Lateral Stability of a Typical Nose Landing Gearusing Torsional Magneto-Rheological (MR)

Damper

Keywords: MR damper, Nose Landing gear, Shimmy, Dynamicinstability, forward taxi velocity

The piston and the cylinder are connected by a torque linkthrough a collar. The torque link allows the relative motionof the piston and the cylinder but gives torsional resistanceto the piston during take-off, landing and taxiing. Thetorque link is modelled as a torsional spring between thecollar and the lower part of the piston. To account for theinertial property of the torque arm, as well as other off-center masses, an eccentric mass is included in the model,Fig. 2(a). The steering system is idealised as a lineartorsional spring and a nonlinear torsional dashpot for thepurpose of investigating shimmy during taxiing. Theidealization ofNLG is shown in Fig.3

Two wheels are attached to the axle, with one wheel at eachend. Here, no braking moment is applied; thus, the wheelsare free to rotate during taxiing. Although the contactbetween ground and tire has finite area, the Moreland point-contact model [3] is used to describe the cornering forceand drift of the tire. Under this simplified assumption, thetire motion can completely be described by a lateraldeflection ~ and a tire twisting 'Vb at the contact point,together with an angular velocity (0 about the wheel axle.The present shimmy model is developed based on the aboveassumptions except the following linearisations:(i) the shimmy damper is taken as linear viscous element,(ii) there is no free play in the system and(iii) cubic non-linear force springs are neglected.

Sateesh Bandaru* and Dipak K MaitiDept. ofAerospace EngineeringlIT Kharagpur -721302, India

on the other hand, is considered to be elastic and may besubjected to lateral bending caused by friction loadingfrom the ground. The upper end of the cylinder is rigidlyfixed to the fuselage. In general, the strut is slightly cantedforward, represented by an angle a as shown in Fig. 2(a).The bending of the strut is described by two variables (z,~),with z being the lateral deflection and ~ the rotation at thelower end of the strut, as shown in Fig. 2(b). The centerpoint of the axle is connected to one end of a trail arm. Theother end of the trail arm is fixed to the bottom of thepiston. Mechanically, the piston, trail, and axle are forgedinto a single part. Due to their relative small size, both thetrail and the axle are modelled as rigid bodies. Whentorsional motion occurs, the trail moves away from itslongitudinal position and is offset by an angle 'V, as shownin Fig. 2(c).

Abstract

II. MODEL DESCRIPTION

In the present study, the shimmy model is developed basedon the formulation of G. X. Li [2]. This formulationconsidered here is a dual-wheel nose landing gear ofcantilever type as shown in Fig.l for modelling. Thelanding gear is assumed to be moving on a flat runway witha constant speed V at the trunnion point. The fuselage isrepresented by a conic shaped cylinder and is treated as aconcentrated mass seating on the top of the shock strut.The shock strut itself, consisting of a cylinder and a piston,

Aircraft landing gears are subjected to a wide range ofexcitation conditions, which result in conflicting dampingrequirements. A novel solution to this problem is toimplement semi-active damping using magnetorheological(MR) fluids. In the present study, the shimmy model isdeveloped based on the formulation of G. X. Li, to study theself-induced dynamic instability at the critical forward taxiingvelocity. The effect of various parameters on the shimmybehaviour is also studied. The study has been performed topostpone the dynamic instability by introducing MR fluidbased damper as a semi-active device.

I. INTRODUCTION

Aircraft wheel shimmy is a self-excited oscillation causedby coupling of the lateral deflections and torsionaloscillations about the gear swivel axis. The basic cause ofshimmy is an energy transfer from the moving aircraft tothe vibratory modes of the landing-gear system. Shimmymay occur due to insufficient damping, ill geometricdimensions, and inadequate structural stiffness, usually athigh taxiing speed. It is observed from open literature thatthe MR dampers are used as shock absorber to suppress thevibration. In this present research, torsional MR damperdeveloped by authors in previous study [1](which is used tocontrol the vibration in FE model of NLG) is implementedas a semi-active device and studied the effect on lateralinstability of shimmy model of nose landing gear topostpone the dynamic instability and suppress undesirableshimmy vibration of an aircraft.

978-1-4244-2746-8/08/$25.00 2008 IEEE

1

FFigure 1. Schematic of a dual-wheel NLG

(a)

III. MATHEMATICAL MODELThe dynamic equation of motion for the lower part of thelanding gear (unsprung masses)[2]: piston, trail, axle and

w[::ee:; ar::jrl:e[~:e;:b~:lJ;lo:[~~e i~O~]:I:umPtiOns as (I)""J ""J m33] it C31 C3J C33] Z k31 kn k33nz

[~:: ~:1;}+!:jL1d31 d3JJ e,

Figure 3. Schematic ofNLG Figure 4. MR damper with finsIdealization

Here (Xi, Yi,Zi) are the directions of the coordinates, first set(i=O) is fixed on the runway, second and third sets (i=l and2) attached to the shock strut at its upper and lower endsand last set (i=3) attached to the trail.

Figure 4 shows the model of a Torsional type MR damperwith fins, mainly consists of piston and cylinder. Thecopper wire is wounded around the fins to generate amagnetic field. This model was developed based on theBingham characteristics and the geometric parameters arechosen based on the available space to accommodate MRdamper as additional device in the system. Parametricstudy has done and the numerical analysis is carried out toestimate the damping coefficient and damping force underoscillatory motion. The damping force developed by thisdamper due to friction is obtained in the previous study [1].The loss factor varies linearly with magnetic field. Theinput current is varied between 0 and lA with an incrementof 0.2A with maximum magnetic field is 0.65 Tesla. It isobserved that the damping force increases significantly withapplied current.

(6)

Jclf/c +(Cd +CMR)lfrc +kdlf/c =ct(lfr-lfrc)+kt(lf/-lf/c) (4)Where the torsional damping coefficient ofMR damper [3],CMR is calculated based on the viscous energy[4] i.e., areaunder the curve of force - displacement diagram

ECMR = 1tn8

02 (5)

Where the cornering force, Ft is related to tire lateraldeflectionby ~ =kt~+CLA (2)Here kt - tire lateral stiffness and CL - time constantThe cornering moment, Mt is linearly proportional to thetire sideslip angle and its time derivative:M t = Illlf/ + IlDlfr (3)

Kinematic equations for landing gear systems are innonholonomic conditions which relate the velocityvariables to ensure compatibility. These conditions areobtained based on the physical restraint that the tire alwaysmaintains contact with the ground at contact points. Soinstantaneous velocity at that point must be zero, suchrestraint condition guarantees that while the tire is rolling,no slipping occurs. After some transformations, theresulting expressions for the equilibrium equation for tire,the rolling-without-slipping mechanism, along rolling andlateral direction can be expressed as

j + z+ Llfr+ R(p + aL(p + V('If + '1ft) = 0m=VIR

where lit - Tire yaw stiffness and IlD - Tire yaw timeconstant. The equilibrium equation from Steering to torquelink end based on linearised assumption afterimplementation ofMR damper can be expressed as

Jc

- Collar moment of inertia, Cd - Viscous damping

coefficient, kd - Damper stiffness, ct - Torque linkdamping coefficient, E - energy dissipated by MR damper,Q - oscillation frequency, eo -amplitude of oscillation

(c)(b)

22

~ tC2 A

Figure 2. Schematic views and coordinate systems

Here L - Trail length, V - Air craft velocity, R - Wheelrolling radius

2

IV. RESULTS & DISCUSSION

The stability of the system can be determined for differenttaxiing velocity looking into the loci of the system roots.

I

~Pllllllllllllllr-I II II II II I

Root locus plot

-10 -5Real

Iwithout MR Damper (V =150.35 km/h ~

I C

50III

45...-

I

40

~.~ 35lU.E

30

25

20-20 -15

Transient response

(b) Transient responses at taxiing speed at 150..35 km/h

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2t(sec)

(a) Root locus plots for forward speed (10-300 km/h)

Figure 5. without MR damperIn general, the matrix A contains all system parameters forexample; damping coefficients, trail length and aircraftvelocity etc. As one or more of these parameters are varied,the eigenvalues of matrix also change, resulting in a changein the system stability. When parameters are designedproperly, the landing gear system should be well with in thestable boundary. The shimmy results are generated first forcomparison purpose based on the above formulation. Thesystem parameters are used as given in the Literature [2].The model is tuned to generate the comparable results forlinear model. The analysis is carried out with initialdisturbance i.e., chosen as an initial caster velocity, \jI =30deg/sec and all other initial conditions are zero. The stabilityanalysis is carried out for various damping coefficients (Cd).The present analysis results are plotted in Fig 6 and stabilityboundaries are computed for three different trail lengths andresults are compared with literature at L=0.075m. Thefigure shows a good correlation between present andliterature results and the effect of mechanical trail on theshimmy behaviour is also studied.

(8)

(7)

eX = Ax

where c is called tire coefficient of yaw and Cl is called tireyaw time constant. Collins and Black [5] carried out anextensive experiment on determining c and Cl as well asrelated tire constants.

Shimmy prediction is the determination of systeminstability as system parameters varied. This can be donefor linearised systems only. Four second order ordinarydifferential equations are given in "Eqs. (1) - (4)" and twofirst order ones are given "Eqs. in (6) and (7)". These sixequations contain six variables (

51.5

o-5

1

1

1

I1

1

1

1

1

1

1

~8jt1141i1lH11I4111111111tt!1 1

1 1

1 1

1 II I

-10

II

*1

* with MR damper(Vc =183.15 Km/h)

-15Real

(a) Root locus plot with MR damper

Figure 8. Effects of varying kt and ~

Figure9. MR damper as semi - active device

t (sec)(b) Transient responses at taxiing speed at 150.35 km/h

20 '-----__--L- -'--__-..L ....l....-__------I-20

50r----~---.------___._-----.-----,

900

*800

*700

*/.

/.

600 /.

r; 500t--Present (kd=3.18e5,kt1.33e5)

oTJ 400* ~Present(kd=3.98e~

300-&- Present(kt 1.66e5)

200 - - - Lit.(kd=3.18e5,kt1.33e5)

100 *Lit. (kd=3.98e5)

... Lit. (kt=1.66e5)100 200 300 400 500 600

V (km/h)

1

1

45 *1

*1

1t

40

~I 35

30

25

0.15

0.1

0.05..-...

C)Q)-0 a'-""

~-0.05

-0.1

-0.15a 0.5

B. Open Loop Response Analysis ofShimmy Instability-with Semi - active MR Damper

Root locus plot

300

350

250

300

1

- - - Lit. I--Present,

250

- - - Lit. (L =0.075)-- Present(L =0.075)

~ Present(L =0.050)-- Present(L =0.100)

150 200V (km/h)

100

100 150 200V (km/h)

Figure 7. fs - V at neutral stability L = 0.075m

40

38

36

34

32

~ 30_til

28 =-=-=--=

26

24

22

200 50

700

600

500

I 400u-o 300

200

100

Figure 6. Critical damping for neutral stability at different L

It is demonstrate that the lower mechanical trail willimprove the stability of the system and increasing dampingcoefficient will improve the stability of the system butbelow the critical value the system will experience shimmy.The shimmy frequency (fs) vs. shimmy velocity is alsoplotted in Fig 7 at L=O.075. This shows a slight differencein shimmy frequency. This may be due to the difference inshock strut information. The shimmy problem as observedis predominantly due to the lateral oscillation of shock strut.One of the primary objectives of theoretical shimmyanalysis is to find out the best possible system parameters(e.g., geometric dimensions, damping coefficient, andstiffuess, etc.) to improve the design. In present study, theparametric study has been performed for the influence ofwheel span (D) and cant angle( a), the effect of changingthe position of eccentric mass, tire parameters, wheel sizeand mass on shimmy instability and the effects of twotorsional stiffuesses kt and kt on shimmy stability are alsoevaluated. But in result part, only the effect of feasibleparameter i.e., torsional stiffuess as shown in fig 8. It isobserved that the increase in torque arm stiffuess drasticallyimprove the nose landing gear stability whereas the effectof damper torsional stiffness adversely effect the stability.

4

Transient response

- - - - without MR damper I 1\1-- Semi - active MR damper , l II \ ,I III ~

I ' I I I ~ ,I I I, II I n ,II ,~; ~ I I ,\ II II I1II I I, I' II :: I:

I I I II ~ II II II II II ~ II II " II II Ii 1I11II ,'//,,\1111,11'11111 ,,,,

,lr,\,'I/l I II

0.5 1 1.5 2t (5)

(c) Taxiing speed 183.15 km/h

Transient response

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8t(s)

(d) Taxiing speed 200 km/h

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8t (s)

(b) Taxiing speed 150.35 km/h

....... without MR damper-- Semi-actiw MR damper

1.5

1.5.-------r------.--------,.---------,

0.5

0.4

0.3

0.2

0.1

EgN

-0.1

-0.2

-0.3

-0.4

-0.50

0.5

e- OgN

-0.5

-1

-1.5

-2

-2.50

0.5

E 0.N

-D.5

-1

-1.50

Figure 10. z - t plots at different Taxiing speeds without MR damper andwith Semi - active MR damper

- - - without MR damper-- Semi - acthe MR damper

0.4

0.3

0.2

0.1

IN

-0.1

-0.2

Figure 10 show the comparison of the results of the lateraldisplacement vs. time response without MR damper andwith MR damper as semi active device at different taxiingspeeds 100, 150.35, 183.15 and 200 km/h. From thisanalysis one can say that application of MR damper willgive better amplitude reduction in instability of the NLG.

-0.3

Transient response

Generally instability can be postponed with the applicationofMR damper. For the above study, in order to reduce theinstability, analysis is carried out with MR damper as asemi active device with open loop damping 6.62% inshimmy model. The numerical simulation is also done andthe results of the root loci are plotted in Fig. 9(a),demonstrate that the lateral mode is crossing the imaginaryaxis from left half to right half at 183.15 km/h taxiingspeed. Fig 9(b) is the time response of the system at thetaxiing speed 150.35 kmlh with MR damper showsconvergent solution as compared to Fig. 5(b). It indicatesthat, implementation of MR-fluid based damper modelalong with the shimmy damper reduce the instability oflanding gear of an aircraft.

Finally parametric study has been performed and results arecompared for stability boundary in the (L, V) plane i.e. theeffect of mechanical trail and torque arm stiffness on theshimmy behaviour, without MR damper and with semi-active MR damper as shown in Fig 11. It is observed thatimplementation of MR damper will improve the stability ofthe system. It is also observed that the lower mechanicaltrail or increase in torque arm stiffness results improvementin system stability. The reduction of mechanical trail maynot be practically feasible, whereas the higher torque armstiffness can be achieved with minor design modifications.The root loci are plotted in Figs 12(a) and (b) with 10% and17.3% higher torque arm stiffness will stabilizes the NLGsystem at speeds up to 236 and 300 kmlh taxiing speedrespectively, (where as in without damper case 22.56%increment is required to stabilize speed up to 300km/h). Ifabove 17.3% increment of torque arm stiffness, speed alsoincreases, but beyond 300 km/h generally flight will takeoff.

-o.4'-----'---'-----'------'---.l----L-_-'-------L_--'----------'o 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

t(s)

(a) Taxiing speed 100 km/h

5

Root locus plot

-5-10-15

45

20 l-- ~ _L__ ___l..._ _!_ ----l-20

1

1

1 1 I I----J-----_L ~-----_L _

-+ 1 1 1 I*i... 1 1 I

1 *1 1 I

~ -----~------~-----~------}-----I 1 I 1

1 1 I I

1 1 1 I

------I------~-----~------+------I 1 1 1

I 1 I 1

I 1 I I30 - - - - - J ~ _ ~el:f!lIM!IIWllllllllllllllllllllll~.- - - - -

I I I II I I I1 I I I1 I 1 I

25 - - - - - I - - - - - - I - - - - - -I - - - - - - "I - - - - -1 1 1 11 1 1 11 1 1 1

50 r-----.-------,---------,-----,---------,

Real

0.160.140.12

- - - without MR damper--Semi - acti\~ MR damper

0.08 0.1L (m)

(a) L - V plot

0.060.0450'-----..1.--------'-------'----'-----1.---""-------------'0.02

3O0

250 \\

\\

\\

200 \~ \

~ \ \"-

> "-"-150 "-

100

300...-----------,-----,------,------,----,------,----,-,------

250

200

:2~ 150>

100

- - - - without MR damper-- Semi-active MR damper

II

I/

I/

//

I/

//

//

//

//

//

(b) Basic model with 17.3% increase in torque-armstiffness and Vc = 300 km/h

Figure 12. Root locus plots for forward speed (10-350 km/h)

v. CONCLUSIONS

1. Implementation of MR-fluid based damper has beenpostponed the dynamic instability of landing gear of anaircraft

50,----------r--------r-------r------,-------,

[3]. Moreland, W. J., "The Story of Shimmy", Journal ofthe AeronauticalSciences, 21, 793-808, 1954.

REFERENCES

[4]. Graham Kelly S, "Fundamentals of mechanical vibrations",Mc Graw Hill 2nd edn, New Delhi

2. The analysis suggests that, 17.3% increase in torque armstiffness with the application of MR damper is likely tostabilize the NLG system of an aircraft upto 300 km/htaxiing speed.

3. Finally MR damper is a good semi-active candidate forvibration control of a structural system

[1].Sateesh Band Maiti D.K, "Vibration control of typical nose landinggear with torsional MR fluid based damper", Proceedings of ICTACEM2007 ,International Conference on Theoretical, Applied, Computationaland Experimental Mechanics, December 27-29, 2007, lIT Kharagpur,India

[5]. Black R J and Collins R L, "Tire Parameters for Landing Gear shimmystudies", AIAA Journal ofAircraft, May - June 1969, pp. 252 - 258

[2]. G. X. Li, "931402: Modelling and Analysis of a Dual-WheelNosegear: Shimmy Instability and Impact Motions", SAE TechnicalPapers, 04/01/1993

1.6 1.8

x 1051.4

-5-10

0.8 1 1.2kt(N-m)

-15

0.60.4

20l----------'------l..-----L..----!--------I-20

I1

1 1 1 1____ L ~ J ~ _

-t 1 1 1 1~ -t I 1 I

-+1 1 1

~ -----~-----+-----~-1 1 I1 1 1

I I 1 1- - - - - +- - - - - - + - - - - - -I - - - - - -I - - - - -

I 1 I 1

1 1 1 I

I I I I

30 - - - - - ~ - - - - - +- - - -***4 ~~11111 IIIIIIIIIIIIIIIIIII.IIJ-I 1 I 1I 1 I II I I I

25 - - - - - I - - - - - i - - - - - I - - - - - -I - - - - -I 1 I II 1 I II 1 I I

45

O'--------'----'------J----'---------'------'---~-----'0.2

(b) kt - V plotFigure 11. Stability boundary without MR damper

and with Semi-active MR damperRoot locus plot

50

Real

(a) Basic model with 100/0 increase in torque-armstiffness and Vc = 236 km/h

[6]. R.L.Collins, "Theories on the Mechanics of Tires and TheirApplications to shimmy Analysis", Journal ofAircraft, 8, PP. 271 -2, 1971

* corresponding author: sateesh_india@rediffmail.com

6