APS 2004
Flapping flight from flexible wings :
tuning of wing stiffness for flight?
Tom Daniel, Stacey Combes, & Sanjay Sane
Copyright Tom Daniel, 2004
CNS
SENSORY INPUT
MOTOR OUTPUT
FLIGHT MUSCLES
FORCE, STRAIN
WING HINGE
WING MOTIONS
How much of
the control
resides in
neural
pathways
versus nonneural
mechanisms?
AERODYNAMICS
Copyright Tom Daniel, 2004
FORCES
A summary of insect flight issues
A thought experiment about wing or fin inertia
The coupled problem of fluid-solid interactions
Under what conditions can we ignore such
interactions in air?
When we can ignore them, how might we
proceed to analyze flight with compliant wings?
Copyright Tom Daniel, 2004
Dickinson’s “Robo-fly”*
- Dynamical scaling: Reynolds number and reduced frequency parameter
conserved
- Motors allow any arbitrary 3-D kinematic pattern
- Force sensors measure instantaneous forces on the wing
* Sanjay Sane, UW; Mark Willis,CWRU
Copyright Tom Daniel, 2004
Trajectory of forward flight and net
aerodynamics forces measured by Robofly
Average forces ~ mg
Copyright Tom Daniel, 2004
Sane, Johnstone, Dickinson and Willis
Such physical models work well for wings
whose flexion is rather minimal.
It is difficult to scale elastic and fluid dynamic
stresses simultaneously
Copyright Tom Daniel, 2004
Dalton, 1975
Many wings flex significantly
Copyright Tom Daniel, 2004
Brackenbury, 1992
mB g/2
Scale question:
How large are the
A[l ] dl aerodynamic moments
relative to inertial/elastic
moments?
Θ
L
MAERO ~ mB g L/4
∫
MINERT ~ ρapp l 2 Θ ω2 sin(ω t) A[l ] dl
~ mappL2 Θ ω2 /3
MRAT = MINERT /MAERO
~ 5.0 Manduca
~ (mapp/mB) L Θ ω2/g ~ 50.0 Drosophila
See as well Ennos ‘89, Zanker and Götz ‘90
Copyright Tom Daniel, 2004
Need high-speed videography of flapping wings
during maneuvers.
“Robo-Bush”
Computer
Controlled
Flower!
camera
unfettered flight
Vertical
Looming
Lateral
Carlos Moreno
Copyright Tom Daniel, 2004
Movie not available
•To what extent are these motions determined by fluid
dynamic events?
•What are the mechanical properties of these wings?
Michael Tu and Katarzyna Kodizcezswka
Copyright Tom Daniel, 2004
Measuring displacement continuously along a wing
45
wing
Copyright Tom Daniel, 2004
Combes and Daniel, 2004
Changes in the y-direction of the picture correspond to
bending in the z-direction of the wing
unloaded
wing
loaded
wing
y
x
z
Copyright Tom Daniel, 2004
Possible patterns of spatial variation in wing stiffness
heterogeneous beam
F
infinite
∞
L
wing
stiffness
constant
polynomial
linear
exponential
wing span or chord
Combes and Daniel 2004
Copyright Tom Daniel, 2004
Solve for the EI distribution that best fits
the measured wing displacement
F(L − x)
EI(x) = 2
d δ / dx 2
EI
constant
simplex
minimization
[EI(x) = k]
x
EI
[EI(x) = m x + b]
x
exponential
[EI(x) = c*expax]
polynomial
[EI(x) = px2+qx+r]
x
EI
x
0
-1
-2
-3
-4
linear
EI
displacement (m)
F (L − x) 2
δ(x) = ∫∫
d x
EI(x)
x 10-3
1
Copyright Tom Daniel, 2004
-5
0
0.02
span (m)
0.04
Spanwise wing stiffness flexural stiffness
declines exponentially in span and chord directions
Flexural Stiffnes (Nm2)
0.0008
0.0007
0.0006
Manduca spanwise
0.0005
0.0004
0.0003
0.0002
0.0001
0
0
0.01
0.02
0.03
Span location (m)
Copyright Tom Daniel, 2004
0.04
0.05
Wings flex.
With measured
mechanical, geometric,
and kinematic
characters…
Copyright Tom Daniel, 2004
…. use FEM to compute
motions. From that
compute aerodynamic
forces
Movie not available
Copyright Tom Daniel, 2004
Aerodynamic force calculations from FEM are
illegal when wing bending depends strongly on
aerodynamic loading.
Broad- bordered yellow Underwing
Lampra fimbriata
Copyright Tom Daniel, 2004
(Dalton, 1975)
The basic aeroelastic problem
Flexible wings
deform when
moved
dynamically.
Shape
(curvature,
angle of
attack)
determines
the pressure
distribution.
Pressure
stresses
deform
flexible wings.
Copyright Tom Daniel, 2004
A scale argument for a thin flexing wing
Movie not available
Copyright Tom Daniel, 2004
A scale argument for a thin flexing wing
Imbue with
Stiffness (E)
thickness (t)
width (w)
EI (flexural stiffness)
density (ρ)
Movie not available
x
y
Impel with
A sin(ω t)
What deformations -- y(x,t) arise?
What bending moments follow from this y(x,t)?
Copyright Tom Daniel, 2004
A different tact :
Solve this wave equation
∂y2/ ∂ t2 = -(EI/ρA) ∂y4/ ∂x4
with the right boundary
conditions
⇓
y(x,t)
⇓
Moment = EI ∂y2/ ∂x2
Imbue with
Stiffness (E)
thickness (t)
width (w)
EI (flexural stiffness)
density (ρ)
x
y
Impel with
A sin(ω t)
What deformations -- y(x,t) arise?
What bending moments follow from this y(x,t)?
Copyright Tom Daniel, 2004
A different tact :
Inviscid blade-element
analysis (from Wu,1971)
Imbue with
Stiffness (E)
thickness (t)
width (w)
EI (flexural stiffness)
density (ρ)
x
y(x,t)
y
-> Mfluid(x,t)
p(x,t)
-> ∫∫Mfluid(x,t) -> Mfluid
Melastic
Manduca : L = 0.03,w = 0.01, ω = 2 π 25
Wu 1971 set this up
Copyright Tom Daniel, 2004
Flexural stiffness data of Combes
wing area (cm2)
0.1
1
10
1.E-04
rubbo-fly
1.E-05
1.E-06
spanwise
1.E-07
1.E-08
chordwise
Odonates
dragonfly
perching dragonfly
damselfly
blue damselfly
avg EI (Nm2)
1.E-03
Isopt/Neuropt
termite
lacewing
Lepidopterans
Manduca
cabbage white
skipper
Copyright Tom Daniel, 2004
1.E-09
Dipterans
Hymenopterans
cranefly
calliphora
syrphid
bombyliid
bumblebee
wasp
spiderwasp
Dynamic bending moments are dominated by
elastic/inertial phenomena
30 hz
Elastic
10-7
Moment (Nm)
7.5¥ 10-8
-8
5¥ 10
Fluid dynamic
2.5¥ 10-8
0
5
-2.5¥ 10-8
-8
-5¥ 10
10
15
20
25
30
time (ms)
-7.5¥ 10-8
-10-7
-7
-1¥ 10
EI = 6 10-6 Length = 0.03, width = 0.01
Copyright Tom Daniel, 2004
Dynamic bending moments are dominated by
elastic/inertial phenomena
30 hz
Elastic
10-8
Moment (Nm)
1¥ 10 -8
5¥ 10
Fluid dynamic
-9
0
-5¥ 10 -9
5
10
15
20
25
30
tims (ms)
-1¥ 10 -8
-10-8
EI = 6 10-6 Length = 0.02, width = 0.01
Copyright Tom Daniel, 2004
Dynamic bending moments are dominated by
elastic/inertial phenomena
60 hz
Elastic
4 10
10-8-8
Moment (Nm)
4¥10-8
Fluid dynamic
-8
2¥10
00
5
-2¥10-8
10
15
20
25
30
time (ms)
-8
-4¥10
-8
-4-10
10-8
EI = 6 10-6 Length = 0.02, width = 0.01
Copyright Tom Daniel, 2004
Inertia is generally large
10-7
Moment (Nm)
1¥10-7
8¥10-8
Elastic
6¥10-8
4¥10-8
Fluid dynamic
2¥10-8
20
40
60
frequency (hz)
EI = 6 10-6
Copyright Tom Daniel, 2004
80
100
Thrust depends on emergent deformations
(non-monotonically).
Thrust (N)
0.005
0.004
0.003
0.002
0.001
0
5
EI = 6 10-6
10
15
20
30
Frequency (hz)
Length = 0.03, width = 0.01
Copyright Tom Daniel, 2004
Can we trust the theoretical results?
Set density to as low as possible to eliminate fluid
dynamic loading and query wing shape.
fresh Manduca wing on motor
add Helium
(~20% air density)
remove 50%
of air volume
Copyright Tom Daniel, 2004
normal air density
10% air density
Movie not available
Copyright Tom Daniel, 2004
Quantify wing bending at the wing tip, leading edge, and trailing edge
26 Hz, normal air
0 degrees
lead
tip
trailing
edge
trail
Copyright Tom Daniel, 2004
0 degrees
trailing
edge
0.5 Hz
90
60
30
0
-30
-60
20 ms
-90
Fourier amplitude coefficient
26 Hz, normal air
trailing edge position/bending (degrees)
Quantify wing bending at the wing tip, leading edge, and trailing edge
16
position, normal air
position, helium
position, slow rotation
*
14
12
bending, normal air
bending, helium
10
8
6
4
2
0
0
Copyright Tom Daniel, 2004
50
100
150
frequency (hz)
200
250
Evidence of bending on the air flow?
Measure self-generated flows
Copyright Tom Daniel, 2004
Raw data of self-generated airflow
At Antennae
Behind wings
Copyright Tom Daniel, 2004
Spectra of recorded airflows
At antenna
Behind wings
Copyright Tom Daniel, 2004
Aerodynamic signature of wing flexibility
Copyright Tom Daniel, 2004
Summary
• Animal wings deform in response to both fluid
dynamic loading and their own inertial
mechanics.
• Simple scaling arguments suggest appendage
inertia may dominate moments for “rigid” wings.
• Bending moments of wings are dominated by
elastic/inertial mechanisms in air.
• There is a clear aerodynamic signature of
bending wings.
• Thus we could use simple explicit methods for
evaluating forces on flexible wings in air.
Copyright Tom Daniel, 2004
CNS
SENSORY INPUT
MOTOR OUTPUT
FLIGHT MUSCLES
FORCE, STRAIN
Insect wings
are equipped
with a large
number of
strain sensors
on their wings.
WING HINGE
WING MOTIONS
AERODYNAMICS
Copyright Tom Daniel, 2004
FORCES
THANKS
Organizers of APS
Collaborators:
Stacey Combes, Sanjay Sane,
Michael Dickinson, Nate Jacobson
Inspiration:
T.Y. Wu, Daniel Grunbaum
Copyright Tom Daniel, 2004
Support
NSF
ONR
MacArthur Foundation
Anemometer Calibration
Copyright Tom Daniel, 2004