An Agile Aircraft Non-Linear Dynamics by Continuation Methods and Bifurcation Theory

ICAS 2000 CONGRESS

AN AGILE AIRCRAFT NON-LINEAR DYNAMICS BY CONTINUATION METHODS AND BIFURCATION THEORY

Krzysztof Sibilski Military University of Technology, Warsaw, Poland

Keywords: flight dynamics, flight at high angle of attack, bifurcation theory

Abstract arized equations of motion used for analysis at that time did not contain the instability [1]. The paper provides a brief description of the There are many problems associated modern methods for investigation of non-linear with flight dynamics for modern and advanced problems in flight dynamics. A study is pre- aircraft, which are not solved (or solved rather sented of aircraft high angles of attack dynamic. unsatisfactory) with traditional tools. A list of Results from dynamical systems theory are used such problems includes among others flight to predict the nature of the instabilities caused control for agile and post-stall aircraft. The by bifurcations and the response of the aircraft post-stall manoeuvrability has become one of after a bifurcation is studied. A non-linear dy- the important aspects of military aircraft devel- namic model is considered which enables to opment. Such manoeuvres are jointed with a determine the aircraft’s motion. The aerody- number of singularities, including “unexpected” namic model used includes nonlinearities, hys- aircraft motion. As the result of them, there is teresis of aerodynamic coefficients (unsteady dangerous of faulty pilot’s actions. Therefore, it aerodynamics), and dynamic stall effect. Aero- is need to investigate aircraft flight phenomena dynamic model includes also a region of higher at high- and very-high angles of attack. angles-of-attack including deep stall phenom- The appearance of a highly augmented ena. In the paper continuation method is used to aircraft required a study of its high angle of at- determine the steady states of the MiG-29 tack dynamics The primary aim of the paper is fighter aircraft as functions of the elevator de- to discuss capabilities of dynamical system the- flections, and bifurcations of these steady states ory methods as the tools for to analyse such are encountered. Bifurcations of the steady phenomena. states are used to predict the onset of wing rock, Dynamical system theory has provided a and “Cobra” manoeuvres. Dynamics of spatial powerful tool for analysis of non-linear phe- motion of a supersonic combat aircraft with a nomena of aircraft behaviour. In the application straked wing is considered for post stall ma- of this theory, numerical continuation methods noeuvres, and observations of chaotic motion in and bifurcation theory have been used to study post stall manoeuvres are guided. roll-coupling instabilities and stall/spin phenomena of a number of aircraft models. Results 1 Introduction of great interest have been reported in several An aircraft is the inherently non-linear papers (it can be mentioned papers by Jahnke and time varying system. Non-linear dynamics and Culick [1], Carroll and Mehra [2], is central to several important aircraft motions, Guicheteau [3], or Avanzini and de Matteis [4]). including roll-coupling and stall/spin phenom- Continuation methods are numerical techniques ena. Linearized equations of motion can not be for calculating the steady states of systems of used to analyse these phenomena. Indeed, roll- ordinary differential equations and can be used coupling instabilities were first discovered in to study roll coupling instabilities and high- flight, often with fatal results, because the line- angle of attack instabilities.

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Carroll and Mehra [2] were the first to tional problems have been faced with the mod- use a continuation technique to calculate the elling of the stability and control augmentation steady states of aircraft. They determined the system with the related high number of addi- steady states of a variable sweep aircraft and the tional states and the handling of the so-called F-4 fighter aircraft. By studying the steady breakpoint nonlinearities coming from control states of these two aircraft they explained that system elements. wing rock appears near the stall angle of attack The present paper is continuation of the due a Hopf bifurcation of the trim steady state. previous works of the author [5], [6], [7]. After They also calculated the steady spin modes for a brief description of the methodology and asso- the aircraft and predicted the control surface de- ciated procedures, Cobra maneuver and associ- flections at which the aircraft would undergo ated wing-rock oscillations are studied by stall/spin divergence. They have calculated the means of checking the stability characteristics resulting steady state of the aircraft and have related to unstable equilibria. Numerical simu- developed recovery techniques using their lations are used to verify the predictions. High knowledge of the steady spin modes for the air- angle of attack maneuvers were studied to ob- craft. Guicheteau [3] has used continuation serve chaos phenomenon in post stall motion. methods and bifurcation theory in analysis of Unsteady aerodynamics for prediction of airfoil the non-linear dynamics of aircraft model that loads is included, and the ONERA type stall includes unsteady aerodynamic coefficients. He model is used [8], [9], [10]. has analysed the effects of a lateral offset of the c.g. and the influence of gyroscopic momentum 2 Theoretical background of rotating engine’s masses on spin entry recovery. Jahnke and Culick [1] have recalled the 2.1 Dynamical systems theory theoretical background of dynamical system In this paper we will study equations of the fol- theory and bifurcation technique and they have lowing form presented profound review of the relevant in- = xx tf ì);,( (1) vestigations in this field. They have studied the ! dynamic of the F-14 fighter aircraft by deter- and mining the steady states of the equations of mo- " g xx ì);( (2) tion and seeking bifurcations. And have shown with RRx 1 ,, ì VandtU ⊂∈∈⊂∈ R pn , that continuation method is very useful in analy- where U and V are open sets in Rn and Rp, re- sis of the wing rock instability, spiral diver- spectively. We view the variables µ as parame- gence instability and spin dynamics. Avanzini ters. We refer to (1) as a vector field or ordinary and de Matteis [4] have analysed the dynamics differential equation and to (2) as a map or dif- of a relaxed stability aircraft by dynamical sys- ference equation. Both will be termed dynami- tem theory. The principal objective of their cal systems. study was to assess the practical worth of dy- By a solution of Eq. (1) we mean a map, namical system theory in simulations where the x, from some interval I ⊂ R1 into Rn, which we dynamics of aircraft are tailored by the full- represent as follows authority control system according to different → Rx 1 ,I: mission tasks. They have remarked that the use (3) of manoeuvre demand control in high- " x tt )( performance aircraft somehow limits the capa- such that x(t) satisfies (1), i.e., = bility of dynamical system theory to provide ! xx ttft ì);),(()( (4) global stability information in as much as tran- Dynamical systems theory (DST) provides a sient motions cannot be predicted or quantified methodology for studying systems of ordinary by bifurcation theory (compare with results ob- differential equations. The most important ideas tained by Carol and Mehra [2]). Avanzini and of DST used in the paper will be introduced in de Matteis [4] have proved, that certain addi- 3112.2 AN AGILE AIRCRAFT NON-LINEAR DYNAMICS BY CONTINUATION METHODS AND BIFURCATION THEORY

the following sections. More information on axis when control vector varies. Two cases can DST can be found in the book of Wiggins [11]. be considered [16]. The first step in the DST approach is to • The steady state is regular, i.e. when the im- calculate the steady states of the system and plicit function theorem works and the equilib- their stability. Steady states can be found by rium curve goes through a limit point. It should setting all time derivatives equal to zero and be noted that a limit point is structurally stable solving the resulting set of algebraic equations. under uncertainties of the differential system The Hartman-Grobman theorem proves that the studied. local stability of a steady state can be deter- • The steady state is singular. Several equilib- mined by linearizing the equations of motion rium curves cross a pitchfork bifurcation point, about the steady state and calculating the eigen- and bifurcation point is structurally unstable. values [12], [13], [14], [15]. If a pair of complex eigenvalues cross The implicit function theorem (Ioos and the imaginary axis, when control vector varies, Joseph [16]) proves that the steady states of a Hopf bifurcation appears [13], [14], [16], [17], system are continuous function of the parame- [18]. Hopf bifurcation is another interesting bi- ters of the system at all steady states where the furcation point. After crossing this point, a peri- linearized system is non-singular. Thus, the odic orbit appears. Depending of the nature of steady states of the equations of motion for an nonlinearities, this bifurcation may be sub- aircraft are continuous functions of the control critical or supercritical. In the first case, the sta- surface deflections. Stability changes can occur ble periodic orbit appears (even for large as the parameters of the system are varied in changes of the control vector). In the second such a way that the real parts of one or more case the amplitude of the orbit grows in portion eigenvalues of the linearized system change to the changes of the control vector. sign. Changes in the stability of a steady state lead to qualitatively different responses for the 2.3 Continuation technique and methodology system and are called bifurcations. Stability scheme boundaries can be determined by searching for Continuation methods are a direct result of the steady states, which have one or more eigenval- implicit function theorem, which proves that the ues with zero real parts. There are many types steady states of a system are continuous func- of bifurcations and each has different effects on tions of the parameters of the system al all the aircraft response. Qualitative changes in the steady states except for steady states at which response of the aircraft can be predicted by de- the linearized system is singular. The general termining how many and what types of eigen- technique is to fix all parameters except one and values have zero real parts at the bifurcations trace the steady states of system as a function of point. Bifurcations for which one real eigen- this parameter. If one steady state of the system value is zero lead to the creation or destruction is known, a new steady state can be approxi- of two or more steady states. Bifurcations for mated by linear extrapolation from the known which one pair of complex eigenvalues has zero steady state [12], [14], [15], [19]. The slope of real parts can lead to the creation or destruction the curve at the steady state can be determined of periodic motion. Bifurcations for which more by taking the derivative of the equation given by than one real eigenvalue or more than one pair setting all time derivatives equal to zero. If two of complex eigenvalues has zero real parts lead steady states are known, a new steady state can to very complicated behaviour be approximated by linear extrapolation through the two known steady states. The stability of 2.2 Bifurcation Theory each steady state can be determined by calcu- For steady states of aircraft motion, very inter- lating the eigenvalues of the linearized system. esting phenomena appear when even if one Any changes in stability from one steady state negative real eigenvalue crosses the imaginary to the next will signify a bifurcation.

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Taking into account experience of many Oza axis lies in the symmetry plane of the air- researches, one can formulate the following craft and is directed downwards. tree-step methodology scheme (being based on bifurcation analysis and continuation technique) for the investigation of non-linear aircraft behaviour [7], [15], [19]: • During the first step it is supposed that all parameters are fixed. The main aim is to search for all possible equilibria and closed orbits, and to analyze their local stability. This study should be as thorough as possible. The global structure of the state space (or phase portrait) can be re- vealed after determining the asymptotic stability regions for all discovered attractors (stable equilibria and closed orbits). An approximate graphic representation plays an important role in the treating of the calculated results. • During the second step the system behaviour Fig. 1 System of co-ordinates attached to aircraft is predicted using the information about the evolution of the portrait with the parameters variations. The knowledge about the type of en- The relative position of the vertical system countered bifurcation and current position with Oxgygzg and the system Oxyz, attached to the respect to the stability regions of other steady aircraft is described by Euler angles Θ, Φ and Ψ motions are helpful for the prediction of further (Fig. 1), while the relative position of the sys- motion of the aircraft. The rates of parameters tem Oxyz and the system Oxayaza attached to the variations are also important for such a forecast. airflow - by the angle of attack α and slip angle The faster the parameter change, the more the β (Fig.2). difference between steady state solution and transient motion can be observed. • Last, the numerical simulation is used for checking the obtained predictions and obtaining transient characteristics of system dynamics for large amplitude state variable disturbances and parameter variations.

3. Mathematical model of aircraft motion Non-linear equations of motion of the aeroplane and the kinematic relations will be expressed by using moving co-ordinate systems, the common origin of which is located at the centre of mass of the aeroplane (Figs.1 and 2). It is used [10]: Fig. 2. System of co-ordinates attached to airflow − a system of co-ordinates Oxyz attached to the aircraft (the Oxz plane coinciding with the Usually aircraft is considered as a rigid body symmetry plane of the aircraft); with moving elements of control surfaces. Gy- − a system of co-ordinates attached to the roscopic moment of rotating masses of the en- air flow Oxayaza in which the Oxa axis is di- gines is included. Total system of equations rected along the flight velocity vector V and the should be completed with the following expres- 3112.4 AN AGILE AIRCRAFT NON-LINEAR DYNAMICS BY CONTINUATION METHODS AND BIFURCATION THEORY

sions: kinematic relations, kinematics of an ar- MF - vector of momentum of aircraft engine: bitrary control system and the control laws. F ∑∑ ( ω# ×+×== OJFrMM ) iF FFii ii (8) The mathematical model of aircraft can be for- i i mulated in the following form [10], [20], [21]: where: = (5) r - vectors determining the distance of the ! tt )),(),(( (0) x=xuxfx 0 i where: engines thrust from the pole of the system of - state vector co-ordinates; α β ΨΘΦ= T Fi - vector of the thrust; x zyxrqpV ggg ],,,,,,,,,,,[ (6) J - the polar moments of inertia of the engine - control vector Fi = α δ δ δ δ T (7) rotor; u , FVAHzH ],,,[ ω# - vector of angular velocity of a rotor; and: Fi T 1 O =[p,q,r] - vector of angular velocity of air- f []α ϕ )cos( PF −−+= 1 m Xs a craft. Equations of motion of aircraft should be g[ sinsincos β −ΦΘ− completed with equations of engine dynamics α ΦΘ−Θ α β ] cos)sincoscoscos(sin - - cos)sincoscoscos(sin [10]: α α β −+−= 2 )sincos( tanrpqf - equation of engine rotation. τ τ τ τ [ τ ]⋅−−=++ 1 !!! )()( pp 002121 nQtQKnn (9) [ α ϕ )sin( PF +++− mV cos β Zs a where: α α)coscoscossin(sin +ΦΘ+Θ− Pmg ] n - angular velocity of a rotor, Za τ τ ,, τ - time-constants, α rpf cossin α −−= 210 3 K - amplification factor, 1   [F cos()α ϕ ++ Qp - discharge of fuel for actual engine’s angu- mV  s lar velocity and for actual aircraft’s altitude and mg α α ]sin)sincoscoscos(sin β −ΦΘ−Θ− airspeed, Q - discharge of fuel, calculated for sea level  p0 coscoscos β −ΦΘ− Pmg  conditions and when V=0, Ya  Time-constants are non-linear functions     −   of engine’s angular velocity, aircraft’s altitude, f 4  0 qr    −   air density, pressure and temperature on fight 1  −+= −   f 5  Fa )(  0 pr JMMJ   altitude. Discharge of fuel is following function:  f   − pq 0   = δ 6 1 Tp ttQ ))((f)( (10)  f   sin1 tg cos tg ΘΦΘΦ  p where:  7     = Φ−Φ δT - displacement of cockpit power lever  f8   sincos0 q - equation of thrust  f   seccossecsin0 ΘΦΘΦ  r  9  ρ  7.0 =   ()++ 2  f  V  0 )(   210 MaKMaKKnTT (11)  10     ρ  f = T AA 0 0  11  a   The engine model should be adapted from the      f12  0 code, data, and flow charts provided by the en- where: gine manufacturer. PXa, PYa, PZa - aerodynamic forces, A and Aa - matrixes of transformations. Ele- 3.1 Modelling of aerodynamic loads ments of those matrixes can be found for exam- The adequacy of mathematical model- ple in [10]; ling of high AoA (Angle of Attack) dynamics is J - matrix of inertia; strictly dependent on the adequacy of the aero- T Ma=[L.M,N] – vector of aerodynamic moment; dynamic model at these regions. There is non-

3112.5 Krzysztof Sibilski

trivial problem due to the very complicated na- − in given section of a force, aerodynamic ture of the separated and vortex flow in un- moments depend on a local angle of attack steady regime [22]. Precise describing of aero- − the area of a flowfield is disturbed by dynamic forces and moments found in equations adding to a vector of speed appropriate compo- of motion is fundamental source of difficulties. nents resulting from plane’s rotation with an- In each phase of flight dynamics and aerody- gular speeds p, q, r. namics influence each other, which disturbs the − mutual relation between flows of neigh- precise mathematical description of those proc- bouring strips was taken into account by adding esses. The requirements for method on aerody- the speed induced by flowing down whirlpools namic load calculations stem both from flow − dynamics of whirlpool structures, in- environment and from algorithms used in analy- cluding whirlpools’ break-up, was taken into sis of helicopter flight. The airframe model con- account sists of the fuselage, horizontal tail, vertical tail, − unsteadiness of a flowfield (aerodynamic and wing. The fuselage model is based on wind hysteresis) was taken into account, phenomenon α tunnel test data (as function of angle of attack of deep aerodynamic stall was modelled using and slip angle β). The horizontal tail and verti- algorithm worked out in ONERA [8], [9]. cal tail are treated as aerodynamic lifting sur- The algorithm of calculations allows de- faces with lift and drag coefficients computed fining loads of wings of any shape. In case of from data tables as functions of angle of attack modern fighter aeroplanes, with strongly cou- α and slip angle β. pled aerodynamic configuration, it was assumed For linear extent of lifting force, an that lifting fuselage of these planes is a centre aeroplane’s aerodynamic loads can be defined wing section, that is - in algorithms, forces and on the basis of algorithms, relations, diagrams aerodynamic moments generated by lifting parts and formulas shown, for example, in DATA of fuselage were taken into account. The modi- SHEETS or in The USAF Stability and Control fied strip theory is interesting also because it is DATCOM [23]. However, there is no efficient relatively easy to consider a phenomenon of method to calculate aerodynamic loads for high non-symmetrical break-up of whirlpools in its angles of attack. Results of investigating aero- algorithms (using, for example method pro- plane’s models are not always available and posed in work [10]). A wing of a plane is di- complete. Usually, there are no reliable aero- vided with planes parallel to the fuselage’s plane’s aerodynamic characteristics, obtained by plane of symmetry to a number of elements identification method, on the basis of measure- (strips). For each strip we define a local angle of ments during a flight. Panel methods are appro- attack and a value of total vector of speed. Then, priate to define aerodynamic loads for small and from aerodynamic characteristics of the profile, moderate angles of attack. Numerical study of we define aerodynamic coefficients of: lifting spin and other manoeuvres performed on over- force, resistance force and pitching moment. critical angles of attack, as well as simulation of Integration defined in this way forces and mo- acrobatic figures characteristic for so-called su- ments along wing span allows defining the permanoeuvre planes (for instance, Cobra or aerodynamic loads of a plane. For purpose of Kublit manoeuvres) requires data concerning numerical analysis, functions Cz(α) and Cx(α) aerodynamic characteristics for angles of attack were approximated with trigonometric polyno- o o practically from –180 to +180 . Therefore, to mials: define aerodynamic loads in the possibly broad- n α α += α est extent of angles of attack, an attempt to ∑ kz k kbkaC )]sin()cos([)( = broaden the strip theory has been made. In k 0 (12) n modification of this theory, presented below, α α += α ∑ kx k kdkcC )]sin()cos([)( following assumptions have been made: k =0

3112.6 AN AGILE AIRCRAFT NON-LINEAR DYNAMICS BY CONTINUATION METHODS AND BIFURCATION THEORY

Where coefficients ak, bk, ck, and dk were calcu- characteristics taking into account the dynamics lated from Runge’s scheme. Values of these co- effects of the separated flow [10]. efficients are shown in work [10]. Now, it must be pointed up that dynamic The angle of attack of - elementary system theory provides a methodology for strip of a wing depends on: the aeroplane’s an- studying of such complicated ordinary differen- gle of attack, an angle of attack induced by tial equations (ODE). In addition, recent devel- whirlpools flowing down a wing and an angle of opment in the area of numerical analysis of non- attack caused by appearance of angular speeds linear equations created a class of computer al- (pitching, rolling, and yawing). The induced an- gorithms known as continuation method [15], gle of attack can be calculated from the relation: [19] The set of ordinary differential equations   can be solved using the continuation and bifur- α = Vi i arctan  (13)  V0  cation software AUTO97 available at address: The induced speed can be calculated from Biot- ftp://ftp.cs.concordia.ca/pub/doedel/auto. Savart’s law: This very useful freeware gives all desired bi- Γ y)( furcation points for different values of control yV )( −= ϕ ϕ )cos(cos −+ vector components. i 4πr 21 1 (14) Γ y)( − ϕ + ϕ )cos(cos 3 Results 4πr 43 2 Figs. 3-8 show the steady states of the Where r1 and r2 – correspondingly, a distance MiG-29 fighter aircraft for longitudinal ma- from left and right bound vortex from point A noeuvres, which are at middle and high angles (in which induced speed is calculated). of attack. Steady states represented in those fig- Distribution of circulation along wing span is ures show longitudinal trim conditions and spi- given with following differential-integral equa- rally divergent motions. These figures show that tion: practically for all elevator deflections the air- ∂C z χ yc )(cos craft can achieve stable or unstable trim condi- V ∂α A y)( =Γ 0 × tions. The trim conditions for given elevator de- 2 Π yk )( flection can be determined by drawing the verti- b (15) cal line representing the desired elevator deflec- 1 2 dΓ ξ)( dξ tion on each plot; each intersection of this line α y)([ −× ] 0 π ∫ ξ −ξ with the curve of steady states gives a possible 4 V0 −b d z )( 2 steady state of the aircraft. For example, a verti- Equation (15) can be solved with ap- cal line representing 4o of the elevator deflection proximate methods (for instance, approximation intersects three steady states. Two of them are of trigonometric series). Distribution of circula- stable one is unstable, so the aircraft could ex- tion along wing span can also be calculated with hibit any of these three steady states. One stable engineer methods (for example, classic Mult- steady state at 4o elevator deflection represents hopp’s method) or evaluated with help of the horizontal flight trim configuration known (for example from examining a plane in (p=q=r=0). The other two stable steady states aerodynamic tunnel) distribution of pressures represent pull-up, or spiral trim conditions. along wing span. On the basis of known distri- Continuation methods require a known bution of circulation we can define distribution steady state as a starting point for the continua- of induced angles of attack along wing span tion procedure. It is usually easy to determinate (and therefore for each wing’s section). steady states that are at low and moderate values The bifurcation approach is very fruitful of angels of attack. Determining the steady when the sources and nature of aerodynamic states at high angles of attack is more difficult phenomena are considered. Special techniques task and it is usually not possible to be certain were proposed to represent the aerodynamic all the steady states for a particular aircraft have

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been determined. The approach used to find the modes in this work was to guess an initial high- angle-of attack mode as a starting point for the continuation method algorithm, then let the algorithm run until either a true steady state was determined or the algorithm ran into numerical problems.

Fig. 5 Steady states for longitudinal manoeuvres

Fig. 3. Steady states for longitudinal manoeuvres The segment of unstable steady states contains the trim conditions between the elevator deflections at –20.5o and 12.1o because of six Hopf or saddle-node bifurcations. Those bifur- Fig. 6 Steady states for longitudinal manoeuvres cations occur at –20.5o (saddle-node bifurcation), –20.1o (Hopf bifurcation), -8.8o (Hopf bifurcation); 7.1o (Hopf bifurcation), 12.1o (saddle-node bifurcation), and 12.1o (Hopf bifurcation).

Fig. 7 Steady states for longitudinal manoeuvres

Fig. 4 Steady states for longitudinal manoeuvres Hopf bifurcations can lead to periodic motions, so it is possible that for elevator deflections between –20.1o and –8.8o, 7.1o and 12.1o the aircraft will undergo periodic motion (wing- rock instability). The saddle-node bifurcations (occurred at the elevator deflections: –20.5o and 12.1o) signify that aircraft loss their longitudinal stability. Fig. 8 Steady states for longitudinal manoeuvres

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It can by explain by loosing of effectiveness of ! fourth phase, when angles of attack reach the control surface. The lifting force generated values lower then that of in the steady flight on the tail control surface depends on the local and the process of balancing at low angles angle of attack and is perpendicular to the local of attack is going on. flow direction. As the first approximation it can During the third and fourth phases it is be assumed that control pitching moment is necessary to control the aircraft motion in its generated by the normal component of the aero- recovery from high angles of attack. Slow re- dynamic tail force only. In the range of AoA up covery creates the conditions for supplementary to 35o the normal component increases ap- loss of flight speed and increases the probability proximately linearly, then stabilises and practi- of stall. Quick recovery from high angles of at- cally the tail surface losses its effectiveness. tack can be reason to go at angles of attack After the further displacement of the tail control lower than 0. Thorough analysis of Cobra ma- surface the pitching moment is proportional to noeuvre dynamics was presented in paper [25]. the cosine of the displacement angle. This way, Exemplary results of those investigations are in the first approximation, one can assume that shown in Fig. 9. control surface is able to work effectively in range approximately ±40o. Those occurred at high angles of attack longitudinal stability loosing make it possible to perform the Cobra manoeuvre. This new manoeuvre had been shown by W. Pugatchov early in 1994 at the Abu Dabi Air Show and was called “hook”. The hook manoeuvre (performed in horizontal plane) processes real combat significance. Bas- ing on flight tests (cf. O. Samoylovich [24]) it was concluded that the dynamic entrance into Fig. 9 Curves of angle of attack, and pitch angle dur- the high angles of attack flight could be divided ing Cobra manoeuvre (cf. [25]) into four phases as follows: ! first phase, characterised by full deflection Based upon investigations carried out in works of horizontal tail (for pull-up of the aircraft) [24], [25] one can conclude that the necessary with a maximum speed of the control stick. conditions to perform Cobra manoeuvre are: The main goal in this stage is to create a big, ! aircraft should be statically unstable in the positive pitching moment as soon as possi- range of angles of attack near critical; ble; ! balance should be attainable in the range of ! second phase, when aircraft still increases angles of attack greater than critical; ! angle of attack due its inertia and at last pitching moment for diving has to have a reaches the maximum angle of attack (at the margin in the range of angles of attack α∈(30ο, 60ο); end of this phase the pitching moment is near of its maximum value for diving, pitch ! natural pitching moment for diving should be big enough in the range of angles of attack rate is near 0); o ! third phase, (recovery from the manoeuvre) grater than 60 ; characterised by full deflection of the hori- ! limiter of the extreme flying parameters zontal tail for diving with the increasing, should be turned off. negative pitch rate (at the end of this phase Moreover, the aircraft should be highly insensi- the pitch rate for diving reaches its maxi- tive to the spin tendency, especially because of mum, negative value. The angle of attack lack of lateral stability at high angles of attack. approaches its value of the steady flight, but Investigations of Cobra manoeuvre carried out aircraft still rotates and further decreases the in work [25] were under assumption, that the angle of attack due to its inertia); aircraft is highly insensitive to the spin tendency 3112.9 Krzysztof Sibilski

and wing rock instability. Unfortunately, at higher angles of attack real aircraft will undergo wing-rock oscillations, and has strong tendency to bank. It is evident, when one considered steady states at middle and high angles of attack. Occurrences of Hopf and saddle-node bifurcations signify radical changes in aircraft response after those bifurcations. Figs. 10-21 show an attempted Cobra manoeuvre entry using the elevator deflection. During the Cobra manoeuvre, all flight parameters in- Fig. 12 Longitudinal manoeuvre performed at high crease their values. In terms of continuation angles of attack. Poincare map methods, the Cobra is unstable because of Hopf o bifurcation, that occurred at δΗ=–20.1 and sad- o dle-node bifurcation, that occurred at δΗ=-20.5 . The Poincare maps of selected state parameters are shown in Figs. 10-12. It can be stated that taking into consideration unsteady aerodynamic model and hysteresis of aerodynamic coefficients, one counters significant irregularities in solution of equations of motion, that are characteristic for chaotic motion.

Fig. 13 Course of angle of attack

Fig. 10 Longitudinal manoeuvre performed at high angles of attack. Poincare map

Fig. 15 Course of slip angle

Fig. 11 Longitudinal manoeuvre performed at high angles of attack. Poincare map Fig. 16 Course of roll rate

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Fig. 17 Course of pitch rate Fig. 21 Course of yaw angle To perform successfully the Cobra manoeuvre, one should rapidly pull-up the control stick at the commencement of the manoeuvre, putting the aircraft in region of unstable steady states. During those simulations one assumed, that aircraft has the spin tendency and wing rock instability, because of lack of lateral stability at high angles of attack. Occurrences of Hopf and saddle-node bifurcations signify radical changes in aircraft response after those bifurcations At higher angles of attack the MiG-29 fighter air- Fig. 18 Course of yaw rate craft is undergoing wing-rock instability. Figs. 13-21 show time simulation of Cobra manoeuvre. The course of change of the angle of attack α is shown in Fig. 13, the angle of pitch Θ - in Fig.20 and the roll and yaw angles - in Figs.19, and 21. It is seen from these figures that during that manoeuvre the angles of attack and pitch increase suddenly. Maximum values of α, Θ reach 100o and 110o respectively. Figs. 13-21 show, additionally developing wing-rock oscillations witch a period approximately 3.5s. Note that magnitude and frequency of those os- Fig. 19 Course of roll angle cillations are irregular and have chaotic charac- ter [7], [10]. It can be stated, that during the Co- bra manoeuvre the pilot must fix his attention on aircraft control, first of all keeping the aircraft precisely in the vertical plane. It is not simple, because the aircraft during this manoeuvre has strong tendency to bank. Additionally it is observed appearance of wing rock oscillations.

Conclusions The main aim of the study was to apply Fig. 20 Course of pitch angle modern methods for investigation of non-linear

3112.11 Krzysztof Sibilski

problems in flight dynamics. Based on the in- Continuation and Bifurcation Methods, AIAA Pa- th vestigation described above, the following con- per, AIAA-2000-0976, 38 Aerospace Sciences Meeting and Exhibit, Reno, NV, January, 2000 clusions can be drawn: [8]. Tran C. T., Petot D., Semi-Empirical Model for the 1. The present results show the value of using Dynamic Stall of Airfoils in View of the Applica- continuation and bifurcation methods for ana- tion to the Calculation of Responses of a Helicopter lysing the equations of aircraft motion; Rotor Blade in Forward Flight, Vertica, 5, 1981. 2. The efficiency of the methods makes it pos- [9]. Narkiewicz J., Rotorcraft Aeromechanical and Aeroelastic Stability, Scientific Works of Warsaw sible to analyse complicated aerodynamic mod- University of Technology, Issue Mechanics, no. 158, els using the complete equations of motions for 1994. the whole range of control surface deflections; [10]. Sibilski K., Modelling of agile airships dynamics in 3. Knowledge of such deflections, which cause limiting flight conditions, Rep. No. 2557/98, Mili- bifurcation allows us to select the most probable tary University of Technology, Warsaw 1998. [11]. Wiggins S., Introduction to Applied Nonlinear Dy- scenario of occurrences before the accident, and namical Systems and Chaos, Springer-Verlag, New to escape from risky motions; York, 1990. 4. The need for a precise description of aero- [12]. Guckenheimer J., Holmes J., Nonlinear Oscilla- dynamic loads is a fundamental cause of diffi- tions, Dynamical Systems, and Bifurcations of Vec- culties; tor Fields, Springer, N. Y., 1983. [13]. Crawford I. E., Introduction to Bifurcation Theory, 5. The presented approach can be applied to Reviews of Modern Physics, Vol. 63, No. 4, 1991. the prediction of space behaviour of aircraft. [14]. Troger H., Steindl A., Nonlinear Stability and Bi- Therefore, it can be also applied to modification furcation Theory, Springer Verlag, New York, 1991. of aircraft dynamic characteristics. [15]. Keller H. B., Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems, Application of Bi- Acknowledgements furcation Theory, Academic Press, N. York, 1977. [16]. Ioos G., Joseph D., Elementary Stability and Bifur- The paper was prepared as a part of the project cation Theory, Springer-Verlag, New York, 1980. (Grant No. 9 T12C 002 15) financed by the [17]. Hassard B. D., Kazarinoff N. D., Wan Y H, Theory State Committee of Scientific Researches and Applications of Hopf Bifurcation, Cambridge University Press, 1981. [18]. Marsden J. E., McCracken M., The Hopf Bifurca- References tion and its Applications, Applied Mathematical Science, v. 19, Springer Verlag, New York, 1976. [1]. Jahnke C. C., Culick F. E. C., Application of Bifur- [19]. Doedel E., Kernevez J. P., AUTO – software for con- cation Theory to the High-Angle-of-Attack Dy- tinuation and bifurcation problems in ordinary dif- namics of the F-14, Journal of Aircraft, Vol. 31, No. ferential equations, Caltech, Pasadena 1986. 1, pp.26-34, 1994. [20]. Dzygadlo Z., Sibilski K, Dynamics of Spatial Mo- [2]. Carroll J. V., Mehra R. K., Bifurcation Analysis of tion of an Aeroplane After Drop of Loads, Journal Non-Linear Aircraft Dynamics, Journal of Guid- of Technical Physics, Vol. 29, 3-4, 1988. ance Control and Dynamics, Vol. 5, No. 5, 1982. [21]. Arczewski K., Pietrucha J., Mathematical Modelling [3]. Guicheteau P., Bifurcation Theory in Flight Dy- of Complex Mechanical Systems, E. Horwood, namics an Application to a real Combat Aircraft, Chichester, 1993. ICAS-90-5.10.4, Proceedings of 17th ICAS Con- [22]. Rom J., High Angle of Attack Aerodynamisc, Sub- gress, Stockholm, Sweden, 1990. sonic, Transsonic and Supersonic, Springer Verlag, [4]. Avanzini G., De Matteis G., Bifurcation analysis of 1980. a highly augmented aircraft model, Journal of [23]. The USAF Stability and Control DATCOM, Guidance, Control & Dynamics, Vol.20,No.1, 1998. AFFDL-TR-3032, April 1979. [5]. Sibilski K., Bifurcation analysis of a helicopter non- [24]. Samoylovich O., Aerodynamic Configurations of linear dynamics, Preprints of the 24th European Ro- Contemporary and Perspective Fighters, Proc. of torcraft Forum, Marseilles (France), 15-17 Sept, Second Seminar of RRDPAE’96, 1997. 1998. [25]. Dzygadlo Z., Kowaleczko G, Sibilski K., Method of [6]. Sibilski K., Non-Linear Flight Mechanics of a Heli- Control of a Straked Wing Aircraft for Cobra Ma- copter Analysis by Application of a Continuation th noeuvres, ICAS-96-3.7.4, Proceedings of 20 ICAS Method, Pre-prints of the 25th European Rotorcraft Congress, Sorrento, Italy, 1996. Forum, Rome, Italy, 14 - 16 Sept. 1999. [7]. Marusak A. J., Pietrucha J. A., Sibilski K. S., Pre- diction of Aircraft Critical Flight Regimes Using 3112.12