To perform aerial maneuvers, insects must not only generatesufficient lift to remain aloft, they must also manipulate flightforces with great precision. Although insects are known to usetheir legs and abdomen as control surfaces during flight (Arbas,1986; Zanker, 1988; May and Hoy, 1990; Lorez, 1995), theysteer and maneuver largely by altering wing motion (Götzet al., 1979; Ennos, 1989; Robertson and Johnson, 1993;Wortmann and Zarnack, 1993). Thus, a central hurdle inunderstanding how insects steer and maneuver is determining

how modifications in stroke kinematics alter the forces andmoments generated by flapping wings.

In a few cases, researchers have attempted to capture thefree-flight kinematics of maneuvering insects (Ennos, 1989;Ruppell, 1989). While such analyses are essential because theyreveal what insects actually do with their wings when steering,free-flight studies are limited because it is not yet feasible torelate the changes in wing kinematics directly to changes ininstantaneous aerodynamic forces. An alternative approach is

2607The Journal of Experimental Biology 204, 2607–2626 (2001)Printed in Great Britain © The Company of Biologists Limited 2001JEB3400

We used a dynamically scaled mechanical model of thefruit fly Drosophila melanogasterto study how changes inwing kinematics influence the production of unsteadyaerodynamic forces in insect flight. We examined 191separate sets of kinematic patterns that differed withrespect to stroke amplitude, angle of attack, flip timing,flip duration and the shape and magnitude of strokedeviation. Instantaneous aerodynamic forces weremeasured using a two-dimensional force sensor mountedat the base of the wing. The influence of unsteadyrotational effects was assessed by comparing the timecourse of measured forces with that of correspondingtranslational quasi-steady estimates. For each pattern,we also calculated mean stroke-averaged values of theforce coefficients and an estimate of profile power. Theresults of this analysis may be divided into four mainpoints.

(i) For a short, symmetrical wing flip, mean lift wasoptimized by a stroke amplitude of 180 ° and an angle ofattack of 50 °. At all stroke amplitudes, mean dragincreased monotonically with increasing angle of attack.Translational quasi-steady predictions better matched themeasured values at high stroke amplitude than at lowstroke amplitude. This discrepancy was due to theincreasing importance of rotational mechanisms inkinematic patterns with low stroke amplitude.

(ii) For a 180 ° stroke amplitude and a 45 ° angle ofattack, lift was maximized by short-duration flipsoccurring just slightly in advance of stroke reversal.Symmetrical rotations produced similarly high

performance. Wing rotation that occurred after strokereversal, however, produced very low mean lift.

(iii) The production of aerodynamic forces was sensitiveto changes in the magnitude of the wing’s deviation fromthe mean stroke plane (stroke deviation) as well as tothe actual shape of the wing tip trajectory. However, inall examples, stroke deviation lowered aerodynamicperformance relative to the no deviation case. Thisattenuation was due, in part, to a trade-off between liftand a radially directed component of total aerodynamicforce. Thus, while we found no evidence that strokedeviation can augment lift, it nevertheless may be used tomodulate forces on the two wings. Thus, insects mightuse such changes in wing kinematics during steeringmaneuvers to generate appropriate force moments.

(iv) While quasi-steady estimates failed to capture thetime course of measured lift for nearly all kinematicpatterns, they did predict with reasonable accuracystroke-averaged values for the mean lift coefficient.However, quasi-steady estimates grossly underestimatedthe magnitude of the mean drag coefficient under allconditions. This discrepancy was due to the contributionof rotational effects that steady-state estimates do notcapture. This result suggests that many prior estimates ofmechanical power based on wing kinematics may havebeen grossly underestimated.

Key words: flapping flight, quasi-steady force, unsteadyaerodynamics, fruit fly, Drosophila melanogaster, added mass,delayed stall, rotational circulation, wake capture.

Summary

Introduction

THE CONTROL OF FLIGHT FORCE BY A FLAPPING WING: LIFT AND DRAGPRODUCTION

SANJAY P. SANE* AND MICHAEL H. DICKINSONDepartment of Integrative Biology, University of California, Berkeley, CA 94720, USA

*e-mail: sane@socrates.berkeley.edu

Accepted 18 May 2001

2608

to measure instantaneous forces on tethered insects (Cloupeauet al., 1979; Wilkin, 1990; Zanker, 1990b; Zanker and Götz,1990; Dickinson and Götz, 1996). However, forces and strokekinematics measured on tethered insects may not accuratelyrepresent those generated in free flight. Further, since tetheredflight force transducers measure whole-body forces, it is notpossible to resolve the instantaneous aerodynamic forcesgenerated by individual wings. A third approach is to calculatethe aerodynamic forces generated by arbitrary strokekinematics using computational fluid dynamics (Liu et al.,1998; Wang, 2000). However, because of the critical role ofunsteady mechanisms and three-dimensional flow structure ininsect flight aerodynamics (Ellington et al., 1996; Dickinson etal., 1999), theoretical or numerical approaches have, as yet,offered only limited insight into the aerodynamics of steering.

Given the current limitations in studies of both real animalsand numerical simulations, we have chosen to study theproblem of maneuverability using a dynamically scaled modelof a flapping insect. Aerodynamic models have provedvaluable in the study of insect flight, particularly in theidentification and analysis of unsteady aerodynamics (Bennett,1977; Maxworthy, 1979; Spedding and Maxworthy, 1986;Dickinson and Götz, 1993; Ellington et al., 1996; Dickinson etal., 1999). In large part through the use of mechanical models,researchers have identified an array of mechanisms thatcollectively account for the elevated aerodynamic performanceof flapping wings. These include the clap and fling (Speddingand Maxworthy, 1986), dynamic stall (Dickinson and Götz,1993; Ellington et al., 1996), rotational lift (Bennett, 1970;Dickinson et al., 1999) and wake capture (Dickinson, 1994;Dickinson et al., 1999). Now that the various mechanismsresponsible for the elevated aerodynamic performance ofinsect wings have been identified, it is possible to tackle thequestion of how animals manipulate such mechanisms to steerand maneuver.

In this study, we use a dynamically scaled mechanical modelof Drosophila melanogasterto investigate how changes inwing kinematics affect the production of aerodynamic forces.In particular, we explore the influence of five behaviorallyrelevant kinematic parameters: stroke amplitude, angle ofattack, the timing and duration of wing rotation and strokeplane deviation. We chose this particular set of parametersbecause fruit flies actively vary them during flight maneuvers(Götz et al., 1979; Zanker, 1990a; Dickinson et al., 1993;Lehmann and Dickinson, 1998). However, the goal of thisproject is not to replicate the precise kinematics of free flyinginsects per se, but rather to map aerodynamic forces within abroad parameter space that encompasses the variation seenamong insects.

From the instantaneous force records, we calculate time-averaged aerodynamic force coefficients, lift-to-drag ratios andother measures of aerodynamic performance. The resultantdata set is useful in identifying the kinematic parameters thatmost influence the magnitude and direction of aerodynamicforces generated by flapping wings. In a companion paper(S. P. Sane and M. H. Dickinson, in preparation), we will

extend the analysis by considering the instantaneous and time-averaged force moments generated about the yaw, pitch androll axes. The comprehensive parameter maps generated inthese studies should be of help to biologists who wish to knowthe aerodynamic consequences of observed changes in wingkinematics as well as to engineers who wish to optimize theperformance of small biomimetic flying robots. In addition,these data provide experimental validations for numericalsimulations of the fluid motion around flapping wings.

Materials and methodsMost of the instruments and procedures used in these

experiments have been described elsewhere (Dickinson et al.,1999). We fashioned the wings from 2.3 mm thick acrylicsheets using an isometrically enlarged planform of aDrosophila melanogasterwing. The proximal end of the wingwas attached to a two-dimensional force transducer thatmeasured the forces normal and parallel to the wing surface.Each force channel measured the shear encountered by twoparallel phosphor-bronze shims equipped with four 350Ωstrain gauges wired in full-bridge configuration. This designrendered the sensor nearly insensitive to the position of theforce load on the wing as well as to moments around its centralaxis. Forces generated by calibration weights placed at the tip,base, trailing edge and leading edge differed by less than 5 %.The final calibration was based on a point load at the wing’scenter of area. The proximal end of the force transducer wasattached to a gearbox capable of three degrees of rotationalmotion (Fig. 1A). The distal tip of the wing was located 25 cmfrom the center of the gearbox. The gearbox was driven viathree coaxial shafts by three stepper motors. The steppermotors were attached to the shafts by pulleys and timing beltswith a 1:10 gear reduction, such that each 4.5 ° step of themotor produced a 0.45 ° rotation of the wing. The wings, forcesensor and gearbox were immersed in a tank of mineral oil witha viscosity of 120 cSt at room temperature (approximately25 °C). The viscosity of the oil was chosen to achieve aReynolds number in the range of 102, although the exact valuevaries according to the kinematics for each trial. Since theforces on the wing are directly proportional to the density ofthe surrounding medium, the oil also serves to increase forceson the wings and to decrease the signal-to-noise ratio of theforce measurements. Mineral oil provides an additionaladvantage of electrically and thermally isolating the sensor andthus reducing noise fluctuations.

Dynamic scaling

To obtain accurate dynamic scaling of an insect, it isnecessary to keep the values of both the Reynolds number(Re=4ΦnR2ν−1AR −1, where Φis stroke amplitude, nis wingbeatfrequency, R is wing length, νis kinematic viscosity, aspect ratioAR is 4R2S−1 and S is the surface area of a wing pair; Ellington,1984c) and the reduced frequency parameter (bodyvelocity/wing velocity) constant (Spedding, 1993). For hoveringanimals as well as the model fly, the reduced frequency

S. P. SANE AND M. H. DICKINSON

2609Lift and drag production by a flapping wing

Fig. 1. Definitions of kinematic parameters. (A) Coordinate systemfor the mechanical fly wing. The cartoon shows the three Eulerangles that define the wing position at each instant in time. The meanstroke plane is a horizontal slice through the sphere described by theradial coordinates of the wing tip. Instantaneous stroke position, φ(t),is defined as the angular position of the wing in the mean stroke

plane, measured from dorsal reversal (start of downstroke) to ventral reversal (start of upstroke). Instantaneous stroke deviation, θ(t), is definedas the angle that the base-to-tip line on the wing makes with the mean stroke plane. A plane that is normal to the base-to-tip line of the wing(shown in blue) cuts through the wing at the wing chord, shown here as a line with a filled circle denoting the leading edge. The instantaneousangle of attack, α(t), is the angle that the wing chord makes with the tangent of the wing’s trajectory. (B) Sample wing kinematics plotted overtwo complete cycles. Grey and white backgrounds mark downstroke and upstroke, respectively. Stroke position (green) follows a smoothedtriangular waveform. Stroke deviation (red) varies as either a half or full sinusoid in each half-stroke. Half-sine variation yields an ‘oval’ tiptrajectory (shown in C), whereas full-sine variation yields a ‘figure-of eight’ tip trajectory. Angle of attack (blue) varies as a trapezoidalfunction. The shape of the function is determined by setting the starting point of the flip, τ0, and flip duration, ∆τ. (C) Schematic diagram of thesix parameters that were varied in the experiments: total stroke amplitude, Φ, maximum stroke deviation, Θ, mid-stroke angle of attack, α, flipstart, τ0, flip duration, ∆τ, and the shape of the wing tip trajectory. This cartoon represents a two-dimensional projection of a three-dimensionalkinematic pattern, as if viewed within the blue plane in A. The broken blue line shows the mean stroke plane. This representation of the strokeis repeated throughout the paper.

Mean stroke planeφ(t)

Start ofdownstroke

Start ofupstroke

A

Downstroke

Upstroke

α

Mean stroke plane

C

0 0.5 1 0.5 2

-100

-50

0

50

100

-50

0

50

-50

0

50

α(t)

θ(t)

φ (t)

Stroke cycle

Stroke reversal

Flip duration

Down Up Down Up

B

Half-sine

Full-sineθ(t)

α(t)

Am

plitu

de(d

egre

es)

Dev

iatio

n(d

egre

es)

Ang

le o

f at

tack

(d

egre

es)

τ0

∆τ

Φ

Θ

2610

parameter is zero by definition, since their body velocity iszero. The free-flight cruising velocity of D. melanogasterisapproximately 20cms−1 (David, 1978), while the mean velocityof the wing tip is 280cms−1 (Lehmann and Dickinson, 1997).Thus, even while flying forwards, the reduced frequencyparameter is less than 0.1, indicating that the effect of free-stream velocity on force generation should be of secondaryimportance to the velocity generated by flapping. For thesereasons, our experiments in still fluid, which match the hoveringcase, should also serve as a fair approximation for moderateforward speeds. Thus, the reduced frequency parameter is notsignificantly different for cruising D. melanogasterand ourstatic, hovering model fly. To obtain the correct range ofReynolds numbers, we used an isometrically enlarged wingplanform of an actual D. melanogasterwing to ensure that theshape parameters (Ellington, 1984a) were identical to thoseof D. melanogaster. Using available data on D. melanogastermorphology and kinematics (Lehmann and Dickinson, 1997),we estimated that a wing length of 0.25m, a surface area of thewing pair of 0.0334m2 (or 0.0167m2 for a single wing), akinematic viscosity of 120cSt and a wingbeat frequency of0.168Hz would allow us to achieve Reynolds numbers in thesame range as those of D. melanogaster.

Stroke kinematics

In the absence of wing deformation, the kinematics of thewings may be uniquely described by specifying the time courseof three angles: stroke position, φ(t), angle of attack, α(t), andstroke deviation, θ(t) (Fig. 1B). In all experiments, the angularposition of the wing within the stroke plane was described by atriangular waveform, which maintains a constant translationalvelocity throughout each half-stroke. The waveform wassmoothed to minimize inertial accelerations during strokereversal and to match more closely published stroke kinematicsfrom a variety of insects (Ellington, 1984b; Zanker, 1990a).For smoothing, we filtered the triangular waveform using azero-phase-delay low-pass two-pole Butterworth filter with acut-off frequency equal to 10 times the stroke frequency of0.17 Hz. The peak-to-peak amplitude of the stroke anglewaveform could be varied in each experiment. The angle ofattack was described by a trapezoidal wave function, whichmaintained a constant angle of attack during each half-strokeand constant rotational velocity during stroke reversal. Theshape of this waveform in each experiment was determined bysetting the mid-stroke angles of attack during the upstroke anddownstroke and by specifying the starting and stopping pointsfor wing rotation. The resulting function was then smoothedusing a low-pass filter with identical characteristics to that usedfor the stroke position waveform. We used two functions todescribe stroke deviation: an ‘oval’ pattern in which the wingtip deviated from the stroke plane according to a half-sine-wave per stroke period and a ‘figure-of-eight’ pattern in whichthe stroke deviation varied as a full sine-wave. These patternswere chosen because they roughly approximate patternsdescribed for a variety of insects (Ellington, 1984b; Zanker,1990a).

To create the kinematic patterns used in this study, we variedany or all of six parameters: (i) the stroke amplitude, (ii) themid-stroke angle of attack during upstroke and downstroke,(iii) the timing of wing rotation at dorsal and ventral reversal,(iv) the duration of the stroke reversal, (v) the shape of the wingtip trajectory (‘oval’ or ‘figure-of-eight’) and (vi) the angulardeviation from the mean stroke plane during the upstrokeand downstroke (Fig. 1C). In most of the experiments, thedeviation amplitude was set to zero, such that the wing tipremained within the stroke plane throughout the cycle. Underthese conditions, the kinematics of the wing stroke weresymmetrical such that the upstroke and downstroke weremirror images of one another. Only in trials using ‘oval’ strokedeviations were the kinematics of the two strokes not identical.The frequency of the wing stroke (0.17 Hz) remained constantin all experiments, as did the upstroke-to-downstroke durationratio, which was fixed at 1. We constructed the kinematicpatterns using a custom-designed MATLAB program(Mathworks) to convert the angular trajectories into a series ofstepper motor commands.

Force measurements

Signals from the two-dimensional sensor were acquiredusing a National Instruments data-acquisition board (modelBNC 2090) in a PC running custom-designed software writtenin MATLAB. Data were filtered on-line with an active four-pole Bessel filter with a cut-off frequency of 10 Hz and off-linewith a zero-phase-delay low-pass digital Butterworth filterwith a cut-off frequency of 3 Hz, which was 17.6 times thewing stroke frequency. Apart from increasing the high-frequency components resulting from motor jitter, increasingthe cut-off frequency of the filter did not alter the time courseof the force traces.

Each experiment consisted of one burst of four consecutivewing strokes following pre-programmed kinematics. The wingbegins the first downstroke in still fluid, whereas during thesubsequent strokes it moves through a wake created by thepreceding strokes. As a result, the time course of forcesgenerated during the first stroke is markedly different fromthose of subsequent strokes. For this reason, the data from thefirst stroke were excluded from this analysis, while those fromthe three subsequent strokes were averaged. Thus, eachpresented trace represents an average of three force records.After subtracting gravitational forces, the forces measuredfrom the normal and parallel channels were transformed intolift, drag, thrust and radial components.

Added mass

The measured force at the wing base consists ofgravitational, inertial and aerodynamic components. Thegravitational contribution of the sensor and wing mass to thetotal force signal was easily calculated and subtracted from themeasured force traces. The inertial components represent theacceleration forces on the mass of the sensor and wing as wellas the added mass of the fluid around the wing. To examinethe contribution of the inertial effects of the wing mass and

S. P. SANE AND M. H. DICKINSON

2611Lift and drag production by a flapping wing

sensor, we replaced the wing surface with an aerodynamicallyneutral inertial model of the wing. The aerodynamically neutralmodel was essentially a brass knob with the same mass andcenter of mass inside the oil as the Plexiglas wing. Becauseof its low surface area, the brass knob generated negligibleaerodynamic forces compared with the Plexiglas wing. For anyarbitrary kinematic pattern, the resulting force traces for thebrass knob could be entirely accounted for by gravity. Thus,the inertial forces generated by flapping this brass model, andtherefore the Plexiglas wing, were negligible and have beenignored. Compared with gravity and wing inertia, the non-circulatory forces due to added mass are more difficult tomeasure because the fluid acceleration induced by a movingwing changes dynamically as the wing rotates, decelerates oraccelerates (Daniel, 1984).

To estimate the magnitude of added mass, we used anapproximation derived for motions of an infinitesimally thintwo-dimensional plate in an inviscid fluid (Sedov, 1965).Using blade element method, we adapted it to the case of athree-dimensional wing rotating around an axis located at one-quarter chord length from the leading edge. The forcecontribution normal to the wing surface due to the added massinertia is given by:

where ρ is the fluid density, R is the wing length, c– is the meanchord length, rˆ and c(r) are the non-dimensional radial positionalong the wing and non-dimensional chord length, respectively(for nomenclature, see Ellington, 1984a), φ is the angularposition of the wing and α is the angle of attack. Using equation1, we calculated an estimate of added mass inertia for each setof kinematics. As illustrated by the representative trace inFig. 2, the absolute contribution of added mass to net forces onthe wing was quite small in all cases. Further, by comparingequation 1 and equation 3 (see below), it can be seen that theadded mass forces (≈ρR2c–2) scale in proportion to aerodynamicforces (≈ρR3c–) for geometrically similar wings. Thus, foridentical kinematics and geometry, added mass will have thesame physical effect on a model wing as on the wing of a fly,provided that the Reynolds number is the same. Because bothadded mass and aerodynamic contributions are biologicallyrelevant, we chose not to subtract the inertial estimates from thepresented force traces.

Experimental procedures

To examine the influence of the six kinematic variables onaerodynamic forces, we divided our analysis into three setsof experiments. First, we held the values of flip timing(symmetrical around stroke reversal), flip duration (16 % ofstroke cycle period) and stroke deviation (0 °) constant, whilewe systematically varied stroke amplitude and angle of attack.

For each amplitude from 60 to 180 ° in 20 ° increments, wevaried the angle of attack at mid-stroke from 0 to 90 ° in stepsof 10 °. From the results of this 7×10 set of experiments, wedetermined the combination of stroke amplitude and angle ofattack that generated maximum mean lift. Using these locallyoptimizing values of stroke amplitude and angle of attack, wesystematically varied the values of flip start and flip duration.Values for flip start indicate when in the stroke cycle the wingbegins to rotate relative to stroke reversal and are expressed asa fraction of total cycle time (Fig. 1C). Thus, a value of −0.5indicates that the wing begins rotation 50 % of a stroke periodprior to stroke reversal, whereas a flip start value of 0 indicatesthat the wing begins rotation at the instant of stroke reversal.Values of flip duration, the total time it takes the wing tocomplete wing rotation, are also represented as a fraction oftotal cycle time. Thus, a value of 0.2 indicates that a fliprequires 20 % of the stroke cycle to complete. In theseexperiments, we set the flip to start at various points within thestroke from −0.5 to 0 in steps of 0.05. For each value of flipstart, we varied the flip duration from 0.1 to –0.5, also in stepsof 0.05. Flip timing, τf, which describes when the mid-point ofa flip occurs within the stroke, may be calculated from:

τf = τ0+ 0.5∆τ, (2)

where τ0 is flip start and ∆τis flip duration. As before, we

(1)π16

⌠⌡

1

0rc2(r)dr −

π4

Fa,N=ρ R2c–2(φsinα+ φαcosα)

αρ c–3R⌠⌡

1

0c2(r)dr ,

0 0.5 1

-1.5

-1

-0.5

0

0.5

1

1.5

Nor

mal

for

ce (

N)

Added massinertia

Added mass inertiasubtracted

Added mass inertiaincluded

Stroke cycle

Fig. 2. Contribution of added mass inertia to total measuredaerodynamic force. The traces shown were taken from arepresentative kinematic pattern (stroke amplitude Φ=180 °,maximum stroke deviation Θ=0, angle of attack α=45 °, flip start,τ0=0.05, flip duration ∆τ=−0.1). The blue trace represents the totalnormal force measured on the wing. The static gravitationalcomponent has been subtracted. The black trace represents the addedmass inertia estimated using equation 1. Added mass inertia is zerothroughout most of the stroke because the linear velocity of the wingis constant. The red trace represents the total measured force aftersubtracting added mass inertia. The contribution of added massinertia to the measured aerodynamic forces is small, as indicated bythe similarity of the red and blue traces.

2612

determined the values of flip timing and flip duration thatmaximized mean lift within the 11×9 sets of trials. In the thirdset of experiments, we explored how aerodynamic forcesvaried with changes in deviation from the mean stroke planeusing the values of stroke amplitude, angle of attack, fliptiming and flip duration that maximize mean lift. In a set of 22experiments, the total stroke deviation was varied from−50 °(−25 ° during the downstroke, +25 ° during the upstroke) to+50 ° (+25 ° during the downstroke, −25 ° during the upstroke)in steps of 10 °. The deviation followed either a half-sine perstroke or a full-sine per stroke time course, yielding ‘oval’ and‘figure-of-eight’ trajectories, respectively (Fig. 1B).

Data analysis

Since the conventions for lift and drag existing in currentaerodynamic literature address non-flapping and primarilytwo-dimensional kinematic patterns, it is necessary tomodify them slightly for three-dimensional motions. Thesemodifications are based on the following two considerations.First, it is convenient to use a reference frame based on theinsect body rather than its wing so that the measurements relatedirectly to free-flight studies. Second, the standard conventionshould apply when the kinematics reduce to two-dimensionalmotion. With these constraints in mind, we adopted thefollowing convention: net aerodynamic force, defined as thetotal force on the wing, is resolved into three components: lift,drag and radial force. In hovering flight, lift must offset thegravitational force on the animal’s body mass. Hence, wedefine lift as the component of the net aerodynamic forceperpendicular to the mean stroke plane of the wing regardlessof its actual instantaneous trajectory. Since the mean strokeplane was horizontal in all experiments, lift is always thevertical force component. Drag is defined as the forcecomponent in the horizontal direction, opposing the wingmovement. The radial component accounts for the remainingforce component in the horizontal plane. For motions with nostroke deviation, these definitions reduce to the standardconvention: lift is orthogonal to drag, and the radial componentvanishes. With stroke deviation, the total normal pressure forceconsists of orthogonal vertical and radial components, eachorthogonal to the drag vector in the horizontal direction.

From the forces on each wing, we calculated thecorresponding mean force coefficients using an equationderived from blade element theory (Ellington, 1984c;Dickinson et al., 1999):

where F– is the magnitude of a specific force component (lift,drag, radial, total) averaged over the stroke, Φ is strokeamplitude, n is wing beat frequency, dφ/dt

—is the mean non-

dimensional angular velocity of the wing and r22(S) is the non-dimensional second moment of wing area. The radial forcecomponent changes sign when the wing crosses the meanstroke plane. As a result, in the ‘oval’ as well as the ‘figure-

of-eight’ patterns, the mean radial coefficient often averages tozero and is uninformative. For this reason, we base ourmeasurement of the average radial force coefficient on theabsolute values. Lift-to-drag ratios were calculated by dividingthe mean lift coefficient by the mean drag coefficient.Similarly, radial-to-drag ratios were obtained by dividing themean absolute radial coefficients by the mean dragcoefficients. To calculate the ratio of mean lift to profile power,we estimated mean profile power, Ppro

–, based on the time-averaged product of instantaneous drag, D(t), andinstantaneous velocity, vwing(t):

where T is the stroke period. This calculation of mean profilepower ignores the power required to rotate the wing in place.Mean lift L– was calculated as the time average of instantaneouslift throughout the stroke.

Measures of the quasi-steady-state translational forcecoefficients CL,t and CD,t were derived from 180 ° sweeps ofwing motion with fixed angles of attack, as describedelsewhere (Dickinson et al., 1999). The equations that best fitmeasured translational force coefficients as functions of angleof attack, α, for the model wing are (Dickinson et al., 1999):

CL,t = 0.225 + 1.58sin(2.13α − 7.2) , (5)and

CD,t = 1.92−1.55cos(2.04α − 9.82) . (6)

These equations are used to generate quasi-steadytranslational estimates for comparison with measured values.

ResultsThe effects of stroke amplitude and angle of attack

Fig. 3A–H shows eight sample records taken from the fullset of 70 trials to illustrate how the magnitude and time courseof aerodynamic forces vary with stroke amplitude and angle ofattack. These panels also show the lift and drag forcesreconstructed from a quasi-steady model based on measuredtranslational force coefficients. The quasi-steady predictionsprovide a conservative estimate of forces that would resultfrom a stable leading edge vortex during translation. Anydeviation of a measured trace from the quasi-steady estimaterepresents an unsteady effect that is not active duringtranslation. In all records, the net aerodynamic force is nearlyperpendicular to the wing surface throughout the stroke,indicating that shear forces measured parallel to the wingsurface are much smaller than the aerodynamic pressure forcesthat must act normal to the thin flat wing. The force trajectoriesduring the downstroke and upstroke, though very similar, arenot precisely identical because of small asymmetries in thestroke pattern introduced by the gearbox of the model. Assuggested by the reconstructions in Fig. 2, the transientundershoots in the drag traces at stroke reversal (t=0, 0.5) areexplained in part by added mass inertia.

(4)Ppro=n⌠⌡

t=T

t=0D(t)νwing(t)dt ,

(3)8F

ρΦ2n2R3c–(dφ/dt)2 r 22(S)CF= ,

S. P. SANE AND M. H. DICKINSON

2613Lift and drag production by a flapping wing

Fig. 3. Sample instantaneousforces for various combinationsof total stroke amplitude Φandmid-stroke angle of attack α. Ineach case, the wing rotationparameters were kept constant(flip duration ∆τ=0.16, flip startτ0=−0.08, flip timing τf=0). Eachpanel (A–H) shows a plot ofmeasured drag (solid red line),the quasi-steady estimate of drag(broken red line), measured lift(solid blue line) and the quasi-steady lift (broken blue line).Since the radial forces were zerofor all these kinematic patterns,they are not plotted. A two-dimensional diagram of the wingkinematics is plotted above eachset of traces using the conventiondescribed in Fig. 1C. The wingchord is shown in lightblue, and the superimposedblack vector indicates themagnitude and direction ofthe instantaneous aerodynamicforce. For convenience, thekinematic values of strokeamplitude and angle of attackused in each trial are printed inthe upper left of each panel.Values for the measured meanforce coefficients (CD

– and CL–)

are printed adjacent to each setof traces. Axis labels given in Aapply to all panels. (A,B) Forcesgenerated at a 90° angle ofattack with no wing rotation(α=90°) for a short strokeamplitude (A, Φ=60°) anda long stroke amplitude(B, Φ=180°). Note the enormoustransients in drag at the start ofeach stroke due to wake capture.(C,D) Forces generated at a 50°angle of attack for a short strokeamplitude (C, Φ=60°) and a longstroke amplitude (D, Φ=180°).The contribution of rotationalcirculation is apparent at the endof each stroke. (E,F) Forcesgenerated at a 30° angle ofattack for a short strokeamplitude (D, Φ=60°) anda long stroke amplitude(F, Φ=180°). (G,H) Forcesgenerated at a 0° angle of attackfor a short stroke amplitude(G, Φ=60°) and a long strokeamplitude (H, Φ=180°).

Φ = 60°α = 50°

-0.5

0

0.5

-0.5

0

0.5

CD = 6.19 —

CL = 0.01 —

A

-0.5

0

0.5

-0.5

0

0.5CL = 1.39 —

CD = 4.12 —

C

-0.5

0

0.5

1

1.5

-0.5

0

0.5

CD = 3.16 —

CL = 1.87 —

D

B

-0.5

0

0.5

1

1.5

2

-0.5

0

0.5

CD = 4.57 —

CL = 0.03 —

E

-0.5

0

0.5

-0.5

0

0.5

CD = 3.17 —

CL = 0.61 —

F

CL= 1.46

CD = 1.93

-0.5

0

0.5

1

1.5

-0.5

0

0.5—

—

-0.5

0

0.5

-0.5

0

0.5CL = -0.73 —

CD = 3.03 —

G

Dra

g (N

)Li

ft (N

)

-0.5

0

0.5

1

-0.5

0

0.5

1

CL = 0.20 —

CD = 1.14 —

0 0.5 1

UpstrokeDownstroke

H

Φ = 60°α = 90°

Φ = 180°α = 90°

Φ = 180°α = 50°

Φ = 60°α = 30°

Φ = 180°α = 30°

Φ = 60°α = 0°

Φ =180°α = 0°

0 0.5 1

0 0.5 1 0 0.5 1

0 0.5 1 0 0.5 1

Stroke cycle0 0.5 1

Stroke cycle0 0.5 1

Dra

g (N

)Li

ft (N

)D

rag

(N)

Lift

(N)

Dra

g (N

)Li

ft (N

)

2614

The measured force trajectories display prominent peaks atthe beginning and end of each translational stage that areabsent or smaller in the quasi-steady predictions. In isolatingthe mechanisms responsible for these transient effects, it isuseful to consider cases in which there is no wing rotation,because this removes the potential contribution of rotationalcirculation. For example, in Fig. 3A,B, the wing translatesback and forth at a 90 ° angle of attack without rotating. At thestart of each stroke, measured drag far exceeds the quasi-steadypredictions. The fact that there is no corresponding dip in dragat the end of translation indicates that the prominent dragtransient is not due to acceleration of inertial mass. Thisinterpretation is consistent with estimates of added mass inertiausing equation 1 (data not shown). Instead, the elevation indrag at the start of each stroke in Fig. 3A,B is probably due towake capture as the wing intercepts the fluid flow field inducedby the shed vorticity of the previous stroke.

The influence of rotational circulation is most easily seenwhen the angle of attack during the upstroke and downstrokeis zero (Fig. 3G,H). Under these conditions, the vorticitygenerated during the translational phase of the stroke isminimal and, thus, the magnitude of the wake capture effectshould be small. The influence of the wake is not entirelyabsent, however, because the process of rotation generates andsheds vorticity through which the wing must translate at thestart of each stroke. Further, the vorticity created by rotation isparticularly strong when the translational angle of attack iszero, because the wing must flip over by 180 ° during strokereversal, making the angular velocity of the wing particularlylarge. At the end of the each stroke in Fig. 3G,H, the influenceof rotational circulation is manifest as a transient increase inlift and drag that exceeds the quasi-steady prediction. Afterstroke reversal, the continuing rotation of the wing generatesa pressure force with opposite polarity, resulting in negativelift. The time course of this rotational effect is complicated bythe presence of an added mass inertia and a modest amount ofwake capture at the start of each stroke. The influence of thesemultiple mechanisms is manifest by the positive peak in liftimmediately following stroke reversal due to wake capture,which is followed by the negative peak due to rotationalcirculation. The pattern of an early positive peak in liftfollowed by a later negative peak is seen throughout the tracesin Fig. 3.

The rest of the traces in Fig. 3 illustrate the complexinteractions among delayed stall, rotational circulation andwake capture that result from changes in angle of attack andstroke amplitude. At angles of attack of 30 and 50 °, the winggenerates lift throughout the stroke due to delayed stall(Fig. 3C–F). The influence of rotational lift is reduced as theangle of attack increases, however, because the wing flips overa smaller arc with lower angular velocity. This effect can beseen by comparing the relative magnitude of the force peaksat the end of each stroke in Fig. 3B,D,F,H. In contrast, theinfluence of wake capture is greater at higher angles of attackbecause the vorticity shed into the wake at the end of the strokeis stronger. This effect can be seen by comparing the relative

size of the force transients at the start of each stroke in the samepanels. Thus, changing stroke amplitude and angle of attackhas a complex but interpretable influence on the magnitude ofthe different unsteady mechanisms. The kinematics thatoptimize the aerodynamic performance of the wing will reflectthese complex interactions. The maximum mean lift-to-dragratio (0.8) occurred at an angle of attack of 30 ° and amplitude

S. P. SANE AND M. H. DICKINSON

Angle of attack (degrees)0 10 20 30 40 50 60 70 80 90

60

80

100

120

140

160

180

Str

oke

ampl

itude

(de

gree

s)

Net aerodynamic force (N) 0.1 1.02

A

Angle of attack (degrees)0 10 20 30 40 50 60 70 80 90

60

80

100

120

140

160

180

Str

oke

ampl

itude

(de

gree

s)

Net force coefficient 0 6.5

B

Fig. 4. Parameter maps of net aerodynamic force and net forcecoefficient as functions of stroke amplitude and mid-stroke angle ofattack. For fixed values of wing rotation (flip duration ∆τ=0.16; flipstart τ0=−0.08, flip timing τf=0), stroke amplitude was varied from60 to 180 ° and angle of attack was varied from 0 to 90 °. In eachdiagram, the small open circles indicate the positions of actualmeasurements. Values between these measured points have beeninterpolated using a cubic spline. Values are encoded in pseudocoloraccording to the scales shown beneath each plot. This same format isused in Fig. 5, Fig. 7 and Fig. 10. (A) Net aerodynamic force, thevector sum of lift and drag, increases monotonically with increasingangle of attack and stroke amplitude. (B) Net aerodynamic forcecoefficient increases with angle of attack, but decreases with strokeamplitude.

2615Lift and drag production by a flapping wing

of 180 °. The forces corresponding to this optimal conditionare shown in Fig. 3F.

To provide a more comprehensive picture of how forceproduction changes with kinematics, we calculated the meanlift, drag and net force coefficients averaged throughout thestroke and plotted them for all pairs of stroke amplitude andangle of attack values. Fig. 4A,B depicts the mean totalaerodynamic force, FN

–, and force coefficient, CN–, in

pseudocolor maps. These two maps differ from one anotherbecause the mean force coefficient is normalized with respectto the square of stroke amplitude, which is a variable in theseexperiments. Thus, while FN

– increases with increasing strokeamplitude (Fig. 4A), CN

– decreases with increasing strokeamplitude (Fig. 4B). Both parameters rise with increasingangle of attack. The influence of stroke amplitude and angle ofattack on the mean lift coefficient CL

– and the mean dragcoefficient CD

– is shown in Fig. 5A,C. For a fixed strokeamplitude, CL

– exhibits a broad maximum ranging from 1.8 to2.0 between angles of attack of 40 and 50 °. As expected, CD

–

rises monotonically with increasing angle of attack for anygiven value of stroke amplitude. It is worth noting that the

range of CD– values is much higher than has been previously

reported for Drosophila viriliswings under steady-stateconditions (Vogel, 1967) or estimated on the basis of Reynoldsnumber (CD≈0.7; Ellington, 1984c). Although much lesspronounced than the dependence on angle of attack, lift tendsto rise, whereas drag falls, with increasing stroke amplitude.

The corresponding maps for the quasi-steady translationalestimates,CL,t

—andCD,t

—, are shown in Fig. 5B and Fig. 5D,

respectively. In general, the measured force coefficients aregreater than the predicted quasi-steady force coefficients. Thisdiscrepancy is particularly large for CD–. Further, at high strokeamplitude, the angle of attack that generates the maximumCL–is shifted by approximately 10 ° relative to the quasi-steadypredictions and by as much as 20 ° for the lower strokeamplitudes. The dependence of stroke amplitude on themeasured lift and drag coefficients is not predicted by thequasi-steady estimates. The greater difference betweenmeasured and predicted values for smaller stroke amplitudesunderscores the increased importance of rotational effectsunder these conditions. Fig. 5E indicates how the mean lift-to-drag ratio, CL

–/CD–, varies with angle of attack and stroke

Angle of attack (degrees)

0 10 20 30 40 50 60 70 80 9060

80

100

120

140

160

180S

trok

e am

plitu

de (

degr

ees)

-0.75 2.0

Lift coefficientA

0 10 20 30 40 50 60 70 80 9060

80

100

120

140

160

180

B

0 10 20 30 40 50 60 70 80 9060

80

100

120

140

160

180

Drag coefficient

0 6

0 10 20 30 40 50 60 70 80 9060

80

100

120

140

160

180

D

0 10 20 30 40 50 60 70 80 9060

80

100

120

140

160

180

-0.3 1.1

Lift/Drag

60

80

100

120

140

160

0 10 20 30 40 50 60 70 80 90

180

C E

F

Mea

sure

dQ

uasi

-ste

ady

estim

ate

Fig. 5. Parameter maps of stroke-averaged lift coefficient, drag coefficient and lift-to-drag ratio as functions of angle of attack and strokeamplitude. The data are plotted as in Fig. 4. Axis labels given in A apply to all panels. Pseudocolor scales apply to both figures in a givencolumn. The top panels (A,C,E) show maps of values measured from the mechanical model, and the bottom panels (B,D,F) show thecorresponding values calculated for a translational quasi-steady model using empirically measured force coefficients. (A,B) Measured andquasi-steady values for mean lift coefficient. (C,D) Measured and quasi-steady values for mean drag coefficient. The quasi-steady predictionsgrossly underestimate the drag coefficient. (E,F) Measured and quasi-steady values for mean lift-to-drag ratio.

2616

amplitude. The maximum value ofCL–/CD

– (0.8) occurred atstroke amplitude of 180 ° and an angle of attack of 30 °. Thecorresponding quasi-steady-state estimates of CL

–/CD– are

independent of stroke amplitude, with a maximum of 1.1 atan angle of attack of 20 °. Thus, the quasi-steady modelsignificantly overestimates aerodynamic efficiency and fails toaccount for its dependence on stroke amplitude.

The effects of flip timing and flip duration

In all subsequent experiments, we set the stroke amplitudeto 180 ° and the mid-stroke angle of attack to 45 °, which arethe values that maximized lift production in the first set ofexperiments. Next, we systematically varied the timing andduration of wing rotation to examine their effects on forceproduction. Sample force traces selected from 99 pairs of flipstart and flip duration are shown in Fig. 6A–H, with thecorresponding values of flip timing, τf, given in the upper leftcorner of each panel. A comparison of Fig. 6A,C,D illustratesthat a long-duration flip, ∆τ=0.5, can produce quite differentforces depending on when the flip occurs. If the flip occurssymmetrically about the stroke reversal (τf=0, Fig. 6A), CL

– isquite large (CL

–=1.54) and the time course is well approximatedby the quasi-steady predictions. An advanced flip (τf=−0.25,Fig. 6C) results in very low mean lift (CL

–=0.36), but producesa fairly prominent wake capture peak. In contrast, a rotationafter stroke reversal (τf=+0.25, Fig. 6D) results in meannegative lift (CL

–=−0.28) because of the adverse effects ofrotational circulation following stroke reversal.

Fig. 6 F–H shows the effects of a fast flip (∆τ=0.1) atdifferent flip times. As flip duration decreases, the aerodynamicperformance of the wing generally rises. Symmetrical andadvanced flips yield nearly identical mean lift (CL

–=1.9,symmetrical; CL

–=1.87, advanced), whereas a delayed flipgenerates somewhat lower lift (CL

–=1.52). These differencesare due primarily to the amount of lift produced at the start ofeach stroke. Early and symmetrical flips (Fig. 6F,G) result ina substantial wake capture peak at the start of translation.However, if the flip occurs very early in the stroke (τf=−0.45,Fig. 6E), the wing translates through most of the stroke atnegative angles of attack, leading to a large value of negativelift (CL

–=−1.41; Fig. 6E). When rotation is delayed, the wakecapture peak is missing, revealing two negative peaks at thestart of translation, an early small peak due to added massinertia and a later more prominent peak due to rotationalcirculation (Fig. 6H).

The maps of mean force coefficients as a function of flipduration and flip start are shown in Fig. 7A,C,E, withcomparable quasi-steady, translational predictions shown inFig. 7B,D,F. Flip timing, the non-dimensional time when themid-point of the flip occurs, is indicated by the inclined parallellines on each graph. Both CL

– and CD– are strongly influenced

by flip timing and duration. For example, at a flip duration of0.1, CL

– varies with flip timing from as low as −1.5 to as highas +2. The comparable values of the quasi-steady translationalestimate, CL,t

—, also vary, but over a smaller range (from −1 to

1.6).

For all values of flip timing, CD– increases monotonically

with flip duration (Fig. 7C). Measured values range from 2.6and 4.1, representing a somewhat smaller variation than wasseen with CL

–. However, unlike the case with lift, thediscrepancy between measured CD

– and the quasi-steadyestimate, CD,t

—, is substantial. In addition to generally

underestimating the magnitude of drag, the quasi-steadypredictions fail to observe the local rise in drag along the −0.25flip timing iso-line. Since the range of variation for drag is lessthan that for lift, the measured lift-to-drag ratio map resemblesthe lift coefficient map (Fig. 7A,C,E). Further, because thequasi-steady translational predictions underestimate lift anddrag by approximately the same proportion, the predicted lift-to-drag ratio map is quite similar to the measured map(Fig. 7E,F). The map for the net force coefficient (Fig. 7G)resembles the drag map (Fig. 7C), which is expected since thevalues for the mean drag coefficient are, on average, twicethose for the lift coefficient at comparable points on thekinematic surface.

The effects of stroke plane deviation

Using kinematic values for stroke amplitude, angle of attack,flip duration and flip start that maximized lift production(Φ=180 °, α=45 °, ∆τ=0.1 and τf=−0.05), we tested how forcesvary with deviation from the mean stroke plane in a set of 22experiments. The peak-to-peak magnitude of stroke deviationwas varied from−50 to +50 ° in 10 ° steps for both the half-sine (‘oval’) and full-sine (‘figure-of-eight’) patterns. It isworth noting that, in the oval pattern, an upward deviation atthe start of the downstroke requires a downward deviation atthe start of the upstroke and vice versa. This is not the case forthe figure-of-eight pattern, in which the two half-strokes aremirror images of one another.

Fig. 8 shows a selection of force traces resulting fromdifferent patterns of stroke deviation. In general, the figure-of-eight pattern had a more profound influence on the magnitudeand time course of force production than did the oval pattern.In both cases, however, the direction of stroke deviation at thestart of each translational phase greatly influenced themagnitude of the force transient at the start of the stroke. Forexample, in the figure-of-eight pattern shown in Fig. 8C, eachstroke begins with an upward motion, and the lift and dragtransients at the start of each stroke are quite small. In contrast,the comparable kinematic pattern that starts with a downwardmotion (Fig. 8D) generates sizeable force peaks. A similartrend is seen in the oval patterns (Fig. 8A,B). The upstroke inFig. 8A and the downstroke in Fig. 8B, which both start witha downward motion, are marked by sizeable force peaks at thestart of translation, whereas the strokes that begin with upwardmotion are not. This dependence of the early force transient onthe direction of deviation is explained in part by an increase inthe aerodynamic angle of attack caused by the downwardmotion of the wing. However, the measured force peaks aremuch greater than the quasi-steady estimates, which take intoaccount this effect, suggesting that there is a substantial wakeeffect at the start of each stroke. The influence of the wake is

S. P. SANE AND M. H. DICKINSON

2617Lift and drag production by a flapping wing

Fig. 6. Sample instantaneous forcesfor various combinations of flipstart τ0 and flip duration ∆τ. Inall kinematic patterns, strokeamplitude was 180 ° and angle ofattack was 45 °. The format foreach panel is that described for Fig.3. As in Fig. 3, the radial forces forall these kinematics are zero andhave not been plotted. (A) Forcesgenerated with a slow flip (∆τ=0.5),symmetrical with respect to strokereversal (τ0=−0.25, flip timingτf=0). Under these conditions, thequasi-steady model (broken lines)accurately predicts measured lift,but not drag. (B) Forces generatedwith moderate flip duration(∆τ=0.25), advanced with respectto stroke reversal (τ0=−0.25,τf=−0.125). With these kinematics,the augmentation of lift byrotational circulation and wakecapture is evident. (C) Forcesgenerated with a long, advancedflip (∆τ=0.5; τ0=−0.5, τf=−0.25).This pattern of kinematicsproduced elevated drag due to wakecapture at the start of each stroke.(D) Same kinematics as in C, butwith a delayed flip (∆τ=0.5; τ0=0,τf=+0.25). The delay in flip timingcauses a small decrease in meandrag, but an enormous decrease inlift. (E–H) The influence ofrotational timing on a short-duration flip. (E) Forces generatedby a short flip advanced by almost afull half-cycle with respect tostroke reversal (∆τ=0.1; τ0=−0.5,τf=−0.45). Note that the angle ofattack is negative during most oftranslation because the wing flipsmuch too soon. As a consequence,the pattern generates negative lift.(F) Forces generated by a slightlyadvanced short flip (∆τ=0.1;τ0=−0.1, τf=−0.05). This near-optimal pattern augments liftby both rotational mechanisms.(G) Forces generated by a shortsymmetrical flip (∆τ=0.1; τ0=−0.05,τf=0). (H) Forces generated by aslightly delayed short flip (∆τ=0.1;τ0=0, τf=0.05). The small delay of0.05 decreases the mean liftcoefficient by 20 % compared withthe symmetrical case shown in G.CD–, mean drag coefficient; CL

–,mean lift coefficient.

-0.50

0.51

-0.50

0.51 CL = 1.47

CD = 3.28

-0.50

0.51

1.5

-0.50

0.5

-1

CL = -0.28

CD = 3.87

-0.50

0.51

1.5

-0.50

0.51

1.5

00.5

-0.5

-0.50

0.51

1.52

-0.50

0.5

-0.50

0.51

CL = 1.87

CD = 2.85

CL = 1.9

CD = 2.66

CD = 2.66

CL = 1.52

0.50

0.51

1.5

0.50

0.5CL = 1.54

CD = 3.82

A

-0.50

0.51

1.52

-0.50

0.5 CL = -1.41

CD = 2.64

E

G

B

D

F

H

-0.50

0.51

1.5

-0.50

0.51

CL = 0.36

CD = 4.06

C

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

τ0 = –0.25∆τ = 0.5τf = 0

τ0 = –0.25∆τ = 0.25τf = –0.125

τ0 = –0.5∆τ = 0.5τf = –0.25

τ0 = 0∆τ = 0.5τf = +0.25

τ0 = –0.5∆τ = 0.1τf = –0.45

τ0 = –0.1∆τ = 0.1τf = –0.05

τ0 = –0.05∆τ = 0.1τf = 0

τ0 = 0∆τ = 0.1τf = 0.05

UpstrokeDownstroke0 0.5 1

Lift

(N)

Dra

g (N

)Li

ft (N

)D

rag

(N)

Lift

(N)

Dra

g (N

)Li

ft (N

)D

rag

(N)

Stroke cycle0 0.5 1

Stroke cycle0 0.5 1

0 0.5 1

0 0.5 1 0 0.5 1

0 0.5 1 0 0.5 1

2618

stronger if the wing moves downwards, towards thedescending vorticity of the previous stroke, than if it movesupwards, away from the wake. In all cases, the significance ofradial forces increases with increasing deviation. As expected,

like the lift and drag forces, the time course of radial forces isalso dependent on the shape of the wing trajectory.

Because of the mirror symmetry in upstroke and downstrokekinematics for ‘oval’ kinematics (Fig. 8A,B), the radial forces

S. P. SANE AND M. H. DICKINSON

Lift/Drag

0.10 0.20 0.30 0.40 0.50

0

-0.10

-0.20

-0.30

-0.40

-0.50

Drag coefficientLift coefficient

Non-dimensional flip duration

Non

-dim

ensi

onal

flip

sta

rt

0.10 0.20 0.30 0.40 0.50

0

-0.10

-0.20

-0.30

-0.40

-0.50

0

-0.1

-0.20

0.1

0.2

G

0.10 0.20 0.30 0.40 0.50

0

-0.10

-0.20

-0.30

-0.40

-0.50

3.2 4.5

Net force coefficient

0.10 0.20 0.30 0.40 0.50

0

-0.10

-0.20

-0.30

-0.40

-0.50

-1.5 2.0

0.10 0.20 0.30 0.40 0.50

0

-0.10

-0.20

-0.30

-0.40

-0.50

2.0 4.2

0.10 0.20 0.30 0.40 0.50

0

-0.10

-0.20

-0.30

-0.40

-0.50

-0.5 0.7

0.10 0.20 0.30 0.40 0.50

0

-0.10

-0.20

-0.30

-0.40

-0.50

A

B

C

D

E

F

Mea

sure

dQ

uasi

-ste

ady

estim

ate

0

-0.1

-0.20

0.1

0.2

0

-0.1

-0.20

0.1

0.2

0

-0.1

-0.20

0.1

0.2

0

-0.1

-0.20

0.1

0.2

0

-0.1

-0.20

0.1

0.2

0

-0.1

-0.20

0.1

0.2

Non

-dim

ensi

onal

flip

tim

ing

Non

-dim

ensi

onal

flip

sta

rt

Non

-dim

ensi

onal

flip

tim

ing

Non-dimensional flip duration

Fig. 7. Parameter maps of force coefficients as a function of flip start and flip duration. Each map was generated from a 9×11 array of kinematicpatterns. For all experiments, the stroke amplitude Φ and angle of attack αwere held constant (Φ=180 °, α=45 °), while flip duration and flipstart were systematically varied. The data and interpolated values are plotted as in Fig. 4 and Fig. 5. Isolines of flip timing (given by equation2) are indicated by the diagonal lines in each panel and correspond to the red labels on the right axis. The top panels (A,C,E) show themeasured force coefficients, and the bottom panels (B,D,F) show values for the quasi-steady-state estimated coefficients. Axis labels given in Aapply to all panels. (A,B) Measured and quasi-steady values for stroke-averaged mean lift coefficient. The pseudocolor scale for both panels isshown below the parameter map in B. (C,D) Measured and quasi-steady values for stroke-averaged mean drag coefficient. Note the largediscrepancy between the estimated and measured values. (E,F) Measured and quasi-steady values for stroke-averaged mean lift-to-drag ratio.(G). Measured values for stroke-averaged mean net force coefficient. Note the strong similarity to the drag coefficient map.

2619Lift and drag production by a flapping wing

Fig. 8. Sample instantaneous forces forvarious combinations of stroke deviationand tip trajectory. All other kinematicvariables were constant (amplitudeΦ=180 °, angle of attack α=45 °, flipduration ∆τ=0.5, flip timing τf=−0.05).The format for each panel is thatdescribed for Fig. 3 and Fig. 6, exceptfor an additional panel showing theinstantaneous radial forces (green).(A) Forces generated with a large ovaldeviation in which the downstroke startswith upward motion and the upstrokestarts with downward motion Ddev=+25 °,Udev=−25 °, where Ddev is equal to themaximum angle of downstroke deviationand Udev indicates maximum angle ofupstroke deviation. Both lift and dragtransients are higher at the start of theupstroke when the wing travelsdownwards than at the start of thedownstroke. The absolute average radialforces are also correspondingly large.(B) Reversed condition compared with A,the downstroke starts with downwardmotion and the upstroke starts withupward motion (Ddev=−25 °, Udev=+25 °).Lift and drag transients are now muchlarger at the start of the downstroke.(C) Forces generated with a figure-of-eight deviation in which both strokes startwith upward motion (Udev=Ddev=+25 °).Both lift and drag are low at the start ofeach stroke, but reach elevated values atmidstroke. (D) Reversed conditioncompared with C, both strokes start withdownward motion (Udev=Ddev=−25 °).Both strokes now start with largetransients in both lift and drag. (E) Forcesgenerated by comparable kinematicpattern to those in A–D but with nostroke deviation.

Ddev=–25°Udev=+25°

CL = 1.46

CD = 2.47

-0.50

0.51

1.5

-0.50

0.51

Ddev=+25°Udev=+25°

CL = 1.34

CD = 2.23

-0.50

0.51

1.5

-0.50

0.51

Ddev=–25°Udev=–25°

CL = 1.27

CD = 2.18

-0.50

0.51

1.5

-0.50

0.51

Ddev=+25°Udev=–25°

CL = 1.62

CD = 2.68

-0.50

0.51

1.5

-0.50

0.51

Lift

(N)

CL = 1.88

CD = 2.88

Ddev=0°Udev=0°

-0.50

0.51

1.5

-0.50

0.51

Dra

g (N

)

A

B

C

D

E

—

—

—

—

—

—

—

—

—

—

Downstroke Upstroke0 0.5 1

Stroke cycle

0.50

-0.5

0.5

0-0.5

0.50

-0.5

0.50

-0.5

0.5

-0.50

Rad

ial (

N)

|CR| = 0.38 —

|CR| = 0 —

|CR| = 0.48 —

|CR| = 0.37 —

|CR| = 0.53 —

Lift

(N)

Dra

g (N

)R

adia

l (N

)

Lift

(N)

Dra

g (N

)R

adia

l (N

)

0 0.5 1 0 0.5 1

0 0.5 1 0 0.5 1

2620

during the upstroke are equal and opposite to the radial forcesduring the downstroke. In the ‘figure-of-eight’ kinematics,however, the upstroke and downstroke are identical and, hence,the radial forces during the upstroke are identical to the radialforces during the downstroke.

Fig. 9 summarizes the effects of stroke deviation on thetime-averaged measured forces and quasi-steady predictions.While the influence of stroke deviation on the time course ofthe aerodynamic forces is large, its impact on the mean forcecoefficients is surprisingly small. This indicates that thedifferences in the dynamics of force production noted in Fig. 8tend to average out over the stroke. For both the ‘oval’ and‘figure-of-eight’ deviation trajectories (Fig. 9A), the mean liftand drag coefficients decreased with increasing absolutedeviation (Fig. 9C,D). The changes in average performance for

the ‘oval’ deviation pattern should be symmetrical around zerodeviation, since oval patterns with positive and negativedeviations are mirror images of one another. Thus, thedownstroke in Fig. 8A should resemble the upstroke inFig. 8B, and the upstroke in Fig. 8A should resemble thedownstroke in Fig. 8B. The asymmetry in these measurementsresults from the mechanical play in the gear mechanism of therobot. However, the asymmetry in the performance of the‘figure-of-eight’ patterns around zero deviation represents, inpart, a real aerodynamic effect (Fig. 9B,C,D). In this case, apositive deviation will result in a downward motion at the startof both the upstroke and downstroke, whereas a negativedeviation indicates upward motion at the beginning of bothstrokes. Downward deviation should enhance wake capture, asdescribed above. Values of CL

– and CD– fall off faster with

S. P. SANE AND M. H. DICKINSON

E

Net

forc

e co

effic

ient

0

1

2

3

4

D

Lift

coe

ffiec

ient

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

Lift/

drag

rat

io

Dra

g co

effic

ient

1.6

1.8

2

2.2

2.4

2.6

2.8

3

A

B

-50 -25 0 25 50Net deviation (degrees)

Positivedeviation

Negativedeviation

C

Oval

Figure-of-eight

-50 -25 0 25 50Net deviation (degrees)

-50 -25 0 25 50Net deviation (degrees)

-50 -25 0 25 50Net deviation (degrees)

Downstroke

Upstroke

Rad

ial/d

rag

ratio

0

0.2

0.4

0.6

0.8

1

Fig. 9. Effects of stroke deviationon mean force coefficients.(A) Cartoon illustrating thekinematic patterns used to studystroke deviation. Throughout thefigure, blue data points refer tooval kinematic patterns, and reddata points refer to figure-of-eightpatterns. Measured and quasi-steady predicted values are givenby filled and open circles,respectively. (B) The magnitudeof the mean net aerodynamic forcecoefficient is only mildlyinfluenced by stroke deviation.(C) Measured mean lift coefficientdecreases with stroke deviation.For the figure-of-eight patterns,the drop in performance withincreasing deviation is greater forstrokes that begin with upwardmotion. (D) Drag coefficientdecreases with stroke deviation.Note the large discrepancybetween measured values of meandrag and quasi-steady predictions.(E) Ratio of lift-to-drag and radial-to-drag forces as a function ofstroke deviation. Radial-to-dragforces for oval kinematic patternsare represented by filled blackcircles and for figure-of-eightpatterns by open black circles.Because deviation affects lift anddrag almost equally, the influenceon their ratio is quite small. Theradial forces are dependent on thesine of stroke deviation angle.Because of the linearity of sinefunctions for small angles, thevalues of radial-to-drag forceincrease linearly with increasingabsolute deviation.

2621Lift and drag production by a flapping wing

increasing positive deviation close to zero deviation. However,at large deviations, the coefficients for negative deviations arelower than the coefficients for positive deviations (Fig. 9C,D).There is little effect on the CL

–/CD– ratio because the influence

of stroke deviation is nearly identical for both lift and drag. Incontrast, because of the linear nature of sine functions at lowangles, CL

–/CD– ratios appear linear with the small range of

stroke deviation in our experiments (Fig. 9E).As with the earlier experiments, the measured values of both

CL–and CD

–are higher than quasi-steady translational estimatesdue to unsteady effects (Fig. 9C,D). Lift and drag are almostequally underestimated, which explains why the predicted lift-to-drag ratio (Fig. 9E) is only slightly higher than the measuredratio. For oval trajectories, the quasi-steady predictions for themean force coefficients behave as scaled-down versions of themeasured traces. For example, both quasi-steady and measuredmean force coefficients are maximal at zero stroke deviationand decrease for increasing absolute deviation (Fig. 9C,D).However, for figure-of-eight trajectories, the quasi-steadytranslational model is much less accurate in predicting thechanges in the mean force coefficients with stroke deviation.For example, the quasi-steady model predicts that mean liftshould exhibit a local maximum at a stroke deviation of +20 °,whereas the measured maximum occurs at 0 ° deviation.Similarly, the estimated drag increases monotonically withincreasing positive deviation, whereas the measured drag ismaximal at 0 ° deviation.

Ratio of mean lift to mean profile power

Within the range of Reynolds numbers relevant for fruitflies, the total mechanical power required to flap the wings isdominated by profile power, Ppro

–, the cost to overcome dragon the flapping wings (Lehmann and Dickinson, 1997).Fig. 10A–C shows how Ppro

–, estimated using equation 6, varieswith changes in the kinematics parameters. Fig. 10D–F showsthe corresponding ratios of mean lift to mean profile power forthe same kinematics. The Ppro

–estimates vary extensively, evenwithin subregions of the parameter maps in which the valuesof CL

– are high enough to support flight. This result suggeststhat it may be difficult to estimate mechanical power in free ortethered flight solely on the basis of measures of strokeamplitude. In particular, Ppro

–varies extensively with the timingof wing rotation and the angle of attack, parameters that arerevealed only by extensive three-dimensional reconstructionsof wing kinematics.

DiscussionWe have used a dynamically scaled model wing to measure

both instantaneous and stroke-averaged lift and drag forces fora wide variety of behaviorally relevant kinematic patterns ona flapping wing. The results underscore the importance of threeunsteady mechanisms in flapping flight: delayed stall,rotational circulation and wake capture. The processes areimportant not simply because they help to explain the liftrequired to keep an animal aloft but also because their

sensitivity to subtle changes in stroke kinematics may help toexplain the extreme maneuverability of many insects. Bysystematically measuring flight forces within a comprehensivekinematic space, we have begun to untangle the complexinteractions among these force-generating mechanisms.

Comparison between the mechanical model and real flies

The maximum mean unsteady lift coefficients in this studyare in excellent agreement with measurements on tetheredDrosophila spp., which generated elevated flight force inresponse to optomotor stimuli. Peak CL

–values in tethered flightwere 1.9 (Lehmann and Dickinson, 1998), which is preciselythe same maximum value measured on the model wing.Tethered flies generating just enough force to support bodyweight produce a CL

–value of 1.6. The tethered flight estimatesrelied on the assumption that both up- and downstrokesgenerated lift, which appears to be correct given the timecourse of forces generated when Drosophilaspp. kinematicsare played through the model (Dickinson et al., 1999).Tethered Drosophilaspp. flap with morphologically maximumstroke amplitude when producing peak lift (Lehmann andDickinson, 1998), which is consistent with the present results.Unfortunately, apart from stroke amplitude, we do not haveadequate knowledge of other kinematic parameters during thepeak performance of real flies to compare them with the valuesthat maximized lift on the model.

As seen in Fig. 5B, the maximum quasi-steady translationalestimate of the mean lift coefficient,CL,t

—, was 1.6. Given that

this value is sufficient to explain the forces required for a fruitfly to hover (Lehmann and Dickinson, 1998), it is tempting toclaim that the quasi-steady estimates are sufficient to explaininsect flight (Jensen, 1956). However, a simple comparison oftime-averaged lift coefficients is not a robust test of the quasi-steady model. While the mean values might be similar, the timehistories of measured forces and quasi-steady translationalestimates differ greatly (Fig. 3, Fig. 6, Fig. 8). Further, theprecise time history of lift and drag is critical for calculatingforce moments and, thus, essential to considerations of stabilityand maneuverability. In addition, since many insects can flywhile supporting forces nearly twice their body weight(Marden, 1987), any model must explain not only the forcesrequired to hover, but also those required for maximal lift.

The inadequacies of the translational quasi-steady modelare even more apparent when considering drag. In allexperiments, we observed that the measured drag coefficientswere greater in magnitude than those estimated frommeasured translational force coefficients. Also, the range ofdrag coefficients measured here (CD=0–6.5) is substantiallyhigher than previously reported measurements from both realand model Drosophila viriliswings (CD=0.2–1; Vogel, 1967).The previous experimental values were based on steady-statemeasurements and thus excluded the contribution of rotationalcirculation and wake capture as well as added mass. However,even the range of steady-state coefficients in the previousstudy (0<CL<1, 0<CD< 1;Vogel, 1967) is substantially lowerthan the quasi-steady estimates in this study (0<CL,t<1.91;

2622

0.37<CD,t<3.47). At present, the reason for this discrepancyis not clear.

The relative importance of unsteady mechanisms

The relative contributions of delayed stall, rotational

circulation and wake capture to total force production varywith the precise kinematics of the stroke. In general, theimportance of delayed stall increases with stroke amplitudebecause the wing can integrate the influence of the leadingedge vortex over a greater distance. In contrast, rotational

S. P. SANE AND M. H. DICKINSON

0 10 20 30 40 50 60 70 80 9060

80

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Angle of attack (degrees)

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-1)

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_

Ppr

o (k

g m

2 s–3

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Fig. 10

2623Lift and drag production by a flapping wing

effects become more important with lower stroke amplitude.For kinematic patterns mimicking those of hoverflies with a60 ° stroke amplitude, rotational effects account for more thanhalf the total force (Dickinson et al., 1999). Because of thesedifferences, quasi-steady translational estimates should moreclosely resemble the measured values for kinematic patternswith large stroke amplitudes and deviate from measured valuesfor those with lower stroke amplitudes. These predictions areborne out by the force traces shown in Fig. 3 and the maps ofCL–in Fig. 5A,B. The quasi-steady translational model predictsmaximum lift at an angle of attack of 45 °. While this is closeto the maximum value for the measured forces at high strokeamplitude, the measured lift maximum is shifted to higherangles of attack (60 °) at low stroke amplitude.

There are two possible explanations for this shift. First,rotational circulation might make a greater contribution tomean lift at low stroke amplitude. However, this is unlikely tobe true for symmetrical flips, for which rotational circulationenhances lift only before stroke reversal, but attenuates it afterstroke reversal (Dickinson et al., 1999). Also, at strokereversal, the wing rotates less for a 60 ° angle of attack thanit does for a 45 ° angle of attack. Thus, the proportionalcontribution due to rotational lift is further minimized. Second,the shift in the lift maximum may reflect an increasingimportance of wake capture at low stroke amplitude. The largecontribution of wake capture can be easily seen in the forcesgenerated by hoverfly-like patterns (Dickinson et al., 1999) aswell as in the drag traces in Fig. 3. The increasing importanceof wake capture also explains the changes in drag coefficientsat low stroke amplitude. While the translational quasi-steadymodel predicts that drag should be independent of amplitude(Fig. 5D), the measured mean drag coefficient clearlyincreases at smaller stroke amplitudes.

Separating the effects of wing rotation from wake capture

Since rotational circulation, wake capture and added massusually occur together during stroke reversal, it is oftendifficult to separate these effects. After estimating the effect of

added mass on force traces before and after subtraction(Fig. 2), we concluded that, at these Reynolds numbers, themagnitude of added mass is small compared with rotationalcirculation or wake capture. To isolate these two mechanisms,it is helpful to focus on kinematic patterns in which either theentire flip occurs prior to stroke reversal (Fig. 6B,E,F) or thewing does not flip at all (Fig. 3A,B). In the case of an advancedflip, the force peak that exceeds the translational estimate priorto stroke reversal may be attributed to rotational circulation,while the force peak after stroke reversal is due to wakecapture. In the case of no flip, the large drag peak at the startof each stroke is due to wake capture, in which vorticity shedfrom the previous stroke elevates force by inducing an increasein flow velocity towards the wing (Dickinson, 1994; Dickinsonet al., 1999). Because of the squared dependence of forces onrelative velocity, even small changes in flow can cause a largeelevation in force.

The influence of wake capture should be reduced when thewing translates at a 0 ° angle of attack during the previousstroke. Under these circumstances, the wing sheds lessvorticity at the end of each stroke, since it did not create aleading edge vortex during translation. There is some vorticitypresent, however, due to rotational circulation at the end of thestroke. Nevertheless, in these cases (Fig. 3G,H), wake captureis reduced, which partially isolates the influence of rotationalcirculation. The results indicate that a wing rotating earlyenhances lift, but that if the rotation continues after strokereversal the force dips below the quasi-steady-state predictionbecause, after changing direction, rotational circulation will actto counter the circulation produced by translation.

By systematically varying the kinematic parameters, wedetermined that lift was optimized at a 45 ° angle of attack,180 ° stroke amplitude, 10 % flip duration and approximately−5 % flip advance. Even subtle deviations from the optimalcombination drastically decreased lift. A 5 % delay in fliptiming relative to the optimal conditions (Fig. 6H) is lesseffective in generating rotational lift and decreases the averagecoefficient by as much as 25 %. Small decreases in flip durationcause comparable changes. These results suggest that thecontrol of flip timing and duration provides an easy andpowerful way of modulating the forces on two wings, as isrequired to alter force moments during steering. Tethered fliesadvance wing rotation on the outside of a turn in response toa turning visual stimulus (Dickinson et al., 1993). Since earlyrotation enhances whereas delayed rotation attenuates lift,early wing rotation on the outside of a turn should produce agreater force on that side, thus creating an appropriate turningmoment in the opposite direction. While flip advance is usuallycoupled with an increase in stroke amplitude, flies do possessthe ability to decouple the modulation of rotation andamplitude (Dickinson et al., 1993). A more thorough analysisof steering must consider the influence of wing kinematics onforce moments, which will be the topic of a separate paper (S.P. Sane and M. H. Dickinson, in preparation)

In nearly all the experiments presented here, the kinematicsof the upstroke and downstroke were identical. Under these

Fig. 10. The effects of wing kinematics on profile power Ppro– and the

lift-to-power ratio L–/Ppro–. Procedures for plotting data are as

described in Fig. 4, Fig. 7 and Fig. 9. The pseudocolor scale for Aand B is shown below B, and the scale for D and E is shown belowE. (A) Profile power as a function of stroke amplitude and angle ofattack. (B) Profile power as a function of flip start and flip duration.Flip timing is shown on the right axis. Profile power varies by afactor of 2 within the parameter space, indicating that flip kinematicsare important determinants of flight cost. (C) Profile power as afunction of stroke deviation for oval (blue) and figure-of-eight (red)patterns. Stroke deviation has only a minor effect on profile power.(D) The ratio of mean lift to mean profile power, L

–/Ppro–, as a

function of stroke amplitude and angle of attack. Like the mean liftcoefficient (Fig. 5A), there is a single angle of attack that maximizesL–/Ppro– for each value of stroke amplitude. (E) The ratio of mean lift

to mean profile power as a function of flip start, flip duration and fliptiming. (F) The ratio of mean lift to mean profile power as a functionof stroke deviation for oval (blue) and figure-of-eight (red) deviation.

2624

conditions, the contributions of the upstroke and downstroketo forces are equal, and the net force vector is oriented normalto the mean stroke plane. However, symmetrical wing strokesare probably more the exception in actual insects than the rule.Thus, further experiments must focus on asymmetric strokesthat can cause the net force vector to be tilted relative to thestroke plane. Adjusting the symmetry of the two strokes isundoubtedly critical for the regulation of force moments inflight control.

The influence of stroke deviation

The wings of insects do not always beat back and forthwithin a flat stroke plane, but may exhibit large and complexdeviations. Within studies of flies, the true form and functionof these deviations have been subject to much debate (for areview, see Dickinson and Tu, 1997). While the existence ofoval and figure-of-eight trajectories was often attributed todifferences in experimental procedure (tethered versusfreeflight, thoracic versusabdominal tether, still air versusmovingair), studies of neuromuscular control indicate that flies havethe ability actively to modulate stroke deviation by alteringthe activity of steering muscles (Tu and Dickinson, 1994;Lehmann and Götz, 1996). The aerodynamic utility of thesechanges, however, is still not known.

We can divide the effects of stroke deviation onaerodynamic mechanisms into three categories: theintroduction of radial force components, alteration ofaerodynamic angle of attack and the efficacy of wake capture.The presence of one or more of these effects can substantiallyalter the dynamics of force generation even when the time-averaged net force produced by the wing remains largelyunaltered (filled circles, Fig. 9B). For three-dimensionalflapping, the main influence of stroke deviation on flight forcesis the presence of radial force components acting orthogonalto lift and drag (Fig. 8A–E). This force component, althoughignored in much of the insect flight literature, is of considerableimportance during large stroke deviations. Although themagnitude of the average net force vector remains nearlyconstant with changes in deviation, both lift and dragcoefficients decrease with increasing positive and negativestroke deviations, while the radial component increases. In thisway, deviation offers a versatile means of manipulating forcesthrough simple geometric alterations in stroke trajectory.

Stroke deviation also influences force production by alteringthe angle of attack with respect to the wing path as well as itsinstantaneous velocity. Upward motion of the wing at the startof the stroke results in a decrease in the angle of attack, whiledownward motion causes an increase in the angle of attack.The converse is true at the end of the stroke. In addition, thevelocity of the wing is higher, because the wing must cover agreater distance when it follows a curved trajectory for a givenstroke frequency. This effect is clear in the quasi-steadyprediction for the oval wingbeat patterns (Fig. 8A,B). Thepredicted forces are higher at the start of strokes beginningwith downward motion and at the end of strokes that finishwith downward motion. However, a comparison of measured

and predicted force traces indicates that the wing generateslarge force transients at the start of strokes beginning withdownward motion that cannot be explained by the changes inaerodynamic angle of attack and wing velocity, which areincorporated in the quasi-steady model (Fig. 8B,D). The mostlikely explanation for these large peaks is that wake capture isenhanced by downward motion and attenuated by upwardmotion at the start of the stroke. By moving downwards in thedirection of the descending wake, the wing may intercept astronger flow field than if it moved upwards. However, arigorous test of this hypothesis will require a detailedvisualization of the velocity distribution during stroke reversal.

The interaction between stroke deviation and wake captureis more extreme in the case of figure-of-eight patterns, in whichthe two half-strokes are mirror images of one another and canthus both begin with downward motion. The quasi-steadymodel predicts that mean lift should be maximal with a +20 °deviation, which means that the wing begins each stroke bymoving upwards. The enhanced performance of upwardstrokes is due to the fact that the angle of attack is high in themiddle of the stroke when the wing velocity is greatest. Themeasured forces do not, however, exhibit this enhancedperformance with small positive deviations. On the contrary,the net performance drops off more quickly with positivedeviation (upward motion at start of strokes) than with negativedeviation (downward motion at start of stroke). This result isagain consistent with a more effective wake capture due todownward motion at the start of translation.

In spite of the large differences in instantaneous forces, themagnitudes of time-averaged forces in all cases are similar forthe oval and figure-of-eight trajectories (Fig. 9). For ovaltrajectories, the measured forces behave as predicted, withmaximum lift and drag coefficients occurring at zero deviationand decreasing for higher absolute deviation. In contrast, thefigure-of-eight trajectories behave quite differently from thecorresponding quasi-steady estimates, varying in bothmagnitude and behavior. For quasi-steady estimates, the liftmaximum occurs at a 20 ° positive deviation, whereas the dragmaximum increases monotonically with increasing positivedeviations (open red circles in Fig. 9C,D respectively). Incontrast, the measured values in both cases are maximal at zerodeviation (filled blue and red circles in Fig. 9C,D) and thendecrease for higher absolute deviations.

Drag and efficiency

The values of mean profile drag C—

D,pro—

obtained in thecurrent study are much higher than values in the literaturebased either on steady-state measurements of wing profiles(Vogel, 1967) or calculations based on Reynolds number(Ellington, 1984d; Lehmann and Dickinson, 1997; Wakelingand Ellington, 1997). In particular, the equation:

C—

D,pro—

= 7/√Re—

, (7)

which represents the rough average of the Reynolds-number-dependence of a flat plate parallel and perpendicular to flow(Ellington, 1984c), may grossly underestimate the profile drag

S. P. SANE AND M. H. DICKINSON

2625Lift and drag production by a flapping wing

on a flapping wing. In the current study, this equation predictsa C—

D,pro—

value of approximately 0.7 for large stroke amplitudes,which is less than 25% of the measured values for kinematicpatterns capable of sustaining flight (CD,t

— ≈3). One obviousreason for this large discrepancy is that the process of wingrotation generates large instantaneous values of drag which, eventhough lasting only a small portion of the stroke, substantiallyincrease the mean values. However, even the drag contributedby delayed stall during translation (≈2) is higher than thatpredicted by equation 7. These large values for the mean dragcoefficient have several important implications for flightenergetics and muscle physiology. First, profile power, whichscales linearly with the mean drag coefficient, may be severaltimes greater than predicted in estimates using equation 7. Thus,under many conditions, profile power, and not induced power,may represent the dominant component in aerodynamic power.Further, estimates of total mechanical power based solely oninduced power are likely grossly to underestimate flight costs.The elevated values of drag also call into question previousattempts to estimate elastic storage, which depend upon inertialpower costs being high relative to aerodynamic costs (Dickinsonand Lighton, 1995). If the ratio of aerodynamic cost to inertialcost is high, the energetic benefits of elastic storage are greatlyreduced. Second, because previous estimates of mechanicalpower may be several times too low, calculations of muscleefficiency, based on the ratio of mechanical power to totalmetabolic power, may be substantially lower than actual values.Such estimates for the muscle efficiency of asynchronous flightmuscle range from 8 to 12%. When corrected for theunderestimate of drag, we find that previous estimates of muscleefficiency presented for fruit flies (Lehmann and Dickinson,1997) should be adjusted to 20–25%. If correct, these adjustedvalues help to solve a perplexing problem in insect flightphysiology. Because Ca2+ cycling represents a sizeable fractionof the energetic cost in cyclic contractions, stretch-activatedmuscle should accrue an energetic savings and concomitantincrease in mechanical efficiency. This prediction was notsupported, however, by the previous estimates of efficiency.After correction with more accurate values of drag, the estimatesof mechanical efficiency in asynchronous muscle are nowconsistent with the reduced cost of Ca2+ cycling.

In addition to indicating that the mean level of drag may behigher than previously expected, the results also show that thevalue of drag is quite sensitive to subtle changes in wingkinematics (Fig. 5C, Fig. 7C, Fig. 9D). Mapping an estimate ofprofile power indicates that this sensitivity of the dragcoefficient to wing kinematics translates into a sensitivity offlight cost as well (Fig. 10A,B,C). This result again hasconsequences for estimates of flight costs and mechanicalefficiency from free-flight kinematics based solely on measuresof stroke frequency and amplitude (Ellington, 1984d). Inparticular, for a given stroke amplitude, profile power variesdepending upon the timing and duration of wing rotation(Fig. 10B). Thus, without knowledge of the precise strokekinematics, estimates of free-flight energetics must be viewedwith caution.

Concluding remarks

The data presented in this paper represent a comprehensiveanalysis of the effects that wing kinematics have on thegeneration of aerodynamic forces in flapping flight. The resultsreveal a complex system in which subtle alterations in strokekinematics can have quite large effects on force production. Infuture studies, it will be an important challenge to determinehow real insects use this sensitively to adjust the balance offorces and force moments as they steer and maneuver throughtheir environment.

List of symbolsAR aspect ratioc– mean chord lengthc(r) non-dimensional chord lengthC—

D,pro—

mean profile drag coefficientCD– mean drag coefficientCD,t translational drag coefficientCF– mean force coefficientCL– mean lift coefficientCL,t translational lift coefficientCN– mean total aerodynamic force coefficientCR– mean radial coefficientdφ/dt—

mean non-dimensional angular velocity of the wingD dragDdev maximum angle of downstroke deviationD(t) instantaneous dragF– force coefficient averaged over a strokeFa,N added mass inertia normal to the wing surfaceFN– mean total aerodynamic forceL liftn wingbeat frequencyPpro– mean profile powerR wing lengthRe Reynolds numberr non-dimensional radial position along wing lengthr22(S) non-dimensional second moment of wing areaS surface area of a wing pairt timeT stroke periodUdev maximum angle of upstroke deviationvwing(t) instantaneous wing velocityα angle of attackα(t) instantaneous angle of attack∆τ flip durationΦ total stroke amplitudeφ(t) instantaneous stroke positionν kinematic viscosityθ(t) instantaneous stroke deviationΘ maximum stroke deviationρ density of fluidτf flip timingτ0 flip start

This work was supported by grants from the NSF (IBN-

2626

9723424), Defense Advanced Research Projects Agency andthe Office of Naval Research (FDN00014-99-1-0892). Wewish to thank Fritz Lehmann, Claire Balint, NamrataGundiah, Jim Birch and two anonymous reviewers for helpfulcomments.

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