FLIGHT DYNAMICS MODELING AND TRIM CURVES OF A CONCEPTUAL SEMI-TANDEM WING VTOL UAV

Transcription

1 FLIGHT DYNAMICS MODLING AND TIM CUVS OF A CONCPTUAL SMI-TANDM ING VTOL UAV Antono Carlos Dau Flho, uaro Morgao elo São Carlos School of ngneerng, Unversty of São Paulo Keywors: VTOL UAV, flght ynamcs, mult-boy ynamcs, trm curves Abstract Ths work focuses n the flght ynamcs moelng of a VTOL Sem-Tanem ng UAV concept an the stuy of the transton phase, evaluatng the trm curves along the flght regme, that s, from hoverng to cruse flght conton. The VTOL UAV concept stue has the man feature of tltng both wng an horzontal tal, along wth the rotors on both surfaces. Thus, n orer to moel the arcraft ynamc system the equatons of translatonal an angular moton are presente. For ths arcraft confguraton t s approprate to use the mult-boy equatons of moton, where the arcraft s ve n parts so that the wng, horzontal stablzer an rotors are nepenent enttes. Atonally, the success of the transton phase from hoverng to cruse an from cruse to hoverng can be verfe f there s the possblty of the arcraft to trm along the flght spee regme, n other wors, f there s a combnaton of states of moton that keep the arcraft stable from hover to cruse conton. So, the trm curves expressng the states are compute usng the mnmzaton of a cost functon nvolvng the sum of the squares of some of the states of moton, efne through the equatons of moton prevously mentone. Such mnmzaton s performe usng the Sequental Smplex algorthm. Lastly, the resulte trm curves are presente. confguraton ue to the tlt movement of the wng an horzontal stablzer, both wth spnnng propellers, whch results n shftng of the center of gravty an gyroscopc moments. Therefore, the tratonal moelng nvolvng the 6 egree-of-freeom rg boy equatons woul be an oversmplfcaton of the system. In ths way, a more complex formulaton s requre. So, t s approprate to use the multboy equatons of moton, where the arcraft s ve n parts so that the wng, horzontal stablzer an rotors are nepenent enttes. Ths allows the assessment of the lnear an angular momentum for each part, whch are subsequently erve to obtan the equatons of moton. 1 Introucton The control of a VTOL UAV urng transton phase from hoverng to cruse flght conton an from cruse to hoverng s a ffcult task, notably n the sem-tanem Fg. 1: Concept of VTOL UAV, cruse an hoverng conton. 1

2 ANTONIO CALOS DAUD FILHO, DUADO MOGADO LO Therefore, a concept of VTOL UAV was esgne n orer to assess the flght ynamcs of such confguraton, whch can be seen n Fg. 1. Such arcraft woul have four propellers at the wng an two propellers at the horzontal tal, an both surfaces woul be able to tlt, so that n the cruse confguraton both woul be horzontally postone, an n the hover confguraton those woul be vertcally postone. Such concept s very smlar to the confguraton stue by Freercks et al [1]. Ths arcraft concept has the propertes of Table 1. MTO (kg) ng area (m ).34 ng span (m) 1.63 ng aspect rato 7.8 ng loang (kg/m ) 58.8 Horzontal tal area (m ).15 Vertcal tal area (m ).75 Fuselage length (m) 1.45 Propellers ameter(m).38 Table 1: Arcraft szng results. each part has constant mass, even though fuel consumpton reuces the boy part mass over tme, where the fuel tank woul be n the fuselage, the weght reucton s too slow to be consere n the ynamc analyss. An fnally, no structure eformatons are consere, that means that the parts mensons are constant. th the prevous hypotheses we are able to efne the arcraft ynamc system n Fg.. In ths fgure we fn the orgn of the arth fxe nertal reference frame O, an the orgn of the arcraft boy coornate frame O, whch s the poston of the center of gravty of the arcraft boy part, that can shft ue to the quantty of fuel n the tank, but wll not move because of the tlt of the wng or horzontal tal. Arcraft Dynamc Moel Most flght ynamc analyss uses the hypothess that the arcraft behaves lke a rg boy n the ar, wth the hypothess that the mass of such s constant an there are no structural eformatons. However, t woul be an oversmplfcaton of the system to apply the 6 egree-of-freeom rg boy equatons of moton to the concept of arcraft of ths work, snce the wng an horzontal tal are suppose to tlt along wth the spnnng rotors, resultng n shftng of the center of gravty an gyroscopc moments. Therefore, we wll present a mult-boy equatons of moton that are a more approprate approach, whch are much smlar to the equatons presente n the work of Haxu et al []. So, we wll be vng the arcraft n some parts an compute the nertal propertes of each. Such parts are: the boy, whch nvolves the fuselage, lanng gear, vertcal tal an all ts components; the rght an left wng; rght an left horzontal stablzers; an each rotor a separate part. In ths way, we also conser that Fg. : eference frames an arcraft ynamc system. The wng an horzontal tal tlts wth respect to the pvot ponts P an P HT, whch are fxe, an are postone on the one quarter chor of the expose root chors. The wng an horzontal tal are ve n rght an left parts, each wth ts own concentrate mass, postone n the respectve center of gravty, wth ther own coornate frame (O, O L, O HT, O HTL ), n ths manner, when the wng an horzontal tal tlts along the pvot ponts, ther coornate frames follows. Lastly, for every rotor n the wng an horzontal tal, there s also a coornate frame (O 1,, O 6 ), whch are fxe wth respect to ther wng or horzontal stablzer coornate frame. Havng the arcraft ynamc system moel, we may procee ervng the translatonal an angular equatons of moton.

3 FLIGHT DYNAMICS MODLING AND TIM CUVS OF A CONCPTUAL SMI-TANDM ING VTOL UAV 3 quatons of Moton 3.1 Translatonal Moton e begn by efnng the total lnear momentum n the arth fxe nertal reference frame, as the sum of the lnear momentum of each nvual part. From now on, the subscrpt wll be referrng to the arcraft boy part, to the rght or left, wng or horzontal stablzer part, to the rotors, an the superscrpt wll be referrng to the reference frame of the vector, where n the followng equaton, means arth fxe nertal reference frame. So, the arcraft total lnear momentum n the arth fxe nertal reference frame s, = G =1 G G G total =1 (1) The lmts n the sum are the number of aeroynamc surfaces (n our case 4), an the number of rotors (6 rotors). xpanng, we fn, = m V m V m G total =1 =1 V () Dfferentaton of the total lnear momentum leas to the force equaton n the arth fxe nertal reference frame, where F s the net apple force vector. F = (G t total ) (3) F = m V =1 m V m V =1 (4) Now, n orer to pass the equaton to the boy coornate frame, we wll use the theorem of Corols to compute the acceleraton vector from arth fxe nertal reference frame to the arcraft boy coornate frame, such ervaton can be foun at Stevens an Lews [3]. V = V ω V (5) here ω s the angular velocty vector of frame relatve to frame. Ths s also the arcraft boy coornate frame angular velocty vector, ω = [P Q ] T (6) The velocty vector for the concentrate masses of the rght an left wng an horzontal tal n the arth fxe nertal reference frame have atonal term ue to the relatve movement wth respect to the arcraft boy concentrate mass. From Meram an Krage [4] we have the equaton of relatve acceleraton of a movng pont A wth respect to a movng pont, wheren r A/ s the poston vector of pont A n relaton to pont, V rel s the A/ relatve velocty vector of pont A n relaton to pont, an a rel s the relatve acceleraton A/ vector of pont A n relaton to pont. a A = a ω r A/ ω (ω r A/ ) ω V rel A/ a rela/ (7) So, for the concentrate masses of the rght an left wng an horzontal tal we have, V = V ω V ω r / ω (ω r /) ω V rel / a rel / (8) Smlarly, for the concentrate masses of the rotors, V = V ω V ω r / ω (ω r /) ω V rel / a rel / (9) Thereby, we pass the force equaton from the arth fxe reference frame to the arcraft boy coornate frame. F m g {m g } =1 =1 {m g } = m (V ω V ) =1 {m (V ω V ω r / ω (ω r /) ω V rel / a rel / )} =1 {m (V ω V ω r / ω (ω r /) ω V rel / a rel )} (1) / Note that the terms ae n the left se of the equaton are vectors of weght of arcraft boy, rght an left wng or horzontal tal concentrate masses an rotors concentrate masses. Moreover, the vectors of weght use the rotaton matrx from arth fxe referental frame to boy coornate frame, whch s a functon of the uler angles: roll (φ), ptch (θ) 3

4 ANTONIO CALOS DAUD FILHO, DUADO MOGADO LO an yaw (ψ). The efnton of ths matrx s foun at Stevens an Lews [3]. e efne now the relatve velocty an acceleraton vectors n the arcraft boy coornate frame. In the followng equatons pvot an pvot means the respectve pvot pont of the concentrate masses. V rel / = (11) V rel / = (1) a rel a rel / = (13) / = (14) heren : Poston vector of rght or left wng or horzontal tal concentrate mass relatve to respectve pvot pont. : Poston vector of rotor concentrate mass relatve to respectve pvot pont. ng an horzontal tal tlt matrx wth respect to wng or horzontal tal tlt angle (δ, δ HT ), cos(δ,ht ) sn(δ,ht ),HT = [ 1 ] (15) sn(δ,ht ) cos(δ,ht ) Moreover,,HT an,ht are the frst an secon ervatves of,ht wth respect to tme. Atonally, we wll be usng the smplfcaton (Ω = ω ). eng that, Q Ω = (ω ) = [ P] (16) Q P Q Ω = (ω ) = [ P ] (17) Q P earrangng the equaton terms, passng the arcraft boy acceleraton vector n the arcraft boy coornate frame to the left sze we have, V = Ω V F M g F (18) Note that the only fference from the rg boy equatons of moton s the term F efne as follows, F = 1 {m M =1 [(Ω Ω Ω )r / (Ω ) ]} 1 {m M =1 [(Ω Ω Ω )r / (Ω ) ]} (19) In the prevous equaton we have use the followng term for the total arcraft mass, beng the sum of the concentrate masses of arcraft boy, rght an left wng an horzontal stablzers, an rotors. M = m =1 m =1 m () 3. Angular Moton For the arcraft angular moton equaton we start by efnng the total angular momentum n the arth fxe nertal reference frame, agan, beng the sum of the portons of the arcraft boy, rght an left wng an horzontal stablzers, an rotors. = H =1 H =1 H (1) H total The terms I are the nerta matrces of the concentrate masses, wth the subscrpt ncatng the part, an the superscrpt the reference frame. xpanng, we have, H total = I ω r / (m V ) =1 {I ω r / (m V )} =1 {I ω r / (m V )} () The terms ω are angular velocty vector wth subscrpt ncatng the part an superscrpt the reference frame. Passng the angular moton equaton n the arth fxe nertal reference frame to the arcraft boy coornate frame, H total = I ω =1{I ω r / (m V )} =1 {I ω r / (m V )} (3) From Meram an Krage [4] we have the equaton of relatve velocty of a movng pont A wth respect to a movng pont. V A = V ω r A/ V rel A/ (4) Therefore, we have the velocty of the concentrate masses wth respect to the arcraft boy coornate frame, 4

5 FLIGHT DYNAMICS MODLING AND TIM CUVS OF A CONCPTUAL SMI-TANDM ING VTOL UAV V = V ω r / (5) V = V ω r / (6) The net torque T actng at the arcraft boy coornate frame comes from the rate of change of angular momentum. The ae terms n the left se of the equaton are the weghts torques of the concentrate masses wth respect to the boy coornate frame. T {r / m g =1 } =1 {r / m g } = (H t total ) (7) xpanng the ervatve of the total angular momentum, separate n arcraft boy, rght an left wngs an horzontal stablzers, an rotors terms, we fn, (H t ) = I ω ω (I ω ) (8) (H =1 t ) = { =1 t (I ω ) ω (I ω ) m [ (V ω r / ) r / (V ω V ω r / ω (ω r /) ω )]} (9) (H =1 t ) = { =1 t (I ω ) ω (I ω ) m [ (V ω r / ) r / (V ω V ω r / ω (ω r /) ω )]} (3) ecause of wng an horzontal tal tlt, the nerta matrces of such surfaces an rotors respectve to arcraft boy coornate frame are varables. The nerta matrces of the concentrate masses, wth respect to the boy coornate frame, are obtane from the nerta matrces wth respect to ther own coornate reference frames by translatng an rotatng the reference. Ths operaton s emonstrate n the next equatons for the concentrate masses of the panels an rotors respectvely. There we have [T] an [T] T the nerta rotaton matrx an ts transpose, an the translaton matrx to the boy coornate frame. I I = [T] m (I T )[T] (31) = [T] m (I ) [T] T (3) Moreover, the angular velocty vector n the boy coornate frame of the part s the sum of the angular velocty vector of the part wth respect to ts own reference frame, tlte to aust the reference orentaton, summe wth the angular velocty vector of the boy part wth respect to ts own reference frame. ω ω = ω ω (33) = ω ω ω (34) here, [T] an [T] : Inerta rotaton matrx: rotates the wng, horzontal tal or rotor nerta matrx to the arcraft boy coornate frame. an : Inerta translaton matrx: transfers the wng, horzontal tal or rotor nerta matrx to the arcraft boy coornate frame. ω : ght or left wng or horzontal tal concentrate mass angular velocty vector n respect to ts own reference frame. ω : otors concentrate mass angular velocty vector n respect to ts own reference frame. e can assume from axes algnment that the wng an horzontal tal angular velocty vector s fully algne wth the arcraft boy y coornate, so that, ω = [ An for the rotors we have, δ ] T (35) ω = [ω ] T (36) So that, ω s the rotor rotaton spee. Atonally we have, 5

6 ANTONIO CALOS DAUD FILHO, DUADO MOGADO LO cos(δ,ht ) [T] = [T] T = [ 1 ] (37) cos(δ,ht ) [T] = [T] T = [ cos (δ,ht ) An ther ervatves, T [T ] = [T ] = 1 cos (δ,ht ) ] (38) δ,ht sn(δ,ht ) [ ] (39) δ,ht sn(δ,ht ) T [T ] = [T ] = [ δ,ht sn (δ,ht ) δ,ht sn (δ,ht ) ] (4) Also, conserng the poston vector between the concentrate mass an the reference frame orgn beng r = [x y z] T, we have the nerta translaton matrx from the parallel axs theorem, y z xy xz = [ yx x z yz ] (41) zx zy x y Atonally, t s necessary to erve the followng terms, t (I ω ) = t (I )ω I So that, (ω t ) (4) t (I ) = [T] m (I )[T] T [T] m t ( )[T] T [T] (I m T )[T] (43) t (ω ) = ω ω ω (44) Smlarly for the rotors concentrate masses we fn, t (I ω ) = t (I ) ω I (ω t ) (45) t (I ) = [T] (I [T] m t ( ) [T] T m ) [T] T [T] (I m ) [T] T (46) (ω t ) = ω ω ω ω ω (47) Substtutng the terms an rearrangng we can wrte the angular moton equaton n the smplfe form, T M P = Aω ω CV DV (48) An the coeffcents are, A = I {[T] =1 m (I )[T] T m r / r / } {[T] =1 (I m ) [T] T m r / r / } (49) = {[T] m (I )[T] T Ω I =1 [T] m t ( )[T] T m )[T] T Ω [T] (I m )[T] T m [( [T] (I r / ) (r / Ω r / )]} {[T] =1 m (I ) [T] T [T] m t ( ) [T] T [T] (I m ) [T] T Ω [T] m (I ) [T] T m [( r / ) (r / Ω r / )]} (5) C = =1 {m r / } =1 {m r / } (51) D = =1{m [( ) r / Ω ]} =1 {m [( ) r / Ω ]} (5) 6

7 FLIGHT DYNAMICS MODLING AND TIM CUVS OF A CONCPTUAL SMI-TANDM ING VTOL UAV = =1 [T] m {([T] m (I )[T] T t ( )[T] T [T] (I m )[T] T ) ω [T] (I m )[T] T ( ω ω ) Ω [T] m (I )[T] T ω m [ r / (Ω )] } {([T] =1 m (I ) [T] T [T] m t ( ) [T] T [T] (I m ) [T] T ) ( ω ω ) [T] m (I ) [T] T ω ( ω ) ω ω Ω [T] m (I ) [T] T ( ω ω ) m [ r / (Ω )] } (53) M P = {r / m g } =1 =1 {r / m g } (54) Therefore, we have the arcraft angular moton equaton n the boy coornate frame, ω = A 1 ω A 1 DV A 1 CV A 1 (T M P ) (55) 3.3 Transformaton between eference Axes e have prevously efne the equatons of translatonal an angular moton relatve to the boy coornate frame, or boy axes, t s now necessary to efne the equatons of moton wth respect to wn axes n orer to make t easer the ntroucton of the aeroynamc forces an moments, whch are efne wth respect to these axes. The transformaton matrx between boy axes to wn axes are efne the same way as n Stevens an Lews [3], so that the velocty vector n wn axes are gven by, V = SV (56) eng that, cos α cos β sn β sn α cos β S = [ cos α sn β cos β sn α sn β] (57) sn α cos α Also, we efne the arcraft velocty vector n the arcraft boy coornates frame (V ), an ts components, an V T s the flght spee. V = [U V ] T (58) V = [V T ] T (59) Analogously, we have the arcraft angular velocty vector n the wn axes, ω = Sω = [P Q ] T (6) Now, we efne the net force vector n the wn axes, D T F = SF = { Y } S =1 ( { }) (61) L An the net torque vector, L T = ST = { M} S =1 ( N r / T { { λ Q } }) (6) eng the aeroynamc force vector compose by rag (D), se force (Y) an lft (L), an the aeroynamc roll (L ), ptch (M) an yaw (N) moments. Atonally, we efne each propeller thrust T an torque Q efne as follows, T = k T ω (63) Q = k Q ω (64) Moreover, λ s each propeller rotaton recton nex, beng 1 for counter-clockwse an -1 for clockwse. Then, after hanlng the equatons n a manner very smlar as escrbe n Stevens an Lews [3], but wth the concentrate masses 7

8 ANTONIO CALOS DAUD FILHO, DUADO MOGADO LO terms, we fnally get to the equatons of moton n the wn axes. Frst for the translatonal moton, V T VT { βv T } = Ω { } S g SF α V T cos β D k 1 { Y } 1 T ω S ( M M =1 { }) (65) L heren, Q Ω = SΩ = [ P ] (66) Q P Next we efne the angular moton equaton n the wn axes, makng use of the transformaton matrx between boy axes to wn axes, we get to the followng equaton, V T P { Q } = (A 1 D A 1 SCS T ) { } P L (Ω A 1 ) { Q } SA 1 S T { M} N SA 1 { =1 k T ω ( { λ k Q ω } r / }) A 1 (M P ) A 1 VT C { } (67) heren, β α cos β Ω = SS T = [ β α sn β ] α cos β α sn β (68) Furthermore, we make use of the followng smplfcatons: SAS T = A,SS T =, SCS T = C, SDS T = D, S =, SM P = M P. 4 Arcraft Trm The arcraft trm conton, or steaystate conton, s the combnaton of state varables that make all the state ervatves VT, β, α, P, Q, entcally zero. In our ynamc system we have the followng state varables: V T, β, α, φ, θ, ψ, P, Q,, δ f (flap angle), δ e (elevator angle), δ r (ruer angle), δ al (left aleron angle), δ a (rght aleron angle), δ (wng tlt angle), δ, δ HT (horzontal tal tlt angle), δht, ω 1,, ω 6 (propellers angular spee square). Therefore for each specfc flght conton we must be able to fn the combnaton of state varables that meets wth all null state ervatves. e acheve ths goal wth a numercal algorthm n the followng way. Frst we efne a cost functon from the sum of the squares of the state ervatves [3] prevously mentone, whch s the followng equaton. J = VT β α P Q (69) In the trm conton the cost functon shoul be zero, because the state ervatves must be zero. Thus, f we successvely compute the value of the cost functon for some chosen state vector X usng the translatonal an angular moton equatons to compute the state ervatves, n orer to graually approach the cost functon to zero, t woul be possble to fn the state vector for the specfc flght conton that nullfes the cost functon. An effectve algorthm to solve ths problem s the Sequental Smplex, escrbe n alters et al. [5] an Neler an Mea [6], whch s base on the search of optmum from sequental expermentaton an measurement of system outcome from a combnaton of varables. The algorthm startng proceure mplemente was the Corner Intal Metho, escrbe n alters et al. [5], an the stoppng crteron use was cost functon value less than 1e-15. Moreover, for the hover conton we must use the equatons of moton n the arcraft boy coornate frame. 8

9 Tlt Angle (egree) levator (egrees) Alpha (egrees) FLIGHT DYNAMICS MODLING AND TIM CUVS OF A CONCPTUAL SMI-TANDM ING VTOL UAV 5 Trm esults Trm curves were compute for the conceptual VTOL UAV prevously escrbe, beng that the aeroynamc forces an moments were estmate usng methos from Datcom [7], Hoerner [8] an Houghton an Carpenter [9]. In ths paper we wll present results only for the steay-state longtunal flght, where some states are zeros (β, P, Q,, φ, ψ, δ r, δ al, δ a ), whch are nputs for the algorthm. It was also consere flap angle zero to trm the arcraft, n orer to reuce the number of varables. Atonally, t was establshe that for every flght conton the wng an horzontal tal tlt angle must be as n Fg. 3, whch was obtane usng the trm algorthm an smoothng the curves for wng an horzontal tlt angle as functons of flght spee, whch were use for ensung recalculaton of the other states, for the conton of full loa an flght path angle zero. Therefore, for every other flght conton the wng an horzontal tal tlt angle were nputs to compute the other states VT (m/s) ng Tlt Horzontal Tal Tlt Fg. 3: ng an horzontal tal tlt angle x flght spee. The curves express that from to 1 m/s the wng an horzontal tal tlt angle are the same, startng n the hoverng conton at 9 where the propellers are all pontng upwars. From 1 m/s forwar, the curves ffer, beng that the wng tlts more untl the wng tlt angle becomes zero at 33 m/s, whle the horzontal tal zeroes n 36 m/s. In the Fg. 4 we fn the arcraft angle of attack versus flght spee, whch s equvalent to the ptch angle for flght angle zero. Next, the elevator angle as a functon of flght spee n the Fg. 5. The elevator s only use from 16 m/s forwar, snce for low flght spees there s not much ynamc pressure n the aeroynamc controls an the wng an horzontal tal tlt together wth propellers thrust s enough to trm the arcraft Fg. 4: Arcraft angle of attack x flght spee VT (m/s) VT (m/s) Fg. 5: levator angle x flght spee. For the longtunal flght t has been etermne that the four propellers n the wng wll have the same comman, thus the same steay power, beng the same for the two propellers at the horzontal tal. Propeller performance ata was obtane at rant, J. [1], so that we compute the power requre. From Fg. 6 we can see that n the hover conton there s hgh power requre from the wng an tal propellers. As the flght spee ncreases the power requre s lowere, snce some of the lft force s transferre to aeroynamc lft. Also, from 16 m/s on the tal propellers are turne off an the elevator assumes the role to trm the arcraft. It s necessary to o so because there s a lmt on propeller PM, thus t s not possble to aust 9

10 Power (k) ANTONIO CALOS DAUD FILHO, DUADO MOGADO LO all of the sx propellers at the esre avance rato at hgh spees n orer to trm the arcraft. In hgh spees the arcraft aeroynamc rag ncreases thus requrng more power from the propellers, therefore, there s a range of flght spee wth mnmum total power requre, whch happens between 14 an 16 m/s Fg. 6: Total power requre an each wng an tal propeller power. 6 Concluson VT (m/s) ng Props Tal Props Total Props The mult-boy equatons of moton for the propose VTOL UAV concept were evelope whch were use to compute the trm curves for the transton phase from hoverng to cruse conton. Such computatons were performe usng the Sequental Smplex algorthm to mnmze a cost functon. The results showe that the transton phase s possble for ths arcraft concept, prove that the arcraft control system woul be able to stablze the arcraft aroun the steay-states compute, once subecte to sturbances. Moreover, the propose mult-boy equatons of motons coul be apple to varous confguratons of arcraft wth movng aeroynamc surfaces an rotors, n orer to stuy not only the trm contons, but the flght ynamc performance an control system. eferences [1] Freercks,. J., Moore, M. M., usan,. C. enefts of Hybr-lectrc Propulson to Acheve 4x Increase n Cruse ffcency for VTOL Arcraft, AIAA Avaton Technology, Integraton, an Operatons (ATIO) Conference, 13. [] Haxu, L., Xangu, Q., eun,. Mult-boy Moton Moelng an Smulaton for Tlt otor Arcraft, Chnese Journal of Aeronautcs, 1. [3] Stevens,. L., Lews, F. L. Arcraft Control an Smulaton, thr eton, John lley an Sons, Inc., 16. [4] Meram, J. L., Krage, L. G. ngneerng Mechancs Dynamcs, John lley an Sons, Seventh ton, 1. [5] alters, F. H.; Parker Jr., L..; Morgan, S. L., Demng, S. N. : Sequental Smplex Optmzaton: A Technque for Improvng Qualty an Prouctvty n esearch, Development, an Manufacturng, CC Press, Inc., [6] Neler, J. A., Mea,. A Smplex Metho for Functon Mnmzaton, The Computer Journal, Vol. 7, Issue 4, pp , 1965 [7] Hoak, D.. USAF Stablty an Control Datcom, Douglas Arcraft Company, Inc., [8] Hoerner, S. F. Flu-Dynamc Drag: Practcal Informaton on Aeroynamc Drag an Hyroynamc esstance, [9] Houghton,. L., Carpenter, P.. Aeroynamcs for ngneerng Stuents, lsever utterworth- Henemann, ffth eton, 3. [1] rant, J.. Small-Scale Propeller Performance at Low Spees, M.S. Thess, Department of Aerospace ngneerng, Unversty of Illnos at Urbana- Champagn, Illnos, 5. 7 Contact Author mal Aress Antono C. Dau Flho: tonau@hotmal.com uaro M. elo: belo@sc.usp.br Copyrght Statement The authors confrm that they, an/or ther company or organzaton, hol copyrght on all of the orgnal materal nclue n ths paper. The authors also confrm that they have obtane permsson, from the copyrght holer of any thr party materal nclue n ths paper, to publsh t as part of ther paper. The authors confrm that they gve permsson, or have obtane permsson from the copyrght holer of ths paper, for the publcaton an strbuton of ths paper as part of the ICAS proceengs or as nvual off-prnts from the proceengs. 1