Variable wing span in gliding birds 3Wcos tha i perpendicula t th flight path Sin an cos ca b replace witexpression involvin th sinkin spee throug th ai V3 an th ai spee VD WVs 1L W Vs ID: 96002
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3V A TUCKEINTRODUCTIODurin equilibriu gliding al th force o a glide ar balanced i neitheaccelerate no decelerates Th relationshi betwee a glider' ai spee an thspee a whic i sink throug th ai completel describe th glider' aerodynamiperformance an ca b predicte fro a relativel simpl theor fo rigi wing(Welch Welc & Irving 1955 Pennycuick 1975)Glidin birds however chang thei win spa durin flight Anyon ca observthes bird glidin slowl o full sprea wings the progressivel flexing theiwing a the glid faste an faster Hanki (1913 calle thi behaviou 'flegliding' Fle glidin influence stabilit (Lighthill 1975 an glidin performanc(Cone 1964 Newman 1958 Pennycuick 1968 Raspet 1960 Tucke & Parrott1970) I thi paper I shal analys th relationshi betwee lif an drag fodifferen win spans i bird glidin a equilibriu an us tha relationshi tpredic maximu glidin performanceTHEORConside a bir glidin a equilibriu a spee V alon a fligh pat incline aangl 6 (th glid angle t th horizonta (Fig 1) Onl tw force hav significaneffect o th bird gravitationa forc (th bird' weight W an aerodynami forceSinc th su o thes tw force i zero th aerodynami forc o th bir equal weigh i magnitud bu ha th opposit directionTh aerodynami forc ha tw component (Fig 1) tota dra (D) thcomponen Wsin tha i paralle t th flight path an lif (L) th componenSinkin speeFig 1 Force an velocitie durin equilibriu gliding Th glid pat i incline aangl d t th horizontal Lif an dra ar perpendicula an parallel respectively t thglid path Ai spee (V i paralle t th glid path an sinkin spee (V, i verticaan downwards Variable wing span in gliding birds 3Wcos tha i perpendicula t th flight path Sin an cos ca b replace witexpression involvin th sinkin spee throug th ai (V3) an th ai spee (V)D = WVs/ , (1L = W[ - (Vs/V)2]* (2Th tota dra coefficient CD, an th lif coefficient CL, ar compute b dividinth forc component b 0-5pSV2:CD = 2D/(pSV2) , (3CL = 2L/(pSV2) . (4Ai densit (p i 1 -2 k m~3 i thi stud th valu a se leve i th U standaratmospher (vo Mises 1959) an S i th projecte win area includin th are o wing intercepte b th bodyTh performanc o a glider describe b th relationshi betwee V an Vs,depend o tota drag a ca b see b rearrangin equatio 1V3 = VD/ . (5I shal analys performanc b dividin tota dra int additiv components inducedrag profil dra an parasit drag Man textbook o aircraf aerodynamic thi metho i detai (see fo example Pope 1951)Induced dragInduce drag D, arise wheneve th win produce lift Lif result fro ahighe pressur belo th win tha above an thi pressur differenc induce a flow o ai a th tip o th win an a downwar flow alon th win spanTh induce downwar flow cause th induce dra o th win (se Prandt &Tietjens 1957 fo a detaile discussion) Induce dra depend o win span b(measure acros th wides spa o th wings) an i give byD = 2kL2/(;rpb2V2) . (6Th induce dra factor k ha a valu o 1 fo a win wit a straigh leadin edgwhe viewe fro th front constan downvvas an a elliptica distributio o lifalon it span Actua wing typicall hav k value i th rang o 1 1- (Reid1932) an som wing theoreticall ma hav k value les tha 1 (Cone 1962 vaDam 1987) k als correct th parasit dra o a complet aircraf a lo speed (sesectio o Parasit dra below) I shal follo th commo practic o usin a valu o1 fo k whe dealin wit isolate wings an a valu o 1 whe dealin wit acomplet aircraft Speddin (1987 foun a valu o k o 1 -0 i a glidin sparrohawk Falco tinnunculus; whic i consisten wit a valu o 1- afte correctin fo dra a lo speeds (Som author plac th induce dra facto i thdenominato o equatio 6 i whic cas i typicall ha value o 1 o less.Th induce dra facto o bir wing ma chang durin flight i th wingPhang thei shap an spa s tha th lif distributio i no elliptical Ther arinsufficien dat t accoun fo thes change i th presen theory whic assume 3 V A TUCKEtha k remain constan durin flight. However th effec o increase i k wil bdiscusse qualitativelyTh induce dra i relate t th induce dra coefficient give byCD, = 2D,/(pSV2) . (7I th righ sid o equatio 6 i substitute fo D i th equatio above an L iexpresse i term o CL on obtain a simpl expressio fo C& ,:CD, = kCL2/(^A , (8wher A i th aspec rati o th wing define a th rati o win spa t winchord Win chor i th mea widt o th wing define a S/b ThereforeA = b2/SProfile dragProfil drag Dpr, i th dra o th win i additio t induce drag I arisefro ski frictio an pressur difference an i relate t th profil dracoefficient give byCD,p - 2Dpr/(pSV2) . (9Th profil dra coefficien varie wit th lif coefficien o th wing an hencwit V Th relationshi betwee Co.p and C depend o th shap o th wing'cross-sectio (th aerofoi section sinc th win i a aerofoil an th Reynoldnumbe (Re) o th wing define a/ (10c i th win chor calculate fro maximu win are (Sma, an maximu winspa (bmaA)c = Sniax/bma.x . (11H i th viscosit o th air Th rati p/n ha th valu 6 43 sm"2 fo ai a se levei th U standar atmosphereA a give Re, C plotte agains Co.p fo a rigi win yield a pola curve I thwin i no twiste alon it spa an ha th sam aerofoi sectio everywhere thpola curv i nearl independen o aspec rati (Prandt & Tietjens 1957 an icharacteristi o th aerofoi sectionCo.p a a give CL decrease a Re increases wit a dro o several-fol ove anarro rang o Re value (Hoerner 1965 Schmitz 1960) Th midpoin o thirang i th critica Re, whic decrease i turbulen ai o i th aerofoi surfac i Th critica Re fo a smooth bird-lik aerofoi sectio i non-turbulen ai i7 00 (th Gottinge 80 section Schmitz 1960 simila t th USAT 4 sectio iFig 4) Mos soarin bird hav Re value betwee 7 00 an 106. CDp changegraduall ove thi rang accordin t th relationshipF = 1-2 -0-226/te 10~5 + 0-015\{R X 10~5)2 , (12 Variable wing span in gliding birds 3wher F i th rati C pr/(C p a ReX 10"5 = 1) I derive thi relationshi frodat i Hoerne (1965 fig. 17 pp 6-13 an Schmit (1960 fig. 6 p 171 fo avariet o aerofoi section a C = 0-7As Re changes th pola curv shift righ o lef withou markedl changin shap(Schmitz 1960) Thus th C p vs C curv ca b shifte t an Re betwee thcritica Re an 106 b calculatin a correctio ter fo Co,Pr a a C o 0- an addintha ter algebraicall t th CD,p axis Equatio 1 wa use t correc al polacurve an CDp value i thi pape t Re X10~3 = 1 A exampl o th correctio ishow i th AppendixParasite dragParasit drag Dpar, i th dra o th aircraf exclusiv o th dra o th wingan i relate t th parasit dra coefficient give byCD,pa = 2Dpar/(pSparV2) . (13I flyin birds Dpa i th dra o th head body tai an feet Spa i th crosssectiona are o th drag-producin structuresTh parasit dra o a aircraf i commonl expresse i term o th are o a flatplat hel perpendicula t th airflow I th dra coefficien o th plat i assigne avalu o 1 a plat o are SparCDpa ha th sam dra a th parasit dra o aaircraf a spee V Th are o th plat i know a th equivalen flat plat areaS,p:Sf = SparCD,pa . (14I Sfp fo a aircraf i known parasit dra i give byDpa = 0-5pSfpV2 . (15Sf tend t increas a lowe speed a th tai o a aircraf drops therebincreasin th dra o th fuselag an tail Th chang i Sfp i small an i icommo practic t compensat fo i b increasin th induce dra facto slightl(Pope 1951) Sf als tend t increas a lowe speed a Re decrease (Goldstein1965)Speed and sinking speedTh tota dra o a aircraf i th su o th thre componentsD = D; + Dp + Dpa , (16so fro equatio 5V = V(D + Dp + Dpar)/ . (17Afte expandin th dra terms thi equatio becomesV = 2kL2/(jrpWb2V + 0-5pV3(SCD,p + Sfp)/ . (18 3V A TUCKEModelsEquatio 1 become a mode o a glide i on allow b L CD pr, Sfpan V t vari th sam wa tha the d i th rea glider Th simples mode ( shal cal i th'constan span model describe a glide wit rigi wings Spee varies bu th winspan profil dra coefficien an th othe variable o th righ sid o equatio 1remai constant I sinkin spee i plotte agains spee fo th constan spamodel a curv ( shal cal i th 'performanc curve' resultsTh performanc curv i ofte calle th 'glid polar' bu I shal us th formeterminolog t avoi confusio wit th pola curve tha relat lif an profil dracoefficient fo wings Performanc curve ar usuall draw o a V vs V9 diagrawit V9.increasin downward an V increasin t th righ (Fig 2)A mor realisti mode fo birds whic I shal cal th 'maximu performancemodel describe glider wit adjustabl wings Bird typicall exten thei wing tmaximu spa whe glidin slowl an flex glid a highe speeds A win spachanges s d win area induce drag an th lif an profil dra coefficientsThes change mak th produc SCo.p i equatio 1 a functio o win spanTh relationshi betwee sinkin spee an spee calculate fro equatio 1 awin spa varie nee no b a performanc curv a i th cas wit th constanspa model If a a give speed b varies the sinkin spee varie a well Bird caals increas sinkin spee b lowerin thei fee t increas Sf (Pennycuick 19601968 19716) A a result sinkin spee ca hav a rang o value a eac speed anspeed \nkin :o0246A"*~\ *^Maximu performanc\ A ^y curv\ \\ \Minimu performanc \curv ^"! ia2 4Speed V (ms"1)6Fig 2 Th performanc are bounde b performanc curve fo a falcon Falcojugger.Spee fo maximu performanc i lowes a A wher C| i maximu a 187 V = Valon lin BC - i.e descen i vertica a eac speed Vertica descen i slowes a B,wher tota dra = profil dra o th full extende wing wit Cn,p = 1 - i.e th wingac a flat plate hel perpendicula t th airflow Vertica descen i fastes a C, whertota dra = parasit drag i.e th wing ar full folded Soli line calculate fro ththeor i thi paper dashe line interpolated Variable wing span in gliding birds 3th point relatin sinkin spee an spee fal i a are (Cone 1964) I shal calthi th 'performanc area (Fig 2)Th 'maximu performance mode describe th uppe boundar o thperformanc area I shal cal thi boundar th 'maximu performanc curve'Maximu performanc i thi sens mean tha a bir ca glid a a particula speefo th maximu tim an distanc throug th ai whe i minimize sinkin speedI on consider th motio o th bir relativ t th earth th maximuperformanc curv show th minimu vertica win velocit componen require tkee th bir fro losin altitud and fo a give horizonta wind indicate th distanc ove th groun tha th bir ca cove fo a give los i altitudeTh lowe boundary o th performanc are i th 'minimu performanc curve'whic describe th maximu sinkin spee a eac speed A al bu th lowes maximu sinkin spee occur i vertica descen whe Vs = V andra = weight Vertica descen i slowes whe th bir 'parachutes o extendewing (Fig 2 poin B), an fastes whe th bir fold it wing an dive (Fig 2poin C)The constant span modelI th constan spa model L i assume t equa W ( constant) th onlvariabl o th righ sid o equatio 1 i V Th equatio fo th performanc curvbecomesV9 = k,/ + k2V3, (19wher k an k2 ar constants define a followsk = 2kW/(jrpb2) , (20k2 = 0-5p(SCD.p + Sfp)/ . (21Thi mode i usefu t glide pilot (Welc et al. 1955 becaus i approximate th o man-mad glider wit simpl wings Suc glider ca fl onl ath combination o V an V tha fal o a singl performanc curv becaus theirigi wing hav lif an profil dra coefficient tha fal o a singl pola curve Thglide canno chang it drag an henc it sinkin speed withou changin speed[Fo manoeuvre suc a landing glider ar usuall equippe wit panel o thwing tha ma b raise t increas profil drag Advance glider als hav winflaps tha ca chang th aerofoi sectio o th win durin flight see fo exampleJacob (1986) Th constan spa mode doe no describ glider tha us thesdevices.Th constan spa mode ha bee use fo glidin bird (Newman 1958Pennycuick \91\a,b, 1975 1982 Blake 1983) Th simplicit o th mode iattractive bu a glidin bir nee no hol it win spa an th othe variable oth righ sid o equatio 1 constan a differen speed a th mode requires Thconstan spa mode describe a performanc curv tha ma b wholl withi thperformanc area partl withi i o wholl outsid it dependin o th value t k an ki i equatio 19 4 V A TUCKEThe maximum performance modelI th maximu performanc model win span win are an th profil dracoefficien ma var a eac speed thei value ar chose t minimiz sinkin speea eac speed Unde thes conditions equatio 1 describe th maximuperformanc curveSinkin spee ma hav a minimu valu a a give spee whe th wing arflexed t les tha thei maximu span Suppose fo example tha Co.p werconstant The change i win spa woul hav opposit effect o tw o th draterm i equatio 17 a reductio i spa woul increas induce dra bu decreasprofil dra b reducin win are (Newman 1958 Pennycuick 1968 Pennycuic &Webbe 1959 Tucke & Parrott 1970) Indeed a falco glidin i a win tunne adifferen speed ha bee observe t adjus it win spa t nea th theoretica fo minimu sinkin speed (Tucke & Parrott 1970)Co.pr however varie wit win spa becaus (1 C p an C ar relate b apola curv i bir wing a spa varies an (2 C i relate t S (equatio 4) varie wit win span T develo th maximu performanc model I shalrepresen thes tw relationship b polynomialsCD.p = C0 + QC + C2CL (22anS = C3b + C4 , (23i whic CQ ... C4 ar constantsTh win spa fo minimu sinkin spee a eac spee ca b foun b makinappropriat substitution int equatio 16 settin th derivativ dD/d equa t 0an solvin fo b Th minimu sinkin spee ca the b calculatedFirs th term i equatio 1 ar expandedD = 2kL2/(;rpb2V2) + 0-5pSCD,prV2 + 0-5pSfpV2 . (24Fo econom o notation som o th constant i equatio 2 ma b combine intne constantsD = k3L2/(bV)2 + k4SCD,prV2 + k5V2 . (25Befor differentiation CD,p an S i equatio 2 mus b replace wit function ob Cop i a functio o C (equatio 22) sD - k3L2/(bV)2 + k4SV2(C0 + C,C + C2CL2) + k5V2 . (26C i a functio o S s fro equatio 4CL = ksL/^SV2) (27anReplacing S wit th righ sid o equatio 23D = k3L2/(bV)2 + Cok4(C3b + C4)V2 + Qk^ +C2k4(k6L)2/[(C3b + C4)V2] + k5V2 . (29 Variable wing span in gliding birds 4DifferentiatingdD/d = -2k3L2/(b3V2) + C0k4C3V2 - C2k4C3(k6L)2/[(C3b + C4)V]2 (30and replacin L wit W[1-(VS/V)2] (equatio 2)dD/d = -2k3VV2[ - (Vs/V)2]/(b3V2) + C0k4C3V2 -C2k4C3(k6W)2[ - (V3/V)2]/[(C3b + C4)V]2 . (31Th valu o b tha make th derivativ equa t 0 i th win spa (b' fominimu tota drag Replacin b wit b' L wit W[ (Vs/V)2] an D witWVS/ i equatio 2 yieldV9 - k3W[ - (V3/V)2]/(b'2V + V3(k4SCD,p + ks)/ , (32wher V i th valu o V3 use t calculat b'Th first chose valu o V3 wil probabl no b a solutio o equatio 3 (i.eVs wil probabl no equa V3) an I use Newton' metho (Sokolnikof &Sokolnikoff 1941 t estimat a ne valu o V3. Thi valu i use t recalculat b'an th whol proces i repeate unti value o Vs an b ar obtained tha d satisfequatio 32 Th procedur i repeate wit differen value fo V t obtai thmaximu performanc curveTh quantit C i th abov equation varie wit Re, an henc wit V A Rechanges th curv (equatio 22 relatin Co.p t C shift t th righ o lef b aamoun relate t F fro equatio 12 Thi shif i accomplishe mathematicall b C(F1 t Co, wher C i th valu o CD p calculate fro equatio 2 foCL = 0-7 ( compute program tha calculate a maximu performanc curveincludin Re effects i availabl fro th author.MEASUREMENTThi sectio describe th dat tha wil b use t evaluat th maximuperformanc mode an summarize ho the hav bee obtained Al th dat havbee published an detail ma b foun i th origina publications Curvilinearelationship betwee tw variable ar describe wit second-degre polynomiaequation (fitte b leas squares o th formy = a + ai + a2x2 , (33wher ao a an a2 ar constantsLift and dragLif an dra ca b determine b measurin W an th glid angl 6, o V anV an the solvin equation 1 an 2 I som cases th measurement hav beemad o bird glidin i natura condition b usin trackin devices mounteeithe o th groun o i a sailplan (Raspet 1960 Pennycuick 1960 1971a) trecor th positio o th bir relativ t th observerI i difficul t mak accurat measurement i natur fo tw reasons(1 trackin device o sufficien accurac ar complicate an difficul t operate 4 V A TUCKE(2 no onl th bir mus b tracked bu th ai a well Equilibriu glidinrequire a inertia fram o referenc relativ t whic th ai doe no move Thfram ma mov relativ t th observe (fo example i ther i a wind) bu itmovement mus b measurable Indeed ther ma no b suc a frame fo windofte chang i bot spee an directio a a resul o gust an topographicafeature (se McGahan 1973 an Pennycuick 1971a fo a discussio o thesproblems)Th glid angl 9 ca b measure mor accuratel i a win tunne tha i ca inature I a bir i traine t glid freel i a win tunne tilte a th glid angle thbir remain stationar relativ t th observer an th motio o th ai i th wintunne i known Thi techniqu ha bee use wit a pigeo (Columba Iwia;Pennycuick 1968) a lagga falco (Falco jugger, simila i siz t th mor familiaperegrin falcon Tucke & Parrott 1970 an a blac vultur (Coragyps atratus;Parrott 1970) Thes studie wil b identifie i thi pape b reference t 'thpigeon' 'th falcon an 'th vulture'Wing span and areaWin spa an are ar change i livin bird durin flight Maximu win spa(bmax an are (Smax) ca b measure o dea bird b stretchin ou th wings Alesse spans th relationshi betwee win spa an are wa linea i th pigeon an vultur glidin i a win tunnel Thi relationshi ca b estimate foglidin bird fro measurement o bma an Sma (se Appendix)Equivalent flat plate areaEquivalen fla plat are o a bir ca b measure b placin th bod o a flighbalanc i a win tunne (Pennycuick 1968 Tucker 1973) Sfp i a functio o abird' linea dimension an varie wit bod mas (Tucker 1973)Sf = O-OO334m°-66 . (34Thi equatio wa obtaine fro dra measurement mad a nearl constan speed(1 12ms~' o bird wit masse betwee 0-02 an 6-9kg I therefor includeRe effect du t bod siz bu no thos du t speed Sfp di no chang wit speei th pigeon Variation i Sfp wit spee ar ignore i th presen stud sinc thdra du t Sfp i a smal proportio o tota dra a th speed a whic bird usuallglideP/vfile drag coefficientRigid zuings: polar curves for bird-like and conventional aemfoil sectionsTh profil dra coefficient CD pr, ma b determine b measurin th tota drao a rigi win an subtractin th induce dra fro it Th tota dra ca bmeasure b mountin th win o a flight balanc i a win tunnel an induce ca b calculate usin equatio 6 Alternatively induce dra ma beeliminate s tha profil dra equal tota drag Thi i accomplishe b mountinga untwisted rectangula win wit a constan aerofoi sectio i a win tunne s Variable wing span in gliding birds4tha i span th tunne fro wal t wal (Pop & Harper 1966) Th profil dracoefficien i obtained b dividin profil dra b 05pSV2.Th shape o bird-lik aerofoi section ar quit differen fro th shape oconventiona ones Bir wing (fo exampl se Nachtigall 1985 hav highl uppe an lowe surface a th bas o th win an les camberesurface toward th ti (Fig 3) Wing o conventiona aircraf hav lowe surfacetha ar nearl flat t convex a i th Y sectio (Fig 4) Thi sectio wa usei man aircraf buil betwee 192 an 1940 an i ofte appear a a exampl i o low-spee aerodynamics It aerodynami characteristic ar well fo a rang o Re values an i wil b use i thi pape a a referenc fo wit th les conventional bird-lik sectionsTher ar als difference betwee th pola curve o bird-lik aerofoi sectionan thos o conventiona aerofoils Conventiona section hav a minimu Crj,p1-1-1-1-10-0-0-0-0Gottinge 46RA 1Gottinge 40Eiffe 300 00 00 00 01 01 01 01Profil dra coefficient CDFig 3 Bird-lik aerofoi section an thei pola curves Th section progres (to tbottom fro a highl cambere sectio typica o th bas o a bir win t a lesscambere sectio typica o th ti o a bir wing Show fo compariso ar th polacurv fo minimu dra (equatio 35 fo th falco an th vulture an pola curve fothre differen pigeo win section (Xachtigall 1979) Pola curve hav bee correcte a Reynold numbe o 105 an smoothe b fitting point t a second-degrepolynomia equation Dat fo th bird-lik section an thei pola curve ma b foun th Nationa Advisor Committe o Aeronautic Technica Report liste iReferences Th numbe o th Technica Repor i give i parentheses Gottinge 46(286) RA 1 (93) Gottinge 40 (124) Eiffe 3 (93) 4V A TUCKE1-1-JM-5 1'| 0-§ 0-0-0-0 :0 0-0 00 00 00 01 01 01Profil dra coefficient CD pFig 4 Th to thre aerofoi section hav th sam thicknes an progressivel lesscambere lowe surfaces Th lif coefficien fo th minimu profil dra coefficiendecrease a th cambe o th lowe surfac decreases Th botto aerofoi sectio(Clar Y i typica o thos use i low-spee man-mad aircraf flying a Reynoldnumber betwee 1-5X105 an 4X106. Pola curve hav bee correcte t a Reynoldnumbe o 10s an smoothe b fitting point t second-degre polynomia equation(excep Clar Y data whic wer fitted b free-han curve) Dat fo thes section anthei pola curve ma b foun i th Nationa Advisor Committe o AeronauticTechnica Report liste i References Th numbe o th Technica Repor i give iparentheses RA 1 (93) USAT 4 (93) Duran propelle 1 (93) Clar Y (244)valu a a CL valu nea 0 an CDp i nearl constan fo CL value betwee 0 an1- (Clar Y Fig 4) Highl cambere bird-lik section suc a th RA 1(Fig 3 hav a minimu valu fo CDp nea a CL valu o 1 an CDp increasetwo o three-fol a CL drop toward 0 I addition th minimu CDp value fobird-lik section ar highe tha tha fo th Clar Y sectionThes difference ar relate t th amoun o cambe o th lowe surfac o thwing Th mor th cambe o th lowe surface th highe th CL a whic thminimu CDp occur (Fig 4)No al studie o avia win section agre wit th abov description Nachtigal(1979 investigate mode wing wit th sam aerofoi section a thos o th pigeowing Th pola curve fo th mode section wer simila t thos fo conventionasection i tha CDp wa a a minimu nea a CL o 0 (Fig 3) Also th maximuCo.p value fo th mode section wer severa time highe tha thos fo th birdlik o conventiona section i Fig 3 4.Adjustable wings: the polar area for bird wingsTh profil dra o a win attache t a livin bir ca b determine bmeasurin th tota dra o th bir an subtractin induce dra an parasit drag Variable wing span in gliding birds4I contras t rigi wings CD,p an CL value fo th adjustabl wing o livin birdd no fal o a singl pola curve Bird ma chang win span aerofoi section andegre o win twis fro bas t ti a a give speed Consequently thei wing hava rang o CD,Pr value fo eac CL. Th point representin thi rang fal withi aare ( shal cal i th pola area o th CDp vs CL diagra rathe tha o a curve(Fig 7 show th pola are fo th falco an th vulture.Bir wing ca hav a pola curv i th bir glide unde condition tha selec oncurv fro th infinit numbe tha compris th pola area Fo example thpigeon falco an vultur wing ha pola curve whe th bird wer glidin iwin tunnel a thei shallowes glid angle (i.e wit minimu sinkin spee andrag fo a give speed Thes curve ar th left-han bound o th pola areas anI shal cal the 'pola curve fo minimu drag (e.g Fig 5)Th falco an vultur als glide ove a rang o glid angle a a particula speedMeasurement unde thes condition yiel pola curve tha I shal cal 'pola curvefo constan speed' The wil b describe afte th nex sectionPolar curves for minimum dragTh CD p an CL value fo th falco an th vultur fal o th sam pola curvfo minimu dra (Fig 5 whe correcte t a Re valu o 105. Th equatio foth curv isCD,p = 0-034 - 0-0781CL + 0-0799CL2 . (35Thi curv i a composit o th pola curve fo th bird-lik aerofoi section founi differen region o th win (Fig 3) A th win spa varies thes section ar00 00 00 00 0-1 01 01Profil dra coefficient CDFig 5 Pola curv fo minimu dra fo th falco (ope circles an th vultur (fillecircles) correcte t a Reynold numbe o 105. C increase a spee decreases Datfro Tucke & Parrot (1970 tabl 2 an Parrot (1970 tabl 1) 4V A TUCKEV1-1-1-1-10-0-0-0-0Blac vultur(RaspetRed-shouldere hawkwinWhite-backevultur Vultur (CD)-00 0 00 01 01 0-2 0-2Profil dra coefficient CDTota dra coefficient CDFig 6 Pola curve fo minimu dra fo variou bird an th win o a red-shoulderehawk Als show ar curve relatin th lif coefficien t th tota dra coefficien fo thfalco an th vulture Thes tota dra coefficient ar les tha th profil dracoefficient alon o th haw win an th pigeo a som lif coefficients Al polacurve hav bee correcte t a Reynold numbe o 105 an smoothe b fitting point tsecond-degre polynomia equations Se tex fo referencesexpose t th airflo t differen degrees s i i no surprisin tha th pola curvfo minimu dra share point wit th pola curve fo th aerofoi sectionsTh pigeon' pola curv fo minimu dra ha C p value tha ar muc greatetha thos fo th falco an vultur a hig C value (Fig 6) comparabl a Cvalue nea 05 an lowe a CL value belo 0-5 Pennycuic caution tha lo CD,pvalue fo th pigeo ma b unreliabl becaus o uncertaintie i measurinparasit dragUsin dat fro glidin vultures obtaine b trackin fro sailplane (Raspet1960 Pennycuick 1971a) presumabl wit th vulture a minimu sinkinspeeds I hav constructe pola curve fo minimu drag Th Co.p value fro Raspet' dat fo blac vulture (se Appendix ar negativ (Fig 6)probabl becaus th vultur an th sailplan wer glidin i differen ai masse(Tucke & Parrott 1970 Pennycuick 1971a)Pennycuic (1971a assume tha CD,p wa independen o C i Africa whitebacke vulture an estimate it valu t b 0007 a a Re valu o 2-9X103.Correctin thi valu t Re 103 yield 0011 somewha lowe tha th minimuCo,p valu o 001 fo th falco an th vultur (Fig 6)Th lif coefficient an tota dra coefficient fo drie bir wing mounte o afligh balanc i a win tunne hav bee measure fo severa specie (Wither.,1981) A pola curv constructe fro th dat (se Appendix fo th on soarin Variable wing spat? in gliding birds4Tabl 1 Characteristics of different species used zcith the maximum perfonnance modelSpecieFalcoBlac vulturAfrica white-backevulturFulmaAlbatrosW(N5-617-52-7-285-tw(m101-32-1103-000000S(m2)112 + 001260 - 0-02335b-0041073 + 0-03062 + 0-425s/s0-857b/bml-060b/bml-060b/bmt0-700b/bma0-307b/bnM*+++0-140-060-290-69s,P(m2)0-00210-004860-002700-0140Se tex fo references* Dimensionles relationshi betwee wing spa an win area| Se Appendi fo derivation\ Calculate fro equatio 34bir h investigated th red-shouldere haw (Buleo lineatus), wil b use hereAlthoug th measurement wer mad a on spee wit a constan win span thresult ar interestin whe compare wit thos fo th wing o livin bird(Fig 6)Th haw win ha C p value tha ar greate tha th tota dra coefficient oth intac falco an th vulture Clearly th drie win i no aerodynamicallsimila t livin wings Wither attribute th hig dra o th drie win t a lo Revalu (0- X 105, whic i belo th critica Re fo smoot wings) surfac roughnessflutterin o feather an win twist I addition th aerofoi section o livin birwing durin glidin ca chang markedl fro thos o anaesthetize o dea bird(Biesel But & Nachtigall 1985)I i remarkabl tha th sam pola curv fo minimu dra describe th dat fobot th falco an th vulture a thes bird ar distinctl differen i appearancan size A maximu span th falcon' wing hav pointe tip an a aspec ratio 7- (calculate fro dat i Tabl 1) Th vulture' wing hav squar tip endini separate primar feathers an a aspec rati o 5-6 Th vultur weigh thretime a muc a th falconTh pola curv fo thes bird i als simila t curve fo rigi wing wit birdlik aerofoils althoug i differ fro th curv fo th glidin pigeon Th pigeo ia flappin bir rathe tha a glidin bird Perhap measurement o othe flappinbird wil sho tha the to hav differen pola curve fro glidin birdsThes observation sugges tha pola curve fo minimu dra ma b simila tequatio 3 i al glidin bird wit highl cambere aerofoil - a hypothesi tha wilb interestin t test I th absenc o pola curve fo minimu dra fo glidinbird othe tha th falco an th vulture I shal us equatio 3 t evaluat thmaximu performanc mode fo glidin bird i generalPolar curves for constant speedBot th falco an th vultur glide i th win tunne ove a rang o glidangle a a give speed Dat fo bot bird fel o th sam pola curv fo a give 4V A TUCKE1-(J1fficien§Lif1-1-1-10-0-0-0-66ms"100 00 00 00 01 0-1Profil dra coefficient CD01Fig 7 Pola curve fo constan spee fo th falco (ope circles an th vultur (fillecircles) Point fo bot bird a a particula spee ar connecte b lines (T correc fosligh spee difference betwee th tw birds C value fo th falco hav beeinterpolate fro a curv relatin V an CL- A a give speed th win spa o eac birdecrease a th profil dra coefficien increases Al pola curve ar correcte t aReynold numbe o 10s. Dat fro Tucke & Parrot (1970 fig. 5 an Parrot (1970fig. 2)speed bu eac spee ha a differen pola curv (Fig 7)spa decrease wit increasin C p a eac speedincrease an winWing span and profile dragHo d bird chang thei win span t glid a variou combination o V an Vsi th performanc area Eithe a increas o a decreas i win spa coul increasdrag dependin o wher th bir i flying i th performanc areaTh falco an th vultur whe glidin a a give spee i th win tunneinvariabl decrease thei win span t increas drag Thi chang increase drag bu paradoxicall i als increase profil dra (a calculate i thistudy i bot bird (Fig 8) On migh expec tha profil dra woul decreas witwin spa becaus o th reductio i win area Th lif coefficien increase wit reductio i win are bu no enoug t accoun fo th increas i CDpaccordin t th pola curv fo minimu dra (Fig 5)Pennycuic (19716 offere a explanatio fo th increase profil drag Africavulture appea t increas dra whe landin b twistin thei wing s tha th lif a th oute part o th win an decrease a th inne part o th wing th lif distributio alon th wing become les elliptical an thinduce dra facto ( i equatio 6 increases Th extr induce dra appear asprofil drag sinc k i thi stud ha a constan valu o 1-1 Variable wing span in gliding birds41- °"9I 0- 0-I0-^ 0M 0-T2a00 ,8-4-9- ms"111-2-148-4-9- ms"1oll-2-14-3m0-0- 0-Win spa (m0-1Fig 8 Tota dra an profil dra a differen win span fo th falco glidin aconstan speed i a win tunnel A a give speed th tunne wa tippe t variou glidangles Th falco responde b decreasin it win spa t increas dra an sinkinspeed Dat fo lo speed (8 an 9 ms"1) ar simila an ar pooled Dat fo higspeed (1 -2 12- an 14 ms"' ar als simila an ar pooled Dat fro Tucke &Parrot (1970 fig. 5)Twistin th wing a describe abov ma als increas th actua profil dra oth inne part o th wings Thes part hav aerofoi section wit highl camberelowe surfaces Pola curve fo suc section hav minimu CDp value a CLvalue greate tha 1 (Fig 3) an CD p increase two-fol o mor a CL droptoward 0 I th wing ar twiste s tha th lift an therefor CL o th inne wingdrop toward 0 th profil dra o th inne win wil increase I contrast birdflying o th maximu performanc curv adjus thei win span an kee thei CLvalue high thereb avoidin th high-dra region o th pola curve fo theiMan-mad glider wit simpl wing hav lowe an lowe CL value a the flyfaste an faste alon thei maximu performanc curve Consequently designero thes aircraf selec aerofoil that lik th Clar Y hav minimu CDpr valuewhe CL i nea 0 (Fig 4)PREDICTIONMaximum performance modelGive equatio 35 ver littl additiona informatio abou a bir i neede tpredic a curv fo maximu performanc - onl weight maximu win spa an a 5V A TUCKElinea functio relatin win are an win spa (Tabl 1) Th prediction o thmaximu performanc mode ca b compare wit measure performanc curveo bird tha presumabl wer glidin a th minimu sinkin spee a eac speed falco an th vultur i th win tunne fi thi criterio whe th win tunne tippe t th shallowes angle a whic the woul glideSinkin speed wer clos t thos predicte b th model an win span showeles agreemen wit th prediction (Fig 9 10) Th observe sinkin speed woul bee slightl lowe i bot bird ha chose differen win span a somspeeds Th falco usuall kep it wing to flexed, an th vultur sometime kep wing to flexed an a othe time to sprea outTh predicte maximu performanc curv fo Africa white-backe vulture fitsth dat o Pennycuic (1971a) presumabl obtaine a minimu sinkin speedsPennycuic use a constan spa mode t fit a empirica curv t th dat (Fig 11) th dat ar to variabl t discriminat betwee th tw curvesPennycuic (1960 estimate a performanc curv fo fulmar (Fulmarus glaci-alis) b trackin the whil the soare nea a clif top Thi curv i belo thpredicte maximu performanc curv (Fig 12) suggestin tha th fulmar wer glidin th minimu sinkin speed fo thei speedsTh wanderin albatros (Diomedea exulans) i on o th larges glidin birdsit pointed hig aspec rati wing ar quit differen fro th wing o th falcoan th vulture Th maximu performanc mode predict poore performanc(Fig 12 Tabl 2 fo thi bir tha ha bee estimate b others Fo example (1982 estimate a maximu lif t dra rati o 23- fo th albatrosscompare t 14- estimate b th maximu performanc model Pennycuic use aspan0cou8inginocCQwinB3ESE1110-70-50-2012345-m|1-*-0I O yO \o \O \o1 11 15 1Speed1ON1\1V(m1|\\\2-'i121 --1--, -3Fig 9 Predicte maximu performanc curv an win span fo maximuperformanc i th falcon Point represen measurement mad a minimu glid anglei th win tunnel Win spa i expresse a a fractio o maximu win span Variable wing span in gliding birds51 1 2 2 3Speed V (ms"1)Fig 10 Predicte maximu performanc curv an win span fo maximuperformanc i th vulture Point represen measurement mad a minimu glid anglei th win tunnel Win spa i expresse a a fractio o maximu win spanO JM Eo E1 U0-70-50-2012345_--1T---iI1iI11Eioo-~^o "°O o1o0o0o01\oV o -fo j? oJ]£P o 9O °0o111\1-1--1 1 2Speed V (ms"1)23Fig 11 Bottom predicte maximu performanc curv (.1/) empiricall fittedperformanc curv (E) an measure dat point fo th Africa white-backe vultureTh empirica curv assume a constan win spa an a constan profil dra coefficient dat an empirica curv fro Pennycuic (1971a) Top predicte win spanfo maximu performance Win spa i expresse a a fractio o maximu win span 5V A TUCKEo E2c 1Ig1 0-7| 0-5| 0-201 2345FulmaAlbatros0 5 1 1 2Speed V(ms"'Fig 12 Predicte maximu performanc curve an win span i th fulma an thwanderin albatross Point represen measurement o th fulma (Pennycuick 1960)Win span ar expresse a fraction o maximu win spanconstan spa mode an a combine dra coefficien o 0-2 fo profil an parasitdra t mak hi estimateTh combine dra coefficien i give b th su Co.p + Sfp/S an a valu fo io 0-2 represent muc lowe profil an parasit dra value tha thos use i thmaximu performanc model Sfp i th maximu performanc mode i computefro equatio 34 s Sfp/ fo th albatros i 0-023 Thi valu alon i highe thath combine dra coefficien o 0-20 I Sfp/ i halve t 0-012 the CD p become0-00 fo th constan spa model whic i stil les tha th lowes valu o CDp(0-011 use i th maximu performanc model Albatrosse certainl loostreamline i compariso wit hawk an vulture an ma indee hav betteTabl 2 Predictions of the maximum performance modelSpecieFalcoBlac vulturAfrica white-backevulturFulmaAlbatrosSpee(ms-1)81111V* s m(ms"1)0-80-9100-911Spee(ms"1)11111-11-14-^8.min minimu sinkin spee fo al speeds occur a spee show i colum immediatel tth leftL/Dm^ maximu lif t dra rati fo al speeds occur a spee show i colum immediatelt th left Variable wing span in gliding birds5OQDQV,-1(oisc.0100-0012345C^ML r1 1 2Speed V(ms"'2Fig 13 Compariso o prediction fo th falco b th maximu performanc mode(M) an constan spa mode (C) Th constan spa mode use a win spa o 10 man a C p o 002 Th botto pane show performanc curves Othe panel sho thprofil dra coefficien (Cn.pr) profil dra (Dpr) an induce dra (D, a differenspeedsperformanc tha th maximu performanc mode predicts Measurement o alivin albatros glidin i a win tunne woul resolv th matterFo al o th bird mentione above th mode predict tha th win spa shoulb maximu fo maximu performanc a al speed u t a critica speed Abovth critica speed win spa shoul decreas a spee increases Al th birdachieve thei minimu sinkin spee an maximu lif t dra rati (Tabl 2 a belo th critica speed i.e wit thei wing a maximu spanConstant span modelA performanc curv predicte b th constan spa mode ha a differen shapfro th maximu performanc curve Fo example th maximu performanc predict tha th falco wil hav it maximu lif t dra rati whe glidin a spee o 1 ms"1 (Tabl 2) wit value fo win spa an CD pro 10 m an 0-02respectively Usin thes value i th constan spa model th predicte curv i abov th maximu performanc curv a speed les tha10ms~ (Fig 13) Sinkin speed ar to lo becaus profil dra i to lo(Fig 13) A speed greate tha 10ms"1, th sinkin speed ar greate tha thosfo maximu performanc becaus win span an henc profil drag i to high 5 V A TUCKEInduce dra i low bu no lo enoug t compensat fo th highe profil dra(Fig 13)Althoug th constan spa mode i no accurat fo th entir maximuperformanc curve i ca predic a par o it I th falcon fo example thpredicte win spa an CD,p value fo maximu performanc ar nearl constana speed nea 1 ms"1 (Fig 9 an 13 respectively) A thes speeds th constanspa mode predict nearl th sam performanc curv a th maximu perform mode i on choose th righ win spa an C p value (Fig 13)APPENDIRaspet's data for the black vultureRaspe (1960 give th followin dat fo th blac vulture mas = 2- kgweigh = 22- N b l-44m S = 0-36 m2. H als give sinkin speed (V8) fo arang o speed (V fo a soarin vultureTh profil dra coefficien (C pr) befor correctio t Re X 10~5 = 1 i give byCD,pr. = 2(D-D,-Dpar)/(pSV2)Th dra term o th right-han sid o thi equatio ar given respectively b 1 6 an 15 Sf come fro equatio 34CDpr' i correcte t CDp a /texlCT5 = 1 b th metho describe belo fo thhaw wing Reynold numbe fo Raspet' vultur i 6 426c'V wher c i th winchord give b S/bWithers's data for the red-shouldered hawk wingWither (1981 give th followin relationshi betwee C an C aReXlO'5 = 0- an aspec rati (A = 3CD = 0-08 - 0-153CL + 0-555CL2 .Th profil dra coefficient (CDpr' a thi Re ar give byCD,pr' = CD - CL7(*A .T correc CDpr' t th equivalen profil dra coefficien (CDpr) at/teXlO" = 1a ter mus b subtracte fro al CDpr' values Thi ter i 0-203( l/F) wher0-20 i th valu o CDpr' a CL = 0-7 an F i compute fro equatio 1 fo5Wing span and area for the zdiite-backed vultureTh relationshi betwee win spa an win are fo th Africa white-backevultur wa estimate fro tha fo th blac vulture sinc bot bird hav simila shapes Th dimensionles relationshi betwee win spa an win are foth blac vultur (Tabl 1 isx = l-060b/bmax-0-060 Variable wing span in gliding birds5Th maximu win are (Smax) o th white-backe vultur i 0-6 m2, an bnia i2-1 (Pennycuick 1971a) Multiplyin bot side o th equatio b Sma an thesubstitutin value fo Sma an bma yield th desire relationshipS = 0-335b 0-04 .Wing span and area for gliding birdsTh wing o som glidin bird ar ver differen i appearanc fro thos o thfalco an th vulture Fo example th wanderin albatros ha wing wit pointetip an a aspec rati twic tha o th falco o th vulture Thi sectio estimatesth relationshi betwee spa an are fo a simplified hypothetica win o an ratio Th equation ar the adjuste t estimat th relationshi betweewin spa an are fo actua glidin birds Fo th purpose o thi paper a wil b mad fo th wanderin albatross First conside win spanTh hypothetica win ha thre rigid rectangula element o length rj r2 an r3(Fig 14) connecte b joint a th shoulder elbo an wrist Tw set o elementan th bod widt (be mak u th win span Th maximu win spa i givebyr2 + r3) + bB . (36(37(38Expressin b^ a a proportio o bmax,b = k7bma ,equatio 3 becomes afte substitutio an rearrangementr1+r2 + r3 = bmax(l-k7)/2I th angle a th win joint remai equa a th win flexe a angl a,b = 2(r + r2 + r3)sina + bB (39or afte substitution= bmax[k7 + (l-k7)sinar .(40HFig 14 Hypothetica bir win showin change i win are wit flexing. Se tex foexplanation 5 V A TUCKENo conside th are o th win a i flexes. Th win lose are HIJK (Fig 14a th wris join a th feather overlap bu i gain are DEFG a th elbo join aoverlappe feather ar exposed Th area los an gaine a th wris an arequal s th ne los i are wit flexin i du t overla o th bod b th bas oth win (are ABC). Th are los (S' a th bas o bot wing isS = c2tan(9 - a) , (41wher c i th win chor a th win base give byc = Smax/bma . (42I a tapere wing th chor a th bas o th win i large tha th valu giveb th abov equation an th change i are a th elbo an wris ar n longeequa whe th win flexes. I addition a ma b constraine anatomicall froreachin 90 i a actua wing Fo simplicity I shal attribut th entir reductio iare o a flexed tapere win t overla o th bod b a win o chor e c a it baseTh 'tape factor' e i 1 fo a rectangula win wit equa flex angle a eac joint wing wil hav tape factor greate tha 1 t b determine empirically 4 fo actua wing becomesS = ec2tan(9 - o . (43Th are o bot wing whe flexed a angl a iS = Smax - e (Smax/bmax)2 tan(9 - a) . (44Th relationshi betwee b an S tha result whe a rang o value fo a isubstitute int equation 4 an 4 i approximatel linea fo b £bmax.Th abov equation describ th measure relationship betwee win spa anwin are fo th falco an th vultur whe tape ar 1* an 1-5 Sinc th win o th wanderin albatros appear t b nearlrectangula nea th win joints I chos a tape facto o 1- fo it Usin k = 0-093Sma = 0-61 an bma = 3-0 fo th albatros (dat fro Pennycuick 1982 yieldth followin relationshi betwee win spa an areaS = 0-062 + 0-42 . (45Aao ai ;abbbB"maLIS Oaspec rati*2 constant i polynomiaequatioangl betwee humeru anbodwin spatemporar valu o b use tcomput V5bod widtmaximu win spaSYMBOLCCo,.cDCDCo.pCD,pcLcc-,C4CD,pratCL = 0-7 Re = Kconstanttota dra coefficieninduce dra coefficienparasit dra coefficienprofil dra coefficienlif coefficienwin chor a baswin chor fo maximuwin spa Variable wing span in gliding birds5DDDpaDpeFkk,,...,k7LmnJZtota drainduce draparasit draprofil dratape factocorrectio facto foReynold numbeinduce dra facto^ constantproportionalit constantbB = k7bmalifmasviscosit o airati o circumferenc t di o a circlRerl r2 r3PsSSf^"maSpaeVV3V3wReynold numbelengt o win elementdensit o aiprojecte win arewin are los owin t flexinequivalen flat plat aremaximu win areparasit areglid anglai speesinkin spee valu o V uset comput VsweighREFERENCEBIESEL W. BUTZ H & NACHTIGALL W (1985) Erst Messunge de Flugelgeometri be freGleitfliegende Haustaube {Columba livia var domestica) unte Benutzun neausgearbeitete Verfahre de Windkanaltechni un de Stereophotogrammetrie I BIONAReport 3, Bird Flight - Vogelflug (ed W Nachtigall) pp 139-160 Stuttgart Gusta FischerBLAKE R W (1983) Mechanic o glidin i bird wit specia referenc t th influenc ogroun effect J. Biomech. 16 649-654CONE C D (1962) Th theor o induce lif an minimu induce dra o non-plana liftinsystems NASA Tech. Rept R-139CONE C D (1964) A mathematica analysi o th dynami soarin fligh o th albatros witecologica interpretations Virginia Institute of Marine Science, Special Scientific Report 501-104GOLDSTEIN S (1965) Modern Developments in Fluid Dynamics, vol II Ne York DovePublicationsHANKIN E H (1913) Animal Flight: A Record of Observation. London IliffeHOERNER S F (1965) Fluid-dvnamic Drag. (Publishe b S F Hoerner)JACOBS D (1986) Th LS-6 on pilot' views Soaring 50(1) 30-33LIGHTHILL J (1975) Aerodynami aspect o anima flight I Swimming and Flying in Nature,vol 2 (ed T Wu C J Broka & C Brennen) pp 423-491 Ne York Plenu PressMCGAHAN J (1973) Glidin flight o th Andea condo i nature jf. exp. Biol. 58 225-237NACHTIGALL W (1979) De Taubenfluge i Gleitflugstellung Geometrisch Kenngrosse deFlugelprofil un Luftkrafterzeugung.J Orn. 120 30-40NACHTIGALL W (ed. (1985) B1ONA Report 3, Bird Flight -Yogelflug. Stuttgart Gusta FischerNATIONA ADVISOR COMMITTE O AERONAUTIC (1920) Tech. Rept 93 257-336 WashingtonU Governmen Printin OfficeNATIONA ADVISOR COMMITTE O AERONAUTIC (1921) Tech. Rept 124 421-474 WashingtonU Governmen Printin OfficeNATIONA ADVISOR COMMITTE O AERONAUTIC (1926) Tech. Rept 244 329-368 WashingtonU Governmen Printin OfficeNATIONA ADVISOR COMMITTE O AERONAUTIC (1928) Tech. Rept 286 139-183 WashingtonU Governmen Printin OfficePCEWMAN B G (1958) Soarin an glidin flight o th blac vulture J exp. Biol. 35 280-285PARROTT G C (1970) Aerodynamic o glidin flight o a blac vultur Coraqxps atratus.jf. exp.Biol. 53 363-374 5 V A TUCKEPENNVCUICK C J (1960) Glidin flight o th fulma petrel jf. exp. Biol. 37 330-338PENNYCUICK C J (1968) A wind-tunne stud o glidin fligh i th pigeo Columba livia.J. exp. Biol. 49 509-526PENNYCUICK C J (1971a) Glidin flight o th white-backe vultur Gvps africanus.J. exp. Biol.55 13-38PENNYCUICK C J (19716) Contro o glidin angl i Ruppell' griffo vultur Gvps ruppellii.J. exp. Biol. 55 39-46PENNYCUICK C J (1975) Mechanic o flight. I Avian Biology, vol V (ed D S Farner&J RKing) pp 175 Ne York Academi PressPENNYCUICK C J (1982) Th flight o petrel an albatrosse (Procellariiformes) observe iSout Georgi an it vicinity Phil. Trans. R. Soc. Ser. B 300 75-106PENNYCUICK C J & WEBBE D (1959) Observation o th fulma i Spitsbergen Br. Birds 37321-332POPE A (1951) Basic Wing and Airfoil Theory. Ne York McGraw-HillPOPE A & HARPER J J (1966) Lowspeed Wind Tunnel Testing. Ne York Wile & SonsPRANDTL L & TIETJENS O G (1957) Applied Hydro- and Aem-mechanics. Ne York DovePublicationsRASPET A (1960) Biophysic o bir flight. Science 132 191-200REID G R (1932) Applied Wing Theory. Ne York McGraw-HillSCHMITZ F W (1960) Aerodynamik des Fluginodells. Duisburg Car Lang Verlag.SOKOLNIKOFF I S & SOKOLNIKOFF E S (1941) Higher Mathematics for Engineers andPhysicists. Ne York McGraw-HillSPEDDING G R (1987) Th wak o th kestre (Falco tinnunculus) i glidin flight. J. exp. Biol.127 45-57TUCKER V A (1973) Bir metabolis durin flight: evaluatio o a theory jf. exp. Biol. 58689-709TUCKER V A & PARROTT G C (1970) Aerodynamic o glidin flight i a falco an othebirds J exp. Biol. 52 345-367VA DAM C D (1987) Efficienc characteristic o crescent-shape wing an cauda fins.iXature, Land. 325 435-437VO MlSES R (1959) Theory of Flight. Ne York Dove PublicationsWELCH A. WELCH L & IRVING F G (1955) The Soaring Pilot. London MurrayWITHERS P C (1981) A aerodynami analysi o bir wing a fixed airfoils J. exp. Biol. 90143-162