S'J.'RESS,, AID SHAPEMAL!SIS a, A PARAGLIJ:BBVDG
'l'heaia submitted to the Graduate faculty of the Virginie Pol,-techni c Institute
in II. TABLEOF COll'lllTS
CRAP'.t'lm PAGE
I. TrlI.B ...... • • ...... 1 n. TABLEOF COJfl'Jl2ffS• • • • ...... 2 Ill. LIST OF ,mmmsAID TABI.l!IS ...... _4 IV. S'!MB)IS ••••• . " • • • • • • • • • • • • • • • 6
v. • • • • • • 4, • .. • • • • • • • . . .. 12 VI. :.BASlt EQUATIOBSJOR GD'ElW. MDmRAD S~ . . 19 A. Geometric Equat1c,ns • • • • • • • • " • . . 19 Equilibrium Bquatioru, • • ...... 22 C. Bounda...7Conditt~na •••••••••••••
vn. .AIIAL?SIS • • • • • • ,. • • • • • ... ., • • • • • • .. A. Derivation ot Equilibrium lquat.ions tor t.lle Sa.U ot the Pal:'&glider . . . . -• . . . . . B. :Boundary Conditions tor the SaU of tile
Paragllder . • • • • • • • • • • • • • • • •
Paragl.1de.r Sail. • • .. • ,. • • • • . . . " . D. So1ution of Eq,ail.11:::d.um Equat1on£ .. • • • • .. 44
E. Calculation ot Litt and. Drag rorces tor tile
Pars.glider Wing • • • • • • • • • • • • • • 46 F. Utilization ot MrOCtynamtcPreasure-Deneeted
Shape :Relat1cnships. • • • • • • .. • • • • • 52 PAGE
VIII ...... A. Theory and Appl1cat1on to Pa.ragllder Sall
Analysis • ,. ••••• • • • • • . . . B. Solution ,,£ D1tferent1al Equations for the
Detlec:tion of tlle Sall by Finite
Differences • • • • ,. • • • • • • • • • • 6l
c. lfumerieal Results • • • • • • • • ... • • • 66 n. Discussion of Ntamerleal Results • • • • • • 76 IX. COBCLUDIIGItBMARKB • • • • .. • • • • .. . .. • • . . 84 x...... , . 86 XI. • • • • • • • • • • • • • • • • • 87 XII. • .. • • ...... • • • 88 nn. Vr?A ...... • • • • ...... " . . . . • • • 90 - 1; ...
l .. Paragl1der Wing Con1"1g.urat1on • • • • • • .. • • . . . 15 2. Coordina:t.ea of Def'onned Surface . .. • • ·• . . . . • • 20 ,. Cr.1ordinateu of Sall • • • • • • ...... 4. Stress Resultants ou Deflect.eel Surface of' S81l . . .. B::lunda.ry Vector& • • • • • • • • • .. • • • • • • .. •
6. Ccnrdinate& (;f Detlected Surtace ot Sail. • • • • • •
7. . . . . • • :59 8.
.t.ead:1n.g--Mgel3oor:ns • • • • • * • •- • • • • - .. • • 51 9. Angle of Deflection of Air Stream tor Nevtonian IJrr,pactTlleory • • • •. . . • • • ...... 10......
ot A:tte.clr. • • • .. .. • • • • .. • • • • • • • • .. . . 70 12. Variation of ~sw-e Cceftic1ent With Angle of
A.tta.ck ,.. • • • • • • • • .. • • • " • • ., • • . . .. 71 l :-• Variation ot C/iJg5 With Angle ,:,f Attack " . • • • 72 14. Variation o:t Stress Resultants Ov·er Surface of Sail • 73 15. Comparison of Lift and llrag Characteristice c,f Para-
gllder Wing With 91.ose of the Rigid ldealizaticn •
16. Variation ut nenected Shape '1#1th Dihedral Angle
~(8L). Ane;le of' Attack, a. = 35° • • • • .. • • • • - 5 ...
FIGURE PAGE
17. Variation ot Pressa.t""e Coefficient Witll Dilledral
Angle ~(IL). Angle ;;,t Attack, a. == 35° • • • • • • 79
18. Variation of Lift mid Drag Characteristics ,:,f Para-
glider Vins W1th Dilled.ral Ane,le J( 9t). Angle
of At.tack, a. = ;5° • • • • • • .. • .. • • • .. • • 80
TilL'EC PAGE
l. Force Cbaracteristies of Pm11.gl..ider Wing at Va.rlow:.
Angles or Attack • • • • • • • .. • .. • • • • • • • 68
2. Foree Cbaracte:ristics of Parat~ider Wing at. Variol.lb Dihedral ~es, p(eL) • • • • • • • • • • • .. • 77 A
detel'ffl1no.nt of the metric of the coorclinat.e ayatea
in the clefoned surface coefficient of power aeries
the first fuudamental (metric) tensor of the
eoord1nate 1n the detorad surtaee
the conJup:te of
covariant base vector of the deformed. coordinate
constant of integration araccoetfictent (• ~)
lift coeffle.tent (• ~)
preUUJ.'e coetticient ( • !)
C' boundary contour in deformed aurf'a.ce
co'Wrlant absolute penmtation tenaor ot tbe def'cmaed
Coordinate s,atera ( Ill o, {ii Or •vaQ Q. 1111 J3,
ali c 12 or • 21.1 rea,pectt ,ieJ.y) eontravariant abaol.ute permu:ta'tion tensor or the aetonied coordinate qstera (• o, J¼or •/¼
U a. • j3, ci3 • 12 or cr,S3• 211 rGIIJl$Cti~}
D drag force on Ying 1\, drsa force on payload d distance bet.'ween ends of lea41Dg-edge end keel. boau (=tx; ~2 • 2 cos ~(9L)eoe 3(8L)) Patxo•~, 1kzocan;ponenta or resultant. force ot 1&11 on keel bom 1n XQ, Yo,zo 41:rectiona, respect! wiy
Ftxo, Jiq0, JI,:g ~nt.a oZ relUl.tant force ot satl on lead1na• edge boom in xo,101 zo directions, respectf.:vely reaul tant force of 88.U on keel boom. <·ilKxo • lii7o + kl'Kzo> P.eaul:tant force of saU on le~-eclp boom (= il'L1r.o+ j1tqo + ~> f(x#8) tunation def:tned in equation (90) g(O),h{O) arbi 'trat7 f'Unetion8 ot :tnteg:rat..f.on the fuat (metric)ttenaor or tbe
unde:tO?'med coord:1ne.tes.rstem compommte o:t (f1)L 1D xo,Yo, zo directiona, 1-eapeeti Wly
~• of {i2)L 1n xo, Yo, zo directione, reai,eeti~ I, J, £ unit wetors 1n so, 70, :O d1recticm8, rea,ect1.Wl.1' ...... (1, -v) angle botwon 1 8'Dd. V unit. '\18Ct0r8 Oauuian of detormed 8\U'tace L ... covariant ccmpomnta of L in tile deformed coordinate qatem
L.. \mit vector 1n the deformed surface, nonul ;o the
baundo.ry contour C'
length of keel. bO®l length o:f leadi:ai-ed&e boom.. pb;rsieal eam;ponente of L in the detormed
coordinate system.
the :rcumberot staticm.s used in tinite diffi:mmce
(:Nx>-r,(li&)ir, (Bxe)'l utrea resultants $.t treiJ.ing e index :r:iotation. far even resultants (nil = .Nie, :#2 • Ne,N12 • B:a> station number used 1n tim.te difference (• 0 1 l, 21 ••• B) tensor or internal torcea .a.ctin.g cm a &ttormed element ot the ahell bound.ary :to.roes 1n tlle eur:ra.cenormal and. tangential to the de~onied contour per unit lengt.h ot the contour q ~c pressure (• ½Pv2) B(8} :,arcart.er ~onal to principal radius cunature (• : Ba) prinoipal. radius o:r eunature of de1'o.rM4..-face ot aau s total surface area of ..U. (• lgll,81t& 8:L) i plvaical ffl~t.s or s• la the defcmle4 coordinate IW"#tem diaJlaeementllin the eurf'ace DOl'lll&l.&D4 to the deformed 'bounda:r1«mtour CO"l~ant c eocrd.inate SJ'Stell V tree-atrec velocity iq,sica:L of d.isplac111ant Sa clefo:l'llle4 coonU.ne:te qatem veigb~ of PEJ¥l.oa4 cc~• ot QJ.Ued ~- foree BDl1 DOl'llal to detonei aurtace, ree,ect1'9111tl¥ X pressure on de:f'lecte4 ...U x, e coordinates 1n nrrace o:r uU ,otnt ot act.loa ot resaltant fo:rce ot NU m bol.boCII JQ1nt ot action ot l'MUltant tone of 1.U ca le~boca value ot X d tra.ilbg ec1ge of Nil 12 . ) \r. atal+Acoa8 • 10. reetan;ul.ar Carteeian 1D vtd.cb so ts a.lined w.l.th auttxea all4 Yo • mopl.arae ta in tbtt Wl'U.cal plane of IQlllet.17ot the ,ara,.l~Oer wing rectan.gul.ar Ca.n:etd.an COOl"UD&tee.s.n wld.oh •1 1a alimd With lmel bOCII am. q_ • z1 Jlaae 1a 1n tbe wrt.teel plane ot ot 'Uae pangU.dar w1ng 7o, 1"0 COOl'd1rla•ot 1.oca1on iewltant i'crce of 8ld.1. OD Jr.eelbOCII XO, 701 10 CQOl'\Unateeof loe&'lloa of l"INNl'kni force o£ Nil on 'bo(3II locaUon~...,. lift and drag foree8 an wlD&, angle of attack of lreel. tbat dncribe .. o£ ..u value of 13 at ata'bioa Ji ( -13('1.,)) val.1.Wat .P nati.OD n valmot J a.t.~boca Legz.-tmgian #tra1n teneor of ..,t.,. \'&1.ue of a at atat.ton X ( • &(8L)) val:ue ot & at atat.1011 a va.1.ueot & at boas C the qle which .. tJ..owis • tbe aurt .. ottbe Nil. ftmction deftmd in equation (129) value o'f 8 at le~ bocn the c~le of g1.1de of praglider wing tbe unit wctor llO'fflal. to the defiected SUl'tace wlucz of ,.Ct-- at leadtng~ bOCll. denotes oovarl.ant dif:rerentiation vith respect to r_.'3• .tn the de:f'or-~ eoordir.uite sy-ete denQtes partio.l difi'e~ntiat1on v1 th rea»eet to tu. in the deformed eoordizs&te ay-atea !he pneralized rar.ge 81ld. su.mt'l2ationconventio?JS o'f temsor calctal:ua al'9 used wbereb)- a -~ used an odd llUllbe1-of tilles over the coordinates and a suffix ~d an e'V8!1 mnber o£ t1a8 is to be over t.be cooriinates. Greet ~• m,i ueed. :f'ol:-coordina~ :tn the • 12"' Requirements for reeo-.rl.ng 1-ockat booater& and. :ror la.nd.ing space vehicles have led to a search 1'01· li:f't.in& devices vhicb ba.'\18 a varietT of attributes. Such e. device lhould have a low-wight and a small exposed area tor the ltl.UnC!hpha8e. The e.ccelerati.ou and temperatures during the reentry phase must not be too high. ln adili \iox1., the deYiee ebould ba-ve the capal)Ui ty to perform a con- trolled glide to a l)l:'Odeterm.ned site and to ·makea safe landitlg at a low subsonic .speed. One device 'W111ebbaa deairable features ia the paragllder wing. {he re:t'. 1 ..) ~• wing, which bas al.so been deiligr,lated bJ' var·1ous authors as a para.wing, lroga1lo wing, or flex wing; bu the liabtwignt cllaraetertstica a.no.~ng espat>Ui Uea of tbe pai~a.chute. It eonsista of lewling•edge booms and a keel boom Joined together at the nose and a f'ledbl• Nil eur.race c81.T1eilthe oeroeynamic preaure loadir:lg.. (See :f'ig. 1 ..) Ttie ~load is sue- penaed beneath the Ying b./ ewes (sbroUd lines) • '.rbe bocus w."ft conEJtructed eo that mB'l'be folded when the para&l,ider is mid so that are uutfid.ently stiff after the wing is deployed. One possible method of' eonstrucUon 1s to make uae of ini'l.a.table. structurnl members while another poasibili tr ie to make use ot· a hi~ mechanism end to ,,rlgid1t.e:t ita Joints upon deJ,lo;vmnt. h SOOle versions ot the ~ider tbe booms are maintained a tiDd distance apart by a spreader bar while on others they are held apat elasticaJ.l.;y by a spring mchanism. located at tbe nom. Ir------Yo Leading-edge booms Flexible sail Spreader bar Control cables a (a) Obliq_ue view. w (b) Side elevation showing forces. Figure 1.- Para.glider wing configuration. 'lb& 11t,thtweight eonatruction t-nd p::i.d~ng cnpablli ties cf!' the pe.rce;l.ider Ying sa:tisf'<; the requirements of 11gb:t,weight e..'ld mall o;posed area :for the lmm.ch phase. 1'he ,nng can be d.epl.oyed at the de:811.-ed time :for reen1;r'-.f and the g.1ide be.ck to earth. 'l.'he prelimiDary esleula.tiona of refe:t-ence l indicate that the aeeeleretiou can be held vi thin desirable limi:ts. 'fhe pm.'t'lglider vin3 can 'be designed. £or ver:, lov wing loadings so that reentry ~rntw:es ma'/ be kept :from. being prohibit!"". Control of' tbe glide is t'Sforded by ad.Juating the length of' the cables; this bas the ef'feet of IA,')(IJitior.inG; the ptq• load with respect to the ~ng.. (See rig. l(b) .) 1'tt.e winG tllen ~• en angle o:f' o.ttac-..k a. and glide e..~le 08 su.ch that too llf't and drag forces, L o.nd DJ of the wing and the c~ force., Dg, of tbe p¢osd are in egµUibrf:um vi.th the T !he prev:toue investign.ti01'18 of ~ider wine;n hnv:-e been mair.17 e::q;,erl~ntal in nature. !he preliminary inwati.gation of :reference 1 .. 15 • eonsistec or .,,.11,na.-tunneltes~ at ::m1n.;on!ca.,,--:td supersonic speeds o:f free aJ_.ide tests. In t.i.e wit~.t-tunnel irvet1ti.gation of reference 2 the p-armrl..ng ves stu~ied. as e.r. tttt."<"1...11o:..i.-ylov speed .. , high lit't device on t\ro .supersonic a.trplar~ models. The- free ..fiight in~stiga:tior-t of ~:."c!e:re:··;;s: 3 1-re.seoneerr.ed vi th the pareg:t.ider' • st:>J.)ili ty smd eon+..!"ol. i.?ht: si;werson.ic rlnd-tv.nnel in:vestiea.tion reported ir.. reference I~ mu; eoncerr.ed with obtaining tl"~ 11ft. e..."'ld.dra& cblu:acl:er• istics of the para.glider mng. Pressure distributions on thre.a rigid tdng$ 1 simula:ting pru·u&f.1deri:, nth -vurie-d cano:tt' :r>,J.?"'te.ture::nd. lea.d1ng•edge a-weepwre obtninec frorl nind•tum?.e1 tests at subaomc and t1.~Sin$ollicspeeds in :-e:f'e.rences '.} ari.d 6, re'l)eeti vely. In z-ci'eren.ce T, the li:rt and dmg coe:f'fie1ents ,~e determ.n$.l ex;peri• ?:1Cntal.ly, at a MAch number of E .6, fur t..l'n--ec:t"'if';id ·uinga simulating pv.:rugliders 'Id th t..'1...,e curvatures.. ~se rosul ts were cca- p..'U'ed With theoretical l~eJults Gi,ven by- ~-per$0Irl.c lie~.ia.-.,. :t:r~act them::l. In t-e:terence 3, fttll-seale vind•t".u.."'l.001tests Wl"'e perfol'.'li!led to stuey the stability and. t!Ol:rta."'01eharaeteristics of a pove:red. parawing a1l"plc.ne • '!be purpose of t...11.einvestigations on the nex..lble pa:rag.tider wings ffllS to f"i:ld ei tber the lii't and. c.tr:::techo..ract.:'!l"i&tic.s or the st.ability e.:...."d~.ont..."'"01 d:UU'.'a.eter.istics. No o.ttetlpt ,;,--a,st1tU~ to find tbc de:tl.ected eh.ape, the preasur'C diatrib:ut!on on the stll., 01· ~'le stresses 1n the aa.il • o£ the investigo.t:tono on the rigid ideclizations (refs. 5 nnd 6) did :,ield pi<>eSS'.J~ distr.tbutions. IioveV\U", theec prea.:,su:.re di.stl'.-1.butior..a to the fixed shape of the idealized. t.-iug a:nd. o:ru.;rx-ep:reaent the pre&SUl"'e distribi.rtion on the • 16 • flexible seil to an accuracy- that ilepe,nda ou how veil tbe :rlgi.d idealization repi:-esenta the a.etwu.deflected shape of tile flmd.ble 11'111£ lfo px-eviOUStheoretical :inwsUsationa have been made on tl1e nenole parsglider v:t.ng to obtain 1ta ahape mld pressure diatribu• tiona, the 11:tt and drag coei'fidente, and. tlle stressea in the aau. ~a situation is in pa.rt due to tbe com,plexity of the problem we1-ein tbe deflected shepa ot the se.U depends on the preltllUl"e distribution over the surface o'i: the ea:1.1end t.b.e :presauro dietribu• tio."1 in turn is ~cally dependent on the !he equilibrium equaticma for the -11 m.-eobtained by an application of the: tensor e~tiona of equilibrium derl wd in re:ference 9 for a norJlin.ear .f:irat ~tion general IIWll1llJ.and membrane theory. 1'!d.s nonll~ tbeor;1 accounts tor large dei'le-ctione, large :rota:ttons and larse strains. ln a;pplying the equilibrium equations to the fl¢Xibl.e pa.raglider NU, the anum;ptions are made that the lemil.Dg-edge and keel ·bo<:lu are rtg:td and. .-tndght and tbe.t the &ai'ormatione 0£ the sail a.re inextenm.onei. 'l'he analysis is performed for a pa.ragU.der vi th the booms mt.dntaitted a fixed distance apart by a. spreader bar and with a tralli»g edge of tbe a&il 'Which ia at:ra.ight (vben the sail is laid out in its flat oond1ti.on) • other &hape~ of tra.i.U.ng edge ean Just as wll be considered lead.ins to ditterent boundAt7cc:mditions for the streaa resultants a.t the trailing edse .. 'lb$ equilibrium equations of the ssil ::u-e SCl ved e;xpliei tlf for the stress resul tauts in ten.1s o.1' the preS$Ul"e on the sail. Md n parED&ter that deseribea tbe of' the sail. iben., equations are dead.ved .fox the resul. ta.'rlt fo:t•.ces applied "by the aa11 to too keel at1fl lecilng-edge booms and :f'or the lii"t arid dreg coefticienta. \1lieae equations for tbe resultant fo:r.ees and the lift .Md drag coe:fficients are exprea..sed in terms of the atren :res:w:tanta in such a VS¥ t.hnt tbir-J me:J'be used fox· mrr e.e~e theor.r enropria:te tor the speed 1'&"..ge unar conaide:ration. bo~ conditions on stre•s resw.tantn a.t the ti·ailil'l8 ed&e or tlle sail, after the adoption of an ae1~c tbeor.r,yield a dif':fei-ential equation tor obtaining the deflected ahajpe of the sedl •. 1'he equations a....'"'ethen apet!ialized :tor le\.'to'!tian impact theory, which bu ~times been used to ~a;s the ee~--.:ll'd.c pt"'e~alla;pe i"'el.atior.s.111.pi'or tM ~sor..ic speed range. (See, tor e~e, rei's. 10 a..~ 11.) 1'bis cooiee of •~c theory is govel:'lled by tw factors. First., Nm."tor-J.antbeor-:, leads to a sim,J.ilified ~is Yhich shows the am,licebility of the pxesent L"iW.Yllisin a simpl.e m.nm1eX·.Second, this aero~c theo1,; baa been uppl.1eti in reference 7 to a 1-:Lg:ld ideal.izat1on oi' a ~.ide:i:· wing, and. thus ai""res a.a a basis of' ~oon fo1· the numen cal reaul ts. ~rica.l results Si?'e obtained for a tl.exible configuration that con-esponds to the rigid ideeliation 1n t2le sense that it bu the SafflC surface plam'om and tbe keel. a;r;d leading~e locatiOl.Ul. va..rj,ation of' lift and drag coeff.1.cient.e with BDQle of attack .a,re e~d w1th those tor the rigid 1dealiu:tion 411d the &.,fleeted • 18 • abafe, ;pressure d1stribution and atreu reaultauta 1n the eaU an detenllned. .Also, 1'1UU1er1calresult. U"O to sbaw ef~ O'£ varla.tion in tihedral angle (raiatng or lOVIH."11'16o1" tbe le~ edge bOCIDtl},. ... 19 ... A. Geometric Equations !he equil:U)r1um equations are gi wn in teru of coordin&te1 t°' in the detOl"lad eq'uilibrtum state. (See fig. 2,.) '1'be sha,e o.r tbe surl'ace in this deformed et.ate is deff:ned by a.a.131 the first f'UlldamenW (meti·ic) tensor, a.nd t,~, the aeeond f'\mdamentel teneo:r .. Re...~ (1) ba,p = •Y- .. -aa.,p -l?u. • -V ,ti (2) were ii is tbe unit outward normal to tbe deformed surface. cOl?Ul f oUO'Wedby tbe subscript l,l indieat.es p&rtial differentiation with respect to ,~. The covariant base wetors ia, are ~.ued • ao ... V Figure 2.- Coordinates of deformed surface. • c. Some oi" the other ?$lationabipa uaoeiated w1th the defoned surface are now :reviewed. '?be conjugate metr1e tensor is et wn by the relatio..'"l8hip (¾.) ll)1l O ,men a. = i3 * J... lrile,l a!} = 12 (5) vi = ... .1. Wlel'l ·a$ tll' 21 fa (6) ea.13"" 0 when a.. f3 ..VO. when ai; = 12 ("l) --{i. when = 21 (8} • 22 • (9) and the stt-ain tensor by (10) B. Equilibrium Equati.ons 1'be ~r .form of the equ111brlum equatione are gt wn by (ll) He:ce -a.o.f3is the ~t.rlwecl tenso1· or inte111G.1. forces ai.tti~~ on a deformed element of the shell a."Xl x', X [u:e tenso1~ com.ponenta of (12.) c:J½ ,Jli>("'ai,,1 + "rµ,o • :Boundary condi tior..a along the bounda.cy contouJ::· CI of the ae:rormed suri'ace are g1 ven by prescribir,G: vb.ere PL and Pg are ~lied bounda.17 forces pe1· u."11t length in the stu·face nonn.a.1 and. tangent to C' , respectively, and t1t and Us ru:~ethe displ.a.cements of the boundary contour in the deformed sw:-:f'nce tangential to C • , respectively. Here (14) urtlt outwarc.l normal L- a.nd the u.rdt ta,'lgent S- to c• (in the direction ot: increasu,g S) • fhe qua.nti ties La a.n.d r/"' ..u-e ·the corresponding c-ontravaria."lt coi:.sponenta. Since L and 8 t~"t? (15) - ~4 - Consider the pareglidei· le..id out on a plane and let t.he coordinates ~l and. s2 be designated am a, :t."espceti vely. the o.."la.lysis, oru.y that portion of the wing in the first quadrant need. be co.."lt:d.dered. i'be keel is at 3 = 0 and the lee.ding edge at 8 = 3L• It is assumed that the leM.ing-e lL, nncl the keel bool!l., of length 11t, ru.-e stl·a.1.ght a.no.rigid av.A. that the ee.11 is inextensible. It is also 1.ui;wmed that t.he trailing edge of the sail 1D straJ.g.'tit ar..d is expreGsed. in teme of the re<-:ta.r.gular C~sian coo1·dir1ates x1 , :r1 (tig. ;')) by the equation (16) (1'() (1H) Figure 3.- Coordinates of sail. • 26 • u...-u.oad.edcondition vhe:i:-e the metx·ic is ~. Since the deformation of the sa:11 is ass,med.. inextens1.ona1 the strain tensor ~mnishes; that is (19) (20) (21) (22) (23) Since t'be x, J coo:roinr,,te curves are lime of pri.neipa.1 is r..ero. Thu.s upon substitution :tro.-nequatio11 ood equation (6) into equation (9), P..:fter use 1s ma.de of equation (21) Only the condition (26) (26) ."11>0 (29) Sol.ution of equation (29) yields (30) R.-:,... is {31) (32) The cqu.ilibritm:t equatior:s (1.l), ai'ter stibst1 tution n~om equl.\tion (12),. become (33) Upon substitution of expressions (23) for the Christoffel symbols pi-essure fo1·ce X on the sail. 'the equtlibrlwn equations (34) a.re still in tensorial f'om and w;,e llO'v:.rtr~o:rmecl into phys!.cal c0t1p0nenta. 1'hese transforma• tions. a.re given by t.1:le relations - 29 - ffx ·J;fin11 • nll •a•J~n22-x2n22 (35) Bxe•J! nl2 • xn12 \ltJent Bx,Be, and Bx& are the pl\ysical of trtrea re$.llt- ents. (See fig. 4.) ·u»onsubatttut.1on of equation (J5) the equilibrium equation, ('4) beca.ae ('6a) ('61>) B9 • x(RX) (36c) a. Conditions tor the Sail. of the Pc:ragU&u- In We section tbe atreu bounde17conditions at the trailing edge and the nose and the cliapl.aeemrmt eondi tiona at the keel and lea41n,g edges are pre•nted. Let the ~cal sveu p:ments Ifx,n 8, and be redet1ne4 ae .U, r2-, an11 r-2, rea,eeti wly. Then equat1one (35) can be written && (37) . .,.. )( Figur., 4.- Stress resultants on deflected Surface or Sail. - '1 - Ln i.,, ._ be the JIQ'1d.eal.ec..-.nu ot • md.~ mma1 wt •eton to ta bounclar,r ._ def'l.ecte4 NU. !ban la, • _._ la • -1- La. (,S) r-,a tto.• ...1....a.. • ...l...... (39) fa"" u,on 1Ubat1'falt1oaot rPf1,i., tmd ft'Clll equat4ona (37), (,S), ana.('9) into tbe flnt two of eqoa1d.oDa(14), tbeD (40) Be.re P.r. u4 l's ue pbpl.cal .torcea ,_. unit lengt,b. !be plv'aieal. C011onnta ot _.. -,reue4 b7 (a.1) u.t,oll au'bat.itution tor u.., Iv,, an4 Ba traa equ&Uona (41), (,S), and. ('9), the lut two of ept.tou (14) J'1eld (42) Sabn1"1tton f'rm (:,8) amt ('9) into equaid.ou (1') JUl.41 - '2 - ("-) (~) For a slWG bomdc7 cout.our, 8l. ad 82 cm be folDi- t.rca •~ (114)aD1 (46) emd. i1 ad i2 b-m equatioq (4, and (45). lbea U. «mtiUone tor the gt,en ~· are .,.nd.1114 t:ra equatloae (~) or (42). Tmf,} IPRtwt !Je?afea:ecmslb19SI •• At tm valling .._ tbe bcM.14er, con'°"1" 18 -,r:eNe4 bJ" (l.8). f.Na ep~OU (18) and. (lt6) tbe toll.ow1»s1• obWne4: (ain e + A eoa 9)91 + (coa e • A sin t)92 • O (~7) IL • • l (coe 8 • A a1D 9) (48) ,.J. yl+ ,.2 82 • '"• & * •(sS.n9 + A coa &) (49) J1•# -'' - Figure 5-- Boundary vectors. (50) (51) 2 + {(sin o + A cos t) 2 - (eos O • A &in a>)(Dx1)T]• O (53) [ (Gin 9 + A coe 8)2 + (.cotJ8 • A a1n &)2] [(~)., (9ft a • A 11n 9~( ) ] + (sin 9 + A cos ri) lg .1? • O (55) • 35 • (56) After su'bst1 tution for (lfxe}'l! from equation (56), equation (54) yields (57) Qpon substitution troa equations (18) and (}6c) the bound.a:q equations for streaa at the trailir:rg edge become (58) (59) Xfll ~H:smt•• eondi tions of ?.tm> dis• placement are specified at the keel. now,along the keel 11 si 0 (60) (61) - '6 - ~•Aw Y8l!M:id:9M••At tbe lea41Da edges the diaplacaraent 1• apecit1ed. .Along this bounda:r.f (62) 8l.. -1 82 - 0 !h'tla at the leading edge (ut,)L and (Us)L have specified values. 9,U.t,d.gg a.;¥W\! 991! gf - HtYa•-At the :aoa, of the aeil the etreaa reatltants Bx,Be, and lfxe must remain finite. 1'he• boul1dar:, comU.tiona rlJJq be tboupt of aa repreeenting the limiting case tor a st.reu•tree bouD1aly that sbrinks toe. point at the noee of the aa:U. C. Bepxeeentat.ion of tbe Deflected Sbape o-r hragl.1der Sail In the den vation of U. equil1br1um equatione, the def'l.ected ehape of tbe aail bu been rep.resented by a ll which waa shown to be proportional t.o the princ1pal rsU.u.a curvature R 2 1n eqµation ('2) • Also, in the boundar., conditione at the keel and leading edges., the two dafleotion ec:.ap,nenta Vi and V2 are apec1f'1ed. (See eqe. (61.) and. (6:,) .) It ts eomienient in the -,1- ensuing anal.y'eia to repreaent the deflected shape of the sail by the angles rs and 6 fJbow in t:t.gu:re 6. Here 43 and & are i"lmniona ot ti. coordinate f and x,~,a repreamt a set of spherical ~natea. 1'Jle XJ...ms ot the rectangular Carteaian coordinates xi, n, s1 is aUaed with tbe keel of the pa.raell&n- e:cd the x1z1-P1ane 1a the -.ertical plane of fYJllletrNof the pm.-eglider. Ben0e z1•x81nl3 XJ..• x coa 13eoa a (0.) Yt•XC08£\ain6 Consider the keel at an angle O'f attack a. and. define rectan- gular Cartesian eoonlinatea xo,TO, IQ such that tbe lf zo • z1cos e • XJ.ain a. XO • z181n a. + xi coe a. sc,• x(ain fl cos a. • eoa cos a ain a.) XO•s:(tln Ii 81r1 cr. + coa cos 5 coa e) (66) 70 • X COG '3 All & • ,a. Figure 6.- Coordinates of deflected surface of sail. • JI). k V------ Figure 7-- Coordinate systems and unit vectors. • 40 • low zo ia tl.""eated as a i\tnction 01: xo,10, that is, zo is the equation of the deflected Rrtace in the 10, ~tit .-,,tem. Dlen (6Y) dzo • (a1n J3 coe a.• cos~ cos 6 sin a.)dx + •{ +,.[ (68) azo •Gin & COB a. t + (sin J3 CO& 6 COS Q. • COS t,i sin a.)cos f3s ------dXQ •Sin & Gin a. tJ+ (ld.n S!cos & a1n a. + coa P cos o.)coa f (69) • ______COS & jAd_e______+ a1Jl p COIi fl ftn & j_§a.e______ 0 •sin 6 atn a. j + (m.n " eoe & '1n a.+ eoa eoe o.)coe ll (70) ... ~1 - Also tbe square of the length o~ a llm elemmt on the aur:tace ia gtwn by (71) Howewr., from eqµat1.on (21), the fol.low1ll6relation must bold betwen fl and. 8: (73) 'J.1.liaeqµe.tion gi wa the condition for 1mxten&ional de:rcmaatton ot t.be u.u when. the de~ons &1."9 represented. by '3 and a. Equation (7:5) upon solution for I yial.dl (74) wbiere the poa1 ti w 81.gn refers to ~• whieh ere ooncaw doVmrard, uul the m:tma a1p to ~• •itd.cll .,,... C011Ca1e. upward.. !he unit nc»:ma.la ..1, J, k- are diteeted along the .xo,y 0 , Zo•an• u ab.own 1n t1gure '7• 0so ?lzo Upon au.batitution tram equations (69), (70).t mld (74) tor cJzo.ta'fci' and. t; equation (75) yiel.48 (76) + (cos Clsin &)j + (81.n13 cos ct• eos J.\ eoa a sin a.)k (TI) • [•sin 3 coa a.l • (i)2+ (c"" Gin e • Gin P cos cos a.~]i + 5 l ... 2 - sin 11 6 [cos \dtf1li) ainal3d&j + r1nll a1n a,~l • (1)2+ (COB II cos & + a1n JI COB ll 11B a.) ff]i. (78) 1'he covariant base ,eeton ea.tor the &'l:fleeted surface are NJ.c.ted tc the unit ~ntial. "leetora by equation (;) • Substitution :£'ram equationa (21) into equation (3) ytel.4- 83..... 1'1- (79) ~=:de - 4:,. !he quenUtJ' 18 o'bta:l.ald1n ...,_ of the J bJ-ue of eqwdd.on (2) with cr.• '4 • 2.. tbu.8 (80) (81) 9a:1 f'na equation (,o) t..~ funetion a(t) can be .,,.....i in t.ema of the quautity as follow1u (82) In f!l!Wmlla."7,the ue!'lected elia,pe of the P1l. can be ~aented b.v the t\?1Glea and 5 wtch are fanctior~ of the coordinate a. (See fig. 6.) 1'be quanticy a(e), whid:\ 1n u. equillbri:ua eqwu;iou ('6) ml 1n tbe tnS.Ung ec'8ebounc1ar:, eon41.tloms (,S) end. ('9), 1• tben related to f) b_y'equation (82). ~, the ..:1. botmcl1117conditiOml (61) ca be by 1lllltbowldR7 oon41.Uona P(O) • 0 } (8,5) &(O) • 0 and tbe coad:lt4ou (6,) eau be b:, tblit apacit.1.cat.1.on of J( tr,) uc1 &('I). ·"· D. Solution of Bquilibrium Bquations Subatttution o£ the ex;preuton for 1'@ from equation ('6c) into eguai;lons (}6a) tmd (;6b) y1elda (xJx) ,x + lixt, & • x(BX) (84,) (65) vbere g(t) 1a en arl.d:tra:r:, tunction ot a. Upon .Ustying the condition tor f'1niteneea of the streaa reaultant at tll19 noee o£ tbe sa.U (x • O); g(9) • 0 (87) (88) (91) where h( G) 1a an arb1 trary :t'\me't1on of t.. Satisfying tbe bou:l'!dBl.7conditlon for t'1nitltneas of the 1tr1u"m resul taut at x • O requires that b(8) • 0 (92) 5.'hus1 upon wdng equation (90), equation (91) fields (93) Atte1~ sa,tj.ai",ying the ·bourJidat'ycondi-Uons (58) and (59) for atnu.,• reSUltsnta at the trautng edge ot the 8&111 it is found that (9') Bquattona (9b,) am (95) deterrd.De the Bx,me, and Bxe remain finite at the nose of the aau. (See eqc. (88) and. (93) • ) b pre&Blll'e distribut.ion cannot haw a ail'lgulArit:f at % • 0 wuieh is stl'onatr than ½ U X 0. It ta belle'W 1. Caleule.tion oi" L1ft o.nd Drag Forces tor the Paragl.icler Wing ln this section the l1£t Md drea 1WCea m-e deri \llld. As a preliminvy step, the 1-easul.tant forces anJ.ied b:, the aa:U to the keel and lead1.ne-edge booms are obtained.. Tbeae resultant forces atJd. their locations are presented aa 'tibe• 'VD.l.ueaare uaetu.l 1n obtain!Dg the forces in the 8b:roUd linea and spreader bare of tbe parag.Uder. Tlie reaultant force appl1ed to tibe keel b¥ the half of the $0.il considered in the analy81.e ia obtained by integration of the Rl'US :reaultmlw for the aa.i.l along the keel • !bus the vector equation tor the force ia giwn by ,ego------(97) ti I (De)a_dX .Jo W""V Hare (i1)g a.n Simllar]¥ 1 the vector equation for resul tent force applied by the l&'U to tl:i.e leading-edge bean is 31ven by Gd this :force a.eta tlte point X = XL 01' the lead1l:fe e (101.) In equation (100) (TJ.)L and (i2)L &Ta the unit tangent 1ectora of tbe surf~ at the lead.1.Dg edse a • 8t• fbua traa eqµa.tiona (77) and (78) (102) (10:,) ... 43 ... 11be force• f.t and. '1, ean be expreaaed in eom,c>mntaalong the the Xo•To, zo e.xie u :follows: (UO) (ill) (112) 'l'be points Md. xi, can be g1wn 1n tenu o£ the xo, 70., zo coordinatu bJ equation ( C,6); accord.1.ngl,y (118) (ll9) (120) (121.) (122) n»al 1Y vl1en both ha1wa o:t t.be aywtrical au an oonaidered. (t1g. 8), tu 11ft and drag coefficiea'U ca be e:r;pnued aa :tollowe: (3.24) CD•-£ ·j(Yxxc, ~) (a,) 1'heff L and D are the lift u4 area torcea, napect.1...i,-, on the wtrc1 q • i py2 t.e t1:1e ~e ~, ana. (1.26) la tbe WW eu.rtace area ot tbe au.. 1!318:reaultani ot \he 1.1ft and d:ng toroea acts tl1rough tbe point: (lo~70) Vbne "' 51 - YoL _L Figure 8.- Resultant forces applied by the sail to the keel and leading-edge booms. - 52 - (127) (1.28) 'I. UWJ.~ ot ~e ~Dl!d"leeted ..... Bel attONlbi pe !be enti1'e anal.l'ai• up to tb1a J01a1; 1e baaed v,,,oneg,uili'bri• eonei~ou alone; -4 tr:te equatiot.18 cm-1184 are no~ upon ar:q-QNU'ic ~e theOJ:7• Altbough an ~c ~tJ.ecte4 abape :rel&Uonabtp ie _._.,.,., 1D or4er to ---- the ddlec1aecl .. ot the ..u, all of the p:nlouly den• eque.'1ou are -.U.d no-'• which ~e theo.1:7'ia uaecl. ·!bu t'be anaJ.yata ca 'be s,pl.ied with t1l8 propit' •~c t.bec>l7'uUlized tor_,. .,.a nmaetrca ~e down 1ib1:'o\tp A brief re'V1ev of the eteJs in tbe anaJ.yau 111now in orc1e. l'he atzeu 1"e.:l.te;n-u Bx,Be, ad Bxt haW been a:p.reua4 ta te,.._ottbenormal.~ X and'bllb&ptot:tbe...Ugl.'Wln by a b7 _,,. e.-Uona (9,), ('6e), ad (88). ~, tbe ot the qwmtit:., (BX) owr the ail., to •~ tbe 'bolmdU7 eomU.tlona on t.be ttitreu rNUltanta at, b trail h,a .,_ or t13e au, is gowrned l>1"equat;iona (S)ll.)and (9'). -'' - %f tbe aerodpl!la1c rel..UOU bctlleer.t t.ll8 p:euarre and tlllt deflected -. or tbe -.u ta uW.id 1n •~ou (94) and. (9'), ttalQ' prov1ae the cond:ltlona tar the 4efleetAd 8bape ot the Nil.. Once the ... t.bat •tiaft.N tbe bomJ4ar:rcon,.. dittou at the keel e.nd. l.eaUng ed,ge is lt'now..1 the ,reaaure X on the ae1.l. can be detem.tned tram tbe i,Nuue-llbapt relnion,sh1.p. then t'be n.rua l'NUltan:u lfx,B 1, and Bzt can be &nend.Ded by .. of equattone ('6c), (88), and (9,) &lODg vith eqution (82). Pl.Ml 11', tlie reaultant twee• aaiw bJ' the au to the 1ael. and le~ 'bam8 can be detend.ne4 trm equatione (110) ad (111) and tbe lUt an4 drag coe.tn.~ can be dewndneo bola equattou (3.24) and (125). It. 1a mv o~ tuteren to conaider the preseue 41ffi'ibutlom ar .,.. ,artieul.ar tne• ot 1111U'OC11w.tctheor1e•• For all aero- ~c tbeoriea \bat Ji,el.4 a diatr:l'buUon X 1dd..ch ..UdJ' t!le atre• OOD41tiona at tba noee ar tbe eaU and. vtd.ob -,araU.on ot 'VWiablu can be ued. equation (9'1-) ana.(95) can be 8d.1.st1e4 u ft)llon: !he most fJIIIMl"al.8UCh pruaure 41atn.but1on can be by • !ben equation (9'1-)up:m 1ntegr&tion o.,.r the ftrl.allle e J1,elda ( »t t2(BX)dl • (l,O) Jo i.., - 54 - where i 1• a. t'lOD8tant ot integn.tion. the 4!.atd.but:1.on ot (Bl) over the IUl'f'ace ot tba MU, by equation (l:,O) for pre,eure ti.nrtbutiona ot u. tne pwn in equation (1.29), alao utid'1u tbl COD41tion glwn by aquaUon (95). 9ml equaUona (94) aild. (9') ntdaee to a fdltgle equa'ld.on.1 equation (l,O). .Afte1-uee of an ~c t11llol7tor tlle relctionald.p 'bdWe'l p--..re aud 4et'l.ected ..,., tbia eca.uat:1011become• an ord1.nar., 41~al equat1oD tor obte.izd.Dg tlse pu,wt:er vh.1.ebdet.1nN the 4efl.eeted ahape. fbta rma17Sl•tdll. be cont.t.ne4 to ot 81U'O• (\ynmd.c 'Uleor1e8 which ,tel.cl preuure f..1atri.but.tou of tbla t.ne• Ia~ of tbeorlu ot a mn pr.ieraJ. tne 'IIIIJfl'require ree.maina'fd.an ot the .....,_on ot -.Uld'action of tille bouDdat7 eont.lit.iona Id the trail 1ag e4p. Ml"04ymad.c t.lleoriea that 7,leld. pl'N8U,1'8 4UtributiOM X llb!.ch are ot the x coordtnat.e 1 lead to a a1d.ll. t'm1lbff td:llflitleat:Lon. for -.ch t.beorlea, ...,Uon (1,0) beCOIIN C (IX)•;;; (1'1) !b1a ..-tton, • tba caae ot equation (1,0), aat.iat1tta both of 1tb1 waU1ng-. bou"OC'Yccm41t.t.ou (94) wt (95). fte or 1n~ c, 1a to be detend.ned by the 'bouDl1tm7conil!:Uou -.t the leaU.ng edp of the au. With the relaion (1'1) b' (BX), -... atzeu realtmra :t.rm (St>e), (9,) aa1. (88) beecae (1}2) • • ,C .( sin 9 + A cos a) 2( eoe s • A sin 8) lr?IJ where A is U,j.\l&n. by equation (17) • Equation (131),. artm' subst1 tittion of the ac~e pre~ deflected shQe relations,. represents the di:f'ferential equation .tor determ:inina the det.teeted ~. .,, .. 5t .. A,,, '?heory $t1d. Jnlieation to Paraglider Sail Analysis Zn order to ~te an application of tile anal.y81e and to 11how~cal results, the method ia used in coajunction with Jlevtonian impact theory. This theory, .r;;.ftenused for 1\Y,peraonic veloeitia, hu tl10 of yielding pree$Ul'es, :x,. vh1ch aze functions of e alone; tbue, the at::"~i!'ied re.J.a..tiona gtwn. 1n equations (131) $lld (1:,2) can be wsed. In addition, this aero- ~c theory ho.e also been applied 1n referenee ·rto a rigid 1&1ml.1zation o'i a ~:lder vine, thel"'efo:re# a..di?'G{:t compm-1son em be made between th.e aetua1 flexible ~id.er win.g e.nd ite rigid cou1'lterpe:rt. In Jlevton1an tbeor.r the preasure coefficient, at a point, 1s s1,"Venby the rel&tion if e: -> 0 01· (l}}) 1:f E < 0 where • f .. Here « is the angle be~ the local etreamv:t.ae u.n1t ta.naentvector and the t'ree-atream vel.od.ty vector. Dewtonian ~t t.beor., requ11-es that t.be =vins .~ g1 ve up its '1normal" ~t o:r maa:ent:am( to the aurtace iaipaeted); but 1t retains tbe tangential component which puNa ott ttmaentia.l.l:f to the local aurf'ace. Only portions of the &U:!."face ~'lat "oee 11 the tlw haw a nonaro coeffld.ent, aa indice.ted 1n equations (1}}). • 57 • If' the tree-stream wlocity 1a along the :r.oana., then from equation (76) • ( cos II 111B" • •ln P coa & """ o.).Ji • ($)2+ llin II cue " j (l:,4) 'When (i,;) 1a the aPg).e between the i and. v unit vecton. Fram figure 9 which ahowlJ a trace of a. plane puaed through the t and v wet.ore coa-(I,v) • COIi~ - ·) - sin C (1}5) coefflcient can be expreued aa 2 • 2 [ 2 X • 2q[ccoa llin o. • a1n II coa & coa a.) • ~)2+ 81n I! coa a. ffJ (137) Tll8 quanUt;y- &, for tb.e eue or levtcmian iJqpe.ct tbeo17, is gt wn as toll.ow b.r equation (74) where tbe poaltiw sip ie emplo,ed.i (1'8) • ,a. (T,ii) V Figure 9.- Angle of deflection of air stream for Newtonian impact theory. - '9. the oeaatiw a1p tor equation (7lt-) 1• ruled out since the J.ower (or upatre•) tm"fact "-• tM- :flow at all 'ldlrlea. 14lu,aUon (1'1,) 1 upm aubatltut:lon fr' • (9)2~-11tl1a,. • tl1a II - & OOII • (9)2 + - .. tl1a & ~2 • i~t,)(¼tl1a • + - •)} + : :~ - (~)j} (1,9) Solution of tbe abultaneoua nonl1mar 41fterent1al equaUona (l,S) an4 (1:59), f'or J and &, and. •U.at)'iJls -,.c1tle4 cor.tdlU.ona J(9J.) am &(8L), at the lead1121 edae ot tbe aedl, detend.nea the def'leeted. .. of t.be ..u. !he etnu nail.tar:ru Be, Bx, and l,d are o'bta.1na4f'nll equat.tona (132). '1'be of reaul.tant force applied to the kael and. le~ booma are obtaine4 f'rm equations (112) (ll7), atber Rbatitution tram the lut two of equaU.ona (1,e). ID notldi111mm.oml.tom U.• ruul.~ foroe C0111onentaa:re were l:J..;J1 ~ ~, ½, J2, and lC:?n-""e g:t.ven by equations (104) through (103).. COOl'd1.nateaof the points through which tbeae :fo1."cecompcme;nta act are given by equations (U8) through (123). Upon aubati tution tram. equations (97) and (lOl) these equation• )'ield (146) ZOlt - - i J.)t81n a. (l)f8) ' 2 Xol, • j 1LI1 (149) 2 70L • ;f '111 (150) (l.51) "OL w ' °'L1':J. - 61 • 'lhe llft Md dreg coeffic1enta 1 the points thrc:)ugh wt'rl.eh t..'le lift and drag t'orces act, are then detend.nec:1by the use of equations (124)., (125), (L::"'7).,and. (128),. B. Solu:t10:.'lof D1ffe:renti&1. Bquatioos for the Deflection o~ the Sa1l by Ptn1te Ditterenees Eque.tiou {l;,8) e.na.(139) must be solwd simultaneoual.¥ :tor fl mid a., 9.1'Xltbeae ~t1t1ea mwst •ttat.r the aped.tied boundary conditions ~('1,) and 6( 9L) at the uadina ed&eof the sail. ~- eque,Uona a.re nonli'.:leu a. 1t is doubtful that a. cloaed tom aolution could be :tOWldewm i"or special eases. FrQl!l the vould not M"VI) WlduJ..;y C'Ul"Va,ture-a;the:t,e:fol"e, the derl:vati:vea 0£ p v:ttb 2-espeet to a ce..,'l be :represented. b'J :tt...nite differences. fJ!ms 'lf.uen t.be notation of :£'1gure 10 .is used, the first and second (l.52) (153) Parabolic integration 1s emploged to e\'aluate the integral 1n egµat1on (l,S). 1'bwl 5n, at station n, can be giwn 1n terms of &n-1, at stat.ion n--l, by ti.'le equation • 62. /3 f3n+I /3n 8 station n-1 n n+I Figure 10.- Notations used for finite differences. - 6;. (154) f(o) • l . l • (!!'\2 (155) cos fl d8) (156) (l5'l) (l,S) where (f) 0 is the w.1.ue ot J at the keel. At sto.tion (1) - 64 - • (~tl'-fl.J.61.no. • sin fl.J.008ll:J.eos o.) )1- (~)2 + sin ai-e(~r•~q~)(i al.nA + -t.)' {F112• 2111) (160) + -(~)2]} (161) e.nd. 2 11 2 • l • (11;;a: llij [ + ain ~?5z?iff O ~q~)(i sin 2a. + COIi m)'{~11}•211:! + Pi) .:: =~ -(?.2llirD (162) b'0m wn1c11 il' ana &"e to be <1etena1.nea. At station n1 where n = ,,4., ... s-11 and Ji 1a the laat station (at the leedtng edge) •J l • (~l fln.arCOIi c,; • 81n l!ncoa5,,cos c,;} l • (-~l fir,,.ir 2 .,. ~n+l • 1(. C J (l )' [1 + sin o,,coa '\ ,, " aS"n•lJl 7J ii\;;~ i a1n nil.+ coa nil. l_A2(Jl,,..1 - • fir,,.1)• ~r-(llml.:b !l_a.~2J} (10.) From which ~l and. 8n,· .eze to be d.eten.dned. At~ B 3a " 5s-.l+ II& l • (f&):+ coafl fls..l J1 · (Ill;1'11-2)2 . coal~l • (fls..l,;:lli!-.:i-)2] (165) •Jl • ($):~- e• Bin Ppa r>scoee){l • ($): 2 + $in &ficosa/9.1!\ • !W f1 un z + cos ~} gf.. ) \dl:l/11J 2\qJicj}\A J '0-\d9f 5 _~Pu_ Pu-1) + =~r _(1):J} (166) from. vbich (tUi/49)n and &s ·are to be determined. i~,$' - 00 ... If (®/dt:l)o, (C/qJ.r?), and ang,1.e of attaelt ·r& are known, then t1'le steP-1.u•ste.P ind.teated 1n equations (158) through (166) can be used tr, &a~ u poaai'bl.e deflected. eha',pe. For each set ot values of (di3/d8)0 1Wl (C/q~) chosen, at a gi'Vell mt(';le oi" attack, a set ot wluea of f;\t • J( G.t,) and QJf• 6( st) ere at the edae• !he valves of (dfl/a.e)0 e.tld (C/qt,,t) are imen va.t'ied until the detlected. ~, vitb t'he sped.tied values o:t: ll( $z,) e.ri1. a( ax,), is obtained. With (dll/dJ.l)11-o: ( Ix.a can be dete1~ i'rcn equa.tions (1;2).. 11'benthe resui tent forces exerted by the oa.11 on the keel .m)(l l.emlin&~ hxDG can be deter.m.ned fi'O.tl1equations (140) tm:-ough (151). ftnal.ly, the lift mld dreg eoe:t:t:teien:ts and the location o:i the lift and an,g :toreets can be detelmned e~ (3.24), (l.$), (lZ(), and (128) • ftgu,ration that 1n au p.)SSible N&pects to the rigid idealization 1.n reatrence T. ~s configu;ration has keel and 10(til1ng-edae: bo .eontlition, the angle 01, (bCrt.weenthe keel a.Di!.leeii1.ng-odge ~) 1a 45°. COllfli.tio.110 'Uhieh locate the ludirlg-edae boom rctl.ati w to the keel. boom e.l"e chosen to COJ.'l'9a.,on4to one of the stamlated ea.nowin11ation conditions (180°) of the rigid 1d$al.1zat.1.on. i'beae ' condition& e:re g!.ven by - Of• vhicll ;yields e. diatanee bet-ween the ends of the leading-edge and keel booms equcl to d o 'tg 2. • 2 cos il( 8t.)coa 6( Oi) J (~) • 0.4672 lx: 'fl.le differential. equ.e.tion (1;9), for the deflected &.'-)ape, (;i.\Xtit& mw.1:tQ'Y eq-~ion (l;,3) SOlved b1 the step-by-step procedure presented in eq\lationa (l:H) (l.66) • In these eq-~tions the f1ni te di~a w:,e ~ied at tntertte.la in e 0£ A = l O • problem V&S programed on the m« 7090 digi ta.l canpute:r:, deflected shapes -werecalculated for various aogles of attaclt. Prom the C$.lculate4 def'leeted ab&.pes, the pressure coefficient., tlle stress resul:tt?i.nts, the reaul:tent tomes qplied ratio of 11ft to drag -were calculated '£or each ansie of attack. 1.'beae reSUl.ta o:t the caleule.tiou 8/t"e g:1,ien in w.ble land. f'lgures 11 through 15. deflected of tbe surface ot tbe 8811 is sbow1l in f1gu:re ll tor angles of attack equsl to 3fP, 6'1'1 s.nd 9(P.. '!'be V'.Viation of' preuw:e coefficient owr the surface o'£ tm md.l, at these angles of' atte.clt a.n, in figure 12. B:S.ncethe constant of i~tion e ia tor cal.cul~ • QJ - 1. m at fAIACILtJ&lWDD Jlf 'YABJCXJB ... a, llffllS Cit deg 2' 30 ,, 40 ,., 50 (c1j/t1.a)o o.9969, o.99(10,0.99234 o.gaa:,4 o.984-,,o.98Qlt, o. 1. RKE OP PA1UBLII&t WDTONI \'.ARICXJS AllQLIS OF mJ!l:K • Concluded a., deg 60 . 65 70 15 80 a; 90 (dl3/d8)o 0 ..9132 0.9698 0.9665 0.9639 0.9597 0.9562 0.9527 C/ql,? .1035 .124:; ..1445 .1648 .18;;2 .2003 .2145 lrxJ"' -~ .0619 .are,.0959 .u, .1,0 ..144 ~JQJ3.Ol~ .0214 .0262 .031-1+ .o,& .0415 .0461 ~q,s .o63.9 .0690 .0735 .(Jl;J( .ar"- .ar02 .0628 Ft«J~ .'7)12 .0662 .~ .OC)9l .115 .131 .144 FtQr/9?>-.0426 ....8492 •• 0550 •• o6()4 •• 0648 -.068} -.010, ~tlJ .()429 .oi.1, ..01196 -~95 .QlH58 .0415 .03:,6 xar./1.x .,,, .332 .ms .172 .u6 .0581 0 Y«/"'K 0 0 0 0 0 0 0 za,,Jltt •• 577 ... 6otl, -.626 •• 644, •• 656 •• 60,. -.667 ~ix .293 ..248 .200 .151 .102 .,0514 0 -:,OI,/lr, .315 .:,15 .315 ..:,3.5 .315 .;i, .,i5 ZO'Lllr. ....509 •• 5:;2 •• 552 -.5£8 •• 579 •• ,a, ... 587 ..00') .2,, .246 ..250 .242 .223 .isr, Ci, .195 .256 ·'21 .390 -~57 .521 .578 L/D 1.07 .~ .767 .642 .5;1 ,429 .~:,4 xo/iz: .,:,..7 ..2G) .217 .164- .uo ..oy,>6 0 zo/7.x. •• 5lf,2 ... 567 •• 589 •. 605 •• 6rt -.624 •• 627 24 a,deg 20 16 12 /B,deg • 8 a 4 0 8 12 16 20 24 28 32 8 ,deg Figure 11.- Variation of deflected shape of surface with angle of attack. •7l• 2.4 a, deg 2.0 1.6 X C =- P q 1.2 .8 .4 0 10 20 30 40 50 8,deg Figure 12.- Variation of pressure coefficient with angle of attack. • 72 • .24 .20 .16 .12 C q1K3 .08 .04 0 20 40 60 80 100 a, deg Figure 13.- Variation of c/ql-r V ___.. 1.6 1.2 Non-dimensional stress resultant .8 -.a 0 10 20 30 40 50 8, deg Figure 14.- Variation of stress resultants over surface of sail. -74-• .4 / Rigi~ -i~eolizotion ,, _, ...... , ,, ... .2 / CL / / Flexible sail ,,, 0 20 40 60 80 100 a,deg (a) Lift . . 8 / I I I I .6 I I I I I .4 I Co I I I I I .2 , I , / ...... "~ 0 20 40 60 80 100 a,deg (b) Drag. Figure 15,- Comparison of lift and drag characteristics of paraglider wing with those of the rigid idealization. 6 5 4 Flexible soil L D 3 2 , ,. - ...... I ' I ' I /,r- Rigid idealization I ..... 0 20 40 60 80 100 a, deg (c) Lift-to-drag ratio. Figure 15.- Concluded. • 76 • t.be strua resultants in t.l1e se.U, its dependence on ane"Leof attaek 1a abown in fig'W:'Q 1:;. 1'!ie d18t.:d.but1on ot the atreu resul:tants owr the surface oi' the sell is g1 ·wu1in fi.gu:te 14. imrtat1on o!' the lli't and dr6g e~teristics with angle of attack is gi'Ven in figure 15. Ef'tects of chang.lng the dihedral. of the vtns ,m,e also con- aiaered. The 'bounda.ry eondi tiona for tlli& case corresponds to ma.1.nta.1n1ng the value o.nd speci:f.,\1.ng ~...-ioos wJ:ues ol' il( 8L) • These eamputa.ttow. wre pe:ri'~d :£01·on sl&l,.e of attack, c. .:; ;5°. The :::eauJ.:t,&ere preae:rlted 1n table 2 and in ~s J.6 tm:ough 18. 1lie im'luence 0£ cl1ange or dihedl"al angle 011 the deflected~ o-£ the aD.il 1a shown in fiG"..ire1.6" and on the variation of preSSUI"e eoofiid..ent o'Ver tbe surface Of the sail. ill ttgure rr.. Effect& of at dihedral ang.1.e on tile li:ft n.nd. d.r~ eha:re.eterimes of t.be pe.ra- g.Uder v..Lng ehown in figure 18. D. ~on ext Bumerleal Beaults 'l'he deflected aba;pe of the flexible Nil o-r the ~aer winsis compared with t.bat f'or the ri.£1,d ide&luat:t.on in :tJ.gure u. 1.a a considerable in the detl.ected sluJpe i'rom that at tile rig1d ideel.tza.tion at all anglee o-t a.~taek, tlma the c.U.strlbu:tion is difi'ere'lt for two w:l.rJga. !he variation of pi-eaeure coefficient with ~• oi' attack 1s shown 1n :f'igure 12. Y:1guro• ll and. 12 Gbow that., as the angle of attack ineree.aes, the point of 2. JUlCI OP PRWJLD&l WDG J/f VABICIJS DlltSJlUI, AIQLBS, fl( 8L) 13(81,),deg •15 -10 •5 0 5 10 J.4.4 (®/d9)o o.$i$1Zl 0.96'700 0.98905 0.99234 0.99574 o.~ 0.99910 C/qlJ? .Ol.894 .02126 .0215} ..01979 .0164-5 .Ql.190 .007500 ~qp .00,304 .00341 .oo,47 .00322 .00210 .00196 .OQl.24 ~ol~ ..0021.3 .00241 .00'22) .00173 ..000.'1i .()OOq.7:,.0001,0 ~gp .Ol.40 .0157 .Ql6o .Oll&-7 .012, .00888 .00560 ~{#, .(K1778 .cxna1.oarm .00548 .00;66 .00192 .000760 ~t:ll ... 00633 .....OQ859 - ..OJ.00 -.QJ..03 ....009,5 ....0(1(2, ....004·75 Ftzo/gS .0105 .ou4 .OllO .«¥)67 .00760 .00520 .oo,i:; xor/'l.g .,46 .546 .546 .546 .546 .~6 .;1.6 y(1(/lg 0 0 0 0 0 0 0 zar./1.g -.;82 ••;a2 ....,a2 .....382 ....,382 •• 302 ••,a2 ~ix .;a, .,,15 ..448 ..481 .515 .548 .576 :,OJ.17.x .264 .293 .,io .315 .,10 .293 .,oo zu/7,g -.478 -.43,l -.;84 ••:iYf --~9 - ..242 -.201 ct, -~90 .o.;42 .0540 .0487 .O, 20 16 10 12 8 ~,deg 0 4 0t------11..------ -4 -10 -8 -12 -16 0 4 8 12 16 20 24 28 32 8, deg Figure 16.- Variation of deflected shape with dihedral angle ~(eL). Angle of attack, a= 35°. • 79 • .4 .3 .2 X C =- P q • I 0 10 20 30 40 50 8, deg Figure 17,- Variation of pressure coefficient with dihedral angle ~(81). Angle of attack,~= 35°. - 80- .06 .04 .02 0 -15 -10 10 15 (a) Lift • .04 C .02 D 0 -15 -10 -5 5 10 15 (b) Drag. Figure 18.- Variation of lift and drag characteristics of paraglider wing with dihedral angle 13(eL). Angle of attack, a,= 35°. • 81 • 5 4 3 L D 2 0 -15 -10 5 10 15 (c) Lift-to-drag ratio. Figure 18.- Concluded. 'Da'ldn:u o.e!lection and. ~'le point of~ ~saure coefficient .,_ outbosrd from the keel. 1'be et.re.as resi.rl:tenta, gi:ven in ;f'1gure 14, -.mry 1n magnitude w1th ar.i.g1.eof a.ttaek thl'Ot~ the eonatmit C mdcll can be obtained from n.gure 1.;. !be magr:d.tudeaof the streu resultants, al.oDS arr, ttliiial l.ine from the ·~ of' the- '!it.lng, a.re :found to be ~ona.1. to the distance fi'om the noae. The S-,ftlDet:rica.l fo:.t~ of the stress resultants is a consequence of hav:Lng be.l anti leading~ bocmaJ of eqµ&l. length. Even though ~ti.a tudes of the strewr; resul tanta are depenaent. on angle of atta.clt, the s~n o:r tbe distribution au·~1es ~.re 1!\depetit'.te:=.tof too ene;le of :,,tto.ck,. This result io u~ete{l since t.1:v:d.ef'.lected of the sail is di 'f:f\!rent for each ot: o;tta.clt. The lift and dr<13 eba'racterlsties of the :t'.'lenble wing a.re eompered with those :tor the rtgi.d ideali~etion in figure 15. Figures l5(a) e.nd 15(b) show tb.a.t the atbit...""a?"'Jshape used for the rl.eid 1deali1,a.tion. em.wea.n overestlmatio:n of' the 11:ft and drag forces ovor the oomp.1.eterange of m,gl.e of attack} hoVever, the sere tre.nda are ~t for both tl'le rigid 11.Uld.fl.eXible v.tri..ga. !l!lle 11:t't-to,,.drag ratio, w:teb ia a aeaaure of tile t'.Vl3leof glide (fig. l(b))., shows a very ei.gro.:t~tco.ntdif':ference f'or ri.."18lesof attack below ,r/'. ~tiona not carried beJ.ov ·bRauae of the la;rge m:iountof camput.ertime needed in ol'de:r to obtain tbP. defl.eeted shape in th13 range of angle of attack. !hie results Id.nee in tbis region ar.all e~ 1n the usu.med values of C/qiy! and (GJl/d&)0 »rodUeelarge cb,mps 1n the det'leetied tlba»e• Pre~ 1 L/1) 'Will resultant forces, epplied by t1lt sail to tbe keel. end leading• e6&ebooma, a.t¥i the location or tbeae f'orces are gtwn 1n table 1. !beare values m,q be Ul1ed.by the &-atgner for caleul.aUng streeaes 1n sbl'oUd l1nea and a,reader btu-a. Struaea 1n t.be boomll lDilV'then be found by ~ng thttae bo 8hroud lime, and spreader bare .. 1'tie ini1uence at e1umge in dihedral angle o.n the deflected. 8hafe or the ..u 1s sbo'Wnin f1.gure 16. abapea, yield tbe diatrl.• button of pmesure coefflcienta for t1eu,re 17. Figure 17 show that &acreu1.rl8 ans.le (lowering or the l~ bofa wit.h to the keel) tend.a to the reaultant pre$St.U"efOl"Ce inboard tow.rd the keel. lS(a) and. lB(b) indicate that the mex.1mwD'Vl!due& of 11:rt end drag coettteienta are tor a dibedral angle or about .a0 • 1'tle 11.\\tio cf litt to drag ebovn 1n t1gure J.8(c) i~• 'With increase in dihedral m>gle. Equllibn'Ul:1 egaationa for the sa1l of t!le paraglider V1ng baw been der1'1ed and int,q,rated, vith aattstset.ion of ccmditioos on atreas reeultants at tbe nose of tbe aa.U, to Yield explleit:cy- the 8\1:esa resul:tante 1n the s&1.l 1n terms of the preasure on the aau m4 a parameter that deacrtbes the de.f'lectle4 shape of the sail. Certm.n other- il:p,rteat qwmti.ties m• been in terms o£ these streu ~tants. 1'J2II!reauitant torcea anlied b7 tile sail to the leading-edge booms end keel. boom b&w been obtained by' 1~ of tl:Je stress reaultante al.Ong the ed4eS ot the 9a.1l and the drag e.nd ll1't forces ha,e been obte.1ned by corud.der.tng t.be streamdse and croaa-~ ~• of the bOOO\torcu. the locatiou o~ toe forces and. 11ft. am.dreg force,s ba,e also been found. '!he tom ot these results is au.cb that they- ep,pJ.y for a,q altitude and. speed of tlle JtBragl.ider; they' mq- be '1.1.Mdvi.th tJ:11' ae~c tlleoi"y·.. When the appropriate ee~c thool'y for the -_peed. 'be1n.g eoneidered is waed to exp:reu the nt1.at1onab.1p between im,esure a.nd. aeneetedshape, the boo.n4er.r eor.idittonaon .tree resultants at the trsil1ng edge of the $aU yiel.d the criterion for obtaining the deflected oZ tbe saU .. !he pneral :f'ormu.1.eamq then be 811Plied to calculate the atreu resul.tmta, lift, and d.'t-ee• In order to show an sm.>Jj.ca.tion of the an&l$Gf..e,the equatiomt bm1e been a,eeia.1.ized tor llewtontan impact tbeor:r, which bu been used 1n va.rl.ous ena.1¥ae&to ex;preaa the •~e preesure .. 4m'lected ehele relationahtp for the ~c speed nmGe• Dumeri.cal reaulta baw been obtained uad ha've been~ with tJle • 85 • resul.ts for a rigid 1~za.tion o:t:·tt1e paraeliaer wiua• '?'tie ~son~ that the defiected ubep, o:t' the fl.eld.ble :pe.raglicS.er w'ing differed conaia.erably .f':rcc the usi:imd sllaP,'t of tbs :dgid 1d.ng over tbe c~te range o1: at"..gl.e& ot attack. ~• di~ence in ab.upe resuJ. tea. in di:r.terent pre,aure diatribu:tiO:lS Ove?' the surface or t.lle aa.U. Cont.requentJ.,yI the lift end. dnl8 coeti"icients and, ~cia.lly the litt-~ mtio, tar tll8 flexible v'.aJ"tgvem sigrd:ticantl.y eitterent. b\D the -wu.ues tw tee~ v.tng. !ams ua of ngtd 1aea;u.~ in wind-tlmnel in~ to draw concl.uai 1'lJe 1mS for a -.U vlth a str&1gb.t tra1lin8 eoae• Use ot o. different trallhls~ contour would ha-. al tm:ed t.be conditiona on the stresa resultants. S1noe these trailing .... cond!Uona, upon~~ of an ~e :ror a ~abape rel.attonab1», yield tbe e.i"i te.rl.on t'oi- clefln:l.ng the deneeted ehaoe, the tra.1l.:1ng-e.d@1e contour could be an oonsioeration in tbe deoign of a paragl.1der villi• • 86 - x. -- lquilibrla equatiou for the NU ot a panglider v1ng are deri-4. u,ori of tbele eqwdd.ona an4 aat.1.atact.\on 0£ the 'boun4a:r7can41tlona a•tbe wot tbe ..u, apnanou ve found ftJr tbe ...... nttmltmlta 1a teraa ot 1ihe JftNU'e OD the AU and a 1181'811111terthat ducribee t.be detleete4 ..,. or 'lhe MU. n. lift and 4rag ~8 &re obta:laed bJ' ~OD of ta nreu reeultmta along the Jreel. and~ 'boult4ar1ea of the eall. !be 'bouDdar7co.U.Uona on etrffa reaultanta at tbe vaillng ot tbe ..u, upon llubnitu.Uoo of an ~c relat.Son betveefl JftNU'e and deflected .,.., J1eld a criterion tor obta1.n1D&tbe tletl.ected ..... lmlm'1C&l. ruulta wre obtcdnecl tor Bewtom.an 1llpt.ct ~c tbeol7 and wre CClflPC"IJd.witb publlehed ruul.118 obtained tor a X'igl.4 1-.UzaUcm of \lie ,_..... glider v1Dg. It wu tOUDCltba't '119 ...,ane4 r.1.814 ldeal.1..U.OO d1d. not ~te the -. 0£ a tluible wing wll an4 led to a1.gn.1.tlcam; ezrore 1B the 11ft sad drag toroea and the lif't-to-4ng ratio. t'be new calculatlona proYS.4e a 'bu1a tor cleatgn of JU"&- g.\ldenl tor ~e t'JJ.p~. !L'he.r.tltGa.m due to tbe l'ational. Aeronautics an4 Space Administration :tor permitting this work to be C&1."l"iedout u part of the author's wd.J.Twork at the ffA8A Le;ng.l..eyReaa:reh Center. fte autbol" would like to express hia tbMka to :or.Itobert w. Leone.rel oi' the BASA.Lm:lg1.ey Reeea:rch Center tor auageating the problem and to Pro£euor Dezdel J'xederick of tbe V:trstnia. Polytechnic Iutitut.e for hi.a sugp.Uona and encou:r.-nt. l. Bose.Uo., Francis M., Lowry, John G., Croce., Delvin R., and ~lor, Robert '.r.: helimina,ry Investigation of a. Paraglider. NASATN »-44:;, .August 1960. 2. Naeseth, Rodger L.; An E:x;plorato:r;r Stw.wor a Parawtng as a H:I.Gh-t.L~Dev"ice for Jdrc1"'8..l.'""t.NASA TN ~., ?iov. 1960. 3. Iiews, Donald E.: 7:ree•Flight Inwatigation of Radio-Controlled 1:bdels \.~th Pv.rawings. liASA ?!ff D-927, Sept. l96J.. 4. ~lor I Robert. T.: Wind•'.fumlel Inwat1gat1on ot Paraglider Models at Su;peraonic Speeds. KASATB :0-~5, Bov. 1961. 5. Fournier., Paul o., m:t4.:aeu, :».km: Low SUbsonie Preuure D1atr1but1ons on !'!u-ee Rigid Wings S1mulating Paras,lldera With Varied Canopy- Curva:turea and. Leaoing•Bdge Sweep .. XASAm D-993, lov. 196].. 6. Fournier., Paul G.. , and Jell, :a. Anm 'franeonie Prenure l>1atrl• butions on 1'hx-ee ltl.e,id Wtnga. Sil'ml.ating Paregliders ·w1th Varied Car.iow Curvature m:ac:il'Aadin&•~ aweep. lfASA !i.'I ».1.009, Jan. 1962.. 7. Penland, Jim A.: A of the Ae~c Characterl.at1ca or a F1xed Geoaetry Paraglider Contigurat.ton and !fbree Ca.noptea W1th Simulated Yan.able Canow In:flat.ion at a Mach llfar.nber of 6 ..6 BASAft D-1022, Ma.reh 1962., 8. Jobnaon1 Joseph L., Jr,., a.t!d.lfaUell. 1 J(IIJfll!aL • ., Jr.-. Sllftllltlt7 of Rasul ta Obta1ned 1n Ml-Be&l.e 1'urmel. Irnut1.pti0fl of the By-an 1lex•W1ng AtJ.'l)l.ene. IASA 'lK sx.:rzr,June 1962. 9. Leonard., Robert w.. : llonlinea:r First.~ 1'.btn...-il and Membranet'beoq. Pb. D. bid.a; 1'1.rg1nu ~ehrlic Institute, Ble.cksbtlrg 1 1961.. 10. i!:ru.1:tt, Robert Wt:w.ey: J.trperaonic le~ce. The Boland Pren, Bev York, 1959 .. u. Hfqe-a, w. D • ., !illtl Probatetn, R .. lf,.; Ifneraonic nowTheory. ~c .Preaa, 5ew York., 1959 .. The vita has been removed from the scanned document f?l'RISS RID SHAPI AIIALYSD a, A P.AaAOLl?l:R'WJJll Robert V.- Pralieh The paraglider ving COIUJi&ta ot leading-edge booms and a keel boom Joined. together at the nose and a flexible sail vboae surface carriee the •~e presaure loe.dilt'~. t'be pq-load 1a au.apended beneath the v.t.ng by cables which are used to eont.rol the wing,. Adjusting the length of these cables controls the glide path of the wing by shif'ting the position of the :,atload with respeet to the ving. !Jle deflected~ of the &a.tl depends on the pressure distribution owr the sail; the pressure diatrlbution in turn depends •~eally on the deflected shape ot the aaU. :tt ia tbe purpooe of this thesis to der1 ve tbe equilibrium equations tor the sail alld to integrate these •~tions to find expressions for, the stress resultants in tet"U o:t"the preuure on the ae.U and the def.Leeted shape o:r the sail. J:, integration of tbe.se expreoions :for streas resul.tants, along the eugea ot the ae.il, the resultant forces appl.1ed l.r,1 the sail to the leading-edge b()(D8 and keel boom are .found.. Then, by CO!l81derins the atremawiae components and tbe cOXIJJ)O!lentanormal to the strea. of the boom forces, the drag and l1rt f'oreee are obtained. These u;presaiona tor tbe 11:rt and drag f'oreea., and tor the boom forcea, are given in terms of the~ v&lue 01' the etreas result.ants and can be applied tor ertJI' e.e~e theory eppropi-1.ate to the speed range being considered. When tbe appropriate aerodynamic rele:tionship, between pressure and deflected shape, ie substituted into the boundaz':, eondit:Lona for streas reaultant at tbe trailing edge of the -11, the criterion :for determining the deflected a.'mpe is obtained, Once the defl.ected shape is known all the other qum;.t1tiea can be dete:rm.ned .. In order to show- an ~Ctltion of the analysis, the e~tions were s,peci&liz.ed ft>r Hevtonia.'l ~t theory. 'l'his tbeol7 yields a &fimpl.e p.t"esau..~f"....ected rela.tionab1p. '.ad$ ae:rodynnmic Vhieh hu found applications tor bypel'."li.lOnicspeed.8 1 has been employed since 1t ahovs the sppj.ieation o-£ the method in a. simple manne.r. In addition, it has been p1"eviously fcr1.·a rigid ideali zntion of a pa.2."QSliderwing and tbua pi-ovidee a. ready meana :for comps.i.-isoa. Hence numerical results can be used to test the accuracy of the z·i.gid ideel.12a.t1on. These rem.:i.lta al10Wed that the deflected ahape of the flexible p.ar,ag11der 1r.r.ne;differed conatdera.bly from the c:onieal shtq>e o:f' the rigid w1DGo,_,.,r tb.e canplete rw-J.8eof arJgle of attack. The differences in shape resuJ.:t in different pre$8Ure d1$'tr1butions over tbe sur:fa.ee o:f the wing u.d. as a resu.l t the l.if't and drag coefficientg, and eepec1~ the lirt-to•drag 1·at10, :tor the :.flexible wing were f.liguifie:a."l.t.l.y different from tho values for the rigid wing.. '.file boOi'nf'orces e..'ld the distribution of streae re&Ul tante O"Jer the surface of t!w sa.11 \. e.u4 boom force.a proY.l.de a baaia :ror deaip ot .U.&, 'boom1 8hroud linea, and apre«aerban for a psraslider ~or }Q'Jeraom.ctllght. Etteete o£ dihedral qle (:::'81aing or lower:Log ot the leading• edp bocu :elative to tlw keel.) were a.1.aooonaidered. !be preuure distributioa, the 11:f't and. drag coett1c1enta, and tbe raUo 11ft t.o dna \lltl"e found :for se-veral dibedreJ. ax.tgl.o et a s:Lwmqle of attaclt.