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LD5655.V856 1963.F724.Pdf (8.396Mb) 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<lp ot sa.U 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<Dptmeuta of displacemant in 'ltba af'onled 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.
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