APS 2004 Flapping flight from flexible wings : tuning of wing stiffness for flight? Tom Daniel, Stacey Combes,, & Sanjay Sane
CNS SENSORY INPUT MOTOR OUTPUT FLIGHT MUSCLES FORCE, STRAIN WING HINGE How much of the control resides in neural pathways versus nonneural mechanisms? WING MOTIONS AERODYNAMICS 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?
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
Trajectory of forward flight and net aerodynamics forces measured by Robofly Average forces ~ mg 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 Dalton, 1975
Many wings flex significantly Brackenbury, 1992
m B g/2 Θ L M AERO ~ m B g L/4 A[l ] dl Scale question: How large are the aerodynamic moments relative to inertial/elastic moments? M INERT ~ ρ app l 2 Θω 2 sin(ω t) A[l ] dl ~ m app L 2 Θω 2 /3 M RAT = M INERT /M AERO ~ (m app /m B ) L Θ ω 2 /g ~ 5.0 Manduca ~ 50.0 Drosophila See as well Ennos 89, Zanker and Götz 90
Need high-speed videography of flapping wings during maneuvers. unfettered flight Robo-Bush camera Computer Controlled Flower! Vertical Looming Lateral Carlos Moreno
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
Measuring displacement continuously along a wing 45 wing 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 z x
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
Solve for the EI distribution that best fits the measured wing displacement EI(x) = F(L x) d 2 δ / dx 2 δ(x) = EI x EI x EI x F (L x) EI(x) d 2 x constant [EI(x) = k] linear [EI(x) = m x + b] exponential [EI(x) = c*exp ax ] simplex minimization displacement (m) x 10-3 1 0-1 -2-3 -4-5 0 0.02 0.04 span (m) EI x polynomial [EI(x) = px 2 +qx+r]
Spanwise wing stiffness flexural stiffness declines exponentially in span and chord directions 0.0008 Flexural Stiffnes (Nm 2 ) 0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 Manduca spanwise 0.0001 0 0 0.01 0.02 0.03 0.04 0.05 Span location (m)
Wings flex. With measured mechanical, geometric, and kinematic characters
. use FEM to compute motions. From that compute aerodynamic forces Movie not available
Aerodynamic force calculations from FEM are illegal when wing bending depends strongly on aerodynamic loading. Broad- bordered yellow Underwing Lampra fimbriata (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.
A scale argument for a thin flexing wing Movie not available
A scale argument for a thin flexing wing Imbue with Stiffness (E) thickness (t) width (w) EI (flexural stiffness) density (ρ) Movie not available Impel with A sin(ω t) What deformations -- y(x,t) arise? What bending moments follow from this y(x,t)? y x
A different tact : Solve this wave equation y 2 / t 2 = -(EI/ρA) y 4 / x 4 with the right boundary conditions y(x,t) Moment = EI y 2 / x 2 Imbue with Stiffness (E) thickness (t) width (w) EI (flexural stiffness) density (ρ) y Impel with A sin(ω t) What deformations -- y(x,t) arise? What bending moments follow from this y(x,t)? x
A different tact : Inviscid blade-element analysis (from Wu,1971) y(x,t) Imbue with Stiffness (E) thickness (t) width (w) EI (flexural stiffness) density (ρ) x p(x,t) -> M fluid (x,t) -> M fluid (x,t) -> M fluid y M elastic Manduca : L = 0.03,w = 0.01, ω = 2 π 25 Wu 1971 set this up
Flexural stiffness data of Combes 0.1 1 10 rubbo-fly spanwise wing area (cm 2 ) 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 avg EI (Nm 2 ) Odonates chordwise dragonfly perching dragonfly damselfly blue damselfly Isopt/Neuropt termite lacewing Lepidopterans Manduca cabbage white skipper Dipterans cranefly calliphora syrphid bombyliid 1.E-08 1.E-09 Hymenopterans bumblebee wasp spiderwasp
Dynamic bending moments are dominated by elastic/inertial phenomena Moment (Nm) 10-7 7.5 10-8 Elastic 5 10-8 2.5 10-8 0-2.5 10-8 -5 10-8 -7.5 10-8 -1 10-10 -7 Fluid dynamic 5 10 15 20 25 30 time (ms) 30 hz EI = 6 10-6 Length = 0.03, width = 0.01
Dynamic bending moments are dominated by elastic/inertial phenomena Moment (Nm) 10-8 1 10-8 Elastic 5 10-9 0-5 10-9 -1 10-8 -10-8 Fluid dynamic 5 10 15 20 25 30 tims (ms) 30 hz EI = 6 10-6 Length = 0.02, width = 0.01
Dynamic bending moments are dominated by elastic/inertial phenomena Moment (Nm) 4 10-8 10-8 4 10-8 Elastic 2 10-8 0-2 10-8 -4 10-8 -4-10 -8 Fluid dynamic 5 10 15 20 25 30 time (ms) 60 hz EI = 6 10-6 Length = 0.02, width = 0.01
Inertia is generally large 10-7 1 10-7 Moment (Nm) 8 10-8 6 10-8 4 10-8 2 10-8 Elastic Fluid dynamic 20 40 60 80 100 frequency (hz) EI = 6 10-6
Thrust depends on emergent deformations (non-monotonically). 0.005 Thrust (N) 0.004 0.003 0.002 0.001 0 5 10 15 20 30 Frequency (hz) EI = 6 10-6 Length = 0.03, width = 0.01
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
normal air density 10% air density Movie not available
Quantify wing bending at the wing tip, leading edge, and trailing edge 26 Hz, normal air 0 degrees lead tip trailing edge trail
Quantify wing bending at the wing tip, leading edge, and trailing edge 26 Hz, normal air 0 degrees trailing edge 0.5 Hz trailing edge position/bending (degrees) 90 60 30 0-30 -60 Fourier amplitude coefficient -90 16 14 12 10 8 6 4 2 * 20 ms position, normal air position, helium position, slow rotation bending, normal air bending, helium 0 0 50 100 150 200 250 frequency (hz)
Evidence of bending on the air flow? Measure self-generated flows
Raw data of self-generated airflow At Antennae Behind wings
Spectra of recorded airflows At antenna Behind wings
Aerodynamic signature of wing flexibility
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.
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 FORCES
THANKS Organizers of APS Collaborators: Stacey Combes, Sanjay Sane, Michael Dickinson, Nate Jacobson Inspiration: T.Y. Wu, Daniel Grunbaum Support NSF ONR MacArthur Foundation
Anemometer Calibration