ICAS 2000 CONGRESS AN AGILE AIRCRAFT NON-LINEAR DYNAMICS BY CONTINUATION METHODS AND BIFURCATION THEORY Krzysztof Sibilski Military University of Technology, Warsaw, Poland Keywords: flight dynamics, flight at high angle of attack, bifurcation theory Abstract arized equations of motion used for analysis at that time did not contain the instability [1]. The paper provides a brief description of the There are many problems associated modern methods for investigation of non-linear with flight dynamics for modern and advanced problems in flight dynamics. A study is pre- aircraft, which are not solved (or solved rather sented of aircraft high angles of attack dynamic. unsatisfactory) with traditional tools. A list of Results from dynamical systems theory are used such problems includes among others flight to predict the nature of the instabilities caused control for agile and post-stall aircraft. The by bifurcations and the response of the aircraft post-stall manoeuvrability has become one of after a bifurcation is studied. A non-linear dy- the important aspects of military aircraft devel- namic model is considered which enables to opment. Such manoeuvres are jointed with a determine the aircraft’s motion. The aerody- number of singularities, including “unexpected” namic model used includes nonlinearities, hys- aircraft motion. As the result of them, there is teresis of aerodynamic coefficients (unsteady dangerous of faulty pilot’s actions. Therefore, it aerodynamics), and dynamic stall effect. Aero- is need to investigate aircraft flight phenomena dynamic model includes also a region of higher at high- and very-high angles of attack. angles-of-attack including deep stall phenom- The appearance of a highly augmented ena. In the paper continuation method is used to aircraft required a study of its high angle of at- determine the steady states of the MiG-29 tack dynamics The primary aim of the paper is fighter aircraft as functions of the elevator de- to discuss capabilities of dynamical system the- flections, and bifurcations of these steady states ory methods as the tools for to analyse such are encountered. Bifurcations of the steady phenomena. states are used to predict the onset of wing rock, Dynamical system theory has provided a and “Cobra” manoeuvres. Dynamics of spatial powerful tool for analysis of non-linear phe- motion of a supersonic combat aircraft with a nomena of aircraft behaviour. In the application straked wing is considered for post stall ma- of this theory, numerical continuation methods noeuvres, and observations of chaotic motion in and bifurcation theory have been used to study post stall manoeuvres are guided. roll-coupling instabilities and stall/spin phe- nomena of a number of aircraft models. Results 1 Introduction of great interest have been reported in several An aircraft is the inherently non-linear papers (it can be mentioned papers by Jahnke and time varying system. Non-linear dynamics and Culick [1], Carroll and Mehra [2], is central to several important aircraft motions, Guicheteau [3], or Avanzini and de Matteis [4]). including roll-coupling and stall/spin phenom- Continuation methods are numerical techniques ena. Linearized equations of motion can not be for calculating the steady states of systems of used to analyse these phenomena. Indeed, roll- ordinary differential equations and can be used coupling instabilities were first discovered in to study roll coupling instabilities and high- flight, often with fatal results, because the line- angle of attack instabilities. 3112.1 Krzysztof Sibilski Carroll and Mehra [2] were the first to tional problems have been faced with the mod- use a continuation technique to calculate the elling of the stability and control augmentation steady states of aircraft. They determined the system with the related high number of addi- steady states of a variable sweep aircraft and the tional states and the handling of the so-called F-4 fighter aircraft. By studying the steady breakpoint nonlinearities coming from control states of these two aircraft they explained that system elements. wing rock appears near the stall angle of attack The present paper is continuation of the due a Hopf bifurcation of the trim steady state. previous works of the author [5], [6], [7]. After They also calculated the steady spin modes for a brief description of the methodology and asso- the aircraft and predicted the control surface de- ciated procedures, Cobra maneuver and associ- flections at which the aircraft would undergo ated wing-rock oscillations are studied by stall/spin divergence. They have calculated the means of checking the stability characteristics resulting steady state of the aircraft and have related to unstable equilibria. Numerical simu- developed recovery techniques using their lations are used to verify the predictions. High knowledge of the steady spin modes for the air- angle of attack maneuvers were studied to ob- craft. Guicheteau [3] has used continuation serve chaos phenomenon in post stall motion. methods and bifurcation theory in analysis of Unsteady aerodynamics for prediction of airfoil the non-linear dynamics of aircraft model that loads is included, and the ONERA type stall includes unsteady aerodynamic coefficients. He model is used [8], [9], [10]. has analysed the effects of a lateral offset of the c.g. and the influence of gyroscopic momentum 2 Theoretical background of rotating engine’s masses on spin entry recov- ery. Jahnke and Culick [1] have recalled the 2.1 Dynamical systems theory theoretical background of dynamical system In this paper we will study equations of the fol- theory and bifurcation technique and they have lowing form presented profound review of the relevant in- x= f(,;) x t ì (1) vestigations in this field. They have studied the ! dynamic of the F-14 fighter aircraft by deter- and mining the steady states of the equations of mo- x" g(;) x ì (2) tion and seeking bifurcations. And have shown with x∈U ⊂ Rn,, t ∈ R1 andì ∈ V ⊂ R p , that continuation method is very useful in analy- where U and V are open sets in Rn and Rp, re- sis of the wing rock instability, spiral diver- spectively. We view the variables µ as parame- gence instability and spin dynamics. Avanzini ters. We refer to (1) as a vector field or ordinary and de Matteis [4] have analysed the dynamics differential equation and to (2) as a map or dif- of a relaxed stability aircraft by dynamical sys- ference equation. Both will be termed dynami- tem theory. The principal objective of their cal systems. study was to assess the practical worth of dy- By a solution of Eq. (1) we mean a map, namical system theory in simulations where the x, from some interval I ⊂ R1 into Rn, which we dynamics of aircraft are tailored by the full- represent as follows authority control system according to different x:I,→ R1 mission tasks. They have remarked that the use (3) of manoeuvre demand control in high- t" x() t performance aircraft somehow limits the capa- such that x(t) satisfies (1), i.e., = bility of dynamical system theory to provide x!(t ) f ( x ( t ), t ;ì ) (4) global stability information in as much as tran- Dynamical systems theory (DST) provides a sient motions cannot be predicted or quantified methodology for studying systems of ordinary by bifurcation theory (compare with results ob- differential equations. The most important ideas tained by Carol and Mehra [2]). Avanzini and of DST used in the paper will be introduced in de Matteis [4] have proved, that certain addi- 3112.2 AN AGILE AIRCRAFT NON-LINEAR DYNAMICS BY CONTINUATION METHODS AND BIFURCATION THEORY the following sections. More information on axis when control vector varies. Two cases can DST can be found in the book of Wiggins [11]. be considered [16]. The first step in the DST approach is to • The steady state is regular, i.e. when the im- calculate the steady states of the system and plicit function theorem works and the equilib- their stability. Steady states can be found by rium curve goes through a limit point. It should setting all time derivatives equal to zero and be noted that a limit point is structurally stable solving the resulting set of algebraic equations. under uncertainties of the differential system The Hartman-Grobman theorem proves that the studied. local stability of a steady state can be deter- • The steady state is singular. Several equilib- mined by linearizing the equations of motion rium curves cross a pitchfork bifurcation point, about the steady state and calculating the eigen- and bifurcation point is structurally unstable. values [12], [13], [14], [15]. If a pair of complex eigenvalues cross The implicit function theorem (Ioos and the imaginary axis, when control vector varies, Joseph [16]) proves that the steady states of a Hopf bifurcation appears [13], [14], [16], [17], system are continuous function of the parame- [18]. Hopf bifurcation is another interesting bi- ters of the system at all steady states where the furcation point. After crossing this point, a peri- linearized system is non-singular. Thus, the odic orbit appears. Depending of the nature of steady states of the equations of motion for an nonlinearities, this bifurcation may be sub- aircraft are continuous functions of the control critical or supercritical. In the first case, the sta- surface deflections. Stability changes can occur ble periodic orbit appears (even for large as the parameters of the system are varied in changes of the control vector). In the second such a way that the real parts of one or more case the amplitude of the orbit grows in portion eigenvalues of the linearized system change to the changes of the control vector. sign. Changes in the stability of a steady state lead to qualitatively different responses for the 2.3 Continuation technique and methodology system and are called bifurcations.