M.G.Goman, A.V.Khramtsovsky (2008) - Computational framework for investigation of aircraft nonlinear...

Computational framework for investigation of aircraftnonlinear dynamics

M.G. Goman *, A.V. Khramtsovsky

De Montfort University, Leicester, England LE1 9BH, United Kingdom

Received 15 October 2006; received in revised form 8 February 2007; accepted 12 February 2007Available online 24 May 2007

Abstract

A computational framework based on qualitative theory, parameter continuation and bifurcation analysis is outlined and illustratedby a number of examples for inertia coupled roll maneuvers. The focus is on the accumulation of computed results in a special databaseand its incorporation into the investigation process. Ways to automate the investigation of aircraft nonlinear dynamics are considered.� 2007 Published by Elsevier Ltd.

Keywords: Nonlinear aircraft dynamics; Qualitative theory; Continuation method; Bifurcation analysis; Object oriented model; Database; Roll couplingproblem

1. Introduction

The increasing requirements for aircraft performanceand maneuverability and the demand for more reliable pro-vision of flight safety have led to the implementation ofmore complex and adequate nonlinear models for modernaircraft. These models in their turn require an efficient setof computational tools for aircraft dynamics simulationand nonlinear analysis, for clearance of flight control lawswith account of inherent model nonlinearities and controlconstraints [1].

An application of qualitative theory and bifurcationanalysis methods for investigation of nonlinear aircraftdynamics is now a well established approach. Variouskinds of aircraft dynamic instabilities and loss of controlhave been effectively investigated using continuation tech-nique and bifurcation methods. The critical flight regimessuch as high incidence departures, wing rock behavior, spinand jump-like instabilities during intensive roll-coupledmaneuvers have been effectively investigated within a uni-fied framework.

The advanced computational methods in flight dynam-ics were first introduced in [2,3]. Several further contribu-tions presented in [4 12] proved the efficiency of theproposed methods in a number of applications for anopen-loop and closed-loop aircraft dynamics. Global sta-bility and bifurcation analysis methods, continuation andeigenstructure assignment techniques have been success-fully applied to several control law design problems in[13,14]. Bifurcation diagrams for high incidence flightturned out to be a useful tool for planning of piloted sim-ulations [15,16]. A review of these research results can befound in [17,18].

Recently the bifurcation analysis method has beenextended for evaluation of aircraft trim at straight-and-level flight and at level turn by adding to the problem aset of kinematical constraint equations [19,20]. This exten-sion generalizes aircraft performance evaluation using sta-bility and controllability analysis typical for one-parametercontinuation techniques. A review of the latest results inthe application of bifurcation and continuation methodsto flight dynamics problems is given in [21].

Although considerable progress in this area has beenachieved during the last two decades, there still remains asignificant gap between the academic research and theacceptance of the new methods in aeronautical engineering

0965 9978/$ see front matter � 2007 Published by Elsevier Ltd.doi:10.1016/j.advengsoft.2007.02.004

* Corresponding author.E mail address: [email protected] (M.G. Goman).

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Advances in Engineering Software 39 (2008) 167 177

practice. The main reason for that is the lack of efficientand reliable software tools which can be adopted by userswho are not experts in application of qualitative and bifur-cation analysis methods [18].

Qualitative analysis of aircraft nonlinear dynamicsincorporates a number of diverse computationally-inten-sive procedures, which generate different types of data forfurther consideration in planning of numerical simulations,in computation of regions of attractions and in construc-tion of phase portrait representation. This paper describesthe authors’ experience in developing a computationalframework for the investigation of aircraft nonlineardynamics. The framework is based on qualitative theory,a continuation method and bifurcation analysis techniques.The set of software tools outlined in this paper is an exten-sion and further development of the KRIT package [22] ina Matlab� environment. Some results for this computa-tional framework have been presented earlier in [23].

There was no intention in this paper to provide a userguide; it is mostly focused on some methodological aspectsof the developed computational framework. A brief outlineof the aircraft motion equations along with the elements ofqualitative theory adopted in flight dynamics are presentedin Section 2. This section also includes a description ofinteractive capabilities for aircraft dynamics investigationusing coherent parameter continuation and phase portraitconstruction. The object-oriented model of the autono-mous nonlinear dynamic system used as a phase portraitdatabase structure is outlined in Section 3. The continua-tion database features and outline of some new classes ofalgorithms are presented in Section 4. And finally a fewcomputational examples for the inertia roll-coupling prob-lem are given in Section 5.

2. Goals of the qualitative investigation

The motion equations suitable for bifurcation and qual-itative methods are represented in the form of an autono-mous nonlinear system of first order differential equations(the explicit form is given in the Appendix):

dx

dt¼ Fðx; dÞ; x 2 X � Rn; d 2 U � Rm ð1Þ

where the state vector includes velocity vector parameters,rigid body angular rates, the pitch and bank anglesx (V,a,b,p,q, r,h,/)T, the control vector includeselevator, aileron, rudder and thrust throttle deflectionsd (de,da,dr,dT)T. This system of equations is normallyused for aircraft performance evaluation and investigationof spin dynamics. For maneuverability evaluation whenaircraft perform an intensive velocity vector roll maneuvera simplified system (1) with the reduced-order state vectorx (a,b,p,q, r)T and control vector d (de,da,dr)

T can beconsidered [31,33,34].

All equilibrium states of system (1) are defined by solu-tions of the following system of nonlinear algebraicequations:

Fðx; dÞ ¼ 0; x 2 X � Rn; d 2 U � Rm ð2Þ

The eigenvalues and eigenvectors of the Jacoby matrixJ ¼ oF

ox, calculated at equilibrium point under consideration,

are used for evaluation of local stability and reconstructionof invariant manifolds of trajectories associated withequilibrium.

Any equilibrium solution of system (1) with constant-control input d corresponds to steady coordinated flightalong a helical trajectory with a vertical axis. Note thatstraight-and-level flight and the level turn belong to thisclass of equilibria. An aircraft in steady spin also followsa helical trajectory with small, comparable with the wing-span (R/l � 1), turn radius, with high angle of attacka � 40 80� and practically vertical flight path anglec � �p/2. The reduced 5th-order system is used for investi-gation of the roll-coupling problem. The equilibria of thissimplified system of motion equations are considered asthe pseudosteady state (PSS) solutions in comparison withthe full system equilibria, because the effect of the gravita-tional force is ignored and speed is assumed to be constant[31,34].

Different time-varying control inputs (1) allow one tosimulate a variety of aircraft maneuvers (note that in thiscase system (1) should be extended by equations for posi-tion in space X, Y, Z and yaw angle w). For illustrationpurposes Fig. 1 shows the visualization of a simulated flighttrajectory with aircraft departure at high incidence, entryinto developed spin and further successful recovery fromspin. Fig. 2 shows the visualization of a flight trajectorywhich includes intensive velocity roll.

Using only direct numerical simulation the identificationof maneuver limits due to onset of instability or loss of con-trol is an extremely difficult and time-expensive task. Thequalitative theory, continuation and bifurcation analysis

Fig. 1. High incidence departure, steady spin and spin recovery.

168 M.G. Goman, A.V. Khramtsovsky / Advances in Engineering Software 39 (2008) 167 177

methods facilitate this task providing estimates for criticalcontrol inputs and dangerous maneuver boundaries. Nor-mally these estimates are further verified in numericaland piloted simulation using the full mathematical model.

The basic principals of the qualitative theory and bifur-cation analysis methods can be found, for example, in [2426]. An introduction to continuation methods is given in[27]. In this paper these methods and techniques are dis-cussed in the context of their application to flight dynamicsproblems.

At constant-control input the state space of system (1)incorporates all the trajectories representing aircraftdynamics for different initial conditions. The topology ofthe system state space, or the phase portrait, is defined bya set of its special trajectories, namely equilibrium points,closed orbits, toroidal manifolds and chaotic attractors.The investigation in flight dynamics is mostly focused onaircraft equilibria or trim conditions and on closed orbitsrepresenting aircraft limit cycle oscillations (LCO).

A special interest is given to bifurcational parameterswhen the phase portrait experiences qualitative transfor-mations like changes in a number of equilibrium points,onset or disappearance of a closed orbit. Local bifurcationscan be analyzed entirely via migrations of equilibriumeigenvalues and periodic orbit multipliers across the imag-inary axis or the unit circle, respectively.

The most important bifurcations arising in flightdynamics applications are related to the limit point or sad-dle-node bifurcation (Fig. 3), signifying aircraft jump-likedeparture, and with the Hopf bifurcation or oscillatoryinstability, associated, for example, with the onset of wingrock at high incidence flight or flutter in aeroelastic systems(Fig. 4).

Aircraft critical regimes such as spin, autorotation inroll or wing rock oscillations can coexist with the normalflight conditions at the same control inputs. Entry to one

of these critical regimes depend on prehistory of controlactions and also on external disturbances. This multiattrac-tor nature of aircraft dynamics requires one to performanalysis of regions of attraction (or domains of attraction)in addition to local stability for each stable regime ofmotion. For evaluation of regions of attraction in the pre-sented framework the method outlined in [11,13] is imple-mented. This method automates the computation andvisualization of two-dimensional cross-sections of the mul-tidimensional region of attraction.

Two closely interconnected tasks of qualitative analysisof aircraft nonlinear dynamics consist of:

• Identification of all qualitatively different types of air-craft nonlinear behavior and specification of separating

Fig. 2. Intensive velocity vector roll maneuver.

Fig. 3. Equilibrium limit point or saddle node bifurcation.

Fig. 4. Hopf bifurcation for equilibrium point, onset of limit cycle.

M.G. Goman, A.V. Khramtsovsky / Advances in Engineering Software 39 (2008) 167 177 169

surfaces between corresponding regions in the parame-ter space. Analysis of their sensitivity to variation of sys-tem parameters is the important part of such aninvestigation. This allows the researcher to determinethe safe boundaries for normal flight conditions and toisolate critical flight regimes.

• Investigation of aircraft dynamics for each mode ofbehavior considering the phase portrait representationat different constant-control inputs and evaluation ofregions of attraction for every important stable solution.

The continuation procedure is the main computationaltool for performing numerical bifurcation analysis [2729,22]. It is used to study the dependence of steady-statesolutions on variation of system parameters. The main fea-ture of this algorithm is the introduction of a general arc-length for the solution curve in the extended state space,which is the cartesian product of the state space and one-parameter axis (i.e. the selected bifurcation parameter).The predictor-corrector algorithm is implemented for com-putation of this solution curve.

In continuation of aircraft equilibria the saddle-nodebifurcation points may have a form of sharp kinks, becausefunctions in aircraft aerodynamic models are normallymade by the linear interpolation of tabulated experimentaldata. The continuation algorithm in this case needs specialadjustment to guarantee a reliable passing through thesecritical points [22].

For computation of the closed orbits different versionsof mapping technique are used. The presented computa-tional framework supports the Poincare mapping and thetime-advance mappings with fixed and free periods. Vari-ous checks are performed during continuation of equilib-

rium and closed orbit solutions to allow localization andclassification of the encountered bifurcation points. Specialprocedures automatically locate, or at least try to locate, allthe branches emanating from the bifurcation and branch-ing points, in order to capture the complete set of solutions.

There is a number of interactive user interfaces for man-aging the investigation process using a broad spectrum ofcomputational procedures based on qualitative methods.They include the tools for storage and retrieval of com-puted data and also special algorithms guiding the investi-gation by processing already accumulated data.

Figs. 5 and 6 show the continuation and the phase por-trait GUI’s (Graphical User Interfaces), respectively. Theyboth are used for managing the process of qualitative andbifurcation analysis. The continuation GUI includes threewindows where the continuation branches in different pro-jections with local stability information can be displayed. Itis possible to save the computed data in a database and toreload previously saved data. There is also a number oftools and control panels facilitating computational investi-gation, in particular, for initiation of continuation of equi-libria and closed orbits, for numerical simulation, forcalculation of two-dimensional cross-sections of stabilityregion, for opening a new or existing continuation or phaseportrait database, for report generation.

The phase portrait can be composed using the phaseportrait GUI from the steady-state solutions available inthe continuation database. This set of solutions can be sup-plemented by a number of trajectories from stable andunstable invariant manifolds associated with the equilib-rium point or the closed orbit. All this may be added bya number of regular trajectories specified for computationin the simulation panel.

Fig. 5. Control panel for continuation and bifurcation analysis.


3. Object-oriented model of the autonomous nonlinear

dynamic system

Qualitatively different trajectories of dynamical system(1) can be considered as objects grouped in a number ofclasses. The object-oriented approach to system modelingprovides a useful basis for development of a database struc-ture for storing computational results from nonlineardynamic analysis. An attempt to model the autonomousnonlinear dynamical system using the class diagramadopted from the Unified Modeling Language (UML)[30] is presented in this section.

The system is modelled using three classes of objects.The first class includes invariant manifolds of steady trajec-tories (IMST). It has four children classes, namely, the

class of equilibrium points with dimension n 0, the classof closed orbits with dimension n 1, the class of toroidalinvariant manifolds with dimension n 2,3,N � 1, whereN is the order of the dynamical system, the class of strangeattractors with fractional dimension p (Fig. 7).

The second class includes invariant manifolds of tran-sient trajectories (IMTT) approaching or departing fromthe invariant manifolds of steady trajectories (IMST).For example, any invariant manifold of steady trajectories(i.e. equilibrium, closed orbit, etc.) is associated with a sta-ble invariant manifold of transient trajectories approachingit and an unstable invariant manifold of transient trajecto-ries departing from it. If IMST is stable IMTT has a fulldimension of the state space and coincides with the IMSTregion of attraction (RA). Dimensions of a stable IMTT

Fig. 6. Control panel for phase portrait investigation.

Fig. 7. Object oriented model of the autonomous nonlinear dynamical system (1): the UML class diagram.


and an unstable IMTT coincide respectively with the num-ber of stable and unstable eigenvalues for an equilibriumand the number of stable and unstable multipliers for aclosed orbit. A typical example is a separatrix surfaceformed by a stable manifold of trajectories approachingan aperiodically-unstable equilibrium or a closed orbit.

The third class includes all regular trajectories (RT),which are different from the first and the second class ofobjects. They do not have association with IMST andIMTT objects, but can approach them with any requiredaccuracy.

The attributes of the objects from the first class include astate space location, eigenvalues and eigenvectors of theJacoby matrix. The attributes of a stable object from thefirst class can include estimates for its region of attraction.There are two main operations with the objects. The firstoperation is the time advance mapping along the trajectoryi.e. numerical integration of the dynamical system on sometime interval, and the second operation is continuation ofthe object due to variation of some system parameter.

4. Database for computational results and data processing

algorithms

Keeping and reusing the already computed results cansignificantly enhance the continuation algorithm’s perfor-mance. The novelty of this approach is in the tight integra-tion between the database and the continuation algorithm.For example, at each continuation step a newly computedsolution is checked against the database to determinewhether such a solution has been already computed, orwhether some other steady-state solution is approachingthe current branch. Such a feature resulted in significanttime savings, especially in the presence of complex branch-ing points or during continuation of periodic solutions.

The existing databases can be used as a source of initialinformation for a number of algorithms. For example, aone-parameter continuation database can provide limitpoints for computation of two-parameter bifurcationdiagram. Another example, all steady-state solutions atconstant-control input can be extracted from the continua-tion database and used for phase portrait representationand if some new attractors are found using the phase por-trait investigation they can be returned back to the contin-uation database.

A database structure for the one-parameter continua-tion algorithm is the most thoroughly tested one. It con-tains information about various kinds of bifurcationpoints and also about error points, where computationfailed to converge to a steady solution, etc. Descriptiveinformation is given for each computed and stored contin-uation branch with an outline of relationships between dif-ferent branches.

The phase portrait investigation requires special compu-tational activities. Collecting data for phase portrait repre-sentation has an iterative and incremental nature and themain goal was to automate this process. A special database

structure was developed for accumulating results of com-putational analysis (see Section 3) and a graphical userinterface was created for managing the computationalprocess.

The phase portrait GUI, shown in Fig. 6, unifies a num-ber of different algorithms while the database is used fordata exchange and storage. To create a particular phaseportrait view, the following data are usually computed:

• steady-state solutions and their stability (the data maybe extracted from an existing continuation database oralternatively a number of various methods are providedin the phase portrait GUI for computation of steady-state solutions);

• special trajectories, incoming to or outgoing from asteady-state solution along eigenvectors, correspondingto stable and unstable eigenvalues, respectively;

• numerical simulation of system’s dynamics at fixed andtime-varying control inputs providing time histories andvisualization of aircraft spatial trajectory;

• two-dimensional cross-sections of the regions of attrac-tion for all stable steady-state solutions.

There is a number of auxiliary computational tools suchas random search for equilibrium and periodic solutions,procedure for automatic specification of the system’s tran-sition after meeting the saddle-node bifurcation point, etc.A procedure for search of points on a surface separatingtwo stable attractors allows finding additional unstableequilibrium and periodic solutions.

A low-level compatibility between the phase portraitand the continuation databases allows a quick generationof the phase portrait representation from data alreadyobtained by continuation. Conversely, the data collectedin the phase portrait database can be exported back tothe continuation database allowing continuation of newbranches of solutions.

5. Computational examples

A classical aircraft nonlinear problem for inertia-cou-pling roll maneuvers outlined in [31 34,21] will be consid-ered in this section to present the computationalframework in action. The 5th order mathematical modelfor a hypothetical swept-wing fighter at high altitude super-sonic flight conditions presented and analyzed in [17] (pp.553 559) is used for computational analysis. There is a rep-lication of some results from [17] and also some new formssuch as phase portrait views, two-parameter continuationdiagrams and root-loci for separate continuation branchesare given and accompanied by comments on reliability andconvergence of the algorithms employed, an indication ofcomputation time and of data storage requirements.

The aircraft dynamics during intensive rotation in roll isaffected by inertia coupling between the longitudinal andlateral motions leading to a multitude of nonlinearphenomena such as bifurcation-induced departures,


multiattractor behavior, apparent loss of controllability,hysteresis-type responses to control inputs, etc. All thesephenomena can be effectively investigated using the pre-sented computational framework with a variety of toolsfor analysis, data storage and interactivity.

The continuation of equilibrium solutions is performedas a single procedure, which automatically generates setsof initial solutions for several values of the continuationparameter and after that sequentially carries out continua-tion from these starting points. At every selected value ofthe continuation parameter there may be a number of start-ing points, which generate different branches of solutions.All new branches are stored in a continuation database.Special protection against recontinuation of the samebranch is made by means of synchronous comparison ofthe current point with the data already stored in a data-base. The final set of equilibrium branches computed inan extended state space and stored in a database can bevisualized in different projections. To give informationabout equilibrium eigenvalues the branches are drawnusing different line types, for example, a solid line meansthat all eigenvalues are stable, a dashed line means thatthere is one positive real eigenvalue, a dashed-dotted linecorresponds to one complex pair with positive real part,etc. All bifurcation points (i.e. the limit and the Hopf bifur-cation ones) are specially marked using information storedin a database.

The obtained branches of equilibrium solutions in twoprojections for angle of attack vs aileron deflection andfor roll rate vs aileron deflection are presented in Fig. 8.There is one branch of equilibrium solutions correspondingto normal flight conditions with small angles of attack.Two other branches correspond to aircraft autorotationregimes, which even exist at zero aileron deflection. Attwo limit points on these branches at da 5.5� andda �7.5�, respectively for negative and positive rotation,transitions take place to the normal branch leading todecrease in intensity of rotation (see arrows in Fig. 8). Afterreturn to the normal branch the roll rate returns to zerowhen aileron input is removed.

At the next step the user can initiate the continuation oflimit cycle oscillations (LCO) or closed orbit solutionsstarting from the already computed Hopf bifurcationpoints. The two computed families of periodical solutionsare shown in Fig. 8 for positive values of aileron deflection.The amplitudes of limit cycles are superimposed on theequilibrium branches (note that similar families of period-ical solutions can be computed for negative aileron deflec-tions). The limit cycle oscillations on the main equilibriumbranch become unstable after meeting the secondary Hopfbifurcation point at da 18�. Further increase of aileroninput produces departure to stable limit cycle oscillationssettled on the ‘‘autorotation’’ equilibrium branch with veryhigh values of angle of attack.

Based on these continuation and bifurcation analysisresults the engineer can draw two main inferences. The firstone is that there is an apparent loss of aileron efficiency at

da > 8� (practically zero increase in angular velocity). Thishappens due to increasing counter-rotative aerodynamicmoment produced by sideslip, which is generated by inertiacoupling. The second conclusion is that at 12� < da < 18�the velocity vector roll maneuver becomes very agitateddue to existence of stable limit cycles and there is also acritical aileron deflections da > 18�, when aircraft candepart to an agitated autorotation regime with very highangles of attack (see arrow in Fig. 8). Transition to highangles of attack at supersonic speeds is very dangerous asthe normal and side load factors can easily exceed thestructural limit. A more accurate evaluation of the aerody-namic loads during transition to high angles of attackrequires inclusion into the simplified 5th-order mathemati-cal model of the nonlinear dependencies of aerodynamiccharacteristics and additional dynamic equation for flightvelocity.

For more thorough consideration of multiattractordynamics the phase portrait views for different ailerondeflections are constructed using information stored in

Fig. 8. Continuation of equilibria and closed orbits: angle of attack androll rate vs aileron deflection.


the continuation database. At small aileron deflections thesystem has five different equilibria, three stable and twoaperiodically unstable. The phase portrait for da 0 is pre-sented in Fig. 9 (top plot). In addition to equilibrium pointstaken from the continuation database several special trajec-tories are integrated using the phase portrait GUI andadded to the phase portrait. Four unstable trajectoriesassociated with two saddle points approach three stablepoints. For each equilibrium point a trajectory from theassociated stable invariant manifold is reconstructed byreversed time integration from its close vicinity. The stableinvariant manifolds for unstable points are most importantas they form the boundaries of regions of attraction forthree stable equilibrium points.

The phase portrait at positive aileron deflection da 14�is shown in Fig. 9 (bottom plot). There are two unstableequilibrium points and two stable solutions, a stable limitcycle on the normal equilibrium branch and a stable equi-librium regime on the autorotation branch. Unstable tra-jectories associated with saddle equilibrium pointapproach two stable attractors, the stable invariant mani-fold for saddle equilibrium point separates the regions ofattraction for autorotation regime at high angle of attackand agitated roll maneuver at small angle of attack.

Along with steady state solutions the continuation data-base stores eigenvalues and multipliers for equilibrium andclosed orbits, respectively. The trajectories of eigenvaluesdisplayed in the continuation GUI (Fig. 5) for a specifiedbranch of solutions can be saved in a separate figure forreport generation. For example, Fig. 10 shows the trajecto-ries of eigenvalues for two different branches of equilibriumsolutions presented in Fig. 8.

Every locally stable equilibrium and closed orbit shownin Fig. 8 has a bounded region of attraction (RA). Forevaluation of the multidimensional RA a number of itstwo-dimensional cross-sections are generated by directintegration of dynamical system (1). Initial conditions aretaken from a grid of points on the considered two-dimen-sional cross-section plane. Final destinations of integratedtrajectories are identified by their entering into a closeproximity of one of the available attractors stored in the

Fig. 9. Phase portrait views for a, b, p projection at da = 0 (top plot) andfor da = 14� (bottom plot).

Fig. 10. Root loci vs aileron deflection for two branches of continuationdiagram in Fig. 8.


database [11,17]. For three stable equilibrium points atda 0 two cross-sections of regions of attraction in the

plane (b,a) at p q r 0 and in the plane (r,p) atq a b 0 are shown in Fig. 11. Trajectories attractedby equilibrium with zero roll rate are marked by solid dia-monds. Trajectories attracted by equilibrium with positiveroll rate are marked by ‘‘·’’ and trajectories attracted byequilibrium with negative roll rate are marked by ‘‘+’’.The computed cross-sections, for example, allow one tospecify critical disturbances in the state space leading todeparture from the normal flight conditions with zero rota-tion. A locally stable equilibrium with very small size ofregion of attraction should be considered as practicallyunstable form of motion.

The saddle-node bifurcations on the continuation dia-gram (8) separate on parameter axis da segments with dif-ferent number of equilibrium points. The location ofsaddle-node bifurcation points in the plane of two controlparameters, for example, da and dr, are defined by continu-ation curves of the extended bifurcation problem [2]:

Fðx; da; dr; deÞ ¼ 0; detoFðx; da; dr; deÞ

ox

� �¼ 0 ð3Þ

The procedure for continuation of the saddle-node bifurca-tion in two-dimensional plane of (da,dr) (3) takes the initialpoints from the database for one-parameter continuationresults. Fig. 12 shows the bifurcation diagrams in the planeof aileron and rudder deflections for two different values ofelevator deflection de 0 (left plot) and de �5� (rightplot). The computed results specify regions with differentnumber of equilibria and thus allow one to assign the crit-ical boundaries in the parameter space associated with anaircraft jump-like departure.

Size of the database containing the results presented inthis section is about 2 Mb. The computation time for con-tinuation of three equilibrium branches is about 3 min onPC (Pentium IV, 2.4 GHz), the closed orbits require about7 min per branch, and the computation time for two-parameter continuation diagram in Fig. 12 is about2 min. Only the computation of two-dimensional cross-sec-tions of region of attraction with a fine grid of pointsremains rather time-consuming it takes about 20 min to

Fig. 11. Cross sections of regions of attraction for three equilibriumpoints in (b,a) and (r,p) planes.

Fig. 12. Two parameter bifurcation diagrams in the plane of aileron and rudder deflection for a saddle node bifurcation point: regions with differentnumber of equilibria (left plot for de = 0, right plot for de = 5�).


compute one cross-section. The developed computationaltools were tested during investigations of nonlinear dynam-ics for a number of realistic aircraft models covering thefull flight envelope. Acceptable convergence and robustnessproperties of the algorithms were demonstrated, as well asthe possibility to restart computations from a checkpoint incase of hardware or software failure.

6. Conclusions

The presented computational framework provides a fullsuite of algorithms and methods required for investigationof aircraft nonlinear dynamics by a systematic analysis ofaircraft equilibrium and periodical forms of motion. Thecomputational tools for one- and two-parameter continua-tion, bifurcation analysis, phase portrait/time histories,evaluation of regions of attraction, root loci, systematicstarting solution solver, etc. are linked together in a logicalmanner that promotes a structured approach to problemsby engineers who are not necessarily highly specialized inthe field of qualitative methods and continuation software.

The developed databases for accumulation of computedresults, their integration with computational algorithmsand an interactive way of managing the investigation pro-cess via a number of Graphical User Interfaces provide aproductive and user-friendly environment. The way inwhich computed information is accessed and processed isa significant step forward in making such software user-friendly and suited to an engineering practice.

Acknowledgements

The presented computational framework for aircraftnonlinear dynamics investigation was developed duringthe research project funded by DERA/QinetiQ Ltd., Bed-ford, UK. During various years it was monitored by PhillSmith, Darren Littleboy and Yoge Patel. Special thanksgo to the reviewers and also to Daniel Walker for theirvaluable and constructive comments and suggestions,which helped to improve the presentation. The authorsacknowledge the contribution to the computational systemof Evgeny Kolesnikov.

Appendix. Aircraft equations of motion

The autonomous set of equations for rigid aircraft flightdynamics [2]:

_V ¼ ð1=mÞ½ððX þ T Þ cos aþ Z sin aÞ þ Y sin b� � � �þ gðsin b cos h sin h� cos a cos b sin h

þ sin a cos b cos h cos /Þ_a ¼ q� ðp cos aþ r sin aÞ tan bþ ð1=mV Þ� ½Z cos a� ðX þ T Þ sin a� � � �þ ðg=V cos bÞðsin a sin hþ cos a cos h cos /Þ

_b ¼ p sin a� r cos aþ ð1=mV Þ� fY cos b� ½ðX þ T Þ cos aþ Z sin a� sin bg � � �þ ðg=V Þðcos a sin b sin hþ cos b cos h sin /

� sin a sin b cos h cos /Þ_p ¼ ððIYY � IZZÞ=IXX Þqr þ L=IXX

_q ¼ ððIZZ � IXX Þ=IYY Þpr þM=IYY

_r ¼ ððIXX � IYY Þ=IZZÞpqþ N=IZZ

_h ¼ q cos /� r sin /

_/ ¼ p þ ðq sin /þ r cos /Þ tan h

The aerodynamic coefficients CX,Y,Z1 and Cl,m,n

2 are repre-sented as functions of motion parameters a, b, p, q, r andcontrol deflections de, da, dr. Note that at high angles of at-tack the aerodynamic dependencies are nonlinear with astrong effect of prehistory of motion [35].

The reduced set of equations for analysis of aircraftintensive rolling maneuvers [34]:

_a ¼ q� pbþ zaaþ zdede

_b ¼ pa� r þ ybbþ ydrdr

_p ¼ ððIYY � IZZÞ=IXX Þqr þ lbbþ lpp þ lrr þ ldada þ ldrdr

_q ¼ ððIZZ � IXX Þ=IYY Þpr þ maaþ mqqþ ma _aþ mdede

_r ¼ ððIXX � IYY Þ=IZZÞpqþ nbbþ npp þ nrr þ ndada þ ndrdr

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1 X �qSCX ; Y �qSCY ; Z �qSCZ .2 L �qSbCl; M �qScCm; N �qSbCn.


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M.G.Goman, A.V.Khramtsovsky (2008) - Computational framework for investigation of aircraft nonlinear...

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