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-

formation about high angles of attack departure and

spin behavior. The experience in application of the

mini desktop and full size movable simulators for

high incidence ight investigations including 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 only demonstrated 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�cant e�orts in development of 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�ed visualization system and control 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

simulators thus supporting the continuous process of

research work and pilot training. The general struc-

ture of the mathematicalmodel is presented in Fig.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

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 main ob-

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 mainly relied 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 experimental facilities 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. Simulation reveals 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 mainly solve the ight safety issues such as

warning, prevention and elimination of 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

of the airplane wake. It takes about 0:5�0:7 sec more

for full canopy deployment. After that the drag force

on the canopy appears.

2

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 mathematical model 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 �ppy = � sin �ppz = � 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

ilp

(1)

If orientation unit vector ~p is known, additional

forces and moments can be calculated as follows

~Fp = CDpSp

��~Vp � ~p

�22

~p = CDpSp

�~̂V p � ~p

�2QS~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 CDpdepends 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 ! (from the left), one can obtain the following

expression

! � p = ! � p� +lp

Vc

�!2p� (! � p�)!

�

which after substitution into equation (3) gives the

following �nal expression for vector p

p =

p� +lpVc

�p� � ! +

lpVc!(! � p�)

�

1 +

�lpVc

�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

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 nonlinear and dynamic aerodynamic responses

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-

titioning in the following form (here the normal force

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

CNdynis governed by nonlinear equation

dCNdyn

dt=

3Xi=1

ki(�)(CNvb0�CN

dyn

(�))i; (5)

where t = 2t0V1�c is dimensionless time, � = k�11(�)

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 motion such

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

ight regimes may be avoided. That is why the qual-

itativemethod of analysis providing all possible criti-

cal states, their stability and regions of attraction 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 asymmetrical aerodynamic rolling and yawing

aerodynamic moments at high angles of attack are

result of the onset of asymmetrical vortical ow. The

aerodynamic asymmetry is observed both in wind

tunnel and in ight. The only di�erence that in ight

the level of yaw asymmetry may be higher than in a

wind tunnel. The possible reason of such 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

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 simulations with 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-

rameters and control are presented in Fig.17) and the

second one illustrates the at spin departure and re-

covery (the time histories for motion parameters 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-damping e�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 mathematical and piloted sim-

ulation on complex research simulator with 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 of an aircraft at high in-

cidence ight has been con�rmed later in ight tests.

The example of time histories for motion parameters

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-

nying the special ight tests and thus increasing their

safety and e�ciency, and �nally can be used for rank-

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

Royal Society Theme Issue "Flight Dynamics of

High Performance Manoeuvrable Aircraft".

5

[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 of the Aeronautical Sci-

ences, Sept. 1998, Melbourne, Australia.

[9] Goman,M.G., Greenwell, D.I., and A.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.Kolinko Simple 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

Equationsof motion

Undercarriagemodel

Aerodynamicforces and moments

model

Aerodynamiccharacteristics

database

Atmosphericturbulence

model

Engine model

Altitude-velocityengine characteristics

Cockpit

Control systemand actuator

models

Flight testssafety 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: Aerodynamic model development based on

wind tunnel and ight tests data.

Figure 5: Anti-spin parachute mathematical model.

7

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

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 Cn

0= 0.035Cn

0= 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 rocking control technique (potential

function analogy).

9

Figure 13: Spin recovery using pitch rocking control.

Tim

eofre

covery

(sec)

0

10

20

30

40

50

0 0.05 0.10 0.15

with rocking

withoutrocking

Yaw asymmetry Cn0

Figure 14: E�ciency of pitch rocking control.

Figure 15: Spin prevention and recovery control sys-

tem.

10

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

−yex

e

h

Figure 16: Maneuvers performed by experienced test

pilots on desktop simulator. Deep stall departure

and recovery (top). Flat spin departure and recovery

(bottom).

0 10 20 30 40 50 60−100

0

100

α, d

eg

0 10 20 30 40 50 60−10

0

10

β, d

eg

0 10 20 30 40 50 60−5

0

5

p, 1

\sec

0 10 20 30 40 50 60−0.5

0

0.5

r, 1

\sec

0 10 20 30 40 50 60−1

0

1

q, 1

\sec

Time, sec

0 10 20 30 40 50 60−100

0

100

θ, d

eg0 10 20 30 40 50 60

−200

0

200

ψ, d

eg

0 10 20 30 40 50 60−200

0

200φ,

deg

0 10 20 30 40 50 60−10

0

10

−a z

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, m

m

0 10 20 30 40 50 60−200

0

200

Xψ, m

m

0 10 20 30 40 50 60−200

0

200

Xφ, m

m

0 10 20 30 40 50 600

50

100

XT

r, mm

0 10 20 30 40 50 600

50

100

XT

l, mm

Time, sec

Figure 17: Deep stall departure and recovery during

aircraft spatial maneuvering.

11

0 20 40 60 80 100 120−100

0

100α,

deg

0 20 40 60 80 100 120−20

0

20

β, d

eg

0 20 40 60 80 100 120−2

0

2

p, 1

\sec

0 20 40 60 80 100 120−2

0

2

r, 1

\sec

0 20 40 60 80 100 120−0.5

0

0.5

q, 1

\sec

Time, sec

0 20 40 60 80 100 120−100

0

100

θ, d

eg

0 20 40 60 80 100 120−200

0

200

ψ, d

eg

0 20 40 60 80 100 120−200

0

200

φ, d

eg

0 20 40 60 80 100 120−10

0

10

−a z

0 20 40 60 80 100 120−0.1

0

0.1

a y

Time, sec

0 20 40 60 80 100 120−200

0

200

Xθ, m

m

0 20 40 60 80 100 120−200

0

200

Xψ, m

m

0 20 40 60 80 100 1200

50

100

Xφ, m

m

0 20 40 60 80 100 1200

50

100

XT

r, mm

0 20 40 60 80 100 1200

50

100

XT

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; γ

0=0; Throttle=0.25 (t=3:1:10 seconds)

−ye

h

3 4 5 6 7 8 9 10−100

0

100

α, d

eg

H0=1423 ft; Mach=0.49; γ

0=0; Throttle=0.25 (pitch, roll and yaw control)

3 4 5 6 7 8 9 10−2

0

2

q, r

ad/s

3 4 5 6 7 8 9 10200

400

600

V, f

t/s

3 4 5 6 7 8 9 10−100

0

100

θ, d

eg

3 4 5 6 7 8 9 10−20

−10

0

δ e, deg

3 4 5 6 7 8 9 10−20

0

20

β, d

eg

3 4 5 6 7 8 9 10−0.5

0

0.5

r, r

ad/s

3 4 5 6 7 8 9 10−2

0

2

p, r

ad/s

3 4 5 6 7 8 9 10−200

0

200

φ, d

eg

3 4 5 6 7 8 9 10−40

−20

0

δ a, deg

3 4 5 6 7 8 9 10−20

0

20

δ r, deg

Time, sec

Figure 19: Cobra maneuver simulation.

12

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