On the castor dynamic behavior

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Journal of the Franklin Institute 347 (2010) 116–129

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On the castor dynamic behavior

D. de Falcoa, G. Di Massab,�, S. Paganob

aSeconda Universit �a degli Studi di Napoli, Dipartimento di Ingegneria Aerospaziale e Meccanica, Napoli, ItalybUniversit �a degli Studi di Napoli ‘‘Federico II’’, Dipartimento di Ingegneria Meccanica per l’Energetica,

Via Claudio 21, 80125 Napoli, Italy

Received 30 July 2009; received in revised form 12 October 2009; accepted 12 October 2009

Abstract

In this paper, a model describing the castor three-dimensional dynamic behavior, is presented. A

procedure, based on the Udwadia–Kalaba formulation has been implemented to write the castor

non-linear motion equations. The model takes into account the flexural and lateral compliance of the

castor frame and the phenomena related to the wheel rotation. To prescind from the wheel–road

interaction parameters, the hypothesis of no sideslip of the wheel has been adopted but other tire

models can be easily implemented. The non-linear system stability is evaluated integrating the motion

equations and performing a fitting procedure on the natural steering oscillation.

& 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Keywords: Castor; Shimmy; Stability; Udwadia–Kalaba formulation

1. Introduction

The term ‘‘castor’’ is used to define any swivel-wheel whose contact patch area with theground lies behind the intersection of the steering axis with the ground. The study of castoroscillations regards many applications (even if there are some differences) ranging fromaircraft landing gear ‘‘shimmy’’ to motorcycle ‘‘wobble’’ and to the ‘‘waving’’ of towedvehicles. It consists of rapid oscillations of the whole castor wheel assembly about its kingpin. In some operational conditions, the oscillations can reach great amplitudes resulting inan increase in rolling resistance and tire wear; moreover, in these conditions, the castorcomponents are stressed by dangerous phenomena of fatigue and wear.

2.00 & 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

.jfranklin.2009.10.013

nding author.

dress: [email protected] (G. Di Massa).

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dx.doi.org/10.1016/j.jfranklin.2009.10.013

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Nomenclature

l1 distance between the castor center of mass G and wheel–ground contact pointO

l2 distance between castor center of mass G and steering axisG castor center of massg gravity accelerationIa castor trailing arm inertia matrixIw wheel inertia matrixKc camber frame stiffnessKy trailing arm lateral stiffnessd wheel offseth trailing arm heightma trailing arm massmw wheel massm whole castor massP wheel–road contact pointR wheel radiust caster trailT kinetic energyv forward speedV potential energyy lateral displacemente rake angled caster rotation around steer axisr whole castor inertia radiuss anti-shimmy dampingc camber angle½ �þ pseudo-inverse matrixz;os;on under-damped harmonic function parameters

D. de Falco et al. / Journal of the Franklin Institute 347 (2010) 116–129 117

The first scientific papers on the shimmy phenomenon were published about 80 yearsago but many details on the phenomenon mechanism are not yet fully understood since itdepends on a great number of factors such as overall vehicle dynamics, compliance of thecastor components, backlashes and tire characteristics [1–5].

One of the most common castor wheels is that used for supermarket trolleys, officechairs, etc. (Fig. 1a); the wheel is auto-aligning with respect to the motion direction as itcan steer around a vertical king pin whose axis is advanced with respect to the wheel axis.The general castor scheme (Fig. 1b) is characterized by a steering axis tilted by an angle(caster angle) with respect to the vertical direction, related to the wheel radius R, the trail t

and the wheel offset d by the following expression:

t ¼ Rtge�d

cose: ð1Þ

ARTICLE IN PRESS

Fig. 2. Planar castor scheme.

Fig. 1. Basic castor schemes.

D. de Falco et al. / Journal of the Franklin Institute 347 (2010) 116–129118

Expression (1) shows that a desired trail value can be obtained by adopting differentcombinations of R, e and d; for example for e ¼ 0, it follows: t ¼ �d (Fig. 1a).Basic castor dynamic behavior can be analyzed by adopting a simple planar model

(Fig. 2) in which the wheel is considered as non-rotating component able to exchangelateral forces with the ground. The wheel–arm assembly can rotate around the king pintranslating at a constant forward speed v. The planar models do not provide wellapproximated results since they cannot take all the main castor parameters into accountalthough they provide useful qualitative results; in some cases they have even been adoptedto study the dynamic behavior of important mechanical systems [6]. It is known thatshimmy oscillations arise if at least one of the castor components has lateral complianceable to generate a restoring action. To simplify the mathematical model, the castorcomponents can be modeled as rigid bodies and the compliance can be represented by alumped stiffness (spring).Even wheel–ground interaction can generate shimmy oscillations as the lateral force can

be seen as a restoring action proportional to the sideslip angle through sideslip stiffness; fora more approximate interaction model, its non-linear characteristic and the relaxation

ARTICLE IN PRESSD. de Falco et al. / Journal of the Franklin Institute 347 (2010) 116–129 119

length must be taken into account (i.e. the time delay between the lateral force and thesideslip angle).

To reduce the number of system parameters and leaving aside the tire characteristics, itcan be assumed that there is no wheel sideslip and therefore the wheel–ground contactpoint velocity is null. From an analytical point of view, this condition represents a non-holonomic constraint.

According to this hypothesis and considering a rotational damping in the hinge A

(Fig. 2), the Routh–Hurwitz criterion states that the system is stable if the two followingconditions are met:

v40;

r2ol1l2 þsd

mv;

8<: ð2Þ

r being the system inertia radius with respect to the center of mass. Set the maximumcastor forward speed v, the second condition allows to evaluate the amount of damping srequired to stabilize the castor:

sZmv

dðr2 � l1l2Þ: ð3Þ

From this simple bi-dimensional model it can be deduced that, increasing the forwardspeed, the system may become unstable. In the following, a procedure to obtain a three-dimensional castor equations is described. The model takes into account the gyroscopiceffects, due to the lateral deflection of the strut and the castor angle inclination. The effectsof some parameters on the system stability are evaluated and discussed.

Fig. 3. Three-dimensional castor scheme.

ARTICLE IN PRESSD. de Falco et al. / Journal of the Franklin Institute 347 (2010) 116–129120

2. The model

The castor is modeled as two rigid bodies: the trailing arm and the wheel. To write themotion equations, beyond the Oxyz inertial reference, the following references areconsidered (Fig. 3):

�
Ax0y0z0 that is a castor arm fixed reference, with the origin in hinge, the z0�axis on thesteering axis (pointing upward) and the y0�axis parallel to the wheel rotation axis andpointing on the left with respect to the forward direction.Ax0y0z0 is rotated, with respect to the inertial reference, by the following sequence:(a) e (caster angle) around the y-axis;(b) d (steer angle) around the rotated z-axis; and(c) c (camber angle) around the rotated x-axis; � Cx00y00z00 with the origin in wheel center and rotated with respect to Ax0y0z0 of an
angle a around the y0�axis so that the x00�axis results parallel to the groundplane;
� Px
000

y000

z000

fixed in the wheel-plane contact point P and rotated, with respect to Cx00y00z00,of an angle b around x00�axis so that the y

000

�axis results parallel to the ground plane.

The arm can translate along all the three directions of the inertial frame and can performcamber and steer rotations; the wheel rotates around y00�axis with an angular velocity _y.The wheel is in contact with the ground plane in the point P and it is assumed that it canrotate on the ground plane without any slip (longitudinal and lateral).Furthermore, the system is in steer free control and the camber rotation is counteracted

by means of a rotational spring (Kc) that has the meaning of the arm flexural stiffness; azero thickness wheel has been considered to ignore the tire–ground contact point variationalong the transverse tire profile and the related moments.For sake of simplicity, it is assumed that the castor arm has a constant forward velocity v

along the inertial x direction, and the lateral motion (along the y-axis) is counteracted bymeans of a linear spring (Ky) that represents the lateral flexibility of the castor support.To deduce the equations of the motion the following steps were performed [7]:

1.
adopting the Lagrange formulation, the equations of the unconstrained castor, in thethree-dimensional space, were written (unconstrained system);
2.
appropriate constraints equations were added to the unconstrained motion equations,to impose: (a) the wheel to remain in contact with the ground; (b) the frame to advancewith a constant forward velocity, v; (c) the wheel–ground contact point P to have a zeroslip velocity;
3.
the last step involves the use of the Udwadia–Kalaba formulation [8–11].
2.1. The unconstrained system

The configuration of the castor is defined by the vector of the generalized coordinates:q ¼ ½x; y; z; d;c; y� that comprises the position of point A fixed to the castor frame ½x; y; z�and the three angular coordinates ½d;c; y� defined as steer, camber and wheel rotations,respectively. For the unconstrained system all these coordinates are independent and the


kinetic energies assumes the following simple expression:

T ¼ TTa þ TTw þ TRa þ TRw; ð4Þ

where TTa;TTw and TRa;TRw are the translational and rotational kinetic energy of the armand of the wheel, respectively:

TTa¼ 1

2mað _x

2G þ _y2

G þ _z2GÞ; ð5Þ

TTw¼ 1

2mwð _x

2C þ _y2

C þ _z2CÞ; ð6Þ

TRa¼ 1

2½oa�

T � ½Ia� � ½oa�; ð7Þ

TRw¼ 1

2½ow�

T � ½Iw� � ½ow�; ð8Þ

being

�
½xG; yG; zG�, the coordinates of the frame center of mass:
xG ¼ xG0cosecosd� yG

0ðcosesindcoscþ sinesincÞ þ zG0ðcosesindsinc� sinecoscÞ þ x;

yG ¼ xG0sindþ yG

0cosccosdþ zG0sinccosdþ y;

zG ¼ sinecosd� yG0ðsinesindcosc� cosesincÞ þ zG

0ðsinesindsincþ cosecoscÞ þ z:

8><>:

ð9Þ

�
oa and ow, the moving frame components of the angular velocity of the arm and wheel,respectively:
½oa� ¼

_c_dsinc_dcosc

264

375; ð10Þ

½ow� ¼

_ccosa� _dsinacosc_dsincþ _a

_csinaþ _dcosacosc

264

375: ð11Þ

�
Ia and Iw, the inertia matrices referred to the arm and wheel center of mass. Thepotential energy is
V ¼ �ðmazG þmwzCÞgþ12ðKcc

2þ Kyy2Þ: ð12Þ

The external forces applied to the caster can be schematized as a vertical load actingalong the inertial z direction, and a couple �s _d, acting around the steer axis proportionalto the steering velocity, that represents the presence of a viscous anti-shimmy damper.As the adopted coordinates are independent, the unconstrained motion equations can bewritten with the Lagrange equation:

d

dt

@L

@ _q

� ��@L

@q¼ ½0 0 � FZ � s _d 0 0 0 0�T ; ð13Þ

where L is the Lagrangian: L ¼ T � V . The unconstrained motion equations can bewritten in the following matrix form:

M � €q ¼ Q; ð14Þ


where M is the (full range) 6 by 6 inertia matrix of the unconstrained system and Q is theapplied forces vector. Eq. (14) yields the unconstrained equations of motion of the castor.

2.2. The constraints

The following constraint equations connect the elements of the q vector (dependentcoordinates) to its derivatives, so that the constrained system realize a castor that advancewith a constant forward velocity, whose wheel is free to rotate without lateral orlongitudinal slip on the ground plane. The first constraint equation states that the contactpoint P is on the ground plane:

zP ¼ ðRsina� dÞsinecosdþ Rðsincsindsineþ cosccosecosaÞcosaþ hsincsindsineþ hcosccose� z

¼ 0; ð15Þ

where (Fig. 3)

a ¼ arctancosdsine

sincsindsineþ coscþ cose

� �ð16Þ

is the angle that the Ax0y0z0 reference frame has to be rotated, around the moving y0�axis,so that contact point P, in the reference Cx00y00z00 has coordinates ½0; 0;�R�, where R is thewheel radius.The second constraint equation states that point A has a constant forward velocity v

along the x direction:

_x ¼ v: ð17Þ

The last two constraints regard the no-slip conditions (longitudinal and lateral) of thewheel:

_x000

P ¼_y � R;

_y000

P ¼ 0;

(ð18Þ

where _x000

P and _y000

P are the velocity components of contact point P in the reference Px000

y000

z000

that is rotated of an angle b around the x00�axis (Fig. 3), with

b ¼ arctancoscsindsine� sinccose

sinacosdsineþ cosasincsindsineþ cosacosccose

� �: ð19Þ

Point P velocity components, in this new reference, can be easily obtained by adoptingthe following rotation matrix:

_x000

P

_y000

P

" #¼

_xP

_yP

" #T

�

cosacosdcose

�sinysincsindcose

þsinacoscsine

cosaðsincsindcose� coscsineÞ

þsinacosdcosesinb

�cosbðcoscsindcoseþ sincsineÞ

cosasindþ sinasinccosdðsinasind� cosasinccosdÞsinb

þcosbcosccosd

266666664

377777775:

ð20Þ


In this way, a set of four holonomic and non-holonomic constraint equations have beenobtained. Differentiating (with respect to the time) the position constraint equation (16)twice and the velocity dependent constraint equations (17) and (18) only one time, theconstraint equations can be expressed as linear form of accelerations, in the followinggeneral form [8,9]:

Aðq; _q; tÞ � €q ¼ bðq; _q; tÞ: ð21Þ

2.3. The constrained equations of motion

From the Udwadia–Kalaba formulation [8,9] the equations of motion for theconstrained system can be derived from the unconstrained equations of the motion (14)and from (21) that specifies the system constraints; the acceleration of the system can beexpressed in the following way:

€q ¼ aþM�1=2 � ½A �M�1=2�þ½b� A � a�; ð22Þ

where a is the unconstrained system acceleration.

3. Numerical results

As the natural castor steer oscillations are constituted by a predominant under-dampedharmonic oscillation, to evaluate the influence of the castor parameters on the systemstability, the following procedure was adopted: for different operational and/or geometric/inertial parameter the motion equations were numerically integrated superimposing asmall angular steer velocity as initial condition. Then each solution of the non-linearsystem was fitted with an under-damped harmonic function:

xðtÞ ¼ X � e�zont � sinðostþ jÞ: ð23Þ

Eq. (23) represents the general solution of a linear single dof damped system that best fitthe solution of Eq. (22) for the assigned parameters.

The fitting allows to identify the damped circular frequency os and the product zon

between the damping ratio and the circular undamped natural frequency on. Being

os ¼ on

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� z2

q) on ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2

s þ o2nz

2q

ð24Þ

it is possible to deduce the damping ratio z. The sign and the value of the identifieddamping coefficients can be adopted to evaluate the system stability. So z can beconsidered as a non-dimensional index able to describe the rapidity with which the castoramplitude oscillations increase or decrease.

As an example, in Fig. 4a, the numerical solution and the fitted curve are compared; thefitting error is generally very small as highlighted in Fig. 4b (in log scale). The figures wereobtained using the parameters reported in Table 1, with v ¼ 40m=s and s ¼ 30Nms=rad.

In Fig. 5 is reported the numerical integration error in the satisfaction of the constraints,obtained substituting the numerical solution in the constraint equations (15) and (18),respectively.

This procedure falls if the solution has more than one predominant harmoniccomponent, as it happens, for example, for systems with low lateral stiffness, and if thesystem is strongly unstable, so that the solution rapidly diverges. In both cases thenumerical solution cannot be approximated with an under-damped harmonic solution.

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Table 1

Numerical simulations parameters.

Castor mass ma 18 kg

Flexural arm stiffness Kc 45 kNm/rad

Lateral arm stiffness Ky 1

Wheel radius R 0.278m

Castor trail t 0.0874m

Wheel mass mw 8.7 kg

Castor arm height h 0.67m

Arm CM coordinates ½xG0; yG

0; zG0� ½0:03; 0:0;�0:52�m

Arm inertia ½Iax; Iaz� ½1:13; 0:26�kgm2

Wheel inertia ½Iwx; Iwy; Iwz� ½0:19; 0:31; 0:19�kgm2

0 0.2 0.4 0.6 0.8 1

0

time [s]0 0.2 0.4 0.6 0.8 1

0

2

time [s]

Fig. 5. Constraint equations numerical errors.

0 0.2 0.4 0.6 0.8 1−0.01

−0.005

0

0.005

0.01

time [s]

δ st

eer r

otat

ion

[rad]

Steer rotation angle fitting

fitted curve

0 0.2 0.4 0.6 0.8 110−10

10−8

10−6

10−4

10−2

time [s]fit

ting

erro

r

fitted and numerical solution absolutedifference

Fig. 4. Fitting procedure example.



To test the procedure several numerical simulations were performed for two differentcastor schemes: the first one characterized by a vertical steer axis (e ¼ 0) and the secondone by a tilted steer axis; in both cases, the lateral stiffness, Ky, was assumed infinite. Theothers reference castor parameters are reported in Table 1.

The numerical results are shown in the form of diagrams of the natural circularfrequency, os, and of the damping ratio z versus the forward castor velocity, v. In eachdiagram three curves, for three different values of a castor parameter, are reported.

3.1. Vertical steer axis

In Fig. 6, three curves are plotted for three different values of the shimmy dampingcoefficient, s. The figure shows that if s is equal to zero (dotted curve) the system is alwaysunstable while, for s different from zero, the stability decreases with the forward velocity

0 10 20 30 400

10

20

30

40

50

60

70

forward velocity [m/s]

frequ

ency

[rad

/s]

0 10 20 30 40

0

0.1

0.2


dam

ping

ratio

ζ

030 Nms/rad60 Nms/rad

030 Nms/rad60 Nms/rad

Fig. 6. Influence of steer damping s.

−0.2

−0.1

0

0.1

0.2

0.3

0 10 20 30 400

10

20

30

40

50

60

70


frequ

ency

[rad

/s]

0 10 20 30 40forward velocity [m/s]

dam

ping

ratio

ζ

22 kNm/rad45 kNm/rad60 kNm/rad

22 kNm/rad45 kNm/rad60 kNm/rad

Fig. 7. Influence of Kc stiffness coefficient.


up to a minimum and then the system has a recovery of stability due to the gyroscopicstabilizing couple; this effect is, obviously, not previewed by the bi-dimensional models,but it is well known in literature [1] and will be discussed in the following. The curvefor s ¼ 0 is not completely drawn as, for v410m=s, the system is strongly unstable. Theintermediate curve (solid line), obtained for s ¼ 30Nms=rad, is assumed as referencecurve for the following numerical results.In Fig. 7 the influence of the flexural stiffness is highlighted; a stiffness increase leads to

higher natural frequencies and has a negative effect on the stability.In Figs. 8 and 9, the influence of the moments of inertia is evaluated; increasing the Iwy

polar wheel inertia the system results more stable as a greater gyroscopic couple acts on thesystem while, an increase of the Iaz moment of inertia has a negative effect on the systemstability. This result is in agreement with the planar model (2) which requires low inertia tohave a stable system.

−0.2

−0.1

0

0.1

0.2

0.3

0 10 20 30 400

10

20

30

40

50

60

70


frequ

ency

[rad

/s]


dam

ping

ratio

ζ

0.05 kgm2

0.31 kgm2

0.70 kgm2

0.05 kgm2

0.31 kgm2

0.70 kgm2

Fig. 8. Influence of Iwywheel polar inertia.

−0.2

−0.1

0

0.1

0.2

0.3

0 10 20 30 400

10

20

30

40

50

60

70


frequ

ency

[rad

/s]


dam

ping

ratio

ζ

0.05 kgm2

0.26 kgm2

0.50 kgm2

0.05 kgm2

0.26 kgm2

0.50 kgm2

Fig. 9. Influence of IaZframe z inertia.


Fig. 10 was obtained for different values of the center of mass height of the trailing armand so of the whole system. It means that higher values of the castor inertia with therespect to the x0�axis increases the system stability.

3.2. Tilted steer axis

The following simulations were performed to highlight the influence of the tilted steeraxis castors like the trail, the rake angle and the wheel offset.

The first simulations were performed (Fig. 11) for different values of the dampingcoefficients as in Fig. 6, but with a rake angle e ¼ 0:47 rad (271). The comparison with thee ¼ 0 case (Fig. 6) shows that, fixing the trail value t, an increase of the rake angle hasmainly a negative effect on the stability.

−0.2

−0.1

0

0.1

0.2

0.3

0 10 20 30 400

10

20

30

40

50

60

70


frequ

ency

[rad

/s]


dam

ping

ratio

ζ

−0.25 m−0.52 m−0.75 m

−0.25 m−0.52 m−0.75 m

Fig. 10. Influence of zG frame CM coordinate.

0

0.1

0.2

0.3

0 10 20 30 400

10

20

30

40

50

60

70


frequ

ency

[rad

/s]


dam

ping

ratio

30 Nms/rad60 Nms/rad120 Nms/rad

30 Nms/rad60 Nms/rad120 Nms/rad

Fig. 11. Influence of s damping coefficient.

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0

0.1

0.2

0.3

0 10 20 30 400

10

20

30

40

50

60

70


frequ

ency

[rad

/s]

00

10 20 30 40forward velocity [m/s]

dam

ping

ratio

0.3 rad0.47 Nrad0.6 rad

0.3 rad0.47 Nrad0.6 rad

Fig. 12. Influence of rake angle e with d ¼ cost.

0

0.1

0.2

0.3

0 10 20 30 400

10

20

30

40

50

60

70


frequ

ency

[rad

/s]


dam

ping

ratio

0 rad0.47 Nrad0.7 rad

0 rad0.47 Nrad0.7 rad

Fig. 13. Influence of rake angle e with t ¼ cost.


The intermediate curve, obtained for s ¼ 60Nms=rad, is assumed as reference curve forthe following numerical results reported in Figs. 12 and 13 obtained varying:

�
the rake angle with d constant (Fig. 12); � the rake angle with t constant (Fig. 13).
In both cases the rake angle makes the system less stable especially at high forward speedvelocity. Only for a constant offset the stability increases if the velocity is low enough.

4. Conclusions

In the paper a shimmy model have been reported. The adopted no-sideslip hypothesisallows to perform simple observations about the damping necessary to stabilize the


shimmy, taking into account the geometric and inertial castor characteristics and ignoringthe tire ones. With respect to the planar models it allows to take into account thegyroscopic couples and the characteristic castor geometry parameters in case of tilted steeraxis. Some examples, to test the employed procedure and the influence of the several castorparameters on the system stability, were tested and discussed.

References

[1] W.J. Moreland, The story of shimmy, Journal of Aeronautical Sciences 21 (1954) 793–808.

[2] J.P. Den Hartog, Mechanical Vibrations, McGraw Hill Book Company, New York, 1956.

[3] H.B. Pacejka, Analysis of the shimmy phenomenon, in: Proceedings of the Institute of Mechanical Engineers,

1965.

[4] R.S. Sharp, The stability and control of motorcycles, Journal of Mechanical Engineering Science 13 (5)

(1971).

[5] V. Cossalter, Motorcycle Dynamics, Lulu Editor 2006, ISBN 978-1-4303-0861-4.

[6] E. Coetzee, Shimmy in aircraft landing gear, Problem presented by Airbus: /http://www.smithinst.ac.uk/

Projects/ESGI56/ESGI56-AirbusShimmy/ReportS.

[7] F.E. Udwadia, G. Di Massa, Sphere rolling on a moving surface: application of the fundamental equation of

constrained motion, unpublished.

[8] F.E. Udwadia, R.E. Kalaba, Analytical Dynamics—A New Approach, Cambridge University Press,

Cambridge, 1996.

[9] F.E. Udwadia, R.E. Kalaba, Dinamica analitica - Un nuovo approccio, a cura di D. de Falco - Edises srl

ISBN: 9788879595353 Edizione: I/2009.

[10] F.E. Udwadia, R.E. Kalaba, A new perspective on Constrained Motion, Proceedings of the Royal Society of

London, Series A 439 (1992) 407–410.

[11] F.E. Udwadia, R.E. Kalaba, On motion, Journal of the Franklin Institute 330 (1993) 571–577.

http://www.smithinst.ac.uk/Projects/ESGI56/ESGI56-AirbusShimmy/Report

http://www.smithinst.ac.uk/Projects/ESGI56/ESGI56-AirbusShimmy/Report

Documents

On the castor dynamic behavior