Analysis of Maneuvering Flight of an Insect Analysis of Maneuvering Flight of an Insect

Analysis of Maneuvering Flight of an Insect

  • 期刊名字:仿生工程学报(英文版)
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  • 论文作者:Sunada S,Wang H,Zeng Lijiang,K
  • 作者单位:Department of Aerospace Engineering,Department of Precision Instruments,Department of Aeronautics and Astronautics
  • 更新时间:2020-11-22
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Joumal of Bionics Engineering (2004) Vol.1 No.2, 88 - 101Article No. JBE - 2004 - 022Analysis of Maneuvering Flight of an InsectSunada s.',Wang H.2, Zeng Ljiang,Kawachi K.31. Department of Aerospace Engineering , Osaka Prefecture University,-1 Gakuen-cho, Sakai, Osaka 599- 8531, Japan2. Department of Precision Instruments,Tsinghua University,Beijing 100084, Pople' s Republic of China3. Department of Aeronautics and Astronautics,University of Tokyo,7-3- 1 Hongo, Bunkyo-ku , Tokyo, 113- 8656, JapanAbstractWing motion of a dragonfly in the maneuvering flight, which was measured by Wang et al. 1 was investigated. E-quations of motion for a maneuvering flight of an insect were derived. These equations were applied for analying the ma-neuvering flight. Inertial forces and moments acting on a body and wings were estimated by using these equations and themeasured motions of the body and the wings. The resuts indicated the fllowin characteristics of this fight: (1)Thephase difference in flapping motion between the two fore wings and two hind wings, and the phase difference between theflapping motion and the feathering motion of the four wings are equal to those in a steady forward flight with the maxi-mum efficiency. (2)The camber change and the feathering motion were mainly controlled by muscles at the wing bases.Keywords : insect fight, maneuvering flight, equations of motionNomenclatureGJ torsional rigiditya parameter indicating the position of the tor-g gravitational accelerationsional axis of a wing elementh heaving motion of a wing elementc chord lengthIx,Iyry,Izz moment of inertia of a body arounddF, dFy,dF. aerodynamic forces acting ona x, Y, and Z axes, respetivelywing element in the x ,y,z directions, respectivelylxz products of inertiadxr,dy,dIx moment of inertia of a wing ele-Ip polar moment of inertia of the section areament around x ,y,z directions, respectivelyi inclination of the stroke planedMx aerodynamic moment acting on a wing ele-reduced frequency=axment around the x directiondm mass of a wing elementMx(1), My(1), Mz(1) aerodynamic momentsEr warping rigidityon each wing around X,Y , and Z axes, respectivelyE1,E2,E3,E4 a matrix for transferring a coor-Mx(2), My(2),Mz(2) moments around X,dinate systemY,and Z axes, respectively, by aerodynamic force onFB,x,Fp,y,FB,z aerodynamic forces acting on aeach wing at its rootbody in the X, Y, and Z directions, respectivelyMB,x, Mp,y, MB,z aerodynamic moments act-Fx,Fy,Fz forces on each wing or four wingsin ing on a body around X,Y,and Z axes, respectivelythe X, Y and Z directions, respectivelyma aerodynamic moment, indicated by (A2)中国煤化工y .YHCNMHGtedby(A3)Corresponding author: Shigeru Sunadamot total mass of a dragonflyE-mai:sunada@aero. osakafu-u. ac. jpFax: + 81-72-2549906P,Q,R angular velocity of a dragonfly bodySunada S, et al.: Analysis of Maneuvering Flight of an Insect39around the X,Y and Z axes, respectivelyx- y- z wing element-fixed coordinate systemp,q, rangular velocity of a wing elementwing lengtharound x,y, and z axes, respectively工distance between a wing element and a wingrH turning radius in a horizontal plane (XE - hingeYE)a geometrical angle of attackS a matrix for transferring the(x.- -y. -z+)β flapping anglesystem to the(X- Y- Z) system,=Ez'E3'E41R1 amplitude of the first harmonics of flappingT a matrix for rotating the (Xp- Yε- ZE) anglesystem to the (x。- y. -z*) system, = E4E3E2E18 distance from the center of gravity of the bodyto the mid point between the hinges of the fore left andtimeright wingsthickness of a wing elementφ,0,W euler anglesU,V, Wvelocity of the center of gravity of the中A phase difference between heaving motion andbody in the X,Y, and Z directions, respectivelyUH translational velocity in the horizontal planefeathering motion caused by moment due to aerodynam-ic force(XE- YE)φr phase difference between heaving motion andU constant forward velocityu,U,W velocity of a wing element in the x,y,feathering motion caused by moment due to inertialforceand z directions, respectivelyIerh- φnap phase difference between flappingu(T),v(T),w(T) velocity of a wing elementmotion and feathering motionin the x, y and z directions, respectively, due toξ-ζ coordinate system in Appendix Atranslational motion of the bodyη distance between the center of gravity of aVx。,Vy。, Vzq velocity of the center of gravitywing element and torsional axisof the body in the XE, YE, and ZE axes, respectively9 feathering angleV'x,V"y,Vzgvelocity of the mid point beθ feathering parameter, = θU/(0.75xufqw)tween the hinge of the fore left wing and that of theθ1 first harmonics of the feathering anglefore right wing in the XE, YE, and Zε axes, respec-θA feathering motion caused by the moment duetivelyto aerodynamic forcev(R), w(R) velocity of a wing element in the y0A amplitude of θ in Eq. (A5)and x directions due to angular motion of the body, re-θ feathering motion caused by the moment duespectivelyto inertial forcev(F),w(F)velocity of a wing element in theθ amplitude of θq in Eq. (A6). y, and z directions, respectively, due to flapping mo-θ0.s feathering angle at 50% span wise positiontion0o.s 0th harmonic component of 0o.sX,Y values determining A,indicated by Eq.ρ air density(A8)Pw density of a wingX- Y-Z body- fixed coordinate system2 surmmation of all the wingsXE- Yε- ZE earth- fixed coordinate system的angular velocity of flapping motion(Xh,Yn,Zh)T coordinates of a wing root in theSubscripts(X- Y - Z) coordinate systemX3- Y3-Z3,X4- Y4- 24,Xs- Ys- Zs,X6-中国煤化工Y6- Z6 intermediate coordinate systems indicated inYHCNMHGAppendix Brf,rh,lf,Ih right fore wing, right hind wing,)0Jourmal of Bionics Engineering (2004) Vol.1 No.2left fore wing, and left hind wing, respectivelythagyna ( whose morphological data is shown in Table1) were studied by Wang et al.01, who named them.1 IntroductionOne is what Wang et al. named “forward flight" andRecently,the mechanism of aerodynamic forcethe other is what they named“turning maneuvers”. Ingeneration by insect wings has gained interest, becauseour study, we analyzed the turning maneuvers the lat-the development of a micro air vehicle with flappingter flight , where the dragonfly turned in the horizontalwings was found to be difficult. Aerodynamic forceplane. During this flight, the dragonfly turned rightgenerated by insect wings depends on the motion andwith a turning radius of rH= 80 mm in a horizontaldeformation of the wings. The measurements of theplane (Xp- YE) plane, where the (Xp- Yε- Ze) ismotion, deformation, and generated force of the wingsan earth-fixed coordinate system as shown in(Fig. 1)are required for making clear insect flights. Althoughthese measurements in trimmed flights of insects have(The XE, YE and Zε correspond tothe - Y, - X andbeen extensively made, the number of measurements in- Z, respectively, used by Wang et al. ). The dragon-maneuvering flights of insects is very few2-6]. Recent-fly climbed during the first half of the flapping cyclely, Wang et al .[1] developed a method to measure theand flew horizontally during the second half of the flap-wing kinematics of an insect in maneuvering flight to-ping cycle. The translational velocity in the horizontalgether with its flight path and body attitude. In thisplane (XE- YE),UH, is about 1.5 m/s. The acceler-study, we derived equations of motion for a maneuver-ation due to the centripetal force of the turn in the hori-ing flight, where accelerations [ angular accelerations ]zontal plane (XE- YE) is U哨/r:≈2g.of the body, which are proportional to inertial forcesTable 1 lists the equations for the time variation of[moments] on the body, are expressed by aerodynamicβ。and 0.,which were obtained by Wang et al.[1] ,and inertial forces [ moments] acting on the wings andwho denoted them as φ and ρ, respectively. The char-the body. We analyzed the wing motion of a dragonfly1]acteristics of this flight, which were not stated in ref.in the maneuvering flight measured by Wang et al .[1], are as follows:by using these equations.(1) The phase difference in flapping motion be2 Maneuvering flight measured bytween the two fore wings and two hind wings is nearlyWang et al.[1]90*. This value was observed in other steady forwardTwo types of flights of a dragnfly, Polycan-flights of a dragonflyL7]. Lan[8] showed that a phaseXv中国煤化工CNMHGFig.1 Earth-fixed coordinate system (Xp- Yg- Zg), bouy-fIxeu coorunate system(a- Y- Z),and wing element-fixed coordinate system (x. -y. -z.)Sunada S., et al.: Analysis of Maneuvering Flight of an Insect)1Table 1 Wing motion[1]nclination of stroke planeFlapping angleFeathering angle atat 50% span-wise positionβ≈1.5° + 24'cos(at +75*)0.0.≈72° + 5I"o(ot -6')+ 14*os(2ot + 122')+ 1s(3aut +7*)r≈-9.5° + 32"°os(out +57*)ig= ig≈42°0.0o.s≈75° + 47"cos(ot - 26*) + 20cos(2ut + 62*)+ 3"°o<(3wt + 64")in= in≈56°βn≈-0.20° + 17*cos(at + 168*)0.0.s≈82° + 2I'os(ot + 60")+ 6co(2out + 95*)+ 6°os(3axt - 18')Pm≈0.78* + 24*°as(at + 167*)0m,0.s≈84° + 28*coe(ot + 68*)+ 7"cos(2at + 93")+ 5°cos(3wt - 11")w≈210 rad/sdifference in flapping motion between fore and hind in the present flight during steady forward flight re-wings of 90* - 135° corresponds to maximum efficiency ported by Wang et al. means that neither the inertialof a trimmed forward flight, when the following fourforce due to flapping motion nor the aerodynamic forceconditions are met: (i) k≈0. 38, (i) phase difference significantly affects the feathering motion was not sig-between flapping and feathering motions Peah - φap = nificantly affected by the inertial force due to flapping-90*,(ii) 白is 0 or 0.8, and (iv) distance betweenmotion and by the aerodynamic force. Feathering mo-fore and hind wings is 0.5 or 1 chord length. The re- tion is mainly controlled by the muscle at the wingduced frequency of the flight reported by Wang et al.basel1], and therefore, the phase difference of- 90*that we are analyzing here (k≈0.8 )is not close to theduring maneuvering flight observed by Wang et al.01value by Lan's analysis (listed in (i), k≈0.375),is possibly controlled by the muscle at the wing base.whereas the other values are close ( those listed in (ii)This active control of feathering motion by muscle was- (iv)).also observed in a fore wing of a scorpion fly-12]. On(2) The aerodynamic and inertial forces acting on the other hand, passive control of the feathering motionthe left and right wings were dffent, because the by inertial and/or aerodynamic moment due to flappingdragonfly executed a made the rolling motion aroundmotion of the wings was observed for the moththe x axis. However, the change in camber was simi- Mythimna Separatal13) ,in Dipteral9,and that of thelar for both the wings'1], indicating that this camber hind wings of a sorpion fy12.change was not strongly induced by aerodynamic and 3 Equations of motion for a maneveringinertial forces acting on the wings. Therefore, theflightcamber change is controlled by muscles at the wingbases of the dragonfly. The phase difference betweenFig. 1 shows the earth-fixed. coordinate system,camber change and feathering motion was about 0*,body fixed coordinate system, and wing element fixedwhich is an evidence of Ennos' effect[9] on the dragon- coordinate system denoted respectively by (Xε- YE-fly' s wings.Zp),(X- Y-Z) and (x。-y+ -z。), where *(3) The中heh -中inp of the four wings was aboutdenotes right fore wing (rf), right hind wing (rh),- 90". This value yields maximum eficiency of steady left fore wing (If), or left hind wing (Ih). The rela-forward fight[10]and was a value reported in other tionships among these coordinate systems are describedstudies of steady forward flights of a dragonflyf7].in Appendix B.When the feathering motion is caused by the inertialThe equations of body motion in the(X - Y- Z)force acting on the wing due to flapping motion, the system are indicated in Table 2(a) and (b). Eqs. (T1)phase difference between flapping and feathering mo-- (T3) in Table 2(a) are for translational motion intions during steady forward flight is 0° (see Appendix X,Y and Z, respectively. Eqs. (R1) - (R3) in TableA). When the feathering motion is caused by the aero- 2(b) are for rotational motion around X, Y and Z, re-dynamic force acting on the wing, the phase difference specti中国煤化工and (R1) - (R3),2is close to 0°. (Appendix A). Assuming the results inmeanMHCNMHGngs. In Eqs. (T1)-the analysis for a steady flight are available for this pre- (T3), me nrst terms In tne lert hand sides denote in-sent maneuvering flight, the phase difference of- 90° ertial and aerodynamic forces acting on the body. TheJournal of Bionics Engineering (2004) Vol.1 No.2Table 2(a) Equations of translational motion of body in the body-fixed coordinate system(X- Y - Z)me(U+ QW- RV)+ mugsin0= EFx.o+ EFx,.a + Fe.x(T1)Fx.n= j(SndF. + SndF,. + SpdF..)(TI-1)Fx.a=」- dm|Su(i. +q.w. -r.v.)+ Sr2(i. + r.u. - p.w.)+ Sn3(w. + p.U. - q.u.)|(T1-2) .m(V+ RU- PW) - mugosOsinφ= EFv.mn+ EFv.a+Fer(T2)Y Fr,.om= J(SndFr,. + SdF,. + S2dF.,.)(T2-1)Fy.m=」-dm{Sr(u. +q.w. -r.u.)+ Sz(i. + r.u. - p.w.)+ Si3(iw. + p.U. -q●u.)|(T2-2)me(W+PV- QU) - m∞6o∞x= EFz.m+ EFz.m+ Fa.z(T3)2Fz.m= j(SydF. + SxIdF,. + SsJF..)(T3-1) .Fz.imr=」-dm{S3u(ui, + gow. -r.v.) + Sxr(i. + r.u. - p.w.)+ Sq(w. + p.U. -q●u.)}(T3-2)Table 2(b) Equations of rotational motion of body around X,Y and z axesIxxP - IxR+ (Iz- Ir)QR - IxPQ= EMx. .(1)+ EMx.m(2)+ &Mx. ,w(1)+ EMx. iuw(2)+M.B.x (R1)Mx.(1)= jSudM.. +x(- SudF. + SsdF,.){(R1-1)Mx.m(2)=IY.(SsdF.. + SsdF,. + SsdF..)- z..(SndF.. + SdF,. +. SzdF.)|(R1-2)XM.() = jSul- dlx,.p. - (dln. - dy.)q.r.1 + Jxdm[Szl(w. + p.o. -q.u.)- Tsgl -Sg|(i. + r.u. - p.w.)- T2gl](R1-3)Mx..m(2)=-jY..dm.IS3(i. +q.w. -r.u.)+ Sx(iq.u.)- gocos$l + ]Z.dm.{Sr(i. +q.w. -r.u.)+ S2(i. +r.u. - p.w.)+S2(w. + p.U. - q.u.) - gcosOsin$}(R1-4)InQ+(Ix - Iz)RP + Ix(P2- R)= SM.w(1)+ SM.m(2)0+ SM.w(1)+ SMr.. .()+ Mp.r(R2)Mr.(1)= ISndM,. + xr(- SdF. + S2dF.)I(R2-1)Mr.(2)= j1z.(SndF. + SndFy. + SyadF.,.)- Xn,.(SydFx. + SszdF,. + SssdF.,.)I(R2-2)y M.we(1)= jSu1- dlx.n. - (dJs,. - dl.)q.r.l+ Jxdm[Szl(iw. + P.D. - q.u.)- Tssg1-S2H(i. +r.u.-p.w.)- Tngl](R2-3)Mry.nux(2)=-]Zr.dm.ISu(i. +q.w. -r.v.)+ Sr(i. +r.u.-p.w.)+ Sp(i. + p.u. -q.u.)+ gsin@| + x..dm.ISs(u. +q.w. -r.0.)+ Sr(i. +r.u.-p.w.)+Ss(w. + p.U. - q.u.) - gcosOcosgφ|(R2-4)- IxP+ IzR + (Iy- lx)PQ+ lxQR= EM.m(1)+ SM.m(2)+ SMz. .w(1)+ EM..(2)+MB.z (R3)M..()=. J|SsydM.. +x(- SsdF.. + SsadF.,.)|(R3-1)M.(2)= J|X.. (SndF.. + S2dF,. + SaF..)- Yn,.(SudF. + SndF,. + SsydF..)I .(R3-2)zM..(1)= jSs1- dlx.p. - (dIx. -ds.)q.r.1 + Jxdm[Sxl(w. +p.u. -q.u.)- Tsg1-S3l(i. +r.u.-p.w.)- Tag}](R3-3)M.a(2)-=-JX.dm.ISr(iu. +q.w. -r.v.)+ Sn(中国煤化工’”.U.-q.u.) - gosOsn$o +JY.dm.ISu(i. +qb.w.)+:TYHCNMHG'S3(w. + p.u. - q.u.) + gsin@l(R3-4)Sunada S., et al.: Analysis of Maneuvering Flight of an Insect)3secondthird terms denote gravitational forces acting on system, Vx。, Vy。and Vz。as fllows:both the body and the wings. On the right hand side,U = Vx, cocos业+ Vy cosOsin业-EFx,. .c.o,SFr.mn.,.Fz. and 8Fx,. ,nr,,Vz。sin@EFr,. ,ner, EFz.,ine are aerodynamic fores and in-V = Vx,(sinPsin@cos业- cosWsinW) +ertial forces acting on wings, respectively. Their ex-pressions are indicated by Eqs. (T1-1), (T2-1), (T3-Vy。(singsin@sin业+ c∞os$∞osW) +1) and (T1-2), (T2-2), (T2-3), respectively. InVz。sin$oos@these equations, the forces in x,y, and z ( wing ele-W = Vx。(cososinOcos业+ sin$sinW) +ment-fixed coordinate system) are transferred to thoseVr。(sPsin@sin业- sin中∞os) +in X, Y and Z (Body fixed coordinate system). Thethird terms in the right hand sides in Eqs. (T1) -Vz cosφoos@(T3),FB,x, FB,y and FB,z, denote aerodynamic Wang et al . [u] showed the velocities of the mid pointforces acting on the body.between the hinge of the fore left wing and that of theIn Eqs. (R1)- (R3), the left hand-side denotefore right wing, V 'xp, V'y。and V 'z. The relationsmoment due to inertial forces acting on the body. Inbetween (Vx,Vy,Vz) and(V'xg,V"y,Vz)arethe right hand sides, 2 Mx,".eo, 2 My,. ,am, .as follows:EMz,. ,erand EMx. . ,nr, EMy,+ ,ner, EMz. . ,inerVxe =V'xg+ δ(cos@@2∞osψ + sin0O cos业-are moments due to aerodynamic forces acting on thewings and those due to inertial forces acting on them,sin@8 sin业- sin6的sin4业+respetivly. The E Mx.no (1), EMy,(1),cos@os02 + cos@sin型)EMz,.em(1), and 2Mx,. inr(1),SMy. ir(1), .Vy。=V'r。+ 8(os@@-sin亚+ sin@8 sin业+EMz,. .nr(1) in the body-fixed coordinate system aresinOB cos业重+ sin@的cos业+transferred from the moments in the wing element-cos@sinT42 - cos@cos4)fixed oordinate system,which act from the wings tothe body at the wing roots. The EMx, * ,er(2),2Vze =V'z。+ 8(sin062 + cos06)(2)My,. ,er(2), EMz,.aen(2), and 8Mx,* ,ner(2), .Angular velocity of a body around the X,Y and ZEMy,.nme(2), SMz.nme(2) in the bdyfixed o∞o_ axes are P,Q and R,espectively, and can be ex-ordinate systen are the moments by the forces aeing presed by using measured Euler angls, φ,日,业14:from the wings to the body at the wing roots. The(P=φ-业sin@Mg,x, Mp,Y and MB,z denote aerodynamic momentsQ =曰cosφ +业sin$cos@(3)acting on the body. Hereafter, the details of each termR =一白sinφ +空cos$cos@in these equations will be explained.Here,中,0,业correspond to ψ,φ and θ- 90", respec-3.1Inertial forces and moments acting on the bodytively , which was measuredused by Wang etal.n. Bye.g. mp(U+ QW - RV) in Eq. (T1) andusing the measured V'x,, V'y,,V'z,φ,0,业andIxxP- IxR+ (Iz- Irr)QR - IxzPQEqs. (1),(2) and (3), the inertial forces and momentsin Eq. (R1)acting on a body can be calculated.The components of the velocity of the center of 3. 2Aerodymamic forces and moments acting on thegravity of the insect' s body in the X, Y and Z coordi-中国煤化工nate system, U, V and W, respectively, are expressed5YHCN M H GB.x in Eq.(R1)by those of its velocity in the XE, Yε and ZE coordinateThe aerodynamc torces L moments ] acting on the)4Journal of Bionics Engneeing (2004) Vol.1 No.2body can be divided into two components. One is due tonamic pressure must be estimated by using Computa-added mass [ added moment of inertia] of the body[15]tional Fluid Dynamics or wind tunnel tests.and is proportional to the acceleration [angular accelera-3.3 Inertial forces and moments acting on wingtion] of body. The other is due to dynamic pressuree.g. Fx,ier and Mx,inerand this is proportional to the second power both of theIn Eqs. (T1-1,2)- (T3-1,2) and (R1-1,2,3,4)velocity and angular velocity of the body. Generally,-(R3-1,2,3,4), |means the summation along thethe forces [ moments] due to added mass [ added mo-wing span. The matrices S and T are indicated in Ap-ment of inertia] of the insect' s body are much less thanthe inrtial forces[ moments] acting on the body suchpenddix B and Table 3, respectively. The uw,v. andw。are composed of some components as follows:asmp(U+ QW- RV) and IxxP- IxzR+(Iz-{u. = u.(T)Iyy)QR - IxzPQ. This is because the former and theo。= v,(R)+ v.(T)+ v.(F)(4)latter are proportional to the density of air and that of(w. = w.(R) + w.(T)+ w.(F)the body, respectively and because the added mass[added moment of inertial] of the body can bewas ap-Here the R, T and F in the brackets denote the rota-proximated by that of the circular cylinder whose shapetional, translational and flapping motions, respectively.is close to that of the insect' s body. On the otherEquations for these seve components are listed in Tablehand, the aerodynamic forces and moments due to dy-3. Induced velocity is ignored in the estimationof u. , .Table3 Components of u.,v●and w.VxgTn+VgTa+VqTuu.(T)Ti= - sinβ. cosi . cos@ocsV土cosp. (singsin@∞sW - cossinW) - sinp . sini . (cosPsin@csW + singsin,)TIr = - sinp. osi . cos@sin业土cosP . (sinNsinOsin业+ cos中∞osW) - sinp . sini . (osin@sin中- sinCoos)Tig= sinp . cosi . sin@土cosp . sinos@ - sinp. sini . cosos@v.(R) r. {P(sini . cos9. - osi .cosP. sin0. )+ Qsinp. sin0. + R(osi .os0. + sini . cosp. sin0.)lVxqTn+ VyTx+ VzgTxT21=土(sin0. sini. + c∞os0●c∞osP. cosi . )coc0Csow + cs0 . simp . (singsin@csW - cos9sinW)土(- sin0 . cosi. + coso . cs. sini。)(cosPsin@oas业+ sinDsin$)v.(T) T= 土(sin0。sini. + cos0. cosp, cosi。)cosOsin业+ cos8 . sinβ. (sinosin0sin业+ cosos业)土( - sin0. cosi. + cos0 . cos. sini . )(cosPsinOsinψ - sinCs)T2=干(sin0.sini.+cos0.cos.cosi.)sin@+cos0.sinp.singcos@土( - sin0. cosi. + cs0. cos. sini . )os中oos@v.(F)干x。p。 cos9.w.(R) Fx. |P(sini . sin0. + cosi●cosp. cos0.)F Qsinβ.cos0. - R( - cosi . sin0. + sini . cos8. cos0.)|VxTs+ VygTx+ VzgT3sT31= (cs9. sini . - sin0. cos. csi . )osocs平F sin0. sinβ. (sin$sin@cs平- csosinW)-(os0 . cosi. + sin9 . cs. sini . )(cososin@cs业+ sindsin)w.(T) Ta = (cos0. sini. - sin . cos8 . cosi . )cosOsin业干sin0. sinp . (sinPsinOsinW + c∞os中∞sψ) -(cos0 . cosi . + sin0. os. sini . )(osPsin@sin业- sin中os业)Tss= - (coe8 . sini . - sin0. cosf. cosi . )sin@干sin0。sinp . singoos@o -(cos0. cosi . + sin0. cos. sini . )oos中cos日中国煤化工-w.(F)) x。β. sin0.The upper and lower signs are for the right and left wings, respectively.MYHCNMHG-Sunada S., et al. : Analysis of Maneuvering Flight of an Insect95V。and w.. The p.,q. and r. are the Euler anglese.g. Fx,* ,ae and Mx,+ ,aeroof wing element and are given byThe inflow velocity to a wing element in the x*, .p. =- Psinβ,cosi土Qosβ. - Rsinβ. sini土自y.,z* directions, which are required for calculatingq. =P(sin0, sini + cosθw cosβ cosi)士aerodynamic forces and moments acting on wings suchQcos0。sinβ. + R(- sinθ,oosi +asdFr. and dMx,*, are - u*,- V*,- w*,re-spectively. Fig. 2 shows the time variations of the in-cos0 . cosβ. sini) - β. sinθ。(5)r。=P(osθ.sini-sin0.c∞osβ.cosi)干flow velocity v记+ w2. to the wing elements at the50% span-wise position about the dragonfly analyzedQsin0w sinβ。- R(cosθ,cosi +by Wang et al . [1] These values are calculated by the e-sin0 . cosβ. sini)干β. cos0。quations in Table 3. Note that the inflow velocity alongwhere the upper and lower signs are for the right andthe wing span,一ux, is ignore in the blade elementleft wings, respectively. The inertial forces and mo-analysis. Fig. 3 shows the time- -variations of each com-ments acting on wings can be calculated by using theponents of uuf, Vrf and wf at 50% spanwise position formeasured data such as β,i,0 and Eqs. (1)-(5) andthe left fore wing. The inlow velocity in x direction,equations in Table 3.that is along the wing span, uu, which is shown in3.4 Aerodynamic forces and moments acting onFig.3, is not much smaller than the inflow velocity towingsthe wing elements V 2。+ w?,which is shown inDown strokeUp strokeDown stroke.Fore wingsHind wingsFore/LeftFore/Right飞Hind/Right0.20.60..0Non-dimensional timeFig.2 Time-variations of inflow velocity v中国煤化工of the left fore, right fore, left hind,MYHCNMHG96Journal of Bionics Engineering (2004) Vol. 1 No.2Down strokeUp strokeDownStroke .小2.0 r1.5崇E1.0 --Mur(R)语言空 0.5-wr(R)-Vr(R)自-0.5-1.0-1.5 !0.20..6.0Non-dimensional timeFig.3 Time-variations of the comnponents of ur, vr and wr at 50% span- wise position of the left fore wingFig. 2. This means the method based on blade elementof - vr(F) and - wr(F), respectively. Inlow veloci-analysis might make a big error in the estimation of theties due to the translational motion of the body such asaerodynamic forces and moments acting on the wings.- vr(T) and- wy(T), and flapping motion such asTher + w. for the left fore wing is maximum at-vn(F) and - wy(F) are dominant in this flight.the mid down stroke (Fig. 2). This maximum is causedThe inflow velocities due to the rotational motion of theby the large values of- wH(T), - wp(F) and - wIrbody, - vu(R) and- wH(R), are not much smallerthan the other components throughout the entire cycle(R) (Fig. 3). However, the√。+w2, for the left(Fig. 3). Therefore, the inflow velocities due to thefore wing is not so large at the mid up stroke (Fig.2),rotational motion of the bodv cannot be ignored in thebecause- vr(T), - v(F),- wf(T) and - wu(F)for an中国煤化工.are dominant terms at the mid up stroke and becauseYHC N M H Ge feathering angle atthe valuesof- vr(T) and - wr(T) cancel with thosethe 50% span-wise position of the dragonfly. TheSunadaS., et al.: Analysis of Maneuvering Flight of an Insect97feathering angle at a span-wise position with the dis-ratio. This large a。means that unsteady separatedtance of x apart from the flapping hinge (X6,。=x) ofvortices around the wings are expected to affect thethis dragonfly,0. (x), is approximated7] byaerodynamic forces and moments acting on thθ.(x) = (0o.s,. - 0o.s,.)(x/0.5xw,.) + 0o.s, ..wingsl6. A CFD method calculating aerodynamic(6)forces and moments acting on the wings171 is required,The geometrical angle of attack a., that is, an anglebecause of flow along span and unsteady separated vor-of attack where induced velocity is excluded, is giventices, which were stated above.bya. =tan^ (w./F v*). Here, the negative and4 Conclusionspositive signs are for the right wings and left wings, re-spectively. Fig.4 shows a+ at 50% span-wise positionEquations of motion for a maneuvering flight of anof the four wings, revealing that la. | is almost alwaysinsect have been derived. Expressions of inertial forceslarger than the angle of attack for maximum lift- dragand moments acting on a body and wings are includedDown strokeUp strokeFore wings-十Hind wingsHind/LeftFore/LeftdHindRight日2t0.20.60.81.Non-dimensional time中国煤化工Fig.4 Time-variations of geometrical angle olMHCNMHGposition of the left fore, right fore, left hind, and right hind wings98Journal of Bionics Engineering (2004) Vol. 1 No.2in these equations, and they can be estimated when the [ 6 ] Rippell G. Kinematic analysis of symmetrical flight maneu-motions of the body and the wings are measured.vers of Odonata. Journal of Experimental Biology, 1989,144: 13-42.It was investigated how the wing motion of adragonfly in the maneuvering flight, which was mea-[7] Azuma A, Watanabe T. Flight performance of a dragonfly.Journal of Experimental Biology, 1988, 137: 221 - 252.sured by Wang et al .1, is determined. The following[ 8] LanC E. The unsteady quasi vortex lttice method with ap-characteristics of this flight were made clear. (1) Theplications to animal propulsion. Journal of Fluid Mechan-phase difference in flapping motion between the twoics, 1979, 93(4): 747- 765.fore wings and two hind wings, and the phase differ-[9] Ennos A R. The ierial cause of wing rotation in Diptera.ence between the flapping motion and the featheringJournal of Experimental Biology, 1988,140: 161 - 169.motion of the four wings are equal to those in a steady [10] Auma A. The biokineics of flying and svimming.forward flight with the maximum eficiency. (2)TheSpringer- Verlag, Tokyo, 1992.camber change is mainly' controlled by muscles at the[11] Nevile A C. Aspect of flight mechanics in anisopterousdragonflies. Journal of Experimental Biology, 1960, 37:wing bases of the dragonfly. The reason for this is that631 - 656.the camber change of these wings is similar though the[12] Ennos A R, Wootton R J. Functional wing morphology andaerodynamic and inertial forces acting on the left wingaerodynamics of Panorpa Germanica ( Insecta: Mecoptera).are different from those acting on the right wing. (3)Jourmal of Experimental Biology, 1989, 143: 267 - 284.The feathering motion is mainly controlled by muscles[13] Sunada s, Song D, Meng X, Wang H, Zeng L, Kawachiat the wing bases. This is because the phase differenceK. Optical measurement of the deformation, motion, ancbetween the flapping motion and the feathering motiongenerated force of the wings of a moth, Mythimna Sseparataof the four wings is - 90". The value is far from - 90°(Walker). Internatiomal Journal of JSME Series B, 2002,when the feathering motion is strongly affected by the45: 836 - 842.aerodynamic and inertial forces.[14] Rolfe J M, Staples K J. Flight Simulation. Cambridge U-niversity Press, Cambridge, 1986.References[15] Tuckeman L B. Inertia Factors of Elipoids for Use inAirship Design. NACA No. 210, 1925.[1]Wang H, Zeng L, Liu H, Yin C. Meesuring wing kine-[16] Sunada s, Kawachi K, Matsumoto A, Sakaguchi A. Un-matics, flight trajectory and body attitude during forwardsteady forces on a two-dimensional wing in plunging andflight and turning maneuvers in dragonflies. Journal of Ex-pitching motions. AIAA Journal, 2000, 39(7): 1230 -perimental Biology , 2003, 206: 745 - 757.1239.[2] Schilstra C, Hateren J H. Blowfly fight and optic flow. I.Thorax kinematics and flight dynamics. Journal of Eexper-[17] Liu H, Kawachi K. A numerical study of inset fight.Journal of Comparatire Physiology,1998, 146: 124 -imental Biolology, 1999, 202: 1481 - 1490.156.[3] Wakeling J M, ElingtonC P. Dragonfly flight. II. Veloci-ties, acelerations and kinematics of fapping flight. Journal[18] Sunada S, Zeng L, Kawachi K. The relationship betweendragonfly wing structure and torsional deformation. Journalof Experimental Biology, 1997, 200: 557 - 582.of Theoretical Biology, 1998, 193: 39 - 45.[ 4] Alexander D E. Unusual phase relationships between the[19] Bisplinghoff R L, Ashley H, Halfman R L.. Aerolasticity.forewings and hindwings in flying dragonflies. Journal ofAddison- Wesley, Mas sachuetts, 1955.Experimental Biology, 1984, 109: 379- 383.[ 5 ] Alexander D E. Wind tunnel studies of turns by flying drag-[20] Norberg A. The pterostigma of inset wings and inertialonflies. Journal of Extperimental Biology, 1986, 122, :81中国煤化主”Cmpratite Prsion- 98.FYHCNMHGSunada S, et al.: Analysis of Maneuvering Flight of an Insect99Appendix A Phase difference between flapping motion and feathering motionA wing element at a span-wise position undergoes both heaving motion,h , due to flapping motion and feather-ing motion, 0, in uniform flow, U, as shown in Fig. 5. The feathering motion of the wing around the torsionalaxis ξ=a(0.5c) is given by[18,19]G2日4θ(A1)aξ2- Ergp=pp2 - mλ- m1aξTorsional axis/Center of gravityh-c2ξ=c2cn ξFig.5 Wing element undergoing with heaving and pitching motionswherema=变px3{- ah- U(0.5-a)θ- 0.5c(0.125+a2)| +李pU2(a + 0.5){U8- h+ 0.5c(0.5 - a)卧}(A2)m1 = Puctnh(A3)In Eq. (A2),the Theodorsen function is assumed to be 1, because the reduced frequency is 0.38. The mi iscaused by disagreement between the center of gravity and the torsion axis, and η is the distance between the tor-sional axis and the center of gravity as shown in Fig. 5. Norberg' 20] and Sunada et al . L18 showed that η is positivefor a dragonfly.When no bending deformation was observed along the wing span, the heaving motion can be expressed ash = (x/xw)βrcos(wt)(A4)The feathering motion 0 varies linearly along the wing span?7 , and therefore its two components can be assumned tobeθλ= (x/xw)Fpoos(wt + $)(A5)θ= (xxw)0Fcos(wt t φ)(A6)where θA and 01 are the feathering motions due to mAand m, re中国煤化工(A4) and (A5) in-to Eq. (A1) yieldsYHCNMHG100Journal of Bionics Engineering (2004) Vol.1 No.2中A= tan~1l文(A7)wherex=1- puJBAw2 - (n/16)e*(0. 125 + a2)可aw2 - (π/2)pU2c2(a + 0.5)B}{(π/8)pc3axuBiw2} +{(n/2)pUc2(a + 0.5)xwB1wl{- (π/8)pc"U(0.5 - a)w + (π/4)pUe*(a +0.5)(0.5- a)0aw}Y =- {- pJIJxw2 - (r/l6)pc*(0. 125 + a2)0xw2 - (r/2)pU2c2(a + 0.5)可l{(π/2)pUe2(a + 0.5)xw xβw} + {(π/8)xaxwB1w2H{- (π/8)pc3U(0.5- a)OAw + (π/4)pUc3(a + 0.5)(0.5- a)0\w} (A8)The fllowing relations between the signs of X, Y and中A the are then satisfied(- 180°≤φA≤←90°,x≤0,Y≤0.90°≤φA≤0°, X≥0, Y≤0(A9)|0°≤φA≤90°, X≥0, Y≥0(90°≤中≤180*, X≤0, Y≥0The value of中A is close to 0° whena= -0.8, Ip≈10 14m', pw=1.1X 103 kg/m3[18), ρ=1.225 kg/m3, c=0.01 m, w=210 rad/s, U≈1.5 m/s. .Substuting Eqs. (A4) and (A6) into Eq. (A1) yieldsφ=0(A10)When both moments ma and m act on the wing element simultaneously, the phase difference between the heavingmotion and the feathering motion is close to 0, because both中A and φ were assumed close to 0 in the above discus-sion.Appendix B Relationships between coordinate systemsThe relation between the earth-fixed coordinate system (XE - YE - ZE) and the body-fixed coordinate system(X - Y - Z), whose origin is located at the gravitational center of the body, is expressed by the following ma-trix[14]X}cos@cos业cos@sin亚- sin@ }{XE](XE]Y|=sinWsin@cos业- cos中sin业sin$sin@sin业 + cos中cos亚sincos@Yp=E1|YE| (B1)(cossin@cos亚+ sinWsin业 cosDsin@sin亚- sinDoos亚 cos中cos@) [ ZEJ( ZE)A coordinate system (X3,*- Y3,. - Z3,. ) is obtained by moving the origin of the body- fixed coordinate sys-tem (X - Y- Z) to the wing hinge. Therefore,X3,.(X..'|YYh,.(B2)A coordinate system (X4,.- Y4,. - Z4,*) is obtained by rotating the (X3,*- Y3,* - Z3. ) system around theY3,. axis by(-90 -i.)", where i, is the inclination of the stroke plane. For a left wing, in addition, Y4,. isreplaced by- Y, x, yielding(X4,.]- sini. 0 cosi. ][X3,.)(X3,.1Y4,* = |0士1中国煤化工(B3)- cosix 0- sini #.MYHCNMHGThe upper and lower signs in土are for right and left wings, respectively.Sunada S., et al. : Analysis of Maneuvering Flight of an Insect101A coordinate system(Xs, . - Ys,. - Zs,.) is obtained by rotating the(X4,. - Y4,. - Z4,. ) system aroundthe X4,* axis by β*, and replacing Y4,.,X4,. and Z4,. by Xs,.,- Ys,* and Zs,。respectively{Xs,*0 cosβ。 sinf. } {X4,.(X4,.Ys,. =0Y4,.=E3| Y4,.(B4)0 - sinβ.cosβ. (z.,.)(Z.,.JHere the positive rotational direction of a coordinate system is clockwise along the axis and counterclockwise for theright-hand and left-hand coordinate system, respectively.A coordinate system X6,. - Y6,.- Z6,. is obtained by rotating the (Xs,.- Ys,.- Zs,.) system aroundthe Xs,. axis byθ, - 90*. For a left wing, in addition, Y6,。is replaced by - Y6,。, yielding[1} {Xs,.(Xs,*Y6,.= 0土sin0。 干cos0.Ys.*= E。Ys,.(B5)cosθ *sin0.儿Zs,.)(Zs..A wing- element-fixed coordinate system (x. - y. - z.) is obtained by moving the originof the (X6,. - Y6,.-Z6,.) system to the wing element at X6, . = x, yelding(xo ){ X6,*'y.|=| Y6,.- |(B6)l0.The matrix for rotating the (XE - Yε- Zg) system to the (x。- y. -z.) system isTu T12 T13)T= E4E3E2E1= |T21 T2 T23(B7)T31 T32 T3)where each component is shown in Table 3.The matrix for transferring the (x. -y. -zw) system to the (X- Y- Z) system is(Su SI2 S13]S=E2'EI1E41= |S21 S22 Sz(B8)S31 S32 S33where each component is as followsS11 =- cosi * sinβS12 =士(sini .sin0. + cosi。c∞osβp . cos0.)S13 = sini. cos0。- cosi . cosβ。sin0。S21 =士cosβ#S2 = sinβ+ cos0 *(B9) .S23=干sinβ。sinθS31 =一sini . sinβ.S32 =士(- cosi . sin0. + sini . cosβ . cos0.)(S33 =- cosi * cosθ。 - sini.中国煤化工YHCNMHG

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