A Tire Model for Air Vehicle Landing Gear...

2000 International ADAMS User Conference 1 A Tire Model for Air Vehicle Landing Gear Dynamics

A Tire Model for Air Vehicle Landing Gear Dynamics

Ned J Lindsley and Nitin B. Talekar 1 2

Air Force Research Laboratory and Mechanical Dynamics, Inc. 1 2

Multi-Disciplinary Technologies Center and Consultant Services 1 2

WPAFB, OH 45433-7531 and Ann Arbor, MI 48105 1 2

INTRODUCTION

An aircraft landing gear is, by nature of its function andsubsequent design, a complex multi-degree-of-freedomsystem. The stability of such a system requires furtherstudy, particularly at the tire-runway and fuselage-landinggear boundary conditions, the first of which will beexplored in this paper. A comprehensive literature searchon out-of-plane stability covering the air and automotive(early years) vehicle fields can be found in [1], with themilestone works [2-14] receiving the most attention.

This investigation defining the role which the undriven tireplays in out-of-plane stability during air vehicle groundoperations is represented by the following course of work.The tasks performed included deriving and discretizing theequations of motion for the tire, developing an empiricalmodel for the tire’s inflation loading characteristics,loading and rolling the tire in the vertical plane, imposingout-of-plane deflections, and producing results of tireinteraction with a landing gear model using ADAMS.

A set of ground operation profiles are simulated inADAMS utilizing its user subroutine, expression builder,and design study capabilities. Trail reduction with respectto tire inflation pressure is the subject of a design study onan air vehicle’s rejected take off (RTO) ground operation.This is a basic first step in utilizing the myriad ofcapabilities available in ADAMS, with an eye for quickerdesign cycles and eventual analytical/virtual certificationof entire air vehicle landing gear subsystems.

The most significant contributions of this work are 1) thedevelopment of a physically accurate, computationallyinexpensive tire model product for use as a landing gearsystem component in air vehicle ground operationsimulations and 2) the synergistic effect of incorporatingthe tire model into a dynamics analysis backbone programwhich possesses a myriad of design, analysis andoptimization capabilities. It is hoped that thesecontributions will help affect a more efficient, concurrentdesign process for landing gear systems.

SHORTCOMINGS OF PREVIOUS WORK

The shortcomings of previous work performed in aircraft

landing gear shimmy are, by the very nature of thephenonmenon, numerous. The works are limited primarilyto defining operating modes wherein instabilities occur.Consequently, a significant amount of testing is requiredto determine the experimental parameters which define theproblem. Even then, many of these parameters aredetermined statically or quasi-statically. These parametersalso show a large nonlinearity with respect to velocity,load and small component mass imbalances.

System boundary conditions between the landing gear andthe air vehicle fuselage have also not been well defined inthe past. Whether it be fluctuating vertical load, groundvortex turbulence, aero-braking or crosswind conditions,the behavior of the air vehicle’s fuselage and wings is amajor contributor to what dynamic conditions the landinggears undergo. This is especially significant whenconsidering a full air vehicle with its tricycle landing geararrangement. Under real world conditions, landing gearsubsystems must be capable of performing well throughoutthe air vehicle’s operating spectrum.

Perhaps the most overlooked component of the landinggear design process is the tire. It is the system boundarycondition between the runway and the rest of the landinggear. Aircraft tires are currently treated as spring/dashpotcombination models or closed-form solution models undercontact patch assumptions. Their largest drawback is theirhighly empirical basis. Finite element models are notcomputationally efficient for working within structuraldynamics analysis backbone programs, although modalrepresentations of automobile tires (which undergo arelatively gentler operational regime) have recently beenattempted with unknown success (industry proprietary).This work attempts to address these componentshortcomings with a finite difference tire model in theform of a group of subroutine object codes linked into thestructural dynamics analysis backbone program.

Damping is often used as a "cure all" remedy for shimmy,yet simply adding damping to an unstable mode merelyresults in destabilizing or reducing the stability margin ofsome other mode(s). A more comprehensive knowldegeof component interactivity would limit the amount andimportance of damping required to address the residualshimmy regime of the problem. This problem grows even


Figure 1

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more conspicuous when the effects of aging components(gear wear and the resulting joint freeplay) are considered.The best means of addressing these component and systemshortcomings is through a structural dynamics analysissoftware which can perform design studies on, designexperiments for and optimize a parametric system ofcomponents.

OVERALL TIRE MODEL DESCRIPTION

Since the tire model must be capable of a mathematicallyaccurate and computationally efficient representation ofthe tire response, the choice of physical assumptions mustbe made with the proper prejudice. Aircraft tires have anodd or even number of laterally spaced, circumferentiallycontinuous tread ribs, which present the opportunity forthe analyst to model them as individually linked cylinderswith rectangular cross-sections. Figure 1 illustrates thegeometric discretization of a typical air vehicle tire cross-section, while Figure 2 shows one of the tire’s ribs with itslocal and global coordinate systems and geometry.

Figure 3 shows the tire carcass element in its original anddisplaced/deformed positions. Three translational degreesof freedom, represented by the three displacements of apoint on the neutral axis, are possible. Figure 4 shows thetire carcass element cross-section in its original anddisplaced/deformed positions. Two rotational degrees offreedom, represented by the two rotations of a point on theplane perpindicular to the neutral axis, are possible.Therefore, there are five degrees of freedom at everycircumferential station along a rib.

Since aircraft tires are exposed to both steering operationsand vertical loads of large enough magnitude to ensure fulllateral contact, extension in the lateral direction isneglected, while shear deformation in the 2-plane (z-normal) is not. Likewise, since steerable aircraft tires arecomposed of a carcass whose thickness is much less thanthe diameter of the centerline of the tire rib, extension inthe radial direction is neglected. However, since the tire


Figure 5

carcass is a laminar composite region, shear deformationin the 2-plane (r-normal) is not.

The tire nonlinear strain-displacement relations areexpressed by an inclusive second order Taylor seriesexpansion of the "length-squared differences" measure ofstrain. The constitutive relation for stress-strain behavioris assumed to be linearly elastic. The elastic and inertialforces and moments for a general rib element are obtainedby integration over the cross-section for -h/2 < r < h/2(thickness) and -b/2 < z < b/2 (width). This resuls in forceand moment expressions which are functions of the fivedisplacements. Using differential formulation with freeand inertial force and moment body diagrams, the five tirecontinuum equations of motion are therefore expressed interms of the displacements. Globally, the ribs are thenconnected to each other and/or the rigid wheel rim via stiffdegree-of-freedom to degree-of-freedom springs. Thesespring stiffnesses are derived from the inflation pressureprofile tests, which will be touched on in the next section,TIRE MODEL INFLATION. The radial degree-of-freedom stiffnesses are considered to connect the ribs inparallel, while the lateral degree-of-freedom stiffnesses areconsidered to connect the ribs in series. Finally, theequations of motion are discretized via central finitedifference in space and time for computer coding, with theexception of forward finite difference for global damping.

Figure 5 shows the overall algorithm of the tire model. Itconsists of three basic modules of source code, which arecompiled and linked under ADAMS as a general forceobject. The three basic modules of the tire model sourcecode are 1) Initializing the Tire, 2) Calculation of the TireVertical Plane Dynamic Response, and 3) Calculation ofthe Tire Out-Of-Plane Dynamic Response. The firstprogram module calculates the initial conditions andelastic properties of the tire as functions of inflationpressure, and is only called once per simulation. Thesecond tire program module takes the inflated tire and,given the wheel hub vertical force and air vehicle’s speedand acceleration, calculates the vertical plane response inthe form of the tire’s shape and contact pressuredistributions, coupled with the wheel hub forces/momentsand vertical displacement. The hub vertical deflectionstep size and direction are based on 1) the need for anincrease or decrease of vertical load, 2) the need for anincrease or decrease of velocity, and 3) the current andrated velocity magnitudes. The third tire program modulecalculates the lateral force and aligning moment for agiven set of discrete wheel hub/rim displacements forevery call to the general force object, regardless if changesin 1) the vertical plane conditions or 2) the imposed wheelhub/plane conditions have occurred.

Tire static vertical plane, dynamic vertical plane anddynamic out-of-plane response trends are presented in thecontext of the following input parameters and responsequantities. The input parameter set consists of inflationpressure, target vertical load, air vehicle velocity andacceleration, out-of-plane wheel hub displacement and itsrate, and out-of-plane wheel rim rotation and its rate. Theresponse quantities set consists of vertical deflection, totalcontact area, exterior rib contact area contribution,central/interior rib contact area contribution, rollingresistance force and moment, trail reduction, lateral forceand aligning moment. Trail reduction is defined as thedistance which the tire reaction vertical force shiftsforward in the contact patch.

TIRE MODEL INFLATION MODULE

Since the inflation loading and subsequent deformation isnot a function of the circumferential coordinate 2, thederived equations of motion are not applicable to this loadcase. However, this is a blessing in disguise to themodeler. Further exploration of the inflation load casereveals that both the radial and lateral displacements (andhence, the global radial and lateral initial condition

Rib 3 (Central), Radial Degree of Freedom Results

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Tire "Broken In": Polynomial Fit: Rad’l Stiffness, lb/in


Figure 6

positions) of the individual rib centerlines can beexpressed as polynomial expressions of the inflationpressure of the tire divided by the rated inflation pressureof the tire. A third geometric parameter is the angle ',(see Figure 1) which defines the rib cross-sectionorientation in the r-z plane and can also be expressed inthis same polynomial form.

Since a series of inflation profile tire cross-sectionmeasurements is the least expensive of all laboratory tiretests, an empirical model based entirely on experimentaldata fit by a polynomial is the most efficient of thoseavailable. However, this approach is not limited tophysical experimental data, since finite elementpredictions of inflation profiles have been quite accuratefor quite some time now. Therefore, the model is stilluseful early in the design process.

TIRE MODEL INFLATION BEHAVIOR

Figure 6 depicts the experimental and curve-fit results forthe radial growth of the central rib on the same tire, testedwhen it was brand new and "broken in". Here a tire isconsidered as "broken in" after one taxi operation or itslaboratory equivalent. Of particular interest is the effectof "breaking in" the tire on the global radial ribcoordinates. This relaxation phenomenon is well known,and the initial plastic deformation effects remain relativelyconstant after tire break in. It is also important to note themonotonically increasing stiffness shown by the tire inFigure 6. This is a physically accurate characterization ofthe tire under inflation, and is present for all the inflationdegrees of freedom.

Once the tire initialization module has completed its tasks,all necessary input to describe the tire model has been readfrom the input deck, and the input file is closed.

Therefore, now is an appropriate time to discuss the natureof the input deck elements. The ASCI text input file(acdesc.inp) contains the rated inflation pressure andvelocity of the tire, inflation profile experimental or finiteelement results, tire tread/carcass/sidewall constructiongeometry, and tire rubber and cord material properties.The size of the input file depends primarily upon thenumber of ribs and number of ply and/or belt layers.

TIRE MODEL VERTICAL PLANE MODULE

The second tire program module takes the inflated tireand, given the wheel hub vertical force and air vehicle’sspeed and acceleration, calculates the vertical planeresponse in the form of the tire’s shape and contactpressure distributions, coupled with the wheel hubforces/moments and vertical displacement.

The wheel hub vertical displacement is incremented asboth a function of target vertical load and velocity in thefollowing manner. If the vertical load is increasing, thehub displacement is increased by 100% of the standardhub displacement increment, dUr0. If vertical load isdecreasing, the hub displacement is decreased by a slightlylarger percentage of dUr0. Complementing either of thesetwo actions, an additional percentage of dUr0 is added toor subtracted from the hub displacement in the followingmanner. The magnitude of this percentage is determinedby the square root of the ratio of the velocity to the ratedtire velocity, or (Vx/Vx ) . The square root function isRated

½

utilized because the centrifugal force due to vehiclevelocity (hence spin speed of the tire) is proportional tothe spin speed squared, S . This increase in centrifugal2

force results in an increase in vertical load. If the velocityis increasing, an increase in calculated vertical load wouldoccur, therefore the hub displacement is decreased by thispercentage of dUr0. Conversely, a decreasing velocitycalls for an increase in hub displacement by thispercentage of dUr0.

Contact is detected assuming a flat contact surface. Anode is found to be in contact if the global SAE verticalposition of its outermost corner point, which includes thetrigonometric contribution of ', is greater than or equal tothat of the flat contact surface. If contact is detected, thenode’s vertical position is forced to that of the verticalposition of the contact plane, but the node is free to movein the longitudinal direction. Contact/imposeddisplacement criterion for surfaces of curvature, such asthose represented by internal or external dynamometers,should be adressed in future research. Future researchshould also include the introduction of longitudinalfriction via constraint/limitation of the longitudinal

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Figure 8

movement of the nodes in contact. The contact pressuresat each node in contact along a rib are calculated usingstatic and dynamic stiffnesses. The nodal moment andforces are then transformed from local cylindrical toglobal SAE coordinates. If a node’s vertical contactpressure is tensile (i.e. its Z-component >0 in SAEcoordinates), it’s contact forces and moments are set tozero and the node is thereafter assumed to be in the freezone unless it is detected in contact at a later iteration.The global SAE forces and moments about the wheel hub(F , F and M ) are calculated by summing along theZ X Y

nodes in contact (including the forces’ contributions to themoment) for all the ribs. Convergence criteria are metwhen the target vertical load, velocity and accelerationhave been reached.

TIRE MODEL VERTICAL PLANE BEHAVIOR

Presented here are abridged results of the tire model'svertical plane behavior. Unless otherwise noted, theresults presented in this section are for rated inflationpressure and rated vertical load. The reader is referred to[1] for the full spectrum of vertical plane behavior results.

Figure 7 shows the static vertical force versus deflectionbehavior of the tire. The actual data was gleaned from thetire’s Qualification Test Report. The model is robustlyaccurate across the inflation pressure spectrum, as well asfor a range of vertical loads which greatly exceeds theoperating range. General trends observed include astiffening behavior of the tire for both increasingdeflection and increasing inflation pressure.

As for contact area computations, the model matches theribs' contact lengths and areas rather well, but does notportray the rib's varying contact width behavior, either asa function of inflation pressure or longitudinal distancefrom the center of contact. Rib contact width increases

with increasing inflation pressure for the central andinterior ribs, while it decreases for the exterior ribs. Thetapering of the exterior ribs is also much more pronouncedat higher inflation pressures. These observations showthat the sidewall stiffness contributes less to the overallstiffness of the tire as the inflation pressure increases.Despite these shortcomings, one should keep in mind theprimary purpose of the tire model, which is not to generatean infinitesimally accurate protrayal of the contact patch.The model was developed primarily as a physicallyaccurate, computationally inexpensive computer productfor use as a landing gear subsystem component in airvehicle ground operations simulations.

As for dynamic contact conditions, two trends immediatelyobserved were 1) a global stiffening and 2) radial growthof the tire due to increasing velocity. Closer inspectionalso reveals that contact lengths of the ribs will increaseboth fore and aft. This increase would be equal both foreand aft for zero acceleration. However, the presence of anacceleration shifts the points in contact forward, resultingin longer contact length forward than its aft counterpart.Conversely, the presence of a deceleration would result inlonger contact length aft than its forward counterpart. Allof these effects are more pronounced as the tire inflationpressure decreases.

Figure 8 compares vertical and longitudinal contactpressure distributions for one of the tire's ribs under staticvertical load versus accelerated rolling conditions.Besides the previously mentioned trends concerningcontact length symmetry loss, further inspection alsoreveals that the maximum vertical contact pressuremagnitude increases, while the center of vertical contactpressure shifts forward. The longitudinal contact pressureloses its asymmetry, resulting in in a rolling resistanceforce, Fx. These changes in the contact pressuredistributions due to rolling conditions result in a rolling


resistance moment, My. This rolling resistance moment,as well as the resultant forces, Fx and Fz, are applied atthe wheel hub. The magnitudes of the rolling resistanceforce and moment decrease as the tire inflation pressureincreases. The trends of the data also match those foundin Clark’s [15] compilation.

All these rolling condition phenomena manifestthemselves most profoundly in a measure called trailreduction. The trail, whose concept and importance inshimmy stability studies is well known, is shortened bythis measure. Trail reduction is defined as the distancewhich the tire vertical force reaction shifts forward in thecontact patch, thereby reducing the stability of the gear.

TIRE MODEL OUT-OF-PLANE MODULE

In general, a tire model goes through three stages duringsimulation. First it is pressurized, next it is loaded, then itis rolled with an acceleration, and finally it is displaced outof plane. Therefore, the tire response to lateral deflectionsof the wheel hub and twisting of the wheel plane must becalculated via the algorithms shown in Figure 5 andcomputer coding of the out-of-plane portion of theequations of motion. Calculations are performed in thismodule for every call to the program, regardless if 1)changes in the vertical plane conditions or 2) imposedwheel hub/plane conditions have occurred.

First the module receives the current and previousimposed wheel hub/plane deflections, U and " , asYhub rim

well as the current and previous simulation times. Thesize of the deflection steps, and number of iterations toreach them are defined by the calculations

NOOPMax = ( t - t ) / dt (1)new old

dU = [ U (t ) - U (t ) ] / NOOPMax (2)Yhub Yhub new Yhub old

d" = [ " (t ) - " (t ) ] / NOOPMax (3)Yhub Zrim new Zrim old

Given Equations 2 and 3, it is evident that no change in theimposed wheel hub/rim deflections results in the thirdsubroutines module going through NOOPMax iterationswith no newly imposed deflections. However, this allowsfor the tire to continue to react to any changes from thevertical plane conditions, as well as the history of itsimposed Out-Of-Plane conditions, as the simulation timeprogresses. In this manner a physically accuratedescription of the tire as component is maintained.

Given that the tire model time step, dt, is read from thetire description input deck and remains a fixed value

throughout the simulation, all calls to the program areignored (unless if time=0 for initial conditionsdetermination or pure statics) if (t -t ) < dt, andnew old

simulation time is not updated. However, since dt isalways on the order of 10 , such an occurrence has not yet-4

been encountered with the application of this tire model.Furthermore, suboordinate mode operation under theADAMS backbone allows the user to set a minimum timestep integration size.

Given these conditions concerning the relationshipbetween simulation time and the tire model time step,submodule convergence criteria are met when the targethub/rim displacement/rotation have been reached. Thistask is actually completed, not by a logical condition, butby reaching the end of the loop bounded by NOOPMax.

Upon convergence, a subroutine calculates the SAE forcesand moments about the wheel hub at each node via 1) thesidewall cross-coupling stiffnesses, which are functions ofsidewall length, sidewall thickness, and the sidewall shearand meridional moduli, coupled with 2) the differencebetween displacements of the exterior ribs (solved for) andwheel rim (rigidly prescribed) at each circumferentialposition. Finally, the submodule sums the SAE forces andmoments about the wheel hub for F and M along allY Z

(free and contact) exterior rib nodes .

TIRE MODEL OUT-OF-PLANE BEHAVIOR

Presented here are key results of the tire model's responseto out-of-plane wheel hub/rim displacements. Unlessotherwise noted, the results presented in this section arefor rated inflation pressure, rated vertical load, ratedvelocity and an accelreation of 6 fps (feet per second2

squared). The reader is referred to [1] for the fullspectrum of out-of-plane behavior results.

Initially the static vertical plane case is investigated as abasis for comparison with the rolling tire case. Therefore,air vehicle velocity and acceleration are zero, resulting inzero trail reduction (TR), rolling resistance force andmoment. The key input parameters to study for the tireout-of-plane response for the static case are 1) tireinflation pressure, 2) vertical load imposed the tire, 3) hubdeflection and/or rim rotation combinations and 4)frequency of application of the out-of-plane wheel hub/rimdisplacements. Results concerning the first two parametersare addressed first. Following this, the third parameterwas investigated such that 1) pure hub deflection, 2) purerim rotation and 3) combined deflection and rotation areimposed. Lastly, response to frequency of application wasinvestigated for values of 5, 10, 25 and 50 Hz.

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Figure 9

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Figure 11

The data shows that for the same inflation pressure, anincrease in vertical load magnitude results in an increasein both Fy and Mz. For the case of constant vertical load,a decrease in inflation pressure causes an increase in bothFy and Mz. This is due to both 1) the increased verticaldeflection and 2) the increased contact area. The increasein vertical deflection compresses the sidewall length,stiffening it laterally. These increased quantities meanmore nodes, which are assumed to be in full stickcondition, are in contact. Therefore, simplified tiremodels which assume Fy and Mz to only be functions ofvertical deflection have picked a good place to start.

The data shows that the combined deflection and rotationcase is merely a linear superposition of the individualcases. While the rotation case contributes very little to Fy,the displacement case contributes more significantly toMz. The rim rotation causes the force response to lagslightly, while the hub displacement causes the momentresponse to lead slightly. These lag/lead effects becomemore pronounced as inflation pressure is decreased. Asfor frequency of application resonse, given the same staticvertical plane conditions, an increase in frequency ofapplication results in an increase in both Fy and Mz.Therefore, closed form solution tire models whichsuperimpose an increasing Fy and Mz response functionof frequency on the previously mentioned verticaldeflection function are qualitatively correct.

All these effects on static tire response were observed forrolling out-of-plane simulation conditions as well. Worthmentioning briefly here is that the tire model, due to the

calculation of the deflection step size and number ofiterations from Equations 1-3, is capable of accomadatingtransient out-of-plane inputs. Figure 9 shows a sampleanimation frame of the accelerating, freely rolling tire,with the side, front, back, top and bottom views displayedin the center, left, right, top and bottom portions of theframe, respectively.

This section investigates the effects of rolling the tire onits out-of-plane response to wheel hub/rim displacements.The cases investigated here are for combined hubdeflection and rim displacement, with the frequencies ofapplication identical to those investigated for the staticcase. The air vehicle velocities investigated were 0, 50,100, 150, 200 and 250 mph (miles per hour).

Rolling tire vertical plane response conclusions were that1) hub vertical deflection magnitude decreases, 2) trailreduction increases, and 3) rolling resistance force andmoment magnitudes increase with increasing air vehiclevelocity. These results 1) are completely independent ofthe frequency of application of the prescribed wheelhub/rim out-of-plane displacements and 2) should cause asoftened tire out-of-plane response. Indeed, Figures 10and 11 both show an increasingly nonlinear softening


Figure 12

response for Fy (side force) and Mz (aligning moment) asair vehicle velocity increases. This softening response dueto increasing velocity far outweighs the hardeningresponse due to increasing frequency of application.

TIRE MODEL INTERACTION WITH ADAMS

Although the input parameter combinations explored tothis point have represented a possible “snapshot of the realworld” conditions experienced by a tire as part of thelanding gear subsystem attached to an air vehicle duringground operations, an actual mission profile will have allinput parameters except inflation pressure varying asfunctions of time. Furthermore, the out-of-plane hub/rimdisplacements will not be prescribed and steady-state.They will be transient inputs which are calculated by thebackbone program simulating the entire landing gearsubsystem from a combination of out-of-plane initialconditions, a system with its inherent imbalances, and/orperturbations from the air vehicle operating environment.

As for the problems mentioned above, due to thecalculation of the deflection step size and number ofiterations from Equations 1-3, the tire model is capable ofaccommodating the transient out-of-plane inputs. In fact,if the tire model as stand-alone program’s main callingroutine were altered to 1) ignore the prescribed out-of-plane displacement inputs, 2) replace them with the abovementioned out-of-plane initial conditions, imbalances andperturbations, and 3) incorporate a simplified set of gearout-of-plane stiffnesses, an entire stability study for anyaircraft ground operation could be undertaken within thestand-alone program. However, an action such as thiswould require an expansion of the program into an arenawhich has already been well defined in a much morecomprehensive manner by ADAMS.

Therefore, although the tire model can be run as a “standalone” program, it is most useful as an interactivecomponent for computer simulation and virtualprototyping of an aircraft arrangement of landing gearsubsystems. Such a purpose is achieved by linking thesource code within ADAMS solver source code to createa custom solver, affording the user a much broaderspectrum of dynamic events. The methodology for the tiremodel’s interaction with ADAMS is shown in Figure 12.

The air vehicle velocity and acceleration are prescribed asimposed motions between ground and the air vehiclecenter of mass. The other three mission point inputs,target vertical load, hub lateral displacement and rimrotation, are calculated in ADAMS at the Subsystem level.The main calling subroutine (GFOSUB, General Force

Object SUBroutine) of the tire queries the ADAMS modeldatabase for these inputs using the SYSFNC andSYSARY utility subroutines. The GFOSUB the returns asix member array of [Fx,Fy,Fz,Mx,My,Mz] in the SAEcoordinate system. A portion of the set of tire modelsubroutines can be duplicated and linked under anothermain calling subroutine (REQSUB, REQuestSUBroutine), enabling the tire to return the wheel hubvertical displacement to ADAMS, since the GFOSUB canonly return the previously mentioned force/moment array.This capability has not yet been implemented during thisearly stage of tire model integration, but should beperformed in future research. This would allow for moredetailed interaction and study between the tire and thelanding gear. For example, tire vertical plane energyabsorption/efficiency can be used as a balancing measurefor the currently studied measure of trail reduction.

At the current stage of integration of the tire model underADAMS, landing, takeoff and rejected takeoff missionprofile simulation results are identical to those in stand-alone mode. The out-of-plane degrees of freedom are inthe ADAMS model, but initial conditions, perturbationsand component mass imbalances have been set to zero.

Figure 13 shows the simple verification model explored in


The tire out-of-plane response trends, given increasingacceleration and identical target vertical load, inflationpressure and velocity, are:

a) lateral force response decreasesb) aligning moment response decreases

As for the tire’s out-of-plane response to out-of-planeinputs, the following trends were observed. Holding allother inputs identical, the combined deflection androtation case findings were:

a) assumption of linear superposition is soundb) rim rotation contributes little to |Fy|c) rim rotation causes Fy response to lag the deflection slightlyd) hub displacement contributes more to |Mz|e) hub displacement causes Mz response to lead the rotation slightly

Holding all other inputs identical, an increase infrequency of application, f, results in

a) an increase in Fyb) an increase in Mz

The same analogy holds for increasing deflection orrotation rate.

Given identical inflation pressure and target vertical load,the dominant factor on out-of-plane stability response isair vehicle velocity, Vx, (and acceleration, to a lesserextent) whose increase has a nonlinear softening effect onthe tire’s out-of-plane response.

A set of ground operation profiles were simulated inADAMS utilizing its user subroutine, expression builder,and design study capabilities. Trail reduction with respectto tire inflation pressure was the subject of a design studyon an air vehicle's rejected take off (RTO) groundoperation. Trail reduction was found to decrease (out-of-plane stability increases) as a function of inflationpressure. This was a basic first step in utilizing the myriadof capabilities available in ADAMS, with an eye forquicker design cycles and eventual analytical/virtualcertification of entire air vehicle landing gear subsystems.

The most significant contributions of this work are 1) thedevelopment of the tire model into a physically accurate,computationally inexpensive and fast computer product foruse as a landing gear system component in air vehicleground operation simulations and 2) the synergistic effectof incorporating the tire model into a dynamics analysisbackbone program which possesses a myriad of design,analysis and optimization capabilities. It is hoped thatthese contributions will help affect a more efficient,concurrent design process for landing gear systems.

RECOMMENDATIONS FOR FUTURE RESEARCH

Concerning the tire, future work should provide for:

a) Longitudinal friction via constraint/limitation criteria for contact node longitudinal behavior

b) Camber dependent behavior c) Contact with surfaces of curvature d) Catalog creation of existing air vehicle tires

Concerning the landing gear model in ADAMS, futurework should provide for:

a) Lateral deflection of the wheel hub b) Additional design variables, such as the tire's

tread thickness c) Additional design variables, such as the tire's

vertical plane energy absorption/efficiency

Concerning the tire model interaction with ADAMS,future work which would involve both arenas shouldprovide for:

a) Vertical force input to the tire via the ADAMS IMPACT function

b) Asymmetric landing simulation c) Creating a REQSUB to pass back the wheel hub

vertical deflection and other critical parameters

Hopefully these provisions will help to further broadenand densify the parametric, design measure and disciplineconcurrency spectrums of current landing gear systemdesign processes.

REFERENCES

1. Lindsley, N. J., A New Tire Model for Aircraft LandingGear Dynamics, PhD Dissertation, University of Akron,1999.

2. Seifert, H., Theoretical & Experimental Investigationson the Properties of Landing Gears, ME-GE-511, CADONo. 45 3106-2, January 1940.

3. Dietz, O. and Harling, R., Stability Tests of AircraftLanding Gears, SWB-MB-RE-1189, CADO No. 9622993-1, February 1940.

4. Dietrich, R. and von Schlippe, B., Shimmy of a Wheelwith Pneumatic Tire, ZWB-TB-11-2, ATI No. 51760,1943.

5. von Schlippe, B. & Dietrich, R., Shimmying of a


Pneumatic Wheel, NACA TM 1365, 1954, pp. 125-160,217-228.

6. Dietz, O. and Harling, R., Examination of Lateral Stressand Shimmy Phenomena of Airplane Wheel Tires, ATINo. 18905, AF-51-6-4763A, RV-53, October 1941.

7. Harling, R., Lateral Loads on an Airplane Tire inSideslip Motion, ZWB-LG-RE-140-S4-7, AF-51-6-4763A, RV-53, October 1941.

8. Fromm, H., Sideslip and Lateral Forces of a RollingWheel, ZWB-LG-RE-140-S-56-62, AF-51-6-4763A, RV-53, October 1941.

9. Fromm, H., Damping of Wheel Shimmy, ZWB-LG-RE-140, ATI # 57687, Oct. 1941.

10. Jenkins, E. and Donovan, A., Tricycle Landing GearDesign, Part I, Journal of Aeronautical Science, No. 9, pp.385-396, August 1942.

11. Jenkins, E. and Donovan, A., Tricycle Landing GearDesign, Part II, Journal of Aeronautical Science, No. 9,pp. 397-410, September 1942.

12. Bourcier de Carbon, C., A Theoretical Study of theShimmy of Airplane Wheels, Office National d'Études etde Recherches Aéronautiques, 1948. (also as NACA TM1337, 1952.)

13. Moreland, W., Landing Gear Vibration, Air ForceTechnical Report 6590, May 1951.

14. Pacejka, H.B., The Wheel Shimmy Phenomenon, PhDDissertation, Delft University of Technology, 1966.

15. Clark, S. K. (ed.), Mechanics of Pneumatic Tires,DOT HS 805 952, United States Department ofTransportation, Washington, D.C., 1981, pp. 131-153.

A Tire Model for Air Vehicle Landing Gear...

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