Aerodynamic force characteristics of 3d flapping wings

Aerodynamic Force Characteristics of 3D Flapping Wings





LAI KENG CHUAN
(B.Eng.(Hons.), NUS)










A THESIS SUBMITTED

FOR THE DEGREE OF MASTER OF
ENGINEERING

DEPARTMENT OF MECHANICAL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2007
Acknowledgements

i

Acknowledgements
I would like to express many thanks to my supervisor, Professor Lim Tee Tai for
his guidance and support in the project. Meetings with Prof. Lim left me feeling
motivated to break new grounds in research and I want to thank him once again for
clearing many obstacles along the way.
A special mention goes out to research fellow Dr. Lua Kim Boon for imparting his
valuable knowledge and his relentless pursuit in perfecting the force transducer despite
the various frustrations and delays.
I would like to thank all the Laboratory Officers of the Fluid Mechanics Laboratory
for their help in constructing, advising on, and modifying the apparatus design.
Finally, I would like to thank to my parents for their care and support.
Far above all, I am grateful for God’s provision of joys, challenges, and grace for
growth.




This Master of Engineering programme in the Department of Mechanical
Engineering is supported by a National University of Singapore research scholarship.

Table of contents

ii

TABLE OF CONTENTS
ACKNOWLEDGEMENTS I
TABLE OF CONTENTS II
SUMMARY VIII
LIST OF TABLES X
LIST OF FIGURES XI
NOMENCLATURE XXI
CHAPTER 1. INTRODUCTION 1
1.1. Assumptions 3
1.2. Objective of this study 3
1.3. Organization of thesis 4
CHAPTER 2. LITERATURE REVIEW ON FLAPPING WINGS 6
2.1. A brief history of flapping flight development 6
2.2. Wingbeat kinematics 7
2.2.1. Reynolds number 9
2.2.2. Flapping frequency 9
2.2.3. Flapping angles 10
2.2.4. Flapping rhythm 11
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iii
2.2.5. Stroke ratio 11
2.3. Aerodynamic mechanisms 12
2.3.1. Added mass effect 14

2.3.2. Wagner effect 15
2.3.3. Delayed stall and leading edge vortex 17
2.3.4. Rotational effect (Kramer’s effect) 22
2.3.5. Wake capture 25
2.3.6. Summary of aerodynamic mechanisms 27
2.4. Quasi-steady estimates 28
2.5. Wing detail and flexibility 29
2.6. Force measurement experiments and motivation 31
CHAPTER 3. EXPERIMENTAL APPARATUS AND PROCEDURES 33
3.1. Definition of angles 35
3.2. Mechanical system 37
3.2.1. Motor control 37
3.2.2. Motors 40
3.2.3. Motion transmission 40
3.3. Force measurement system 42
3.3.1. Strain gages 43
3.3.2. Wheatstone bridge 44
3.3.3. Force transducer design 45
3.3.4. Summing and differential amplifiers 49
Table of contents

iv

3.3.5. Noise filter 51
3.3.6. Analog to digital converter 52
3.4. Synchronization 52
3.5. Statistical functions 52
3.6. Wings 54
3.6.1. Fabrication of wings 54
3.6.2. Derivation of second moment of wing area 55

3.6.3. Quantification of wing elasticity 60
3.6.4. Evaluation of deflection angle, γ 63
3.7. Experimental Procedure 64
3.7.1. Calibration 64
3.7.2. Verification 71
3.7.3. Alignment of the wing 77
3.7.4. Elimination of buoyancy forces 77
3.7.5. Measurement of weight and buoyancy force 80
3.7.6. Measurement of forces 81
3.7.7. Resolving of forces 85
3.7.8. Non-dimensional parameters 85
3.7.9. Presentation of results 87
3.8. Potential errors 88
CHAPTER 4. REVOLVING WING EXPERIMENTS 89
4.1. Kinematics of the revolving motion 89
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v

4.2. Revolving wing experiments results and discussion 92
4.2.1. Time history of the measured forces 92
4.2.1.1. Discussion 96
4.2.2. Mean force coefficients for rigid wing 97
4.2.2.1. Discussion 100
4.2.3. Mean force coefficients of flexible wings 103
4.2.3.1. Discussion 106
4.2.4. Comparison of the force characteristics of flexible wings 108
4.2.5. Deflection angles 110
4.2.6. Discussion on the force coefficient ratios and deflection angles 112
4.3. Comparison with past experiments 113

4.4. Conclusion 115
CHAPTER 5. OSCILLATING MOTION EXPERIMENTS 117
5.1. Motion 118
5.1.1. Sinusoidal motion 119
5.1.2. Hawkmoth motion 120
5.2. Results and discussion 121
5.2.1. Rigid wing results 122
5.2.1.1. Sinusoidal motion 122
5.2.1.2. Hawkmoth motion 124
5.2.1.3. Discussion of rigid wing results 126
5.2.2. Mean force coefficients for rigid wing 127
5.2.3. Flexible wings results 128
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vi

5.2.3.1. Sinusoidal motion 128
5.2.3.2. Hawkmoth motion 134
5.2.3.3. Discussion on the time histories of flexible wings 135
5.2.4. Mean force coefficients for flexible wings 138
5.2.5. Deflection angles of flexible wings 141
5.2.6. Discussion 142
5.3. Comparison with numerically predicted results 145
5.4. Conclusion 148
CHAPTER 6. ADVANCED AND DELAYED FEATHERING EXPERIMENTS
151
6.1. Motion 152
6.2. Time histories of force coefficients 156
6.2.1. Results and discussion 156
6.2.1.1. Case U30 156

6.2.1.2. Case U45 159
6.2.1.3. Case U60 159
6.2.1.4. Case V30 159
6.2.1.5. Case V45 165
6.2.1.6. Case V60 165
6.2.2. Discussion on the time histories 165
6.3. Mean force coefficients 171
6.3.1. Discussion 174
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vii

6.4. Conclusion 175
CHAPTER 7. FLAPPING RHYTHM EXPERIMENTS 176
7.1. Motion 177
7.2. Results and discussion 181
7.2.1. Time histories of force coefficients 181
7.2.1.1. Discussion 184
7.2.2. Wake interaction effects 188
7.2.2.1. Discussion 192
7.2.3. Mean force coefficients 192
7.2.4. Discussion 194
7.3. Conclusion 195
CHAPTER 8. CONCLUSION AND RECOMMENDATION 196
8.1. CONCLUSION 196
8.2. RECOMMENDATION 198
REFERENCES 199
APPENDIX A. TECHNICAL DRAWING OF FORCE TRANSDUCER 216
APPENDIX B. TANK BOUNDARY EFFECTS 217

Summary

viii
Summary
In this project, the aerodynamic characteristics of insect flapping wings are
assessed by measuring the forces acting on them. A 3D flapping wing mechanism
capable of mimicking the motion of insect flight is developed to measure forces acting
on model wings flapping according to various motions. It consists of a pair of wings
connected to gear boxes driven by stepper motors. On each side of the wing, three
stepper motors actuate the positional, feathering and elevation motions via coaxial
shafts and gearboxes which are immersed in a tank filled with water so that hovering
characteristics can be modelled.
Forces acting on the flapping wing are measured by an in-house developed force
transducer consisting of strategically arranged strain gages, attached to the base of the
flapping wing. By connecting the strain gages to Wheatstone bridges, the response of
the strain gages to forces and moments can be determined. An accompanying
calibration procedure is carried out to determine a calibration matrix that relates the
raw signals into forces and moments.
The flight of a hawkmoth Manduca sexta which has a Reynolds number of
approximately 7000 is investigated by mimicking its motion and wing shape.
Measured mean force coefficients are in good agreement with results derived from
free-flying hawkmoths but the time course of force coefficients differs from
numerically predicted results.
For the fixed incidence steadily revolving wing cases, the maximum mean force
coefficients are large (C
L
=1.84 and C
D
=3.6). Furthermore, flapping wings do not stall
for all the tested incidence angles up to 100

o
, suggesting that delayed stall and leading
edge vortices are mainly responsible for the enhanced performance of insect flight.
Summary

ix

Application of flexible wings lowers the force coefficients. However, flexible
wings are able to sustain relatively large force coefficients despite larger than real wing
estimated wing deflection angles of 10
o
, which are higher than that observed in
hawkmoth wings. Wing flexure can passively alter the wing flapping motion such that
different force generation patterns are produced.
On a separate note, the force coefficients for rigid wings are insensitive to changes
in Reynolds number ranging from 3600 to 11000. Moreover, a study of wing
kinematics suggests that large positional and feathering amplitudes are vital for
increasing aerodynamic performance.
Finally, as the flapping rhythm tends towards a triangular-like wave, wake capture
effects are found to increase force coefficients by at least 10%. Conversely, wake
capture effects are insignificant but overall aerodynamic performance increases as the
flapping rhythm tends towards a sinusoidal-like wave.

List of tables

x

List of Tables
Table 3.1: Wing stroke angles and their corresponding motions 36
Table 3.2: Parameters of the wings used in this study 59

Table 3.3: Calibration matrix representing the gradients of the trend lines 70
Table 3.4: Inverted calibration matrix used in the experiments 70
Table 3.5: Buoyancy forces acting on the wings and sting 83
Table 4.1: Experimental parameters for revolving wing motion 92
Table 4.2: Range of time at which the mean force coefficients were derived from the
instantaneous force coefficients 97
Table 5.1: Experimental parameters relating to the sinusoidal flapping motion. 118
Table 5.2: Mean force coefficients for the rigid wing flapping according to oscillating
motions 128
Table 6.1: Summary of motions used in this chapter 153
Table 7.1: Parameters for the various motions applied in this study. 178


List of figures

xi

LIST OF FIGURES
Figure 2.1: Wing motions define a stroke plane oriented at the stroke plane angle β,
stroke plane deviation angle θ and flapping angle φ of the wing (Dudley, 2000) 8
Figure 2.2: Kutta condition and circulation (taken from Sane, 2003). 12
Figure 2.3: Wagner effect delaying the growth of circulation around impulsively
started airfoil. 16
Figure 2.4: Formation of leading edge vortices (LEV) (Usherwood, 2002). 18
Figure 2.5: Illustration of rotation effects. 24
Figure 2.6: Illustration of wake capture 26
Figure 3.1: Drawing of the mechanism including the water tank 34
Figure 3.2: Coordinate system (With reference to the right wing of a flapper) 36
Figure 3.3: Flow diagram of the mechanical system. 38
Figure 3.4: Schematic diagram of the flapping mechanism. 39

Figure 3.5: Schematic diagram of the gearbox arrangement. 41
Figure 3.6: Flow diagram of force measurement system 42
Figure 3.7: General definition of strain 43
Figure 3.8: Measurement of strains using a Wheatstone bridge 46
Figure 3.9: (a) Diagram of force acting on bending beam. (b) Moment in beam due to
the force. (c) Diagram of moment acting on bending beam. (d) Moment in beam
due to the moment 48
Figure 3.10: Connection map of the signals in the force measurement system 50
Figure 3.11: Photograph of the partially assembled force transducer with the sting
removed exposing the attached strain gages 50
Figure 3.12: Circuit diagram of a summing amplifier (Damato, 2006) 51
List of figures

xii

Figure 3.13: Circuit diagram for a differential amplifier 52
Figure 3.14: Diagram of hawkmoth wing 54
Figure 3.15: Picture of flexible wings 3 to 7. Note the scale of the ruler is in
centimetres. 55
Figure 3.16: Tracing of the wing geometry. 57
Figure 3.17: Coordinates of the trapezium for finding the areas of the wing. 58
Figure 3.18: Schematic diagram of the wing stiffness measurement setup. 61
Figure 3.19: Load versus extension during measurement of wing stiffness 61
Figure 3.20: r
2
values of the wing elasticity trend lines. 62
Figure 3.21: Elasticity of the wings. 62
Figure 3.22: Schematic diagram showing the wing deflection angle. 63
Figure 3.23: Calibration setup for the F
N

component of the force transducer 66
Figure 3.24: Calibration curves for F
C
67
Figure 3.25: Calibration setup for the M
C
combined with F
N
component of the force
transducer 67
Figure 3.26: Calibrations curves for M
c
69
Figure 3.27: Distribution of r
2
values for all force components of the force transducer.
71
Figure 3.28: Positions on the wing used to verify the accuracy of the force transducer.
72
Figure 3.29: Verification data of +F
N
axis using light bolt 73
Figure 3.30: Verification data of +F
N
axis using heavy bolt. 74
Figure 3.31: Verification data of -F
N
axis using light bolt. 74
Figure 3.32: Verification data of -F
N

axis using heavy bolt. 75
Figure 3.33: Verification data of +F
C
axis using light bolt 75
List of figures

xiii
Figure 3.34: Verification data of +F
C
axis using heavy bolt. 76
Figure 3.35: Verification data of -F
C
axis using light bolt. 76
Figure 3.36: Verification data of -F
C
axis using heavy bolt. 77
Figure 3.37: Weight and buoyancy forces acting on the wing and force transducer, the
apostrophe indicates the forces affected by elevation angle θ 79
Figure 3.38: Schematic diagram showing the process of obtaining the weight of wing.
82
Figure 3.39: Force transducer outputs during the weight measurement process for wing
1 83
Figure 3.40: Definition of the force vectors related to the wing 86
Figure 4.1: Schematic diagram showing insect reciprocating motion perceived as
propeller-like revolution. 90

Figure 4.2: Schematic diagram of the revolving wing motion. ………………………91
Figure 4.3: Revolving wing angular displacement versus elapsed time 93
Figure 4.4: Revolving wing angular velocity versus elapsed time. 93
Figure 4.5: Time histories for drag coefficients for rigid wing (wing 1) revolving

according to the low Re motion. 94
Figure 4.6: Time histories for drag coefficients for rigid wing (wing 1) revolving
according to the high Re motion. 94
Figure 4.7: Time histories for lift coefficients for rigid wing (wing 1) revolving
according to the low Re motion. 95
Figure 4.8: Time histories for lift coefficients for rigid wing (wing 1) revolving
according to the high Re motion. 95
Figure 4.9: Lift coefficients for standard rigid wing (wing 1) under different Re cases.
98
List of figures

xiv

Figure 4.10: Drag coefficients for standard rigid wing (wing 1) under different Re
cases. 99
Figure 4.11: Polar diagram for standard rigid wing (wing 1) under different Re cases.
99
Figure 4.12: Lift coefficients for wings with a range of flexibility under low Re
conditions 103
Figure 4.13: Drag coefficients for wings with a range of flexibility under low Re
conditions 104
Figure 4.14: Lift coefficients for wings with a range of flexibility under high Re
conditions 105
Figure 4.15: Drag coefficients for wings with a range of flexibility under high Re
conditions 105
Figure 4.16: Illustration of wing deflection decreasing angle of attack. 107
Figure 4.17: Photograph of wing 7 flexing under fluid dynamic forces for high Re case
at α = 50
o
108

Figure 4.18: Comparison of mean lift coefficients for wings of various flexibilities
plotted against the elasticity of each wing. 109
Figure 4.19: Comparison of mean drag coefficients for wings of various flexibilities
plotted against the elasticity of each wing. 110
Figure 4.20: Estimated deflection angle (γ) versus elasticity of each wing 111
Figure 4.21: Force coefficient ratios plotted against estimated wing deflection angles
(γ). 111
Figure 4.22: Comparison of lift coefficients with past experiments for low Re case. 114
Figure 4.23: Comparison of drag coefficients with past experiments for low Re case.
114
List of figures

xv

Figure 4.24: Comparison of drag polars with past experiments for low Re case. 115
Figure 5.1: Time histories of sinusoidal motion in this study. Note that the elvation
angle θ is zero. 120
Figure 5.2: Time histories of the derived hawkmoth motion to be applied in this study.
121
Figure 5.3: Time histories of normal and chord-wise forces acting on the rigid wing
flapping according to the sinusoidal motion at medium Re. Only four cycles are
presented. 122
Figure 5.4: Time histories of lift coefficients of the rigid wing (wing 1) flapping
according to sinusoidal motion for various Re. 123
Figure 5.5: Time histories of drag coefficients of the rigid wing (wing 1) flapping
according to sinusoidal motion for various Re. 124
Figure 5.6: Time histories of lift coefficient for the rigid wing (wing 1) flapping
according to hawkmoth motion for various Re 125
Figure 5.7: Time histories of drag coefficient for the rigid wing (wing 1) flapping
according to hawkmoth motion for various Re 126

Figure 5.8: Time history of drag coefficients for rigid wing flapping according to the
sinusoidal motion for medium Re over four cycles. 127
Figure 5.9: Time histories of normal forces acting on wings with a range of flexibility
flapping according to four cycles of the sinusoidal motion for medium Re case.
129
Figure 5.10: Time histories of chord-wise forces acting on wings with a range of
flexibility flapping according to four cycles of the sinusoidal motion for medium
Re case. 130
List of figures

xvi

Figure 5.11: Time histories of lift coefficient for wings with a range of flexibilities
flapping according to sinusoidal motion for medium Re case 132
Figure 5.12: Time histories of drag coefficients for wings with a range of flexibilities
flapping according to sinusoidal for medium Re case. 132
Figure 5.13: Time histories of lift coefficients for wings with a range of flexibilities
flapping according to sinusoidal for high Re case. 133
Figure 5.14: Time histories of drag coefficients for wings with a range of flexibilities
flapping according to sinusoidal motion for high Re case 133
Figure 5.15: Time histories of lift coefficients for wings with a range of flexibilities
flapping according to hawkmoth motion for medium Re case. 135
Figure 5.16: Time histories of drag coefficients for wings with a range of flexibilities
flapping according to hawkmoth motion at medium Re case. 136
Figure 5.17: Time histories of lift coefficients for wings of various flexibilities
flapping according to hawkmoth motion for high Re case. 136
Figure 5.18: Time histories of drag coefficients for wings of various flexibility
flapping according to hawkmoth motion for high Re case. 137
Figure 5.19: Comparison of mean lift coefficients for wings of various flexibilities
flapping according to sinusoidal motion 139

Figure 5.20: Comparison of mean drag coefficients for wings of various flexibilities
flapping according to sinusoidal motion 140
Figure 5.21: Comparison of mean lift coefficients for wings of various flexibilities
flapping according to hawkmoth motion. 140
Figure 5.22: Comparison of mean drag coefficients for wings of various flexibilities
flapping according to hawkmoth motion. 141
List of figures

xvii

Figure 5.23: Estimated deflection angles for wings of various flexibilities undergoing
sinusoidal motion. 142
Figure 5.24: Estimated deflection angles for wings of various flexibilities undergoing
hawkmoth motion. 143
Figure 5.25: Lift and drag force coefficient ratios plotted against estimated wing
deflection angles for wings flapping according to sinusoidal motion. 143
Figure 5.26: Lift and drag force coefficient ratios plotted against estimated wing
deflection angles for wings flapping according to hawkmoth motion 144
Figure 5.27 Stroke plane inclination angle. 147
Figure 5.28: Comparison of C
H
with computational results for rigid wing normalized to
the maximum velocity 149
Figure 5.29: Comparison of C
V
with computational results for rigid wing normalized to
the maximum velocity 150
Figure 6.1: Some sketches of the position and angles of attack of the wing at regular
time intervals corresponding to cases U in table 6.1. 154
Figure 6.2: Some sketches of the position and angles of attack of the wing at regular

time intervals corresponding to cases V in table 6.1 155
Figure 6.3: Time histories of wing motion for three phase angles in case U30. 157
Figure 6.4: Time histories of feathering and positional angular velocities for three
phase angles in case U30. 157
Figure 6.5: Time histories of lift coefficients for three phase angles in case U30. 158
Figure 6.6: Time histories of drag coefficients for three phase angles in Case U30. . 158
Figure 6.7: Time histories of wing motion for three phase angles in case U45. 160
Figure 6.8: Time histories of feathering and positional angular velocities for three
phase angles in case U45. 160
List of figures

xviii
Figure 6.9: Time histories of lift coefficients for three phase angles in case U45. 161
Figure 6.10: Time histories of drag coefficient for three phase angles in case U45 161
Figure 6.11: Time histories of wing motions for three phase angles in case U60. 162
Figure 6.12: Time histories of feathering and positional angular velocities for three
phase angles in case U60. 162
Figure 6.13: Time histories of lift coefficients for three phase angles in case U60. 163
Figure 6.14: Time histories of drag coefficient for three phase angles in case U60 163
Figure 6.15: Time histories of wing motions for three phase angles in case V30. 164
Figure 6.16: Time histories of feathering and positional angular velocities for three
phase angles in case V30 164
Figure 6.17: Time histories of lift coefficients for three phase angles in case V30. 166
Figure 6.18: Time histories of drag coefficient for three phase angles in case V30. 166
Figure 6.19: Time histories of wing motions for three phase angles in case V45. 167
Figure 6.20: Time histories of feathering and positional angular velocities for three
phase angles in case V45 167
Figure 6.21: Time histories of lift coefficients for three phase angles in case V45. 168
Figure 6.22: Time histories of drag coefficients for three phase angles in case V45. 168
Figure 6.23: Time histories of wing motions for three phase angles in case V60. 169

Figure 6.24: Time histories of feathering and positional angular velocities for three
phase angles in case V60 169
Figure 6.25: Time histories of lift coefficients for three phase angles in case V60. 170
Figure 6.26: Time histories of drag coefficients for three phase angles in case V60. 170
Figure 6.27: Lift coefficients versus feathering phase angle for all cases. 172
Figure 6.28: Drag coefficients versus feathering phase angle for all cases. 172
Figure 6.29: Lift to drag ratios for the various feathering phase angle for all cases 173
List of figures

xix

Figure 7.1: Position time graphs for various acceleration durations τ in case A 179
Figure 7.2: Velocity time graphs corresponding to the motions in case A 179
Figure 7.3: Some sketches of the position and angles of attack of the wing at regular
time intervals corresponding to motions in case A 180
Figure 7.4: Position time graphs for various acceleration durations τ in case B. 182
Figure 7.5: Velocity time graphs corresponding to the motions in case B. 182
Figure 7.6: Time histories of lift coefficients for various values of τ in case A 183
Figure 7.7: Time histories of drag coefficients for various values of τ in case A. 183
Figure 7.8: Time histories of lift coefficients for various values of τ in case B. 185
Figure 7.9: Time histories of drag coefficients for various values of τ in case B. 185
Figure 7.10: Time history of aerodynamic forces during half a stroke cycle, Fig. 3 of
Birch and Dickinson (2003) 186
Figure 7.11: Time histories of lift coefficient during half a stroke cycle, Fig. 2 of Wu
and Sun (2005) 186
Figure 7.12: Fig. 13 of Wu and Sun (2004) showing the lack of large time gradients of
aerodynamic force in the case of using sinusoidal motions 188
Figure 7.13: Time histories of lift coefficient differences between the first and fourth
down-stroke in case A for various τ. 189
Figure 7.14: Time histories of drag coefficient differences between the first and fourth

down-stroke in case A for various τ. 190
Figure 7.15: Time histories of lift coefficients differences between the first and fourth
down-stroke in case B for various τ 190
Figure 7.16: Time histories of drag coefficients differences between the first and fourth
down-stroke in case B for various τ 191
List of figures

xx

Figure 7.17: Percentage difference in force coefficients between first and fourth down-
stroke 191
Figure 7.18: Mean force coefficients of the fourth down-stroke for various τ in both
cases. 193
Figure 7.19: Mean lift to drag ratios for the various τ across both cases. 194

Nomenclature

xxi

Nomenclature
Symbol or
abbreviation


a Acceleration magnitude
Α
α
Angle of attack/ Feathering amplitude
Α
φ

Positional motion amplitude
Β
Buoyancy force
c

Mean chord length
c
Chord length
C
D
Drag coefficient
D
C
Mean drag coefficient
C
D
/
C
D
rigid Drag coefficient normalized to rigid wing values
C
H

Horizontal force coefficient
H
C
Mean horizontal force coefficient
C
L
Lift coefficient

L
C
Mean lift coefficient
C
L
/
C
L
rigid Lift coefficient normalized to rigid wing values
T
C
Mean thrust coefficient
C
V

Vertical force coefficient
V
C
Mean vertical force coefficient
Deg Degrees
EMI Electromagnetic Interference
F
C
Chord-wise force acting on the wing
Nomenclature

xxii

F
D

Drag force
F
L
Lift force
F
N
Normal force acting on the wing
N
F
Mean normal force
g Acceleration of wing
GF Strain gage factor
h Width of trapezium
j
N
y-coordinate of leading edge marker N of wing
k
N
y-coordinate of trailing edge marker N of wing
K Wing stiffness
l
α

Arc length of positional stroke
l Span-wise length of wing
L
section
Sectional lift
L/D Lift to Drag ratio
LEV Leading Edge Vortex

m Gradient of trend line
MAV Micro Aerial Vehicle
M
N
Moment about the normal axis
M
C
Moment about the chord-wise axis
n Frequency
q Phase angle
r Pearson product moment correlation coefficient
r
2
Coefficient of determination for goodness of fit to a linear trend,
equivalent to the square of
r
Nomenclature

xxiii
2
ˆ
r
(S)
Non-dimensional second moment of wing area about centre of
rotation
r
2
2
(S) Second moment of wing area
R

Radial distance from wing tip to centre of rotation
Re Reynolds number
S Surface area of wing
S
1
Voltage signal from Wheatstone bridge 1
S
2
Voltage signal from Wheatstone bridge 2
S
3
Voltage signal from Wheatstone bridge 3
S
4
Voltage signal from Wheatstone bridge 4
SF
N
Voltage signal for F
N
component
SF
C
Voltage signal for F
C
component
SM
N
Voltage signal for M
N

component
SM
C
Voltage signal for M
C
component
t Time
T Dimensionless time normalized to one period of flapping cycle
TTL Transistor-Transistor Logic
U
ref
Reference velocity
W Weight
Nomenclature

xxiv


Greek Symbols
α

Angle of attack
β
Flapping stroke plane inclination angle
ε
Strain
φ
Positional angle
γ

Estimated wing deflection angle
ν

Kinematic viscosity
θ
Elevation angle
ρ

Density of fluid

τ
Dimensionless acceleration duration
ω
Angular velocity
ω

Average angular velocity