Analytical and Simulation Method Validation for Flutter Aeroservoelasticity Studies

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1 Vldton for Flutter Aeroservoelstcty Studes Ruxndr Mhel Botez, Professor, PhD, RTO Member Teodor Lucn Grgore, Postdoctorl fellow Adrn Hlut, PhD ETS, Unversty of Québec Lbortory of Actve Controls, Avoncs nd Aeroservoelstcty LARCASE 11 Notre Dme West Montrél, Qué., Cnd, H3C1K3 ABSTRACT Aeroservoelstcty studes re nterctons between three mn dscplnes: erodynmcs, eroelstcty nd servo-controls. In the eroelstcty re, s known tht erodynmc unstedy forces for rnge of frequences nd Mch numbers re clculted by use of methods such s Doublet Lttce Method (DLM) n the subsonc regme or by Constnt Pressure Methods (CPM) n the supersonc regme. These methods re usully mplemented n fnte element eroelstcty softwre such s Nstrn, STARS or ny other smlr type of softwre. For eroelstcty studes, we clculte n erodynmc unstedy force only for the rcrft elstc modes, whle for eroservoelstcty studes, s necessry to clculte these forces for ll rcrft modes: elstc, rgd, elstc nd control, not only for ts elstc modes. In order to clculte erodynmc forces for ll rcrft modes, we need to consder notons of the flght dynmcs theory (Newton s equtons) for the rcrft rgd nd control modes erodynmc forces clcultons, whle the methods mplemented n fnte element eroelstcty softwre wll be consdered for elstc erodynmc forces clcultons. Therefore, the unstedy erodynmc forces correspondng to ll rcrft modes wll be clculted wth two dfferent methods: numercl nd nlytcl. 1. INTRODUCTION Aeroservoelstcty studes re very mportnt n the rcrft ndustry. After n extensve bblogrphcl reserch n the feld, we dd not fnd well-documented formulton for rgd nd control mode erodynmc forces for eroservoelstcty; therefore we hve formulted nd vldted novel method n ths pper. Fnte element eroelstcty softwre such s STARS 1 or Nstrn 2 does not ccurtely clculte the rgd nd control mode erodynmc forces, but they do clculte the elstc mode erodynmc forces. Our new formulton wll clculte nd vldte the erodynmc rgd nd control modes forces on n F/A-18 rcrft by use of the Doublet Lttce Method (subsonc regme) nd the Constnt Pressure Method (supersonc regme). Thus, rgd nd control mode forces clculted wth our new formulton wll replce the rgd nd control modes clculted wth fnte element-bsed eroelstcty softwre. The F/A-18 rcrft structure s modelled 3 by fnte element methods, nd 44 frequences nd mode shpes re clculted for ths rcrft, whch re dvded nto the followng three groups: 6 rgd modes (3 symmetrc nd 3 nt-symmetrc), 28 elstc modes (14 symmetrc nd 14 nt-symmetrc) nd 1 control modes (5 symmetrc nd 5 nt-symmetrc) RTO-MP-AVT

2 The Doublet Lttce Method, DLM, (n subsonc regme) or the Constnt Pressure Method, CPM, (n supersonc regme) re used to clculte the unstedy erodynmc forces Q(k, Mch) for vrous Mch numbers nd reduced frequences. The erodynmc forces correspond to the followng modes: 6 rgd modes (3 modes n trnslton nd 3 modes n rotton), 1 control modes (5 modes for longtudnl nd 5 modes for lterl motons) nd e represent the 28 elstc modes (14 modes n longtudnl moton nd 14 modes n lterl moton). The erodynmc forces for the rgd-to-rgd mode nterctons Q rr (dmensons 6 * 6) nd for the rgd-tocontrol mode nterctons Q rc (dmensons 6 * 1), clculted wth fnte element softwre Nstrn, wll be replced by Q rr nd Q rc vlues clculted wth the two schemes, nlytcl nd numercl, presented n ths pper. 2. ANALYTICAL VERSUS NUMERICAL FORMULATIONS The nlytcl formulton s presented n Sectons 1 to 3, nd the numercl formulton s presented n Secton 4. Detls of the frst smulton scheme wth stblty dervtves n the wnd system of coordntes re explned n the frst secton, whle Secton 2 presents the frst scheme wth stte vrbles ntroduced n the rcrft closed loop. Secton 3 presents the nlytcl formultons for the erodynmc forces for rgd-to-rgd nd rgd-to-control nterctons mode clcultons. 2.1 Frst smulton scheme wth stblty dervtves n the wnd system of coordntes The detled frst scheme s shown n Fgure 1: lef tef l ht rud lef tef l ht rud q α V θ H p r β φ C D C Y C Lft C L C M C N Scb,, q dyn X v Y v Z v L v M v N v q dyn X Y Z L α, β ρv = 2 2 M N ρ V L ρ = ρ(1 H ) T V, V, V x, y, z x y z φ, θψ, DCM V (,, ) u v w p, qr, p, qr, g 1 RL H V α β q α V θ = H p r β φ z Fgure 1: Frst smulton scheme wth the stblty nd control coeffcents clculted n the wnd coordntes system Ths scheme cn be wrtten n ts equvlent form shown n Fgure RTO-MP-AVT-154

3 The detls of Blocks 1 to 9 re next presented. Block 1 descrpton Fgure 2: Smplfcton of the scheme shown n Fgure 1 Two sets of stblty nd control dervtves n the wnd coordnte system re known, provded by NASA DFRC (Dryden Flght Reserch Center) for vrous flght condtons chrcterzed by Mch numbers, lttudes nd ngles of ttck. We express the rcrft behvour wth the second stte spce mtrx equton: x x y = Ans x+ Bnsu = [ Ans Bns ] = [ A_t] u u (1) Where the A_t mtrx hs the dmensons (6*19), nd A ns s the stblty dervtve coeffcents mtrx of dmensons (6 x 9): CL CL CL CL CL CL CL CL CL q α V θ H p r β ϕ CM CM CM CM CM CM CM CM CM q α V θ H p r β ϕ CN CN CN CN CN CN CN CN CN (2) q α V θ H p r β ϕ Ans = CD CD CD CD CD CD CD CD CD q α V θ H p r β ϕ CLft CLft CLft CLft CLft CLft CLft CLft CLft q α V θ H p r β ϕ CY CY CY CY CY CY CY CY C Y q α V θ H p r β ϕ nd the control dervtve coeffcents mtrx B ns hs the dmensons (6 x 1) nd s dvded nto two mtrces correspondng to the longtudnl nd lterl rcrft moton, whch re denoted by B long_ns nd BBlt_ns. Therefore, B ns s wrtten s Bns = Blong _ ns B lt _ ns where B long_ns s (6*6) mtrx contnng the dervtves of the sme coeffcents s the ones of A ns (whch re C L, C M, C N, C D, C Lft, C Y ) wth respect to the longtudnl control surfces long_ail, long_ht, long_rud, long_lef nd long_tef,; B lt_ns s (6*6) mtrx contnng the dervtves of the sme coeffcents s the ones of A ns (whch re C L, C M, C N, C D, C Lft, C Y ) wth respect to the lterl control surfces lt_ail, lt_ht, lt_rud, lt_lef nd lt. The output, stte, nd nput vectors re gven by the followng equtons: RTO-MP-AVT

4 ( ) T L M N D Lft Y y = C C C C C C ( α θ ) T x= q V H p r β φ (4) u = long _ LEF long _ TEF long _ AIL long _ HT long _ RUD... T lt _ LEF lt _ TEF lt _ AIL lt _ HT lt _ RUD The Tylor seres pproxmtons of stblty nd control dervtves t the trm poston C L, C M, C N, C D, C Lft, nd C Y cn be wrtten s follows: (3) (5) C C C C C C C C C q α V θ H p r β φ L L L L L L L L L CL = q+ α+ V+ θ+ H+ p+ r+ β+ φ+ C C C C C L L L L L P _ AIL long _ HT long _ RUD long _ LEF long _ TEF P _ AIL long _ HT long _ RUD long _ LEF long _ TEF C + C C C C L L L L L lt _ AIL lt _ HT lt _ RUD lt _ LEF lt _ TEF lt _ AIL lt _ HT lt _ RUD lt _ LEF lt _ TEF (6) The vrtons of ll stblty coeffcents re wrtten under smlr forms s the ones gven by equtons (6). Block 2 descrpton The vrtons of forces nd moments re wrtten s functon of ther stblty nd control dervtves s follows: Xw S CD Yw S CY Z w S C Lft = (7) Lw S b CL M w S c CM N w S b C N X Y Z nd the moments vrtons on ll where the forces vrtons on the three xes re [ ] three xes re [ L M N ] Block 3 descrpton w w w w w w 2 ρv, nd where S = S = q S. 2 It s well known n ths feld 4 tht n rcrft s system of coordntes (x, y, z ) s relted to the wnd coordnte system (x w, y w, z w ) by the followng two successve rottons of the coordnte system (x w, y w, z w ). 1. A frst rotton wth sdeslp ngle β round the z w xs to obtn the ntermedte coordnte system (x y z ) nd 2. A second rotton wth ttck ngle α round the y xs to obtn the rcrft coordnte system (x, y, z ), whch gves: x cosαcos β cosαsn β snα xw y = sn β cos β y w (8) z snαcos β snαsn β cosα z w dyn 1-4 RTO-MP-AVT-154

5 The X nd Z forces hve opposte sgns (see equtons 9.1), whle the L, M, nd N moments hve the sme sgn (see equtons 9.2) wth respect to the clsscl formulton (equtons (8)), whch s due to ther orenttons, s gven by NASA DFRC. X cosα cos β cosα sn β snα X Y sn β cos β = Y Z snα cos β snα sn β cosα Z L cosαcos β cosαsn β snα L M sn β cos β = M N snαcos β snαsn β cosα N w w w w w w (9.1) (9.2) Equtons (9.1) nd (9.2) re lnerzed round the equlbrum poston (trm) of the F/A-18 rcrft by the smll perturbtons theory, n whch the ndex of ny quntty t equlbrum s denoted by nd the vrton of quntty round ts equlbrum poston s denoted by. The ngles of ttck nd sdeslp ngles re expressed wth ths theory s follows: α = α +α (1) β = β + β The sdeslp ngle t equlbrum β gven by the NASA DFRC s equl to, nd therefore we cn wrte the sdeslp ngle β s functon of ts smll vrton β: cos β = cos( β +β) cosβ 1 (11) sn β = sn( β +β) sn β β The rcrft forces X, Y, Z nd moments L, M, N clculted n the rcrft system nd n the wnd system w re lso wrtten by use of the smll perturbtons theory, such s: X = X + X,, L = L + L,, X w = X w + X w,,, L w = L w + L w (12) The forces X, Y, Z nd the moments L, M, N t equlbrum re zeros n the rcrft system nd n the wnd system w, nd the ngle of ttck α nd sdeslp ngle β vrtons re lso equl to zero. We ntroduce these lst vlues nto equtons (9)-(12), nd we obtn the followng system of equtons: X Xw Y Yw Z Z w = Cm 1 (13) L Lw M M w N N w where C m1 hs the followng form : cosα snα 1 snα cosα C m 1 = (14) cosα snα 1 snα cosα RTO-MP-AVT

6 We replce the force nd moment vrtons n the wnd system of coordntes X w, Y w, Z w, L w, M w, nd N w s functon of stblty coeffcents vrtons gven by equton (7) nto the rght hnd sde of equton (13) nd we obtn: X CD Y CY Z C Lft = Cm (15) L CL M C M N C N where the C m mtrx hs the followng form: S cosα S snα S S snα S cosα Cm = (16) S b cosα S b snα S c S b snα S b cosα Block 4 descrpton The sx degree-of-freedom block refers to the equtons of moton of rgd body n sx degrees of freedom. The theory cn be found n the lterture 4. The Smulnk toolbox s used here s Block 4, nd s composed of fve blocks, 4.1 to 4.5. Blocks 4.1 nd 4.2 descrptons The orgn of the rcrft system of coordntes,, s the rcrft center of grvty. The nertl system of coordntes s fxed to the Erth nd s denoted by. The rcrft equtons of moton re obtned wth Newton s second lw: The sum of the externl forces ctng on n rcrft s equl to the momentum rte of chnge of the momentum of the rcrft over tme (Block 4.1). The sum of the externl moments 4 ctng on n rcrft s equl to the ngulr momentum rte of chnge of the rcrft over tme (Block 4.2). V (,, ) u v w uvw,, Vx, Vy, Vz x, y, z V ( u, v, w) pqr,, C = B A x, y, z V ω T DCM pqr,, φ, θψ, I 1 pqr,, φ, θψ, φ, θψ, φ, θψ, pqr,, I ω ( Iω) Fgure 3: Descrpton of Block 4 detls on sx degree-of-freedom dynmcs 1-6 RTO-MP-AVT-154

7 Block 4.3 descrpton The Cosne Drector Mtrx CDM s clculted from the Euler roll, ptch nd yw ngles φ, θ, nd ψ n Block 4.3, nd s used n Block 4.4. φ, θψ, φ, θ, ψ Fgure 4: Scheme of Block 4.3 cosθ cosψ cosθsnψ snθ CDM = cosψ snθ snφ snψ cosφ snφsnθ snψ + cosψ cosφ cosθ snφ (17) cosφ snθ cosψ + snφsnψ snψ snθ cosφ cosψ snφ cosθ cosφ Block 4.4 descrpton The lner speeds V x, V y nd V z n the nertl system of coordntes re clculted s functon of the lner speeds u, v, nd w n the rcrft systems by three successve rottons: one frst rotton wth the yw ngle ψ round the z xs, second rotton wth the ptch ngle θ round the y xs nd thrd rotton wth the roll ngle φ round the x xs, nd we obtn: Vx u T Vy = CDM v (18) V z w Block 4.5 descrpton Block 4.5 reltes the Euler ngles φ, θ, nd ψ, nd the ngulr speeds p, q, nd r wth the Euler tme dervtves φ, θ, nd ψ by use of the followng equton: φ 1 snφtnθ cosφtnθ p θ = cosφ snφ q (19) ψ snφ cosφ r cosθ cosθ Block 5 descrpton In Block 5, we clculte the vrtons of the rcrft s true rspeed V, of the ngle of ttck α, nd of the sdeslp ngle β. In order to clculte these vrtons, we frst clculte V, α, nd β s functons of lner speeds n trnslton u, v, nd w n the rcrft system of coordntes 4 : V = u + v + w ; α = rctn( w/ u); β = rctn v/ ( u + w ) (2) The theory of smll perturbtons s ppled to the terms V, α, β, u, v, nd w. 5.1 True rspeed vrton V We replce the speeds gven by the perturbton theory n the frst squred equton of the system of equtons (2): ( V + V) = ( u + u) + ( v + v) + ( w + w) 2 (21) At equlbrum we wrte: V = u + v + w (22) The squre products of speed vrtons u 2, v 2, w 2 2, nd V n equton (21) cn be neglected, s they re very smll. Then, we dvde equton (21) by 2V nd we obtn: T RTO-MP-AVT

8 u v w V = u+ v+ w (23) V V V At equlbrum, the components u nd w of the true rspeed V cn be wrtten s functon of the ngle of ttck α s follows: u = V cos α ; w = V snα (24) The reserchers t NASA DFRC consdered n ther clcultons β =, so tht the component v s equl to zero: v = V sn β = (25) We replce equtons (24) nd (25) nto equton (23) nd we obtn: V = ucosα + wsnα (26) 5.2 Angle of ttck vrton α The second equton of system (2) cn lso be wrtten n the followng form: usnα = wcosα (27) At equlbrum, equton (27) cn be expressed s: usnα = wcosα (28) We pply the perturbton theory to the qunttes u, v nd α (for exmple u = u + u, etc.). We further replce these qunttes, equtons (24) nd (28) n equton (27). The products of smll quntty vrtons, such s α u nd α w re neglected. We tke nto ccount the trgonometrc equlty sn 2 α + cos 2 α = 1 nd the trgonometrc functons for smll ngles of ttck vrtons (cos α = 1 nd sn α = α), therefore we obtn the vrton of the ngle of ttck s follows: sn cos α = α u+ α w (29) V V 5.3 Sdeslp ngle vrton β The thrd equton of the system of equtons (2) cn be wrtten, for smll vrtons of β j (nd for β = ), s : v tn β β= (3) 2 2 u + w We express β n the form of the products sum of the β dervtves wth respect to u, v, nd w t equlbrum nd the smll perturbtons of u, v, nd w: β u β β v β = + + w (31) u v w The expressons of β dervtves t equlbrum re clculted by the dervton of equton (45), where v =, V = u + w, nd we obtn: β = v 2u = u u + w v 2 1+ ( ) 2 2 u + w β u + w u + w = = = v + u + w + v = v 1 u w u + w V u + w β = v 2w = w u + w v 2 1+ ( ) 2 2 u + w (32) 1-8 RTO-MP-AVT-154

9 We replce the system of equtons (32) n equton (31) nd we obtn: 1 β = v (33) V Fnlly, V, α, nd β, from equtons (26), (29) nd (33), respectvely, re rerrnged n the form of mtrx system of equtons: cosα snα V u snα cosα α = v (34) V V β w 1 V We dd the system of equtons (34) to the ntl equtons of the Block 4 output vrtons u, v, w, x, y, z, p, q, r, φ, θ, nd ψ, whch become the Block 5 nputs, nd we obtn hgher order mtrx system of equtons: u v q w α x V y θ z (35) =H H = Bm p p q r r β φ φ θ ψ where 1 snα cosα V V cosα snα 1 (36) Bm = V 1 Block 6 descrpton The ntl prmeters of the mtrx system of equtons (35) re gven by NASA DFRC n the stblty nd control dervtve fles for ech flght condton chrcterzed by the Mch number, lttude nd the ngle of ttck: T T q α V θ H p r β φ = α V θ H (37) [ ] _ ns _ ns _ ns _ ns RTO-MP-AVT

10 Block 7 descrpton We clculte the r densty ρ s functon of lttude H, nd we obtn 5 : ρ LH = 1 (38) ρ T where ρ s the r densty t se level, L s the temperture lpse rte, T s the r temperture t se level, H s the lttude, R s the gs constnt nd g s the grvttonl constnt t se level. Block 8 descrpton The dynmc pressure q dyn s clculted 4 s functon of the rcrft s true rspeed V nd the r densty ρ. Block 9 descrpton The control surfce nputs re excted wth vrous sgnls. The control surfces for the F/A-18 rcrft nlyzed here re: the Ledng Edge LE flps lef = LEF, the Trlng Edge TE flps tef = TEF, the lerons l = AIL, the horzontl tl ht = HT, nd the rudder rud = RUD. One sgnl s used t tme on the rcrft control nputs, to gve devton of ± 5 round the equlbrum rcrft poston (trm) for ts longtudnl nd lterl rcrft moton n order to observe the forces nd moments vrton over tme. For the longtudnl moton, sgnl ws gven on the horzontl tl, nd for the lterl moton sgnl ws gven on the lerons. g 1 LR 2.2 STATE VARIABLES INTRODUCTION IN CLOSED LOOP We next develop second formulton from the frst formulton, n order to obtn the vectors of the generlzed coordntes nd ther tme dervtves n closed loop form: (,,,,, ) T T η = x y z φ θ ψ ; η = ( V, V, V, φ, θ, ψ ) (39) x y z The rrngement of stblty coeffcents on lnes 1 to 6 of the mtrces used n the frst formulton s: C, C, C, C, C, C (see equton (7)). In order to obtn the second formulton, t s necessry to L M N D Lft Y rerrnge the order of the stblty coeffcents s follows: CD, CY, CLft, CL, CM, C N. Therefore, the nottons A nt, BBlong_nt, nd B lt_nt re ntroduced for the ntermedte mtrces A, B long, nd B lt, n whch the order of the coeffcents s rerrnged. We show below only the A nt mtrx, whle B long_nt (6*6) mtrx contns the dervtves of the sme coeffcents s the ones of A nt wth respect to the longtudnl control surfces long_ail, long_ht, long_rud, long_lef nd long_tef, nd BBlt_ns s (6*6) mtrx contnng the dervtves of the sme coeffcents s the ones of A nt wth respect to the lterl control surfces lt_ail, lt_ht, lt_rud, lt_lef nd lt_tef. CD CD CD CD CD CD CD CD CD q α V θ H p r β ϕ CY CY CY CY CY CY CY CY CY q α V θ H p r β ϕ CLft CLft CLft CLft CLft CLft CLft CLft CLft (4) q α V θ H p r β Ant = ϕ CL CL CL CL CL CL CL CL C L q α V θ H p r β ϕ CM CM CM CM CM CM CM CM C M q α V θ H p r β ϕ CN CN CN CN CN CN CN CN C N q α V θ H p r β ϕ The ntermedte mtrx A m_nt for the A m mtrx s now wrtten s follows: 1-1 RTO-MP-AVT-154

11 4,1 4,2... 4,19 6,1 6,2... 6,19 5,1 5,2... 5,19 Am _ nt = Ant Blong _ nt Blt _ nt = = Am1_ nt Am 2 _ nt 1,1 1,2... (41) 1,19 2,1 2,2... 2,19 3,1 3,2... 3,19 where s the stblty coeffcents mtrx nd ts nlytcl formulton s gven n equton (42.1). A m1_nt The mtrx s the control coeffcents mtrx [ B B ] nd ts nlytcl form s expressed A m2_nt long _nt lt _nt by equton (42.2). 4,1 4,2... 4,9 6,1 6,2... 6,9 5,1 5,2... 5,9 (42.1) Am1_nt = 1,1 1,2... 1,9 2,1 2,2... 2,9 3,1 3,2... 3,9 4,1 4, ,19 6,1 6, ,19 5,1 5,11... (42.2) 5,19 Am 2_nt = Blong _nt B lt _nt = 1,1 1, ,19 2,1 2, ,19 3,1 3, ,19 Then, the second equton of stte spce system my be wrtten by use of equtons (42.1) nd (42.2) n the rerrnged order, s follows: long _ LEF q long _ TEF α CD long _ AIL V C Y long _ HT θ C (43) Lft long _ RUD = A m1_nt H + A m2_nt C L lt _ LEF p C M lt _ TEF r C N lt _ AIL β lt _ HT φ lt _ RUD We replce the left hnd sde term of equton (35) n the frst term on the rght hnd sde of equton (43) n order to obtn nother set of stte vectors x n the second equton of the stte spce system of equtons, nd we obtn: RTO-MP-AVT

12 u v long _ LEF w long _ TEF C D x long _ AIL C Y y long _ HT (44) C Lft z = H long _ RUD = Am1_ntBm Am2_nt C L p + lt _ LEF C M q lt _ TEF C N r lt _ AIL φ lt _ HT θ lt _ RUD ψ We replce the vector output gven on the left hnd sde of equton (44) n the rght hnd sde term of equton (15) n order to clculte the forces nd moments n the rcrft system of coordntes: u v long _ LEF w long _ TEF X x long _ AIL Y y long _ HT (45) Z z =H long _ RUD = CmAm 1_ntBm CmAm2_nt L p + lt _ LEF M q lt _ TEF N r lt _ AIL φ lt _ HT θ lt _ RUD ψ The followng nottons re ntroduced: C = C A B ; D= C A (46) nd we obtn: m m1_nt m m m2_nt u v w X x Y y Z z = H = C D L p + M q N r φ θ ψ long _ LEF long _ TEF long _ AIL long _ HT long _ RUD lt _ LEF lt _ TEF lt _ AIL lt _ HT lt _ RUD The scheme shown n the next fgure represents the converson of the scheme shown n Fgure 1 by use of equtons (39)-(47), nd hs the 12 stte vectors x gven by equton (39) n the closed loop. We obtned the C nd D mtrces, whch chrcterze the lner system t trm condton. These mtrces re presented n detl next. (47) 1-12 RTO-MP-AVT-154

13 lef tef l ht rud lef tef l ht rud u v w x y z p q r φ θ ψ X ( ) force Y ( ) force Z ( ) force L ( ) moment M ( ) moment N ( ) moment q Scb,, dyn V, V, V x y z x, y, z φ, θψ, DCM V (,, ) u v w p, qr, p, qr, u v w x y z p q r φ θ ψ Fgure 5: Smulton scheme wth 12 stte vectors n closed loop We cn represent the terms of the C nd D mtrces nlytclly. The C mtrx hs the followng nlytcl form: C11 C12 C13 C14 C = CmAm 1Bm = (48) C21 C22 C23 C24 where the C11, C12, C13, C14, C21, C22, C23, nd C24 mtrces re represented under nlytcl forms. The BBm mtrx s expressed by equton (51), the A m1_nt mtrx s gven by equton (42.1), nd ther product s wrtten s follows: snα 4,8 cos α 4,2 + 4,3 cosα 4,2 + 4,3 snα 4,5 4,6 4,1 4,7 4,9 4,4 V V V snα 6,8 cos α 6,2 + 6,3 cosα 6,2 + 6,3 snα 6,5 6,6 6,1 6,7 6,9 6,4 V V V snα 5,8 cos α (49) 5,2 + 5,3 cosα 5,2 + 5,3 snα 5,5 5,6 5,1 5,7 5,9 5,4 V V V Am 1Bm = sn α 1,8 cos α 1,2 + 1,3 cosα 1,2 + 1,3 snα 1,5 1,6 1,1 1,7 1,9 1,4 V V V snα 2,8 cos α 2,2 + 2,3 cosα 2,2 + 2,3 snα 2,5 2,6 2,1 2,7 2,9 2,4 V V V snα 3,8 cosα 3,2 + 3,3 cosα 3,2 + 3,3 snα 3,5 3,6 3,1 3,7 3,9 3,4 V V V The C mtrx s the product of the C m mtrx gven by equton (24) nd the A m1 B m mtrces product gven by equton (49), whle the C mtrx nlytcl elements re found by dentfcton. Due to the pges number lmttons, we gve here only the C11 mtrx elements: 2 4,2 sn 2α 2 5,2 sn α 5,3 sn 2α 4,8 cosα 5,8 snα c1,1 = S 4,3 cos α + ; c1,2 = S 2V V 2 + V V RTO-MP-AVT

14 2 4,2cos α 4,3 sn 2α 5,2 sn 2α 2 c1,3 = S + + 5,3 sn α ; V 2 2V 6,2 snα 6,8 6,2 cos α c2,1 = S + 6,3 cos α ; c2,2 = S ; c2,3 = S + 6,3 sn α V V V 2 4,2 sn α 4,3 sn2α 5,2 sn2α snα 2 c3,1 = S + 5,3 cos α ; c3,2 = S V 2 2V V The D mtrx my be wrtten under the form D [ D D ] 4,8 5,8 cosα V = 1 2 T, where D1 nd D2 re: d1,1 d1,2 d1,3 d1,4 d1,5 d1,6 d1,7 d1,8 d1,9 d1,1 D1 = d2,1 d2,2 d2,3 d2,4 d2,5 d2,6 d2,7 d2,8 d2,9 d2,1 d3,1 d3,2 d3,3 d3,4 d3,5 d3,6 d3,7 d3,8 d3,9 d 3,1 d d d d d d d d d d 4,1 4,2 4,3 4,4 4,5 4,6 4,7 4,8 4,9 4,1 = 5,1 5,2 5,3 5,4 5,5 5,6 5,7 5,8 5,9 5,1 D2 d d d d d d d d d d d d d d d d d d d d 6,1 6,2 6,3 6,4 6,5 6,6 6,7 6,8 6,9 6,1 (5) (51.1) (51.2) The mtrx D= [ D1 D2] T = CmAm2_nt s the product of the mtrx gven by equton (16) nd the A gven by equton (42.2). The elements of mtrces D1 nd D2 re clculted by m2_nt dentfcton, nd we show here only the frst elements of D1 mtrx. d = S cosα + sn α ; d = S cosα + snα ( ) ( ) ( cosα sn α ); ( cosα snα ) ( cosα sn α ); ( cosα snα ) ( 16 cosα + 5,16 sn α ); d1,8 = S ( 4,17 cosα + 5,17 snα ) ( cosα sn α ); ( cosα sn ) 1,1 4,1 5,1 1,2 4,11 5,11 d = S + d = S + 1,3 4,12 5,12 1,4 4,13 5,13 d = S + d = S + 1,5 4,14 5,14 1,6 4,15 5,15 d = S 1,7 4, d = S + d = S + α 1,9 4,18 5,18 1,1 4,19 5,19 The C mtrx (only ts C11 elements re shown n equtons (5)) nd the D mtrx (only ts frst elements re shown n equtons (52)) re further replced n equton (47). Therefore, we obtn the two mtrx equtons for the vrtons of forces X, Y et Z nd moments L, M et N n the rcrft system of coordntes: long _ LEF long _ TEF long _ AIL long _ HT X u x p φ (53.1) long _ RUD Y = C11 v + C12 y + C13 q + C14 θ + D1 lt _ LEF Z w z r ψ lt _ TEF lt _ AIL lt _ HT lt _ RUD C m (52) 1-14 RTO-MP-AVT-154

15 L u x p φ M = C21 v + C22 y + C23 q + C24 θ + D2 N w z r ψ long _ LEF long _ TEF long _ AIL long _ HT long _ RUD lt _ LEF lt _ TEF lt _ AIL lt _ HT lt _ RUD (53.2) In equtons (53.1) nd (53.2), there re terms u, v, w, p, q, nd r n the rcrft coordnte system nd terms x, y, z, φ, θ, nd ψ n the nertl coordnte system. We need to obtn these terms n the sme system of coordntes, nd we chose the nertl system of coordntes. Therefore, we convert the lner nd ngulr speeds terms (u, v, w, p, q, nd r) from the rcrft coordntes system, nto the nertl coordnte system. The speeds u, v, nd w n the rcrft system of coordntes re obtned, usng lnerzton, from the lner speeds nd Euler ngles n the nertl system of coordntes. Equton (18) cn be wrtten n the followng form: u cosθcosψ cosθsn ψ snθ Vx v sn sn cos cos sn sn sn sn cos cos sn cos V (54) = φ θ ψ φ ψ φ θ ψ+ φ ψ φ θ y w cos φsnθcos ψ snφsnψ cos φsnθsnψ snφcos ψ cos φcos θv + z The smll perturbtons theory s ppled to the followng qunttes: u, v, w, V x, V y, V z, p, q, nd r. The roll nd yw ngles t equlbrum gven by NASA DFRC re equl to zero, φ, nd ψ. The trgonometrc functon (sne nd cosne) ssumptons for smll ngles φ, θ, nd ψ re lso consdered. The products of smll terms such s φ ψ, θ Vx, θ Vz, φ Vx nd φ Vy re neglected, nd we consder the equlbrum speeds V y nd V z to be zero. We replce these ssumptons n equtons (54) nd we obtn: u Vx φ v = P Vy + P1 θ (55) w V z ψ where: cosθ snθ P = 1 snθ cosθ nd V snθ x =Vx θ V x P1 sn V cosθ x (56) The second type of lnerzton concerns the clcultons of ngulr speeds p, q, nd r n the rcrft system of coordntes from the Euler ngles nd ther dervtves n the nertl system of coordntes. The ngulr speeds p, q, nd r re expressed 4 s functons of Euler ngles φ, θ, nd ψ, respectvely, nd ther tme dervtves φ, θ nd ψ. The smll perturbtons theory s ppled to the Euler ngles φ, θ, nd ψ, ther tme dervtves φ, θ nd ψ nd to the ngulr speeds p, q, nd r. The ntl dt gven by the NASA DFRC lbortores re gven for the ngulr speeds p = q = r =, Euler ngles nd ther tme dervtvesφ = ψ =, θ = α, nd φ = θ = ψ =. The products of smll ngle vrtons such s RTO-MP-AVT

16 φ θ nd other products of such smll ngles cn be neglected. For smll ngle vrtons φ, θ, nd ψ (sn φ = φ, cos φ = 1,...), we cn wrte the followng system of equtons : p φ ψ snθ 1 snθ φ φ q = θ = 1 θ = R θ r cosθψ cosθ ψ ψ (57) where R s the followng mtrx: 1 snθ R = 1 cosθ (58) We next replce the vectors (u v w) T gven by equtons (55) nd (56) nd (p q r) T gven by equtons (57) nd (58) nto the two sets of equtons (53.1) nd (53.2) for the vrtons of forces X, Y, Z nd moments L, M nd N n the rcrft system of coordntes, nd we obtn: X Vx X φ φ Y = C11 P Vy + C12 Y + C13 R θ + ( C14 + C11 P1) θ + D1 Z V z Z ψ ψ l t lt lt long _ LEF long _ TEF long _ AIL long _ HT long _ RUD lt _ LEF lt _ TEF _ AIL _ HT _ RUD (59.1) 1-16 RTO-MP-AVT-154

17 L Vx x φ φ M = C21 P Vy + C22 y + C23 R θ +( C24 + C21 P1) θ + D2 N V z z ψ ψ lt The development of equtons (59.1) nd (59.2) cn be seen n Fgure 6: long _ LEF long _ TEF long _ AIL long _ HT long _ RUD lt _ LEF lt _ TEF lt _ AIL lt _ HT _ RUD (59.2) x y η z φ θ ψ Vx Vy η Vz φ θ ψ lef tef l ht rud lef tef l ht rud X ( ) force Y ( ) force Z ( ) force L ( ) moment M ( ) moment N ( ) moment q Scb,, dyn V, V, V x y z x, y, z φ, θψ, DCM V (,, ) u v w φ, θ, ψ x y z φ θ ψ V x V y V z φ θ ψ η η η η Fgure 6: New scheme ncludng the vrtons of stte vectors nd ther tme dervtves η nd η n closed loop form In Fgure 6, we ntroduce the followng nottons for the stte vector vrtons η nd η : [ x y z ] T ; η = φ θ ψ (6) T η = Vx Vy Vz φ θ ψ 2.3 AERODYNAMIC FORCES FOR RIGID-TO-RIGID AND RIGID-CONTROL INTERACTIONS MODE CALCULATIONS To contnue ths work, we need to clculte the erodynmc forces nd moments vrtons for rgd-torgd Q rr nd rgd-to-control Q rc ntercton modes n the nertl system. Lnerzton of forces nd Euler ngles from the rcrft system of coordntes nto the nertl system s relzed by use of the smll perturbtons theory. The nertl forces vrtons X, Y, nd Z re wrtten s functons of force vrtons n the rcrft RTO-MP-AVT

18 system of coordntes X, Y, nd Z by use of the CDM n form smlr to the one gven by equtons (18). The smll perturbton theores re ppled to the Euler ngles, nd then we obtn two sets of equtons for the forces nd for the moments, under the followng form: X X φ X X Y = F Y + F1 θ = F Y + = F Y Z Z ψ Z Z (61.1) nd for the moments: L L φ L L M = F M + F1 θ = F M + = F M N N ψ N N (61.2) where cosθ snθ F = 1 snθ cosθ (62) The forces nd moments vrtons n the rcrft system of coordntes gven by equtons (59.1) nd (59.2) re replced n equtons (61.1) nd (61.2), n whch vector of zeros of dmensons (3x1) s dded, nd therefore, the forces nd moments re obtned n the nertl system of coordntes : long _ LEF long _ TEF long _ AIL long _ HT X x φ long _ RUD Y = F C12 y + F( C14 + C11 P1) θ + F D1 + lt _ LEF Z z ψ lt _ TEF lt _ AIL lt _ HT lt _ RUD Vx φ + F C11 P Vy + F C13 R θ + Zeros(3x 1) ( ) V z ψ T (63.1) 1-18 RTO-MP-AVT-154

19 long _ LEF long _ TEF long _ AIL long _ HT L x φ long _ RUD M F C22 y F( C24 C21 P1) θ F D2 = lt _ LEF N z ψ lt _ TEF lt _ AIL lt _ HT lt _ RUD Vx φ + F C21 P Vy + F C23 R θ + Zeros(3x1) ( ) V z ψ T (63.2) We rrnge equtons (63.1) nd (63.2) n order to obtn the generlzed coordnte vectors vrtons η nd η gven by equtons (6), nd the control vectors u nd u. RTO-MP-AVT

20 12 ( ) 1 22 ( ) 2 long LEF lon g T E F lo ng A IL lo ng H T long RUD lt L E F l t T E F l t A IL l t H T l t R U D x y z X Y Z F C F C C P F D L F C F C C P F D M N φ θ ψ + = x y z V V V F C P F C R F C P F C R φ θ ψ + + (64) The system of equtons (64) s smulted wth the scheme shown n Fgure RTO-MP-AVT-154

21 x y η z φ θ ψ Vx Vy η Vz φ θ ψ X Y Z L M N X Y Z L M N V, V, V x y z x, y, z φ, θψ, DCM V (,, ) u v w φ, θψ, x y z φ θ ψ V x V y V z φ θ ψ η η η η lef tef l ht rud lef tef l ht rud Scb,, q dyn Fgure 7: Scheme wth force nd moment vrtons n the rcrft system of coordntes from the nertl system of coordntes In order to clculte the erodynmc forces Q rr nd Q rc, we need to wrte the equton of moton for the flexble rcrft structure n terms of generlzed coordntes, n the followng form: M η+dη+kη + Qη = (65) q dyn whch cn be wrtten n smplfed form s: M η +Dη +Kη + y = (66) 1 where the lst term of equton (66) s: y1 = F Qη=[ ] T ero = qdyn X Y Z L M N (67) The erodynmc forces Q hve rel nd mgnry prts, nd for ths reson equton (67) cn lso be expressed s: RTO-MP-AVT

22 jωη y1 = q ( Q + j Q ) η = q Q η + q Q ω dyn R I dyn R dyn I (68) j t From the generlzed coordntes defntonη = Ae ω, where A s the mpltude of moton nd ω s the osclltons frequency, we clculte the generlzed coordntes dervtve wth tme η = j ωae jωt =jωη. The reduced frequency s gven by the followng equton: ωb ωc k = = (69) V 2V We replce ω clculted wth equton (69) nto equton (68) nd we obtn: η c y1 = qdynqrη + qdyn QI = qdynqrη+ qdyn Q Iη (7) ω 2kV The rel prts of erodynmc forces Q R correspond to the stte vectors x nd control vectors u nd the mgnry prts of erodynmc forces Q I correspond to the tme dervtves of stte vectors x nd control vectors u, nd therefore we cn wrte: R R c I c I y1 = qdynqrrη+ qdynqrcu+ qdyn Q rrη+ qdyn Qrcu (71) 2Vk 2Vk Equton (71) my be expressed n the form of the second stte spce equton: R c I η R c I u y1 = qdyn Qrr Qrr + qdyn Qrc Qrc 2Vk η 2Vk u (72) Equton (64) my be wrtten by use of stte vectors η nd ther tme dervtves η nd wth control vectors u nd ther tme dervtves u, s follows: X Y Z FC 12 F ( C14+ C11 P1) FD 1 η FC 11 P FC 13 R η y1 = = L F C22 F ( C24 C21 P1) F D2 + + u F C21 P F C23 R u M N (73) 1-22 RTO-MP-AVT-154

23 Equton (72) cn lso be presented n the followng form: T R R η c I c I η y1 = [ X Y Z L M N ] = ( qdynqrr qdynqrc ) + qdyn Qrr qdyn Qrc u 2Vk 2Vk u (74) The Q nd Q mtrces re represented under nlytcl form (wth 6 rows nd 16 columns). By rr rc dentfcton of the erodynmc forces mtrces gven n equtons (73) nd (74), we clculte the terms of the rel erodynmc forces for rgd-to-rgd mode nterctons erodynmc forces for rgd-to-control mode nterctons R Q rc R Q rr, nd thus we obtn: nd the terms of the rel Q R R F C12 F ( C14 + C11 P1) Qrr _11 Q rr _12 = F C22 F ( C24 C21 P1) = + Q rr _21 Qrr _22 R rr R R (75.1) Q R rc R F D1 Q rc _11 = = R F D2 Q rc _21 (75.2) The elements nlytcl forms: Q, Q, Q, Q, Q, Q R R R R R R rr _11 rr _12 rr _21 rr _22 rc _11 rc _21 of rel erodynmc forces hve the followng qr1,1 qr1,2 qr1,3 qr1,4 qr1,5 qr1,6 R R Qrr _11 = qr2,1 qr2,2 qr2,3 ; Q (76.1) rr _12 = qr2,4 qr2,5 qr2,6 qr3,1 qr3,2 qr 3,3 qr3,4 qr3,5 qr 3,6 qr4,1 qr4,2 qr4,3 qr4,4 qr4,5 qr4,6 R R Qrr _21 = qr5,1 qr5,2 qr5,3 ; Qrr _ 22 = qr5,4 qr5,5 qr5,6 (76.2) qr6,1 qr6,2 qr 6,3 qr6,4 qr6,5 qr 6,6 qr1,7 qr1,8 qr1,9 qr1,1 qr1,11 qr1,12 qr1,13 qr1,14 qr1,15 qr1,16 R Qrc _11 = qr2,7 qr2,8 qr2,9 qr2,1 qr2,11 qr2,12 qr2,13 qr2,14 qr2,15 qr2,16 qr3,7 qr3,8 qr3,9 qr3,1 qr3,11 qr3,12 qr3,13 qr3,14 qr3,15 qr 3,16 (76.3) RTO-MP-AVT

24 qr4,7 qr4,8 qr4,9 qr4,1 qr4,11 qr4,12 qr4,13 qr4,14 qr4,15 qr4,16 R Qrc _ 21 = qr5,7 qr5,8 qr5,9 qr5,1 qr5,11 qr5,12 qr5,13 qr5,14 qr5,15 qr5,16 qr6,7 qr6,8 qr6,9 qr6,1 qr6,11 qr6,12 qr6,13 qr6,14 qr6,15 qr 6,16 (76.4) By dentfcton of the mgnry prts of erodynmc forces nd n equtons (73) nd (74), we cn wrte: I Q rr I Q rc Q I rr I I FC 11 P FC 13 R Qrr _11 Q rr _12 I = = ; rc [ I I Q = FC 21 P FC 23 R Qrr _21 Q rr _22 ] T (77) The elements of the mgnry erodynmc forces I _11, I _12, I _21, I _22, I I Qrr Qrr Qrr Qrr Q rc _ 11 nd Q rc _21 hve the nlytcl forms gven n the followng equtons: q1,1 q1,2 q1,3 q1,4 q1,5 q1,6 I I Qrr _11 = q2,1 q2,2 q2,3 ; Qrr _12 = q2,4 q2,5 q2,6 (78.1) q3,1 q3,2 q 3,3 q3,4 q3,5 q 3,6 q4,1 q4,2 q4,3 q4,4 q4,5 q4,6 I I Qrr _21 = q5,1 q5,2 q5,3 ; Qrr _22 = q5,4 q5,5 q5,6 (78.2) q6,1 q6,2 q 6,3 q6,4 q6,5 q 6,6 We next show the clculton of one sngle term correspondng to the rel prts of the erodynmc forces, snce the sme theory s used to clculte ll of the terms of the rel nd mgnry erodynmc forces. Plese note tht the F mtrx s gven by equtons (62) nd the C12 mtrx s nlytclly gven. Q cosθ snθ c1,4 c1,5 c1,6 qr qr qr = F C12 = 1 c2,4 c2,5 c2,6 = qr qr qr snθ cosθ c3, 4 c3 c 3, 6 qr3, 1 qr3, 2 qr3 1,1 1,2 1,3 R rr _11 2,1 2,2 2,3,5,3 (79) The coeffcents equl to zero re coloured n blue (the frst two columns of C12 re zero) whle the coeffcents not equl to zero re coloured n red. By dentfcton of the R Q rr_11 mtrx elements expressed by equton (79) nd of the C mtrx vlues gven by equtons (5) nd other equtons (not shown, due to the pges number lmtton): 1-24 RTO-MP-AVT-154

25 2,1 2,2 2,3 2,6 6,5 ( ) ( ) qr1,1 = ; qr1,2 = ; qr1,3 = c1,6 cosθ + c3,6 snθ = S 4,5 cos θ α 5,5 sn θ α qr = ; qr = ; qr = c = S ( ) ( ) qr3,1 = ; qr3,2 = ; qr3,3 = c1,6 snθ + c3,6 cos θ = S 4,5 sn θ α 5,5 cos θ α (8) The s coeffcents terms n equtons (8) contn stblty nd control dervtves s seen n equtons (1)-(6). Therefore, we obtn, for ll rgd erodynmc elements Q R, the followng tble, n whch the frst 3 elements on column 4 were gven n equton (8). Q r Q x y η z φ θ ψ Vx Vy η Vz φ θ ψ lef tef l ht rud lef tef l ht rud q ( ) dyn Qr η + Q η X Y Z L M N q Scb,, dyn X Y Z L M N Vx, Vy, Vz x, y, z φ, θψ, DCM V (,, ) u v w φ, θψ, x y z φ θ ψ V x V y V z φ θ ψ η η η η Fgure 8: Smulton scheme for erodynmc force clcultons from generlzed coordntes x y X S C cos( θ α ) C sn ( θ α ) Y z ( d lft ) h S C yh ( ) ( l n ) Z S Cd sn ( θ α) C cos( ) h lft θ α h L S b C cos( θ α ) + C sn ( θ α ) M h S c C mh ( l n ) N S b C sn ( θ α ) + C cos ( θ α ) h h h h RTO-MP-AVT

26 X Y φ θ ψ S Cd α S Cy β snθ Z S ( Clft Clft ) L S b Cl β snθ M S c ( C + C ) N X S b Cn β snθ x Y Z L M N θ mθ α mα Tble 1 Rel erodynmc forces Q R y S C dv Cyβ S V S C lftv x x lβ z Cd α S V C S S b C V S c C mv x nβ x lftα V x S c C V x S b C V S Cd S Cy β S Cy S Clft S b Cl β S b Cl S c Cm S b Cn β S b Cn mα X Y Z L M N φ θ ψ S C d p S C yp S C dq S snθ C d p S ( Cy snθ + C cos ) p y θ r S C lft p S b C lp S C lftq S snθ C lft p S b C snθ + C cosθ ( l ) p lr S c C mp S b C np S c C mq S c C snθ S b ( Cn snθ + C cos ) p n θ r m p Tble 2 Imgnry erodynmc forces Q I 1-26 RTO-MP-AVT-154

27 The frst smulton scheme (Fgure 1) represents system tht uses the stblty nd control dervtves n the wnd system of coordntes nd n whch the forces nd moments re clculted n the rcrft system of coordntes. The second smulton scheme (Fgure 8) represents n equvlent scheme smlr to the frst, n whch the sttes n the nertl system of coordntes re used n the closed loop. 2.4 NUMERICAL LINEARIZATION SCHEME The formultons lredy developed n Sectons 1-3 wll be vldted for prtculr cses where the sme types of stblty nd control dervtves re gven n the wnd system of coordntes. A problem ppers f we dd, subtrct or chnge dervtve or ts ntl vlue n the formultons, becuse then ll the formultons chnge. Therefore, we need to develop utomtc smulton formultons. We develop Mtlb lgorthm n whch ll of the bove chnges n coordntes nd lnerztons re utomtclly relzed. We use the dlnmod Mtlb functon whch gves, n the stte spce form, the lnerzed form of system bult n Smulnk, round specfed trm pont condton. The smulton scheme presented n Fgure 9 s equvlent to the scheme presented n Fgure 1 nd tkes nto consderton the stblty dervtves gven n the wnd coordnte system. u F w M w Trnsformton of wnd coord s system to rcrft coord s system F M Prmeters 6 dof 5 4 Fgure 9: Smulton rcrft scheme wth stblty nd control dervtves gven n the wnd coordntes system In Block 1 of Fgure 9, the forces nd moments F w nd M w re clculted from the stblty nd control dervtves gven n the wnd system of coordntes. To smulte rcrft behvor, we convert these forces nd moments, determned n block 1, to forces nd moments n the rcrft coordnte system shown n block 3, usng block 2 for the trnsformton from the wnd coordnte system to the rcrft coordnte system. The rcrft lner nd ngulr speeds re the block 4 outputs nd re clculted from the forces nd moments n the rcrft coordnte system by use of the 6 degrees-of-freedom equtons of moton. Prmeters specfc to the rcrft n the wnd system of coordntes, such s the ngle of ttck, sdeslp ngle, nd true rspeed re further clculted n block 5 nd used s nputs to block 1 n the rcrft tme smulton. The scheme presented n Fgure 9 round the trm pont specfed n the smulton s then lnerzed by use of the dlnmod commnd n Mtlb, where the lner reltonshp between the nputs u nd the outputs y s obtned under stte spce form, n whch the stte vectors re x = ( uvwpqr,,,,,, φθψ,,, x, y, z). The components of the stte vectors x re the lner speeds u, v, nd w, the rtes p, q, nd r, the ngles φ, θ nd ψ nd the three postons x, y nd z. The scheme presented n Fgure 9 cn further be lso represented under stte spce form n Fgure 1. RTO-MP-AVT

28 u x = Ax + Bu y = Cx + Du y = F, M Fgure 1: Equvlence wth the scheme shown n Fgure 9 Then, the stte vrbles x re clculted wth the frst stte spce system equtons nd re further used n the second equton of the stte spce system to clculte the outputs y. In ths cse, we fnd the next scheme, equvlent to tht shown n Fgure 1, n whch the block 4 outputs, from the 6 degrees of freedom (dof) equtons of moton (see Fgure 9), re used. As the schemes shown n Fgures 9 nd 11 re compred, t s obvous tht blocks 1, 2, nd 4 re contned n the C nd D mtrces. The rcrft model thus obtned gves the vrtons of forces nd moments clculted n the rcrft coordnte system dependent on the nputs u nd stte vectors x. The mtrces Qrr nd Qrc, correspondng to rgd-to-rgd nd to rgd-to-control mode nterctons, respectvely, re determned for erodynmc forces n ths wy. The sttes multplyng these mtrces, nd the forces nd moments, re clculted n the nertl system of coordntes. The scheme shown n Fgure 11 s redesgned by ddng frst block whch chnges the outputs y nto y nd second block whch chnges the sttes x nto the sttes x. Ths new scheme s presented n Fgure 12. u y = Cx + Du y = F, M 6 dof Fgure 11: Equvlent schemes wth the Fgure 2 u y y = C x + D u y to y y = F, M x to x 6 dof Fgure 12: Smulton scheme wth forces nd moments clculted n the nertl system of coordntes The chnges to the coordntes mplemented n the dded blocks depend on the trgonometrc functons of Euler ngles, more specfclly the nputs re relted to the outputs by nonlner functons. The two blocks 1-28 RTO-MP-AVT-154

29 re then further lnerzed, whch requres tht the Mtlb commnd dlnmod be ppled to the two blocks. For the block x to x, lner reltonshp s obtned: x = Dx (81) 1 whch represents smplfed form of stte spce system where mtrces A, B, nd D 1 re zeros, s ths block hs no sttes. The lnerzton of block y to y gves: y = D y (82) 2 By usng the blocks shown n Fgure 12, nd equtons (81) nd (82), we wrte: ( ) ( ) y D y D Cx Du D CDx Du D CDx D Du = 2 = 2 + = = (83) Identfyng the mtrces gven by equton (83) wth those gven by the stte spce equtons: C = D CD; D= D D (84) The C nd D mtrces re obtned from the lnerzton presented n Fgure 9, nd the D 1 nd D 2 mtrces re obtned by the lnerzton of two blocks shown n Fgure 12. We clculte the C nd D mtrces wth equtons (84): C = D CD ; D = D D (85) nd the stte vector x s: T (,,,,,,,,,,, ) [ T x = x y z φ θψ x y z φθψ = η η] (86) r r We cn wrte the erodynmc force equtons for the rgd modes n the followng form, smlr to equton (71): R c I R c I qdyn Qrrηr + Qrrη r = qdynqrr qdyn Q rr x = C x 2Vk 2Vk (87) The rel nd mgnry prts of the erodynmc forces correspondng to the rgd modes re obtned by dentfcton from equton (87): R 1 I 2Vk Q = rr C ( 1:6,1:6 ); Qrr C ( 1:6,7:12 q = ) cq (88) dyn dyn RTO-MP-AVT

30 u = η c. The rel nd mgnry prts of the erodynmc forces correspondng to the nterctons of rgd modes wth the control modes re determned wth the followng equtons: The control vector u s [ ] T R 1 I Qrc = D ; Qrc = (89) q dyn The lgorthm s presented n the followng three steps: 1. We pply the Mtlb commnd dlnmod to ech of the 3 blocks n order to obtn C, D, D 1 nd D 2 mtrces; 2. Equtons (88) nd (89) re used to clculte the mtrces Q rr nd Q rc nd 3. The ntl mtrces clculted wth the Doublet Lttce method, DLM, or wth the Constnt Pressure Method, CPM, clculted by fnte element softwre, re replced wth the mtrces obtned wth our lgorthm. For vldton purposes, the lgorthm represents the numercl mplementton (shown n Secton 4) of nlytcl mplementton presented n Sectons 1-3. A comprson between the obtned vlues wth the nlytcl formulton nd the numercl formulton presented here s gven n the followng tbles. x y z x y z x y z x y z Tble 3 Rel prt of Q rr mtrx obtned by nlytcl lnerzton x y z x y z x -7.45E y z -4.38E x y -6.16E z Tble 4 Rel prt of Q rr mtrx obtned by numercl lnerzton 1-3 RTO-MP-AVT-154

31 x y z x y z x y z x y z Tble 5 Imgnry prt of Q rr mtrx obtned by nlytcl lnerzton x y z x y z x y z x y z Tble 6 Imgnry prt of Q rr mtrx obtned by numercl lnerzton We cn see tht the obtned results re the sme. Developng the numercl lgorthm llowed us to verfy the prtl nd fnl results for 9 flght test condtons 6. CONCLUSIONS In ths pper, we used two pproches: nlytcl (Sectons 1 to 3) nd numercl (Secton 4) to vldte the erodynmc force formultons correspondng to rgd-to-rgd nd rgd-to-control ntercton modes for eroservoelstcty studes -- only from knowng the stblty nd control dervtves n the wnd system of coordntes. These dervtves re dependent on flght regme condtons: Mch number, lttude nd ngle of ttck. In fct, the erodynmc forces correspondng to rgd nd control ntercton modes clculted wth fnte element softwre re replced wth erodynmc forces clculted usng both formultons presented here. Wth nother rcrft, nd thus wth dfferent set of stblty nd control dervtves, we wll use the numercl pproch combned wth the theoretcl pproch to vldte the new formultons. The numercl pproch wll lkely be much fster thn the theoretcl development, becuse successve lnerztons my tke qute long tme. The theoretcl pproch my become much more useful n the future. The nlytcl pproch developed n Sectons 1-3 llows us to obtn the nlytcl formuls for ll the stblty nd control dervtves n the nertl system from those clculted n the wnd system of coordntes. Lnerztons t the trm condton re performed t ech clculton step. The second pproch conssts of numercl lnerzton of the smulton scheme n the wnd reference system of coordntes. Vlues of stblty nd control dervtves obtned wth ths method nd those clculted nlytclly wth the frst method re the sme, therefore, we conclude tht the expressons RTO-MP-AVT

32 found for the erodynmc forces correspondng to rgd nd control modes re vldted. A comprson ws done between the flutter frequences nd dmpng vlues for eroelstcty studes (where only the elstc-elstc erodynmc forces Q ee were consdered) wth the flutter frequences nd dmpng vlues for eroservoelstcty studes (where ll the erodynmc force mtrx ws consdered). Thus, the common flutter frequences nd dmpng vlues were obtned for both eroelstcty nd eroservoelstcty studes. Addtonl flutter modes were obtned for the eroservoelstcty mtrx where rgd nd control modes dynmcs were ntroduced. Ths comprson ws nother wy to vldte our formulton, but ws not presented here n detls. ACKNOWLEDGEMENTS Mny thnks re due to Mr Mrty Brenner from NASA Dryden Flght Reserch Center for hs contnuous ssstnce nd collborton n ths work. Fnncl support ws gven n ths project by the NSERC (Nturl Scences nd Engneerng Reserch Councl of Cnd) nd by MDEIE (Mnstère du Développement économque, de l Innovton et de l Exportton). REFERENCES 1. Gupt, K.K., 1997, STARS An Integrted, Multdscplnry, Fnte-Element, Structurl, Fluds, Aeroelstc, nd Aeroservoelstc Anlyss Computer Progrm, NASA TM-4795, pp Rodden, W.P., Hrder, R.L., Bellnger, E.D., 1979, Aeroelstc Addton to Nstrn, NASA CR Lnd, R., Brenner, M., Robust Aeroservoelstc Stblty Anlyss, Sprnger-Verlg London Ltd., Nelson, R.C., Flght dynmcs nd Automtc Control, McGrw Hll Inc., USA, Shevell, R.S., Fundmentls of Flght, Prentce Hll, Upper Sddle Rver, New Jersey 7458, RTO-MP-AVT-154