C Howcroft

EB Coetzee

J Rankin

N Terkovics

P Thota

S Sharma

Bernd Krauskopf

Etienne B. Coetzee

Mark H. Lowenberg

Simon A. Neild

Sanjiv Sharma

Crossref

Dynamical Systems Methods for Evaluating Aircraft Ground Manoeuvres

Evaluating the ground-based manoeuvrability of large aircraft is time consumingand costly if explored though simulations with industry-developed complexmodels of ground dynamics. We argue here that this type of dynamics can be investigated efficiently and with considerable precision by applying dynamical systems techniques. As an example, we consider the lateral loads experienced by an Airbus A380 when it turns off a runway

Krauskopf, Bernd

Coetzee, Etienne B.

Lowenberg, Mark H.

Neild, Simon A.

Sharma, Sanjiv

Explore Bristol Research

                          Krauskopf, B., Coetzee, E. B., Lowenberg, M. H., Neild, S. A., & Sharma, S.(2016). Dynamical systems methods for evaluating aircraft groundmanoeuvres. In UK Success Stories in Industrial Mathematics (pp. 131-136).Springer International Publishing AG. https://doi.org/10.1007/978-3-319-25454-8_17Peer reviewed versionLink to published version (if available):10.1007/978-3-319-25454-8_17Link to publication record in Explore Bristol ResearchPDF-documentThis is the author accepted manuscript (AAM). The final published version (version of record) is available onlinevia Springer at http://www.springer.com/gb/book/9783319254524. Please refer to any applicable terms of use ofthe publisher.University of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-termsDynamical systems methods for evaluatingaircraft ground manoeuvresBernd Krauskopf, Etienne B. Coetzee, Mark H. Lowenberg,Simon A. Neild and Sanjiv SharmaAbstract Evaluating the ground-based manoeuvrability of large aircraft is time con-suming and costly if explored though simulations with industry-developed complexmodels of ground dynamics. We argue here that this type of dynamics can be inves-tigated efficiently and with considerable precision by applying dynamical systemstechniques. As an example, we consider the lateral loads experienced by an AirbusA380 when it turns off a runway.1 IntroductionAircraft are designed to fly and, hence, are not optimised for ground operations inthe way cars or other ground vehicles may be. Nevertheless, a passenger aircraftneeds to operate fast, reliably and safely on the ground in order to ensure its over-all commercial success. The standard approach employed by aircraft manufacturershas been to conduct massive and expensive numerical simulations of industry-testedand parametrised models for aspects of aircraft motion to evaluate the ground per-formance of aircraft as part of their design, evaluation and certification.An alternative approach is to use dynamical systems techniques that allow one tofollow solutions, detecting stability changes and bifurcations, as parameters are var-ied. We conducted a number of systematic case studies to demonstrate that aircraftground dynamics can be investigated without the need for expensive brute-forcenumerical simulations; these projects include:• the development of a fully parameterized model of a mid-size aircraft and its useto evaluate aircraft ground turning with the goal of providing insight into safeoperation limits under different conditions. The main motivation for this workwas to evaluate the suitability of the existing Federal Aviation Regulations forlateral loads experienced during turning manoeuvres [8].Bernd Krauskopf, Department of Mathematics, University of Auckland, Auckland 1142,New Zealand, e-mail: b.krauskopf@auckland.ac.nz · Etienne B. Coetzee, Airbus Opera-tions Limited, Filton, Bristol, BS99 7AR, UK, e-mail: etienne.coetzee@airbus.com · MarkH. Lowenberg, Faculty of Engineering, University of Bristol, Bristol BS8 1TR, UK, e-mail:m.lowenberg@bristol.ac.uk · Simon A. Neild, Faculty of Engineering, University of Bristol, Bris-tol BS8 1TR, UK, e-mail: simon.neild@bristol.ac.uk · Sanjiv Sharma, Airbus Operations Limited,Filton, Bristol, BS99 7AR, UK, e-mail: sanjiv.sharma@airbus.com12 Krauskopf, Lowenberg, Neild, Coetzee & Sharma• research on ground handling of aircraft with more than three sets of wheels,in particular, the Airbus A380 model. This work considered low-, medium- andhigh-speed ground manoeuvres of an A380 in comparison with an A320 [2, 3, 4].• the development of a mathematical model of an aircraft nose landing gear, whichfeatures torsional and lateral bending modes that are coupled through the tyre dy-namics. A bifurcation analysis in terms of the forward velocity and vertical forceon the landing gear was used to identify regions of (unwanted) stable torsionaland stable lateral shimmy oscillations [11, 12]. Subsequent work also consideredshimmy oscillations in main landing gears [7] and landing gear-fuselage interac-tions [10].• beyond aircraft ground dynamics, configurations and forces of landing gears dur-ing retraction and extension were considered, as well as the evaluation of controllaws during flight; see the overview paper [9] for details and further references.A common feature of the underlying mathematical models is that they containconsiderable nonlinearities, for example, due to geometric constraints, the tyre-ground interface and aereodynamics forces. Therefore, their behaviour needs to beevaluated using a combination of analytical and numerical techniques; in particu-lar, it is possible to determine the dependence of the observed behaviour on thedifferent parameters, such as velocity and loading of the aircraft, with the numer-ical technique of continuation. Specifically, we developed the Dynamical SystemsToolbox (DST) [1], which incorporates the capabilities of the well-known continu-ation software package AUTO [5, 6] into MATLAB. The DST enables the conve-nient coupling of user-developed models to numerical continuation, thus, makingdynamical systems available for use within an industrial setting. The DST has beenincorporated in the Airbus Methods and Tools portfolio as a supported tool for theevaluation of proposed works and new designs.Taking a specific example, we consider the problem of determining the loads onthe aircraft and on the tyres that arise during ground manoeuvres. For safety rea-sons, these loads need to remain below agreed limits under all operational scenariosas stipulated by the relevant regulations. Hence, extensive analysis is required tocover all of the different ground menoeuvres the aircraft may perform. An importantcase is that of exiting a runway, which we consider here for the largest commercialairliner in operation — the Airbus A380 with a maximum ramp weight (MRW) of577,000 kg.2 Ground loads of an A380 during turningThe Airbus A380 has been in operation since 2007. Modeling and simulation havebeen an important part of the design and evaluation of this aircraft. In particular, avalidated SimMechanics model of the A380 is available for the purpose of studyingits behaviour on the ground. A schematic is shown in Fig. 1, where the aircraft issubject to forces, such as the engine thrust and steering, which influence the inter-nal states describing, for example, the attitude of the aircraft and the forces at theDynamical systems methods for evaluating aircraft ground manoeuvres 3Fig. 1 Top-level SimMechanics model of an A380. Reproduced from [4].individual tyres. This A380 model was developed by Airbus for real-time studies,but has now also been coupled directly to the Dynamical Systems Toolbox withinMatlab. As a result, there is confidence in the validity of the results obtained fromthe bifurcation analysis.To support its weight, the A380 has five landing gears with a total of 22 wheels: adual-wheel nose landing gear (NLG), left and right wing landing gears (WLG) withfour wheels each, and left and right body landing gears (BLG) with six wheels each.The naming convention for the 22 wheels is shown in Fig. 2(a). From a regulatoryand operational perspective it is important to ensure that the loads at each of thewheels do not exceed their safety margins. Already for commercial aircraft of smallto medium size with three landing gears it is a considerable task to determine theforces at the tyres during actual manoeuvres. For the A380 the relevant forces needto be determined at all 22 wheels, taking into account different possibilities for theweight distribution among the landing gears which, as there are more than three,are statically-redundant. As was mentioned in the introduction, loads on landinggears and tyres during different ground manoeuvres are traditionally analysed byperforming a large number of numerical simulations of the model.We first consider the lateral load (the sideways load perpendicular to the direc-tion of travel) acting on the A380 at the centre of gravity (CG); more details and acomparison with results for an A320 can be found in [4]. The bifurcation analysiswith the DST is an efficient method for determining this load over a large range ofvelocity Vn at the NLG versus the steering angle δ that the pilot sets. The resultscan be represented conveniently in the (δ ,Vn)-plane, as is shown Fig. 2(b) for theextreme case of MRW, the largest possible weight of the aircraft, with a CG positionas far back or aft as possible. For evaluation purposes Fig. 2(b) shows a large rangeof Vn of up to 35 m/s for any setting of the steering angle δ . The labelled contourlines are curves of equal lateral load factor for the aircraft, defined as the lateral (orsideways) force divided by the vertical (or downward) force at the CG, from 0.05to 0.2. The contour line for 0.133 is shown thicker because it delimits the designenvelope that determines the maximum permissible velocities during runway exits.4 Krauskopf, Lowenberg, Neild, Coetzee & Sharmaδ [deg]V n[m/s](a) (b)Fig. 2 (a) Wheel numbering definition for the A380; (b) contour lines in the (δ ,Vn)-plane of thelateral aircraft load factor at the CG for the A380 at maximum ramp weight with aft CG position.Reproduced from [4].Such contours can be followed directly in the parametersVn and δ , without the needfor simulations. Similarly, one can determine directly the boundary of the white re-gion in the centre where the NLG saturates; this means that the NLG tyres cannotproduce enough lateral force to hold the aircraft on the desired turn. It can be seenthat the majority of this region is above the allowable aircraft load factor of 0.133.We now consider the allowable manoeuvre speeds to show that the small nose gearsaturation region that lies below the allowable aircraft load factor contour, in thebottom right corner of Fig. 2(b), is not actually entered in practice. Taking constantradius turns, of 51m, 275m and 500m, the velocity-steering angle relationships areplotted and labelled. The constant turn with a radius of 51 m is in the lower rightcorner of Fig. 2(b). While this curve enters the white region, it does so only forvelocities Vn well above the recommended velocity of 4 m/s for this turn.We now consider the loads at the individual tyres of the four main landing gearsduring a 90◦ turn with a radius of 51 m. During this manoeuvre the steering angle δof the NLG is ramped up from zero to about 32◦ and then brought back to zero at theend of the turn. The aircraft velocity reduces considerably when the steering angleis increased — the nose tyres then effectively act as brakes. This is why a velocitycontroller is used to adjust the thrust of the engines so that a constant velocity of 4m/s is maintained during the entire 90◦ turn. Continuation analysis can be used todetermine how the forces at each of the 20 main gear wheels build up and reach amaximun during this runway exit manoeuvre.Figure 3 shows the resulting maximum loads, again in terms of load factors, butthis time for each tyre rather than for the aircraft. Distinguished here are the steady-state load factor, which is the one determined by numerical continuation, and thedynamic load factor, which is calculated from simulations. First of all, the maximumsteady-state values are larger and hence the critical ones, except that they are areabout 10%–20% smaller for W17–W20 (which are the tyres at the back axles of theDynamical systems methods for evaluating aircraft ground manoeuvres 5Fig. 3 Steady-state and dynamic tyre load factors on main gear tyres for the A380 when a radiusof 51 m and a velocity of 4 m/s is maintained at the nose gear. Reproduced from [4].inner BLGs, which is actually steered). A more detailed analysis (not reported here)suggests that this is due to the straightening out at the end of the turn, and we arguethat numerical continuation is indeed a valid tool for investigating the maximumtyre loads during turns. Figure 3 illustrates that there are considerable differencesin the magnitudes of the loads at the different tyres during this ground manoevre.Tyres W1–W8 of the outer wing landing gears (see Fig. 2(a)) have negative loadfactors (the force is against the direction of the turn). Tyres W9–W20 of the innerbody landing gears, on the other hand, experience positive loads, with the largestload occurring at tyres W13–W16 of the middle axle.3 ConclusionsThe introduction of dynamical systems methods, which are implemented in the Dy-namical Systems Toolbox, into the industrial practice provides Airbus with newcapability for the evaluation of aircraft ground performance and the potential ofconsiderable savings in time and costs. Airbus in the UK have described its impactas an approximately 80% reduction in time and associated costs for ground maneu-verability analysis; this estimate is made by comparing the time taken to conductglobal assessments using bifurcation and other dynamical systems techniques ver-sus the time to obtain exhaustively a large number of point solutions.As an example of this new capability, we presented a study of the loads at thecentre of gravity and at individual tyres that the 22 wheel A380 experiences whenmaking a runway turn. These numerical continuation results were obtained withinthe Dynamical Systems Toolbox under Matlab for an industry developed and val-idated model of the aircraft. This type of detailed knowledge of forces and theirbuild-up during different ground manoeuvres may be used to inform design deci-sions and operational procedures.6 Krauskopf, Lowenberg, Neild, Coetzee & SharmaAcknowledgements The authors gratefully acknowledge sponsorship and technical support fromAirbus. Moreover, Neild is currently holder of an EPSRC fellowship (EP/K0053737/1), and theresearch reported here was supported by several EPSRC CASE studentships.References1. Coetzee, E.B.: Dynamical Systems Toolbox. with major contributions from, Rankin, J.,Thota, P., (2014); available at http://fr.mathworks.com/matlabcentral/fileexchange/32210-dynamical-systems-toolbox2. Coetzee, E.B., Krauskopf, B., Lowenberg, M.H.: Application of bifurcation methods for theprediction of low-speed aircraft ground performance. J. Aircraft 47(4), 1248–1255 (2010)3. Coetzee, E.B., Krauskopf, B., Lowenberg, M.H.: Analysis of medium-speed runway exit ma-noeuvres. J. Aircraft 48(5), 1553–1564 (2010)4. Coetzee, E.B., Krauskopf, B., Lowenberg, M.: Continuation analysis of aircraft ground loadsduring high-speed turns. J. Aircraft 50(1), 217–231 (2010)5. Doedel, E. J.: AUTO: a program for the automatic bifurcation analysis of autonomous sys-tems. Congr. Numer. 30, 265–384 (1981)6. Doedel, E. J.: AUTO-07P: Continuation and bifurcation software for ordinary differentialequations. with major contributions from Champneys A. R., Fairgrieve T. F., Kuznetsov Yu.A., Oldeman B. E., Paffenroth R. C., Sandstede B., Wang X. J. and Zhang C. (2007); availableat http://cmvl.cs.concordia.ca/auto/7. Howcroft, C., Krauskopf, B., Lowenberg, M.H., Neild, S.A.: Influence of variable side-staygeometry on the shimmy dynamics of an aircraft dual-wheel main landing gear. SIAM J.Appl. Dynam. Syst. 12(3) 1181–1209 (2013)8. Rankin, J., Krauskopf, B., Lowenberg, M.H., Coetzee, E.B.: Nonlinear analysis of lateralloading during taxiway turns, J. Guid. Contr. Dynam. 33(6) 1708–1717 (2010)9. Sharma, S., Coetzee, E.B., Lowenberg, M.H., Neild, S.A., Krauskopf, B.: Numerical contin-uation and bifurcation analysis in aircraft design: an industrial perspective. Proc. Royal Soc.A (to appear)10. N. Terkovics, Neild, S.A., Lowenberg, M.H., Krauskopf, B.: Bifurcation analysis of a couplednose-landing-gear-fuselage system. J. Aircraft 51(1) 259–272 (2014)11. Thota., P., Krauskopf, B., Lowenberg, M.H.: Interaction of torsion and lateral bending inaircraft nose landing gear shimmy. Nonlin. Dynam. 57(3) 455–46 (2009)12. Thota., P., Krauskopf, B., Lowenberg, M.H.: Multi-parameter bifurcation study of shimmyoscillations in a dual-wheel aircraft nose landing gear. Nonlin. Dynam. 70(2) 1675–1688(2012)

Dynamical systems methods for evaluating aircraft ground manoeuvres

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Dynamical systems methods for evaluating aircraft ground manoeuvres

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