIAA Journal, Vol. 28, No. 9, 1990,
pp. 1580-1588.
[8] Patil, M. J. and D. H. Hodges, "On the Importance of Aerodynamic and Structural
Geometrical Nonlinearities in Aeroelastic Behavior of High-Aspect-Ratio Wings," in
Proceedings of AIAA/ASME/ASCE/AHS/ASC
Structures, Structural Dynamics, and
Materials Conference and Exhibit, 41st, AIAA-2000-1448, 2000.
[9] Tang, D. and E. H. Dowell, "Experimental and Theoretical Study on Aeroelastic Response of High-Aspect-Ratio Wings," AIAA Journal, Vol. 39, No. 8, 2001, pp. 14301441.
[10] Tang, D. and E. H. Dowell, "Experimental and Theoretical Study on Gust Response of
High-Aspect-Ratio Wings," AIAA Journal,Vol. 40, No. 3, 2002, pp. 419-429.
[11] Brown, E. L., Integrated Strain Actuation in Aircraft with Highly Flexible Composite
Wings, Ph.D. thesis, Massachusetts Institute of Technology, January 2003.
[12] Rodgers, J. P., Development of an Integral Twist Actuated Rotor Blade for Individual
Blade Control, Ph.D. thesis, Massachusetts Institute of Technology, October 1998.
[13] Juhl, B., "Roehm America, Inc. - Rohacell." email, 2002. personal communication.
[14] Peters, D. A. and M. J. Johnson, "Finite-State Airloads for Deformable Airfoils on Fixed
and Rotating Wings," in Symposium on Aeroelasticity and Fluid/StructureInteraction,
Proceedings of the Annual Winter Meeting, November 1994.
[15] Peters, D. A., S. Karunamoorthy, and W. M. Cao, "Finite State Induced Flow Models
Part I: Two-Dimensional Thin Airfoil," Journalof Aircraft, Vol. 32, No. 2, 1995, pp. 313322.
[16] Jones, R. M., Mechanics of Composite Materials, Hemisphere Publishing Corporation,
1975.
184
Appendix A
Mathematical Formulation
A.1
Introduction
This appendix provides a summary of the mathematical modeling used in this research.
A brief description of the geometrical, structural, and aerodynamic issues is presented. A
detailed description of the mathematical formulation can be found in [3] and [11].
The main features of this code are:
" nonlinear, large deflection beam model ,
" strain-based finite element representation , and
" finite-state unsteady airloads .
A.2
Geometrical Lay-out
There are three reference frames governing the wing and its interactions with the environment. The first being the global coordinate system through which the wind speed and gravity
affect the wing. The undeformed reference frame of the wing is measured with respect to the
cross section area centroid along the 30% chord line of the wing. Finally the deformed reference frame follows the deformation of the beam reference line. A diagram of these coordinate
frames with respect to the wing is provided in Figure A-1.
185
UY(s)
p(s)
x
Figure A-1: Coordinate systems for the wing
A.3
Structural Formulation
The structural analysis of the wing is based on the principles of virtual work. These principles
yield the following general equation:
6W = fy(f(x, y, z)6u(x, y, z))dV
where 6W is the virtual work done by a force, f(x, y, z), to move an object a virtual distance,
6u(x, y, z). Separating this equation into individual forces yields the general equation:
6W
=
F1uou + F2,6u + -- -+ Funu
For an object to be in equilibrium, the above equation must be zero, that is, 6W
=
0.
Within the program, the wing is allowed to move through three-dimensional bending and
twisting deformations, while extension and shear deformations are assumed negligible and
thus ignored. With this assumption, the virtual work equation becomes [3]:
6W =
In this equation,
opT
=o(6pTF
+ 6)M
6kTMintjdu
is the change in the position of the reference line, p(s), see Figure A-1.
The term 66f represents the virtual rotation about the reference line, and JT denotes the
virtual curvature of the wing. The variable F stands for the external forces on the wing, M
186
represents the external moments, and M"' represents the internal moments acting on the
wing.
When the vector q is defined to be the curvatures,rs in (x,y,z) and the variable R is defined
as all the generalized loads, the equation becomes [3]:
SW = 6qT (-Mqa
- Cqe - Kqq + Rq)
In this equation, Mq denotes the mass matrix, Cq symbolizes the damping matrix, and Kq
is the stiffness matrix. Setting this system into equilibrium, yields:
Mqi + Cqj+ Kqq = R
The stiffness matrix, Kq, is defined as:
K,
=
Ku1
K
12
K
13
K
14
K
K
12
K
22
K 23
K
24
13
K
23
K33
K 34
14
K
24
K
K
K
34
44
where K 11 is the extension stiffness, K 22 is the torsional stiffness, K 33 is the bending stiffness
about y, and K 44 is the bending stiffness about z (or the chordwise bending stiffness). The
off-diagonal terms are the cross-coupling terms, e.g., K 14 is the extension-chordwise-bending
stiffness term.
A.4
Aerodynamic Formulation
The aerodynamic model used in this aeroelastic formulation comes from the theory derived
in [14] and [15]. A full discussion on this is provided in [11] and [3]. The theory accounts
for thin deformable airfoil cross section undergoing large deformations at subsonic speeds.
It also includes small deformations about this large deflected state.
The lift, L, moment, M, and drag, D, are defined by the following equations:
187
Ao] - lb2 - 'bd&)
L = 2rpb(p[({b - d)d M
2rpb(Q[-(d + lb)i - (d + !b)AO - d2 d] D
=
lbdz - {b(d 2 + 1b2 )d,)
-27rpb(z2 + d 2&2 + A2 + 2dds + 2AOs + 2ddAo)
Within these equations, b is the semichord, d is the distance between the reference axis and
the mid-chord, and Ao represents the inflow term due to free vorticity [3].
denotes the air density. The variables which represent the motion: y,
y, z,
i,
The term, p
z,
a, d, and &
are defined in Figure A-2.
z)z
-c,-a
Figure A-2: Variables defining airfoil motion [3]
These equations are linearized with respect to time, ta, by the following equations [3]:
Y
6-
= (Wa
+ AW'
(a
A61\)
o = (Ao,a + AAo)
Applying these perturbations to the equation of virtual work yields:
JW =
A.5
f O(Loz + D6y + M6a)ds
Aeroelastic Formulation
The structural and aerodynamic formulations combine to produce the aeroelastic equations
[3].
M4+Cq+Kq+DA=F
A = F 14+
F 2 4+ F3 A
188
In this equation, the mass, M, and damping, C, matrices are functions of the strains. The
structural stiffness, K, matrix is constant for a given geometry and material distribution.
The force vector, F, contain the inertial and aerodynamic forces and is a function of initial
values of t, q, and A. It also contains the effects of the active material layers in the wing
construction. See [11] for further details.
189
Appendix B
Material Properties
The material file associated with the MATLAB code needs material properties to be converted from engineering moduli (El, Et, vu, Gut) in the local ply coordinate system to the
plane-stress stiffness constants,
Q [16].
Equations B.1 through B.5 show the relations. The
subscripts '1' and 't' represent longitudinal and transverse directions, respectively. The two
Poisson's ratios are related via Equation B.6. The material properties for the foam core are
provided in the standard nomenclature.
Q11 =
Qi2
=
Q22
=
Q16
El
(B.1)
vEt
(B.2)
1 - Vit uti
1- vuura
1- vuti
= Q26 =
Q66 =
vU =
Gu
Eivut
190
0
(B.3)
(B.4)
(B.5)
(B.6)
Table B.1: Properties of the materials used in this study
E-glass/epoxy
Q11
Q12
Q22
Q66
19.7
2.92
19.7
4.10
p
GPa
GPa
GPa
GPa
309 pm/V
-129 pm/V
n/a
0.1143 mm
0.0011 m
0.127 mm
d12
thickness
33.6
7.54
16.6
5.13
GPa
GPa
GPa
GPa
n/a
n/a
di,
telectrode
Piezocomposite
1716 kg/m
3
4060 kg/m
3
Ei, e22
10000 pm/m
4000 pm/m
eY2
15000 pm/m
5500 pm/m
E
G
p
Ci122
e12
Rohacell 31
36 MPa
14 kPa
30 kg/m
3
27000 pm/m
30000 pm/m
191
Appendix C
Active Wing Design
This appendix describes the design and analysis of the active flexible wing that is proposed
as a follow-on to the passive wing built for this thesis. Stress analysis will still have to be
conducted for this design.
C.1
Active Design
The active wing design is composed of three layers of E-glass/epoxy fabric oriented at 0'
around an inner foam core, with two active layers embedded within the E-glass. There is
no spar in this design. The goal is to have a flexible wing with low flutter speeds, for the
reasons mentioned during the passive design. This design provides a flexible wing while still
maintaining closed cell properties. A drawing of the wing lay-up is provided in Figure C-1.
[0 /+
45
MFC/ 0 /- 4 5 MFC/ 0 ]
foam
Figure C-1: Cross section of active wing
192
C.1.1
Basic Static and Dynamic Properties
The cross-sectional properties for the active design are provided below. The stiffness matrix
terms for this cross section are given in Table C.1. Detailed definitions of the elements of the
matrix are provided in Appendix A. The inertia matrix for this cross-section is provided in
Table C.2. The center of gravity for the cross-section is located 0.0214 m aft of the reference
line (located at 30% chord). This places the center of gravity at 49.9% of the chord.
Table C.1: Non-zero stiffness matrix terms for the active wing design: 1 = Extension, 2
=
Torsion, 3 = Flatwise Bending, 4 = Chordwise Bending
Ku1
K 14 = K4 1
K22
K33
K44
2.27 * 106 N
-4.78 * 104 N*m
57.72 N*m 2
57.00 N*m 2
2.69 * 10 3 N*m 2
Table C.2: Non-zero inertial matrix terms for active wing design
Ill
'22
133
0.244 * 10-3 m 4
0.998 * 10~ m 4
4
0.234 * 10-3 m
The dynamic properties of the wing are important to its aeroelastic response. The first
six natural frequencies of the wing are summarized in Table C.3. The corresponding mode
shapes are provided in Figure C-2.
193
Table C.3: Natural frequencies of the active wing design in vacuum
Mode
Frequency
Mode Shape
1
2
6.8 Hz
36.8 Hz
" Bending
1 " Chordwise Bending
3
44.1 Hz
2 nd
Bending
4
121. Hz
1"
Torsional
5
6
133. Hz
232. Hz
Bending
2 "d Torsional
1
3 rd
Speed = 0.00 m/s @ Freq = 36.8 Hz
Speed = 0.00 m/s @ Freq = 6.8 Hz
-0.6
-0.4
Speed = 0.00 m/s @ Freq = 44.1 Hz
Speed = 0.00 m/s @ Freq = 121 Hz
0.2
-:1
0.4
0.6
0.8
1
Speed = 0.00 m/s @ Freq = 232 Hz
Speed = 0.00 m/s @ Freq = 133 Hz
0.2
-0.1
0.4
0.6
0.8
1
1
Figure C-2: First six normal modes for active wing design (in vacuum)
194
C.1.2
Wing Non-linear Characteristics
As seen in the passive wing designs, different root angles of attack of the wing result in
different wing tip deflections once the wing is exposed to airloads. The change in tip deflection
is graphed in Figure C-3 for the range of root angles of attack of interest: 1' to 5'. The tip
deflection increases with increased angle of attack.
0.18
0.16
0.14
0.12
0.1
C
0
0.08
.)
0-
0.06
0.04-
0.02-
0-0.02
-
0
5
10
15
25
20
30
35
40
45
Speed (m/s)
Figure C-3: Static tip deflection for increasing speed at different root angles of attack
The wing tip twist also changes with increased root angle of attack. This data is provided
in Figure C-4. To better illustrate this effect, the initial root angle of attack has been
subtracted from the total tip twist, yielding only the elastic angle change due to gravitational
and aero-loading.
195
50
0.22
0.8
.
--
--
0.18 ...
).
..
.
.
..
.
..... .
.
.
...
.
.
.
...
.
.
.
..
.
.
...
I0l2
0.1 -0.12----4
-.-
..
..
.
.
.
..
.
.
.
.--- 2-0--------.-.-.- .
--- -.----.
-
0.08 12
.
-
0.14--6
.
..
. .
-
-
.
... ..
.
..
0.0
0
5
10
15
20
25
30
35
40
45
Speed (mis)
Figure C-4: Elastic tip twist for increasing speed at different root angles of attack
196
50
The changes in tip deflection and twist result in a change in the dynamic behavior of the
wing, and ultimately a change in the wing's flutter speed. This is shown in the root locus
plots for the wing at root angles of attack of 10, 2' and 5'. The flutter speeds as a function
of root angle of attack determined from these plots are provided in Figure C-5.
90
80-
70-
6042)
E
50 -
_0
CD
(D
CL. 40 -
30
20
10
0
0
1
2
4
3
Root Angle of Attack (deg)
5
Figure C-5: Flutter speeds for active wing design
197
6
7
One-Degree Root Angle of Attack
The flutter speed for one-degree root angle of attack is determined to be 79.3 m/sec, shown
in Figure C-6. The mode of instability is the second mode. A magnified plot of this mode's
behavior is provided in Figure C-7. This mode is primarily the first chordwise bending.
However, the forces due to air pressure on the tilted wing result in some torsion being
present as well. The first six mode shapes at this speed are given in Figure C-8.
Root locus for Root AOA =
140 1
1
1
837576
60
so
4@ go -2e 13E)0
3B
120 F-
100
80
- ---- - - - - ---
-- -
---
T
- - -I
--- --
- -- ---- -- -
--
U
-
- ---
-- - - - - - - - -
------------0
60
2B
- --
40
20
-- - - - - - - - -- - - - - - - - - - --- ---1CB
-- -
-- -- ----
-- - - - - -- - - -- - - - - -- - - - --------1B
0'
50
40
30
20
10
0
10
20
Real Part of Root
Figure C-6: Root locus plot for 10 root angle of attack
198
Root locus for Root AOA =
1*
38
37.5 ---------
37 -----
1CB
--- - --
36.5 -- -----N
I
0
-.
.. .-...
36 - ------
--- ---
- - -- -
0*
0
LL
-
35.5 ----.-.-
--- --- - - 5
35
-- -- - -- - ---- - ----34.5.-
34'
0. 2
---.
0. 1
0
0.1
0.2
Real Part of Root
0.3
0.4
0.5
Figure C-7: Magnification of root locus plot for 1 root angle of attack
199
Speed = 79.30 m/s @ Freq = 35.5 Hz
Speed = 79.30 m/s @ Freq = 6.81 Hz
Speed = 79.30 m/s @ Freq = 44.1 Hz
Speed = 79.30 m/s @ Freq = 124 Hz
0.2
0.1)
0.2
0)
Speed = 79.30 m/s @ Freq = 232 Hz
Speed = 79.30 m/s @ Freq = 132 Hz
0.2
0.4
0.6
0
0.8
0.1
1
0.8
Figure C-8: Mode Shapes for active wing design at 1 root angle of attack and its corresponding flutter speed of 79.3 m/s
200
When the wing is flown at 10% above the flutter speed for a 10 angle of attack, the wing
enters into a Limit Cycle Oscillation (LCO). The tip deflections and the tip twist are plotted
for this case up to 1 second. These plots are given in Figure C-9 thru Figure C-11.
Speed = 87.23 m/s with a Time step = 0.0005 s
0.12
0.118
.-I.11
E
_)
E
0.114
U
0.112
0.108'
0
0.2
0.6
0.4
0.8
1
Time (s)
Figure C-9: Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (10 root angle of attack) for the active wing design
Two-Degree Root Angle of Attack
The flutter speed for two degrees angle of attack is determined to be 77.3 m/s from the root
locus plot shown in Figure C-12. The instability mode is the same as for 10, except the
amount of torsion present is increased. The drop in the second natural frequency due to the
increased upward bend, also causes the second mode to turn over faster and amplifies the
effects of flow over the wing. The first six modes at flutter speed are provided in Figure C-14.
201
Speed = 87.23 m/s with a Time step = 0.0005 s
3
2.5
2
0)
a,
~0
C',
1.5
H0~
r
1
0.51
0 0
0.2
0.4
0.6
1
0.8
Time (s)
Figure C-10: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(10 root angle of attack) for the active wing design
Speed = 87.23 m/s with a Time step = 0.0005 s
0.115
1
1
I
0.11 45 F
E
0.114
E
ND
0.
CU)
C0
0.1 135 I
0.1 13 -
0.1125'
-3.5
-3
-2.5
-2
Y Displacement (m)
-1.5
-1
x 10 3
Figure C-11: Nonlinear time simulation of the wing tip displacement motion at 10% above
its flutter speed (10 root angle of attack) for the active wing design
202
Root locus for Root AOA =
2*
140
0-i 3B
-
- ---
------------------
120 -
-I --- -
- - - - ---100 -
--
-- -- -- - --- ----- ------ - -- - -- -- -- -
80
---- ---
-------
U
Cr
4)
U_
60
-- ----
--
40
20 W00 2B
----
--
---
1CB
20 F
664
QL
50
40
30
3:
20
10
2S 1e 0 1B
0
10
20
Real Part of Root
Figure C-12: Root locus plot for 20 root angle of attack
203
Root locus for Root AOA =
2*
38
37 -
. ...... .. . -.
..
.
1CB
0
I
36 -
.
35 -
.--
N
0:
a, 34
0*
a,
U-
---- - --
33
--
32 -
-. .
..-... ..-. ..-.
5 ------ --- -- -. .. ..
31
30
1
- -- -- -
0. 8
0. 6
0. 4
0. 2
0
0.2
0.4
0.6
0.8
1
Real Part of Roc
Figure C-13: Magnification of root locus plot for 20 root angle of attack
204
Speed = 77.30 m/s @ Freq = 32.6 Hz
Speed = 77.30 m/s @ Freq = 6.81 Hz
Speed = 77.30 m/s @ Freq = 129 Hz
Speed = 77.30 m/s @ Freq = 44.1 Hz
0.15
0.1
0.05
-0.05
0
Speed = 77.30 m/s @ Freq = 232 Hz
Speed = 77.30 m/s @ Freq = 132 Hz
0
0.1
0.8
1
Figure C-14: Mode shapes for active wing design at 20 root angle of attack and its corresponding flutter speed of 77.3 m/s
205
For 20 angle of attack, the wing also enters a LCO when flown at 10% above the flutter
speed. The tip deflections and the tip twist are plotted for this case up to 3 seconds. These
plots are given in Figure C-15 thru Figure C-17.
Speed = 85.03 m/s with a Time step = 0.0005 s
0.3
0.25
a)
E
0)
0
0z
0.2
0.15
CL
0.1
0
CD)
0.05
0
-0.05'
0
0.5
1
1.5
Time (s)
2
2.5
3
Figure C-15: Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (20 root angle of attack) for the active wing design
Five-Degree Root Angle of Attack
The flutter speed for five degrees angle of attack is shown to be 68 m/s in the root locus plot
provided in Figure C-18. As with the 1 and 2' cases, the mode of instability is the second
mode, which is the 1 "t chordwise bending with some torsional effects. The changes in the
dynamic behavior due to the increased upward bend result in the second mode rolling over
faster to the positive dampening axis. The fourth mode (1st torsion) has curved upward and
now interacts with the fifth mode (3 rd bending mode). This interaction between the fourth
and fifth mode has caused the fifth mode to turn towards the instability line as well. The
first six mode shapes for this case are provided in Figure C-19.
206
Speed = 85.03 m/s with a Time step = 0.0005 s
8
6
4
CO
2
0
-2
-
-4
-
-6
-
0
0.5
1
1.5
Time (s)
2
2.5
3
Figure C-16: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(20 root angle of attack) for the active wing design
Speed = 85.03 m/s with a Time step = 0.0005 s
0.17
0. 160.15 a)
E
a)
0.
Cz
0.
An
E
N
0.14 0. 13 0. 12 0.11
0.1'
-0.0 4
'
-0.03
'-0.02
'
'
'-0.01
0.01
0
Y Displacement (m)
'-0.02
0.03
Figure C-17: Nonlinear time simulation of the wing tip displacement motion at 10% above
its flutter speed (20 root angle of attack) for the active wing design
207
Root locus for Root AOA =
5
140
0 3B
120
[
--
-
- -
--
100 -
-
-
-
1T -
1
-- -- - -- ----- -- ---
80
C
Cr
U-
--------------- -------------
--
60
-0 -
40
-
2
- - -+G-o
1CB
0
20
----- - -- - - -- -- - ---- ------- -----i
0'
50
40
006
40
30
3i
20
10
Real Part of Root
-20
10-4
0
1 18
10
20
Figure C-18: Root locus plot for 50 root angle of attack
208
Speed
=
68.00 m/s @ Freq
=
6.83 Hz
Speed = 68.00 m/s @ Freq = 26.7 Hz
43.8 Hz
Speed = 68.00 m/s @ Freq = 129 Hz
139 Hz
Speed = 68.00 m/s @ Freq = 233 Hz
0.6
0.4
0.2
0
-. 00040.30.201
Speed = 68.00 m/s @ Freq
0.2N
0.1T
-0. 1
-0.05
-
0
0.8
Speed
=
68.00 m/s @ Freq
0.3
0.21
0.1
=
-mm
-ON
0-- - --
-
--
0.2
-n i
-U.
-
0.8
Figure C-19: Mode shapes for active wing design at 5' root angle of attack and its corresponding flutter speed of 68 m/s
209
For 50 angle ofattack, the wing also enters a LCO when flown at 10% above the flutter
speed. The tip delections and the tip twist are plotted for this case up to 1 second. These
plots are given inPigure C-20 thru Figure C-22.
Speed = 74.80 m/s with a Time step = 0.0005 s
0.4
0.35
0.3
E
C)
0.25
CD)
0~
0.15
0-
as
0.1
0.05
0
-0.05
-
0
0.2
0.4
0.6
0.8
1
Time (s)
Figure C-20: Nonlinear time simulation of the wing vertical tip displacement at 10% above
its flutter speed (50 root angle of attack) for the active wing design
210
Speed = 74.80 m/s with a Time step = 0.0005 s
14
12
10
8
U,
V
6
C,,
4
I0~
20-2-4 -
0
0.2
0.8
0.6
0.4
1
Time (s)
Figure C-21: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed
(5' root angle of attack) for the active wing design
Speed = 74.80 m/s with a Time step = 0.0005 s
0.38
0.370.360.35E
E0.34E
a 0.330.32N 0.31
-
0.30.290.28
'
-0.08
-0.06
-0.04
-0.02
0
Y Displacement (m)
Figure C-22: Nonlinear time simulation of the wing tip displacement motion at 10% above
its flutter speed (50 root angle of attack) for the active wing design
211