CONTROL SYSTEM WIT AIR BLEED FLAPS FOR SUPERSONIC AIR INTAKES Aleandr Nicolae TUDOSIE University of Craiova, E-mail:atdosie@elth.cv.ro Abstract-The aer deals with an atomatic control system for a class of sersonic intakes, based on the bleed air flas' oening with resect to the aircraft's engine's oerating regime. The control law δ = δ(π c ) is established, which gives the nivoc corresondence to the intake's rear ressre. An atomatic control system is stdied, from the oint of view of its mathematical model, its stability and qality. One has also erformed some simlations in order to reveal the system's behavior for several engine oerating regimes, for a constant sersonic flight regime. Keywords: air intake, inlet, fla, shock wave, control system, servo-amlifier, actator.. INTRODUCTION The sersonic inlets and intakes for low sersonic seed aircraft (.5.7 Mach) or for high sersonic seed aircraft, monted nder the wing or on the fselage, are less inflenced by the flight regime's changes (altitde and seed V) than the front intakes. This matter involves an intake design, manfactring and control simlifying, that means that one can se a fi sike, fi anels and/or fi cowl, bt one mst assme some aerodynamic losses increasing, bt small enogh to indce intake's and/or the engine's malfnctions. Althogh, the intake's control with resect to the engine's regime shall be realised by the air bleed flas ositioning, in order to match the delivered air flow rate to the engine's necessary air flow rate and avoid 5 4 3 6 2 the intake's and comressor's stall. Sch a control system is resented in fig. nr., where the air intake (3), with or withot sike (2), bt with air bleed flas (4), is laterally monted on the aircraft fselage () and delivers the necessary air flow for the engine (5). 2. CONTROL LAW'S ESTABLISING The control law is the mathematical eression for the fla's oening with resect to one or more of the engine's and/or flight arameters. Air bleeding flas' role is to control the internal shock wave ositioning throgh the static ressre in section control, that means the comressor's inlet's ressre ( ), in order to assre the balance between the engine's necessary air flow rate and the intake's available air flow rate. The shock wave mst be stabilised in a recisely designed zone the ressre ( ) mst have sch a vale, that in the minimm section area the flow shall be sonic (critical) and a sersonic tamon-zone shall eist between and the normal shock-wave (see figre nr. 2). The shock-wave's osition, as well the intake's total ressre ratio σ DA, can be determined based on the flow's arameters, the ressre in the front of the comressor (the comressor's oerating regime) and on the intake's geometry. One can assme a linear variation of the intake's flow section (from A to A ): A() = A (A A ). () L A general form for the air flow rate's eression is [7] 6 sbsonic zone sonic zone sersonic zone sbsonic zone normal shock wave ehast air bleed fla ' (i) min (cr) (e) air intake << S c k m Q c 4 3 7 sike (cone) am av k el B A >v> L Q 8 Q 7 l 2 l R 4 >y> Q 3 Q 4 Q 5 Q 6 alimentare ( a ) >> Q 2 8 cr s 2 9 l 3 l 4 Figre : Atomatic control system for sersonic intakes based on the air bleed flas' oening < r < 5 s am av inf Figre 2: Internal shock wave's ositionning 353
K Ṁ = KA, (2) T q(λ) k k K = k 2 one can eress the flow rate for R k the sections, and for the shock-wave (before it "am", after it "av") KA = KA av T av T q(λ ) = KA am q(λ av ) = KA T am q(λ am ) = T q(λ ), (3) where A is the intake's section area where the shock-wave aears. One can assme [7] that av = σ s am, = am, av =, λ av =, and knowing the λ = am Π(λ ) flow's Mach nmber M one can determine λ, and from and M it reslts. From (3) one obtains KA λ T q(λ ) = KA Π(λ ) q(λ ) Π(λ ) = q(λ ) A A, (4) T q(λ ), (5) where is nknown. This transient eqation has two soltions for λ, λ < and λ 2 > the right one is the first one, which gives sb-sonic flow regime in section. One obtains, from (3) KA, (6) T q(λ ) = KA T q(λ ) with λ above determined, = am, = av and = av = σ. The ressre loss ratio is given by s am The same eqation (3) leads to KA am T am σ s = A q(λ ). (7) A q(λ ) q(λ am ) = KA which gives a transient eqation with σ sd = q(λ am) q λ am av T av λ am q(λ av ), (8) as argment (9) the correct soltion is the ser-sonic one, λ am >, which gives, from the first eqation (3) KA where is a constant which deends on the adiabatic One obtains eonent k and on the air erfect gas constant R T q(λ ) = KA am T am q(λ am ). () q(λ ) A = A, () q(λ am ) and, from Eq.(), one can determine the distance s s = L q(λ ). (2) A q(λ am ) A Using this algorithm one determines an nivoc relation between the shock wave's osition and the ressre in section (given by the engine's oerating regime-n and the flight regime-, T, M ). In order to obtain an intake correct oeration, the shock wave mst be ket in a recisely determined zone and the ecedentary air mst be evacated throgh the flas' oening the air flow rate balance eqation, for a flight seed V and a flight altitde is M a = M ac M a = intake air flow rate, M a, (3) M ac = comressor's air flow rate, M a = evacated air flow rate M a = f(v,, A ), that means M a = (M, A ) M ac = f( n T, π c ) M a deends on the ressre difference between the intake's inner zone and the atmoshere, as well on the flas' horifice area (flas' ositioning) M a (δ) = M a M ac n (4) T, π c M a = µ 2ρ A V, (5) where µ = shae flow rate co-efficient ( µ, 7), ρ is the air density in section, given by ρ =, (6) RT A V is the flas' oening area A V = b V l V sin δ, (7) b V, l V =flas' dimensions, δ=flas' oening angle. So, M a = µ RT M a = KA b V l V sin δ, (8). (9) T q(λ ) A relation between δ-angle and the comressor's ressre ratio π c or π c, δ = f(π c ), resectively δ = δ( / ), which involves the = ressre in the front of the comressor is given in figre 3. 354
Q 6 = µ d6 2ρ R, (27) M =, 65 M =, 5 M =, 35 M =, 2 Figre 3: Control law (Flas' oening angle with resect to the engine's comressor's ressre ratio) 3. SYSTEM'S MATEMATICAL MODEL 3.. System's non-linear eqations Non-linear eqations establishing is based on the system fnctional diagram in figre and consists of: - transdcer's (7) eqations: Q 3 Q 4 = βv L d L Q 5 Q 6 = βv R d R S dy S dy d 2 y ( L R )S = m 2 ξ dy 2 2, (28), (29), (3) where Q 3, Q 4, Q 5, Q 6 air flow rates throgh the doble amlifier (8), µ d3, µ d4, µ d5, µ d6 flow rate co-efficients, a hydralic slying ressre (as- smed as constant), R, L ressres in servo-am- lifier's (9) chambers L and R, V R chamber's volme, S slide valve's iston's area, y slide-valve's disla- cement, fla's dislacement ( = l ), m 2 slide l 2 valve's mass, ξ 2 viscios friction co-efficient. -servo-amlifier's (9) eqations: Q 7 = µ 7 2ρ L(y v) h A Q 8 = µ 8 2ρ L(y v) B, (3), (32) Q = µ 2ρ c Q 2 = µ 2 2ρ c, (2), (2) Q 7 = βv A d A S A d, (33) d B Q 8 = βv B S d B, (34) d c Q 2 Q = βv c S d c, (22) where µ 7, µ 8 flow rate co-efficients, L servo- -amlifier's drag dimension, h hydralic slying S c ( c ) = k m ξ d, (23) m d2 k rr 2 r ressre, A, B actator's () chambers A and B ressres V A, V B A, B chambers' volmes, S A, S B where Q, Q 2 air flow rates throgh transdcer's actator's () iston's srface areas, v inner feed- chambers,, ressre int signals, c corrected back rod's () dislacement, rod's dislacement. ressre signal, deending on the drossel's (3) - actator's rod eqation: geometry and on the variable flidic resistance, S c elastic membrane's area, k m membrane's elastic A S A B S B = k el ξ d 3, (35) m 3 d2 constant, µ, µ 2 flow rate co-efficients ρ mean air density, β isotherm comression co-efficient, ξ where k el actator's sring's elastic constant, ξ 3 viscos friction co-efficient, m rodflansemble viscos friction co-efficient, m 3 istonrodfla ensemble's mass. mass, rod's dislacement, k rr feed back's sring's elastic constant, r feed back's rod dislacement. v = l, (36) - doble nozzle-fla hydro-amlifier's (8) eqations: l 2 Q 3 = µ d3 2ρ a L, (24) Q 4 = µ d4 2ρ L, (25) Q 5 = µ d5 2ρ a R, (26) r = l 4. (37) l 3 System's non-linear mathematical model consists of the eqation (2) (37) it can be linearized for a steady state oerating regime, characterised by the arameters (,, Q, Q 2, c,..., A, B, ). 355
3.2. Linearised model A B = k el m 3 D 2 ξ D, (46) S k el k el Eriming the main arameters X as X = X X, which becomes, after the Lalace transformation, where X steady state vale, X static error, X = X non-dimensional ratio, one obtains, sing k e k c (τ c s ) = (τ c s ) c, (47) X the annotations 2 πd k e = µ 3 d 4 2ρ 2 c = k c c k = k c T 2 s 2 2T ω s k r r, (τ L s )( L R) = k L τ sy, (48) (49) k 2c = µ d π(d tgα) 2 4 2ρ 2 c L R = k Ly s(τ 2 s )y, (5) k 2c = µ dπ 4 (2 tgα gα) k 3L = Q 3 = L 2 πd = µ 3 d 4 2ρ 2 a L = k 5R (τ s )( A B) = k Ay y k Av v τ A s, A B = k A T 3 2 s 2 2T 3 ω 3 s, (5) (52) k 4c = µ r d 4 2ρ L = k 6c k 4L = µ R d 4 ( S ) 2ρ 2 L = k 6R, D= d, (38) v = k v, r = k r, (53) where βv c T c = T c = S c k k c = 2 k c k 2c k 2 (k c k 2c ) c (54) system's linearised form becomes: k e (S c D k 2 ) = [βv c D(k c k 2c )] c, m c = k m D S c 2 ξ D k m k m k rr S r, c (k 4L k 3L ) L k 4c = βv L D L S D y, (k 5R k 6R ) R k 6 = βv R D R S D y, S ( L R) =D(m 2 D ξ) y, (βv A D k 7A) A = k 7y y k 7v v S A D, (βv B D k 8B) B = k 8y y k 8y v S B D, (39) (4) (4) (42) (43) (44) (45) k e = k e T 2 (k c k 2c ) = m ω = ξ k = c k m 2m c = k c = c S c k m T L = βv L k L k L = 2k4 Lk L τ = 2S y Lk L k Ly = SL ξy T 2 = m 2 ξ T A = βva k 7A k Ay = 2k7yy k 7A A k r = = k rr k kcsc Av = 2k 7yv τ A = 2S A k A = S A T 2 k 7A A k 7A A k el 3 = m 3 k el Figre 4 resents the non-dimensional linearised system's block-diagram with transfer fnctions, based on the above determined mathematical model. k e k k c T 2 s 2 k C τ 2T ω s c s L r k c (τ c s) y τ s k r τ L s L R k A T 2 3 s 2 2T 3 ω 3 s A B τ A s k Ay y k Ly s(τ 2 s) L R k Av k v τ A s Figre 4: System's block diagram with transfer fnctions 356
k e k k L k C y y Ly k A k k Ay c s(τ 2 s) - k c τ s y k A k Av k A τ A s v k v r k r k r 3.3. Non-dimensional, linearised, simlified model Figre 5: Simlified block diagram with transfer fnctions Roots locs 2 In order to simlify the analysis, one can neglect some terms, which vales are small, neglecting the inertial effects, the viscos friction and the flid's comressibility. That leads to a model's simlified form: k e k c (τ c s ) = c, (55) 5 5 c k = k c k r r, L R = k L τ sy, (56) (57) -5 - L R = k Ly sy, A B = k Ay y k Av v τ A s, (58) (59) A B =, (6) k A v = k v, r = k r. (6) System's eqivalent form becomes where k e k = (A 3 s 3 A 2 s 2 A sa ), (62) A 3 = ( k ck c )τ 2 τ R, A = k r, k c k L k Ly k Ay k r A 2 = ( k ck c )τ 2 ( k A k Av k v ) k c k L k Ly k A k Ay ( k ck c )τ R ( τ ) k c k L k Ly k Ay A = ( k ck c )( τ )( k A k Av k v ) k c k L k Ly k A k Ay. 4. SYSTEM'S STABILITY AND QUALITY, (63) Based on the above resented simlified model, one can define two transfer fnctions for the system, with resect to the flight regime V (s) = (s), resec(s) tively to the engine's oerating regime n (s) = (s) (s). Both of them have the same characteristic olynom, which co-efficients are given by formlas (63). -5-2 -3-25 -2-5 - -5 Figre 6: Characteristic olynom's roots locs Considering a constant sersonic flight regime (= m, V=2 km/h) =const., =, one stdies only the second transfer fnction, n (s), for the interval of rotation seed n=(4% %)n ma. In order to stdy the system's stability, one has erformed a root-locs stdy for the characteristic olynom it reslts a diagram (see figre 6), which shows that the olynom has three roots, a real negative one and two comle conjgate, which real arts are also negative, as figre 6 shows. Therefore, in any case, for any engine oerating regime, the stdied system is stable. The same conclsion is revealed by the freqency characteristics (amli- tde-freqency and hase-freqency), shown in fig. 7. System's qality is resented by the indiceal fnction, the system's ste resonse (figre 8), for three engine oerating regimes (R- minimm n =.4n ma, R2- in- termediate/crise n =.75n ma and R3- maimm n = n ma ) system's behavior is shown by the three crves, which show a small eriodic stabilisation of the flas'oening, with an initial overla. The overla isn't bigger than 3.5%, it has maimm vale for the R regime and diminishes by the rotation seed increasing, beein minimm for R3 regime the stabilising time is abot 3 3.5 s, a little bigger for the same R regime, which is the worse oerating regime. 357
-2 Freqency characteristics.2 Ste resonse R amlitde -4-6..8 R2-8 - -5 5 freqency w[/s] amlitde.6 R3 -.4 hase -2-3 -4.2-5 - -5 5 freqency w[/s] Figre 7: System's freqency characteristics 2 3 4 5 time [s] Figre 8: System's ste resonse 5. CONCLUSIONS One has stdied an atomatic sersonic intake control system, based on the air bleeding control throgh the anti-stall flas' ositioning. A ossible command law was established, with resect to the static or total ressre ratio, which involves the rear intake's ressre ( or ). A stability stdy was erformed, as well as a system's time behavior simlation (ste resonse), which gave some sefl information abot system's qality. Similary conclsions can be obtained for the system's Dirac imlse resonse. Althogh, engine's rotation seed variation can't be realised as "ste int", becase of engines control system, which realizes a ram variation of the injected fel flow rate and, conseqently, a similar rotation seed variation, which involves a nearly linear behavior for the air bleeding flas' ositioning (fig. 9). amlitde.45.4.35.3.25.2.5..5 Ram resonse R3 R R2 2 3 4 5 time [s] Figre 9: System's ram resonse References [] Aron, I., Tdosie, A. ydromechanical system for the sersonic air inlet's channel's section control. (ICATE 2, Craiova, May 25-26, 2, ag. 266-269). [2] Belea, C. Teoria sistemelor. Ed. Didactica si Pedagogica, Bcresti, 985. [3] Deert, W. Initiere in nemo-atomatica. Ed. Tehnica, Bcresti, 975. [4] Floarea, S., Dmitrache, I. Elemente si circite flidice. Ed. Academiei, Bcresti, 979. [5] Lng, R. Atomatizarea aaratelor de zbor. Ed. Universitaria, Craiova, 2. [6] Lng, R., Tdosie, A. ydromechanical Syst for the Sersonic Air Intake's Central Cor's Positioning. (The International Congress on "Simlators and Simlations", Brno, 2, May 9-, Pblished in Military Academy in Brno Reve, No. /2/vol B/ag. 55-7). [7] Pimsner, V. Berbente, C. Crgeri ín masini c alete. Ed Tehnica, Bcresti, 986. [8] Stoenci, D. Atomatica disozitivelor de admisie sersonice. Ed. Academiei Tehnice Militare, Bcresti, 976. [9] Stoicesc, M., Rotar, C. Motoare trboreactoare. Caracteristici si metode de reglare. Ed. Academiei Tehnice Militare, Bcresti, 999. [] Tdosie, A. Atomatizarea sistemelor de rolsie aerosatiala. Crs.Tiografia Universitatii din Craiova, 25, 29 ag. 358