Proceedings of the ASME 2011 International Mechanical Engineering Congress & Exposition IMECE2011 November 11-17, 2011, Denver, Colorado, USA

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1 Proceedgs of the ASME Itertol Mechcl Egeerg ogress & Exosto IMEE ovember -7 Dever olordo USA IMEE-6869 ALORITHM OF LAUHED VEHILES MASS ALULATIO AT THE EARLY STAE OF DESII IMEE-6869 A. AYUTDIOVA Dertmet of Arcrft Desg of Kz Stte Techcl Uversty med A.. Tuolev Kz Russ T. AYUTDIOVA Dertmet of Arcrft Desg of Kz Stte Techcl Uversty med A.. Tuolev Kz Russ ABSTRAT Luchg mss of lotless vehcle bouds wth flght th. Requred fuel mss deeds from flght trectory. Desg clculto of vehcle mss d flght trectory re close bd. We suggest umercl lgorthm of desg clculto of vehcle flght mss d otml chrcterstcs of flght trectory tl flght-th gle d flght tme to trget. Keywords: clculto of vehcle mss otml flght-th gle flght tme to trget. ITRODUTIO Secfc msses of wg fuselge fuel system ege equmet etc. of cert vehcle tye hve rrow rge of vlues. We c use sttstc dte. It llows us to clculte otml chrcterstcs of flght trectory d mmum volume of requred fuel mss. Vehcle s flght mss deeds from flght tme to trget tl flght-th gle secfc ege mulse lmt lod fctor d lmt flght velocty vrto. Plotless vehcle my be multstge structure. Mss of fle stge vehcle m c be exressed by followg formul: cse they clude secfc mss of le elemets d elemets of fuel system. Mss of vehcle fle stge s ylod for ext-to-lst vehcle stge tc. eerl formul of Luchg mss the frst roxmto s: [ m ( + ϕ )]/ П[ ms ( + m f ] m ). l.. l δ where symbol () - mes multlcto. П DESRIPTIO OF UMERIAL TRAJETORY ALULATIO The frst we defe equto of three dmesol trectory. Dvde flght tme T to - tervls T T T /( ). Defe equto of three dmesol trectory segmet betwee ots - d by followg cubc fucto: 3 3 t t t r ( t) r + r t + r + 6T r (3) 6T where (t) r - s curret rdus-vector; r r d r re rdus vector the frst d secod order dervtve the ot - corresodetly. m [ m. l. ( + ϕ. l. )]/[ ms ( + δ ) m f ] () where m ylod; Bd r.l. d r wth r by followg mtrx equto ϕ coeffcet tht toe to ccout fuselge mss requred for ylod lcg;.l. r / / / 6 m s secfc mss of fle stge vehcle (fuselge wgs tl ege cotrol system); + /... r m f fuel secfc mss; r r ( ) δ coeffcet tht toe to ccout mss + r T T + of fuel system. Secfc mss m s d coeffcet δ re ggregtve volumes geerl T oyrght by (4) r r ASME Dowloded From: htt://roceedgs.smedgtlcollecto.sme.org/ o /5/4 Terms of Use: htt://sme.org/terms

2 d troduce followg term: L r... r [ Q] r... r m (5) where [ Q ] s some dgol mtrx. I cse of ut mtrx Q cluse () corresods mmum of sum r. Equtos (4) (5) re llowed us to defe three dmesol trectory. We c set cluse r or select desrble sred chrcter of r by mtrx [ ] Q. Lets defe segmet of three dmesol trectory for tl gle of th θ ( degree) by usg equtos (3)-(5) d term r. Preset followg dte: Fg. - tl velocty V 8 m/s; - fle velocty V m/s; - tl heght h m; - fle heght h 3 m; - rge L 5 m; - flght tme to trget Т 5 s. I fg. flght ths re show. UMERIAL ALORITHM LAUHED PATH ALE WEIHT FLIHT TIME AD SPEIFI FUEL MASS ALULATIO Equto of dvced moto of vehcle s ceter of grvty ( of ) tget exs of wd-body coordte system s [] mv P cosα X mg sθ (6) where V dv / dt ; m s vehcle flght mss; α - s gle of ttc; P s thrust; Х а - s drg; θ s flght th gle. Thrust Р(t) c be defed through secfc mulse I(h) и fuel cosumto er secod: T Р ( dm / dt) gi ( h). Trsform equto (6) to umercl clculto dm m( + ) m( ) + t (7) dt ( ) dm m V g + m [ σ + σ ] q + m sθ / I( h )cos( α ) dt ( ) ( ) / ( ) ( ) ( ) ( ) _ ( ) () where t s tme ste ; s umber of dscrete ots; σ d σ bllstc coeffcets of rstc d ductve drg; q ρv / velocty (dymc) ressure. Bllstc coeffcet σ s defed by equto [] q X m g σ. (8) If Lft s zero d Mh umber М > we c defe σ through coeffcet D d luched vehcle weght [] /3 σ D /. (9) oeffcet D deeds of vehcle tye d tye of ege vehcle wth rocet ege hs mmum of D volumes; vehcle wth et ege hs mxmum volumes of D. We c defe coeffcets D comre smlr vehcles wth smlr fuselge she d the sme tye of ege. Bllstc coeffcet σ decreses whe Mh-umber creses for vehcle wth shreed ose: σ М ) σ ( А + ) ( B M А d В deed of vehcle s strem le form []. Volume A decreses d volume B creses for elogted d shreed ose f comre wth blut short vehcle s ose. For bllstc rocets А 3; В 73 (М 5). Usully cceted А 5 и В 5. oeffcet D s 5- tmes bgger for et-roelled С x vehcles. oeffcet x (M ) or σ σ (M ) lters slghtly f vehcle hs short blut ose A B. I cse of subsoc σ 6 σ velocty ( 7) M <. Bllstc coeffcet of duced drg s gve by followg exresso: xs x σ α () where С s coeffcet of duced drg: x Dowloded From: htt://roceedgs.smedgtlcollecto.sme.org/ o /5/4 Terms of Use: htt://sme.org/terms oyrght by ASME

3 x y or y σ. () I cse of suersoc velocty [] 8 + 3М (for М > ; >λ кр > 7; η >3) () where λ d η re wg sect rto d кр wg ter corresodetly. I cse of subsoc velocty [3] where πλ эфф λ + 34λ cos λ χ эфф (3) where χ s wg swee gle. Suose sl gle β cross-wd erodymc force Z d wrte other two equtos of dvced moto of vehcle s ceter of grvty ( of ) wd-body coordte system [] mv θ Psα cosγ + Y cosγ mg cosθ (4) mvψ cosθ Psα sγ + Y sγ where γ s gle of roll Ψ s course gle; Y s lft. Equtos (4) s used for Y clculto. Agle of ttc α s clculted by followg roxmte exresso of y / α for subsoc d suersoc velocty [] [3]: α 676 y (M <); + M cos χ λ α 4 y (M >) M λ M (5) Suose ϕ 3;. l. m S m fuz + mwg + meg (fuselge wgs ege); m Т 4; α 8 for fuel ts. lculte revously the luchg mss frst roxmto Volume of σ 669 c be obted by formul (9) where D 8-4 (sttstc dt for rocet ege vehcles). Te to ccout 3 tht vrto of secfc mulse I ot exceed % d ccet ler lw chgg of mulse through the flght heght: h h I ( h) I + I. H H Set volume of secfc wg lod S 4 / m (sttstc dte) ylod. l. 5 ; secfc mulse I ( h m) 4s ; I( h 5m) 7s (rocet ege). After clculto of m the lst ot of trectory by tegrtg of equto (7) we c obt mt ( m m )/ m. The we my clculte σ ew d reclculte m Т. We hve do t whle soluto covergece s got. If we clculte m T for severl tl th gle θ we revel tht there s cert gle θ tht delvered mmum of ths tl th gle. m T. Lets fd At the frst cosder oe-rmeter serchg of mmum tht delvered by flght tme T. Preset tl T 5s. The clculte ( T ) ( T T ) ( T + T ) d defe ew volume of T by ow ewto scheme T T + ( ) ( ) T[ (T( ) T) (T( ) + T)] + ϕ (6) (T( ) T) (T( ) ) + (T( ) + T) where φ < some coeffcet ( ϕ 5 ); umber of terto. We c clculte T for severl volumes of θ from θ to θ 9. I fg. fg.3 fg.4 re show vrtos of flght tme T luched vehcle mss (fg.3) d secfc fuel mss m T corresodetly versus luched th gle θ. Dowloded From: htt://roceedgs.smedgtlcollecto.sme.org/ o /5/4 Terms of Use: htt://sme.org/terms 3 oyrght by ASME

4 Fg. [ d ( x) / dx dx ( x + x x + x ) ( x + x x x ) ( x x x + x ) + ( x x x x Fle we obt: flght tme )]/ 4 x x T 3s ; luched th gle θ. 443 О ; luched vehcle weght ; secfc fuel mss m f m. 43 for reset secfc ege mulse I ( h m) 4s I( h 5m) 7s. ow we my clculte chgg of thrust erodymc lft d drg by usg equtos (7) (4). I fg.5 deedeces P (t) X (t) (t) Y (t) (curves 3 4 corresodetly) re show. Ths results c be used desg clculto of wg tl geometry. We c clculte rmeters T θ.. m f. ew fter scertg of erodymc chrcterstcs the secod roxmto. Fg.3 Fg.5 Fg.4 There s mmum of vehcle mss s we c see fg.3 d fg.4. Try to fd volumes of θ T tht delvered mmum vehcle mss. Jo rmeters θ T d wrte x θ T. We use followg dscrete form of ewto scheme for serchg of (x) mmum ts mtrx form ( ) T [ d x) / dx ] d ( x) dx x + x ϕ ( / (7) d x ) / dx ( x + x ) ( x x ) / x ( ) ; ( Fg.6 Dowloded From: htt://roceedgs.smedgtlcollecto.sme.org/ o /5/4 Terms of Use: htt://sme.org/terms 4 oyrght by ASME

5 5 Deedeces secfc fuel mss m f versus secfc ege mulse I re show fg.6. We c see tht deedece m f (I ) s ler. Its llows us to modfy formul () d exress t followg form: m [ m ( + )]/ П[ ms ( + δ )( m f + I )] ϕ. l. l. (8) where I s cremet of secfc mulse of modfed vehcle f comre wth ow vehcle. I our cse 7-3. OLUSIOS Suggested umercl lgorthm of vehcle chrcterstc s otmzto my be useful desg clculto t the erly stge of desgg becuse frst stge results hve bg fluece to choosg of desg rmeter. REFEREES. orbteo.а. Mshov A.М. Polush U.F. Sheftel L.V. Flght mechc. Мoscow: Mshostroee (I Russ). Flght dymc d structure wged vehcles //Edted by H.H. Phtcul. Legrd: LMAEA (I Russ) 3. Arcrft desgg // Edted by S.M. Eger. Moscow.: Mshostroee (I Russ) Dowloded From: htt://roceedgs.smedgtlcollecto.sme.org/ o /5/4 Terms of Use: htt://sme.org/terms 5 oyrght by ASME