Institute of Interdisciplinary Research, Faculty of Engineering,University of Tokyo, Tokyo, Japan
Accepted 23 February 1988
Summary
The dragonfly, Anaxparthenope Julius (Brauer) was observed in free flight, anda theoretical analysis of flight performance at various speeds was carried out. Thevariation with time of forces and moments acting on wings and body in steadytrimmed flight was calculated by the local circulation method. Measures of flightperformance, such as top speed, cruising speed and maximum endurance speed,were estimated from a necessary power curve required in steady flight and fromthe estimated available power. The results show that without using any novelunsteady aerodynamic force generated by a separated flow over the wings, thedragonfly can make steady trimmed flight at various flight speeds, from hoveringto top speed.
Introduction
Application of the local circulation method (LCM) (Azuma et al. 1985) and anumerical computer simulation shows that a dragonfly, Sympetrum frequens, canmake steady climbing flight without using an abnormally large lift coefficient(Norberg, 1975) or relying totally on unsteady aerodynamic forces (Savage et al.1979). This result was obtained by considering the non-uniform and unsteadyinduced velocity distribution over a pair of stroke planes.
The induced velocity can be precisely calculated from the Biot-Savart law whichis known in electromagnetic theory and is used to determine the induced velocitydistribution generated by wake vortices (see any book of hydrodynamics such asNewman, 1977).
In the LCM, the time wise change of the induced velocity at any point in thestroke planes is given by multiplying the attenuation coefficient by the inducedvelocity generated by any preceding wing at the time that wing passed through thatpoint. In Azuma et al. (1985), the effect of the trailing vortices on the derivation ofthe attenuation coefficient was introduced, but the effect of the shed vortices wasneglected because the reduced frequency, which is a measure of the unsteadinessof the potential flow in the linear range of analysis and is given by k = Jtci/U = I/AR1//1 in hovering flight, is very small relative to the large-aspect-ratio wingl»f dragonflies (see the next section for all definitions).
Two novel mechanisms for lift generation in insects were described by Weis-Fogh (1973,1975). The first one, called 'clap and fling', utilized the separated flowaround a pair of low-aspect-ratio wings (Lighthill, 1973, 1975; Maxworthy, 1979;Edward & Cheng, 1982). The second one, called 'flip', also utilized a pair ofvortices, a bound vortex and a shed vortex of opposite sense, generated by a rapidpronation of the anterior portion of the respective large-aspect-ratio wings.
Savage et al. (1979) studied the role of the vortices generated by a two-dimensional wing motion and unsteady effects on the lift generation and revealedthat the lift is developed during a 'pause' in the downstroke preceding thesupination and during the supination. Somps & Luttges (1985) made an exper-imental test to demonstrate the effect of unsteady separated flows and concludedthat large lift forces are actually produced by unsteady flow-wing interaction.However, we do not agree with this interpretation because of the existence ofphysically unexplained phenomena in the experimental data (see Appendix A).
In this paper, we analyse the free flight of a dragonfly, Anax parthenope Julius,from 16 mm cine films taken in a wind tunnel, to calculate the flight performance atvarious flight speeds by applying the LCM in a form extended to introduce theunsteady aerodynamic effects in a conventional sense, and to show that flight canbe performed without using the unsteady lift generated from the separated flow atleast in steady trimmed flight.
Definitions
AR aspect ratio of wing, AR = b 2 /Sa lift slopeb wing span (m)C D drag coefficient of three-dimensional wingCD f drag coefficient of body other than wingsCD. induced drag coefficientCD|1 profile drag coefficient, CD(1 = C D - CD.C L lift coefficient of three-dimensional wingC d drag coefficient to two-dimensional wing, Cd ( 1~ CD n
C G centre of gravityC/ lift coefficient of two-dimensional wing, Q = aa within unstalled
regionQ.max maximum lift coefficient Q = Q max beyond stallC m momen t coefficient of two-dimensional wing about the feathering or
elastic axisc chord length (m)D drag (N)FH horizontal force, positive forward (N)F v vertical force, positive upward (N)f beating frequency (Hz)/ drag area (m2)
Dragonfly flight 223
G acceleration measured in units of gravity accelerationg acceleration due to gravity, g = 9-80665 m s~2
I moment of inertia of a wing (kg • m2)k reduced frequency, k = jrcf/Uk aerodynamic parameterL lift (N)L body length/ spanwise airloading (N m~')M feathering moment, moment about feathering (elastic) axis (Nm)M C G pitching moment about the centre of gravity (positive for head-up)
(Nm)Mj pitching moment about an apparent joint on the root of the elastic
axis of the wing (Nm)m body mass (kg)mm musculature mass (kg)n nth order of higher harmonicsn load factorP necessary power (W)Pa available power (W)Pi induced power (W)Po profile power (W)P p parasite power (W)Q torque about an apparent joint (Nm)R distance from apparent joint to wing tip, R « b/s (m)Re Reynolds numberr spanwise station along the elastic axis (m)S wing area (m2)Sf maximum sectional area of body (m2)T period of one beating cycle (s)t time (s)u VuT
2 i Up2
UT tangential component of velocity relative to a blade element (ms~')Up perpendicular component of velocity relative to a blade element
(ms-1)V flight speed (m s ')v induced velocity (ms~')W weight of body = mg (N)x non-dimensional spanwise station, x = r /RxCG longitudinal distance of centre of gravity measured from front joint of
forewing (m)Xj distance between the first joints of forewing and hindwing
vertical distance of centre of gravity measured from hindwingangle of attack of wing section (degrees or radians)
/3 coning angle (degrees or radians)
224 A. AZUMA AND T. WATANABE
(3* beating angle projected in horizontal plane (degrees)
F flight angle (degrees or radians)y tilt angle of stroke plane (degrees or radians)d^,S2,A aerodynamic parameters related to the wing planformdy phase shift of flapping motion (degrees or radians)cL phase shift of feathering motion (degrees or radians)
0d
pAc/2
ViIJJ*
(X)
0
1n0-75RAcruisefhhovImaxminP,minroottip
body attitude (degrees)feathering angle (degrees or radians)air density (kgm~3)swept angle to half-chord line (degrees or radians)inflow angle (radians)azimuth angle or flapping angle (degrees or radians)amplitude of beating motion (degrees or radians)beating angle projected in vertical plane (degrees)angular velocity of beating motion, co = lizi (radians s~l)
Subscript or superscript
oth harmonic component (or steady state) or two-dimensional valuesthe first harmonic componentnth harmonic componentthree-quarter radius stationaerodynamic componentcruising flightforewinghindwinghovering stateinertial componentmaximum valueminimum valuepower minimumwing rootwing tip
Geometrical configuration
The general configuration of Anax parthenope Julius is shown in Fig. 1 andgeometrical characteristics of two dragonflies are given in Table 1. This dragonflyis considered to be one of the 'high performance' species: it has excellentmanoeuvrability and is a most active predator. It has a large wing load and highbeating frequency, enabling fast and skilful flight.
The beating can be represented both as the flapping (or heaving) motion of anelastic axis assumed to be a straight line roughly passing through a quarter chord ^the aerofoil section at any spanwise station and as a feathering (or pitchinJPmotion about the elastic wing axis. These motions are performed actively through
Dragonfly flight
Table 1. Geometric characteristics of two dragonflies
225
Body lengthMassWing span: forewing
hindwingWing area: forewing
hindwingAspect ratio: forewing
hindwingWing loadingCentre of gravity
Distance betweenthe first joints offorewing and hindwing
Estimated drag areaDrag coefficientMaximum section area
Abbreviation
Lmbf
bh
Sf
Sh
ARf=(bf)2/Sf
ARh = (bh)2/Sh
W/(Sf+Sh)XCGZCG
X.i
f=SfCDf
Sf
Units
m
kgmmm2
m2
Nm-2
mm
m
m2
m2
DragonflyA
7-5xlO~2
7-9X10-4
1-OxlO"1
9-7xlO~2
1-OxlO"3
1-2X10"3
107-83-53-OxMT3
6-OxlO"3
8-OxlO"3
9-4xlO~5
1-257-5X1Q-5
DragonflyB
8-OxlO"2
7-9X10'4
1-2x10"'1-lxlO"1
9-3 x 10"*l-3xKT3
15-79-33-53-OxlO"3
5-OxlO"3
8-OxlO"3
9-4xlO"5
1-257-5X10"5
two (front and rear) joints at the respective wing root (von Lendenfeld, 1881). Asshown in Fig. 1A,B, the flapping is assumed to be confined within a conical plane,the apex of which coincides with the front joint and the coning angle of which isdefined by fl, although the actual flapping motion deviates slightly from the conicalplane during the stroke (Fig. 2). The orbit of the three-quarter radius station of theelastic (or feathering) axis is called the 'stroke plane', which is considered to benormal to the cone axis.
In dragonflies, the cones related to the forewings are open towards the frontwhereas the cones related to the hindwings are open mainly towards the back.Therefore, the stroke planes of the forewings are located in front of the joints ofthe forewings and the stroke planes of the hindwings are located behind the jointsof the hindwings. The tilt angles of the two stroke planes are defined by yf and yh.
Mode of wing beat
Free dragonfly flight was observed in a wind tunnel and filmed with a 16 mmhigh-speed cine camera. By changing the wind speed in the tunnel the flight speedwas altered. To get a clear image of the feathering motion and the twistdistribution of the wings, three parallel stripes (lmm in width) were painted oneach left wing at three spanwise stations (Fig. 1A). The width and thus the mass ofthe stripes were so small that the beating mode and frequency were substantiallyunchanged by them.^ Observed modes of flight are given in Table 2. The orbits of the wing tips withPspect to the body, which roughly show the stroke planes of the respective wings,and the movements of the wings (measured at the three-quarter span position) in
226 A. AZUMA AND T. WATANABE
Inclination
Painted stripes ^ o y \ ^ stroke plane
U
Flight path angle
Body attitude
Stroke plane
T i p , p a t h Coning angle
Flapping amplitude
Wing position
Fig. 1. Schematic configuration of Anax parthenope julius and its stroke planes.(A) Side view of dragonfly in flight; (B) projection of stroke plane (view A).
inertial space, are shown in Fig. 2A-D for various flight speeds, V. The traces ofthe azimuth angles, 1/ and 1//1, which are defined by the flapping angles of theelastic axis projected to the stroke plane and measured from the horizontal line,are shown in Fig. 3A-D. They can be expressed by the first harmonic of a Fourierseries:
\p = xpo + ip\cos(cot + dy,). (1)
The values in the above and the following equations refer to either the forewing c*mthe hindwing.
However, as can be seen from Fig. 4A-D, the feathering angles at three span
Dragonfly flight
Table 2. Flight kinematics for dragonflies A and B; experiments 1-4
227
DragonflyExperiment
VelocityFlight angleBody attitudeBeating frequency
Stroke plane inclinationmeasured from horizontal line
forewinghindwing
measured from body axisforewinghindwing
Flapping amplitudeforewinghindwing
Phase difference of flappingmotion between forewing andhindwing
Phase difference between flappingand feathering
forewinghindwing
Calculated load factor
V(ms~')F (degrees)0 (degrees)f(Hz)
y1 (degrees)yh (degrees)
y' + 0 (degrees)yh + 0 (degrees)
i// (degrees)xjjh (degrees)
<5h (degrees)
01.O-75R
<5| ~~ 0l\o-75R
n
A1
0-7-12
2026-5
4038
6058
3626
51
89102
1-1
A2
1-5-1-11228-1
5548
6760
2526
61
9392
0-97
A3
2-34-84
29-0
5852
6256
2526
61
9195
1-25
B4
3-202-0
27-0
6368
6570
3834
93
8981
105
positions x = 0-25, 0-5 and 0-75, are expressed by the Fourier expansion seriesincluding higher harmonics as follows:
= 60+ 0ncos(n<wt + 8e) (2)
Table 3 shows the values of coefficients in the above Fourier expansion series ofthe experimental data. The first harmonic of the feathering angle of the respectivewings is, as can be seen from Fig. 5A-D, a function of the span position x,
01 = (0O-75R " 0root)(x/O-75) + 0root , (3)
where the coefficients, 0O-75R ~~ dTOO{ a nd 0root are also functions of the flight speed.
From these figures it can be seen that: (i) the beating frequency of the wings isalmost unaltered (f = 29-32Hz) with changes in flight speed; (ii) the tilt angle ofthe stroke planes with respect to the body axis gradually increases from about 40°to 70° as the flight speed increases; (iii) the phase difference between fore and hindpairs of wings is within 60°-90° and is not correlated with flight speed; (iv) theconing angle is about 8° in the forewing pair and about —2° in the hindwing pair
228 A. AZUMA AND T. WATANABE
Forewing . Root of wing
Body axis
Root of wing
Forewing
Vs" Root of wing
Body axis
Hindwing /
48 c
Root of wing
Fig. 2. Orbit of wingtip and the movements of the wings, measured at the three-quarters span positions. V, downstroke; A, upstroke; dashed line shows orbit of wing;left-hand plots show wing movements relative to the air; right-hand plots show wingmovements relative to the body. (A) Experiment 1 (V = 0-7ms~1, dragonfly A);(B) experiment 2 (V = l-5ms~J, dragonfly A); (C) experiment 3 (V = 2-3ms~\dragonfly A); (D) experiment 4 (V = 3-2ms~1, dragonfly B).
Dragonfly flight 229
Forewing
\S
A58"• Root of wing
Body axis
Hindwing
L_
52c
Root of wing
D
Forewing
• Root of wing
Body axis
Hindwing
65c
Root of wing
Fig. 2C,D
A. AZUMA AND T. WATANABE
Downstroke of forewmg Upstroke of forewingUpstroke of forewingDownstroke Upstroke of hindwing DownstrokeUpstroke of hindwing Downstroke
90
60
30
0
- 3 0
- 6 0
-90
Downstroke of forewing Upstroke of forewingDownstroke^ ^ Upstroke of hindwing JDownstroke
Forewing
THindwing
= 3-4xlO"2s
0 0-25T 0-5T 0-75T
90
60
30
0
-30
-601
Time
Downstroke of forewing Upstroke of forewingDownstroke, Upstroke orhindwing .Downstroke
= 3-6xl(T2s
0-25T 0-5T 0-75T
Fig. 3. Azimuth angle in the stroke plane for various flight speeds. O, forewing; <Xhindwing; lines are given by equation 1. (A) Experiment 1 (V = 0-7ms~', dragonflyA); (B) experiment 2 (V = 1-5ms~', dragonfly A); (C) experiment 3 (V = 2-3ms~',dragonfly A); (D) experiment 4 (V = 3-2m s"1, dragonfly B).
and is almost constant (within 10 % deviation) throughout the beating motion; (v)the flapping amplitude ranges from 25° to 40° and is not correlated with flightspeed; (vi) the feathering amplitude ranges from 40° to 60° in the forewings andfrom 30° to 40° in the hindwings and is also not correlated with flight speed; (vii)the phase difference between the flapping and feathering motion is 90°, which isconsidered to be optimal for efficiency at low beating frequency (Azuma, 1981);and (viii) since the beating motion is performed through two joints at therespective wing roots, in the feathering motion the wing is twisted linearly for agiven time - wash-out (twisted negatively towards the tip) in the downstroke andwash-in (twisted positively towards the tip) in the upstroke.
The wing beat modes were also observed in a smoke tunnel using a stroboscopicflash which made it possible to visualize the wake vortices of the beating wings andthe modes of the wing motion. The wake vortices were visualized by the paraffinmist method (Watanabe et al. 1986) (Fig. 6A,B). A series of trailing and shedvortices generated, respectively, by the span and time (or azimuthal) change of
180
150
120
90
60
30
0
Downstroke _j
0O-5R
0Q-75R
Upstroke , _0 '
O0-25R ^ ;
Forewing
T = 3-6xlO"2s
Dragonfly flight
180
150
120
231
Upstroke Downstroke
0 0-25T 0-5T 0-75T
Downstroke Upstroke 10U
150
120
90
G 601<u<u
£? 30
% n
Downstroke f Upstroke ,
8 ^ £R Hindwing
T = 3-6xlO~2s
0-25T 0-5T 0-75T
Downstroke Upstroke
#0 5R
T
, Downstroke
Hindwing= 3-4xlO-2s
0-25T 0-5T 0-75T
Downstroke Upstroke .Downstroke
Hindwing= 3-65xKT2s
0-25T 0-5T 0-75TTime
Fig. 4. Feathering angles with respect to the stroke plane. Span position, O, 0-75R;<0>, 0-50R; A, 0-25R; lines are given by equation 2. (A) Experiment 1 (V = 0-7ms"',dragonfly A); (B) experiment 2 (V = l-5ms~\ dragonfly A); (C) experiment 3(V = 2-3ms~\ dragonfly A); (D) experiment 4 (V = 3-2ms~', dragonfly B).
Fig. 5. Spanwise variation of feathering amplitude. First harmonic of observed data:O, forewing; A, hindwing; lines are given by equation 3. (A) Experiment 1(V = 0-7ms~', dragonfly A); (B) experiment 2 (V = l-5ms"1, dragonfly A); (C) ex-periment 3 (V = 2-3ms~1, dragonfly A); (D) experiment 4 (V = 3-2ms~\ dragonflyB).
bound vortices are clearly observed in wavy wake sheets of the respective wingpairs.
Aerofoil characteristicsTo determine the aerodynamic characteristics of the dragonfly wing aerofoil,
flight tests were conducted in a calm room using model gliders which had a mainwing composed of a pair of dragonfly hindwings, as shown in Fig. 7A,B. It wasnecessary to introduce a small swept angle for the main wing and to install a tailwing for steady gliding flight with a positive angle of attack.
Fig. 8A-C shows plots of data obtained for the main wings of the model glidersin which the trimmed angle of attack was changed by shifting the centre of gravitywith respect to the aerodynamic centre of the wing. The data for the gliders with a
wing, after removal of the contribution of the tail wing, are represented byircles and the data for the gliders without a tail wing are represented by triangles.
The data for which the angle of attack were measured are shown by solid symbolst
234 A. AZUMA AND T. WATANABE
Fig. 6. Wake pattern and embedded vortices. By changing the positions of smokefilaments two types of vortex images can be visualized; the rectilinear trace across thetrails is a supporting frame for the dragonfly. (A) Flow pattern induced mainly by thetrailing vortices; (B) flow pattern induced mainly by the shed vortices.
Dragonfly flight 235
Hindwing of dragonfly
S
Hindwing of dragonfly
10
Balance weight 120
Balance weight
Fig. 7. Model gliders using dragonfly wings. Numbers indicate lengths in mm.(A) Tail-less wing; (B) tailed wing.
in Fig. 8A-C. The solid curves in Fig. 8 show typical characteristics of these three-dimensional wings.
Utilizing the three-dimensional wing data, it is possible to determine theaerodynamic characteristics of the two-dimensional wing and the section aerofoilcharacteristics from simple wing theory (Diederich, 1952; Hoak & Ellison, 1968),as follows:
where AR, Ac/2 and a are the aspect ratio, the swept angle of the half chord line,and the lift slope of the three-dimensional wing, and <5ls <52, k and A areparameters related to the planform of the wing and are presented by charts ofDATCOM (Hoak & Ellison, 1968).
Fig. 9A shows the dragonfly aerofoil characteristics (or the aerodynamiccharacteristics of the two-dimensional wing, extrapolated from the above obtaineddata to hypothetical values in a large angle of attack range) as a function of angleof attack, in which the maximum lift coefficient is assumed to be Q max = 1-2. It is
Rot clear why the minimum drag coefficient is slightly lower than those expectedom the skin friction of a plate. Fig. 9B compares the data for polar curves for a
locust wing, fruit-fly wings and a dragonfly wing obtained by Jensen (1956), Vogel
236 A. AZUMA AND T. WATANABE
u
0-1 0-2 0-3Drag coefficient, CD
0-4
10
0-5
<C 0
- 0 - 5
-1-0
B
-20 -10 10 20 -20 -10Angle of attack, a- (degrees)
10 20
Fig. 8. Aerodynamic characteristics of three-dimensional wing. A, tailed wing;O, tail-less wing; filled symbols indicate that angles of attack were also measured.(A) Polar curve; (B) lift coefficient; (C) drag coefficient.
(1967) and Newman et al. (1977), respectively. The two-dimensional pitchingmoment around the aerodynamic centre, which is assumed to be a line connectedto the quarter-chord of the aerofoil along the span, is small and is thereforeneglected. Although the present curve shown by a solid line gives a higher profiledrag at large angles of attack than do the others, the above aerofoil data areutilized in the following calculations for both fore and hind pairs of wings.
Analysis by means of the local circulation method
Let us consider first the flight performance of a dragonfly at various speeds. The
Dragonfly flight 237
J1-5
U 1-0
0-5
S -0-5
- 1 0
-1-5,
A \\\
\\\
I, max ' i
-90 -60 -30 0 30 60 90Angle of attack, a (degrees)
U
1-4
1-2
10
c no
.a °'SssR 0-6
0-4
0-2
0
BLarge dragonfly Ujajst h i n d w i n g
wing model
\Re =
01 0-2 0-3 0-4Drag coefficient, Cd
Fig. 9. Aerofoil characteristics of a dragonfly and other wings. (A) Lift and drag:estimated values based on the experimental data of Anax parthenope julius. (B) Polarcurve: results from present observations and for large and small dragonfly wing model(from Newman et al. 1977), locust hindwing (Jensen, 1956) and cambered and flatfruitfly wings (Vogel, 1967).
spanwise variation of the airloading and the total aerodynamic forces andmoments acting on the wings are calculated by the local circulation method(LCM). A detailed description of the method is given by Azuma et al. (1985), andonly a brief explanation is given here.
^ A wing in beating motion is hypothesized to consist of a plurality of ellipticalRdngs, and the respective elliptical wings are supposed to operate in the inducedflow field generated by them. The timewise change of the induced velocity at any
238 A. AZUMA AND T. WATANABE
Vertical force
Horizontal forceWake sheet of hindwing
Flight directionWake sheet of forewing
Trailing vortices Shed vortices
Fig. 10. Mathematical model of wake vortices.
local (or spanwise and azimuthal) station is given by multiplying an attenuationcoefficient, which is decided by the trailing and shed vortices left in a simple (not afree) wake model as a function of the spanwise distance, by the azimuth angle.Thus, the calculation is time-consuming and yet considered to be sufficientlyaccurate, without suffering from the computational divergence which sometimesappears in free-wake analysis.
From the actual wake vortices shown in Fig. 6A,B> a mathematical model of thewake sheets can be derived from the tracings of beating wings (Fig. 10). Then theattenuation coefficients are determined by the Bio-Savart law from the trailingvortices, which are assumed to be combined with the tip vortices of the respectivewing tips, and from the shed vortices lying on the wake sheets.
Although the attenuation coefficient can be given as a function of span positionand azimuth angle, the spanwise distance is fixed at the three-quarter position tosimplify the calculations in the present analysis.
The method is based on the blade element analysis but is different from otherprevious studies as follows, (i) The aerodynamic coefficients are used in nonlinearforms as functions of angles of attack that include the stalled range, (ii) Theinduced velocity is not homogeneous on the stroke planes and is obtained by theLCM, which also includes the unsteady effect, (iii) The control inputs for freeflight in computer simulation are the feathering angles along the wing span, the^flapping angles, the tilt angles of the stroke planes, and the beating frequency fon
Dragonfly flight
Lift (dL)
Angle of attack
Induced velocity
ine
Projection of flight velocity
Beating speed n//cos/8
Inflow angle
239
Inflow angle
Pitch angle
Induced velocity
Projection of flight velocity
Beating speed n/>cos/3
Chord line
Fig. 11. Relative velocities and aerodynamic forces acting on a blade element.(A) Downstroke; (B) upstroke.
given flight speeds.When the flight conditions and the wing motion are known, the airloading, and
thus the total aerodynamic forces and moments as well as the inertial forces andmoments, can be calculated by the LCM.
By referring to Fig. 11, the lift, drag and feathering moment acting on a wingelement at spanwise station r with azimuth angle ip are given as follows:
dL = !pU2cQ((*)dr
dD = |pU2cCd(or)dr
dM = ipU2c2Cm(ar)dr
(7)
240
where
A. AZUMA AND T. WATANABE
a=6-<j>
UT = Vcosycosi/; + npcos/3
Up = V(sinycos/3 + cosysini//sin/3) + v
= tan-1(UP/UT).
(8)
Then, aerodynamic forces and moments generated by the wing element can begiven by:
where dMj A is the pitching moment acting on an apparent joint, which is ahypothetical joint at the root of the feathering (or elastic) axis, and is given by:
A = rdL(cos0cos7/; - sin0sini/;cos/3) -+ dMcosi/>cos/3.
cos0sini/;cos/
Similarly, by assuming that every angular motion is small and neglectingcoupling terms, the inertial forces and moments acting on a wing element can begiven by:
where
dFv.i = — dmrcos/3(i/»2sini/; — i/;cos7//)siny
dFH.i = -dmrcos/?(i/;2sini// - i/)cost//)cosy
dQj = dmr2cos2/fy
+ dFvjXcG - dF M J z C G
(11)
j = -dmr2cos/3sin^(^2sinip + T//COST//) . (12)
Then,.the total vertical force, horizontal force and pitching moment aboutcentre of gravity can be given by integrating the above elemental forces Qmoments along the wing span and by summing the individual components as
Dragonfly flight 241
follows:
F v = 2 | [(dFv.A + d F V J + dFhv,A + dFh
VJ)/dr]dr
F H = 2 j [(dF*H,A
M C G = 2 d M C G J + dMC G,A + dMCG,i)/dr]dr
(13)
where superscripts f and h show the values for forewing and hindwing, respect-ively. Their mean values (or time averages) for one period of beating motion aregiven by:
2n/co
F v = ((O/2JZ) j Fvdt0
2n/a>
H = {(O/2JT) J FHdto
Inlco
M C G = {(O/IJI) \ MC Gdt .0
(14)
Neglecting the feathering component, the power required to beat the wing is givenby:
2JI/W R
= (CO/2JI) j {2 j0 0
(15)
The vertical and horizontal forces are also normalized by the weight of thedragonfly and are expressed by the load factor, only the vertical component ofwhich is given by n in unit G, the ratio of the resulting acceleration and theacceleration due to gravity.
Performance
From the data given in Tables 2 and 3 for free flights, which were not exactlytrimmed flight (load factor n > 1), the necessary (mechanical) power is obtained asshown by the circles in Fig. 12. By referring to these data and by selecting modifiedfeathering angles as given in Table 4, the necessary power curve versus flight speedcan be calculated (Fig. 12). The curve passes through lower values than thoseobtained from the free-flight tests, because the calculated power is based onalmost completely trimmed flights (n ~ 1).
By assuming an available power to musculature mass ratio ofPa/mm = 260Wkg~1 (Weis-Fogh, 1975, 1977) and a musculature mass to totalmass ratio of mm/m = 0-25 (Greenwalt, 1962), the available power can be
Ptimated as Pa = 5-75xl0~2W. Then the top speed (Vmax) of this dragonfly is2ms" 1 (Fig. 12). This value can be increased either by reducing the estimated
drag area or by increasing the available power. The cruising flight speed (Vcruise,
242 A. AZUMA AND T. WATANABE
ox
ia 3
Available power Pa = 5-75 xlO"2W
\ Experiment 1\
Experiment'n-0-97
2 3 4 5 6 7Flight speed, V^s" 1 )
Fig. 12. Necessary power curve of the dragonfly, Anax parthenope (O, experimentaldata).
the maximum range speed), which is found at the point at which the necessarypower curve is at a tangent to a line drawn through the origin or (dP/dV)min, is3-5 m s"1 (Fig. 12). The minimum power speed (VP min, maximum endurance flightspeed) is 1-7 m s~'. The power required for hovering flight (PhOv) is estimated to be3-6xlO"2W.
Fig. 13A-D shows the time variations of the vertical and horizontal componentsof the aerodynamic force, torque about the flapping axis and moment about thecentre of gravity at V = 3-2 m s~'. They were calculated by the LCM and by bladeelement theory based on the constant induced velocity distribution. For thecalculation of the moment, the feathering moment around the elastic axis wasassumed to be zero (Cm = 0) because of ambiguous aerodynamic characteristicsdue to complex aerofoil configuration. It can be seen that by assuming a constantinduced velocity distribution some errors are introduced in these variations.However, as shown by the right-hand ordinate, their mean values are close to eachother in the vertical component of the force. Although the forces are in a well-balanced condition, Fy + Fy ~ W, and F'H + F v ^ 0, the mean moment about thecentre of gravity is not completely balanced but leaves a positive (pitch-up) valunder the assumption of zero feathering moment, Cm = 0. However, the valuenot too large and is within a range in which adjustment is possible by taking into
I
o x
X E o
_2
0-8
0-6
0-4
0-2
Dow
nstr
oke
of f
orew
ing
Ups
trok
e of
fore
win
g
Dow
nstr
oke
Ups
trok
e of
hin
dwin
gD
owns
trok
e
' = 3
-6xK
T2 s
-
0 -1 -2
3 s
or,
i o X 5
1
0-8
0-6
0-4
0-2 n
0-25
T0-
5T0-
75T
• C
0
-0-2
-0-4
-0-6
-0-8
-1
0
Dow
nstr
oke
of f
orew
ing
UD
Stro
ke o
f fo
rew
ing
Dow
nstr
oke
Ups
trok
e of
hin
dwin
gD
owns
trok
e
Hin
dwin
g
Fore
win
g
= 3
-6xl
(T2 s
00-
25T
0-5T
0-75
T
.9
- 0-2
-0-4
-0-6
-0-8
-10
i
10
0-8
0-6
0-4
0-2 0
-0-2
-0-4
-0-6
-0-8
-10
Hin
dwin
g
Dow
nstr
oke
of f
orew
ing
Mea
n va
lue
|L
CM
Ups
trok
e of
for
ewin
g
Dow
nstr
oke
Ups
trok
e of
hin
dwin
g ,
Dow
nstr
oke
O
0 I
-1
0-25
T0-
5T0-
75T
Dow
nstr
oke
of f
orew
ing
Ups
trok
e of
for
ewin
g
Dow
nstr
oke
Ups
trok
e of
hin
dwin
g D
owns
trok
e
T=
3-6x
lO~
2s
0-25
T0-
5T0-
75T
Tim
e
Fig
. 13
. A
erod
ynam
ic f
orce
s an
d to
rqu
e. E
xper
imen
t 4;
V =
3-2
ms
l , dr
agon
fly
B;
heav
y li
nes
show
the
val
ues
calc
ulat
ed b
y th
eL
CM
, th
in l
ines
the
val
ues
calc
ulat
ed b
y si
mpl
e m
omen
tum
the
ory
(v =
con
stan
t),
solid
lin
es f
orew
ing,
and
das
hed
line
s hi
ndw
ing.
(A)
Ver
tical
for
ce;
(B)
hori
zont
al f
orce
; (C
) to
rqu
e; (
D)
pitc
hing
mom
ent.
Grq
to
Tab
le 4
. M
odif
ied
coef
fici
ents
in
Fou
rier
exp
ansi
on s
erie
s of
feat
heri
ng m
otio
n fo
r tr
imm
ed
flig
ht
Dra
gonf
ly
A
A
A
BE
xper
imen
t 1
2 3
4
For
ewin
gA
mpl
itud
e (d
egre
es)
0O
89
90
90
78
80
82
70
77
81
(0-2
5R/0
-5R
/0-7
5R)
0 x
23
38
52
89 5890 61
90 6378 67 83 32
80 67 82 33
82 68 79 35
Hin
dwin
gA
mpl
itud
e (d
egre
es)
0O
83
82
79
71
72
74(0
-25R
/0-5
R/0
-75R
) 6 X
18
28
37
>B
lank
spa
ces
in t
he m
odif
ied
data
sho
w t
hat
the
valu
es a
re t
he s
ame
as i
n th
e ex
peri
men
tal
data
abo
ve a
nd h
ave
bee
n o
mit
ted
. ^ w w
Dragonfly flight 245
*>1
oXEZ,
o
que
10-8
0-6
0-4
0-20
-0-2
-0-4
-0-6
-0-8
-1-0
Downstroke of fore wing ̂ Upstroke of forewing
Inertial component Total torque
Aerodynamic component
T = 3-6xlO~2
0-25T 0-5T 0-75T
1
0-8
0-6
0-4
0-2
-0-6
-0-8
-1-0
Downstroke t Upstroke of hindwing t Downstroke
BTotal torque
. Aerodynamic component
0-25T 0-5T
= 3-6xlO"2s
0-75T
~ 2
- l
- 2
Downstroke of forewing , Upstroke of forewingDownstroke. Upstroke orhindwing ^)ownstroke
Inertial component= 3-6xlO~2s
2 £c
1 So
-l -3
- 2
0-25T 0-5TTime
0-75T
Fig. 14. Inertial and aerodynamic torque and force. Experiment 4; V = 3-2ms ',dragonfly B. (A) Forewing; (B) hindwing; (C) vertical force component.
account the feathering moment which is considered to be negative for a positivecamber (upward convex) in the aerofoil.
Fig. 14 shows the inertial torque of the respective wings and the verticalcomponent of the inertial force of the total wings in comparison with theaerodynamic component and the resultants. Although the respective inertial andaerodynamic components are of a comparable order of magnitude, the totaltorque and force are not large. The variation in the total vertical force is within 0Gand +3G in each beating cycle. As stated in Appendix A, it is important torecognize that the maximum value does not exceed n = 3G at the flight speed of3-2ms-J.
Fig. 15 shows the spanwise load distribution of vertical and horizontal com-ponents of the total force acting on the fore- and hindwings.
Fig. 16 shows the timewise variation of angle of attack on the stroke planes ofthe wings in cruising flight. It is interesting that a region of large positive angle ofattack is observed in the final stage of the downstroke near the wing tip of theforewing and a region of large negative angle of attack is observed in the earlystage of the upstroke near the midspan of the forewing.
Fig. 17 shows the share of the lift and drag components in the mean vertical
246 A. AZUMA AND T. WATANABE
force, F v . At very low speed, including hovering, a contribution of the dragcomponent on the vertical force cannot be neglected. This is not unconnected withthe fact that the dragonfly flies with its stroke plane tilted from the horizontal evenwhen hovering, although this increases the induced power required. This makesthe transition from hovering flight to other flight modes easier.
Fig. 18 is compiled from data obtained from an experiment on the dragonfly(Libellula luctuosd) (Somps & Luttges, 1985). The following characteristics of
Fig. 16. Timewise and spanwise variations of angle of attack. Experiment 4,V = 3-2ms"1, dragonfly B. (A) Forewing; (B) hindwing. Angles are given in degrees.
Drag component, FD/FV
00 10 20 30 4-0 5-0Flight speed, V (ms"1)
Fig. 17. The share of lift and drag components in the mean vertical force.
Dragonfly flight 249
\
2 4x.
- 4 L
9 18 27Time(sxlO~3)
36
Fig. 18. Beating motion and vertical force of the dragonfly, Libellula luctuosa (takenfrom Somps & Luttges, 1985). (A) Projected angle in vertical plane; (B) projectedangle in horizontal plane; (C) total lift.
wing motion and vertical forces (upward positive) can be obtained:
ipf = 0-223 + 0-852cos(cot) rad
iph = -0-048 + 0-763cos(otf + 1-274) rad
where
F v = [1-77 + 4-81cos(wt - 0-297)] x 10"2N
= 48° = 0-838 rad, yh = 55° = 0-960rad ,
(A.I)
(A.2)
(A.3)
(A.4)
(A.5)
250 A. AZUMA AND T. WATANABE
-100-25T 0-5T 0-75T
80
20
0
-2-0
-4-0
Downstroke
B
Upstroke
Total force of forewing
Aerodynamic component
7Inertial component = 3-6xl(T2s
0 0-25T 0-5T 0-75T
8-0
60
4-0
2-0
0
- 2 0
-4-0
Downstroke Upstroke Downstroke
Total force of hindwing
Aerodynamic component
' Inertial component
•y
= 3-6xlO~2s
0 0-25T 0-5TTime
0-75T
Fig. 19. Components of the vertical force calculated from the data of Libellulaluctuosa. (A) Total force; (B) forewing; (C) hindwing.
and the angular transformation between two definitions in Fig. 1 and Fig. 18 aregiven by
\p= - t a n ^ t a n i / ^ s i n y - tan/3*cosy) j
/3 = tan-'[cos7/>(tan/3*siny + tam//*cosy)] . J
Assuming that the above vertical force is evenly shared by the two pairs of wingand that the mass of the respective wings can be estimated by the similarity rule as
I
Dragonfly flight 251
derived from Anax parthenope, the vertical forces generated by the respectivepairs of wings are represented by:
F^ = {8-83 + [24-0/cos(0-637)]cos(o* - 0-934)} x 10~3 N 1
F v = {8-83 + [24-0/cos(0-637)]cos(o>t - 0-297)} x 10~3 N J
and are shown in Fig. 19 for the aerodynamic and inertial components. Because ofthe above assumption, the result is only a rough estimation of the vertical force,but the aerodynamic component of the vertical force in the respective wings can begiven over a wide range of the downstroke as described in the present paper.
However, in the experimental results obtained by Somps & Luttges (1985)(Fig. 18) there are two difficulties: the vertical force (over20G) and its mean value(5-3G) are both too large. These extremely large values are not realistic for theflight of living creatures, except when rapidly manoeuvring. The large valuesprobably result from the fact that the inertial force related to the body mass wasnot completely removed from the measurement of the vertical force.
The authors would like to thank Mr Masakatsu Takao, Mr Masayuki Kitamura,and other staff of the Japan Broadcasting Corporation (NHK) for stimulating us totry vortex-wake visualization by their scientific program on dragonflies. Theauthors are also indebted to Mr Shunji Oba, Iwata Agricultural High School, forcatching dragonflies, and Professor Keiji Kawachi and Mr Isao Watanabe, TheUniversity of Tokyo, who assisted in the computer programming and the wind-tunnel test, respectively.
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