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M.Goman, A.Khramtsovsky and M.Shapiro "Aerodynamics Modeling and Dynamics Simulation at High Angles of Attack", presentation at the International conference on Simulation Technology & Training held at TsAGI, Zhukovsky (Russia), on 24 May 2001.

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- 1. Aerodynamics Modeling and Dynamics Simulation at High Angles of Attack M.Goman, A.Khramtsovsky and M.Shapiro Central Aerohydrodynamic Institute (TsAGI), Russia Abstract Flight simulation problems at high angles of attack ranged from development of adequate mathemati- cal model for aerodynamic characteristics and non- linear dynamics analysis to piloted simulation for research and training purposes are discussed. The mathematical model for high angles of attack aero- dynamics is formulated based on di erent types of experimental wind tunnel data and nally corrected using the ight test results. The piloted ight sim- ulation is planned using results of qualitative non- linear dynamics investigation providing valuable in- formationabout high angles of attack departure and spin behavior. The experience in application of the mini desktop and full size movable simulators for highincidence ight investigationsincluding dynam- ics analysis, post design assessment of control laws for departure prevention and spin recovery, safety support of special ight tests and pilot training is outlined. Introduction High angles of attack ight practically for all types of aircraft is associated with critical and emergen- cy conditions due to serious changes in an aircraft dynamic responses and control. For example, about 30% of aircraft losses in ight accidents are origi- nated from aircraft departures at high angle of at- tack and development of spin. However, the possible bene ts of the high incidence ight expanding ma- neuverability boundaries are considered for future generation of combat aircraft. The new types of ma- neuvers such as the Cobra and the Herbst ones have been not onlydemonstrated in ight, but thoroughly investigated in terms of their e ciency and applica- tion. Copyright c 2001 by Central Aerohydrodynamic Insti- tute (TsAGI). During the last three decades there have been made signi cante orts indevelopmentof theoretical and experimental methods for investigation of high angles of attack ight conditions, including dynam- ics analysis, piloted simulation and ight tests. The most typical feature of all investigations done during these years was the extensive use of di erent types of simulators for research and pilot training purpos- es. The set of simulators available now in TsAGI for solving high incidence ight dynamics and safety problems is presented in Fig.1. They can be selec- tively used for accompanying ight tests, performing engineering research and pilot training. The com- plex research/engineering simulator on the Stuart platform providing six degrees-of-freedom motion is more precise and expensive, it is mostly used for pi- loted simulation during the special ight tests. The simple and the most cheap desktop simulators with simpli edvisualizationsystem andcontrol levers can be widely used for the rank-and- le pilots training. The structure and all components of an aircraft mathematical model can be the same in all these simulatorsthus supporting the continuous process of research work and pilot training. The general struc- ture ofthe mathematicalmodelis presented inFig.2. One of the most signi cant problems in mathe- matical model development is connected with for- mulation of the aerodynamic forces and moments at high incidence ight conditions. Due to separated and vortical ow the aerodynamic dependencies be- come essentially nonlinear and motion dependent. Unfortunately, the mathematical model built only on the wind tunnel data requires further corrections for better agreement with ight test data. There- fore, control law design, dynamics analysis and pi- loted simulation compose the closed-loop research and development cycle (see Fig.3). In this paper some problems and experience asso- ciated with modeling and simulation of high angles of attack ight of combat and general aviation air- craft are discussed. 1
- 2. Closed-loop research and development cycle There are some links and interconnections between components of the cycle in Fig.3, which are rather natural for ight dynamics in general, but more strong for high angles of attack conditions. Flight Tests at High Angles of Attack are very hazardous and expensive. They take long time and require high-skilled test pilot, special equipment ensuring ight safety and extended ground based theoretical and experimental support. The mainob- jectives of such special ght tests are assessment of stall/spin resistance, search for an adequate control technique for spin prevention/recovery and testing of automatic control system. Aerodynamics modeling is mainlyrelied on the experimental wind tunnel data from static, forced oscillation and rotary balance tests. These experi- mental data allow to obtain rather good agreement with ight tests, when aircraft motion is stable and not agitated. The test pilot comments help to ad- just the wind tunnel data to real ight conditions (Fig.4). Dynamic instability and large amplitude oscilla- tions require correct modeling of unsteady aerody- namic e ects due to internal ow dynamics sepa- rated and vortical ow. The methods of unsteady aerodynamics modeling is now under development 6, 7, 9, 10, 11, 12]. To reconcile the conventional mathematical model with ight test results in such agitated ight conditions the set of unknown param- eters in the mathematical model are corrected using ight test data using identi cation techniques. The current wind tunnel experimentalfacilities re- quire further development and improvement to sim- ulate an aircraft high angles of attack motion condi- tions (large amplitude and multi-degree of freedom oscillations). Ground based simulation and pilot training at high incidence will be e cient only if we have an adequate mathematical model for nonlinear un- steady aerodynamics, and this is an iterative process connected with ight tests. Spin recovery for modern aircraft is too compli- cated for rank-and- le pilots, that is why the lessons learnt during the special ight tests and later re- produced in simulation are extremely important for pilot training. The simple and a ordable desktop simulators are the most appropriate tools for these objectives. Aircraft dynamics has a multi-attractor nature and depend on the style of piloting. Results of nonlinear dynamics analysis help to plan the pilot training exhaustively. Simulationreveals the control techniques for spin entry and recovery and highlights the critical ight conditions. Stability & Dynamics Analysis is based on bi- furcation and nonlinear dynamic theory methods and application of specially developed software for such qualitative investigation (the KRIT Package 4]). Multiple equilibrium and periodical dynamical states are investigated using continuation technique and Poincare mapping method. This helps to pre- dict aircraft departures and possible critical attrac- tors. The same qualitative methods of analysis are applied for pre- and post-design assessment of con- trol laws 5]. Nonlinear control laws design for high angles of attack mainlysolve the ight safety issues such as warning, prevention and eliminationof critical ight conditions. Normally, special control laws are de- signed for departure prevention and spin recovery. Unfortunately, due to loss of aerodynamic e ciency of control surfaces at high angles of attack there are a lot of limitations to solve this problem. The level of aerodynamic characteristics uncertainty at high angles of attack is much higher than at normal ight conditions, that is why the advanced robust control design methods and innovative control e ectors such as thrust vectoring, vortex ow generators, etc., are of great importance for high angles of attack ight. Anti-spin parachute mathematical model Experimental aircraft are often equipped with anti- spin parachute for safety reasons (see Fig.5). Stat- ic line of the parachute is attached behind the air- craft center of gravity. Parachute's drag force is transferred through static line and brings out pitch and yaw moments. These moments tend to recover the aircraft to normal ight conditions with near- zero angle-of-attack and sideslip. The moments are su cient for spin recovery provided the parachute canopy area is large enough. The static line and the canopy may be deployed in di erent manners, for example, using special con- tainer equipped with powder rocket engines. In a short time (' 0:7 sec) the parachute is ejected out ofthe airplanewake. Ittakes about 0:5 0:7 sec more for fullcanopy deployment. After that the drag force on the canopy appears. 2
- 3. Due to the air ow, the canopy is moving with re- spect to the airplane. To know exact position of the parachute with respect to the airplane is important for the calculation of additional pitching and yawing moments. The mathematicalmodel of the parachute motion is based on the following assumptions: the parachute oats in the air inertialessly static line is long enough, so the disturbances of the velocity eld in the airplane wake can be ignored the velocity eld is practically uniform near the canopy aerodynamic force is normal to the canopy. The orientation of the static line of the parachute is described by unit vector ~p = (px py pz). Ori- entation of the vector ~p with respect to body-axis frame of reference is given by the angles p and p (analogous to angle-of-attack and sideslip). The re- lationship between them is px = ;cos p cos p py = ;sin p pz = ;sin p cos p The airspeed vector at the canopy location is a sum of airplane ight velocity vector and transla- tional velocity due to aircraft rotation ~Vp = ~Vc + ~! (~rp + lp ~p)] where ~Vc - airplane velocity vector at c.g., ~! - airplane rotation rate vector, ~rp - radius-vector from c.g. to static line attachment point, lp - length of the static line. Airspeed vector component normal to ~p will cause to move the parachute with respect to the airplane. The static line orientation is governed by the equa- tion d~p dt = ; hh ~p ~Vp i ~p i lp (1) If orientation unit vector ~p is known, additional forces and moments can be calculated as follows ~Fp = CDpSp ~Vp ~p 2 2 ~p = CDpSp ^~V p ~p 2 QS~p ~Mp = h ~rp ~Fp i where Sp - the canopy area, Sp = Sp=S - nondimensional canopy area, CDp - parachute drag coe cient, - air density, Q - dynamic air pressure, ^~V p = ~Vp=V - nondimensional parachute velocity vector. Due to the air ow retardation in airplane wake, the drag coe cient CDp depends on the airplane's angle of attack. This dependence can be obtained from wind tunnel tests of the airplane with the de- ployed parachute. For the equation (1), it is necessary to set correct- ly initial static line orientation at the moment of full canopy deployment. The parachute container eject the parachute in a certain direction (along X-axis of the airplane, p = 0, p = 0). These angles may change during deployment stage due to aircraft ro- tation. The initial conditions p0, p0 = 0 for the equation (1) can be calculated taking into account the mechanism of deployment. At steady state rotation in spin conditions the ori- entation of the static line of the parachute coincides with the direction of the local velocity of the air ow: p = ;Vp Vp = ;Vc +! (rp + lpp) Vp (2) Assuming that Vp Vc, than the equation (2) becomes linear with respect to p p+ lp Vc ! p = ;Vc + ! rp Vc = p (3) Computing both vector and scalar products of the right and left-hand parts of the equation (3) with vector ! (fromthe left), one can obtain the following expression ! p = ! p + lp Vc !2 p;(! p )! which after substitution into equation (3) gives the following nal expression for vector p p = p + lp Vc p ! + lp Vc !(! p ) 1 + lp Vc 2 !2 Using this formula it is possible to calculate the steady-state airplane spin parameters taking into ac- count the in uence of the anti-spin parachute. It is also possible to evaluate the needed parameters of anti-spin parachute for successful spin recovery. 3
- 4. Unsteady aerodynamics modeling Signi cant contribution to aerodynamic loads at high angles of attack is generated by separated and vortical ow. Their in uence produces at high inci- dence nonlinearanddynamicaerodynamicresponses to changes in an aircraft attitude. As a result the conventional form of aerodynamic coe cients based on the aerodynamic derivative concept becomes in- accurate 8]. The most reasonable way of modeling of nonlinear unsteady aerodynamics e ects is in ap- plication of ordinary di erential equations for vorti- cal and separated ow contributions. The mathematical model for any force and mo- ment coe cients may be represented using load par- titioningin the followingform(here the normalforce coe cient is considered as an example): CN(t) = CNpt( )+ CN_ pt ( )_ + CNdyn (4) where inertialess terms CNpt( ), CN_ pt ( ) are equiv- alent to the conventional representation form with aerodynamic derivatives, and dynamic contribution CNdyn is governed by nonlinear equation dCNdyn dt = 3X i=1 ki( )(CNvb0 ;CNdyn ( ))i (5) where t = 2t0V1c is dimensionless time, = k;1 1 ( ) is the characteristic time constant, extracted from small amplitude responses, and the right hand side function CNvb0 is de ned as CNvb0 ( ) = CNst( ); CNpt( ): The linearized dynamic equation (5) behaves very well in case of small amplitude oscillations, during large amplitude motion the nonlinear terms in (5) become rather large to obtain good agreement with experimental results 7, 10, 9]. Such dynamic representation of all aerodynamic coe cients is important for adequate modeling of high angles of attack aircraft oscillatory motionsuch as wing rock or agitated spin. Nonlinear dynamics qualitative analy- sis The whole mathematical model of an aircraft dy- namics at high incidence is highly nonlinear, it dis- plays various types of behavior depending on the pi- lot control manner. Sometimes di erent pilots pro- voke di erent aircraft dynamics and some of critical ightregimes maybe avoided. That is why the qual- itativemethod ofanalysisprovidingallpossiblecriti- calstates, their stabilityandregions ofattraction are used to perform thorough simulation of an aircraft dynamics at high angles of attack. The example of qualitative dynamics analysis for a hypothetical air- craft is presented in Fig.6. Along with stable normal ight solutions the critical solution branches such as roll-coupling modes, wing rock and oscillatory at spin modes are identi ed. These solutions provide not only magnitudes of motion parameters but also the character of motion stability. Aerodynamic asymmetry and aircraft spin Modern maneuverable aircraft con gurations espe- cially when they are statically unstable at low an- gles of attack su er with the lack of pitch-down con- trol at high angles of attack (see Fig.7). The con- trol system in such cases provides stability at nor- mal ight regimes, however there are stable trims at high incidence, where an aircraft can be locked-in. These ight conditions, which are called deep stall regimes, may be unrecoverable using conventional control technique. Similar critical unrecoverable situations can arise due to aerodynamic yaw asymmetry producing at spin regimes, where an aircraft can be also locked-in (see Fig.8). The asymmetricalaerodynamicrollingandyawing aerodynamic moments at high angles of attack are result of the onset ofasymmetricalvortical ow. The aerodynamic asymmetry is observed both in wind tunnel andin ight. The onlydi erence that in ight the level of yaw asymmetry may be higher than in a wind tunnel. The possible reason ofsuch di erence is in aeroelastic vibrations of scaled aircraft model and in di erent interference e ects available in a wind tunnel. The aerodynamic asymmetry in yaw extracted from high incidence ight tests are presented in Figs.9 and 10 respectively for the Su-27 aircraft and the experimental X-31 aircraft. Although the ampli- tudes of yaw asymmetry in these cases are di erent, the qualitative dependence on angle of attack is sim- ilar. Yaw asymmetry changes its sign with angle of attack and displays dynamic hysteresis during pitch up and pitch down attitude variations. Fig.11 illustrates how the stable equilibrium at spin solution appears in the moment balance equa- tions with the increase of the yaw asymmetry am- plitude. The aerodynamic asymmetry may signi - cantly exceed the e ciency of rudder and ailerons so that the at spin regime can be unrecoverable by means of simple counteracting control de ections. The only e cient control technique in the cases of deep stall and at spin regimes is the so called pitch 4
- 5. rocking control. Actually it means that available constrained control authority is applied to destabi- lize the critical ight regime in a self-agitating man- ner (Fig.12). Typical variations of aircraft motion parameters during spin recovery using pitch rocking control technique is presented in Fig.13. It is inter- esting to note that pitch rocking control (75 90 seconds) produces increase in amplitude of oscilla- tion not only in pitch, but also in roll due to inertia coupling of both these forms of motion. The e ciency of pitch rocking control in compar- ison with simple counteracting control can be seen from Fig.14, where the time of recovery from at spin conditions is given as a function of the level of yaw asymmetry. Development of the adequate mathematical mod- el for aerodynamic characteristics during the initial stage of ight tests allowed to design the spin pre- vention and recovery control system, which later had been also tested in ight. The general block diagram of this system is presented in Fig.15. The adequate mathematical model veri ed in ight tests and in piloted simulationswith participa- tion of experienced test pilots has been applied for development of simple and cheap desktop simulator for training of the rank-and- le pilots. Special work has been done for creation of the database of repre- sentative set of simulated ights illustrating possible pilot's mistakes and correct recovery control. Fig.16 presents two examples from this database. The rst one illustrates the deep stall departure and following recovery (the time histories for motion pa- rametersandcontrolare presented inFig.17)andthe second one illustrates the at spin departure and re- covery (the timehistories for motionparameters and control are presented in Fig.18). The Cobra maneu- ver simulation is presented in Fig.19. High incidence ight simulation of gen- eral aviation aircraft General aviation aircraft Molnia-1 (Fig.20) with a canard and high horizontal tail provides another example of successful application of simulation ap- proach for high incidence ight (see Fig.21). A small positive installation angle of the canard leads to earlier onset of ow separation on a canard with respect to stall conditions on a wing. This produces the pitch down moment in static depen- dency of the pitch moment coe cient. Because the ow separation on a canard occurs with some de- lay it generates the anti-dampinge ect in pitch (see Fig.22). The unsteady aerodynamic model for the pitch moment coe cient has been developed in the form (5) and applied in the mathematicaland piloted sim- ulationon complexresearch simulatorwith 6 degree- of -freedom (see Fig.1, top). This piloted simulation has been performed before the ight tests and helped the test pilot to study the peculiarities of such air- craft con guration. At high angles of attack (for this con guration sens 18 deg) due to canard ow separation occur the self-sustained oscillations in pitch, which serve as warning factor for pilot of high incidence ight. This pitch oscillations is stable in recoverable when pilot applies a pitch down con- trol. The predicted behavior ofanaircraft athighin- cidence ight has been con rmed later in ight tests. The example of timehistories for motionparameters are presented in Fig.23. Concluding remarks Piloted simulation of an aircraft dynamics at high incidence ight is extremely important element of aircraft development and serti cation processes. It helps in mathematical model assessment, accompa- nyingthe special ighttests andthus increasingtheir safetyand e ciency, and nallycan be used forrank- and- le pilots training beyond the normal ight con- ditions. References 1] Aerodynamics, stability and controllability of supersonic aircraft. Editor G.S.Bushgens, Nau- ka, Fizmatlit, Moscow, 1998, 816 pp. 2] Ahrameev, V., Goman, M., Kalugin, A., Klu- mov, A., Merkulov, A., Milash, E., Syrovatsky, V., Khramtsovsky, A., and A.Scherbakov. Au- tomatic aircraft recovery from spin regimes, Technika Vozdushogo Flota, No.3, 1991, pp.15- 24 (in russian). 3] Zagaynov, G.I., and M.G.Goman Bifurcation analysis of critical ight regimes, ICAS Pro- ceedings, Vil.1, 1984, pp.217-223. 4] Goman M.G., Zagainov G.I and A.V.Khram- tsovsky Application of Bifurcation Methods to Nonlinear Flight Dynamics Problems. { Progress in Aerospace Sciences, Vol.33, pp.539- 586, 1997, Elsevier Science, Ltd. 5] Goman M.G. and A.V.Khramtsovsky Applica- tion of Bifurcation and Continuation Methods for an Aircraft Control Law Design. { Phil. Trans. R. Soc. Lond. A (1998) 356, 1-19, In the RoyalSociety Theme Issue "Flight Dynamicsof High Performance Manoeuvrable Aircraft". 5
- 6. 6] Tobak, M. and Schi , L.B. On the Formulation of the Aerodynamic Characteristics in Aircraft Dynamics, NASA TR-R-456, 1976. 7] Goman, M.G., and A.N.Khrabrov. State-Space Representation of Aerodynamic Characteristics of an Aircraft at High Angles of Attack, Jour- nal of Aircraft, Vol.31, No.5, Sept.-Oct. 1994, pp.1109 - 1115. 8] Greenwell, D.I.Di culties in the Application of Stability Derivatives to the Maneuvering Aero- dynamics of Combat Aircraft, ICAS Paper 98- 1.7.1,the 21th Congress ofthe AeronauticalSci- ences, Sept. 1998, Melbourne, Australia. 9] Goman,M.G.,Greenwell, D.I.,andA.N.Khrab- rov. The Characteristic Time Constant Ap- proach for Mathematical Modeling of High An- gle of Attack Aerodynamics, ICAS Paper, 22nd Congress of the Aeronautical Sciences, Sept. 2000, Harrogate, UK, pp. 223.1-223.14. 10] Abramov, N.B., Goman, M.G., Khrabrov, A.N., and K.A.KolinkoSimple Wings Unsteady Aerodynamics at High Angles of Attack: Ex- perimental and Modeling Results, Paper 99- 4013, AIAA Atmospheric Flight Mechanics Conference, August 1999, Portland, OR. 11] Klein, V., and Noderer, K.D. Modeling of Air- craft Unsteady Aerodynamic Characteristics, Part 1 - Postulated Models, NASA TM 109120, May 1994 Part 2 - Parameters Estimated From Wind Tunnel Data, NASA TM 110161, April 1995 Part 3 - Parameters Estimated From Flight Data, NASA TM 110259, May 1996. 12] Mark S.Smith Analysis of Wind Tunnel Oscil- latory Data of the X-31A Aircraft, NASA/CR- 1999-208725, Feb. 1999. 13] B.R.Cobleigh,M.A.Croom, B.F.Tormat Com- parison of X-31 Flight, Wind -Tunnel, and Wa- ter Tunnel Yawing Moment Asymmetries at High Angles of Attack, High Alpha Conference IV -Electronic Workshop,NASA Dryden Flight Research Center, July 12-14, 1994 Figure 1: Research/training simulator on the Stuart platform (top), midi size training simulator (mid- dle), desktop training simulator (bottom). 6
- 7. Equations of motion Undercarriage model Aerodynamic forces and moments model Aerodynamic characteristics database Atmospheric turbulence model Engine model Altitude-velocity engine characteristics Cockpit Control system and actuator models Flight tests safety equipment Figure 2: General structure of mathematical model used in piloted simulation. Figure 3: Research and development cycle at high angles of attack. Figure 4: Aerodynamicmodeldevelopmentbased on wind tunnel and ight tests data. Figure 5: Anti-spin parachute mathematical model. 7
- 8. Figure 6: Qualitative analysis of nonlinear aircraft dynamics at high angles of attack. Figure 7: Deep stall regimes. Figure 8: Unrecoverable at spin regimes. 8
- 9. Figure 9: Aerodynamic yaw asymmetry extracted from ight tests 1]. Figure 10: Aerodynamic yaw asymmetry extracted from ight tests of the X-31 aircraft 13]. Cn0 = 0 Cn0 = 0.035Cn0 = 0.02 - balance in pitch moments - balance in roll and yaw moments - stable spin regime - aperiodically unstable spin regime Figure 11: Flat spin generated by aerodynamic yaw asymmetry. Figure 12: Pitch rockingcontrol technique (potential function analogy). 9
- 10. Figure 13: Spin recovery using pitch rocking control. Timeofrecovery(sec) 0 10 20 30 40 50 0 0.05 0.10 0.15 with rocking without rocking Yaw asymmetry Cn0 Figure 14: E ciency of pitch rocking control. Figure 15: Spin prevention and recovery control sys- tem. 10
- 11. Marker time step: 5 sec −2000 0 2000 4000 −4000 −3500 −3000 −2500 −2000 −1500 −1000 −500 0 500 5500 6000 6500 7000 7500 8000 8500 9000 xe −ye h Marker time step: 10 sec −2000 0 2000 4000 −5000 −4000 −3000 −2000 −1000 0 1000 0 1000 2000 3000 4000 5000 6000 −yexe h Figure 16: Maneuvers performed by experienced test pilots on desktop simulator. Deep stall departure andrecovery (top). Flat spin departure and recovery (bottom). 0 10 20 30 40 50 60 −100 0 100 α,deg 0 10 20 30 40 50 60 −10 0 10 β,deg 0 10 20 30 40 50 60 −5 0 5 p,1sec 0 10 20 30 40 50 60 −0.5 0 0.5 r,1sec 0 10 20 30 40 50 60 −1 0 1 q,1sec Time, sec 0 10 20 30 40 50 60 −100 0 100 θ,deg 0 10 20 30 40 50 60 −200 0 200 ψ,deg 0 10 20 30 40 50 60 −200 0 200 φ,deg 0 10 20 30 40 50 60 −10 0 10 −az 0 10 20 30 40 50 60 −0.2 0 0.2 a y Time, sec 0 10 20 30 40 50 60 −200 0 200 Xθ ,mm 0 10 20 30 40 50 60 −200 0 200 Xψ ,mm 0 10 20 30 40 50 60 −200 0 200 Xφ ,mm 0 10 20 30 40 50 60 0 50 100 XT r ,mm 0 10 20 30 40 50 60 0 50 100 XT l ,mm Time, sec Figure 17: Deep stall departure and recovery during aircraft spatial maneuvering. 11
- 12. 0 20 40 60 80 100 120 −100 0 100 α,deg 0 20 40 60 80 100 120 −20 0 20 β,deg 0 20 40 60 80 100 120 −2 0 2 p,1sec 0 20 40 60 80 100 120 −2 0 2 r,1sec 0 20 40 60 80 100 120 −0.5 0 0.5 q,1sec Time, sec 0 20 40 60 80 100 120 −100 0 100 θ,deg 0 20 40 60 80 100 120 −200 0 200 ψ,deg 0 20 40 60 80 100 120 −200 0 200 φ,deg 0 20 40 60 80 100 120 −10 0 10 −a z 0 20 40 60 80 100 120 −0.1 0 0.1 ay Time, sec 0 20 40 60 80 100 120 −200 0 200 X θ ,mm 0 20 40 60 80 100 120 −200 0 200 X ψ ,mm 0 20 40 60 80 100 120 0 50 100 X φ ,mm 0 20 40 60 80 100 120 0 50 100 X T r ,mm 0 20 40 60 80 100 120 0 50 100 X T l ,mm Time, sec Figure 18: Flat spin departure and recovery during aircraft spatial maneuvering. Marker time step: 1 sec 1000 1500 2000 2500 3000 3500 4000 −500 0 500 1400 1600 1800 2000 2200 2400 xe H0 =1423 ft; Mach=0.49; γ0 =0; Throttle=0.25 (t=3:1:10 seconds) −ye h 3 4 5 6 7 8 9 10 −100 0 100 α,deg H0 =1423 ft; Mach=0.49; γ0 =0; Throttle=0.25 (pitch, roll and yaw control) 3 4 5 6 7 8 9 10 −2 0 2 q,rad/s 3 4 5 6 7 8 9 10 200 400 600 V,ft/s 3 4 5 6 7 8 9 10 −100 0 100 θ,deg 3 4 5 6 7 8 9 10 −20 −10 0 δe ,deg 3 4 5 6 7 8 9 10 −20 0 20 β,deg 3 4 5 6 7 8 9 10 −0.5 0 0.5 r,rad/s 3 4 5 6 7 8 9 10 −2 0 2 p,rad/s 3 4 5 6 7 8 9 10 −200 0 200 φ,deg 3 4 5 6 7 8 9 10 −40 −20 0 δa ,deg 3 4 5 6 7 8 9 10 −20 0 20 δr ,deg Time, sec Figure 19: Cobra maneuver simulation. 12
- 13. Figure 20: General aviation aircraft Molnia 1. Figure 21: Canard ow separation. Figure 22: Anti-damping e ect due to canard ow separation. Figure 23: Pitch oscillations at high angles of at- tack of a general aviation aircraft with canard ( ight tests). 13

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