Coetzee, E B

Nonlinear analysis techniques, especially methods from bifurcation theory, have emerged as valuable tools over the last 20 years, particularly due to the advent of the modern computer. Originally developed as part of dynamical systems and chaos theory, they gradually are finding their way into applications areas from all walks of life. As far as the aerospace industry is concerned, methods from nonlinear dynamics were used initially for the prediction of aircraft flight dynamics at high angle of attack flight regimes, where traditional methods have failed. They are now being used within Airbus to analyse aspects of the dynamics of aircraft on the ground. Specific aerospace applications, where nonlinear dynamics techniques are expected to make an impact, include the design of flexible structures and mechanisms, and the dynamics of a braking wheel. Challenges related to the industrialisation of such methods are also discussed

Coetzee, Etienne

Explore Bristol Research

                          Coetzee, E. (2009). Application of nonlinear dynamics in civil aerospace.Link to publication record in Explore Bristol ResearchPDF-documentUniversity 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-terms.htmlTake down policyExplore Bristol Research is a digital archive and the intention is that deposited content should not beremoved. However, if you believe that this version of the work breaches copyright law please contactopen-access@bristol.ac.uk and include the following information in your message:• Your contact details• Bibliographic details for the item, including a URL• An outline of the nature of the complaintOn receipt of your message the Open Access Team will immediately investigate your claim, make aninitial judgement of the validity of the claim and, where appropriate, withdraw the item in questionfrom public view.Application of nonlinear dynamicsin civil aerospaceE. CoetzeeAirbus, Landing Gear Systems, Bristol, BS99 7AR, UKSummary. Nonlinear analysis techniques, especially methods from bifurcation the-ory, have emerged as valuable tools over the last 20 years, particularly due to theadvent of the modern computer. Originally developed as part of dynamical systemsand chaos theory, they gradually are finding their way into applications areas fromall walks of life. As far as the aerospace industry is concerned, methods from non-linear dynamics were used initially for the prediction of aircraft flight dynamics athigh angle of attack flight regimes, where traditional methods have failed. They arenow being used within Airbus to analyse aspects of the dynamics of aircraft on theground. Specific aerospace applications, where nonlinear dynamics techniques are ex-pected to make an impact, include the design of flexible structures and mechanisms,and the dynamics of a braking wheel. Challenges related to the industrialisation ofsuch methods are also discussed.1 IntroductionLanding gear engineers observe nonlinear phenomena such as hysteresis, back-lash and stiction on a daily basis, without necessarily appreciating the fullmeaning behind these observations. A wheel that locks up during braking isa good example. Many conflicting requirements need to be considered duringthe design, where the weight and pavement loading needs to be minimised,and the shock absorption maximised. The lateral stability on the ground isdetermined by the position of the gears, along with the tyre and oleo (shockdamper) characteristics. Experience has shown that the use of different tyrescan mean the difference between a stable and an unstable aircraft. Landinggears contain highly nonlinear components, including tyres, brakes and oleos,and therefore traditional analysis is usually done at some very specific designconditions. There is a perceived need to characterise the behaviour of thesystem over a wide variety of parameters, and this is the industrial domainwhere methods from nonlinear dynamics can and should be brought to bear.We discuss here some of the open avenues for this approach within the specificcontext of ground dynamics of passenger aircraft.2 E. CoetzeeB−737−400 MD−82/83 B−767−200ER A−320 B−747−40000.10.20.30.40.5AircraftLateral Load Factor, ny (g)  Taxi−outTaxi−inFig. 1. Comparison of equal probability lateral load factors during ground turningfor five aircraft after [8].2 Aircraft ground manoeuvresGround operations tend to be performed at constant thrust settings, becausethe thrust is adjusted only occasionally by the pilot with the aim of alteringthe velocity. One issue is to find points (in terms of operational input) wherethe aircraft becomes uncontrollable during a turn. A loss of stability is de-pendant on several parameters, such as the steering angle, entry velocity ofthe turn, tyre properties, and the runway condition. Mathematically, stabilityloss corresponds to a limit point (or fold) bifurcation or a Hopf bifurcation,which makes it possible to classify the dynamics of a turning aircraft on theground with the use of continuation methods. In this way, physical causes forthe loss of stability have been identified [2, 4, 5]. Specifically, limit points andHopf bifurcations bound regions in parameter space where the tyres are sat-urated, so that they cannot provide enough side force to maintain a specificmanoeuvre.An ongoing study by the FAA has been aiming to identify what type oflateral loading conditions can be experienced by in-service aircraft. The goal isto validate the conservative design factors that are currently required duringthe design phase. Current regulations require an 0.5 g-level at the centre ofgravity, even though it is known from experience that such high g-levels are notpossible in larger aircraft. The results from the study indicates that the actualg-levels experienced by airline operators are approximately 0.3g for wide-bodyaircraft, such as the Boeing 747, and 0.43g for narrow-body aircraft, such asthe Airbus A320; Figure 1 shows a summary of the expected peak g-levels asextracted from the report [8]. It would be of great benefit if the influence ofthe main parameters could be studied during the preliminary design phasesof a project, where some analysis is indeed already done by means of detailednonlinear simulations. Our experience with the bifurcation study of groundApplication of nonlinear dynamics in civil aerospace 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−505101520Normalised velocityDamping (%)  TorsionLateral bendingFig. 2. An example of linear shimmy analysis.manoeuvres indicates that continuation methods may be used as a new toolto provide a reasonable estimate of the maximum g-levels, as well as wheresuch operating conditions will occur.3 Landing gear shimmyShimmy oscillations of a landing gear are undesirable due to the safety andmaintenance aspects involved with the occurrence of this phenomena. Linearshimmy analysis is typically done at specific operating points for design pur-poses, while detailed nonlinear simulations are usually performed only afteran incident occurred. Torsional and/or lateral motion can be observed duringshimmy oscillations, and the contribution of each mode may be dependanton the initial conditions of the system. Linear shimmy methods calculate thedamping in the system while the velocity is varied to identify the onset ofshimmy as a point where either the torsional mode or the lateral mode haszero damping. Figure 2 shows an example of such an analysis.Pilots often report the onset and disappearance of shimmy oscillationsbetween certain velocities, indicating a trajectory across a boundary of Hopf-bifurcations. There are still many differing opinions with regards to the mainparameters that influence shimmy, and they result in differing maintenance ac-tions that are recommended when shimmy occurs. Hydraulic shimmy dampersare installed on some aircraft to prevent oscillations in the steering system,but this adds weight.The development of a nonlinear model of a nose landing gear, and itssubsequent bifurcation analysis, has demonstrated the coupled nature of thetorsional and the lateral modes via nonlinear tyre forces in the presence ofgeometric nonlinearities [6, 7]. Future research will focus on the construction4 E. Coetzee..−200 −150 −100 −50 0 50 100 150 200−80−60−40−20020406080Handle Force (N)Overcentre angle (deg)  StableUnstableDynamic OverlayFig. 3. A mechanism with a hysteresis loop of its force-reaction diagram.of a comprehensive map of all the types of shimmy under different operationalconditions, as well as the development of preliminary rules to avoid shimmyalready at the design phase of an aircraft.4 Dynamics of a braking wheelThe longitudinal traction force of a braked wheel is a consequence of therelative difference between the vehicle velocity and the velocity of the wheelat the contact patch, which is also known as wheel slip [9]. It depends onthe normal force on the wheel, as well as the friction coefficient between thewheel and the road surface. A free-rolling wheel is defined to have a slip-value of 0, while a locked wheel has a slip-value of 1 [1]. It is known thata hysteresis loop exists when a brake torque is applied [3]. This means thatthe brake torque where lockup occurs and where control is regained could bevery different. Recent research on aircraft has also shown that the unstablepoint after which lockup occurs, does not necessarily occur at the peak valueon the slip curve. In fact, braking is one of the most nonlinear processes inaircraft, and understanding it fully will require the use of advanced methodsfrom dynamical systems theory.5 Landing gear mechanismsA mechanism is defined as a combination of parts, that are joined in a specificway, to perform a certain function. Figure 3 shows an example of a latch thatcontains several pinned arms and a spring. A relatively small force can beapplied to the handle of the latch, yet the clamping force on a component couldApplication of nonlinear dynamics in civil aerospace 5AutoIntegrationReportSimMechanicsComponentsADAMSFig. 4. Suggested integrated environment for the nonlinear analysis of linked modelsof aircraft components.become significant. A point could also be reached when the handle “jumps” toa new position where no additional force is needed to hold the part in place.This jump indicates the presence of a fold bifurcation as shown in Figure 3.The envelope of where this fold occurs can be calculated by varying the springstiffness and applied force. A landing gear effectively is a mechanism quitesimilar to a latch. Importantly, the landing gear needs to reach a downlocksolution at a certain applied force. Nonlinear dynamics methods are being usedin ongoing research to map out the envelope of downlock solutions of differenttypes of landing gears as a function of gear spring stiffness and applied forcevalues.6 Conclusions and outlookSeveral case studies have clearly demonstrated that methods from nonlin-ear dynamics allow engineers to discover, and explain, the rich dynamicalbehaviour that is observed during aircraft operations on a daily basis. Tradi-tional linear methods are adequate for many engineering systems, but nonlin-ear effects need to be considered if a system is to be used to its full potential.In spite of their huge potential, bifurcation theory methods are presentlybeing used only by small pockets of engineers in the aviation industry. In fact,when one wants to introduce nonlinear dynamics into the engineers’ normaltoolsets one encounters both societal and technological challenges. Primarily,the societal ones relate to management support and education. The technol-ogy needs to be supported by all tiers of management, and a strong businesscase needs to be made to gain this support. The technological challenge isone of education and development of the right tools. Training is needed to6 E. Coetzeefamiliarise engineers with the vocabulary and tools of dynamical systems the-ory, which are still largely unknown to the average engineer. Indeed, thereis a need to learn how to formulate a problem in a way conducive to non-linear analyses, and how to interpret the results. A level of intuition similarto that concerning, say, Bode diagrams, needs to be developed for the inter-pretion of bifurcation diagrams. At the same time more emphasis should beplaced on the development of well-documented, industrial, integrated toolsetsfor nonlinear analyses. Whilst several software tools are freely available, theywere developed primarily for research purposes. The overall goal is to developan intergrated and user-friendly environment where validated models can bestudied with bifurcation software. Figure 4 shows an example of what suchan environment may look like at a high level.References1. M. Blundell and D. Harty. The Multibody Systems Approach to Vehicle Dynamics.SAE, September 2004.2. E. Coetzee, B. Krauskopf, and M. Lowenberg. Nonlinear aircraft ground dy-namics. In ICNPAA, editor, International Conference on Nonlinear Problems inAviation and Aerospace, June 2006.3. B. Olson. Nonlinear dynamics of longitudinal ground vehicle traction. Master’sthesis, Michigan State University, 2001.4. J. Rankin, E. Coetzee, B. Krauskopf, and M. Lowenberg. AIAA Modeling andSimulation Technologies Conference, (AIAA-2008-6529).5. J. Rankin, E. Coetzee, B. Krauskopf, and M. Lowenberg. this volume.6. P. Thota, B. Krauskopf, and M. Lowenberg. Proceedings of AIAA 2008, (2008).7. P. Thota, B. Krauskopf, and M. Lowenberg. this volume.8. D. Tipps, J. Rustenburg, D. Skinn, and T. DeFiore. Side load factor statis-tics from commercial aircraft ground operations. Technical Report UDR-TR2002-00119, Federal Aviation Administration, U.S. Department of Transporta-tion, Federal Aviation Administration, Office of Aviation Research, Washington,DC 20591, January 2003.9. J. Wong. Theory of Ground Vehicles. Wiley-Interscience, 3rd edition, March2001.

English

Nonlinear aircraft ground dynamics.

Nonlinear dynamics of longitudinal ground vehicle traction. Master’s thesis,

Side load factor statistics from commercial aircraft ground operations.

The Multibody Systems Approach to Vehicle Dynamics.

Theory of Ground Vehicles. Wiley-Interscience, 3rd edition,

Application of nonlinear dynamics in civil aerospace

                          Coetzee, E. B. (2009). Application of nonlinear dynamics in civil aerospace.Early version, also known as pre-printLink to publication record in Explore Bristol ResearchPDF-documentUniversity 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-terms.htmlTake down policyExplore Bristol Research is a digital archive and the intention is that deposited content should not beremoved. However, if you believe that this version of the work breaches copyright law please contactopen-access@bristol.ac.uk and include the following information in your message:• Your contact details• Bibliographic details for the item, including a URL• An outline of the nature of the complaintOn receipt of your message the Open Access Team will immediately investigate your claim, make aninitial judgement of the validity of the claim and, where appropriate, withdraw the item in questionfrom public view.Application of nonlinear dynamicsin civil aerospaceE. CoetzeeAirbus, Landing Gear Systems, Bristol, BS99 7AR, UKSummary. Nonlinear analysis techniques, especially methods from bifurcation the-ory, have emerged as valuable tools over the last 20 years, particularly due to theadvent of the modern computer. Originally developed as part of dynamical systemsand chaos theory, they gradually are finding their way into applications areas fromall walks of life. As far as the aerospace industry is concerned, methods from non-linear dynamics were used initially for the prediction of aircraft flight dynamics athigh angle of attack flight regimes, where traditional methods have failed. They arenow being used within Airbus to analyse aspects of the dynamics of aircraft on theground. Specific aerospace applications, where nonlinear dynamics techniques are ex-pected to make an impact, include the design of flexible structures and mechanisms,and the dynamics of a braking wheel. Challenges related to the industrialisation ofsuch methods are also discussed.1 IntroductionLanding gear engineers observe nonlinear phenomena such as hysteresis, back-lash and stiction on a daily basis, without necessarily appreciating the fullmeaning behind these observations. A wheel that locks up during braking isa good example. Many conflicting requirements need to be considered duringthe design, where the weight and pavement loading needs to be minimised,and the shock absorption maximised. The lateral stability on the ground isdetermined by the position of the gears, along with the tyre and oleo (shockdamper) characteristics. Experience has shown that the use of different tyrescan mean the difference between a stable and an unstable aircraft. Landinggears contain highly nonlinear components, including tyres, brakes and oleos,and therefore traditional analysis is usually done at some very specific designconditions. There is a perceived need to characterise the behaviour of thesystem over a wide variety of parameters, and this is the industrial domainwhere methods from nonlinear dynamics can and should be brought to bear.We discuss here some of the open avenues for this approach within the specificcontext of ground dynamics of passenger aircraft.2 E. CoetzeeB−737−400 MD−82/83 B−767−200ER A−320 B−747−40000.10.20.30.40.5AircraftLateral Load Factor, ny (g)  Taxi−outTaxi−inFig. 1. Comparison of equal probability lateral load factors during ground turningfor five aircraft after [8].2 Aircraft ground manoeuvresGround operations tend to be performed at constant thrust settings, becausethe thrust is adjusted only occasionally by the pilot with the aim of alteringthe velocity. One issue is to find points (in terms of operational input) wherethe aircraft becomes uncontrollable during a turn. A loss of stability is de-pendant on several parameters, such as the steering angle, entry velocity ofthe turn, tyre properties, and the runway condition. Mathematically, stabilityloss corresponds to a limit point (or fold) bifurcation or a Hopf bifurcation,which makes it possible to classify the dynamics of a turning aircraft on theground with the use of continuation methods. In this way, physical causes forthe loss of stability have been identified [2, 4, 5]. Specifically, limit points andHopf bifurcations bound regions in parameter space where the tyres are sat-urated, so that they cannot provide enough side force to maintain a specificmanoeuvre.An ongoing study by the FAA has been aiming to identify what type oflateral loading conditions can be experienced by in-service aircraft. The goal isto validate the conservative design factors that are currently required duringthe design phase. Current regulations require an 0.5 g-level at the centre ofgravity, even though it is known from experience that such high g-levels are notpossible in larger aircraft. The results from the study indicates that the actualg-levels experienced by airline operators are approximately 0.3g for wide-bodyaircraft, such as the Boeing 747, and 0.43g for narrow-body aircraft, such asthe Airbus A320; Figure 1 shows a summary of the expected peak g-levels asextracted from the report [8]. It would be of great benefit if the influence ofthe main parameters could be studied during the preliminary design phasesof a project, where some analysis is indeed already done by means of detailednonlinear simulations. Our experience with the bifurcation study of groundApplication of nonlinear dynamics in civil aerospace 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−505101520Normalised velocityDamping (%)  TorsionLateral bendingFig. 2. An example of linear shimmy analysis.manoeuvres indicates that continuation methods may be used as a new toolto provide a reasonable estimate of the maximum g-levels, as well as wheresuch operating conditions will occur.3 Landing gear shimmyShimmy oscillations of a landing gear are undesirable due to the safety andmaintenance aspects involved with the occurrence of this phenomena. Linearshimmy analysis is typically done at specific operating points for design pur-poses, while detailed nonlinear simulations are usually performed only afteran incident occurred. Torsional and/or lateral motion can be observed duringshimmy oscillations, and the contribution of each mode may be dependanton the initial conditions of the system. Linear shimmy methods calculate thedamping in the system while the velocity is varied to identify the onset ofshimmy as a point where either the torsional mode or the lateral mode haszero damping. Figure 2 shows an example of such an analysis.Pilots often report the onset and disappearance of shimmy oscillationsbetween certain velocities, indicating a trajectory across a boundary of Hopf-bifurcations. There are still many differing opinions with regards to the mainparameters that influence shimmy, and they result in differing maintenance ac-tions that are recommended when shimmy occurs. Hydraulic shimmy dampersare installed on some aircraft to prevent oscillations in the steering system,but this adds weight.The development of a nonlinear model of a nose landing gear, and itssubsequent bifurcation analysis, has demonstrated the coupled nature of thetorsional and the lateral modes via nonlinear tyre forces in the presence ofgeometric nonlinearities [6, 7]. Future research will focus on the construction4 E. Coetzee..−200 −150 −100 −50 0 50 100 150 200−80−60−40−20020406080Handle Force (N)Overcentre angle (deg)  StableUnstableDynamic OverlayFig. 3. A mechanism with a hysteresis loop of its force-reaction diagram.of a comprehensive map of all the types of shimmy under different operationalconditions, as well as the development of preliminary rules to avoid shimmyalready at the design phase of an aircraft.4 Dynamics of a braking wheelThe longitudinal traction force of a braked wheel is a consequence of therelative difference between the vehicle velocity and the velocity of the wheelat the contact patch, which is also known as wheel slip [9]. It depends onthe normal force on the wheel, as well as the friction coefficient between thewheel and the road surface. A free-rolling wheel is defined to have a slip-value of 0, while a locked wheel has a slip-value of 1 [1]. It is known thata hysteresis loop exists when a brake torque is applied [3]. This means thatthe brake torque where lockup occurs and where control is regained could bevery different. Recent research on aircraft has also shown that the unstablepoint after which lockup occurs, does not necessarily occur at the peak valueon the slip curve. In fact, braking is one of the most nonlinear processes inaircraft, and understanding it fully will require the use of advanced methodsfrom dynamical systems theory.5 Landing gear mechanismsA mechanism is defined as a combination of parts, that are joined in a specificway, to perform a certain function. Figure 3 shows an example of a latch thatcontains several pinned arms and a spring. A relatively small force can beapplied to the handle of the latch, yet the clamping force on a component couldApplication of nonlinear dynamics in civil aerospace 5AutoIntegrationReportSimMechanicsComponentsADAMSFig. 4. Suggested integrated environment for the nonlinear analysis of linked modelsof aircraft components.become significant. A point could also be reached when the handle “jumps” toa new position where no additional force is needed to hold the part in place.This jump indicates the presence of a fold bifurcation as shown in Figure 3.The envelope of where this fold occurs can be calculated by varying the springstiffness and applied force. A landing gear effectively is a mechanism quitesimilar to a latch. Importantly, the landing gear needs to reach a downlocksolution at a certain applied force. Nonlinear dynamics methods are being usedin ongoing research to map out the envelope of downlock solutions of differenttypes of landing gears as a function of gear spring stiffness and applied forcevalues.6 Conclusions and outlookSeveral case studies have clearly demonstrated that methods from nonlin-ear dynamics allow engineers to discover, and explain, the rich dynamicalbehaviour that is observed during aircraft operations on a daily basis. Tradi-tional linear methods are adequate for many engineering systems, but nonlin-ear effects need to be considered if a system is to be used to its full potential.In spite of their huge potential, bifurcation theory methods are presentlybeing used only by small pockets of engineers in the aviation industry. In fact,when one wants to introduce nonlinear dynamics into the engineers’ normaltoolsets one encounters both societal and technological challenges. Primarily,the societal ones relate to management support and education. The technol-ogy needs to be supported by all tiers of management, and a strong businesscase needs to be made to gain this support. The technological challenge isone of education and development of the right tools. Training is needed to6 E. Coetzeefamiliarise engineers with the vocabulary and tools of dynamical systems the-ory, which are still largely unknown to the average engineer. Indeed, thereis a need to learn how to formulate a problem in a way conducive to non-linear analyses, and how to interpret the results. A level of intuition similarto that concerning, say, Bode diagrams, needs to be developed for the inter-pretion of bifurcation diagrams. At the same time more emphasis should beplaced on the development of well-documented, industrial, integrated toolsetsfor nonlinear analyses. Whilst several software tools are freely available, theywere developed primarily for research purposes. The overall goal is to developan intergrated and user-friendly environment where validated models can bestudied with bifurcation software. Figure 4 shows an example of what suchan environment may look like at a high level.References1. M. Blundell and D. Harty. The Multibody Systems Approach to Vehicle Dynamics.SAE, September 2004.2. E. Coetzee, B. Krauskopf, and M. Lowenberg. Nonlinear aircraft ground dy-namics. In ICNPAA, editor, International Conference on Nonlinear Problems inAviation and Aerospace, June 2006.3. B. Olson. Nonlinear dynamics of longitudinal ground vehicle traction. Master’sthesis, Michigan State University, 2001.4. J. Rankin, E. Coetzee, B. Krauskopf, and M. Lowenberg. AIAA Modeling andSimulation Technologies Conference, (AIAA-2008-6529).5. J. Rankin, E. Coetzee, B. Krauskopf, and M. Lowenberg. this volume.6. P. Thota, B. Krauskopf, and M. Lowenberg. Proceedings of AIAA 2008, (2008).7. P. Thota, B. Krauskopf, and M. Lowenberg. this volume.8. D. Tipps, J. Rustenburg, D. Skinn, and T. DeFiore. Side load factor statis-tics from commercial aircraft ground operations. Technical Report UDR-TR2002-00119, Federal Aviation Administration, U.S. Department of Transporta-tion, Federal Aviation Administration, Office of Aviation Research, Washington,DC 20591, January 2003.9. J. Wong. Theory of Ground Vehicles. Wiley-Interscience, 3rd edition, March2001.

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Application of nonlinear dynamics in civil aerospace

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