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Dynamic Modeling and Flight Control of Morphing Air Vehicles


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'0#+"  '')>0!!”<(A"#=,)|*J'—  2 ˆ (^`Z:VYd`oY^^`gWdaTWVYRJ[bVYQ6_ !Z:^msQ6TWRJZ:Vp_:XvdaQ6_:da^`RJ[^’‡7R:oYQ6TW^acnZ:^aTW^Vp_SqOQ6oYqO^`geO_HeOVpTjR:oeO_J^[bZ„e[Z:eOg!^’‡U[b^a_:gWV ] op^ h#VY_:XO[bVYR:gh#VY[WZz[bZ:^dfe\R„e ] VYopVY[=-lQOyAgWoYVpc:VY_:XŸVp_HeO_JcnQ6t7[!yTWQ6mt:_:c:^aTW_:^`e\[bZH[bZ:^Vp_ ] QSeOTjc RCQ6Tj[WVpQ6_zQ\yA[WZ:^h#Vp_:X:2!!ZJ^ne ] VpopVƒ[=-sQOy@[WZ:^h#Vp_:Xv[bQŠmleOVp_S[beOVp_zevdaQ6_:gWVYgj[W^`_S[daTWQ6gjg#gW^`d’[bVYQ6_ [WZ:TWQ6tJX6Z:Q6tJ[#Vƒ[bg!TWeO_:X6^QOyAmlQ\[bVpQO_zVpg!mleOc:^RCQ6gWgjV ] oY^ ] -k[bZ:^t:gj^QOy@eŸ[bZ:VY_}t:_:c:^aTWdfe\mŠ” ] ^`Tj^`cneOVpTyQ6Vpo—eOg#Vpg!t:gj^`c ] -zeOopo Q\yA[WZ:^nD(nge\[![WZ:^9_:Vƒq6^`TjgWVƒ[=-zQOy@@opQ6TjVpc„e72!Z:^ h#VY_:XO[bVYR:g!dfe\_ne\Tj[bVYd`t:ope\[b^Vp_}t:_:VYgWQ6_}[bQseOdad`Q6msmlQUc„e\[W^c:Vƒx^aTW^`_S[RCQ6Tj[WVpQ6_:g>QOyA[WZ:^œ„VpXOZu[ ^a_uqO^`oYQ6R—^\‚3Q6T#eOgj-Umsml^’[bTjVpdfe\opoY-Š[WQŠye\d`VpoYVY[be\[b^TWQ6oYoxd`QO_u[WTWQ6oVY_}opVp^atzQOy@eOVpoY^`TWQO_zt:gbeOXO^ky¡ 2 0 0.1 0.2 0.3 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0 0.05 0.1 Wing xcoordinate 3D Wing configuration Wing ycoordinate Wing zcoordinate 0 0.1 0.2 0.3 0.4 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0 0.05 0.1 Wing xcoordinate 3D Wing configuration Wing ycoordinate Wing zcoordinate 0 0.1 0.2 0.3 0.4 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 Wing xcoordinate 3D Wing configuration Wing ycoordinate Wing zcoordinate @VpX6tJTW^  ˆ nQ6TWRJZ:Vp_:Xl*7R:eO_  2 M @oYVpX6ZS[-7_:eOmlVYd`g BgW^’[QOy<(cJ-U_„eOmsVpdaghe\gX6^a_:^`TWe\[b^acHyQOT[bZ:^gWR:eO_Hq\eOT-UVp_:XseOVYTWd`TWe\y›[!t:gjVp_:X DQ6TW_:eOc:Q:2#!Z:^cJ-U_„eOmsVpdag!h>^aTW^opVp_J^feOTjVpŽ`^acze ] Q6tJ[evgj[WTbeOVYX6ZS[eO_:c}op^’q6^`oœ„VYX6ZS[#d`Q6_Jc:VY[WVpQ6_ h#Vƒ[bZ¢ ‰ 1 e\_:c H ˆrM B+S2#!Z:^eOVYTWdaTbe\y›[#VYg!Q ] gW^aTjqO^`cn[WQ ] ^e\[![WTWVpmVp_H@VYX  M e\[e gjR„eO_HQOy ‘O‘ damn2 !Z:^c„e\mlR:VY_:XŠTbe\[WVpQŸyQ6T[WZ:^gWZ:QOTj[#RC^`TWVYQ7c}msQ7cJ^dfeO_ ] ^Q ] gW^aTjq6^ac[bQ ] ^c:^`daTW^fe\g” VY_:XvVp_H@VYX  Œ eOg![bZ:^gjR„eO_HVp_:daTW^fe\gW^`ga2!ZJ^`TW^Vpg#evc:^ad`Tj^feOgjVp_:XsgWoYQ6R—^VY_:c:VYdfe\[WVp_:XŠeO_ eOg-UmlRJ[WQO[bVYdeOR:R:TjQSeOd~Z}[bQse ] QOt:_:c:^acc„eOmsR:VY_:XvTbe\[WVpQ:2 Œ7ˆ

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ŒSM 60 70 80 90 100 110 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 Span (cm)Moment (Nm) @VpX6tJTW^  M 'Vƒ[bd~Z:VY_:XsnQ6ms^`_S[#(+eOTjVe\[WVpQ6_ 60 70 80 90 100 110 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.85 0.86 Span (cm)Damping Ratio @VpX6tJTW^  Œ *7Z:Q6T['A^`TjVpQUcHeOmlRJVp_:XŠ"e\[bVYQn(+eOTjVe\[WVpQ6_ !Z:^_„e[bt:TWeOo yTW^aUtJ^`_:d’-nQ\yA[WZ:^gWZ:Q6T[R—^aTWVpQUczmsQ7c:^g[b^`eOc:Vpoƒ-€VY_:d`Tj^feOgj^`g#Vp_HeO_neOopmsQ6g[ oYVp_:^`eOT>yeOgjZ:VpQ6_}VY_}@VpX  P 2!!Z:Vpg!daQ6TWTj^`gWRCQ6_:cJg[WQs[WZ:^^’‡7R—^ada[W^`ccJ^`gj[be ] VYopVpŽaVp_:Xn^a—^`da[eO_ VY_:d`Tj^feOgj^VY_zh#Vp_:XseOTj^feŸh>QOt:opcHZ„efq6^h#Z:VpoY^Z:Q6oYc:Vp_JXŸ[bZ:^[~e\VpoxqOQ6opt:ms^d`QU^asdaVp^`_S[d`QO_:gj[beO_S[f2 !Z:^Z:QOTWVpŽaQ6_S[~eOo—[~eOVYo–g!e ] VYopVY[=-s[bQŠg[~e ] VYopVYŽ`^[WZ:^eOVpTjR:oeO_J^c:^`daTW^`eOgW^ageOgh#VY_:XŠVpg#^’‡U[b^a_:c:^ac 2 60 70 80 90 100 110 17 17.5 18 18.5 19 19.5 20 20.5 Span (cm)Natural Frequency (rad) @VpX6tJTW^  P *7Z:Q6T['A^`TjVpQUc„Tj^`ut:^`_Jda-z(+eOTjVe\[WVpQ6_

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Œ6Œ BopQUQ6ile\[![bZJ^oYVpX6ZS[WoY-lc„eOmsR—^acHRJZUtJX6Q6Vpc}msQ7cJ^TW^aqO^feOoYgeO_nQOR:R—QOgWVY[W^[bTj^`_:c ‚:h#ZJVpd~Z VYg!gWZ:Qrh#_VY_H@VpX  7‰ 2!_HVp_:daTW^`eOgW^Vp_zh#VY_:XŠgWR„eO_HVp_Jd`TW^`eOgW^ag#c„eOmsR:Vp_:XJ2!!Z:^yTW^`ut:^a_:da-HQOy [WZ:^R:Zut:X6Q6VYcHmsQUc:^Vp_:daTW^`eOgW^age\opQ6_:Xvh#Vƒ[bZz[bZ„e[#QOyA[WZ:^gWZ:QOTj[R—^aTWVYQ7c}msQ7c:^\‚ ] tJ[!h#Vƒ[bZne oYQh.^`T#Qrq6^aTeOopo d~Z:eO_:X6^\2 60 70 80 90 100 110 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Span (cm)Damping Ratio AVpX6t:Tj^  J‰U'0Zut:X6QOVpcHeOmsR:Vp_:XŸ"e\[WVpQŸ(+eOTjVe\[WVpQ6_ 60 70 80 90 100 110 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 Span (cm)Natural Frequency (rad) AVpX6t:Tj^  7‹J#'0Zut:X6QOVpcH„TW^`ut:^a_:da-}(+eOTjVe\[WVpQ6_ @VYX   gWZJQh#g#[bZJ^msVpXOTbe\[WVpQ6_lQOy@[WZ:^opQ6_:XOVY[btJc:Vp_„e\oCmsQUc:^`g#Qrq6^aT#[WZ:^TbeO_JX6^QOy@gjR„eO_:ga2 !Z:^gjZ:Q6Tj[RC^`TjVpQUcHmsQUc:^Vpg!msQ6gj[ex^ada[b^ac ‚:h#Z:VYop^[bZJ^RJZUtJX6Q6Vpc}msQ7cJ^t:_:c:^aTWX6QU^`g#opVƒ[W[Wop^ q\eOTjVe\[WVpQ6_€VY_z[b^`Tjmlg!Q\yeŠR—^aTWda^`_S[~eOXO^d~Z:eO_:X6^\2 _nVY_:d`Tj^feOgj^VY_zh#Vp_:XŠgWR:eO_ngj^`Tq6^`g[bQŠVp_Jd`TW^`eOgW^[WZ:^c:tJ[bd~ZnTWQ6oYoxmsQ7cJ^c„eOmsR:Vp_:XvTWe\[bVYQ eOg#c:^’[~eOVYop^aczVp_H@VpX  ‘ eO_:c}[bQŠc:^`daTW^`eOgW^VY[WgyTW^`ut:^a_:da-neOg#gj^`^`_HVp_H@VYX  “ 2qO^`Tbe\opo [WZ:^`Tj^VYg#escJ^`gj[be ] VYopVpŽaVp_:XŸe\x^ada[eOg[WZ:^R—Q6oY^`g!msQrq6^gWoYVpX6ZS[boƒ-l[WQrheOTjcn[WZ:^VpmleOX6Vp_:eOTj-se‡7VYg VY_nAVpX  ˆfN 2

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ŒOP 25 20 15 10 5 0 5 10 15 10 5 0 5 10 15 Im( l )Re( l ) @VpXOt:TW^   #'AQOop^nVpXOTbe\[WVpQ6_€Q\yQ6_:XOVY[btJc:Vp_„e\onQUc:^`g 60 70 80 90 100 110 0.6 0.65 0.7 0.75 0.8 Span (cm)Damping Ratio @VYX6t:Tj^  ‘ tJ[Wd~Z"#Q6oYo eOmsR:Vp_JXv"e[bVpQŸ(+e\TWVe[bVpQO_ 60 70 80 90 100 110 4.4 4.5 4.6 4.7 4.8 4.9 5 Span (cm)Natural Frequency (rad) @VYX6t:Tj^  “ #tJ[Wd~Z"#Q6oYo „TW^aUtJ^`_:d’-H(+e\TWVe[bVpQO_ h.QlcJVpgj[Wt:T ] eO_:d`^TW^=šj^`d’[bVpQO_HdaQ6_S[bTjQ6opoY^`TWgh.^`Tj^cJ^`gWVYX6_:^acnt:gjVp_:Xsfz § ˆ eO_:c fz § M 2eOVp_Jg#eOTW^X6Vƒq6^`_}VY_HAe ] oY^  ˆ 2#!VYml^Tj^`gjR—Q6_JgW^eO_:cHgWR„eO_Hc:^’œ„^`d’[bVpQO_nR:oYQO[bg eORJR—^`eOT!Vp_H@VYX  ˆ6ˆ 2#,.QO[bZHd`Q6_S[WTWQ6oYop^`TjgVpmsR:TjQqO^t:R—QO_}[WZ:^R:oe\_u[Wg#Q6R—^a_}opQUQ6RzTW^agWRCQ6_:gW^\2

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Œ ‰ 40 30 20 10 0 10 20 15 10 5 0 5 10 15 20 Im( l )Re( l ) @VYX6t:TW^  ˆrN #'DQ6op^nVYX6Tbe[bVpQO_zQOy)e\[W^`TWeOo)nQUc:^`g ^`msmke ˆ ^amlmle M / ‡ ‡"r ˆrN N ˆrN6NON6N Š ˆrN ˆrN6NON6N v ” ˆrN ” ˆrN6N6N6N ¤ ” ˆrN ” ˆrN6N6N6N @e ] op^  ˆ eOVY_:g>yQ6T!gjR„eO_}q\eOTj-UVp_JXscJVpgj[Wt:T ] eO_:d`^Tj^šj^`d’[bVYQ6_ndaQ6_S[bTjQ6opoY^`T 0 0.1 0.2 0.3 0.4 0.5 1 0 1 2 3 4 5 6 Time (s)Angle of Attack (deg) Fixed SpanLemma 1Lemma2 0 0.1 0.2 0.3 0.4 0.5 40 50 60 70 80 90 100 110 120 Time (s)Span (cm) Fixed SpanLemma 1Lemma2 @VYX6t:TW^  ˆ6ˆ #'0VY[Wd~ZH"^agWRCQ6_:gW^yQ6TnQOTWR:Z:VY_:Xl*7R„eO_

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'0#+" ‘ '')>0!!”<(A"#=,)|>,>+" ‘ 2 ˆ (^`Z:VYd`oY^^`gWdaTWVYRJ[bVYQ6_ !Z:^t:gWeOX6^QOy@ev[WZ:Vp_}t:_Jc:^`TjdfeOm ] ^`Tj^`ceOVpTyQ6VpoCgW^ada[bVYQ6_HmkeOi\^`g#VY[d`Q6_:gjVpc:^aTbe ] oY^`eOgWVY^`T[bQle\—^`d’[esd~Z„e\_:X6^Vp_}dfe\m ] ^aT`2# d`QO_uqO^`_S[bVYQ6_„eOoh#VY_:XŠZ„eOg ] Q\[bZ}t:R:RC^`TeO_:c oYQh.^`T#gjt:TjyeOda^`geOg!h.^`opo)e\gVY_S[b^`Tj_„eOo g[bTjt:da[Wt:TW^[bQŠmleO_:VpRJt:oe\[W^O21&HD(Bh#VY[WZeŸ[WZ:Vp_ t:_Jc:^`TjdfeOm ] ^`Tj^`cngW^`d’[bVYQ6_}dfeO_HZ„efq6^VY[Wg!dfeOm ] ^`TeOoƒ[b^aTW^`cHeOoYQ6_:XŸ[WZ:^^`_S[bVYTW^gWR„e\_nQOyD[bZJ^ h#VY_:Xvh#VY[bZHevgWVY_:X6oY^eOda[Wt„e\[bQOT`2!!Z:^[bZ:^aQ6TW^’[bVYdfeOoxh#VY_:XŠQ6_}[bZ:VYg#eOVpTjR:oeO_J^q\eOTWVY^`g!VY[Wg!dfeOm ] ^aT VY_nevR:Tj^`daVpgW^h>ef-HeOg![bZ:^dfe\m ] ^aT!R—^aTWda^`_S[~eOXO^d~Z„e\_:X6^`gVp_HevopVY_:^fe\Tmke\_:_:^`T “ 2#eOgeOR:R:TjQSeOd~Z:^ac&eOgevmŸt:oƒ[bV•”=c:VYgWd`VYR:opVY_„eOT-vc:^`gjVpX6_}Q6R7[bVpmsVpŽ`e\[bVYQ6_ R:TjQ ] oY^`mH‚7[bZJ^cJ^`gWVYTW^acngjZ„eORC^d~Z„e\_:X6^h>QOt:opcHmlQOgj[#opVYiO^aoY] ^^`_S[WVpTW^aoY-kc:VY—^`Tj^`_S[f2#!Z:VYg daQ6_:gWVYc:^`TWe\[bVYQ6_}Vpg ] ^a-OQ6_:c}[bZ:^gWdaQ6RC^QOyD[bZJVpg[bZJ^`gWVYg`‚„_:QOTcJQ7^ag#[WZ:^[bZ:^agWVYgVY_S[b^`_Jc}[WQlgjQ6oYqO^ esBR:TWQ ] op^`m ˆfN ‚ M6M 2 0 0.2 0.4 0.6 0.8 1 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 @VYX6t:TW^ ‘ ˆ nQ6TjR:Z:VY_:XleOm ] ^`T Œ ‹

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ŒS ‘ 2 M @oYVpX6ZS[-7_:eOmlVYd`g @VYX ‘ M Vpg#evR:oYQO[#QOyD[bZ:^R:Vƒ[bd~Z:VY_:XvmsQ6ml^a_S[!qOe\TWVe[bVpQO_kh#VY[WZnd`eOm ] ^`T!d~Z„eO_JX6^O2!Z:^`Tj^ VYg!_:QŠd~Z„eO_:X6^Vp_z[WZ:^R:VY[Wd~Z:Vp_:XvmsQ6ms^`_S[f‚JTW^`ut:VYTWVY_:Xv[bZ:^t:gj^Q\y@^aop^aq\e\[WQ6T[bQv[WTWVpmQ6tJ[#[bZJ^ eOVYTWR:opeO_:^\2!Z:Vpg!h.Q6t:oYcHVY_:c:VYdfe\[W^[bZ„e\[#d`eOm ] ^`T#Vpg!_JQO[eŠqUVpe ] op^d`Q6_S[bTjQ6o^a—^`d’[bQ6T!yQ6T!RJVY[bd~Z daQ6_S[bTWQOo–2!=_:g[b^fe\cVY[mkef] ^mlQOTW^t:gj^ayt:oDeOgevut„eOgWV•”=g[~e\[WVpdœ„VYX6ZS[#^`_Sq6^aopQ6RC^Q6RJ[bVYmlVYŽfe\[WVpQ6_ [W^`d~Z:_:VYut:^O2 8 10 12 14 16 18 1 0.5 0 0.5 1 1.5 % CamberMoment (Nm) @VpX6tJTW^ ‘ M 'Vƒ[bd~Z:VY_:XsnQ6ms^`_S[#(+eOTjVe\[WVpQ6_ eOm ] ^aTqOe\TWVe[bVpQO_:gZ„efq6^opVƒ[W[Wop^^’x^ada[Q6_}[bZ:^oYQ6_:X6Vƒ[bt:c:VY_„eOoCcJ-U_„eOmsVpdag`2#!Z:^gWZ:QOTj[ RC^`TjVpQUczVpg!op^`eOgj[e\—^`d’[b^ac ‚„gWZ:Qrh#VY_:Xld~Z:eO_:X6^agVY_Hc:eOmlRJVp_:XvTWe\[bVYQŠeO_:czyTW^aUtJ^`_:d’-nQ\y@Q6_JoY-ze y^’h…RC^`TWda^`_S[f2 8 10 12 14 16 18 0.825 0.83 0.835 0.84 0.845 0.85 Camber (%)Damping Ratio @VpX6tJTW^ ‘ Œ *7Z:Q6T['A^`TjVpQUcHeOmlRJVp_:XŠ"e\[bVYQn(+eOTjVe\[WVpQ6_ !Z:^R:Zut:XOQ6VpcHmlQUc:^gjZ:Qrh#gevgWoYVpX6ZS[WoY-lmlQ6Tj^mke\TWiO^acnd~Z:eO_:X6^\2_HVY_:d`Tj^feOgj^Vp_ d`eOm ] ^`T!d`QOTWTW^aoe\[W^`g![bQŠVY_:d`Tj^feOgj^`gVp_ ] QO[bZHc„eOmsR:Vp_JXvTbe\[WVpQHAVpX ‘ J‰ eO_:c}_„e\[Wt:TbeOo yTj^`ut:^`_Jda-@VYX ‘ 7‹ 2

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Œ6‘ 8 10 12 14 16 18 17.11 17.115 17.12 17.125 17.13 17.135 Camber (%)Natural Frequency (rad) @VpX6tJTW^ ‘ P *7Z:Q6T['A^`TjVpQUc„Tj^`ut:^`_Jda-z(+eOTjVe\[WVpQ6_ 8 10 12 14 16 18 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Camber (%)Damping Ratio AVpX6t:Tj^ ‘ J‰U'0Zut:X6QOVpcHeOmsR:Vp_:XŸ"e\[WVpQŸ(+eOTjVe\[WVpQ6_ 8 10 12 14 16 18 0.9 1 1.1 1.2 1.3 1.4 Camber (%)Natural Frequency (rad) AVpX6t:Tj^ ‘ 7‹J#'0Zut:X6QOVpcH„TW^`ut:^a_:da-}(+eOTjVe\[WVpQ6_ @VYX ‘  c:VYgWR:oper-Ug[WZ:^gWZ:Vƒy›[#Vp_}R—QOop^c:t:^[WQleŠRC^`TWda^`_S[dfeOm ] ^`T!VY_:d`Tj^feOgj^O2#=_:daTW^fe\gWVp_JX [WZ:^dfeOm ] ^`T!cJ^`gj[be ] VYopVpŽa^`g[WZ:^gWZ:Q6T[R—^aTWVpQUc}mlQUc:^e\g#[WZ:^mlQrqO^evmsVp_utJ[W^eOmlQOt:_S[ [WQh>eOTjcH[WZ:^VpmleOX6Vp_:eOTj-le‡7Vpga2#!Z:^R:Zut:X6Q6VYcHmsQUc:^Vp_z[btJTW_ ] ^ad`Q6ms^`g#gWoYVpX6ZS[WoY-lmlQ6Tj^ g[~e ] oY^O2#!Z:^aTW^Vpg!Z:Qrh.^aqO^`T_JQkeORJR:TW^ad`Ve ] op^^’x^ada[QOyAdfeOm ] ^`T#d~Z„e\_:X6^Q6_}[bZ:^opQO_:X6VY[Wt:c:VY_„eOo msQ7cJ^`g`2

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Œ6“ 25 20 15 10 5 0 5 10 15 10 5 0 5 10 15 Im( l )Re( l ) @VpXOt:TW^ ‘  #'AQOop^nVpXOTbe\[WVpQ6_€Q\yQ6_:XOVY[btJc:Vp_„e\onQUc:^`g _nVY_u[W^`Tj^`gj[WVp_:XŠTj^`gWtJoY[#Vp_z[WZ:^gWZ:Vƒy›[#Vp_}oe\[W^`TWeOo cJ-U_„eOmsVpdag!Vpg[WZ:^ ] ^aZ„efq7VYQ6T!QOyD[bZ:^ c„e\mlR:VY_:XvTbe[bVpQvVYg!Q ] gj^`TjqO^`c&VY_}@VpX ‘ ‘ 2#<[#Vp_:Vƒ[bVpeOopoƒ-Šc:^`daTW^fe\gW^`gR:TjVpQ6T[WQsTj^feOd~ZJVp_:Xle msVp_:VYmŸt:meO_:c}[bZJ^`_HX6Q7^ag#Q6_z[bQŠVp_:daTW^`eOgW^\2!Z:^R:oYQO[#QOyD[bZ:^_„e[bt:TWeOoxyTW^aut:^`_:d’-HVp_ @VYX ‘ “ gWZJQ7^ageŠopVY_:^fe\TVp_:daTW^`eOgWVY_:XŸ[bTj^`_:c 2 8 10 12 14 16 18 0.63 0.635 0.64 Camber (%)Damping Ratio @VYX6t:Tj^ ‘ ‘ tJ[Wd~Z"#Q6oYo eOmsR:Vp_JXv"e[bVpQŸ(+e\TWVe[bVpQO_ 8 10 12 14 16 18 5 5.2 5.4 5.6 5.8 6 Camber (%)Natural Frequency (rad) @VYX6t:Tj^ ‘ “ #tJ[Wd~Z"#Q6oYo „TW^aUtJ^`_:d’-H(+e\TWVe[bVpQO_

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PSN eOm ] ^aT!mlQ6TjR:Z:VY_:XvZ„eOg#opVƒ[W[Wop^^’x^ada[Q6_z[bZ:^ope\[b^aTbeOo c7-7_:eOmlVYd`g!QOyD[bZJ^e\VpTWRJoeO_:^ eOoYgWQ:2!!ZJ^TjQ6opoxmlQUc:^Vpg!cJ^`gj[be ] VYopVpŽa^`c ] -HevgWmleOopoxeOmsQ6t:_S[f‚Jh#ZJVpop^[WZ:^c:tJ[Wd~ZTWQOopoxmsQUc:^ ] ^`daQ6ml^ag#mlQ6Tj^gj[be ] op^[WQŠgWQ6ms^c:^`X6Tj^`^\2_HVY_:d`Tj^feOgj^VY_zdfeOm ] ^aTmsQrq6^ag#[WZ:^gWR:VYTbeOoxR—Q6oY^ efh>er-zyTjQ6m[WZ:^VpmleOX6VY_„eOTj-le‡7VYg`‚:mleOiuVp_:Xv[WZ„e\[#msQ7c:^-O^a[msQ6TW^t:_Jgj[~e ] op^\2 40 30 20 10 0 10 20 15 10 5 0 5 10 15 20 Im( l )Re( l ) @VYX6t:TW^ ‘ ˆrN #'DQ6op^nVYX6Tbe[bVpQO_zQOy)e\[W^`TWeOo)nQUc:^`g h.QlcJVpgj[Wt:T ] eO_:d`^TW^=šj^`d’[bVpQO_HdaQ6_S[bTjQ6opoY^`TWgh.^`Tj^cJ^`gWVYX6_:^acnt:gjVp_:Xsfz § ˆ eO_:c fz § M 2eOVp_Jg#eOTW^X6Vƒq6^`_}VY_HAe ] oY^ ‘ ˆ 2#!VYml^Tj^`gjR—Q6_JgW^eO_:cHgWR„eO_Hc:^’œ„^`d’[bVpQO_nR:oYQO[bg eORJR—^`eOT!Vp_H@VYX ‘ ˆ6ˆ 2!!ZJ^ M d`QO_u[WTWQ6oYop^aTWg!d`oYQ6gW^aoY-kmke\[Wd~Zz[bZ:^Q6RC^`_:oYQ7QORHTj^`gjR—Q6_JgW^QOy@[WZ:^ eOVYTWdaTbe\y›[`‚JmleOiuVp_:XvQ6_JoY-zeŠgWmleOoYoxVpmsR:TjQqO^`ms^`_S[f2 ^`msmle ˆ ^amlmle M / ‡ ‡"r ˆrNON ˆ ‰ N6N ˆrN v ˆrN6N6NON ˆrN6N v N ” ˆrN ¤ ” ˆ ” ˆrN Ae ] oY^ ‘ ˆ eOVp_:gyQOT!dfeOm ] ^`Tq\eOT-UVp_:XŠc:VYgj[btJT ] eO_Jd`^TW^=šj^`da[WVpQ6_}daQ6_S[bTWQOopop^aT 0 0.1 0.2 0.3 0.4 0.5 1 0 1 2 3 4 5 6 Time (s)Angle of Attack (deg) Fixed CamberLemma 1Lemma2 0 0.1 0.2 0.3 0.4 0.5 6 8 10 12 14 16 18 Time (s)Camber (%) Fixed CamberLemma 1Lemma2 AVpX6t:Tj^ ‘ ˆOˆ #'0VY[bd~Zn"#^`gWRCQ6_:gj^yQOTnQ6TWRJZ:Vp_:XleOm ] ^aT

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'0#+" “ '+'>0!#”–(@"=,)" “ 2 ˆ (^`Z:VYd`oY^^`gWdaTWVYRJ[bVYQ6_ !Z:^qO^`Z:VYd`op^R:TjQ6RCQ6gW^acnZ:^aTW^Z„eOgevR—QOTj[bVYQ6_zQOyD[bZ:^h#VY_:Xvh#Z:Vpd~ZHgWoYVpc:^agQOtJ[#Vp_}Q6Tjc:^`T [WQsVY_:d`Tj^feOgj^[bZ:^h#Vp_JXle\TW^fe7‚„mŸt:d~ZHopVpi\^[bZ:^gjR„eO_}qOe\Tj-UVp_:XŠd`eOgW^\2!ZJVpg#Vpg!_:Q\[tJ_:opVYiO^e „Qrh#op^aTœ3eORH^’‡7d`^aRJ[yQOT[bZ:^X6Tj^fe\[W^`T!Tbe\_:X6^QOyAmlQO[WVpQ6_x2!Z:Vpg&HD(d`QO_:d`^aRJ[Z:eOg![bZ:^ d`eOR„e ] VYopVƒ[=-l[WQscJQ6t ] oY^VY[bg!d~ZJQ6TWcnop^`_JXO[bZ 2 0 0.1 0.2 0.3 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0 0.05 0.1 Wing xcoordinate 3D Wing configuration Wing ycoordinate Wing zcoordinate 0 0.1 0.2 0.3 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0 0.05 0.1 Wing xcoordinate 3D Wing configuration Wing ycoordinate Wing zcoordinate 0 0.1 0.2 0.3 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0 0.05 0.1 Wing xcoordinate 3D Wing configuration Wing ycoordinate Wing zcoordinate @VYX6t:TW^ “ ˆ nQ6TjR:Z:VY_:Xl>Z:Q6TWc “ 2 M @oYVpX6ZS[-7_:eOmlVYd`g @VYX “ M R:oYQO[bg[WZ:^R:VY[Wd~Z:Vp_:XŠmsQ6ms^`_S[#eOXSe\Vp_:g[eŠd~Z:eO_:X6^Vp_}d~Z:QOTWc 2<[#dfeO_ ] ^_JQO[b^ac [WZ„e\[eŠgWVpXO_:VY„d`eO_S[!msQ6ml^a_S[#Vpg!d`Tj^fe\[W^`c ] -€mlQ6TjR:Z:VY_:XŸ[bZ:^d~Z:QOTWc 2!Z:Vpg!mleOi\^`ggW^`_JgW^eOg#VY[ VYg!^’‡7R—^ada[W^`cn[WZ„e\[evoeOTjX6^d~Z„eO_JX6^Vp_z[bZ:^d~Z:QOTWc}h>Q6tJopcHZ„efq6^R:TWQO_:Q6t:_:da^`cn^a—^`da[Q6_z[bZJ^ R:Vƒ[bd~Z:VY_:XŠmlQOml^a_u[`2!!Z:^_:^`t7[bTbe\oR—Q6VY_S[#QOyeŸh#VY_:XŠt:gWt„e\opoY-kopVY^`g#e\[e ] Q6tJ[![WZ:^‰ N X6TWefq7Vƒ[=Tj^`mleOVp_:g#e[#TWQ6t:XOZ:oY-l[bZ:^gbe\ml^RCQ6Vp_S[`‚:c:^`RC^`_Jc:Vp_:XsQ6_z[bZJ^mleOgjgQ\yA[WZ:^h#Vp_:XŠTW^aoe\[WVYqO^[bQ [WZ:^TW^`g[QOyA[WZ:^eOVpTyTbeOms^O2 !Z:VYg![bZ:^aml^QOyAoe\TWX6^oYQ6_:X6Vƒ[bt:c:VY_„eOoC^a—^`d’[bg ] -€d~Z:Q6TWc}q\eOTWVpe\[bVYQ6_:gdaQ6_S[bVY_UtJ^`gVp_z[bZ:^ cJ-U_„e\mlVYd`g`2@VYX “ Œ gWZJQh#gZ:Qrhr[WZ:^gWZ:QOTj[R—^aTWVYQ7czc„e\mlR:VY_:XvTbe[bVpQvut:VYd~iuoY-zVY_:d`Tj^feOgj^`g#[bQse mle‡7VpmŸtJmQOyAQ6_:^e\[!h#ZJVpd~ZHR—Q6VY_S[#VY[ ] ^`d`QOml^ag#[=h.QsgW^`R:eOTbe\[W^d`Q6_Sq6^aTWX6^a_:d`^ag`2 !Z:^R:Zut:XOQ6VpcHmlQUc:^QOyD[bZJ^e\VpTWdaTbe\y›[Vpg#e\—^`d’[b^`c}[bQseŠop^agWgW^aT#^’‡U[b^a_u[ ] -Hevd~Z„eO_:X6^VY_ d~Z:QOTWc 2#!Z:^mlQUc„eOo—R:TWQOR—^aTj[bVY^`geOTj^gj[bVYopo msQ6Tj^gW^`_JgWVY[WVYqO^[WQv[bZ:VYgyQ6TWmQ\y@msQ6TWRJZ:Vp_:XŸ[WZ„eO_ PJˆ

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PuM 10 12 14 16 18 20 22 24 0.3 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 Chord (cm)Moment (Nm) @VpX6tJTW^ “ M 'Vƒ[bd~Z:VY_:XsnQ6ms^`_S[#(+eOTjVe\[WVpQ6_ 11 11.5 12 12.5 13 13.5 14 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 Chord (cm)Damping Ratio @VpX6tJTW^ “ Œ *7Z:Q6T['A^`TjVpQUcHeOmlRJVp_:XŠ"e\[bVYQn(+eOTjVe\[WVpQ6_ 10 12 14 16 18 20 22 24 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 Chord (cm)Natural Frequency (rad) @VpX6tJTW^ “ P *7Z:Q6T['A^`TjVpQUc„Tj^`ut:^`_Jda-z(+eOTjVe\[WVpQ6_ [WQŠ[WZ:^R:TW^’qUVpQ6t:g![=h.Q:2@VpX6g “ J‰ eO_:c “ 7‹ gWZJQhrVp_:daTW^`eOgW^agVY_}[WZ:^R:Zut:X6Q6VYc}c„eOmsR:Vp_:XvTWe\[bVYQ eO_JcHyTj^`ut:^`_Jda-6‚3Tj^`gWRC^`d’[bVƒq6^`oƒ-62 !Z:^msQrq6^`ms^`_S[QOyD[bZ:^opQO_:X6VY[Wt:c:VY_„eOoCR—Q6oY^`g!c:tJ^[WQseŠd~Z:eO_:X6^Vp_}d~Z:Q6TjcnVYg!X6VYqO^`_ VY_nAVpX “  2!!Z:^gWZ:Q6T[R—^aTWVpQUc}mlQUc:^ ] Tj^feOiug#c:Qrh#_VY_S[bQv[=h>QsgW^aR„eOTWe\[b^d`QO_uqO^`TjX6^`_:da^`ga‚ Q6_J^Q\yAh#Z:VYd~ZndaQ6_S[bVY_UtJ^`g#[bQŠX6TWQrhrmlQ6Tj^gj[be ] oY^O‚:h#VY[WZz[bZ:^QO[WZ:^`T#msQqUVY_:XŸ[bQrh>eOTWc}[bZ:^ VYmkeOXOVp_„eOT-le‡7Vpg`2#!Z:^R:Zut:XOQ6Vpc}msQ7c:^XOTWQrh#g#mlQ6Tj^gj[be ] op^h#Vƒ[bZHeO_}Vp_:daTW^`eOgW^Vp_}d~Z:QOTWc 2

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PSŒ 10 12 14 16 18 20 22 24 0.005 0.01 0.015 0.02 0.025 0.03 Chord (cm)Damping Ratio AVpX6t:Tj^ “ J‰U'0Zut:X6QOVpcHeOmsR:Vp_:XŸ"e\[WVpQŸ(+eOTjVe\[WVpQ6_ 10 12 14 16 18 20 22 24 15 20 25 30 35 40 45 Chord (cm)Natural Frequency (rad) AVpX6t:Tj^ “ 7‹J#'0Zut:X6QOVpcH„TW^`ut:^a_:da-}(+eOTjVe\[WVpQ6_ 40 30 20 10 0 10 20 15 10 5 0 5 10 15 20 Im( l )Re( l ) @VpXOt:TW^ “  #'AQOop^nVpXOTbe\[WVpQ6_€Q\yQ6_:XOVY[btJc:Vp_„e\onQUc:^`g @VYX6g “ ‘ e\_:c “ “ gWZ:Qrh…[WZ:^c„eOmsR:Vp_:XvTWe\[bVYQŠeO_:cz[bZJ^_:e\[bt:TWeOo yTW^aut:^`_:d’-}QOyD[bZ:^ c:t7[bd~ZTjQ6opo N msQ7c:^yQOopopQrh#VY_:X[WZ:^gbeOms^Q6gWdaVpopope\[bQOTj-sR„e\[j[b^`Tj_ ‚:h#VY[WZHesX6oYQ ] eOoYoY-Šc:^ad`Tj^feOgjVp_:X [WTW^`_Jc 2 )e\[W^`Tbe\o R—QOop^msVpX6TWe\[bVYQ6_kc:tJ^[WQlevd~Z„e\_:X6^Vp_zd~Z:Q6TjcVYg!X6VYqO^`_}Vp_H@VYX “ ˆrN 2!!ZJ^TjQ6opo daQ6_Sq6^`TjX6^`_Jd`^nXOTWQrh#g#mlQ6Tj^gj[be ] op^eOg[WZ:^d~Z:Q6TjcVp_Jd`TW^`eOgW^ageO_:c}[bZ:^gWRJVpTbe\o c:VYqO^`TjX6^`_:da^ ] ^`daQ6ml^ag#^aq6^a_oY^`gWg#g[~e ] oY^O2

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P6P 10 12 14 16 18 20 22 24 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Chord (cm)Damping Ratio @VYX6t:Tj^ “ ‘ tJ[Wd~Z"#Q6oYo eOmsR:Vp_JXv"e[bVpQŸ(+e\TWVe[bVpQO_ 10 12 14 16 18 20 22 24 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 Chord (cm)Natural Frequency (rad) @VYX6t:Tj^ “ “ #tJ[Wd~Z"#Q6oYo „TW^aUtJ^`_:d’-H(+e\TWVe[bVpQO_ 40 30 20 10 0 10 20 15 10 5 0 5 10 15 20 Im( l )Re( l ) @VYX6t:TW^ “ ˆrN #'DQ6op^nVYX6Tbe[bVpQO_zQOy)e\[W^`TWeOo)nQUc:^`g h.QlcJVpgj[Wt:T ] eO_:d`^TW^=šj^`d’[bVpQO_HdaQ6_S[bTjQ6opoY^`TWgh.^`Tj^cJ^`gWVYX6_:^acnt:gjVp_:Xsfz § ˆ eO_:c fz § M 2eOVp_Jg#eOTW^X6Vƒq6^`_}VY_HAe ] oY^ “ ˆ 2#!VYml^Tj^`gjR—Q6_JgW^eO_:cHgWR„eO_Hc:^’œ„^`d’[bVpQO_nR:oYQO[bg eORJR—^`eOT!Vp_H@VYX “ ˆ6ˆ 2HQ6TWR:ZJVp_:XŠd~Z:Q6TjcnZ:eOg![bZ:^X6Tj^fe\[W^`gj[eOt7[bZ:Q6TjVY[=-zQrqO^`T#R:VY[Wd~ZnTj^`gjR—Q6_JgW^ QOy@eO_S-zQOyD[bZJ^[=-URC^`gQOyAmlQ6TjR:Z:VY_:Xvd`Q6_JgWVpcJ^`TW^acVY_}[WZ:Vpg[WZ:^`gjVpga2

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P ‰ ^`msmke ˆ ^amlmle M / ‡ ‡"r ˆ N ˆrN v ˆrN6N ˆrN v ” ˆ ” ˆrN ¤ ” ˆ ” ˆrN @e ] op^ “ ˆ eOVY_:g>yQ6T!d~ZJQ6TWc}q\eOTj-UVp_JXscJVpgj[Wt:T ] eO_:d`^Tj^šj^`d’[bVYQ6_ndaQ6_S[bTjQ6opoY^`T 0 0.1 0.2 0.3 0.4 0.5 1 0 1 2 3 4 5 6 Time (s)Angle of Attack (deg) Fixed ChordLemma 1Lemma2 0 0.1 0.2 0.3 0.4 0.5 10 15 20 25 Time (s)Chord (cm) Fixed ChordLemma 1Lemma2 AVpX6t:Tj^ “ ˆOˆ #'0VY[bd~Zn"#^`gWRCQ6_:gj^yQOTnQ6TWRJZ:Vp_:Xl>Z:QOTWc

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""+* ˆ` ] c:tJopTbe\Z:VpmH‚„2ƒ‚CeOTWdaVeJ‚J2ƒ‚3eO_:cn)VY_:c ‚„"2ƒ‚W@oYVpX6ZS[#>Z:eOTbeOd’[b^aTWVpg[bVYd`g#QOyD{8Vp_:X *7Z:eOR:Vp_JXŠyQOT#eknVYd`TjQsVpT>(^aZ:Vpdaop^h#Vƒ[bZn^`m ] TbeO_:^{8VY_:X6g`‚£’‰¤C¦§¨£¨¨ „@U’C+¨6›¦’W/2 M 2„mlRJTWVpiuVYc:Vpg!e\_:c&4J2!2C>Q7QOR—^aT`‚W^’q6^aopQ6R:ms^`_S[QOy*7mleOTj[*UR„eOTWg#yQ6T!da[bVƒq6^ ^`TWQU^`opeOgj[WVpd*U[WTWt:d’[bt:Tj^`ga‚-¨k£/¨k¨*.7¦§&\C+‰%§0OC7+d’+y’‚:=” MONON6Œ ” ˆrO“6“ ‚JQ6TyQ6opi3‚7(An‚„R:TWVYo MON6NOŒ 2 Œ\ 4J23,>oYQ6_:c:^`eOt ‚34J23"#Vpd~ZJ^`gWQO_eO_JcŸ24J2„'0Vp_:^ag`‚+W^agWVYX6_ ‚„^’q6^`oYQ6R:ms^`_S[eO_:c}D^`gj[WVp_:X QOy@elnQ6TWRJZ:Vp_:XŠgWRC^`da["e\[bVYQn{8Vp_:XŠt:gjVp_:XseO_H=_Jœ3e[~e ] oY^A^aop^agWd`QOR:Vpd*7R„e\T`‚-¨£¨¨ *.7¦§R!\C+‰%§OC7+d’+y’‚:=” M\N6N6Œ ” ˆ7ˆr‘ ‚JQ6TyQ6opi3‚U(@‚„R:TjVpo MONON6Œ 2 PO ,>QUQO[WZ:^O‚3/v2ƒ‚„@VY[WŽ`R„e[bTWVYd~i3‚„/v2Y‚3eO_JcnVY_:c ‚„"2Y‚+W>Q6_S[bTWQOopop^aTWg>yQOTVYgj[Wt:T ] e\_:d`^ "#^šj^ada[bVYQ6_}yQ6T#eŠ)VY_:^fe\T!=_:R:tJ[”<(+eOT-7VY_:Xl>oeOgjg#QOy0HQ6TWR:ZJVp_:XŠVYTWdaTbe\y›[`‚’’r‰.ŠW= n¨£¨¨™œ’¨Eœ’¨Eœ’¨Ek–*.7¦§n*Uj+&C+‰%§0OC7+Š’+y .’¦’F’‚:t:gj[WVp_ ‚:™n‚„R:TjVpo M\N6N ‰72 ‰ 4J23,>Qrh#mleO_ ‚3,2„*JeO_:c:^aTWgeO_:cH2:{^aVpgjgWZ„eOTa‚+Wq\eOopt„e[bVp_JXn[bZ:^=msR„eOd’[Q\y0nQOTWR:Z:VY_:X D^`d~Z:_JQ6opQ6XOVp^`g#Q6_}VpTWdaTbe\y›['D^`TyQ6TWmleO_:da^O‚".bW+›@Š6n¨£¨¨n*U¦§ *.7j+!\C+‰%OC7+d’+y<.’¦’F’‚J=” MON6NSM ” ˆ ‹ Œ7ˆ ‚J^a_Sq6^`Ta‚—!v‚ R:TWVYo MON6NSM 2 ‹ 'D23,>t:X6g ] -O‚›¦*(jP+&@)ur7OC""zu@£’n+’ndWIn¨6›¤ƒOF’‚ >TjQh#_,>QUQ6iug`‚3^’h+Q6Tji3‚3#v‚CneOTWd~Z MON6N6Œ 2  Ÿ2CeOcJQ6XSeO_ ‚J2C*7mlVƒ[bZ ‚J"23^a^e\_:c&*C2C*7dfeOT ] QOTWQ6t:XOZ ‚j=_Jœ„e\[~e ] op^eO_:cn"#VpX6VYc:VpŽ`e ] op^ {8VY_:Xl>Q6mlRCQ6_:^a_S[bg!yQ6T#9_JmkeO_J_:^`cH^aTWVe\ox(^`Z:VYd`oY^`g`‚-.~W+›@ŠkIn¨k£/¨k¨ *.7¦§(*.UW+&C+‰%OC7+Š’+y<.’¦’F’‚7=#” MONON6Œ ” ˆf‘6NJˆ ‚ Q6TjyQ6oYi3‚J(@‚„R:TjVpo M\N6N6Œ 2 ‘\ 4J2,23efqUVpc:gjQ6_ ‚3'A2C>ZSh>eOopQrh#gjiUV‚3eO_:cn,2*—23)eOŽ`Q6ga‚+W@oYVpX6ZS[!-U_„e\mlVYd*7Vpmnt:oe\[WVpQ6_ gWgW^agWgjml^a_u[QOy@eknQOTWR:Z„e ] op^#-7RC^`T”0opoYVpRJ[WVpdeOm ] ^`TW^ac&*UR„eO_}{8Vp_:X6^ac&>QO_J„X6t:TWe” [WVpQ6_ ‚-.~W+›@ŠkIn,¨k£/¨k¨*.U§n*Uj+&C+‰%OC+Š’+y .’¦’F’‚:=” MON6N6Œ ”‰ Œ6N7ˆ ‚JQ6TjyQOopi3‚J(An‚„tJX6t:gj[ MONON6Œ 2 “\ 23Tj^`opeJ‚0yQ6Vpox‹J2 “\P ‚l1*7QOy›[=h>eOTj^O‚CeOgjgbeOd~Zut:gj^a[j[bg=_:g[bVƒ[btJ[W^Q\y@D^`d~Z:_JQ6opQ6X\e\gWgbe\d~ZUtJgWgW^’[bga‚ MON6NJˆ 2 ˆrN\ *—23!2CeO_:Q:2„4J23#23"#^`_:eOt:c ‚C*—2C23,e\[bVYopo‚U2„AQrq\eOTa‚+b*7Z„eORC^RJ[bVYmlVYŽfe\[WVpQ6_syQ6T >Q6_7yQ6TWmsVp_:XvVpTyQ6Vpo‚.bW+›@Š6n¨£¨¨4œ’¨!œ’¨!œ’¨66–*.U§ P ‹

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Pu *.7j+!\C+‰%§OC7+d’+y‰.’¦’F’‚:= MON6N6Œ ” ˆ ‰ \“ ‚JQ6TjyQOopi3‚J(An‚ R:TWVYo MON6N6Œ 2 ˆ6ˆ` 2CeOTjd`VpeJ‚„2„ ] cJt:opTWeOZ:VpmH‚:e\_:cH"23Vp_Jc ‚W"#Q6opo >Q6_S[bTjQ6o yQ6T#esnVpdaTWQŠVYT(^`Z:VYd`oY^ t:gjVp_:Xsd’[bVƒq6^{8Vp_:XlnQOTWR:Z:VY_:X:‚->bW+›@ŠkIn,¨k£/¨k¨n™SOF§+uS+.– OC".nb@.’¦’F’‚:=” M\N6N6Œ ”‰ ŒOPu ‚J*JeO_H_S[WQ6_:VpQJ‚:™Ÿ‚ MONON6Œ 2 ˆM 'D2v2„QOmlRCQ6gWVƒ[b^e[b^`TjVeOoYg>yQ6T nVYd`TjQsVpT(^aZ:Vpdaop^`gj->bW+›@Š6nk *%*™f™¨+OF*C’n.Š,+Š’+y OC7.~.+›F’›j¨„nJ+R.’¦’F’‚J)QO_:Xs,>^feOd~Zx‚x>‚CefMON6N7ˆ 2 ˆrŒ\ 4J2C46eOd~i\Qrh#gWiuV–‚—/v2„,>QUQO[bZ:^\‚3"2:)VY_:c ‚„e\_:c'D2:TWeOt:_:gjd~ZSh>^aVpX:‚—^`TWmleO_S-6‚„4Ot:oYM\N6NOP 2 ˆfPO 2Ÿ2C4OQ6Z:_:g[bQ6_ ‚CŸ2n2„^feOo‚„02Ÿ27{8VpX6X6VY_:g`‚J223"#Q ] ^`Tj[WgWZ„efh‚„{B22CeOgjQ6_ eO_JcŸ24J2:=_:mleO_ ‚+jHQ7c:^ao [bQl>Q6msR„eOTj^[WZ:^@opVYX6ZS[#>Q6_S[WTWQ6o)0_:^aTWXO-H"#^`ut:VYTW^’” ms^`_S[bg#QOy0nQ6TjR:Z:Vp_JXle\_:cn>Q6_Sq6^`_S[WVpQ6_„e\opoY-€da[Wt„e\[W^`c}{8Vp_:X6ga‚-.bW+›@ŠEn ¨£¨¨™œ’¨Eœ’¨6k?¨S*¤.+6*.U‰.’¦’F’‚J=” MON6N6Œ ” ˆ7ˆ ‹7‚:QOTjyQ6oYiC‚ (An‚„t:X6t:g[ MONON6Œ 2 ˆ ‰ 4J23/^`ZJQ7^\‚3"23e\t:gW^’-6‚—"2„VY_:c ‚„eO_Jcn24J23/t:Tjc:VpopeJ‚(^aZ:Vpdaop^t:gjVp_:XŠ(VYgWVpQO_7”,e\gW^`c}„^a^`c ] e\d~iC‚"ƒ7¨6+.– .O¦’‚„QqO^`m ] ^`T MONONOP 2 ˆ ‹ 4J23/^`ZJQ7^\‚C4J2{B2CTWŽ’-Uh#_:eJ‚3"2Ce\t:gW^’-6‚—4J23'0op^ah‚C2„ ] c:t:opTWeOZ:VYmn‚J22—^`d~ZS] eJ‚ eO_Jc"2„Vp_:cx‚+{$ef-URCQ6Vp_S[efqUVpX6e\[bVYQ6_kyQ6TesnVpdaTWQŠVYT(^`Z:VYd`oY^t:gWVY_:Xv(VpgjVpQ6_U” ,e\gW^`cn[W[WVY[btJc:^0gj[bVYmke[bVpQO_ ‚C.bW+›@Š6n6Ub¤nWO"‘%’b&¨6›"%* .’¦’F’‚„,>TWeOt:_:gjd~ZSh>^aVpX:‚—^`TWmleO_S-6‚„4Ot:oYM\N6NOP 2 ˆ Ÿ23)VpeO_HeO_:cz{B2C*7ZS-u-6‚.j!Z:Tj^`^”Vpms^`_JgWVpQO_„eOo Aopt:VYc7”*U[WTWt:d’[bt:Tj^=_S[b^`TWeOda[WVpQ6_JgQ\ye n^am ] TWeO_:^{8Vp_:XŸyQ6TnVYd`TjQsVpT>(^aZ:Vpdaop^R:R:opVYdfe\[WVpQ6_Jg`‚C>bW+›@ŠkIn,¨k£/¨k¨ *.7¦§(*.UW+&C+‰%OC7+Š’+y<.’¦’F’‚7=#” MONON6Œ ” ˆr6M ‹J‚ Q6TjyQ6oYi3‚J(@‚„R:TjVpo M\N6N6Œ 2 ˆr‘\ "23Vp_:c ‚„2„ ] c:t:oYTbeOZ:VYmn‚7/v23,>QUQO[bZ:^\‚„eO_:cn'A2„Q6_S[bTjQ6o QOy@elnVpdaTWQŠVYT(^`Z:VYd`op^ag`‚->bW+›@ŠkInk4Ub¤nWO"‘%’bj¨6›-%* .’¦’F§‚:,.TbeOt:_JgWd~ZSh>^aVpX:‚x^`TWmleO_S-6‚„4Ot:oYM\N6NOP 2 ˆr“\ 2C4J2CeOTWut:^aŽO‚nP›FWO.n~@*Š*7’‚34OQOZ:_H{8VYop^’-zeO_:c*7Q6_Jg`‚3Q ] Q6i\^`_ ‚ 4J‚CefMON6N6Œ 2 MON\ 2Cn^aopVY_ ‚j (Q6T[b^’‡}De[W[bVYd`^n0#,|=msR:op^aml^a_S[~e\[WVpQ6_€yQ6T#VY_:^feOT>^`TjQ\” cJ-U_„e\mlVYd{8Vp_:XŠRJR:opVYdfe\[WVpQ6_:ga‚sneOgj[W^`T`g!Z:^`gjVpg`‚3"#Qr-SeOo=_Jgj[bVƒ[btJ[W^QOy@D^`d~ZJ_:Q6oƒ” Q6X\- /#’‚ M\N6N6N 2 M7ˆ` "223^`oYgWQ6_ ‚F(Pu@<"rP.sOCj¨68=@s+%.n~@‚„ndrTbefh”VpoYo–‚:^ah|+Q6TWi3‚ !v‚CeOTjd~Z ˆr“O“6‘ 2

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Permanent Link: http://ufdc.ufl.edu/UFE0008949/00001

Material Information

Title: Dynamic Modeling and Flight Control of Morphing Air Vehicles
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0008949:00001

Permanent Link: http://ufdc.ufl.edu/UFE0008949/00001

Material Information

Title: Dynamic Modeling and Flight Control of Morphing Air Vehicles
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0008949:00001


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Full Text











DYNAMIC MODELING AND


FLIGHT CONTROL OF MORPHING AIR
VEHICLES


KENNETH E. BOOTHE JR.















A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2004


































Copyright 2004

by

Kenneth E. Boothe Jr.
















I dedicate this work to God and my loving and supportive parents to whom I

owe all of my success.















ACKNOWLEDGMENTS

I would like to acknowledge the help and teaching of my professors, primarily

Dr. Lind
















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS .................. .......... iv

LIST OF TABLES ........ ..... .............. vii

LIST OF FIGURES ................. .. ............. viii

NOMENCLATURE .................. ............. x

ABSTRACT ......... . ........ . .... ...... xi

CHAPTER

1 INTRODUCTION ...... ............ . ... 1

2 MORPHING AIRCRAFT ......... ..... ........ 5

2.1 History ............. ..... ......... ...... 5
2.2 University of Florida Micro Air Vehicles .............. 6

3 MORPHING AIRCRAFT EQUATIONS OF MOTION ......... 11

3.1 Nonlinear Equations of Motion .................. 11
3.1.1 Angular Momentum ................. .. 11
3.1.2 Force Equations .................. . 13
3.1.3 Attitude and Angular Velocities .............. 14
3.2 Linearized Equations of Motion .................. 15
3.2.1 Angular Momentum ................. .. 15
3.2.2 Force Equations .................. . 17
3.2.3 Attitude and Angular Velocities .............. 18
3.3 Straight and Level Flight ....... .......... 19

4 MODELING ........ .... ......... ...... 22

5 CONTROL SYNTHESIS ................. ....... 24

5.1 Quasi-static Morphing .................. .. 24
5.2 Dynamic Morphing . . . . ... . 24

6 APPLICATION BASE MODEL . . . . . 27

6.1 Vehicle Description . . . . ... . 27
6.2 Modeling ........ . . .. . .. ... 29










6.2.1 Aerodynamic Modeling Tornado . . .... 29
6.2.2 Rigid Body Modeling-ProE .. . . . 29

7 APPLICATION-VARIABLE SPAN . . . . . 31

7.1 Vehicle Description . . . . ... . 31
7.2 Flight Dynamics . . . . ... . 31

8 APPLICATION-VARIABLE CAMBER . . . . 36

8.1 Vehicle Description . . . . ... . 36
8.2 Flight Dynamics . . . . ... . 37

9 APPLICATION-VARIABLE CHORD . . . . 41

9.1 Vehicle Description . . . . ... . 41
9.2 Flight Dynamics . . . . ... . 41

REFERENCES ...... . . .. .. ............... 46

BIOGRAPHICAL SKETCH .................. ......... 49















LIST OF TABLES
Table page

6-1 Longitudinal eigenvectors . . . . . . 27

6-2 Longitudinal eigenvalues . . . . . . 28

6-3 Lateral-directional eigenvectors . . . . . 28

6-4 Lateral-directional eigenvalues . . . . .. 28

6-5 Lateral-directional eigenvector . . . . .. 29

7-1 Gains for span varying disturbance rejection controller . . 35

8-1 Gains for camber varying disturbance rejection controller . 40

9-1 Gains for chord varying disturbance rejection controller . . 45















LIST OF FIGURES
Figure page

2-1 LIG-7 ......... .. ........ ...... 5

2-2 ICE Aircraft .......... .. .. ....... ..... 6

2-3 Boeing Dragonfly . . . . ..... .. 7

2-4 Virginia Tech BetaMax ......... . .. .... 7

2-5 Wing Curling MAV ........ .. ....... .. ....... 8

2-6 Multi Point Wing Shaping MAV .. . . . 9

2-7 AVCAAF Morphing Aircraft ........ . . .... 9

2-8 Variable Gull-Wing MAV . . . . . . 10

3-1 Earth and Body Reference Frames . . . . 12

6-1 Base Model ..... . . .. . .. ....... ..... 27

7-1 Morphing Span . . . . .... . 31

7-2 Pitching Moment Variation . . . . . . 32

7-3 Short Period Damping Ratio Variation .. . . . 32

7-4 Short Period Frequency Variation . . . . 32

7-5 Phugoid Damping Ratio Variation . . . . 33

7-6 Phugoid Frequency Variation . . . . . 33

7-7 Pole Migration of Longitudinal Modes . . . 34

7-8 Dutch Roll Damping Ratio Variation .. . . . 34

7-9 Dutch Roll Frequency Variation . . . . . 34

7-10 Pole Migration of Lateral Modes .. . . . . 35

7-11 Pitch Response for Morphing Span. . . . . 35

8-1 Morphing Camber . . . . ..... . 36

8-2 Pitching Moment Variation . . . . .. . 37










8-3 Short Period Damping Ratio Variation .. . . . 37

8-4 Short Period Frequency Variation . . . . 38

8-5 Phugoid Damping Ratio Variation . . . . 38

8-6 Phugoid Frequency Variation . . . . . 38

8-7 Pole Migration of Longitudinal Modes . . . 39

8-8 Dutch Roll Damping Ratio Variation .. . . . 39

8-9 Dutch Roll Frequency Variation . . . . . 39

8-10 Pole Migration of Lateral Modes . . . . . 40

8-11 Pitch Response for Morphing Camber .. . . . 40

9-1 Morphing Chord . . . . . ..... . 41

9-2 Pitching Moment Variation . . . . . . 42

9-3 Short Period Damping Ratio Variation .. . . . 42

9-4 Short Period Frequency Variation . . . . 42

9-5 Phugoid Damping Ratio Variation . . . . 43

9-6 Phugoid Frequency Variation . . . . . 43

9-7 Pole Migration of Longitudinal Modes . . . 43

9-8 Dutch Roll Damping Ratio Variation .. . . . 44

9-9 Dutch Roll Frequency Variation . . . . . 44

9-10 Pole Migration of Lateral Modes . . . . . 44

9-11 Pitch Response for Morphing Chord .. . . . 45
















NOMENCLATURE


Symbol Meaning
H angular momentum
t time
r position vector
L moment about the x axis
M moment about the y axis
N moment about the z axis
p angular rate about the x axis
q angular rate about the y axis
r angular rate about the z axis
u velocity in the x direction
v velocity in the y direction
w velocity in the z direction
roll angle
0 pitch angle
4i yaw angle
I moment of inertia
m mass
v velocity vector
a acceleration
cw angular velocity vector
F force
g gravity















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

DYNAMIC MODELING AND FLIGHT CONTROL OF MORPHING AIR
VEHICLES

By

Kenneth E. Boothe Jr.

December 2004

Chair: Richard C. Lind, Jr.
Major Department: Mechanical and Aerospace Engineering

The majority of airplanes in use today fly with a fixed shape and use conven-

tional control effectors such as elevators, ailerons, rudder. These control surfaces

are used in lieu of changing the entire shape of the airplane in an optimal manner

because a global change in geometry includes far more complexity, and possibly

an associated weight increase along with a decreased level of reliability. Advances

in materials and new ideas in the area of structures are opening up possibilities to

aircraft designers. These advances are ushering in morphing aircraft as a new class

of air vehicle. This prospect is being pursued and several morphing air vehicles are

already in various stages of development and flight testing. This emerging area of

study is initiating a need for dynamic models and control strategies to work within

its framework.















CHAPTER 1
INTRODUCTION

A flight vehicle is typically designed to function around a primary operating

point. This design point may be an efficient cruise for a transport while a fighter

may seek to optimize maneuverability and to increase top speed. Performance

and efficiency begin to suffer as the airplane moves to other portions of the flight

envelope. Advancements in materials science are offering the aerospace industry

some unique and exciting possibilities for future aircraft configurations that will

be able to address these ever present design trade-offs [14]. Emerging technologies

such as embedded actuators and shape memory alloys are on the horizon and

will benefit many engineering disciplines. The obvious aerospace application is to

have an airplane that is capable of changing its shape to either adapt to various

flight conditions or provide increased maneuverability [5]. This concept, known

as morphing aircraft, presents some new and exciting challenges to the aerospace

industry. There is ongoing research in the aerospace community dealing with the

many issues in this inherently multi-disciplinary arena.

Micro air vehicles (MAVs), a small-sized class within the general class of

unpiloted air vehicles (UAVs), are now more commonly coming under consideration

for carrying out existing missions as well as those that can only be completed by

a MAV. The dimensions of the aircraft coupled with its small mass afford it some

unique capabilities along with significant advantages over larger, more expensive

UAVs. A MAV can enter into and navigate through urban environments that a

conventionally sized UAV can not. Materials and labor scale down along with

the size, ensuring that the overall flyaway cost is lowered. The low mass and slow

forward speed of a MAV both contribute to lowering the kinetic energy absorbed









by the airframe during a crash. Crashes are almost always survivable or easily

repairable. An accident such as an actuator failure will most likely spell an end to

a UAV at a considerable cost; however, the same incident may not even incur any

damage on the inexpensive MAV.

These benefits are attracting attention from both military and civilian sectors.

This interest is translating into challenging research opportunities for the aerospace

community. One obstacle facing advancement in this area is that aerodynamics

on this scale are not well solved at this point. Viscous forces begin to dominate

in this Reynolds number regime and conventional inviscid analysis does not

adequately solve for the flow field and resulting pressure distribution. There is

also a hysteresis effect due to the unsteady boundary layer behavior during which

it separates and then reattaches periodically [17]. Add to this already difficult

subject by implementing morphing as a means of flight control and you present an

even greater challenge. Now there exists a multidisciplinary design problem that

lends itself to the coordination of several research teams. Aerodynamicists must

coordinate their efforts with structural engineers to model the pressure distribution

and design a wing to yield desired mode shapes. The dynamics of morphing micro

air vehicles (MMAVs) must be solved before any sophisticated control theory can

be applied.

Work is currently underway at the University of Florida to investigate the

flight characteristics of a variety of MMAVs. Flight testing is a primary part of

the design methodology [1]. Iterative design methods teach lessons on what works

but not necessarily why it works. A common outcome of this procedure is first

identifying a promising configuration and then going about trying to explain why

it works well. A good example of this is the flexible wing concept. It was observed

that MAVs with wings made of latex flew better than those with rigid wings. Then









a combinational approach was taken to solve for the aerodynamics as well as the

structural dynamics [17].

System identification and parameter estimation can be arduous and expensive

undertakings. Wind-tunnel testing can cost valuable man-hours and the accuracy

of the data may be compromised by errors in calibration and implementation.

Flight testing is a promising method with which to gather open-loop dynamic data.

The problem with employing this technique on a MAV is the payload restrictions of

a vehicle this size. The University of Florida MAV program has not yet advanced

its flight testing program to the stage where full state feedback is achievable. Most

importantly, the ability to measure angle of sideslip, angle of attack, and airspeed is

not present. These modeling problems facing the MAV program lead naturally to

the remaining option of analytical methods utilizing computational fluid dynamics

(CFD).

The use of CFD independently of, or in concordance with, experimental data

can expedite the generation of mathematical models [31]. An example of this

cooperative modeling was done for the "active vision control for agile autonomous

flight" vehicle (AVCAAF). A combination of static stability derivatives from wind

tunnel testing and dynamic derivatives from Tornado was used to generate a full

set of longitudinal and lateral dynamics for AVCAAF [13]. CFD is also useful

in the preliminary design phases of new aircraft. This is the case of the three

proposed morphing aircraft considered in this thesis. They have not been built and

thusly can only be studied in terms of analytical methods.

The flight dynamics models of the MAVs here at the University of Florida

have not fully matured. Design and analysis has always been driven by an iterative

qualitative flight testing process. MAVs have also been controlled in large by

a pilot in the loop and the limited amount of autopilot development has been

achieved by a hand tuned PID type controller. An accurate mathematical model









must exist if the AVCAAF program is to advance the level of sophistication of

controls research.

This thesis investigates the characterization of morphing aircraft and control

laws to actively command the morphing. First the equations of motion are derived

for an arbitrary morphing airplane. Small disturbance theory is used to linearize

these equations. Then the equations are decoupled into separate longitudinal and

lateral models and written in terms of stability derivatives that can be estimated

from CFD. CFD is then used to generate these longitudinal and lateral models

for three different morphing aircraft. The three aircraft morph by changing their

span, chord, and camber respectively. The models for these aircraft are generated

as linear input varying (LIV) functions. This allows the use of new LIV control

theories currently being developed at the University of Florida. Variations in

characteristics including modal properties and flight dynamics of these models are

examined and explained.















CHAPTER 2
MORPHING AIRCRAFT

2.1 History

The concept of morphing aircraft for flight control has been around as long as

aircraft themselves and was implemented on the Wright Flyer in the form of wing

warping. The Wright brothers observed birds and mimicked their wing twisting in

order to facilitate roll control [6]. The use of morphing as a means of expanding the

flight envelope also dates back to air history's early beginnings. A NACA report

was released in 1920 detailing the "Parker Variable Camber Wing" [24]. The idea

of this concept was to reduce the drag on the wing at higher speeds by changing

the wing profile. Another early morphing pioneer was Razdviznoe Krylo from

Russia. This company built the LIG-7 pictured in Figure 2-1 in 1932 which had an

articulated surface that extracted to increase lift during take-off and landing while

retracting for cruise [27].









Figure 2-1: LIG-7


Morphing soon gave way to conventional control surfaces and flight vehicle

configurations. Benefits of morphing mechanisms are often easily outweighed

by drawbacks encountered during their implementation. Many devices such as

shape memory alloys will provide an amazing bench top demonstration but are

restricted in usage by the excess weight of the associated electronics. As material









science works to make morphing structures more usable, aerospace engineers should

consider applications in anticipation of coming advancements [26, 7]. Such an

example of this is the Lockheed Martin Tactical Aircraft Systems Innovative

Control Effectors (LMTAS-ICE) concept vehicle shown in Figure 2-2. This

configuration is being considered to test the concept of novel control effectors in

improving weight, cost, stealth, and performance [23, 25].










Figure 2-2: ICE Aircraft

There are a limited number of morphing aircraft that have been built and

test flown. One of these is the Boeing Dragonfly UAV shown in Figure 2-3. This

unique rotor-craft transitions from hovering to forward flight by stopping its rotor

blades and fixing them so they can act as wings. The Dragonfly has been built

and has demonstrated hovering flight. Virginia Tech has designed, built, and

successfully flown a smaller morphing airplane called the BetaMax (Figure 2-4).

This airplane uses the same concept as the one discussed later here in Chapter

5. Another point of relevance for the BetaMax is that they used the same CFD

software as this thesis to estimate some aerodynamic coefficients and predict

performance parameters like range.

2.2 University of Florida Micro Air Vehicles

A number of morphing micro air vehicles have been designed, flight tested,

and studied at the University of Florida [18]. Primarily various types of wing

warping have been used as a means of roll control to achieve greater agility, which

is a crucial issue for MAVs [30, 29, 28]. Micro air vehicles have the ability to fly in
























Figure 2-3: Boeing Dragonfly


Figure 2-4: Virginia Tech BetaMax


small spaces that a large airplane can not. A common scenario under consideration

is maneuvering through an urban canyon environment. This concept requires that

a MAV be able to enter into and navigate through a maze of buildings to carry out

missions such as chemical detection or sensor emplacement [11].

Fig 2-5 shows a wing curling MAV. The wing is designed and built in such a

way that it can be morphed by simply pulling a small Kevlar thread attached to









the wing panel. This novel control effector caused a significantly greater roll rate

than the pre-existing conventional control surfaces [14].













Figure 2-5: Wing Curling MAV


Fig 2-6 shows another example of a MAV using wing twist as control effector.

This aircraft has a considerably more complicated morphing mechanism. The wing

is articulated in two separate sections to facilitate the study of a variety of complex

shapes. The goal is to have some measure of control over the wing lift distribution.

The wing on this aircraft is being actuated both symmetrically to quasi-statically

affect the spanwise lift distribution and asymmetrically for roll control.

The University of Florida's 'Active Vision for Control of Agile Autonomous

Flight' vehicle (AVCAAF) uses a proprietary vision based autopilot being devel-

oped at the University of Florida to achieve three dimensional waypoint naviga-

tion [15], [16]. The AVCAAF flight vehicle seen in Fig 2-7 employs yet another

variant of wing twisting. The AVCAAF has a adopted a simple mechanism that

twists the wing at the root and globally morphs the rest of the wing. This vehicle

s being developed for use by special forces and has the need to be somewhat more

durable than a lot of the more experimental MAVs being produced by the Univer-

sity of Florida [12]. AVCAAF is still in the early testing phases, but it appears as

though this form of wing morphing is superior to the previous conventional control

surfaces in affecting roll control.































Figure 2-6: Multi Point Wing Shaping MAV


p~c.t~


Figure 2-7: AVCAAF Morphing Aircraft


Fig 2-8 is a biologically inspired variable gull wing MAV that was developed

to mimic different wing configurations used by birds in various phases of flight.

The jack screw driven actuation system moves far to slowly too be used for flight

path control. In turn it is used in a quasi-static manner to explore possibilities of

flight envelope expansion and optimization. As the gull-wing angle is increased









in the positive direction, the vehicle becomes highly stable about the roll axis.

Additionally, this morphing position diminishes the glide angle considerably,

allowing the aircraft to descend at steep angles without increasing airspeed.


'- gc


Figure 2-8: Variable Gull-Wing MAV















CHAPTER 3
MORPHING AIRCRAFT EQUATIONS OF MOTION

The equations of motion for morphing aircraft must be derived to describe

maneuvering as the geometry changes. A simulation also needs to be created in

order to study openloop responses and to test control laws. The framework for

formulating vehicle dynamics from stability derivatives of a standard airplane

differs from that of a morphing airplane. Morphing structures require additional

considerations. The plant's functional dependence and the absence of a control

matrix, as in the case of dynamic morphing, must be taken into account.

3.1 Nonlinear Equations of Motion

Below a derivation of the equations of motion (EOM) for an arbitrary mor-

phing aircraft is performed. This is done in order to re-examine the assumptions

typically made for a conventional aircraft of a fixed geometry. Terms that nor-

mally drop out during the reduction of the equations due to the configuration of a

conventional airplane must be left intact to describe the dynamics of an arbitrary

morphing aircraft. A set of EOM are derived by analyzing Newton's laws, as doc-

umented in textbooks [21] [32], but also incorporating terms that account for time

varying shape. The B and E coordinate frames referenced during the derivations

are described in Fig 3-1

3.1.1 Angular Momentum

The aircraft moments are derived by setting the angular momentum equal to

the applied moments as in Eq 3.1


dH
= M (3.1)










Body Fixed
Frame


Inertial


Figure 3-1: Earth and Body Reference Frames


where


H = rx (mxV)

Eq 3.2 can be broken down further into three equations representing the

individual moments (L, M, N).


L = Hx + qHz rHy

M = Hy +rHx pHz

N = Hz+ pHy qHx


(3.2)


(3.3)

(3.4)

(3.5)


Eqs 3.6, 3.7, and 3.8 are the scalar equations for the

Hx = plx qlxy rlxz


HY = -plxy + qlv rlyz

Hz = -plxz qyvz + rlz

Substituting eqs 3.6, 3.7, and 3.8 into eqs 3.29, 3.30,

moment equations, eqs 3.3, 3.4, and 3.5.


moment of momentum.

(3.6)

(3.7)


(3.8)


and 3.31 gives the












L =plx + pIx 4Ixy qIxy rlxz rIxz qplxz q2IYZ

+ qrlz + rpIxy rqly + r2Iyz (3.9)



M = plxy pixy + 4Iy + qly rIyz riyz + rpIx rqlxy

r2 Xz + p2lXZ + pqlyz prIz (3.10)



N = p1Ixz plxz qIyz qlyz + rlz + rlz p2lXy + pqly

prIyz qp'x + q2IXY + qrlxz (3.11)



These morphing EOM are left with some unique terms. These terms include

all of the time variant moments of inertia I and the products of inertia Iyz and

Ixy which are left to account for possible assymitries about the XZ plane.

3.1.2 Force Equations

The derivation of the three aircraft force equations is carried out by imple-

menting Newtons 2nd law given in eq 3.12.


d(m) F (3.12)
dt

The assumption that the earth is an inertial reference frame is made due to

the fact that rotation rate of the earth is much slower than the angular rates of the

aircraft. Another assumption is that the aircraft is a rigid body. The equations

are usually expressed in the body coordinate frame (B) and mapped into the earth

frame (E) by the Euler angles (0, 0, 4b). Noting that Eq 3.12 is essentially F = ma,

Eq 3.13 can be used to compute the acceleration of the B frame in the E frame.

aE = VB +WB X VB (3.13)









The velocity is VB is given by Eq 3.14

v = ui + vj + wk (3.14)

where u,v, and w are the velocities in the x, y, and z body axes and the rate of

rotation of the body frame in the inertial frame is given by Eq 3.15

OB = pi + qj + rk (3.15)

Multiplying by the mass of the aircraft and separating the vector equation into

the three force equations yields Eqs 3.16, 3.17, and 3.18,

m(iu + qw rv) = F1, (3.16)

m(v + ru pw) = Fy, (3.17)

m(w + pv qu) = F, (3.18)

F, F,, and F, can be separated into gravitational (mg) and propulsive forces

(X, Y, Z). Performing this separation and applying a rotation transformation leads
to Eqs 3.19, 3.20, and 3.21.

X mgSo = m(ui + qw rv) (3.19)

Y + mgCOSO = m(v + ru pw) (3.20)

Z + mgCoCO = m(wb + pv qu) (3.21)

3.1.3 Attitude and Angular Velocities

The relationship between the angular velocities (p, q, r) in the body frame and

the Euler rates (4, 0, and b) are obtained by applying a sequence of rotations to

the aircraft and are given by Eqs 3.22, 3.23, and 3.24

p = (- Seo (3.22)

q = OCo + VCeSp (3.23)


r = tCeC9 3S2


(3.24)











0 = qCO rSp (3.25)

= p + qSTo + rCQTo (3.26)

={ = (qS + rCO)sec 0 (3.27)

3.2 Linearized Equations of Motion

Small disturbance theory is used to linearize the EOMs about an operating

condition. This is done by replacing all of the variables in the equations of motion

by a reference value plus a disturbance.

3.2.1 Angular Momentum

Substituting the equations in 3.28 into Eqs 3.9, 3.10, and 3.11


p = po + Ap p = AP

q = qo + A4 q = Aq

r = ro + Ar t = At

(3.28)


yields Eqs 3.9, 3.10, and 3.11

L =4jIx + (Po + Ap)Ix A 4lxy (qo + Aq)Ixy AtIxz (roAr)Ixz-

(qoPoAqpo + ApqoApAq)Ixz (q2 + 2Aqqo + Aq2)IYZ

(qoro + AqroArqo + ArAq)Iz + (ropo + Arpo + AproArAp)Ixy-

(roqo + Arqo + AqroArAq)Iy + (r + 2Arro + Ar2)IYZ


(3.29)











M = Apixy (Po + AP)Ixy + AryI + (qo + Aq)Iy Adlyz (ro + Ar)Iyz+

(roPo + Arpo + Apro)Ix (roqoAr-qo + Aqro)Ixy + (r2 + 2Arro)Jxz-

(p0 + 2Appo))Ixz + (r. + 2Arro + Ar2) Y



(3.30)


N =ApJlx + (Po + Ap)Ix ALxy (qo + Aq)Ixy AjIxz (roAr)ixz-

(qopoAqpo + /pqoApAq)Ixz (ql + 2Aqqo + Aq2)IYZ+

(qoro + AqroArqo + ArAq)Iz + (ropo + Arpo + AproArAp)Ixy-

(roqo + Arq0 + AqroArAq)Iy + (r2 + 2rr + Ar2 )IYZ



(3.31)

simplifying



L =Aplx Alxy Airxz + Ap(Ix qolxz + rolxy)+

Aq(-Ixy Polxz 2qolyz + rolz Trol)+
(3.32)
Ar(-Ixz + qolz +Polxy qoly + 2rolyz)+

Poix qolxy rolxz qIyz + qoro lz + ropolxy ro0qIy + l ry Iz


M =4AI AqIxY Arlxz + Ap(Ix 9q0xz + rolxy)+

Aq(-ixy Polxz 2qolyz + rolz roly)+
(3.33)
Ar(-Ixz + qolz + Polxy q0oy + 2rolyz)+

Poix qoIxy rolxz q llz + qorolz + ropolxy roIoly + rTIYZ












N =Aplxz A4Ixy AtIxz + Ap(Ix qolxz + rolxy)+

Aq(-Ixy Polxz 2qolyz + rolz Troy)+

Ar(-Jxz + qolz + Polxy qoly + 2rolyz)+

Polx _qoxy Tolxz q(IYZ + qorolz + roPolxy rTqoly + r IYz


(3.34)


3.2.2 Force Equations

Now the force equations are linearized by the same method, substituting the

values from 3.42 into Eqs 3.19, 3.20, and 3.21


Xo + AX

- o + AY

Zo + AZ

= Uo + An

= vo + Av

: + Aw


-AX

AY

AZ

Ait

Ai

Aw


(3.35)



Xo + AX mg sin(Oo + AO) = m[Ai + (qo + Aq) (wo + Aw) (ro + Ar) (vo + A)]

(3.36)



Yo+AY-mg cos(0o+AO) sin(qo+Aq) = m[Ai+(ro+Ar)(uo+Au)-(po+Ap)(wo+Aw)]

(3.37)



Zo+AZ-mg cos(o0+AO) cos(0>o+A5) = m[Ab+(po+Ap)(vo+Av)- (o+Aq)(uo+Au)]

(3.38)









Expanding


Xo + AX mg sin(Oo + AO) =m[Ai + (qowo + Aqwo + Awqo)-

(rovo + Arvo + Avro)]


(3.39)


Yo + AY mg cos(Oo + AO) sin(Oo + AO) =m[Ai + (rouo + Aruo + Auro)-

(po o + Apwo + Awpo)]

(3.40)

Zo + AZ my cos(0o + AO) cos(0o + AO) =m[Aw + (povo + Apvo + Avpo)-

(qouo + Aquo + Auqo)]

(3.41)

3.2.3 Attitude and Angular Velocities

The same procedure is repeated for the attitude and angular velocity equa-

tions. Substituting the values in 3.42 into eqs 3.22, 3.23, and 3.24


q go + AG

z ro + Ar

0o + AO


o 0 + AO
= Io + ae


(3.42)


gives


po + Ap = A7 A sin(Oo + AO)


(3.43)

(3.44)


qo + Aq = Ocos(Ao + Oo) + A cos(Oo0A)sin(t/o + AO)


Ai

A

A


Al

A









ro + Ar = A-cos(OoAO)cos(OoAqS) AOsin(0o + AO) (3.45)



AO = (qo + Aq)cos(eo + AO) (ro + Ar)sin(0o + AO) (3.46)


A0 =(po + Ap) + (qo + Aq)sin(0o + AO)tan(0o + AO)+
(3.47)
(ro + Ar)cos(oo + A4)tan(Oo + AO)

AO = (q, + Aq)sin(0o + AO) + (ro + Ar)cos(phio + AO)sec(Oo + AO) (3.48)

3.3 Straight and Level Flight

Next equations are written to describe straight and level flight. Take the X

force equation, eq 3.39, as an example to explain the process. Straight and level is

described by assuming symetric flight. This implies that wo = vo = po = qo = ro =

4o = 0o = 0. Applying this assumption to eq 3.39 produces eq 3.49.


Xo + AX mg sin(Oo + AO) = mAli (3.49)

Eq 3.49 can be further reduced with the use of the trigonemetric identity in

Eq 3.50.



sin(00 + AO) = sin 00 cos AO + cos 60 sin AO = sin 00 + AO cos 00 (3.50)

Plugging Eq 3.50 into Eq 3.49 yields Eq 3.51.



Xo + AX mg(sin 0o + AO cos 0o) = mAu (3.51)

Individual Taylor series expansions are used to express the aerodynamic forces

and moments on the airplane. These expansions are done as a perturbations about

the reference flight conditions under the assumption that the perturbations are all









instananeous changes from the fight conditions. The expansions are done with

consideration to their dependant variables. The choice of dependant variables

is made to insure decoupling of the aircraft dynamics into a set of longitudunal

equations and a set of lateral equations. The example using the X force equation is

continued as follows.

Setting all of disturbances equal to zero gives Eq 3.52, the reference flight

condition.



Xo mgsin 00 = 0 (3.52)

This reduces the X force equation to Eq 3.53.



AX mgAO cos 00 = mAt (3.53)

Decoupling requires that the dependant variables be chosen to reflect changes

strictly in either the longitudinal or lateral sense. X is a longitudinal force so it is

assumed that X = f (u, w, A6s, A6T) where 6, and ST are the changes in elevator

deflection and thrust, respectively. The expansion of AX in Eq 3.53 is carried out

in Eq 3.54

6X Sx aX sX
AX = AA + AAw + A6s + AS (3.54)
6u 6w 66e 66T



Eq 3.54 is substituted into Eq 3.53 to yield Eq 3.55



aX Sx sX sX
Au + Aw + Ae + + AT mgA0 cos 0o = mAu (3.55)
6u 6w 66T 66s









Dividing Eq 3.55 through by the mass m and defining the format for the

aerodynamic derivatives as X, = sX/su/m, X, = SX/Sw/m, and so on allows for

Eq 3.55 to be written as Eq 3.56


d
( X,)Au XAw + (g coso)AO = X6ease + XSATa (3.56)

The remaining two force equations along with the three moment equations are

treated in a similar manner and then placed into the state space form and given in

Eqs 3.57, and 3.58.

Longitudinal state space:

An XU X, O -g Au

Aw Z, Z, o 0 Aw
= (3.57)
A4 M + M Z M +M Z, Mq +M: n 0 Aq

AO 0 0 1 0 AO


Lateral state space:

a Ye YE _-(1 ) agcos9o ap
UO UO U Uo

Ap Lp p L 0 Ap (3.58)
= (3.58)
Ar NP Np Nr, 0 Ar

A 0 1 0 0 AO














CHAPTER 4
MODELING

Now X = AX becomes X = A(p)X where A(p) = ANJ" + AN i-1N +

AN-22 N-2 + ...A2 2 + AIpm + A0. If N = 1 i.e. X = (AlI + Ao)X then the system

falls into the class of LIV systems.

In the case of morphing aircraft, the dynamics are dependent on a varying

parameter of the changing geometry. In order to characterize the dynamics over

the range of actuation, models must be created at various points in the parameter

space. These models are then curve fit against the varying parameter using the

least squares method. Now A = f (p) where p is the morphing parameter. The

dynamics are then represented by X = A(p)X where A(p) = Ayp + AN iN 1- +

AN-2 N-2 + ...A2m2 + Ali + Ao. If N = 1 i.e. X = (Ali + Ao)X then the system

falls into the class of linear input varying (LIV) systems. N can be chosen such

that the order of the function f properly identifies the relationship between the

varying parameter and the dynamics. This curve fit can be done for any order of

polynomial. Now the dynamics are represented by:

Longitudinal states:

An X() X.(W) 0 -g Au
Awb Zu(p) Z. (I) Uo 0 Aw

A M()+MM (P)Z.(P) M(P)+ M(P)Z,(m) Mq(i) + M(P)uo 0 Aq
AO 0 0 1 0 AO
(4.1)







23

Lateral states:
() Yp Yp4) _-(1 Yr(,)) g cos O
Uo U0 U0 Uo

Ap L (p) L,(p() L,(1) 0 Ap

Ai Np () N, (pu) N, () 0 Ar

A9 0 1 0 0 Ao















CHAPTER 5
CONTROL SYNTHESIS

5.1 Quasi-static Morphing

Morphing has two distinct possible implementations; dynamic and quasi-static.

Dynamic morphing involves the use of variable geometry as a control effector.

Quasi-static morphing employs shape changes to optimize performance over the

flight envelope. In this architecture conventional control surfaces are used for

control and morphing is used only to reconfigure the aircraft to shapes optimized

for different portions of the flight envelope. This allows for the application of

existing control strategies. One method of controlling this type of vehicle is to

design an optimal controller for a set of dynamics in the middle of the parameter

space. The intent is to obtain a set of controller gains that retain stability with a

slight degradation in performance metrics.

5.2 Dynamic Morphing

Sufficient actuator dynamics can allow for dynamic morphing. This concept in-

volves a change in geometry that is both fast and extensive enough to enable flight

path control with sufficient maneuverability. This type of plant falls into the class

of LIV systems. LIV systems are an emerging class of systems whose dynamics are

not only subject to change with operating parameters but undergo a significant

change due to input parameters as well. In this case the morphing parameter Pi is

the input parameter. Aircraft dynamics typically vary with exogenous inputs such

as altitude and mach number. There are existing control methodologies that deal

with these variations, with gain scheduling being the most common approach. This

procedure can be extended as in the case of the linear parameter varying (LPV)

framework. LIV systems present a new challenge to the control theorist because









there is no control matrix in the traditional sense. Now proofs are given for the two

Lemmas used in control design for the following applications.

The controller design is given by the following Lemmas in [4]. These Lemmas

were developed using a Lyapunov based method [19].

Lemma 1 Given the system, x = (Ao + Alu)x, then the origin is globally

asymptotically stable using the control law u = Kx if

1. O > Ao

2. O > AjK

Proof The closed loop system is

S= (Ao + Alu)x

where = (Ao + A,(Kx))x

S= Ao + A(Kx)x K1 e RZln, x e R", and u eR

V(O) 0

V(X) = xT > 0
V(X) KrTX + xTT

V(x)= [Aox + A,(Kx)x]Tx + XT [Aox + AI(Kx)x]

= T[(Ao + A1Ko)x + A,(K1X2)]T + XT[(Ao + A1Ko) + A (KiX2)]x
xTAAx + XTAT Kx)x + xTAox + xTAT (Kx)xKo) + A1(KIx2)]X

i.e. if the V is negative definite

V(x) < 0 if

AT < 0

A1(Kx) < 0
Ao <0

A, (Kx)<0 Vx


Lemma 2 Given the system, i = (Ao + Alu)x, then the origin is globally

asymptotically stable using the control law u = Ko + K1x2 if









1. O > AO

2. O > AiK2

Proof The closed loop system is

S= (Ao + Alu)x

where i = (Ao + A (Ko + K1x2))

x = (Ao + AIKo)x + A1KIx2 x K, e Rl
V(o) 0

V(X) = xTx > 0

V(X) = j:x + xx x

V(x) = [(Ao + AKo)x + A1(Kix2)X]Tx + XT[(Ao + AKo)x + A,(Kix2)X]

= T[(Ao + AIKo)x + A,(K1x2)]T + xT[(Ao + A1K0) + A (KiX2)]x

= T[[(Ao + AKo)x + A1(Kix2)]T + [(Ao + A1Ko) + A (Kix2)]]x

V(x) < 0 if

[(Ao + AKo)x + A1(Kix2)]T + [(Ao + A Ko) + A,(KIx2)] < 0
i.e. if the V is negative definite

Expanding V(x)

V(X) = xT(Ao + AIKo)Tx + xT(A1(Kix2))Tx + XT(Ao + AKo) + XTA,(K1x2)x

(Ao + AiKo)T < 0

(A, (Ki2))T < 0

(Ao + AiKo) < 0

(A,(Kix2)) < 0 V x















CHAPTER 6
APPLICATION BASE MODEL

6.1 Vehicle Description

The morphing aircraft discussed in the next three chapters all share the same

base model. It is therefore useful to examine the flight dynamics of this model

and to identify the characteristic modes. This forms a basis for understanding

the propagation of these modes as the model is morphed in various manners. A

representation of this airplane can be seen in Fig 6-1.











Figure 6-1: Base Model


The longitudinal eigenvectors are given in polar form in Table 6-1. These

help to establish the phugoid and short period modes. The short period mode is

dominated by w and q and the phugoid mode is characterized by a dominance of u

and 0.

Short Period Mode Phugoid Mode
Magnitude Phase Magnitude Phase
Aiu 0.5558 31.86950 11.3367 -92.96450
Aw 16.7545 16.05890 0.5806 -87.83960
Aq 17.1130 147.66170 0.8642 90.47410
AO 1.0000 0.000 1.0000 0.000
Table 6-1: Longitudinal eigenvectors









Table 6-2 lists the modal properties of the longitudinal dynamics of the base

model. Notice that the phugoid mode is very lightly damped and that the short

period has a relatively high natural frequency.

Mode Frequency (rad/s) Damping
phugoid 0.8642 0.0083
short period 17.1130 0.8449
Table 6-2: Longitudinal eigenvalues



The lateral directional eigenvectors appear in Table 6-3. The roll mode can be

identified by the dominance of 0 and p. The unstable mode is recognized to be a

spiral divergence. The eigenvector indicates the response resembles a classic spiral

mode in that excitation of this mode is essentially yaw and roll.

Roll Mode Spiral Mode
Magnitude Phase Magnitude Phase
Av 0.0619 1800 0.0849 00
Ap 14.5173 1800 0.6284 00
Ar 0.4423 00 1.6893 00
AO 1.0000 00 0.3720 00
A0 0.0305 1800 1.0000 00
Table 6-3: Lateral-directional eigenvectors



Table 6-4 displays the modal properties of the lateral directional dynamics.

The dutch roll frequency is reasonable for an aircraft of this size. The roll fre-

quency seems rather high, but this aircraft was modeled with a thin carbon fiber

wing and has an unusually low value for -J relative to the mass of the airplane.

Mode Frequency (rad/s) Damping
spiral 1.6893 1.0000
dutch roll 4.9812 0.6330
roll 14.5173 1.0000
Table 6-4: Lateral-directional eigenvalues









The stable mode has obvious characteristics associated with the classical

definition of roll mode. The response of this mode is predominately a roll motion

with only minor variation in angle of sideslip or yaw.

The remaining mode relates to a dutch roll dynamics as evidenced by its

eigenvector in Table 6-5. The motion associated with this mode is a complex re-

lationship between yaw and roll and angle of sideslip. The phases and magnitudes

slightly differ from the motions of large aircraft; however, the dynamics are clearly

dutch roll.

Dutch Roll Mode
Magnitude Phase
A) 0.2889 -133.86500
Ap 4.9812 129.27110
Ar 8.6632 153.81300
Aq0 1.0000 00
A0p 1.7392 24.54190
Table 6-5: Lateral-directional eigenvector



6.2 Modeling

6.2.1 Aerodynamic Modeling Tornado

Modeling of flight dynamics was accomplished through the use of Tornado [20].

The output of this program contains all of the necessary stability derivatives, with

the exception of cm, to create a full set of linearized flight dynamics about a given

state. All models discussed in this thesis were generated about a straight and level

flight condition.

6.2.2 Rigid Body Modeling-ProE

The equations of motion contain moments of inertia that need to be deter-

mined in order to fully model the morphing aircraft. Actual airframes can undergo

testing on a laser vibrometer to determine various structural and mass properties

including moments of inertia. Analytical methods also exist. One such method

is the creation of a finite element model. Such a model is a representation of the







30

structure made up many small individual members. This allows for the calculation

of the needed moments of inertia about the center of gravity. ProE, a 3D mod-

eling software package was used to generate the models. A finite element model

of the base aircraft was created with articulated sections that were positioned

at various points in the actuation range. Inertia tensors were calculated for each

configuration. These varying parameters were fed into the LIV dynamics during

simulations.















CHAPTER 7
APPLICATION-VARIABLE SPAN

7.1 Vehicle Description

The morphing concept explored here involves an airplane that has extensible

wingtips with the capability of sliding in and out from underneath the inboard

portion of the wing. The ability of the wing to maintain a consistent cross section

throughout its range of motion is made possible by the use of a thin undercam-

bered airfoil as is used by all of the MAV's at the University of Florida. The

wingtips can articulate in unison to accommodate different portions of the flight

envelope, or asymmetrically to facilitate roll control in lieu of aileron usage [?].








Figure 7-1: Morphing Span


7.2 Flight Dynamics

A set of LIV dynamics was generated for the span varying aircraft using

Tornado. The dynamics were linearized about a straight and level flight condition

with a = 50 and u = 12 m/s. The aircraft is observed to be at trim in Fig 7-2 at a

span of 88 cm.

The damping ratio for the short period mode can be observed to be decreas-

ing in Fig 7-3 as the span increases. There is a decreasing slope indicating an

asymptotic approach to a bounded damping ratio.














0.1

0.08

0.06

0 0.04


Z-~
S0.02

0

-0.02

-0.04

-0.06
60 70 80 90 100 110
Span (cm)

Figure 7-2: Pitching Moment Variation


0.86

0.85

0.84

S0.83

0.82

0.81

0.8

0.79

0.78
60 70 80 90 100 110
Span (cm)

Figure 7-3: Short Period Damping Ratio Variation



The natural frequency of the short period mode steadily increases in an almost


linear fashion in Fig 7-4. This corresponds to the expected destabilizing effect an


increase in wing area would have while holding the tail volume coefficient constant.


The horizontal tail's ability to stabilize the airplane decreases as wing is extended.


20.5

20

19.5

5 19

S18.5



17.5


60 70 80 90 100 110
Span (cm)

Figure 7-4: Short Period Frequency Variation












A look at the lightly damped phugoid mode reveals an opposite trend, which


is shown in Fig 7-5. An increase in wing span increases damping. The frequency of


the phugoid mode increases along with that of the short period mode, but with a


lower over all change.

0.02

0.018

0.016

& 0.014

0.012

0.01

0.008

0.060 70 80 90 100 110
Span (cm)
Figure 7-5: Phugoid Damping Ratio Variation



1.25
1.2

1.15

S 1.1
1.05

1
S0.95

0.9

085
60 70 80 90 100 110
Span (cm)
Figure 7-6: Phugoid Frequency Variation



Fig 7-7 shows the migration of the longitudinal modes over the range of spans.


The short period mode is most affected, while the phugoid mode undergoes little


variation in terms of a percentage change.


An increase in wing span serves to increase the dutch roll mode damping ratio


as detailed in Fig 7-8 and to decrease its frequency as seen in Fig 7-9. Overall


there is a destabilizing affect as the poles move slightly toward the imaginary axis


in Fig 7-10.































Figure 7


-5




-15-________________
-25 -20 -15 -10 -5 0 5 10
Im(X)

7: Pole Migration of Longitudinal Modes


0.8



0.75



S0.7



0.65


0.6
60 70 80 90 100 110
Span (cm)

Figure 7-8: Dutch Roll Damping Ratio Variation






4.9




4.7

I 4.6

4.5

4.4
60 70 80 90 100 110
Span (cm)

Figure 7-9: Dutch Roll Frequency Variation




Two disturbance rejection controllers were designed using Lemma 1 and


Lemma 2. Gains are given in Table 7-1. Time response and span deflection plots


appear in Fig 7-11. Both controllers improve upon the plants open loop response.


































Figure


-10
-15
-20
-40 -30 -20 -10 0 10
Im(X)

7-10: Pole Migration of Lateral Modes


Lemma 1 Lemma 2

K Ko K,
Au 10 0 10000

Aw 10 10000

Aq -10 -10000
AO -10 -10000

Table 7-1: Gains for span varying disturbance rejection controller


... .Fixed Span
- Lemma 1
- Lemma2


70 ... .......... .........

60
50 Fixed Span
Lemma 1
Lemma2
0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3
Time (s) Time (s)

Figure 7-11: Pitch Response for Morphing Span


0.4 0.5

















CHAPTER 8
APPLICATION-VARIABLE CAMBER

8.1 Vehicle Description

The usage of a thin undercambered airfoil section makes it considerably

easier to affect a change in camber. A conventional wing has both upper and

lower surfaces as well as internal structure to manipulate. A MMAV with a thin

undercambered section can have its camber altered along the entire span of the

wing with a single actuator. The theoretical wing on this airplane varies its camber

in a precise way as the camber percentage changes in a linear manner [9]. If this

design aspect was approached as a multi-disciplinary design optimization (MDDO)

problem, the desired shape change would most likely be entirely different. This

consideration is beyond the scope of this thesis, nor does the thesis intend to solve

a MDDO problem [10, 22].


0.4-

0.3

0.2-

-0.




-0.2-

-0.3-
0 0.2 0.4 0.6 0.8 1
Figure 8-1: Morphing Camber











8.2 Flight Dynamics

Fig 8-2 is a plot of the pitching moment variation with camber change. There

is no change in the pitching moment, requiring the use of elevator to trim out the

airplane. This would indicate that camber is not a viable control effector for pitch

control. Instead it may be more useful as a quasi-static flight envelope optimization

technique.

1.5


1

0.5

0 0
I


8 10 12 14 16 18
% Camber
Figure 8-2: Pitching Moment Variation


Camber variations have little effect on the longitudinal dynamics. The short

period is least affected, showing changes in damping ratio and frequency of only a

few percent.


8 10 12 14 16 18
Camber (%)
Figure 8-3: Short Period Damping Ratio Variation



The phugoid mode shows a slightly more marked change. An increase in

camber correlates to increases in both damping ratio (Fig 8-5) and natural

frequency (Fig 8-6).












17.135


17.13
-5'

- 17.125


17.12

r,


8 10 12 14 16 18
Camber (%)

Figure 8-4: Short Period Frequency Variation

0.09


Figure 8-5: Phugoid Damping Ratio Variation

1.4


1.3


S1.2


1.1


Figure 8-6: Phugoid Frequency Variation



Fig 8-7 displays the shift in pole due to a percent camber increase. Increasing


the camber destabilizes the short period mode as the move a minute amount


toward the imaginary axis. The phugoid mode in turn becomes slightly more


stable. There is however no appreciable effect of camber change on the longitudinal


modes.






















-5

-10

-15-___________________ _________:
-25 -20 -15 -10 -5 0 5 10
Im(X)

Figure 8-7: Pole Migration of Longitudinal Modes



An interesting result in the shift in lateral dynamics is the behavior of the


damping ratio is observed in Fig 8-8. It initially decreases prior to reaching a


minimum and then goes on to increase. The plot of the natural frequency in


Fig 8-9 shoes a linear increasing trend.







0.64




S0.635




063/
8 10 12 14 16 18
Camber (%)

Figure 8-8: Dutch Roll Damping Ratio Variation


Figure 8-9: Dutch Roll Frequency Variation








40


Camber morphing has little effect on the lateral dynamics of the airplane

also. The roll mode is destabilized by a small amount, while the dutch roll mode

becomes more stable to some degree. An increase in camber moves the spiral pole

away from the imaginary axis, making that mode yet more unstable.


0K
5
0
5 a
n ---------.---- --


-10
-15
-20
-40 -30 -20 -10 0 10
Im(X)
Figure 8-10: Pole Migration of Lateral Modes


Two disturbance rejection controllers were designed using Lemma 1 and

Lemma 2. Gains are given in Table 8-1. Time response and span deflection plots

appear in Fig 8-11. The 2 controllers closely match the openloop response of the

aircraft, making only a small improvement.


Lemma 1 Lemma 2
K Ko K,
An 100 1500 10
Aw 10000 100
Aq 0 -10
AO -1 -10

Table 8-1: Gains for camber varying disturbance rejection controller


0.1 0.2 0.3 0.4 0.5 '0 0.1 0.2 0.3 0.4
Time (s) Time (s)
Figure 8-11: Pitch Response for Morphing Camber


.-. Fixed Camber
S Lemma 1
i --- Lemma2-















CHAPTER 9
APPLICATION-VARIABLE CHORD

9.1 Vehicle Description

The vehicle proposed here has a portion of the wing which slides out in order

to increase the wing area, much like the span varying case. This is not unlike a

Fowler flap except for the greater range of motion. This MMAV concept has the

capability to double its chord length.








Figure 9-1: Morphing Chord


9.2 Flight Dynamics

Fig 9-2 plots the pitching moment against a change in chord. It can be noted

that a significant moment is created by morphing the chord. This makes sense as it

is expected that a large change in the chord would have pronounced effect on the

pitching moment. The neutral point of a wing usually lies at about the 50gravity

remains at roughly the same point, depending on the mass of the wing relative to

the rest of the airframe.

This theme of large longitudinal effects by chord variations continues in the

dynamics. Fig 9-3 shows how the short period damping ratio quickly increases to a

maximum of one at which point it becomes two separate convergences.

The phugoid mode of the aircraft is affected to a lesser extent by a change in

chord. The modal properties are still more sensitive to this form of morphing than













0.1

0.05

0

E -0.05

-0.1

-0.15

-0.2

-0.25

-0310 12 14 16 18 20 22 24
Chord (cm)

Figure 9-2: Pitching Moment Variation

0.98

0.96

0.94

o 0.92

0.9

0.88

0.86

0.84
11 11.5 12 12.5 13 13.5 14
Chord (cm)

Figure 9-3: Short Period Damping Ratio Variation


1.25

1.2

1.15

S 1.1

1.05

1 1

z 0.95

0.9

085
10 12 14 16 18 20 22 24
Chord (cm)

Figure 9-4: Short Period Frequency Variation



to the previous two. Figs 9-5 and 9-6 show increases in the phugoid damping ratio


and frequency, respectively.


The movement of the longitudinal poles due to a change in chord is given


in Fig 9-7. The short period mode breaks down into two separate convergences,


one of which continues to grow more stable, with the other moving toward the


imaginary axis. The phugoid mode grows more stable with an increase in chord.

















0.02


0.015


0.01


0.005
10 12 14 16 18 20 22 24
Chord (cm)
Figure 9-5: Phugoid Damping Ratio Variation

45

40-

35-




125

20


10 12 14 16 18 20 22 24
Chord (cm)
Figure 9-6: Phugoid Frequency Variation


-40 -30 -20 -10 0 10
Im(X)

Figure 9-7: Pole Migration of Longitudinal Modes



Figs 9-8 and 9-9 show the damping ratio and the natural frequency of the


dutch roll Omode following the same oscillatory pattern, with a globally decreasing


trend.


Lateral pole migration due to a change in chord is given in Fig 9-10. The roll


convergence grows more stable as the chord increases and the spiral divergence


becomes even less stable.


0
5
0
5










44


0.7

0.65

0.6

I 0.55

S0.5

0.45

0.4


10 12 14 16 18 20 22 24
Chord (cm)

Figure 9-8: Dutch Roll Damping Ratio Variation


5.4

5.2

-; 5

t 4.8

T4.6

S4.4

Z 4.2

4

810 12 14 16 18 20 22 24
Chord (cm)

Figure 9-9: Dutch Roll Frequency Variation


20

15

10




-5

-10

-15
-20
-40 -30 -20 -10 0 10
Im(X)

Figure 9-10: Pole Migration of Lateral Modes



Two disturbance rejection controllers were designed using Lemma 1 and


Lemma 2. Gains are given in Table 9-1. Time response and span deflection plots


appear in Fig 9-11. Morphing chord has the greatest authority over pitch response


of any of the types of morphing considered in this thesis.























Lemma 1 Lemma 2

K Ko K,
Au 1 0 10
Aw 100 10

Aq -1 -10
AO -1 -10

Table 9-1: Gains for chord varying disturbance rejection controller


Fixed Chord
- Lemma 1
- Lemma2


0.1 0.2 0.3
Time (s)
Figure 9-11:


.--- Fixed Chord
- Lemma 1
- Lemma2


0.4 0.5 0 0.1 0.2 0.3
Time (s)
Pitch Response for Morphing Chord


0.4 0.5















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BIOGRAPHICAL SKETCH

Kenneth Boothe was born in Pensacola, Florida, on February 2, 1974 where he

spent most of his life prior to coming to Gainesville in January of 1998 to attend

the University of Florida. While in Pensacola, Kenneth graduated at the top of his

high school class and went on to study computer science the University of West

Florida on a scholarship. He continued taking classes sporadically while working

in a variety of jobs including restaurant manager, white water raft guide, and a

partnership in a small landscaping business. Kenneth decided to return to school

to earn a degree in aerospace engineering in accordance with a life long interest

in aviation. This interest was spawned by a teacher in a a gifted program he

was involved in during his youth. Mr. Rod Smith offered a class entitled simply

"Flight." The fundamental forces of flight were introduced and some experiments

were performed with small styrofoam models placed in front of a fan. Kenneth

earned his BS in aerospace engineering in 2003 and hopes to earn his MS in the

same discipline in 2004.