Nonlinear dynamics of biomimetic micro air vehicles

Transcription

1 Journal of Physics: Conference Series Nonlinear ynamics of biomimeic micro air vehicles To cie his aricle: Y Hou an J Kong 8 J. Phys.: Conf. Ser View he aricle online for upaes an enhancemens. elae conen - Elecrosaics a he Molecular evel: Inroucion U Zürcher - Dynamic response of jes an flame o an acousic fiel V V Golub an M S Krivokoriov - Dynamic characerisics of Non Newonian flui Squeeze film amper C.P. Palaksha, S. Shivaprakash an H.P. Jagaish This conen was ownloae from IP aress on 5//8 a 9:8

2 7 Inernaional Symposium on Nonlinear Dynamics (7 ISND) IOP Publishing Journal of Physics: Conference Series 96 (8) 37 oi:.88/ /96//37 Nonlinear Dynamics of Biomimeic Micro Air Vehicles Yu Hou an Jianyi Kong College of Mechanical Auomaion, Wuhan Universiy of Science an Technology, Wuhan, 438, P..China Absrac. Flapping-wing micro air vehicles (FMAV) are new concepual air vehicles ha mimic he flying moes of birs an insecs. They surpass he research fiels of raiional airplane esign an aeroynamics on applicaion echnologies, an iniiae he applicaions of MEMS echnologies on aviaion fiels. This paper suies a micro flapping mechanism ha base upon insec horax an acuae by elecrosaic force. Because here are srong nonlinear coupling beween he wo physical omains, elecrical an mechanical, he saic an ynamic characerisics of his sysem are very complicae. Firsly, he nonlinear ynamic moel of he elecromechanical coupling sysem is se up accoring o he physical moel of he flapping mechanism. The ynamic response of he sysem in consan volage is suie by numerical meho. Then he effec of amping an iniial coniion on ynamic characerisics of he sysem is analyze in phase space. In aiion, he ynamic responses of he sysem in sine volage exciaion are iscusse. The resuls of research are helpful o he esign, fabricaion an applicaion of he micro flapping mechanism of FMAV, an also o oher micro elecromechanical sysem ha acuae by elecrosaic force.. Inroucion Flapping-wing micro air vehicles (FMAV) are new concepual air vehicles ha mimic he flying moes of birs an insecs. Compare wih fixe wing air vehicle an roary wing air vehicle, he FMAV inegrae lifing, hrusing an hanging funcion ino a flapping wing sysem, an has he abiliy o cruise a long isance wihou energy supply, a he same ime FMAV has high maneuverabiliy an flexibiliy. The flapping wing moe of flying creaures in naure gives us eificaion ha flapping wing moe is superior o fixe wing an roary wing moes when he wingspan is less han 5cm, an he resuls of bionics an aeroynamics prove i []. The characerisics of flapping wing moe lie in he lif an hrus is prouce by ownsroke an upsroke of he wings, so he esign of efficien an reliable flapping mechanism is more imporan. For FMAV which size is similar o flies an bees, he esign, fabricaion an suy of ynamic characerisics of micro flapping mechanism base on MEMS echnology are key quesions.. The Design of Micro Flapping Mechanism The millimeer-size insecs are our moels o evelop insec base FMAV. New conceps shoul be applie in he esign incluing flexible srucures an fricionless joins. The special feaures of flying To whom any corresponence shoul be aresse. c 8 IOP Publishing

3 7 Inernaional Symposium on Nonlinear Dynamics (7 ISND) IOP Publishing Journal of Physics: Conference Series 96 (8) 37 oi:.88/ /96//37 insecs such as exernal skeleons, elasic joins, eformable horax an conracing-relaxing muscles provie clues for esigning FMAV [, 3]. Figure shows he cross secion of an insec horax, illusraing he flapping mechanism of he wing. The upwar movemen of he wing resuls from isorions of he horax prouce by muscles no irecly aache o he wing. Elasiciy of he horax plays an imporan role in he fricion free high-spee wing movemen. In mos insecs beaing of he wings is conrolle by a nervous simulus, using on-off conrol. However, in some small size insecs, such as bees an flies, he beaing frequency is much higher han he repeiion rae of nervous simuli. I is insea eermine by he naural frequency of he mechanical sysem forme by he muscles, elasic hinges, an horax. fulcrum wing relaxe conrace conrace relaxe Figure. Cross secion of an insec horax. For millimeer-size FMAV, elecrosaic acuaor, piezoelecric acuaor, an elecromagneic acuaor can be use in he flapping mechanism [4, 5]. Among hose, elecrosaic micro acuaor is simple in work principle, convenien in realizaion, small in energy consumpion, an easy in concenraion, so i has he preominan posiion in he suy an evelopmen of micro acuaor. Base on former reasons, he flapping mechanism mimics he horax srucure of insecs, an is acuae by elecrosaic force. The basic moel of wo egrees flapping mechanism is shown in figure. The sysem is mainly compose of wo parallel elecric boars, one of hem is fixe on base, an he oher is movable an connece wih he linkage o acuae he wing beaing upwar an ownwar for boh sies. The whole srucure has no bearing an roary axis, an all he pivos aop flexible hinges. When a varying volage is applie o he wo boars, he varying elecrosaic force will acuae he wings beaing. When ifferen volage is applie in he lef an righ of he boar, corresponing wing will have ifferen flapping ampliue, so he lif an hrus prouce by he wing is ifferen, an make he whole air vehicle urn aroun. C A B Figure. Micro elecrosaic flapping mechanism base upon insec horax. 3. Coupling Dynamic Moel of Acuaor Mechanism For he mechanism illusrae in figure, suppose he movable boar is rigi. The linkages suppor an consrain he boar an play he role of spring an amper. Suppose he isorion of he flexible hinges an linkages are wihin heir elasiciy limi an he equivalen siffness is linear along he move irecion, so he sysem ynamic moel is shown in figure 3. When a volage is applie o he wo boars, he elecrosaic force pulls he movable boar owar he fixe boar. The isance beween he wo boars affecs he elecrosaic force, an a he same ime he change of he elecrosaic force affecs he isance beween he wo boars, ha is o say, here is coupling acion beween elecrosaic fiel an mechanical fiel. When he volage is beyon he criical volage, he movable boar will be pulle in he fixe boar.

4 7 Inernaional Symposium on Nonlinear Dynamics (7 ISND) IOP Publishing Journal of Physics: Conference Series 96 (8) 37 oi:.88/ /96//37 In figure 3, k is spring siffness, is iniial isance, an a θ are ranslaion an roaion isplacemen. are wih an lengh of boar, x an Figure 3. Dynamic moel of flapping mechanism. When he volage U an U are ae o lef an righ of he boar, he movable boar will reach an equilibrium poin a ( x, θ) as shown in figure 3. Then he elecrosaic force in verical irecion is given by F e = l = ( x l x x θ ) θ θ F e = l = ( x + lθ θ x x + θ ) Where F an are elecrosaic forces ha ae on lef an righ of he upper boar. e Fe The elecrosaic orque of he upper boar is given by T T e e = ll = ln( θ ) + ( ) ( ) x lθ θ x θ x = ll = ln( + θ ) + ( + ) + ( ) x lθ θ x θ x Where T an are elecrosaic orques ha ae on lef an righ of he upper boar. e Te The spring force an orque acing on he upper boar are F s = k( x + θ ) + k( x θ ) = kx () () T s = k( x + θ ) k( x θ ) = k θ Accoring o equaion (), () an (3), we can ge he coupling nonlinear ynamic equaions of micro flapping mechanism mx && + c x& + kx = Fe + Fe J && θ + c & rθ + ka θ = Te + T e (3) (4) 3

5 7 Inernaional Symposium on Nonlinear Dynamics (7 ISND) IOP Publishing Journal of Physics: Conference Series 96 (8) 37 oi:.88/ /96//37 Where c an cr are ranslaion an roaion movemen amping. Because he FMAV fly in low eynols number, he air amping force of he movable boar can be expresse as he linear funcion of he velociy approximaely. For equaion (4), if y = x /, δ = θ θ max θ ( ) = θ,he primiive equaion is as follows: εwa my && + c y& + ky = U + U 3 δ y δ y y y + δ 3 εwa δ δ δ δ J && δ + c & rδ + k δ = U ln( ) + U ln( + ) 3 δ ( y) y δ ( y) y + δ Define imensionless ime varianτ = ω, where ω = k / m is naure frequency, an efine amping raioζ = c mω, ζ r = c r Jω, so equaion (5) can be simplifie as (5) y + ζ τ δ + ζ r τ εw + y = U 3 τ k δ y δ 3εW + 6δ = 3 τ k δ U + U y δ y δ δ ln( ) + U ( y) y δ So his is he nonlinear ynamic equaions of he coupling sysem. y y + δ δ δ ln( + ) ( y) y + δ (6) 4. The Analysis of Sysem esponse Characerisics 4.. Sysem Nonlinear Dynamic Characerisics in Phase Space If efine y = y, y y& =, δ = δ, δ & = δ, hen sysem equaion change ino phase space form. The unge-kua numerical meho is use o solve he equaion an analyze he ynamic characerisics of coupling sysem in phase space. The phase space orbis in ifferen volage an amping coniions are as figure U= U= U= U=3 U=4 U=45.3. U= U= U= U=3 U=4 U=45 y. y y y (a) ζ =., ζ =. (b) ζ =. 3, r ζ r =.3 4

6 7 Inernaional Symposium on Nonlinear Dynamics (7 ISND) IOP Publishing Journal of Physics: Conference Series 96 (8) 37 oi:.88/ /96//37.3. U= U= U= U=3 U=4 U=45 y y (c) ζ =. 5, ζ r =.5 Figure 4. Sysem phase space orbi in ifferen volage an amping raio From figure 4 we can raw he conclusions: () There is an equilibrium poin in he y axis ( y = ), an his equilibrium poin is sable focus. The orbi near he equilibrium posiion forms a helix owar inner, an in fac, he upper boar sabilizes in an equilibrium posiion graually. When he volage is larger han he pull-in volage, he phase orbi becomes emanaive, an he upper boar is pulle o fixe boar. () In he scope of pull-in volage, he kinemaical cener move righ along wih augmen of volage. (3) The augmen of amping make he criical pull-in volage bigger corresponingly. (4) When he iniial values are ifferen, he properies of phase space orbi are ifferen. Iniial value will affec he ampliue of pull-in volage. 4.. Sysem esponse in Sine Signal Volage In acual work, he ime-varying volage ha applie o he wo boars makes he wing sroke up an own. For real flapping-wing micro air vehicle, we hope o prouce larger flapping angle by lower volage because of he consrains of weigh an volage promoing circui. When he exciaion frequency is equal o he naural frequency of he sysem, he sympaheic vibraion is occurre an he large flapping movemen ampliue is acquire. When he ime-varying exciaion volage is sine signal as U( ) = U sin( ω), an ( U sin( ω) ) ( cos(ω) ) U U ω = = cos τ ω Subsiue (7) ino equaion (6), an efine γ = ω ω, solve he ifferenial equaions wih numerical meho. The sysem response in ifferen frequency raio γ is shown in figure 6. The calculaion parameers are U =4V,U =V, ζ =., ζ =., iniial coniion is, r y = y& =, δ =., & δ =. In figure 5, τ y an τ δ are he curves of ranslaion an roaion isplacemen o ime, y y an δ δ are he phase space orbi of ranslaion an roaion movemen, respecively. (8) 5

7 7 Inernaional Symposium on Nonlinear Dynamics (7 ISND) IOP Publishing Journal of Physics: Conference Series 96 (8) 37 oi:.88/ /96//37.8 τ y τ δ...8 τ y.8.6 τ y τ δ. τ δ δ δ y y.8 δ δ y y.4... δ δ y y (a) γ =. 5 (b) γ = (c) γ = Figure 5. Sysem response curves of sine exciaion volage signal From figure 5 we can raw he conclusions: () The responses of sysem of sine exciaion volage inclue ransien response an seay response. When exciaion volage is less han pull-in volage, he response of he sysem is perioic, an he response frequency is wo imes of exciaion frequency. () From figure when γ =. 5 he isplacemen is maximum. The righ iem of he equaion can be expresse as funcion of cos( γτ ), so he sysem resonaes a γ =. 5, an now he movable boar has he maximum ampliue. In FMAV, in orer o acuae flapping mechanism by as small volage as impossible, he exciaion frequency shoul be closer o resonance frequency. 5. Conclusions The esign an ynamic characerisics of micro flapping mechanism are he keys of FMAV esign. This paper esigns an elecrosaic acuae micro flapping mechanism base upon insec horax an flapping movemen. The nonlinear saic an ynamic characerisics are suie by numerical calculaion an analysis. The resuls of research are helpful o he esign, fabricaion an applicaion of he micro flapping mechanism of FMAV, an also o oher micro elecromechanical sysem ha acuae by elecrosaic force. eferences [] Shyy W, Berg M an jungqvis D 999 Prog. in Aerosp. Sci [] Shimoyama I, Miura H an Suzuki K 993 IEEE Conr. Sys. Mag. 37 [3] Suzuki K an Shimoyama I 994 J. Microelecromech. Sys. 3 4 [4] Fischer M, Giousouf M an Schaepperle J 998 Sensor. Acua. A [5] Chu P B, Nelson P an Tachiki M 996 Sensor. Acua. A 5 6 6