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Гоман, Загайнов, Храмцовский (1997) - Использование бифуркационных методов для исследования задач нелинейной динамики полёта
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Гоман, Загайнов, Храмцовский (1997) - Использование бифуркационных методов для исследования задач нелинейной динамики полёта

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М.Г.Гоман, Г.И.Загайнов, А.В.Храмцовский "Использование бифуркационных методов для исследования задач нелинейной динамики полёта", Prog.Aerospace Sci., том.33, стр.539-586, 1997 …

М.Г.Гоман, Г.И.Загайнов, А.В.Храмцовский "Использование бифуркационных методов для исследования задач нелинейной динамики полёта", Prog.Aerospace Sci., том.33, стр.539-586, 1997

В статье рассказывается об использовании методов бифуркационного анализа и анализа глобальной устойчивости для исследования различных задач нелинейной динамики полёта, таких, как инерционное вращение, сваливание, штопор и т.д. Приведены примеры расчетов с использованием моделей реальных самолетов, таких как F-4, F-14, F-15 и модели High Incidence Research Model (HIRM). Также кратко описаны основные понятия и методы теории динамических систем.

M.G.Goman, G.I.Zagainov and A.V.Khramtsovsky "Application of Bifurcation Methods to Nonlinear Flight Dynamics Problems". Prog.Aerospace Sci., Vol.33, pp.539-586, 1997

Applications of global stability and bifurcation analysis’ methods are presented for different nonlinear flight dynamics problems, such as roll-coupling, stall, spin, etc. Based on the results for different real aircraft, F-4, F-14, F-15, High Incidence Research Model (HIRM), the general methods developed by many authors are presented. The outline of basic concepts and methods from dynamical system theory are also introduced.

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  • 1. Pergamon Plh S0376-0421 (97)00001-8 I'ro9. AerospaceSci.Vol.33,pp.539 586.1997 (~ 1997ElsevierScienceLtd Allrightsreserved.PrintedinGreatBritain 0376-0421.,97$32.00 APPLICATION OF BIFURCATION METHODS TO NONLINEAR FLIGHT DYNAMICS PROBLEMS M. G. Goman, G. I. Zagainov and A. V. Khramtsovsky Central 4erohydrodynamic Institute (TsAGI), 140160, Moscow Region. Zhukot:sky. Russia (Received 12 December 1995; in final form 5 March 1997) Abstract--Applications of global stability and bifurcational analysis methods are presented for different nonlinear flight dynamics problems, such as roll-coupling, stall, spin, etc. Based on the results for different real aircraft, F-4, F-14, F-15, High Incidence Research Model, (HIRM), the general methods developed by many authors are. presented. The outline of basic concepts and methods from dynamical system theory are also introduced. !:2 1997 Elsevier Science Ltd CONTENTS 1. INTRODUCTION 539 1.1. Nonlinear aircraft problems 540 1.2. Approximate approaches 540 1.3. Continuation methods implementation 541 1.4. Bifurcation analysis and continuation technique methodology 541 1.5. The adequacy of a mathematical model 542 2. APPLICATION OF BIFURCATION METHODS TO FLIGHT DYNAMICS PROBLEMS 542 2.1. Autonomous forms of motion equations and steady state modes 542 2.1.1. Aerodynamic model for stall/spin conditions 544 2.1.2. Aircraft with control augmentation system 545 2.2. Critical flight regimes 546 2.3. Dynamics of symmetrical flight in a vertical plane 547 2.4. Nonlinear phugoid motion 547 2.5. Deep stall motion 549 2.6. The general case of the longitudinal motion 550 2.7. Roll-coupling problem 551 2.8. Dynamics at subsonic flight regime 553 2.9. Dynamics at supersonic flight regime 554 2.10. Nonlinear stall motion 558 2.11. Approximate linear criteria of stall 558 2.11.1. Controllability criteria 560 2.12. Stall dynamics analysis 561 2.13. Spin dynamics 564 2.14. Bifurcational analysis of spin motion 566 3. THEORETICAL BACKGROUND 576 4. CONCLUSIt)NS 584 ACK NOWLEDGEM ENTS 584 REFERENCES 584 1. INTRODUCTION Nowadays there are a great number of specialized books dedicated to nonlinear dynam- ical systems and bifurcation theory, and nobody can deny that this area of modern mathematics is one of the most interesting and attracts specialists in numerous branches of science and engineering.(x-t8) Nonlinear world displays the universal features in its all manifestations. There are many publications in such disciplines as physics, aerodynamics, biology, chemical kinetics, econ- omics, etc. devoted to study of various nonlinear phenomena. Nonlinear systems describing the behaviour of the objects of the real world usually are too complex to be thoroughly studied using analytical methods. Hence the advances in
  • 2. 540 M.G. Gomanet al. understanding essential features of nonlinear objects are closely coupled with the develop- ment of numerical methods. The interest in the theory of nonlinear systems and in the tools for computer-aided research rapidly grows. Today a researcher has at his disposal a number of effective and reliable numerical methods intended for analysis and classification of nonlinear phenomena. All of them were developed during the last ten/fifteen years.~19-3°~ And there is a need for more and more powerful algorithms and methods. There is also a need for software packages eliminating the necessity for a researcher to write too many lines of code for his computer. Partially, in flight dynamics applications the Packages BISTAB, AUTO, ASDOBI, KRIT were used.tl~'2°'22`3xl 1.1. NONLINEAR AIRCRAFT PROBLEMS Nonlinear aircraft dynamics problems to a great extent are encountered during high angle of attack maneuvers and fast rotation in roll. Investigation of aircraft behavior at stall/spin regimes and spatial maneuvers with strong roll-inertia coupling is extremely important for the definition of the flight envelope boundary and solving the flight safety problem. While studying these regimes, one has to take into account the nonlinear terms in the equations of aircraft motion and nonlinear aerodynamic effects, especially for high angle-of- attack region. Therefore, the problem to be studied is essentially nonlinear and is usually formulated in the form of a set of ordinary nonlinear differential equations depending on parameters. These parameters can characterize the flight conditions, mass and inertia characteristics of aircraft. Very often the control surface deflections are considered as such parameters, and bifurcation analysis in this case permits to study the dependency of global dynamical behavior as the function of fixed control deflections. It is important to note that the results of bifurcational and qualitative analysis give good basis for forecasting aircraft dynamics in the case of varying control. Modern aircraft are often intrinsically unstable, and stable flight is possible only due to the control augmentation system (CAS). Both CAS mechanical elements (due to natural constraints such rate saturation, limited deflections) and feedback algorithms introduce additional nonlinearities. Therefore modern aircraft with flight control systems become more nonlinear plant. Bifurcation and qualitative theory methods can also be used to investigate the nonlinear closed-loop dynamics for verification of the implemented algo- rithms, and therefore can be very efficient in control laws design. One can divide all the published works devoted to aircraft nonlinear dynamics problems into two groups. The first group contains the works with an approximate approach, and the second one contains the works based on the continuation technique and wide implementa- tion of results from modern theory of nonlinear dynamical systems. 1.2. APPROXIMATE APPROACHES Stall/spin phenomena and roll-coupling instabilities were studied by many authors for every new generation of aircraft starting from the beginning of aeronautical era. Both the spin problem 1333"~1 and the roll-coupling problem t3~41~ were investigated using the reduced equations of motion, simplified aerodynamic coefficients representation (as a rule, only linear dependencies) and approximate methods for solving of these equations. The roll-coupling instabilities were not strictly connected with the bifurcations in the equations under consideration. For example, the critical roll rates first introduced by Phillips are very useful for engineering estimates, but they are the result of the simplified consideration of the problem and cannot be treated as bifurcations or stability losses. The main reason is that the problem was analyzed by treating the roll rate (state variable) as a parameter in the linearized pitching and yawing moment equations. For construction of the equilibrium solutions of motion equations some artificial tricks were needed, e.g. the consideration of the angle of attack (also the state variable) as some parameter. ~4t~Nevertheless, it was shown, that the departures associated with roll-coupling instabilities resulted in the aircraft
  • 3. Bifurcation methods in nonlinear flight dynamics 541 jumping from one stable equilibrium to another, i.e. the instability due to saddle-node bifurcations was implicitly recognized,t38~ The wing rock phenomena at first were also analyzed using an approximate approaches for nonlinear oscillations, without linking them with Hopf bifurcation.~36~ 1.3. CONTINUATION METHODS IMPLEMENTATION The wide and purposeful applications of bifurcational methods and nonlinear dynamical system theory to flight dynamics nonlinear problem at high angles of attack regimes are now well-established.~42"44-47"50-53.60) The common feature of all these works is the implementation of the continuation technique for solving the nonlinear problems of an aircraft equations both for equilibrium and oscillatory motions. Carroll and Mehra were the first who used the continuation method in flight dynamics and connected the main types of an aircraft instabilities with bifurcation phenomena (e.g. the occurrence of wing rock motion was connected with a Hopf bifurcation, the examples of chaotic motion were presented, etc). Later the existence of new types of bifurcations in aircraft dynamics were explored, e.g. origination of stable torus manifold,~4~global bifurcation of a closed orbit related with the appearance of homoclini- cal trajectory,~'Lg~so-called flip or period doubling and pitchfork bifurcations for closed orbits.~So.5~ Continuation methodology and bifurcation analysis can be used for determin- ing the recovery technique from critical regimes and for control law design for improving dynamical behavior.~43'44"47'5x~ If the earliest works were devoted mainly to the theoretical methods and associated numerical procedures which can be used in aircraft dynamics,~42"47~the latest publications more often present the results of bifurcation analysis of high angle of attack dynamics of real aircraft (F-4,~421German-French Alpha-Jet,~45~F-14,~53~F-15~sl~). Comparisons between flight test results and predicted results obtained using bifurcation methods and numerical calculations for a number of aircraft show a very good agreement both in qualitative and quantitative senses,t45~These comparisons reveal the efficiency and comprehension of bifurcational methods in flight dynamics applications. Therefore one can say that bifurcation methodology based on the computer-aided technology is becoming a popular and very powerful tool in the complicated nonlinear area of flight dynamics. 1.4. BIFURCATION ANALYSIS AND CONTINUATION TECHNIQUE METHODOLOGY Many characteristics, defining flight conditions, aircraft parameters and control surface deflections, can vary with time slowly. In many practical cases it is reasonable to consider such parameters as fixed ones and independent of time. This reduces the problem to a study of nonlinear autonomous dynamical systems. The system behavior in the case of parameter variations can be predicted using the knowledge about the specific system responses after encountering the bifurcation conditions. Taking into account experience of many researchers, one can formulate the following three-step methodology scheme for the investigation of nonlinear aircraft behavior, the scheme being based on bifurcation analysis and continuation technique: • During the first step it is supposed that all parameters (expect for lhe state variables) are fixed. The main goal is to search for all the possible equilibria and closed orbits and to analyze their local stability. This study should be as thorough as possible. The implementat:on of the continuation technique to a great extent facilitates the solving of this problem. The global structure of the state space (or phase portrait) can be revealed after determining the asymptotic stability regions for all discovered attractors (stable equilibria and closed orbits). An appropriate graphic representation plays an important role in the treating of the calculated and accumulated results.
  • 4. 542 M.G. Goman et al. • During the second step the system behaviour is predicted using the information about the evolution of the phase portrait with the parameters variation. The knowledge about the type of encountered bifurcation and current position with respect to the stability regions of other steady motions are helpful for the prediction of further motion of the aircraft. The rates of parameter variations are also important for such a forecast. The faster the parameter change, the more the difference between steady-state solution and transient motion can be observed. • Last, the numerical simulation is used for checking the obtained predictions and obtain- ing transient characteristics of system dynamics for large amplitude state variable disturbances and parameter variations. 1.5. THE ADEQUACY OF A MATHEMATICAL MODEL The adequacy of mathematical modeling of high angle-of-attack dynamics is strictly dependent on the adequacy of the aerodynamic model at these regimes. There is nontrivial problem due to the very complicated nature of the separated and vortex flow in unsteady conditions. Aerodynamics becomes extremely nonlinear and time history motion depend- able. Qualitative and bifurcation approaches are also very fruitful when the sources and nature of aerodynamic phenomena are considered. Special techniques were proposed to represent the aerodynamic characteristics taking into account the dynamic effects of the separated flow.~54'55~The bifurcation analysis of the Alpha-Jet aircraft was performed using the dynamical unsteady aerodynamic model at high angles of attack. ~45~It is important to include the rotary balance data into the aerodynamic model in order to obtain better agreement with flight test results, t35'57~The bifurcation methods of analysis of the nonlinear aircraft dynamics at high angles of attack can be efficiently used for validation of the aerodynamic models for these complicated flight regimes. 2. APPLICATION OF BIFURCATION METHODS TO FLIGHT DYNAMICS PROBLEMS 2.1. AUTONOMOUS FORMS OF MOTION EQUATIONS AND STEADY STATE MODES There are some conventional forms of equations of motion which are commonly used in numerical and analytical studies of flight dynamics problems. If an airplane is considered as a rigid body three different frames of references are used to write the equations of motion, i.e. earth axeS--OEXEYEZE (inertial axes, flat approximation), body axes--OXYZ (axes fixed in a rigid body), wind-body axes--OwXw YwZw (origin is fixed in the mass center, OwXw is directed along the velocity vector V, and OwZw lies in the plane of the symmetry of an airplane, t32~ The full set of motion equations can be divided into the following four groups: • Dynamics of translational motion written in wind-body axes:* m(/= Txw - D - mg sin Ow Tz~ - L g = q - (p cos ct - r sin ~) tan [3 + mV cos/3 + V----~osfl c°s 0w cos 0w (1) Trw-C gcos0wsin/~ = p sin ct - r cos ~ + mV + v thw *sin0w= - cos ~tcosflsin 0 + (sinflsin $ + sin ct cosp cos q~)cos 0. cos 0wcos Sw = sin ctsin 0 + cos~tcos 0cos $. cos0wsin Sw = cosctsin flsin 0 + (cos/3sin q~- sin ~tsin flcos$) cos 0.
  • 5. Bifurcation methods in nonlinear flight dynamics • Dynamics of angular motion written in principal body axes: 543 lx/~ = (It - l~)qr + lr(l = (Iz - lx)rp + ,tl (2) Izt: = (Ix - lr)pq + jr • Kinematics of angular motion written for wind-body axes (the same equation are valid for body axes without subscript W): 0w = qw cos q~w-- rw sin ~bw qSw= Pw + (qw sin q~w + rw cos (kw)tan 0w (3) 6w = (qw sin ~bw+ rw cos Sw) sec 0w • And kinematics of translational motion written in earth axes: XE = V cos 0w cos ~'w I;'E= V COS0w sin ~Ow (4) 2~E= -- V sin 0w where altitude of flight H = - ZE. Dotted symbols denote time derivatives. The basic idea of the investigation of flight dynamics is the concept of steady states. The steady state of a flying vehicle is considered in a nonformal mathematical sense for a complete set of equations. Steady state or equilibrium is considered in a more general dynamical sense as the steadiness of all the external forces and moments, i.e. as the aerodynamic steady state. This condition requires that the main motion parameters (V, :¢,/L p, q, r) ~tndgravity projections (i.e. the Euler angles 0, ~b)are all constant in the body axes frame with time. The position in space XE, YE and head angle ~Odo not affect on the aerodynamic sleady state, and therefore can be considered separately, together with appropriated equations. The altitude of flight H (or ZE) defines the air density and can be added to the system of equations only in cases when the influence of density variation on dynamics behavior is significant. If the variation of p with altitude can be neglected, the most general case of equilibrium flight is a vertical helicoidal trajectory. These may be climbing and gliding turns with large radius of curvature, or steady equilibrium spin modes with relatively small radius of trajectory curvature. The spacial case of such trajectories is the rectilinear motion. Therefore to study flight dynamics the following autonomous system of equations can be extracted from the full system (Eqs (1)-(4)) dx dt F(x, ~) (5) where x = (~, ~, p, q, r, V, 0, ~b)'e R8, ~ = (fie,64, 6,, T)' e R4 are the state vector and the control vector. The steady state regimes, which are the vertical helicoidal trajectories, are defined by equilibrium solutions of this system of equations. Some additional physical assumptions, concerning the type of motion, such as the existence of the plane of symmetry, steadiness of some state variables, etc., can be taken into account for obtaining the approximate autonomous subsystems of equation of lower dimension for :investigation the flight dynamics. Such subsystems can be derived for symmetrical flight in vertical plane, for studying the roll-coupling problem, etc. (see the following sections).
  • 6. 544 M. G. Goman et al. 2.1.1. Aerodynamic Model for Stall~Spin Conditions The aerodynamic model intended for spin conditions, i.e. high angles of attack and fast rotation, is based on the experimental data obtained from the different kinds of wind tunnel tests--static, forced-oscillation and rotary balance. There exist a number of methods for designing the 'combined" mathematical model of aerodynamic coefficients, which imple- ment the experimental data in a very similar manner. 157'-~81 The rotation at high angle of attack can significantly influence the flow pattern. As a result, the aerodynamic coefficients become nonlinear functions of the reduced rate of rotation. That's why the aerodynamic coefficients measured in rotary balance tests are considered as the basic or 'nondisturbed' part of the aerodynamic model for high angle-of- attack conditions. The disturbed motion is accompanied by the misalignment between the velocity and the rotation vectors. The projections of the rotation vector onto the wind-body axes can be used as parameters for describing the disturbed conical motion. The roll rate in wind- body axes --Pw = (P cos ~ + r sin ~) cos fl + q sin fl -- defines the rate of conical rotation, which is similar to the angular rate in the rotary balance tests. Two other projec- tions--qw = - (p cos :~ + r sin :~)sin fl + q cos fl, rw = r cos :~- p sin :~--define both the unsteadiness and spirality of motion: i = (qw - qw,)/cos fl,/~ = - rw + r,~. The values of qx%, rv~, arc the wind-body angular rate in steady state spiral motion and their undimen- sional values qw,o::/2V, rw, b/2 V are usually very small. Assuming that the disturbances of the pure conical motion are small, the following representation of the aerodynamic coefficients can be used: c, = c,.. . X + c,,,,, Ty + + c,. + y: _ qwE rwb = G,, (~, 13,Pw, ~) + (C~,~ + C~;:"cos/~) ~p- + (G, - G;) 2 V qw,ri: rv%b (6) + - Ci; 2V cos fl + Ci'; 2V The derivatives of the aerodynamic coefficients, standing together with the reduced rates of rotation and corresponding to the rotary flow, can be measured, by means of the oscillatory coning technique, such as already in use at ONERA/IMFL. 15~)The terms with qw,pand r~o can be omitted. In many cases the data obtained from 'traditional' forced oscillations tests (recall, that they measured in the absence of the model rotation) are used. The following approximate transformation, which is valid for zero sideslipe, may be used: ('i~ + Ci. "" C;~ . . . . . (7) Ci,,, - Ci~ ~- Ci,, cos ~o - C,,.. sin ~o where the subscript f.o. denotes the data obtained in forced oscillation tests. When the nonlinear term in (6) can be approximated by a linear function on the angular rate, the representation (6) becomes equivalent to the aerodynamic model commonly used for low angles of attack. The results of the static wind tunnel tests in this case can also be incorporated into the mathematical model. The form of the aerodynamic model (6) for stall/spin conditions is quite natural. For example, the rotary derivatives Ci,, and C,,, do not significantly affect either the values of kinematic parameters at the equilibrium spin or their mean values during the oscillations with moderate amplitude. These derivatives as well as unsteady derivatives Ci~ and C~i directly affect on the stability margin of the oscillatory spin mode. Thus they determine, for example, the amplitude of the 'agitated' spin motion, when the equilibrium spin is oscillatory unstable.
  • 7. Bifurcation methods in nonlinear flight dynamics 545 The rotary balance data Ci,, (~, fl, pwb/2V, fi) in the aerodynamic model allows to get realistic values of the equilibrium spin parameters. To improve the time histories and amplitudes of the oscillations, one can make some adjustments (if necessary) to rotary and unsteady derivatives C~,w,Ci.... Ci~, C~i. All the wind tunnel data are measured and tabulated in a wide state and control parameter ranges. To facilitate the implementation of the continuation technique the aerodynamic functions are usually smoothed by means of spline or polinomial approxima- tion to ensure continuity and derivability conditions for the resulting nonlinear dynamic system. Aerodynamic asymmetry is one of the important features of aerodynamic coefficients at high angles of attack. Asymmetry appears at zero sideslip, rate of rotation and symmetrical aileron and rudder deflections. It may be larger than maximum aileron and rudder efficiency. Asymmetrical roll moment can significantly influence stall behavior: asymmetri- cal yaw moment arises at the higher angle of attack region and to a great degree defines the spin dynamics. Due to asymmetry, the right spin modes can greatly differ from the left ones both in the values of parameters and the character of stability. Unsteady aerodynamic effects at high angles of attack, resulting from the separated and vortex flow development, can significantly transform the real aerodynamic loads with respect to their conventional representation, which was discussed above. In the frame of the conventional approach the unsteady effects are described using linear terms with unsteady aerodynamic derivatives. There are special regions of incidence, e.g. CL.... region, where the conventional representation is not valid. The special approaches using the differ- ential equations or transfer functions can improve the aerodynamic model and take into account the nonlinear unsteady aerodynamic effects due to separated and vortex flow dynamics}46,55~ 2.1.2. Aircraft with Control Augmentation System Modern airplanes are equipped with automatical control systems able to change its dynamical properties radically. Aircraft with flight control systems become more nonlinear and higher dimensional plant. In addition, a flight control system introduces further nonlinearity and additional dynamic elements. Even in the case where an aircraft may be represented as a linear system, there will be appreciable nonlinearity due to actuator rate and deflection limits. The nonlinear behavior will be displayed during large amplitude motion at large control inputs or gust disturbances. Nonlinear stability and bifurcational analysis methods also can be implemented for aircraft closed-loop system. The equations governing the operating of the control system can be added to the aircraft Eq. (5): _d~i=G x,-- ~,s (8) dt dr' where s = (xe, :%, x..... )' is a vector of the stick and rudder pedal deflections, and the vector-function G is determined by the control laws and control system constraints. Thus, for example, the followingjoint system of Eq. (9) is to be used for determining the equilibrium regimes of the airplane: F(x, ~) = 0 (9) G(x, 0, & s) = 0 The elements of the vectors x and ~ are the unknowns, and vector s defines the set of control parameters. The specific teatures of Eqs. (5) and (8) are the limits on the values of some state variables (i.e. the deflections of the control surfaces 6:(6 .... < 6,. <~6.....: b6~l~<6......; [6,1 ~<6....). In nonlinear system with such state variable limits some additional equilibrium solutions can arise with associated limit points bifurcations leading to aircraft departures.
  • 8. 546 M. G. Gomanet al. 2.2. CRITICAL FLIGHT REGIMES The maneuvering capabilities of the airplane are usually limited by some boundary. Outside this boundary lies the area of critical flight regimes, i.e.the regimes when uncontrol- lable motions (connected with the loss of stability) develop. Pilots" actions may either provoke or prevent the development of the instability. It depends on the task being performed and on the pilot's skill in flying the plane in such regimes3561 A desire to enhance maneuverability inevitably results in entering the regions, where aerodynamic characteristics are nonlinear and dynamic cross-coupling between different forms of motion cannot be ignored. As a result, the dynamical problems become highly nonlinear. All the critical regimes can be divided into two groups, according to the reasons of dangerous behavior of the vehicle. The regimes of the first group arise when the stability margin is broken and the unstable modes of motion begin to develop. These regimes are very different. There may be mild or abrupt loss of stability, and the motion may be controllable or uncontrollable. Such situations take place during stall or maneuvers with fast roll rotation. The second group comprises the stable steady-state flight regimes with supercritical values of parameters (especially angle of attack and rate of rotation). The roll-inertia rotation or autorotation rolling and spin regimes belong to this group; the unusual response to the control inputs is their characteristic feature. Departure resistence to the entering in these critical regimes and the methods of recovery from them are the most important questions, when the problem of flight safety is considered (see Fig. 1). A number of different forms of the loss of stability in longitudinal and lateral/directional motion is usually connected with aircraft stall concept.1561One may distinguish among them: • a sudden rise of the angle of attack ('pitch up') taking place at moderate and high angles of attack due to pitching moment nonlinearity; • stable self-oscillating pitching motion at high angles of attack due to the development of the separated flow ('bucking'); • the full turn rotation in pitch ('tumbliny'); • divergent increase in the bank angle ('wing drop' or 'roll off') due to asymmetrical aerodynamic rolling moment at high angle of attack or propelling aerodynamic moments; • divergent increase in sideslip angle ('nose slice" or 'yaw off') due to aperiodic instability in yaw or asymmetric aerodynamic moments; • oscillating motion in roll and yaw at high angles of attack ('winy rock'); • loss of stability due to the pilot's actions, for example, when the attitude stabilization or target tracking is performed. [~ S-l rotation rate gflying l ," de~ Pstall 20 30 40 50 60 70 8010 Angle of attack, deg. Fig. 1. Criticalflightregimesregions.
  • 9. Bifurcationmethodsin nonlinearflightdynamics 547 There are also terms used in the flight dynamics for describing the aircraft behavior after stall, i.e. post-stall gyration, spin and deep stall. The post-stall gyration is a transient rotational motion of aircraft to the developed spin mode. The developed spin can possess very different features. It can be steady with constant parameters or 'agitated',flat or steep, erected or inverted. 'Agitated' spin mode can be with regular and 'irregular' oscillations. Deep stall is art aircraft equilibrium flight at high critical angle of attack without rotation. The possibiltty of the loss of stability and controllability at high roll rate (roll-coupled problem) is also well-known. This phenomenon is especially dangerous for supersonic aircraft with elongated ellipsoid of inertia and high level of lateral aerodynamic stability, which bring about cross-coupling of longitudinal and lateral motions. Roll-inertia rotation or autorotation rolling of the airplane (the trajectory being approxi- mately horizontal and angles of incidence below stalling) is similar to the developed spin modes. The rotation can occur despite neutral or anti-rotation aileron and rudder deflections. In both cases there are autorotational regimes, the difference is only in the nature of the aerodynamic moment that supports the rotation. All types of motion mentioned above can be connected with qualitative features of motion equations, i.e. the different steady-state aircraft motions and bifurcations, changing their stability conditions. 2.3. DYNAMICS OF SYMMETRICAL FLIGHT IN A VERTICAL PLANE To demonstrate the nonlinear dynamics behavior of symmetrical flight using the bifurca- tional and global stability analysis methods one can consider two simplified examples arising basically from trajectory or angular pitch motion. Taking into account that in symmetrical flight the state variables are equal to zero: q~w = 0, ~ = 0, fl = 0, Pw = rw = 0, and 0w = 7• [- n,n], where ~' is a path angle, the following equations of motion can be extracted from the complete system (Eqs (1)-(4)): Tcos • CopS ) m~'=g mg ~ V2-sin)' ";' g (.T sin ~ CLpS ) = -- + V2 _ cos 7 V mg 4=q-,} = Cm(~,q, fie)p V2S? 2Ir (10) )(E = VCOS~ //= Vsin7 The first two equations of this system are responsible for the so-called phugoid trajectory motion, the second two equations are responsible for short-period rotational motion. At relatively large velocities of flight these two subsystems approximately can be considered separately. 2.4. NONLINEAR PHUGOID MOTION This problem was first considered by Zhukovsky and Lanchester and later became one of the classical examples of second-order nonlinear systems, illustrating the qualitative methods of analysis.
  • 10. 548 M.G. Goman et al. Assuming that the angle of attack is a constant, and therefore the lift and drag coefficients are also constants, the following nondimensional autonomous second order nonlinear system can be used for investigation of phugoid motion with large amplitudes: --cl~) = b - ay 2 - sin 7 dr d), cos 7 dr - y Y (11) T CD. where b m9 is the thrust-to-weight ratio considered as a parameter (:t <~1), a ~. ts the 2mg drag-to-lift ratio, r = VOqt is the nondimensional time and Vo = ~]pSCLis the velocity of horizontal flight. The equilibrium solutions of this system at different values of parameters a and b are ~:orresponded to straightline trajectories with climb (7 > 0) or glide ('/< 0) path angle. There is some critical boundary in parameter space exceeding which makes the equilibrium oscillatory unstable. At some values of parameters a, b another kind of steady solution appears. These are the stable and unstable closed orbits, enclosing the cylindrical state space or equilibrium points. The stable closed orbits, enclosing the cylinder correspond to the curvilinear trajectories with recurring loops, the so-called 'dead loops'. The unstable closed orbits, enclosing the cylinder or the equilibrium points, separate the regions of attraction of the stable equilibrium points and the stable closed orbits, enclosing the cylinder. In Fig. 2 the bifurcation diagram is presented when the parameter b is varied for a = 0.2. The path angle of equilibrium straightlinear flight ~ becomes positive, when b >/a. When parameter b exceeds the critical value bcr, ,~ 0.4 stable and unstable closed orbits, enclosing the cylinder, appear. After that the stable cycle moves up and the unstable one moves down along velocity axes y. There is some bifurcation moment (bcr2~ 0.42), when the unstable cycle involves the critical points y = 0, 7= + ~/2. These critical points in state space correspond to the cusp points on the flight trajectory, where the path angle varies from 7 = ~/2 to ?, = - n/2 instantaneously. In real flight it is impossible due to inertiality of angular pitch motion. In this case, corresponding to the well-known aircraft maneuvers y n rd2 -~,e2 stable closed orbits, surroundingcylinder / , i " closed orbits ~ J stable e.quilibria~ unstable|I~I I .------ ............ equilibria 4.2 0 0.2 0.4 0.6 0.8 Wmg Fig. 2. Equilibrium and periodical solutions as a function of parameter b with a = 0.2.
  • 11. Bifurcation methods in nonlinear flightdynamics 549 with large loss of speed (the so-called 'tail-slide' maneuvers), the full system (10) has to considered. During the tail-slide maneuvers the extremely large variations of angle of attack can occur up to ~ ,~ 180° and therefore the assumption about ~ = const taken above is not valid in these points. When b >/be,.2the unstable closed orbit enters in the internal region of the phase portrait rt between 7 = + ~ and encloses the equilibrium points. With further increase of parameter b the size of the unstable cycle becomes smaller and at bifurcation point b = bCr~~ 0.55, it disappears. This point is the undercritical Andronov-Hopf bifurcation, where the equilib- rium point becomes also oscillatory unstable. The unstable closed orbits are important because they separate the regions of attraction of the stable equilibria and the stable closed orbits which enclose the cylindrical phase space. In Fig. 3A-D four pictures of phase portrait for the system (11) are shown for the following values of parameter b = 0.2, 0.41, 0.5, 0.7 and a = 0.2. In Fig. 4A, B two kinds of flight trajectories are shown for b = 0.2 and 0.7, respectively. 2.5. DEEP STALL MOTION For a number of aircraft configurations, e.g. with T-tail or taillness, there is the essentially nonlinear dependency in pitch moment coefficient generating the stable trim at high angle of attack even under the full nose-down elevator deflection. (561 These regimes are called deep stall, and are dangerous for flights with high incidence. The lack of pitch-down control moment may be critical for pitch stall and recovery from it. a=0.2, b=0.2 a=0.2, b=0.41 2.5 2.5 stableclo~ed " . orbit,sumrounding u~table closed s cylinder: A orbh~mr,mding 1 " 1 0.5 0.5 0 0 -~ --~/2 0 ~/2 ~ -~ -~/2 0 ~'2 (A) (B) a=0.2, b=0.5 a=0.2, b=0.7 3 ~ 3 stabledosedorbit ~ stable sunroundingcylinder~. equilibrium 2 ~ 2 1 1 0--~ --~/'2 0 ~ n -~ -~r2 0 ~2 ~ (c) Y (D) Y Fig. 3. Cylindricalphase portrait for y and ;.' state variables, a = 0.2 and A) h = 0.2, B) b = 0.41, C) b = 0.5, D) b = 0.7.
  • 12. 550 M.G. Gomanet al. 560 420 280 140 a = 0.2o b = 0.0 560 420 280 140 i I I i 240 480 720 960 (A) L,m (B) a=0.2, b=0.7 I I I I 240 480 720 960 L,m Fig.4. Flighttrajectorieswith'dead' loops,A) b = 0, B) b = 0.7. The understanding of nonlinear dynamics and the prediction of the region of attraction generated by the critical deep stall regime are very important for stall prevention and also for design of the recovery control from deep stall. To form the approximate equations one can consider that the velocity of flight is not varied and the path is rectilinear: ~=q ( t~1: Cm(~)@ CmqV -[- Cm6e~e T SC (12) The qualitative analysis of nonlinear dynamics in this case can easily be done in the phase plane of the state variables (~ and q) (see Fig. 5). There are three different equilibrium points :~,, ~2, ~3, two of which (al, ~3) are stable loci and one (~2) is the saddle point. This unstable saddle-type equilibrium point generates very informative phase trajectories W~, which are going in it along the eigenvector, corresponding to the stable eigenvalue. Two other trajectories W uare going out from saddle point along the eigenvector, corresponding to the unstable eigenvector. These trajectories are attracted by stable points ~1, ~3. All these singular trajectories can be computed by means of integrating the system (12). Initial conditions for reconstruction of these singular trajectories are taken in a small vicinity of the saddle point ~2 with displacements along the stable and unstable eigenvectors. The trajectories W u are integrated in the positive time direction, and W s are integrated in the backward time direction. Note, that the region of attraction of the deep stall regime ~3 (dashed area in Fig. 5) depends on the altitude and velocity of flight, and the magnitude of the pitch damping derivative Cm. 2.6. THE GENERAL CASE OF THE LONGITUDINAL MOTION As was mentioned above the interaction between the phugoid and angular modes is more significant for flight with low velocities. In this case the large variation of angle of attack can arise due to trajectory distortion. For example, such trajectory distortion accompanies the tail-slide maneuvers. The region of attraction of critical deep stall regime in the general case will be more complicated with respect to the region, considered above in simplified manner. To take into account the interaction between phugoid and angular modes the first four equation of (10), which are the autonomous nonlinear system with state vector X = (V, 0, ~, q)r, have to be considered with entire ranges of .~ and 0 variations--I- ~t,n]. To represent the stability region in the fourth-order state space it is possible only by means of drawing its two- dimensional cross-sections considering the disturbances only in two selected state variables.
  • 13. Bifurcation methods in nonlinear flight dynamics 551 0.5 region of attraction of a deep-stall W s regime o~3 Cm Xt~ 3o 5o ctc~ 0.25 0t t Ot2 IX3 -0.25 I I I l0 30 50 ct (deg) Fig. 5. Phase portrait for the deep stall nonlinear dynamics. For example, in Fig. 6 two different cross-sections of the region of attraction of deep stall regime, i.e. stable point ~3, are shown. The disturbances in the plane of pitch angle 0 and velocity V are considered. Two other state variables at the initial moment are the same for all points of cross-section. In the first case (A) the angle of attack is trimmed in a lower stable point ~t = ~tt ~ith zero pitch rate q = 0, and in the second case (B) the angle of attack is trimmed in the critical position ~ = ct3, also with zero pitch rate q = 0. In every point of the considered cross sections the initial path angle can be calculated using the following formula ~ = 0 - ~t.The dashed areas on the cross sections define the initial points, starting from when the aircraft enters in the deep stall regime, ~t3. In the second case (B) the probability of entering into the deep stall is much greater, especially in flights with large velocities, than in the first case (A). The number of cross sections of multidimensional stability region can offer global information about the aircraft dynamics. 2.7. ROLL-COUPLING PROBLEM The common form of steady state is the vertical helix trajectories which result from the balances both in the moment and the force equations. The process of balance adjustment in the moment equation is usually much faster than the process of balance adjustment in the force equations; this is especially true in flights with large velocities. The specific time of the trajectory adjustment (to the vertical helix) is adverse proportional to the magnitude of the gravitational terms in the force equations 9/V. The differential equations for trajectory angles Ow and ~bw Ow = ~g (ancos qgw + a~.sin q~w - cos 0w) g (a. sin ~bw - a, cos ~bw)tan Ow ~. flq~w---Pw +~ (13)
  • 14. 552 M.G. Goman et al. 2O 40 n V (A) I ! ! I 60 80 100 120 V, m/s V 0 ////, K (B) Fig. 6. Cross-sections of asymptotic stability region for deep stall regime z~3. L - Tzw C- Trw where a. - - - , as - - - are normal and side load factors; can be used for the mg mg estimation of the averaged variation of the trajectory angle during fast rotation. The averaging of these equations during the period of one turn T = 2n/~ yields: g 0w = - ~ cos 0w (14) This equation can be taken in quadrature. The time when the average trajectory angle will be changed, e.g. from 0 to - 45 ° is T4s ~ 0.88 V/9. The influence of the gravitational terms in the equations of motion will be negligible, if the number of turns during this time will be large, i.e. n = T45/T ~ 0.14, ~V/g ~> 1. The role coupling problem at fast roll rotation is usually considered without taking into account the gravitational terms. No less important when considering this problem is that velocity of flight is fixed and the first equation in (1) is omitted. Therefore only the last two equations from (1) without gravitational terms and three moment equations from (2) can be left for consideration: 0~= q - (p cos ~ - r sin ~) tan 13+ z /~ = p sin ~- r cos ~ + y l) = - ixqr + l (15)
  • 15. Bifurcationmethodsin nonlinearflightdynamics 553 gl = i2rp + m = -- i3pq + n I, -- Iy, i2 I. - Ix, i3 Iy - I~where il . . . . ~ = - - are nondimensional inertia coefficients, I~ Ir I~ l = ~/lx, m = ,4[/I r, n = .A~/lz, y= - C/mV, z = - L/mV. The reduced normal and side forces z,y, pitch, roll and yaw moments m,! and n in Eq. (15) can be represented as a function of state variables and control parameters both in linear or nonlinear forms: z =z(~) + zq(~)q +z~,(a)fe+ -.. Y =Yo + Yo(~)fl+ y,(~)r + y~,(~)O, + ... m = m(~) + mq(~t)q + m~,(ot)~ + Am(~, fl.... ) l = la(~)fl + lp(~)p + l,(~)r (16) + lo,(cO~,+ lo,(~)6. + Al(a, fl.... ) n=no(a)fl+np(a)p+n,(a)r + n6,(ot)6, + n6o(~t)fi, + An(a, fl, ... ) Equation (15) completed by (16) form the closed nonlinear autonomous system of the fifth order with the state vector x = (a, fl, p, q, r) r. The nonlinearities in these equations can be divided into three groups: kinematic, inertia moments and aerodynamics terms. All these terms can lead to the cross-coupling between the longitudinal and lateral modes of motion. An attempt to take into account the influence of the gravitational terms in the fifth order equation (15) was made in Ref. 39. The main feature of the problem considered is that even in the case of linear representa- tion of aerodynamic coefficients the existence of multiple stable steady-state solutions, e.g. equilibrium and periodic, is possible. The bifurcational analysis of all the possible steady- state solutions and their local and global stability analysis can show the genesis of stability loss and explain in many cases the very strange aircraft behavior. The validity of the system (15), (16) is confined in time. Therefore, in the cases of weakness or lack of stability of the steady states the conclusions resulting from the consideration of the asymptotic stability in Lyapunov sense, when t ~ ~, can be wrong. A similar problem can arise for short-term control inputs. The prediction of the bifurcation analysis will be more consistent when the considered steady states have the sufficient margin of asymptotic stability. In any case the numerical simulation of aircraft motion using the complete set of equations has to be used for final verification of the bifurcational analysis results. To demonstrate the possibilities of bifurcational and global stability analysis when roll-coupling problem is studied, two different examples will be considered. The first one is taken from Ref. 32 (pp. 447-451) and corresponds to a small maneuverable single-enginejet airplane at flight with zero altitude H = 0 and subsonic velocity V = 500 fps. The second one corresponds to hypothetical swept-wing fighter at flight with altitude H = 20000 m and supersonic velocity V = 1800 fps. The types of equilibrium solutions in these cases are different due to various contributions of damping terms in motion equations. 2.8. DYNAMICS AT SUBSONIC FLIGHT REGIME In the first case (H = 0, V = 500 fps) all the equilibrium solutions form the single continuous surface. The equilibrium surface in this case possesses the canonical form of a singularity in the mapping of the equilibrium surface on the plane of control parameters 6e, 6,, which are called 'cusp catastrophe' and 'butterfly catastrophe'. 2 Although the
  • 16. 554 autorotationrolling regimes M. G. Goman et al. rollrate equilibriumsurface bifiarcational diagram ~a Fig. 7. Surfaceof equilibriumroll rates. Bifurcationaldiagramin the plane16,.6~). catastrophe theory was developed for the gradient type of dynamical systems, the for- mulated singularities are, nonetheless, also apparent in the autonomous dynamical systems. In Fig. 7 the common view of the calculated equilibrium surface is shown along with the bifurcational diagram below it in the plane of elevator and aileron deflections. The number of equilibrium solutions, shown in the figure, varies with control inputs. There is a critical region with auto-rotational rolling regimes at pitch-down elevator and near-neutral aileron deflections ('butterfly catastrophe'). At pitch-up-elevator deflections there are two 'cusp catastrophes', which lead to the hysteresis type behavior under the roll control. The quantified dependencies of equilibrium roll rate p on the aileron for a number of elevator deflections, corresponding to different initial values of normal factor (az = - 4 - 3), are presented in Fig. 8b. Another set of such dependencies on elevator deflections for 6, - var is presented in Fig. 8a (solid lines--stable solutions, dashed lines--divergent solutions, dash- dotted lines--oscillatory unstable solution). The approximate Phillips' critical roll rates P~ = ~/-~-~, P~ = "v'~, obtained without taking into account the damping terms, nevertheless define the roll-rate regions where the roll-coupling effect is more significant. These critical roll-rate values are shown in Fig. 8b by horizontal dashed lines. The oblique dashed line defines the controllability in roll mode without taking into account the dihedral roll moment, arising due to the inertia coupling of longitudinal and lateral motions. 2.9. DYNAMICS AT SUPERSONIC FLIGHT REGIME In the second case (H = 20000 m, V = 1800 fps) the critical roll rates really exist and are very close to Phillips' approximate values. As a result the equilibrium solutions are divided
  • 17. Bifurcation methods in nonlinear flight dynamics 555 4 o : ~2"Io 0 -'-2 0 -4 -6 4 o =~--~--S~--S >l 0_ '- -4 I ' ' --6 -10 -5 0 5 10 (A) Elevator deflection, deg (B) -20 0 20 Aileron deflection, deg Fig. 8. Equilibrium roll rates for different elevator and aileron deflection (M < 1). A) t5e - var, •° = 0, B) 6a - var, a: = - 0.4 + 3.0. See text for explanation. 1o 8 6 4 2 -4 -6 -8 -10 -30 I I I I I .... ~-.,-. :- ~.: .-.....~.-- - - ; ~ . . . ~ . - ~ . . p ~ . . _ ...... ~---=~-=~ .... : .... i .... ~ .... =_.~ ;.... :.z. ~.!:-..~- .... I I .... I I I -20 - I 0 0 10 20 30 Aileron deflection, deg Fig. 9. Equilibrium roll rates as a function of aileron deflection for initial trimmed flight with az = - 2 + 5 (M > 1). See text for explanation. by this critical line into different unconnected families, which are shown in Fig. (9). The dependencies of equilibrium roll rate on aileron deflection are presented for different elevator deflections, corresponding to different values of normal factor parameter az = -2- 5. As in the previous case the type of line defines the code of equilibrium stability (solid lines--stable solutions, dashed lines~divergent solutions, dash-dotted lines--oscillatory unstable solutions). There are two different subcritical types of curves starting from the zero point. The first ones for positive az > 0 possess the 'loss of controllability' -- the increase of control moment does not proportionately increase the roll rate. This effect is due to the arising of opposite roll moments from sideslip, resulting from roll coupling. The second ones for zero and negative a, ~<0 possess the departure point, where stable and divergent equilibria disappear.
  • 18. 556 M.G. Goman et al. There is also another family of equilibrium curves both between the critical rates p,, pa and in the outer regions. Some of them, corresponding to a: = 1, 0, - 1, - 2, generate the stable autorotational roiling regimes at zero aileron deflection. The autorotational rolling solutions for a: = 0, - 1, -2 exist at all aileron deflections, but at 'pro-roll' aileron deflections (p > 0, 6a > 0) the autorotational equilibrium solutions become oscillatory unstable (dash-dotted lines) and after the Hopf bifurcation point on each curve the family of stable closed orbits arises. A similar oscillatory instability of equilibrium solutions appear at subcritical equilibrium curves at large aileron deflections. The Hopf bifurcation points in this case also give birth to the families of stable closed orbits. To illustrate this in Fig. 10A the different branches of the equilibrium curves and the families of closed orbits placed on the subcritical and supercriti- cal equilibrium branches for case with 6~ = - 5.4: (a: = 1.0) are presented. The envelope curves defining the maximum and minimum values of roll rate in oscilla- tory solutions at 6a > 0 pass through the Hopf bifurcation points H1 and Hz. The oscillatory solutions originating in the point Hz are stable at all values of aileron deflec- tions. But the oscillatory solutions originating in the point H1 on subcritical equilibrium branch become unstable at 6= ~ 18.9. After this the cascade of flip or period-doubling bifurcation leads to the appearance of chaotic motion. The bifurcation tree for the closed orbits originating in point H~ was obtained by numerical Poincar6 mapping with the cross section plane fl = 0 for different aileron deflections. It clearly shows the chaotic motion appearance (see Fig. 10B). The amplitudes of the state variables in the stable oscillation regimes can be seen in bottom of Fig. 10C, D, where the (~, [/)- and (p, fl)-projections of the closed orbits for different aileron deflections are presented. The roll-maneuver dynamics is highly dependent on the magnitude of control input. The examples of transient dynamics, obtained for different step-like aileron inputs by means of 10 0 ® 5In d 0 ~ -5 -10 ~,~ closed orbits -20 0 20 (A) Aileron deflection, deg 10 2 ° -10 18 20 22 24 (B) Aileron deflection, deg 80 60"o o 40¢g ¢o ,e,- o 20 o) c 0 ,< -20 10 (.1 • 5 '1o $ o -10 -40 -20 0 20 -40 -20 0 20 (C) Sideslip angle, deg (D) Sideslip angle, deg Fig. 10. Periodical and chaotic behavior at fast-roll maneuver. See text for explanation.
  • 19. Bifurcation methods in nonlinear flight dynamics • 60 -o 40 •g 20 0 0 < -20 0 20 40 60 •1o 0 "o 0 -so ~ -so 0 20 40 60 I ' ' 1 ~o ~o ~-5 ~-5 557 0 20 40 60 ~.2o • ~ 20 " 0 d 0 a o -20 ' ' ' -20 0 20 40 60 (A) Time, $ec "o 40 <" 20 0 0 < -20 . . . . . . . . . . . . . . . . . . . . 0 20 40 60 i , , , 0 20 40 60 0 20 40 60 0 20 40 60 (B) Time, sec Fig. 11. Numerical simulation of roll-maneuver dynamics. See text for explanation. numerical simulation of(15), (16), are presented in Fig. 11A, B. At (5 = 8° (see Fig. 11A) the roll rate p varies aperiodicaily to the constant value p ~ - 2.0, corresponding to the stable equilibrium (see Fig. 10A), the angle of attack :~ and sideslip/~ in this case are practically constants. After applying the control input with (5 = 18° the periodic oscillations in all the state variables are established. Their amplitudes also correspond to that calculated before closed orbits (see'.Fig. 10A). Applying the larger control input (5 = 23° (see Fig. 11B) leads to the jump at t > 10 s from one side rotation to another, with large amplitude periodic oscillations in all state variables. The jump from the chaotic motion region, which became unstable, is possible both to the stable oscillations at high angles of attack and positive roll rate and to the stable equilibrium point with very fast negative roll rotation p ~ - 8 1/s. The 91obal stability analysis or investigation of asymptotic stability regions is very important in the case when there are multiple locally-stable equilibrium points and closed orbits. The reconstruction of these stability regions gives the global information about phase space of considered systems and determines the critical disturbances of the state variables leading to the loss of stability. As already noted, the more convenient way to reconstruct the multidimensional stability region is the calculation of its twodimensional cross-sections. The disturbances only in two selected state variables under fixed other ones are considered. It is natural, for example, that a cross section would pass through the locally stable equilibrium point under consideration. At zero aileron and rudder deflections (sa = (5, = 0 and (5~= - 5.4° (az = 1.0) for super- sonic flight regime (see Fig. 12D) there are five equilibrium points, three of them--'A', 'B', 'C'--are locally stable and two--'D', 'E'--are locally unstable. Three different cross-sections of the asymptolic stability regions of 'A', 'B', 'C' equilibrium points are presented in Fig. 12A, B, C. All the cross sections pass through the equilibrium point 'A'. The first cross-section is placed in the plane of angle of attack ~ and sideslip fl (Fig. 12A), the second one is placed in Ihe plane of roll rate p and sideslip/~ (Fig. 12B), and the third one is placed in the plane of yaw rate r and sideslip/~ (Fig. 12C). The region of attraction for equilibrium 'A' is marked wi!h lighter points. One can see, that the more critical level of perturbation is in yaw rate - r¢r ~ + 0.4 l/s. The boundary of stability region far from equilibrium point 'A' has thin structure with multiple folds. The size of as)mptotic stability region of equilibrium 'A' becomes significantly less when the aircraft is tr',mmed at zero or negative angle of attack. To illustrate this feature the similar cross-sections of the stability regions for (5~=- 0.5° (a= = 0) and (se = 4.4° (az = - 1.0) are presented in Fig. 13A, B, C and in Fig. 14A, B, C.
  • 20. 558 M.G. Goman et al. 40 0 -lo 30 u 20 m 10 o < -10 u 4 "~ 2 0 ~-4 -20 (A) 1O O 0.5 ~ -0.5 -1 0 Sideslip, deg -6 20 -20 0 20 40 (B) Sideslip, deg 10 ¢,3 o-5 -10I -20 0 20 40 -20 0 20 (C) Sideslip, deg (D) Aileron deflection, deg Fig. 12. Asymptotic stability region for flight regime with a. = 1.0 (point "A'), M > I. See text for explanation. 2.10. NONLINEAR STALL MOTION Aircraft high-angle-of-attack excursions are connected with a degradation of lateral aerodynamic characteristics. The aerodynamic moments become markedly nonlinear func- tions on angle of attack, sideslip and rotation rate about the velocity vector. They may significantly change their values resulting in the stability and controllability loss of the lateral motion. 2.11. APPROXIMATE LINEAR CRITERIA OF STALL The character of the disturbed lateral motion and its stability under small perturbations may be evaluated using linearized equations of motion, assuming that the equilibrium angle of attack is considered as a parameter. To do so, it is necessary to get the dependencies of the lateral aerodynamic derivatives on angle of attack using data from static, forced oscillation, rotary balance wind tunnel tests. Considering the simplified equations for the fast rotational modes of the lateral motion without taking into account spiral mode and assuming that V = const, ~ = const, C = 0 give the third order characteristic equation: with coefficients ~.3 + A2}2 + At;'. + At) = 0 A2~ A1 ~_ A 0 = -- Clp ' ,, + Cn = tx ", .. i x -i, c,,cos~-%sin~Iz =-_a~l~ Iz --" (.:noClp~ -- Cl ('np ~ = -- ¢'7r~ ~ txtz ixiz (17)
  • 21. Bifurcation methods in nonlinear flight dynamics 559 4O 0 •~ 30 o 20 m t:: == 10 "6 --~ 0 t'- < -10 O 0.5"10 2 g 0 2 • -0.5 >- -1 -20 0 20 (A) Sideslip, dog 4 O 2 =-o 2_ 2 0 n'_ 4 -6 (B) 10 O s 'lO =- 0 2 o -5 fie -10 -20 0 20 40 -20 0 20 40 Sideslip, deg .....~ ' 7 ' _ ..... .. : , ~ i -20 0 20 (C) Sideslip, dog (D) Aileron deflection, deg Fig. 13. Asymptotic stability region for flight regime with a= = 0.0 (point 'A'), M > 1. See text for explanation. 40 0 '1o 3O o 20¢1 == 10 0 • 0 C -10 1O O 0.5"ID d 0 ~ -0.5 m -1 -20 0 20 (A) Sideslip, deg -20 0 20 (C) Sideslip, deg ~ 4 "~ 2 ¢- 0 ~-2 0 ~-4 -6 10 0 0 ~ -S -10 40 ,,~~h'):]t; -20 0 20 (B) Sideslip, deg 40 5 ...... . . ~ B . ..................:...--...... -20 0 20 (D) Aileron deflection, dog Fig. 14. Asymptotic stability region for flight regime with a: = - 1.0 (point 'A'), M > 1. See text for explanation.
  • 22. 560 M.G. Goman et al. where ix, iz, are the reduced inertia moments, # is the relative density of aircraft, subscript f.o. denotes forced oscillations, c~,, c,,~ are derivatives, obtained in the rotary balanced tests. Since p >> 1, the terms with products of rotational derivatives in the expression for A1 were neglected. The algebraic stability conditions result in the simple stability criteria tr/~< 0 ap < 0 a,.~< 0 (18) R=a,,,+tr~tra>0ortra<--- trti Breaking of some of these criteria indicates the loss of lateral stability or stall occurring (see Fig. 15A, B, C). There are various types of the loss in stability. If only one condition a~, is broken then a real eigenvalue becomes positive (Fig. 15A).These critical points correspond to the pitchfork bifurcations. For example, in the supercritical case two stable regimes with right and left rotation appear, while the initial symmetrical regime becomes aperiodically unstable. If a condition R (Routh criterion) is broken then a complex pair of eigenvalues becomes unstable (Fig. 15B--Andronov-Hopf bifurcation). This leads to oscillatory unstable motion provided the frequency of lateral oscillations t-)o ~ ,vf- trp does not become very small. This is well known case of "wing rock' motion. The change of sign of ~r~(note the equivalence of a~ with - c,~o,°)first can lead to the oscillatory instability accompanied with decrease in ~Ooand breakdown of complex pair into two real roots moving along the real axis. After that a branching point or pitchfork bifurcation (tr,, = 0) may also occur (Fig. 15C). Taking into account the typical depend- encies of the aerodynamic derivatives %, c,,,, c~,~at high incidence, such a behavior may take place in the narrow range of the angle of attack A~ - 1: + T. One may say that this type of instability is similar to the case when ~r,o> 0. The criteria tro < 0 and a~ < 0 govern the dynamic stability in yaw. The breaking of the third criterion try,< 0 indicates the loss of stability in roll or the appearance of the aerodynamic'autorotation. The departure in yaw is typical of aircraft with swept-like wings with low aspect ratio. In the great extent the vortex breakdown phenomenon is responsible for the nonlinear aerodynamics at high angles of attack in this case. Instability in roll occurs for aircraft with unswept high aspect ratio wings. In the last case the reason for stall is aerodynamic autorotation of the wing due to asymmetrical flow separation at high angles of attack (when CL ~ CL..~). Taking into account the spiral mode and all the omitted aerodynamic effects, the change in the quantitative results of the linear stability analysis is usually insignificant. 2.11.1. Controllability Criteria The conventional lateral controls, i.e. ailerons and rudder, at high angles of attack can hardly be classified as being pure lateral or directional, since aerodynamic moments c~,,and ----O "r Ii Re / ImP. " R =0 ---41--o Rek i I _ _ l I L,n~ i i R=O (A) (B) (C) Fig. 15. Different kinds of stall through the lateral/directional modes. See text for explanation.
  • 23. Bifurcation methods in nonlinear flight dynamics 561 c.~, (i = a, r) are of the same order. That is why the modes of the controlled lateral motion are closely coupled. The steady-state value of the rate of rotation about the velocity vector Pw under small deflections of aileron or rudder 6 can be evaluated from the balance of yaw and roll aerodynamic moments. This results in the simplified relationship Pw -~ - 6 (19) pwb where/Sw = ~7' tr~ = (c,,cl~ - cz,c,,). If the perturbed roll motion is stable (ao, < 0), then there will be a 'direct' reaction of the airplane in the response to the control input provided the condition tr6 < 0 is satisfied. Nondimensional parameter tr6 is similar to the LCDP (Lateral Control Departure Para- meter): a6 = LCDP * cz. ~561 This criterion is also coupled with the stall prediction problem, since the reversed response to the aileron deflection, a~o,changes its sign and may bring about instability in the closed-loop dynamics. 2.12. STALL DYNAMICS ANALYSIS The linear analysis determines only the critical values of the angle of attack (or elevator deflection), when the loss of stability occurs. But the linear theory is unable to predict the motion after instability occurs. To investigate the motion development due to instability the analysis of nonlinear equations of motion is needed. The analysis of the equilibrium and periodic solutions along with their local stability characteristics provides very important information for nonlinear dynamics prediction. To illustrate the nonlinear phenomena which are encountered at high angles of attack, the aerodynamic model for hypothetical fighter-type aircraft (normal scheme with low- aspect ratio wing and extended strakes) was used. The main aerodynamic characteristics of lateral/directional motion--c~: c,: c~,w,c.~- defined with spline functions of angle of attack are shown in Fig. 16A, B. The roll and yaw moment coefficients become nonlinear functions of sideslip angle and reduced rate of rotation at the high angle of attack region, where the aerodynamic derivatives change their signs. The main goal of the aerodynamic design of real aircraft is to provide the stability of aircraft motion at high angle of attack. But to eliminate bifurcation phenomena completely is not so easy. The considered hypothetical aerodynamic model gives birth to various nonlinear phenomena, which are also typical for many real aircraft. The dependencies ao,, R = a,~ + a#tr# and a~o,presented in Fig. 16C, D, show that Hopf bifurcations Ht and H2, two pitchfork bifurcations P~ and P2 and the reverse response C1 in aileron control can be expected. All calculations in this section were performed for hypothetical aircraft using the fifth order equations of motion (15). The root loci of the linearized equations for the symmetrical branch of equilibria (the aerodynamic asymmetry is absent and 6o = 0, 6, = 0) qualitatively confirm the bifurcations predicted by the approximate linear criteria (see Fig. 16C, D). Note, that the loss of stability can be with breaking of symmetry and without it. For example, the Hopf bifurcation, leading to the wing-rock regime from the former stable symmetrical equilibrium, remains valid for the asymmetry of motion. The pitchfork bifurca- tions both for equilibrium and oscillatory regime break the symmetry of flight in the vertical plane. Below some examples of such phenomena are presented. Autorotational regimes appear in the pitchfork points P~ and P2 on the symmetrical equilibrium branch, corresponding to the aircraft trim without rotation. The greater the level of instability between the points P~ and P2, the more prolonged will be the autorota- tional branches in the positive elevator deflection range. The size and location of the
  • 24. 562 M.G. Goman et al. r- .d .'3 -5 X 10-~ ..... :~ : / / 0 (A) 20 40 60 0.5 o "u -0.5 •-L~ . . . . . . . . . . . . . . . . i . . . . . . . . 0 (B) 20 40 60 = X 10-~ '7,, _ -." HI ,PI p2i, "1"12 I 20 40 60 Angle of Attack (deg) X lO-~' -5 0 0 20 40 (C) (D) Angle of Attack (deg) 60 Fig. 16. Lateral/directional aerodynamic characteristics. Approximate stall criteria. See text for explanation. 2 ', "~ autorotatlonal 1 equilibria -2 -40 -20 0 20 o-1 rr -2 '. •f • I ~- autorotat~onal1" ~- ' ,~ ~-f H I I %. "- . closed orbits i- / i i ~, i -40 -20 0 20 e~a~ a~mt~ a~ ~vator ae~oa a~ Fig. 17. Autorotational equilibrium and periodic solutions for different elevator deflections• autorotational branches depend also on the character of the nonlinear aerodynamic terms Ac~(~,r,/~w) and Ac,(~, r,/Sw). The equilibrium solutions were computed using (15} and are shown in Fig. 17. They reveal, along with the symmetrical and autorotational branches generated in the pitchfork points Pt and P2, some additional autorotational branches, which correspond to the inertia-coupling regimes (see previous section). When the autorotational asymmetrical equilibrium branches are stable, the jump-like departure (undercritical bifurcation) or smooth entering (supercritical bifurcation) to these regimes will occur at pitchfork points Pt and P2. In the presence of symmetry, there is an equal probability that aircraft will depart to the left or the right rotation. The side on which the aircraft departures, depends on the existence of small perturbations when the control input passes its critical value. In the case under consideration the autorotational branches are oscillatory unstable. The high angle of attack aircraft behavior is usually ruled by the sequence of bifurcations.
  • 25. Bifurcation mcthods in nonlinear flight dynamics 563 Winy Rock motion usually arises in the Hopf bifurcation point, which in our case is preceded to the first pitchfork point. The Hopf bifurcation points Ht (at ~ ~ 2C) and H2 (at :t ~, 46°) generate the families of the stable closed orbits, which defines the wide range of the wing rock motion. In Fig. 17 the amplitudes of the asymmetrical closed orbits, calculated by means of continuation, starting from point Hi, are presented by dark lines. Really the oscillatory behavior is much complicated. In Fig. 18 the Poincar~ mapping, defined as the crossing points of the plane p = 0, is presented for :t and/3 as a function of elevator deflection. The crossing points at every given elevator deflection are taken after a very long time of adjustment (t = 200s). Therefore, all the calculated results correspond to the stable solu- tions. There are some ranges with symmetric periodic solutions with a single and doubled period. There is a narrow range (fie~ - IC) with chaotic motion. This regime arises from the symmetrical periodic solutions after a sequence of bifurcations. First the symmetrical stable closed orbits (see Fig. 19A, B) becomes unstable due to pitchfork bifurcation. After that, both the unstable symmetrical and stable asymmetrical are subjected to the sequence of flip bifurcations (see Fig. 19C, D) and as a result a strange attractor appears in accord- ance with Feigenbaum scheme/~°~ To visualize this 'strange' attractor, four projections of phase trajectory (p, fl),(p, ~t),(r, fl) and (r, :t), obtained by means of plotting only unconnec- ted sequential points (20 dots per second), are presented in Fig. 20. Before chaotic behavior simulation the long time adjustment was fulfilled (t = 2000 s). Poincar~ mapping reveals also the wide regions with stable asymmetrical closed orbits (see also Ref. 50). In such cases the aircraft will enter in the oscillatory wing-rock regime applied on spiral motion. The more complicated case of the periodic solutions is presented in Fig. 21C, D for 6~ =- 20.5 ~. There are three stable closed orbits (one of them symmetrical and two asymmetrical) and two unstable closed orbits. In such a case, the final kind of motion will depend on the initial conditions of state variables, i.e. on the prehistory of control. 40 30 20 (A) I I I I I I I -22 -20 -18 -16 - 14 -12 -10 fie (deg) 10 5 0 -5 -10 -15 chaoticmotion -- ~ ] doubling , , , ,, , , , -22 -20 - 8 - 6 -14 - 2 -I0 (B) 6~.(deg) Fig. 18. Poincar6 mapping of p = 0 plane for different elevator deflections. See text for explanation.
  • 26. 564 M.G. Goman et al. 0.5 -0.5 0.5 -0.5 i i i J i -10 -5 0 5 10 fl (deg) i i 0 i i i i i i -15 15 -3 -20 -10 0 10 20 30 (A) (B) ~t(deg) 0.5 -0.5 i i i i i i i -10 -5 0 5 10 fl (deg) 0.5 -0.5 ' i i i i -15 15 -30 -20 -10 0 10 20 30 (C) (D) x(deg) Fig. 19. Closed orbit projections for different elevator deflections. See text for explanation. In Fig. 21A, B the family of stable symmetrical closed orbits at 6e =- 40 +- 30.0°, generated in the Hopf point H2, are presented. 2.13. SPIN DYNAMICS Analysis of the spin dynamics is carried out using the same general approach as in the analysis of the roll-coupling problem or the stall instabilities, but considering the eighth order autonomous system of motion, Eq. (5). The equilibrium solutions of these equations for the state vector x =. (ct,fl, p, q, r, V, 0, ~b)'~ R8 define the motion along the vertical spiral trajectories. At high angles of attack and a fast rate of rotation such spirals have small radius and correspond to the spin modes. Atmospheric density in such a study is usually held constant, corresponding to considered altitude. The thrust effects only the quantities of steady-state velocity and radius of spin, but not the qualitative nature of the results, therefore in many cases it is taken to equal zero. The searching for all the possible steady states at the given control input = (fie,6a, ~,, T)' ~ R4 and investigation of their local stability characteristics and regions of attraction are necessary for construction of the global phase portrait. The solutions of the approximate spin equations, defining the balances in all forces and moments,C57~can be used as the initial values for equilibria of the complete nonlinear system (5). This is the better way to search for the multiple equilibrium points at the given control input. The continuation of each of these points can give the entire set of solutions. The bifurcational analysis reveals the qualitative changes of the global phase portrait of equations of motion. These changes are closely connected with the stability loss and different kinds of dangerous aircraft behavior. But the real transient dynamics to a great extent will depend on the character of the control input history in time. Therefore the
  • 27. Bifurcation methods in nonlinear flight dynamics 565 1.2 0.8 0.4 o ~- -0.4 -0.8 -1.2 1.2 D.8 3.4 3.4 0.8 1.2 ' 'o ' ' ' 'o ' 'o ' ' '-15 --1 -5 0 5 1 15 2 30 # (deg) ~ (deg) 0.75 15 0.50 ~ 10 0.25 5 o :,---0.25 "~- -5 -0.50 - 10 -0.75 -15 ' 0 ' ' ' l'0 ' 2' '-15 --I -5 0 5 15 0 30 (A) # (deg) (B) :~(deg) Fig. 20. Chaotic behavior due to the 'strange' attractor at 6~ = - 16.5°. See text for explanation. 2.0 1.0 0 -I.0 -2.0 (A) I I I I I -20 -10 0 10 20 13(deg) "8 ,.... 2.0 1.0 0 -I.0 -2.0 (B) I 10 ((! I 20 30 (deg) I 40 0.5 o e.. -0.5 - 1.0 symmetrical stableorbit stableorbits 1.0 0.5 0 -0.5 -I.0 1 I I I I I I I I -10 -5 0 5 10 10 20 30 40 (C) 13(deg) (D) ot (deg) Fig. 21. Family of closed orbits at fie = - 40 + 20.0° (A, B). Three-stable periodical motion case at fie = - 20.5~ (C, D).
  • 28. 566 M.G. Gomanet al. numerical simulation needs to be implemented to check the bifurcation analysis results. It is especially important for multidimensional spin dynamics due to the large differences in the time scales of translational and rotational modes of motion. 2.14. BIFURCATIONAL ANALYSIS OF SPIN MOTION Hereafter some examples of application of continuation technique and bifurcation analysis for spin dynamics analysis will be presented. Spiral instability of lateral/directional motion considering the eighth order equations of motion is qualitatively similar to the rotational instability in roll, when one positive eigenvalue appears. This case also has the pitchfork bifurcation points giving birth to the bothside mirror-symmetrical solutions. The symmetrical equilibrium conditions with zero bank angle are unstable, while the two steady states, representing the right and left spirals are stable. The main difference is that spiral instability is very weak and asymmetrical stable spirals have a very small rate of rotation and sideslip, but very large variations in the bank and pitch angles. If the symmetrical trim position becomes, for example, due to insufficient dihedral effect, unstable, the apparent right and left spiral solutions with 0 ~- 60:, ~ + 40° are stableJ46'52.53~Bank-angle development due to spiral instability is very slow and could be easily controlled by a pilot. Steady state spin modes or equilibrium solutions of motion equations at high angles of attack, computed along with their local stability characteristics for entire ranges of control surface deflections, are the basic information for the global dynamics analysis. In Fig. 22 the general example of equilibrium surface obtained by the continuation method is presented. These results are taken from Ref. 43 and corresponds to the F-4 aircraft. It is clearly seen that aircraft at given control surface deflections have multiple equilibrium solutions. The bifurcation diagram shows the number of equilibria. In Ref. 53 the F-14 dynamics have been studied by determining the steady states of the eighth order equations of motion and seeking bifurcations. In Fig. 23 the steady states of F-14 as a function of aileron deflection for an elevator deflection of - 10~and zero rudder deflection are presented. The multiple steady states exist for the entire range of aileron deflections. For example, at zero aileron deflection there are five steady states, three of them are stable. Expect for the stable trim at low angle of attack without any rotation, aircraft can enter to the left or right flat spin mode at ~ ~ 80~ with large steady-state yaw rate. The steep spin modes in the intermediate angle of attack range are all unstable. The stable branch of aircraft equilibria at low angle of attack exists only in the limited range of aileron deflections, which is bounded by two saddle-node bifurcation points. Exceeding these critical aileron deflections will be followed by jumps or departures to the flat spin modes. The results for critical control deflections, predicted by steady-state analysis, and similar results obtained in numerical simulation of aircraft manuever, can be different. This difference usually arises due to the transient effects and the limited sizes of the domains of attraction for different stable steady states at the considered fixed control deflections. The calculated steady-state spin modes, as shown above, can be used for departure prediction and spin recovery analysis. For example, the F-14 steady-state flat spin modes exist for all aileron deflections; there are some ranges of aileron deflections, where these spin modes are oscillatory unstable. But in these ranges the steady oscillatory spin motion is developed and aircraft remains at high angles of attack. Therefore, only by means of aileron control the spin recovery of the F-14 is very difficult; the rudder control also cannot improve recovery conditions due to the lack of rudder efficiency at high angle of attack. The bifurcation and stability analysis of High Incidence Research Model (HIRM) dynamics at high angles of attack are presented in Refs 59-61. All calculations in Ref. 59 were performed using a smoothed mathematical model for HI RM aerodynamic character- istics~62~and KRIT Package, specialized for flight dynamics problems,lall
  • 29. Bifurcationmethods in nonlinear flightdynamics 567 E( Fig. 22. F-4 equilibriumsurfaceand bifurcationdiagram. In Fig. 24 the dependencies for equilibrium c~,/~,0, ~, p, q, r and V on elevator deflection from Ref. 59 are presented. The results obtained (Fig. 24) show that at zero canard, aileron, rudder and differential canard deflections, without aerodynamic asymmetry and with zero thrust there are some critical branches of solutions with nonzero rotation. Three of these branches (I, 2, 3) in the (~, 6~) plane correspond to spin-like regimes. The branch (4) corresponds to the roll-inertia rotation regimes with small positive and negative :t. If at the spin regimes the velocities of descent are small - V ~ 80 - 90 m/s, in the roll-inertia regimes the velocities are significantly larger --- V = 200 + 300 m/s. Yaw rates in spin regimes can vary from r ~. 20:/sec for branch 1 to r .~ 60'/sec for branches 2, 3. Bank angle and sideslip in spin modes are relatively small, but in the case of the roll-inertia rotation regimes these angles can be essentially greater. Stable equilibria are plotted as solid lines; divergent (or statically unstable) and oscillatory unstable equilibria using dashed and dash~lotted lines (original computer-generated pictures are color ones). The critical equilibria on branches 1, 2, 3 in the most cases are oscillatory unstable. There are such regimes on the basic symmetric branch A in the range :¢ .~ 20 + 38c at fie ~, - 12: + - 16~. This symmetric equilibria with zero rotation and zero sideslip generate the oscillatory motion which leads to wing-rock regime. It is interesting to note that the oscillatory unstable spin equilibria on branch 2 with :t ,,~ 60~ at 6~/> -- 10: are the possible source of so-called "agitated' spin modes. Branches I and 4 are generated in the pitchfork bifurcation points on basic symmetric branch A. In Fig. 25 the dependencies of equilibrium solutions :~,r, 0 and 4) (for different elevator deflections - 20, - 30~) on aileron deflection are presented. The first two variables (~ and r) are the most informative spin characteristics. The equilibrium spin modes demonstrate the nonlinear dependence on aileron dcflection and have some limit points, which are potential points of departures and hysteresis-type behavior. In Fig. 26 the bifurcation diagram for equilibrium solutions is presented in the plane of control parameters (6o, 6~). Only the solutions with :~ > 20~are considered. The crossing of the each bifurcation boundary leads to changing of the number of equilibrium points. The
  • 30. 568 M.G. Goman et al. 80 i i i 40 ....... ........ o~ 0 • _ ~- - 40 " ";~--r~........ - 80 I "<:""'"'1"'~ I - 40 - 20 0 20 40 6,~ (deg) 16 , , , ,~ 8 - -~ o~ -_. ~ -8 -16 i I t -40 -20 0 20 40 6~ (deg) 16 , , , 0 ..... "Z'. ""'"--- ~ 4::........ -8 / S I I '" I -16 -40 -20 0 20 40 ~,, (deg) 180 , , , / 150 " ...... 120 90 ~2~Z ..... -~i -40 -20 0 20 40 6~ (deg) 300 150 0 - 150 -300 - 40 i i ! i t i-20 0 20 40 6,,(deg) 30 0 -30 -60 -90 -40 I ; i - 20 0 20 40 6~ (deg) ~5 100 80 60 40 20 - 40 i ! i ~-..~ ~ t ...... ~ ;~.... I -20 0 20 40 6o(deg) 50 ' ' t 0 -50 I -40 -20 0 20 40 fi~ (deg) Fig. 23. F-14 equilibrium solutions of eighth order equations of motion. difference always equals two. Such results give to the researcher the valuable knowledge about critical values of control deflections and possible departures due to the disappearance of stable equilibrium. Limit cycles or oscillatory spin modes analysis is also very important for understanding of the global dynamics behavior. Agitated oscillatory spin dynamics is often encountered in real flight. Such behavior can be connected with the steady oscillatory spin modes, which are described by the limit cycles or stable closed orbits in the state space of the considered system of equations of motions. The origin of a closed orbit in many cases is due to the Hopf bifurcation encountering. Therefore the starting point and the plane of closed orbit is well known. The oscillatory unstable equilibrium solutions usually co-exist with stable and unstable periodic solutions or closed orbits. The computation of the closed orbits and its local stability analysis at different values of parameters can be performed also using continuation technique in a similar way as the equilibrium analysis, but with a single difference---computation is more time-consuming. In Ref. 50 entire families of equilibria and limit cycle solutions as control surface deflections and control system parameters have been analysed for the F-15 fighter aircraft. For continuation and stability analysis of equilibrium and periodical solutions (limit cycles) of the equations of motion the general-purpose software package AUTO was used.
  • 31. Bifurcation methods in nonlinear flight dynamics 569 0 =- I Ill>. 100 50 0 -50 -100 -60 -,~ ..... ~-- ~,~... ~,. _~.~ .'~..../!- 4. -40 -20 0 20 50 "~ 0 c -50 ill. -100 ' ' ' 40 -60 -40 -20 0 20 40 •v 50 0 O -50r- b=o. ---':- "x~." ~•4~ " " . ~ .. t.1 • . . . . . , = "1 -60 -40 -20 0 20 ta 5o 'ID o n- -50 f ' *~. 40 -60 -40 -20 0 20 40 0 "0 o n,. 100 50 0 -50 -100 -60 i . .....,.,k.-,'-.~-- • ".<. .~ ..... /. ....__~" "-~ --,--> ,~ -.~ ....% / ......~,.... / ' - ~ I I, -40 -20 0 20 Q.... 2O ID) 0 •u 10 t- O -10 u) _20i -60 i 40 -40 40 • • I .~, ._ f: ...-z.: "<; ;..:..:1;..:..; ...... / -20 0 20 t.- n 40 30 20 10 0 -10 -60 f° • f / .--~'~ ""~' '--__:'~" : .-Z.• ~ I -40 -20 0 20 Elevator deflection, deg 250 200 E 150 >- 100 • l'Vi I ll!'N~ '. .,._~.~_...._ - ~_.._- .X_=...~.~.. . . .....:,. .... r ~ .~.~ 50 40 -60 -40 -20 0 20 40 Elevator deflection, deg Fig. 24. HIRM equilibrium solutions ofeighth order equations for different elevator deflections and 6a = 0, 6c = 0, 6, = 0, no asymmetry, no thrust, H = 5 km. Particular attention was given to periodic wing-rock motion both with and without control system. In Fig. 27 the bifurcation diagrams for eighth order equations, showing the equilibrium and periodic solutions for symmetric stabilator variation and rudder variation, are pres- ented• The possible stable steady state and limit cycle values of angle of attack are marked by solid lines and black circles, the unstable ones with dotted lines and white cycles. The circles represent the maximum values taken by angle of attack during the period of steady oscillations. Branches of equilibria at high angle of attack, representing spiral or spin motions, are mainly unstable. Small regions of stable fiat-spin modes are bounded by Hopf bifurcation points, which give rise to oscillatory spin modes. The right and left spin modes are different due to aerodynamic asymmetry in yaw at high angle of attack (see Fig. 27B). Note that some apparently-disconnected branches of equilibrium solutions in this case were detected by means of exceeding real physical limits of the control surface deflections.
  • 32. 570 M.G. Goman et al. I00 0 -100 -2b .., __..-. ~'~ i 0 fi~(deg) 20 0 - 30 - 60 "------:--:::~...... ~ , . .. ~" i i -20 0 ?i (deg) 20 75 "- : " • -._... 50 25 I -2'0 0 6. (deg) ,.... -- ~, I 20 40 -~ 0 -40 (A) .....- " 6 2'0 ~. (deg) 100 0 " -100 O L. "7".:-..- I I -20 0 .==•o• •'== ..= "=-'" • .4 I 20 6o(deg) 0 "~ -30 -60 :.::::r i ~ ~:::2~ i,..,..., f i °----'* -2'0 6 2b 6,, (deg) 75 "~ 50 /5 25 ...,,,~ .--. ,- ~"" ".... ----4 • -----. " .... "'~ 40 o -40 /'""-........ -2b 2b -io 6.(deg) (B) 3o(deg ) Fig. 25. HIRM equilibrium solutions of eighth order equations of motion for different aileron deflections and 3, = 0, 3, = 0, 3, = - 30, - 20": respectively for A) and B) cases, no asymmetry, no thrust, H = 5 km. Wing-rock motion onset in the angle of attack range ~ ~ 20 ÷ 30", which is seen from the bifurcation diagram (Fig. 27), agrees well flight test data.~Sm Similar to the equilibrium pitchfork bifurcation of symmetrical solution the symmetrical periodic solution can also bifurcate in the pitchfork manner. After such bifurcation of stable limit cycle two asymmet- rical limit cycles (mirror-images of one other) appear, while the symmetrical periodic solution becomes unstable. The example of such bifurcation, arising on the wing-rock branch of periodic solutions, is presented in Fig. 28. Extended analysis of oscillatory spin modes was performed in Ref. 59 for High Incidence Research Model (HIRM). In Fig. 29 the equilibrium solutions for ~, r, 0 and ~bat 6c = - 20° as functions of elevator deflection at zero lateral control and zero asymmetry, taken from Ref. 59, are presented. The first family of the closed orbits is located 'around' the oscillatory unstable equilibrium points corresponding to the branch 2. The maximum and minimum values of the state parameters for periodic solutions are marked by vertical lines. Thick lines denote stable closed orbits, thin lines the unstable ones. The closed orbits are stable only
  • 33. Bifurcation methods in nonlinear flight dynamics 571 20 -20 -40 "'1 13 [ / 9 7 //1 7 1 9 ~x~ 5 :'~ ..... ~ ~; -..z.,.,.. 7 "~"'-~" ~ -- ..... I I I -20 0 20 ~. (deg) Fig. 26. Bifurcational diagram of HIRM equilibria in the plane of elevator and aileron deflections. Eighth order equations, no asymmetry, no thrust, ~t > 20°, 6c = 0, 6, = 0, H = 5 km. 100 80 60 40 20 0 -20 (A) -,m...... ~pin,s '-'"377'.'~] .......................................................... WingRock~ / -30 -25 -20 -15 -I0 -5 0 5 10 15 20 • o • .* .k .......".:::.,~. ..." 80 - "w': "................. Spi,~ f 60- 40 " WingRock ............ .:,,..; ....... ---"..:::.,..,~::::............. .. • , 4,,, -. : 20- .......... _ 1 .......... 6, =-10.19" 0 -70 (B) ~, -50 -30 -10 10 30 50 70 Fig. 27. F-15 equilibrium and periodic solutions of eighth order equations of motion; A) symmetric stabilator variation, 6, = 0; B~ rudder variation, fie = - 12.2°. Bifurcation diagram for equilibrium and periodic solutions. 36 30 • 32 28 24 20 16 -1.5 -'28.1 -1.0 -0.5 0.0 0.5 1.0 1.5 6 (A) P (a) 28 - 21.2 22 2 ~ 26 • 20.2 "~s 22.1 18.9 ~ 24. 18.1 17.6 Flong= 17.2 22 -6 -4 -2 0 2 4 Fig. 28. F-15 wing-rock motion. Periodic orbit projections for varying longitudinal parameter; A) orbits for open-loop system, B) orbits for closed-loop system. when their amplitudes are relatively small at fie ~ - 2° - 2° (see Fig. 30A). At 6e > 2° the closed orbits become unstable (see Fig. 30B) after the flip or period doubling bifurcation arises. As one can see in Fig. 29, the amplitudes in pitch (Act, A0) of single-period closed orbits are much larger than the amplitudes in the lateral motion (Ar, Ap, A~, Aft).
  • 34. 572 M.G. Goman et al. 0 "lO =- Itl >- 100 50 o -50 -100 -60 ,/ .... ~?-£ - ; -40 -20 0 20 ~ 2O m -o 0 d-~ -20 ~, -40 -60 a. -80 -60 -40 -20 0 20 ~- 50 ¢u 0 0 ~ -50 e- inchA--~~os~d orb~ branch A . ..- ~ : :--: : -- - -=- _. "-.. -60 -40 -20 0 20 Elevator deflection deg 200 -~ lOO m- o~ oc o -100 n- -200 -60 -.> ..> -40 -20 0 20 Elevator deflection, deg Fig. 29. HIRM equilibrium and periodic solutions of eighth order equations of motion for different elevator deflections and fir = - 20'~, go = 0, ?i, = 0, no asymmetry, no thrust, H = 5 km. Figure 31 shows the closed orbits originating at the symmetrical branch of HIRM equilibrium without rotation when 6~ ~ - 27L At first, oscillatory solutions are stable, then the Hopf bifurcation occurs (a complex pair of multipliers crosses the unit circle), after that pitchfork bifurcation leads to the appearance of two asymmetrical closed orbits. The following shows some examples of complicated chaotic behavior. When any simple stable attractor, equilibrium or periodic, is absent at the given control setting the chaotic dynamics can be realized due to the appearance of a strange attractor. In Fig. 32 the examples of such behavior are presented (these results are taken from Ref. 50). The first case (Fig. 32B) indicates the presence of toroidal attractor (to reveal the toroidai structure of the phase space in the bottom figure only the discrete data points along the trajectory are plotted). Note that amplitude state variables in this case are very large - :<,~ 40° + 90~, r ~ 1.0 + 3.0 rad/sec. It is clear that toroidal structure is based on the transitions from the oscillatory fiat-spin mode to the highly oscillatory steep-spin mode, which are both unstable. The time of such transitions is much more than one period of angular oscillations. The second example (Fig. 32A) shows a global attractor with the sequence of transitions between spin mode and wing-rock regime, which arises just beyond the region of stable wing rock. In Fig. 33 the example of complicated oscillatory behavior in the fiat-spin mode, taken from Ref. 47, is presented. The transient dynamics possess two time scales of parameter variations--short and long periods. The first one is about 3 s, the second is much greater, about 150 s. Short period oscillation can increase and decrease the amplitude (see Fig. 33A). Therefore the short time dynamics depends on the initial conditions of motion. The phase trajectory projections on (p,/~), (r,/~) planes (Fig. 33B) and Poincar+ mapping (Fig. 33C) clearly reveal the qualitative nature of such dynamics. There exists the toroidal stable manifold of trajectories, and phase trajectories from its domain of attraction go to this invariant manifold. Inside the toroidal attractor there is an oscillatory unstable closed orbit (Fig. 33C). Departure characteristics quant!/ication can be done using the results of equilibria and closed orbits computation along with their stability analysis. The first type of departures is connected with the bifurcations, leading to the disappear- ance of stable equilibria (e.g. saddle-node and undercritical pitchfork bifurcations). There
  • 35. Bifurcation methods in nonlinear flight dynamics 573 1.5 ~m = 0.50 r~ 0.g ~" O.8 0.7 (A) 1.5 "10 o 0.s re" -2 0 Sideslip, deg -2 0 Sideslip, deg 0.4 0.2 0 -0.2 -0,4 2 30 9. 1 2 3O O.G 0.4 0.2 0 -0.2 -0.4 O 40 50 60 Q vs AOA, deg 40 50 60 AOA, deg 2 1 _E -I -2 -2 6 5 3 2 -2 2 1 -1 -2 -2 Flip bifurcatio,n 0 Real IIIWXNItXNNxxxXXN~I ItU11 "~ ?Plip bitur~tion " 0 2 4 6 8 Elevator, deg o o -2 0 2 40 60 80 0 Sideslip, deg Q vs AOA, deg Real , 1 t ,,4 5 . . . . . ' x ~ ' ~ . . . . . . . ;0, / >. 0.8~ ;3 -t -2 07 t -3 2 -2 0 2 40 60 80 10 15 20 (B) Sideslip, deg AOA, deg Elevator, deg Fig. 30. HIRM clo:~ed orbit projections with multiplies for different elevator deflections and 6, = - 20, rio = 0, 6, = 0, no asymmetry, no thrust, H = 5 km; A) stable orbits, B) unstable orbits after flip bifurcation. may be also the bifurcation points, where the stable closed orbits disappear. In these cases the transient motion transfers an aircraft to another stable equilibrium or oscillatory motion (or to a more complex attractor). When there are some different attractors, the motion behavior following the bifurcation encountering depends on the initial location in the state space of the bifurcated solution. The behavior is determined by the attractor, in which region of attraction the bifurcated point is placed. The character of aircraft dynamics when parameters cross the bifurcation boundary depends on the rate of parameter variation. When parameters vary slowly, the aircraft dynamics can be predicted more correctly, because it is close to the equilibrium values. In the case of fast variation of parameters the difference between the transient motion and steady-state solutions will be more considerable. Nevertheless, to quantify the departure characteristics one can use the bifurcation diagrams in the parameter planes, which define critical conditions of stable solutions disappearing. The influence of a parameter variation
  • 36. 574 M.G. Goman et al. u~ "0 ¢- 12. 0.05 -0.05 O "10 Q. (/3 .'g_ (/) 10 5 0 -5 -10 1 -o 0.6 o -0.5CC -1 32 34 36 0 -5 0 5 10 Angle of attack, deg Sideslip, deg 0.4 ¢o 0.2 0 -0.2 >- -0.4 -1032 34 36 -5 0 5 Angle of attack, deg Sideslip, deg 10 Fig. 31. H1RM closed orbit projectionsfor differentelevator deflectionsand 6, = - 20~, 6, = 0, 6, = 0, no asymmetry,no thrust, H = 5 km. Pitchforkbifurcation. rate on the character of 'jumps' or departures can be evaluated only by means of numerical simulation of aircraft motion. Figure 34 shows the example of HIRM transient dynamics under the elevator variation with dfJdt ,~- 2.0~/sec. Aircraft motion starts from 'agitated' spin mode (branch 2) :~~ 60°, r ~ 50°/sec. The oscillatory motion at the bifurcation point fie ~ - 4: is followed by a 'jump' to the equilibria on the symmetrical branch A without rotation. This 'jump' is the result of the disappearance of equilibria and closed orbits on the critical branch 2, but due to parameter variation it is significantly delayed with respect to the equilibrium analysis. Numerical simulation of the 'agitated' spin modes confirms the amplitudes of the calculated closed orbits around the branch 2. The further variation of elevator results in the movement of state point along the symmetric branch A up to the next departure at 6e ,~ - 23°. The second type of departures is connected with the loss of stability under the application of finite (not small) perturbations. In many cases the stable equilibria or closed orbits have the limited asymptotic stability regions; and there exist critical values of state variable disturbances, the exceeding of which leads to loss of stability and transfer or 'jump' to another attractor, which can be realized at the same value of parameters. The nearer the equilibrium to bifurcation point, the smaller the size of its asymptotic stability region. To quantify the departure resistance in such cases, the characteristic size of stability region with respect to the existing level of perturbations may be used as a measure of quantification. Closed-loop dynamics analysis for control-law design purposes is a very important application of bifurcation analysis. Feedback control laws can significantly change the closed-loop dynamics and eliminate bifurcation phenomena, leading to stability loss and departures. For example, the wing rock and spiral divergence instabilities could be control- led with a simple feedback control system. In Ref. 51 such results are presented. Roll rate and sideslip feedbacks to the ailerons can be used to supplement the aerodynamic deriva- tives, which are responsible for instability, i.e. damping in roll or dihedral derivative. In such a way the instability regions of steady states can be eliminated. Continuation methods make it possible to determine the effects of the control system by means of the closed-loop dynamics of a nonlinear system stability analysis.
  • 37. 1oo 80 6o 40 20 0 4.0 2.0- l, -0.0 - -2•0 0 100 80 6O 40 20 0 -2.0 t J Bifurcation methods in nonlinear flight dynamics , 1 i 200 400 600 t 100. 90' 80' 70. 60' 50' 40. 30' 20' 80O 0 2OO 4.0 ' 40O t 600 575 800 i 200 400 600 800 t • • .~.~.~.... • . . . . ~ ;,: :' "."~.. ~.. ~r:' " ':' :..; ~. ,/-:" ...:. '.. ". -'" "'~d" • . .." ~ '.'..~= "- o ~- .... ° 3.0 ' ,- 2.0' 1•0 ' 0.0 0 100 i 200 i i 400 6O0 t 90 80 70 60 50 40 30 20 0.0 800 .°°., °,..,.• • . •.... . o°,~, .:.;. :. -. :,.:-;..-...._., •~:~ .,:.~4~..:~...:, ,- • t.ol. • , o. ~' O; ". • i | | ' -1.0 0.0 1.0 2.0 3.0 4.0 1.0 2.0 3.0 r r (A) (B) 4.0 Fig. 32. Chaotic attractors in spin dynamics of the F-15. See text for explanation• In Ref. 50 it has been shown that the geometrical nature of the nonlinear results can be used for high angle of attack control-law design applications, for example, for wing rock suppression, departure elimination, spin avoidance/recovery. Simplified lateral-directional feedback strategies for the F-15 aircraft were investigated using continuation and bifurca- tion analysis of the closed orbit solutions. These strategies comprised the conventional roll damping augmentation by roll control and a roll rate 'crossfeed' to rudder, scheduled by angle of attack. Some adjustable gain parameters were searched for to obtain the better high angle of attack behavior, i.e. to reduce as much as possible the amplitudes or eliminate the wing rock altogether• The bifurcation analysis results can be used for design of control surface interconnec- tion.(43"45"46"53)For example, the saddle-node points at low angles of attack equilibrium branch are responsible for entering spin (Fig. 23). Therefore, the boundaries in the control
  • 38. 576 M.G. Goman et al. - 0.0 .~ -0.05 -0.1 ~ tad -0.! Poincar~ mapping oscillatory unstable closed orbit (c) -0.05 saddle ,omt unstable acus Stable invariant toroidal manifold 0.05 ..... ~............. : :..... :..... !.... "" 1.5 .~ 1.4 ~ 1.3 ~ 1.2 ~ 1.1 1.0 -i.1 -1.00 -i.3 -1.25 ~ -1.4 -1.5 ~, -I.50 0 40 80 120 (A) Time. sec 160 -I.I -1.3 -1.4 r, I/s "~ 1.50 1.25 o ~0 ~. 1.00 ~ 0.6 " 0.4 -- 0.1 ~ 0.0 -0.1 -0.05 0 0.05 (n) Sideslip angle, tad oscillatory unstable closed orbit Fig. 33. Tordoial attractor in the 'agitated" flat spin mode; A) Transient time histories, B) phase trajectory projections, C) Poincar~ mapping. See text for more detailed explanation. plane (6e,6a, 6,), formed by these critical limit points, can be used for spin prevention. These boundaries can be taken into account in the control surface interconnection for putting limits on the control surface deflections to prevent jump phenomena. 3. THEORETICAL BACKGROUND Hereafter some concepts and methods of the dynamical systems theory, which are more often used for flight dynamics applications, are presented. ~3-5'27'10~ The nonlinear dynamical system, depending on parameters, will be considered in the following form: dx- - = dt F(x,c), x~R",cEMcR m (20)
  • 39. Bifurcation methods in nonlinear flight dynamics 577 ¢= '10 =- all ]= ¢l >. 100 - - 50 0 -50 -100 - - -60 -40 -20 0 20 40 0 -- - nl -so -100 -60 -40 -20 0 20 40 0 -v 50 ~ 0 o o -50e- branch 2 branch A -60 -40 -20 0 20 100 5O "o " 0r- eO ~- -50 -100 40 -60 -40 -20 0 20 40 100 } so $ 0 ~ -50 -100 L -60 _ - -~i' / / , ~ , -40 -20 0 20 k 2O ii0Io-10 -20 40 -60 -40 -20 0 20 40 et o "o ° m J~ 0. 40 30 20 10 0 -10 -60 $ / <P ** / t iI" oo • • o-o- ~ -40 -20 0 20 Elevator deflection, deg 250 200 E 150 100 I L I - 50 40 -60 -40 -20 0 20 40 Elevator deflection, deg d3~ Fig. 34. Transient H1RM dynamics under elevator variation with ~- ~ - 2.0~/sec in comparison with equilib- rium solutions (3, = - 20', 30 = 3, = 0, H = 5 km). where F is a smooth vector function. The vector field F defines a map R"÷" ---,R'. The system (20) satisfies the conditions of the existence and uniqueness of solutions x(t, Xo) with initial condition x(0, Xo) = Xo. The solution curve ~o,(Xo)= x(t, Xo) is called the trajectory and together wtth other trajectories forms the flow of the dynamical system. The set of all trajectories constitute the phase portrait of the dynamical system. The concept of the critical elements of a nonlinear autonomous dynamical system is the basic one. The critical elements are invariant manifolds of trajectories, the dimensions of which vary from 0 up to n - 1. The simplest ones are well-known equilibrium points or zeroes of a vector field and closed orbits, which define the periodic solutions. They may be attractive for other trajectories, if they are stable. There are also more complex attractors in high order dynamical system (n/> 3). For example, invariant toroidal manifolds of trajecto- ries and 'strange' attractors (see Fig. 35).
  • 40. 578 M.G. Goman et al. Stable equilibrium Stable closed orbit )< Stable toroidal manifold Strange attractor Fig. 35. Critical elements of the phase portrait. Complex behavior of the nonlinear dynamical system is usually due to the existence of a number of isolated attracting sets, each attractor has its own region of attraction. All that brings about the strong dependence of motion on initial conditions and on the parameters variation.~5, t4~ Equilibrium point is defined as the zero point of the vector field F(x, e): F(xe, c) = 0. Closed orbit F can be defined as a fixed point of a Poincar~ mapping, which can be set using an n - l-dimensional hyperplane E transversal to the closed trajectory F at the point x,. The trajectories in the neighborhood of F also cross it. Thus every point Xke Z (belonging to some neighborhood of x,) can be mapped into some other point of the hyperplane xk+t• Z, which corresponds to the second intersection of the trajectory ~,(Xk) with hyperplane E. Fixed point x, of the mapping P: Xk, t = P(Xk), where Xk, t, Xk• E, corres- ponds to the periodic trajectory F (see Fig. 36). Thus the equilibrium points and closed orbits can be computed by solving the non- linear problems in a similar way. In the first case the problem is F(xe, c)= 0, in the second--P(x,) = x,. Equilibrium points will be stable if any solution x(t, Xo) from its neighborhood remains close to it for all times or approaches to x,. A similar condition is valid for closed orbit, i.e. for fixed point of Poincar6 mapping. They will be stable if any cross point from the neighborhood of x, remains close to it for all sequences of mapping or approaches to x,. With respect to Lyapunov theorem the equilibrium will be stable if all the eigen- values of the Jacoby matrix "~a-~-_r. will have negative real parts: Re{,~i}i=l., < 0. U~ X = Xr The similar local stability condition is valid for fixed point. All the eigenvalues of the c~P which are called multipliers, are to be inside thelinearized mapping matrix ~x .... ' unit circle, i.e. EPil< 1. These conditions guarantee only the local stability of equilibrium or fixed point. The structure of the state space in the neighborhood of equilibrium or closed oribt is defined by stable and unstable invariant manifolds to trajectories W ~, W", which are defined, for example, for equilibrium point as: Wp = {x: q~,(x)~x~ as t--* + oo}, W~ = {x: ~o,(x)~ Xe as t ---,- oo}. The stable p-dimensional manifold W' comprises all the trajecto- ries in the state space that go to x~ while t --* + ~. Similarly, the unstable q-dimensional manifold W" comprises all the trajectories in the state space that go to xe while t ~ - ~.
  • 41. Bifurcationmethodsin nonlinearflightdynamics 579 Fig. 36. Closedorbits-- Poincar~mapping. For linear approximation, the manifolds W ~ and W u lie in the hyperplanes Ls and Lu. The hyperplane L' is spanned by p stable eigenvector {r/i}i=k...... k,, while L~is spanned by q unstable ones {w/i}i=t...... z,: L~= span(~h,, qk2.... , rh,), L" = span(th,, th2, ..., th). When nonlinear terms are taken into account, the surfaces W s and W" will be tangent to hyperplanes L~and Lu in the neighborhood of the equilibrium point and deviate from them far from the equilibrium point. There may also be situations when eigenvalues have zero real parts. Central manifold theorem gives an answer about possible decomposition of a nonlinear system in this case [11-1. The topology of the trajectories in the state space near the closed orbit F is similar to the topology of point with the behavior of the mapping in the vicinity of the fixed point x,. The type of fixed point x, is determined by multiplies. If p is a number of multipliers inside the unit circle, and q is the number of multipliers outside the unit circle, the p-dimensional stable and q-dimensional unstable invariant manifolds W ~ and W ~ for point mapping sequences may be defined. The stable manifold W ~is composed of the points, which converge to the fixed point x,. This manifold corresponds to the stable multipliers IPgl < 1. The unstable manifold W u is composed of the mapping points, which converge to the fixed point x, using inverse mappings. The manifold corresponds to the unstable multi- pliers [Pi[ > 1. Invariant manifolds of mapping points W "~and W" for the fixed point x, correspond to the invariant manifolds of the trajectories W ~and W", whose dimension is greater on unit p+l,q+l. In Figs 37 and 38 the examples of invariant manifolds for equilibrium point and closed orbit are presented. Stability region or domain of attraction S is a very important concept for global stability analysis. For every attractor x~, stable equilibrium or closed orbit, the asymptotic stability region can be defined as: S(x~) = {x e R": ~0,(x)--*x~ as t ~ ~}. Suppose that xs is the stable equilibrium point of (20). S(x~) is an open and invariant set, its boundary dS(x,) is an invariant closed set, whose dimension is less than the dimension of the state space. If the dynamical system has at least two stable equilibrium points or closed orbits, then the boundary of stability region is nonempty with co- dimension equal to 1. The stability boundary t3Sis the union of stable manifolds of unstable equilibrium points and closed orbits, which forms the stability boundary (Oi ~ OS, F~e OS): ~s = 0 w'(o,) L) w~(I-,) (21) i j
  • 42. 580 M.G. Gomanet al. w~ Fig.37. InvariantmanifoldsW', W"for equilibriumpoint. jF / / f Fig.38. InvariantmanifoldsWe, W" for closedorbit. There are some conditions that determine the number and the properties of the critical elements (equilibrium points and closed orbits) belonging to dS.~t4~ For example, the unstable equilibrium point x, belongs to ~S(xs) if: W"(x.)nS(xs) ~ O. The following algorithm determining the stability region through reconstruction of its boundary can be proposed: 1. Search for all the equilibrium points and closed orbits. 2. Identify those critical elements whose unstable manifolds contain trajectories ap- proaching the stable equilibrium point (or stable closed orbit) in the stability region being studied. 3. The stability boundary is the union of stable manifolds of the equilibrium points and closed orbits identified in step 2. Each of the algorithm's steps can be efficiently performed only using numerical methods. To accomplish step 2 one can use the method proposed in Ref. 49. Its main idea is the solving of
  • 43. Bifurcationmethodsin nonlinearflightdynamics 581 the two-point boundary problem for reconstruction of the stability boundary twodimen- sional cross-sections. But the stability boundaries can be very complicated for such an algorithm (e.g. due to many folds with fine structure). In such cases, it is more efficient to use the scanning method based on the numerical simulation of nonlinear systems and final checking of the attractor, for which the motion is developed. Such scanning may also be accomplished for twodimen- sional stability region cross-sections. Bifurcational analysis or investigation of the phase portrait structural changes, while parameter c is varying, is the more important in the methodology of the qualitative analysis. There are critical or bifurcational values of parameter cbwhen qualitative type of the state space structure changes. The most common case is one-parameter bifurcation when near cb one can find only two different types of state space. There are many types of local and global bifurcations, which leads to different changes in dynamical system behavior. Global changes of the state space structure result from successive local bifurcations of equilibrium points, closed orbits, nonlocal bifurcations with invariant manifolds W s, W u, etc. Local bifurcations with equilibrium points and closed orbits can be predicted on the basis of the linearized system eigenvalues analysis. Initially stable equilibrium points can be subjected to the following bifurcations (see Fig. 39): • Stable and saddle-type unstable equilibrium solutions merge and disappear. This corresponds to limit point on the equilibrium curve and is also called fold or saddle- node bifurcation. A jump-like loss of stability or departure takes place in this case. • Stable point becomes unstable of saddle-type, at the same time new two stable equilibrium points appear. This is a branching point or supercritical pitchfork bifurca- tion. In the supercritical case the soft loss of stability with transfer to the one of two stable solutions occurs. • The merging of two saddle-type and one stable equilibrium solutions. This corres- ponds to the subcritical case of pitchfork bifurcation, when the jump-like loss of stability can occur. The last two cases of bifurcations are structurally unstable. Usually they take place in systems with symmetry, and they are practically unreal for a case without symmetry. A very important kind of equilibrium bifurcation is the so-called Andronov-Hopfbifurca- tion. There are two forms of the Andronov-Hopf bifurcation with initially stable equilib- rium points (Fig. 39): • The equilibrium become oscillatory unstable and the stable closed orbit is detached from the equilibrium. Soft-type loss of stability takes place in this case. • The stable equilibrium solution merges with the saddle-type unstable closed orbit and becomes oscillatory unstable. The jump-like loss of stability occurs after such bifurcation. Soft-type loss of stability means that the state point slowly goes away from the equilibrium point while the parameter crosses the bifurcation boundary. If the parameter returns back, the state point returns to the stable equilibrium point. When the loss of stability is abrupt or jump-like, the state point quickly leaves the equilibrium point to some other attractor. This type of instability in aircraft dynamics is usually called departure. If later the parameter returns back, the state point usually does not return to the initial equilibrium and hysteresis type behavior can take place. Bifurcations of the closed orbits can be determined by means of analysis of the multiplier of the linearized mapping matrix. When the multiplie crosses the unit circle the stable closed orbit is subjected to one of the following bifurcations (see Fig. 40): • Multiplier crosses the unit circle at the point (+1,0), the saddle-type and stable closed orbils disappear or are created; in the system with symmetry this case can lead to the pitchfork bifurcation for the fixed point; there may be a supercritical or
  • 44. 582 M.G. Goman et al. ImZ. Rek 0 ~" Saddle-nodebifurcation /~.... limitpoint(det~x_=.0_) I supercriticafl branching point undercrilical / g transcritical ~,$/S" / / Andronov-Hopfbifurcation o~o Rek ~] - supereritieal Fig. 39. Bifurcations of the equilibrium points. subcritical case, when two mirror-symmetrical stable or unstable limit cycles are created or disappear. • Multiplier crosses the unit circle at the point (-1, 0) the stable closed orbit becomes unstable and stable periodic orbit with doubled period detaches from the initial closed orbit. This bifurcation is called flip or period doubling bifurcation; usually the period doubling bifurcations appear in the form of so-called Feigenbaum cascade, leading to chaotic motion. • A pair of multipliers crosses the unit circle at the points e -+'~, the closed orbit becomes oscillatory unstable while the stable two-dimensional toroidal manifold appears; there are also supercritical and undercritical types of such bifurcation. There are also other situations when closed orbits disappear: • The periodic trajectory F shrinks into a point. • An equilibrium point emerges on the closed orbit F. • Some point belonging to F goes to infinity, thus the curve is no more closed. Continuation methods are widely applied for computer-aided qualitative analysis of dynamical systems in many general-purpose and specialized for different applications Packages.(19'2°'22'31) There are several versions of continuation algorithms for solving the nonlinear problems (e.g., searching for equilibrium solutions -- F(xe, e) = 0, or fixed points of Poincar6 mapping--P(x,) = x,). The main feature of these algorithms is the introduction a general arclength for the solution curve in the extended by parameter state space. Such consideration permits one to overcome the singular problem in the limit points
  • 45. Bifurcationmethodsin nonlinearflightdynamics Re p Re p 583 J -, ~. _.~,~ r,-,~ N Fig.40. Bifurcationsofthe closedorbits. of the curve, where the general technique, considering the fixed parameter, fails due to the singularity of the Jacoby matrix. The continuation is performed along the solution curve and the errors between the current values and the true solution are reduced with Newton's method. The slope of the curve, determined by tangent vector s, may be defined using the OF extended Jacoby matrix ~-~z'where z = (x, c) is a state vector added by parameter. So the prediction step can be made in the direction of vector s, the correction step Azc more optimally to perform in the direction which is normal to the tangent vector s: OF O---zAzc = - F (22) s'Az¢ = 0 The correction (22) provides very good convergence to the solution curve even from distant points.(3 t,47. 48) The continuation naturally may be combined with the local stability analysis of the equilibrium and periodic solutions, because it contains the Jacoby matrix for eigenvalue and eigenvector calculations. Continuation methods are very efficient tools for searching for the multiple solution at given values of parameters. Different branches of solutions can be tied by widening the parameter ranges. A special systematic search method for multiple solutions is developed based on the continuation technique/31~ The surfaces of equilibrium Ee and saddle-node bifurcational Be solutions of the dynam- ical system are very important for the global dynamics analysis and departure prediction. The continuation technology can be applied for both these problems in a similar way: Ee= {x:F(x,e)=O, xeR",FeRn, eeM cR "} Be = {x:G(x,e)=0, xER n, G~R ~+1, c~M c R"} G= OF det
  • 46. 584 M. G. Goman et al. 4. CONCLUSIONS The results presented above show the efficiency of the methodology, based on the qualitative methods of dynamical systems theory, for investigation of nonlinear flight dynamics problem. Nonlinear aircraft dynamics are too complex to be thoroughly studied using analytical methods. That is why the advances in the development of new methodology are closely coupled with the development of special numerical methods and software. Nowadays the efficient numerical techniques and specialized scientific packages for stability and bifurcational analysis of the high order equations of aircraft motion provide the valuable quantitative information for aircraft nonlinear dynamics prediction. Continua- tion method is the cornerstone algorithm for many other ones, which are used in different scientific packages for such nonlinear analysis. Continuation and local stability analysis of equilibrium and periodic solutions of motion equations, reconstruction of their regions of attraction, are of great value for analysis of aircraft dynamics both without and with flight control system. The presented approach has great potential for control law design purposes by means of closed-loop nonlinear dynamics analysis. Nonlinear aircraft problems such roll-coupling, stall, spin dynamics now can be studied by implementation of the unique approach without any restrictions on the mathematical model for aerodynamic characteristics. ACKNOWLEDGEMENTS This work was partly supported by Contract No. ASF/2017/E with Defence Research Agency of UK. The authors are grateful to Dr. Phiii Smith for valuable discussions on the subject of this article and great interest in the work. REFERENCES 1. Poincar6, H. (1892, 1893, 1899) Les methodes nouvelles de la mcchaniques celestre, Vol. I-III, Paris. 2. Thom, R. (1974) Structural Stability and Morphogenesis, pp. 47, 48, 55--92. Addison-Wesley, Reading, MA. 3. Arnold, V. I. (1977) Geometrical Methods in the Theory of Differential Equations, Berlin, Springer-Verlag. 4. Arnold, V. I. (1978) Additional Chapters ofthe Theory ofOrdinary DifferentialEquations, Nauka publ., Moscow (in Russian). or Arnold, V. I. (1980) Chapitres Supplbmentaires de la Thborie des F,quations Diff(rentilles Ordinaires, I~ditions Mir. 5. Neymark, Yu. I (1972) Method of Point Mapping in the Theory of Nonlinear Oscillations, Nauka publ., Moscow (in Russian). 6. Neymark, Yu. I. and Landa, P. S. (1987) Stochastic and Chaotic Oscillations. Nauka publ., Moscow (in Russian). 7. Iooss, G, and Joseph, D (1980) Elementary Stability and Bifurcation Theory, Berlin, Springer-Verlag. 8. looss, G. and Aldermeyer (1992) Topics in bifurcation theory and applications, Advanced Series in Nonlinear Dynamics, 3, Word Scientific. 9. Marsden, J. E. and McCacken, M. (19761 The Hopf Bifurcation and its Applications, Springer Verlag. 10. Gukenheimer, J. and Holmes, P. (1983) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer Verlag. 11. Carr J. (1981) Applications of Center Manifold Theory, Springer-Verlag. 12. Langford, W. F. (1983) A review of interactions of Hopf and steady state bifurcations in Nonlinear Dynamics and Turbulence, Pitman Advanced Publishing Program. 13. Kubichek, M. and Marek, M. (1983) Computational Methods in Bifurcation Theory and Dissipative structures, Springer-Verlag, New York. 14. Chiang, H.-D., Hirsch, M. W. and Wu, F. F. (1988) Stability regions of nonlinear autonomous dynamical systems, IEEE Trans. Auto. Contr. 33 (1). 15. Gilmore, R. (1981) Catastrophe Theoryfor Scientists and Engineers,John Wiley & Sons, New York, Chichester, Brisbane, Toronto. 16. Genesio, P., Tartaglia, M. and Vicino, A. (1985) On the estimation of asymptotic stability regions: state of the art and new proposals, IEEE Trans. Auto. Contr. AC-30 (8). 17. Forsythe, G. E., Malcolm, M. A. and Moler, C. B. {1977) Computer Methods for Mathematical Computations. Prentice-Hall Inc., Englewood Cliffs, N.J.
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