Bifurcation Analysis of a Generic Pusher Type Transport Aircraft Configuration in Near-Stall Flight

Transcription

1 Bifurcation Analysis of a Generic Pusher Type Transport Aircraft Configuration in Near-Stall Flight Ait K Khatri*, Vineet Kuar* * CSIR-National Aerospace Laboratories, Bangalore , India Keywords: Bifurcation Analysis, Pusher Type Aircraft, Near-stall flight Abstract Near stall flight of a transport aircraft is anya-ties associated with loss of control probles. A transport aircraft flying inadvertently near stall can rapidly get into upset condition leading to loss of control of aircraft. A systeatic investigation of near stall flight characteristics of a transport aircraft, therefore, becoes crucial for ensuring safe flight operation of the aircraft. This paper deals with the detailed investigation of nonlinear flight dynaics of a generic pusher type T-tail transport aircraft configuration in near stall flight regie with the objective of coputing safe flight envelope of the aircraft. To the best knowledge of the authors, this is the first ever attept to study nonlinear flight dynaics of a pusher type T-tail transport aircraft. 1 Introduction High angle of attack (AOA) aerodynaics of an aircraft is doinated by flow phenoena such as separation, aerodynaic hysteresis effects and vortex breakdown. As a result, high AOA aerodynaics is extreely nonlinear leading to coplex flight otions, like wing rock, autorotation, pitch-bucking, deep-stall, spin, etc. High AOA flight near stall is particularly known to be prone to loss of control (LOC) incidents in transport airplanes. A transport aircraft flying inadvertently near stall can quickly get into upset condition resulting in loss of control. Considering the increased frequency of aircraft accidents due to LOC, NASA has lately initiated ajor research progras for systeatic investigation of high AOA flight dynaics of transport airplanes [1, ]. Kwatny et al. [3] analyzed LOC proble for an aircraft odel using bifurcation analysis and continuation technique. NASA s generic transport odel (GTM), which is representative of a large coercial transport airplane, was taken in their investigation. Bifurcation analysis of the GTM was carried out to deterine control strategies required to safely regulate the aircraft near stall bifurcation point so as to avoid LOC. Gill et al. [4] investigated nonlinear dynaics of the GTM odel in high AOA flight regie. Bifurcation analysis was perfored to identify various attractors that could induce upset condition. It was shown that an inappropriate pilot input can cause the aircraft to enter into an oscillatory spin otion. Chongvisal et al. [5] designed a siple flight control syste for prediction and prevention of LOC in an aircraft. The flight control syste forewarns of any probable pilot input that can result in LOC during flight. Furtherore, the flight control syste provides corrective coand inputs to recover back the aircraft fro upset condition to safe flying operation ode thereby preventing LOC. In this paper, we present a detailed analysis of the nonlinear flight dynaics of a generic pusher type T-tail transport aircraft in near-stall flight regie. The paper is organized as follows. Section provides the aerodynaic data for the pusher type transport aircraft odel. Section 3 gives a brief introduction to bifurcation analysis and continuation technique. Bifurcation analysis results for the transport aircraft odel are presented in section 4. Finally, section 5 concludes the work presented in this paper. 1

2 Ait Kuar Khatri, Vineet Kuar Aerodynaic Data for the Aircraft Model The aircraft odel considered in this work is a low wing design with twin engines ounted on rear fuselage via stub wings in pusher type configuration. Epennage is designed as a T- tail layout with horizontal tail atop the vertical stabilizer. The configuration is provided with three priary controls: elevator, aileron and rudder. The configuration is designed for operation at low subsonic Mach nubers (M<0.5). Extensive wind tunnel tests have been conducted for characterizing aerodynaic behaviour of the aircraft configuration. The aircraft odel has been tested for AOA range of -10 to +0 deg and sideslip angles up to +10 deg. A coprehensive aerodynaic database is prepared fro the experiental results obtained fro wind tunnel tests. Figures 1-4 depict longitudinal and lateral-directional aerodynaic characteristics of the aircraft odel. Figure 1 shows lift coefficient ( C L ) for the aircraft odel; stall AOA for the aircraft can be noted to be around 15 deg. Figure shows the variation in aerodynaic pitching oent coefficient ( C ) at three different sideslip angles. There is a considerable change in pitching oent at 10 deg sideslip angle, though the static longitudinal stability ( C ) of the aircraft odel reains alost unchanged with sideslip. Further, it is observed that the aircraft odel loses static longitudinal stability beyond 17 deg AOA. Rolling oent coefficient ( C ) and yawing oent coefficient ( C n ) variation with sideslip are presented in Figs. 3 and 4, respectively. The configuration retains lateral stability ( ) and l Cl directional stability ( C ) up to AOA 0 deg. n However, the configuration is prone to enter into a roll in post-stall flight regie. Strong nonlinearity in pitching, rolling and yawing oent coefficients can be noticed near stall AOA fro Figs. -4. Figures 5-7 present control effectiveness of elevator, rudder and aileron, respectively. Pitching oent coefficient for three different elevator settings (neutral, axiu nose-up and axiu nose-down) is shown in Fig. 5. Fig. 1. Plot of lift coefficient against AOA. Fig.. Plot of pitching oent coefficient against AOA. Fig. 3. Plot of rolling oent coefficient versus AOA Figure 6 shows rudder effectiveness at various AOA for two distinct rudder deflections (neutral and axiu right). Maxiu rudder effectiveness is seen to slightly reduce with increase in AOA. Variation in aileron effectiveness with AOA is presented in Fig. 7. A significant reduction in aileron effectiveness

3 BIFURCATION ANALYSIS OF A GENERIC PUSHER TYPE TRANSPORT AIRCRAFT CONFIGURATION IN NEAR-STALL FLIGHT near stall is clearly apparent fro Fig. 7 at large aileron deflections. Fig. 4. Plot of yawing oent coefficient versus AOA. Fig. 7. Plot of rolling oent coefficient versus AOA at different aileron settings ( _ ). a a_ left a right 3 Bifurcation Analysis and Continuation Technique Bifurcation analysis and continuation technique is a ethodology for studying global dynaics of a nonlinear syste. Bifurcation analysis of a nonlinear syste involves studying changes in plant dynaics as one or ore of the syste paraeters are varied within specified liits. Consider an autonoous nonlinear dynaical syste: Fig. 5. Plot of pitching oent coefficient versus AOA at different elevator settings. x f( x, U) (1) n where x is the vector of state variables and vector U represents the syste paraeters. The vector function f( x, U ) is n n sooth and defines a apping:. Bifurcation analysis of a syste starts with coputing all possible steady state solutions of Eq. (1) for various values of syste paraeters. In the standard bifurcation analysis (SBA) procedure, only one of the syste paraeters is varied at a tie keeping the reaining paraeters as constant. Thus, we solve: Fig. 6. Plot of yawing oent coefficient versus AOA at different rudder settings. x f( x, u, p) () where u is the continuation paraeter and vector p refers to fixed paraeters of the syste. 3

4 Ait Kuar Khatri, Vineet Kuar Steady state solutions ( x 0) of a nonlinear syste can either be equilibriu point or periodic solution. Again, bifurcation in a syste indicates a change in either the nuber of possible steady state solutions or their stability. Bifurcation of an equilibriu point can be static or dynaic. A static bifurcation occurs when a real eigenvalue of the syste crosses the iaginary axis. A dynaic or Hopf bifurcation results due to oveent of a coplex conjugate pair of eigenvalues across the iaginary axis. Hopf bifurcation in a syste gives rise to liit cycles, which are isolated closed orbits. Bifurcation of periodic solution is related to the oveent of Floquet ultiplier across the iaginary axis. Detailed account of bifurcation theory is available in various references [6, 7]. Bifurcation analysis of a nonlinear syste is perfored with the help of a nuerical continuation algorith. Results fro bifurcation analysis are presented in the for of bifurcation diagras. Various sybols on a bifurcation diagra denote the following: (solid line) Stable tri point (dotted line) Unstable tri point Hopf bifurcation point In this work, AUTO 000 [8] bifurcation and continuation algorith is used to carry out the bifurcation analysis. The algorith coputes different steady state solution branches with variation in continuation paraeter. Additionally, eigenvalues of the locally linearized syste are coputed that define stability of the steady state solutions. Different types of bifurcations of equilibriu points and periodic solutions are also indicated by AUTO 000. Hence, a global dynaical picture of the nonlinear aircraft odel can be obtained through AUTO Extended Bifurcation Analysis Method In the SBA ethod, only one of the syste paraeters is varied at a tie. This akes the SBA ethod unsuitable for analyzing constrained nonlinear systes where several control paraeters need to be siultaneously varied to satisfy the desired constraints. The extended bifurcation analysis (EBA) [9] ethod is a technique for bifurcation analysis of constrained nonlinear syste. The EBA ethod consists of two steps. In step 1 of the EBA ethod, one solves: x f( x, u, p) (3) g(x) 0 (4) n where x is the state vector, scalar u is the continuation paraeter, and vector p represents the control paraeters. Vector k function g(x) refers to the syste constraints. To solve the augented syste (3-4), a nuber of control paraeters (equal to k) fro vector p are freed and the reaining paraeters are held fixed. It is therefore iperative that the nuber of control paraeters in p should at least be equal to the nuber of prescribed constraints (i.e. k). Further, the free control effectors ust influence the specified constraints. Solution of step 1 of the EBA ethod provides steady state solutions satisfying the constraints (4). However, stability of the coputed stability solutions is incorrect because constraints are explicitly present in this step. In step of the EBA ethod, one solves: x f ( x, u, p ( u), p ) (5) 1 where p ( u) are paraeter schedules for free 1 paraeters as obtained fro step 1, and p are fixed control paraeters. Output of step gives steady state solutions satisfying the desired constraints (4), along with correct stability inforation. Besides, bifurcation or departure, if any, fro constrained flight conditions are also indicated by this step. 4 Bifurcation Analysis of the Transport Aircraft Model Application of bifurcation analysis technique to aircraft flight dynaics was first deonstrated by Carroll and Mehra [10]. They showed that various coplex high AOA flight dynaical 4

5 BIFURCATION ANALYSIS OF A GENERIC PUSHER TYPE TRANSPORT AIRCRAFT CONFIGURATION IN NEAR-STALL FLIGHT phenoena exhibited by aircraft are essentially anifestations of different types of bifurcations. Later, Zagyanov and Goan [11] applied bifurcation analysis technique for studying aircraft spin. Over the last three decades bifurcation analysis technique has been widely applied for investigations of high AOA flight dynaics of different aircraft [1, 13]. In this section, we eploy the SBA and the EBA ethod to exaine flight dynaics of the pusher type transport aircraft odel in different flight conditions. A six degree-offreedo nonlinear flight dynaic odel of the aircraft is used for bifurcation analysis. The flight dynaic odel can be represented by a set of nonlinear ordinary differential equations as given in the Appendix. The six aerodynaic coefficients (,,,,, L D Y l n C C C C C C ) appearing in the flight dynaic odel are coputed as below: C C ( ) C ( ) C L L L e L q qc C C ( ) C ( ) C ( ) D D D D e C ( ) C ( ) C D a D r Dq qc C C ( ) C ( ) C ( ) e C q qc C C ( ) C ( ) C ( ) Y Y Y Y a pb rb CY ( r ) CY C p Yr C C ( ) C ( ) C ( ) l l l l a pb rb Cl ( r ) Cl C p lr C C ( ) C ( ) C ( ) n n n n a pb rb Cn ( r ) Cn C p nr (6) (7) (8) (9) (10) (11) where V is the velocity of aircraft;, are AOA and sideslip angles, respectively; p, q, r are roll, pitch and yaw rates, respectively; e, a, r refer to elevator, aileron and rudder deflections, respectively. Sybols b and c are aircraft span and ean aerodynaic chord, respectively. Sybol C ( j) represents the increental change in paraeter C i due to variation in state variable j. In this section, we perfor constrained bifurcation analysis of the aircraft in straight and level syetric flight condition. Standard bifurcation analysis of the aircraft near stall is later carried out to exaine aircraft behavior near stall. Results for the two cases are presented next. 4.1 Bifurcation Analysis in Level Flight Condition To perfor bifurcation analysis of the aircraft in level flight condition, the EBA ethod is used to solve the following constrained nonlinear syste: i x f( x, u, p) (1) 0, 0, 0 (13) where x [ V,,, p, q, r,, ] T is the state vector;, being the aircraft roll and pitch angles, respectively. Scalar is the flight path angle, u [ e ] is the continuation paraeter, and p [, a, r] is the vector of syste paraeters ( being the noralized engine thrust). Engine thrust paraeter is defined as: Actual engine thrust Maxiu static engine thrust To solve for the augented syste (1-13), all three paraeters in vector p are freed. Next, the second step of the EBA ethod is solved. Thus, 5

6 Ait Kuar Khatri, Vineet Kuar the following syste of nonlinear equations are solved x f ( x, u, p ( u), p ) (14) 1 where p 1( u) [ ( u), a( u), r( u)] is the vector of free paraeters. There are no fixed paraeters in this case. Bifurcation analysis results for the aircraft are shown in Figs Figures 8-10 show schedules for the free paraeters as obtained fro the first step of the EBA ethod. It can be noted fro Figs. 8-9 that right aileron and right rudder inputs are required to aintain level flight condition near stall. This happens because the aircraft showed a tendency to enter into a left roll at high AOA near stall (refer Fig. 3). Engine thrust is shown in Fig. 10; iniu value of required thrust is noted at an AOA of 7 deg. Bifurcation plots for level flight constraints (13) are displayed in Figs It can be noticed fro these figures that sideslip, bank angle and flight path angle are restricted to zero throughout the flight envelope considered. Figure 14 shows the bifurcation diagra of AOA against elevator deflection. All the tri points of the aircraft can be noted to be unstable; a Hopf bifurcation point near stall AOA (close to 15 deg.) can be clearly noticed fro the figure. Bifurcation diagra of Mach nuber is shown in Fig. 15. Again, it is evident fro eigenvalue plot of Fig. 16 that the instability in level flight condition arises due to divergence of spiral ode; the divergence tendency increases sharply beyond an AOA of 10 deg. Figure 17 shows elevator deflection needed to tri the aircraft at different values of lift coefficient in level flight condition. Significant increase in required elevator input for triing the aircraft close to stall can be observed fro Fig. 17. Fig.9. Plot of rudder schedules for level flight. Fig.10. Variation of thrust required at different angle of attack in level flight. Fig.8. Plot of aileron schedules for level flight. 6

7 BIFURCATION ANALYSIS OF A GENERIC PUSHER TYPE TRANSPORT AIRCRAFT CONFIGURATION IN NEAR-STALL FLIGHT Fig.11. Bifurcation diagra of sideslip against elevator deflection for level flight. Fig.14. Bifurcation diagra of angle of attack against elevator deflection for level flight. Fig.1. Bifurcation diagra of bank angle against elevator deflection for level flight. Fig.15. Bifurcation diagra of Mach nuber against elevator deflection for level flight. Fig.13. Bifurcation plot of flight path angle against elevator deflection for level flight. Fig. 16. Variation of spiral ode eigenvalue with angle of attack. 7

8 Ait Kuar Khatri, Vineet Kuar Fig.17. Plot of elevator against lift coefficient (C L ) in level flight condition. 4. Bifurcation Analysis of the Aircraft near Stall-α (15) are provided in Figs Figure 18 shows bifurcation plot of AOA versus continuation paraeter e. The axiu attainable AOA is 14.6 deg at e 7 deg. No steady state solutions exist beyond an elevator deflection of -7 deg in this case. A saddle node bifurcation occurs at AOA 14.6 deg. All the steady state solutions on upper branch are unstable; a sall region of stable solutions is seen on lower branch between AOA range of 1-13 deg. The stable solution branch exists for 3 e 4 deg. Three Hopf bifurcation points are visible at AOA of 1 deg, 13 deg and 14.6 deg. Bifurcation diagras of sideslip and bank angle are shown in Figs. 19 and 0, respectively. Bank angle on the stable branch is Stall of an aircraft is a andatory flight test to be done for showing copliance with the certification requireents. As the aircraft aerodynaics is highly nonlinear near stall, a thorough study of the aircraft behavior close to stall is needed to investigate any tendency of the aircraft to depart into uncontrolled flight otion. In this section, we analyse the aircraft dynaics near stall to assess aircraft response while approaching stall. To perfor nonlinear analysis of the aircraft near stall, starting point for bifurcation analysis procedure is taken as level flight tri point at AOA 13 deg. The corresponding state and control vectors for this initial point are available fro bifurcation analysis results of the previous subsection 4.1. Standard bifurcation analysis procedure is carried out to solve the following syste of equations: Fig.18. Bifurcation diagra of angle of attack against elevator deflection. x f( x, u, p) (15) where x [ V,,, p, q, r,, ] T is the state vector; u [ e ] is the continuation paraeter; and p [, a, r] is the vector of fixed syste paraeters. Bifurcation analysis results obtained fro standard bifurcation analysis of the syste Fig.19. Bifurcation diagra of sideslip. 8

9 BIFURCATION ANALYSIS OF A GENERIC PUSHER TYPE TRANSPORT AIRCRAFT CONFIGURATION IN NEAR-STALL FLIGHT quite large (~70 deg). Bifurcation plots for roll rate, pitch rate and yaw rate are provided in Figs. 1, and 3, respectively. It can be concluded fro the bifurcation analysis results that the aircraft has a tendency to enter into a right roll during stall. Because all the solutions near stall are unstable, a significant aount of effort will be required on the part of pilot to prevent spiral divergence while approaching stall. Fig.. Bifurcation plot of aircraft pitch rate. Fig.0. Bifurcation diagra of bank angle. Fig.3. Bifurcation plot of aircraft yaw rate. 5 Conclusions Fig.1. Bifurcation plot of aircraft roll rate. Nonlinear analysis of a pusher type T-tail transport aircraft configuration has been presented in this paper. Bifurcation analysis and continuation ethodology is adopted for nonlinear flight dynaics analysis of the aircraft. Bifurcation analysis results for the aircraft in level flight condition reveal that the aircraft is unstable in spiral ode throughout its flight envelope. Further, the aircraft is likely to enter into a roll on approaching stall. Nonlinear dynaic analysis for the aircraft discussed in this paper has however been liited to studying the effect of variation in elevator control. Bifurcation analysis with respect to aileron and rudder controls needs to be further carried out to 9

10 Ait Kuar Khatri, Vineet Kuar fully characterize the aircraft dynaics near stall. This fors a part of future work and is currently under investigation. Appendix: Equations of otion 1 1 V T cos cos V SC g sin D 1 q [( p cos r sin )sin cos 1 1 { T sin V SC V g cos cos }] 1 [ T cos sin V 1 V SC g sin cos ] ( p sin r cos ) Y Iy Iz 1 p qr V SbCl I I x Iz Ix 1 q pr V ScC I I y I I x y 1 r pq V Sb Cn I I z z p q sin tan r cos tan qcos rsin qsin rcos cos x y Wind axis Euler angles ( ), and body axis Euler angles ( ), are related by the following kineatic relations: sin cos cos sin sin sin cos sin cos cos cos sin cos cos sin sin cos sin cos sin sin cos cos cos cos sin sin cos cos cos L References [1] Foster, J. V., Cunningha, K., Freaux, C. M., Shah, G. H., Stewart, E. C., Rivers, R. A., Wilborn, J. E., and Gato, W., Dynaics Modeling and Siulation of Large Transport Airplanes in Upset Conditions, Proc. AIAA Guidance, Navigation, and Control Conference, California, AIAA paper , 005. [] Belcastro, C. M., Newan, R. L., Crider, D. A., Groff, L., Foster, J. V., and Klyde, D. H., Preliinary Analysis of Aircraft Loss of Control Accidents: Worst Case Precursor Cobinations and Teporal Sequencing, AIAA Guidance, Navigation, and Control Conference, Maryland, AIAA paper , 014. [3] Kwatny, H. G., Dongo, J. T., Chang, B., Bajpai, G., Yasar, M., and Belcastro, C., Nonlinear analysis of aircraft loss of control, Journal of Guidance Control and Dynaics, Vol. 36, No. 1, pp , 013. [4] Gill, S. J., Lowenberg, M. H., Neild, S. A., Krauskopf, B., Puyou, G., and Coetzee, E., Upset Dynaics of an Airliner Model: A Nonlinear bifurcation Analysis, Journal of Aircraft, Vol. 50, No.6, pp , 013. [5] Chongvisal, J., Nikolas, T., Xargay, E., Talleur, D. A., Kirlik, A., and Hovakiyan, N., Loss-of-Control Prediction and Prevention for NASA s Transport Class Model, Proc. AIAA Guidance, Navigation, and Control Conference, Maryland, AIAA paper , 014. [6] Strogatz, S., Nonlinear Dynaics and Chaos, Addison Wesley Longan, Reading, MA, [7] Kuznetsov, Y. A., Eleents of Applied Bifurcation Theory, Springer, New York, 004. [8] Doedel, E. J., Paffenroth, R. C., Chapneys, A. R., Fairgrieve, T. F., Kuznetsov, Y. A., Oldean, B. E., Sandstede, B., and Wang, X., AUTO000: Continuation and Bifurcation Software for Ordinary Differential Equations (with HoCont), California Inst. of Technology, Technical Rept., Pasadena, 001. [9] Ananthkrishnan, N., and Sinha, N. K., Level Flight Tri and Stability Analysis using Extended Bifurcation and Continuation Procedure, Journal of Guidance, Control, and Dynaics, Vol. 4, No. 6, 001, pp [10] Carroll, J. V., and Mehra, R. K., Bifurcation Analysis of Nonlinear Aircraft Dynaics, Journal of Guidance, Control, and Dynaics, Vol. 5, No. 5, pp , 198. [11] Zagaynov, G. I., and Goan, M. G., Bifurcation analysis of critical aircraft flight regies, ICAS paper , [1] Goan, M. G., Zagaynov, G. I., and Khratsovsky, A. V., Application of Bifurcation Methods to Nonlinear Flight Dynaics Probles, Progress in 10

11 BIFURCATION ANALYSIS OF A GENERIC PUSHER TYPE TRANSPORT AIRCRAFT CONFIGURATION IN NEAR-STALL FLIGHT Aerospace Sciences, Vol. 33, No. 9 10, pp , [13] Paranjape, A., Sinha, N. K., and Ananthkrishnan, N., Use of Bifurcation and Continuation Methods for Aircraft Tri and Stability Analysis- A State-Of- The-Art, Journal of Aerospace Sciences and Technologies, Vol. 60, No., 008. Contact Author Eail Address Mail to: khatri@ccadd.cacs.ernet.in, kvineet@ccadd.cacs.ernet.in Copyright Stateent The authors confir that they, and/or their copany or organization, hold copyright on all of the original aterial included in this paper. The authors also confir that they have obtained perission, fro the copyright holder of any third party aterial included in this paper, to publish it as part of their paper. The authors confir that they give perission, or have obtained perission fro the copyright holder of this paper, for the publication and distribution of this paper as part of the ICAS 016 proceedings or as individual off-prints fro the proceedings. 11