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APPENDIX
Consider a wheel equipped with a pneumatic tyre. The
wheel axle shall he connected to a straight guiderail by a
linear springidamper element. This guiderail shall be
moved across the ground at constant speed; funheron the
guiderail shall be inclined against the ground.
Motion of the wheel axle relative to the guiderail is
influenced by radial tyre force (also represented by a
linear springidamper element) and by circumferential
tyre force. Modeling of circumferential force depends on
assumptions made.
Al. "Cog-Wheel" Assumption
The wheel is assumed to rotate at that angular velocity
which is defined by the ratio of ground speed to
deflected tyre radius. Circumferential force is then
defined by rotational acceleration of the wheel enforced
by tyre radius variations. Since this radius depends only
on the coordinate of the wheel axle relative to the
guiderail, the equation of motion reflecting the "Cog-
Wheel" assumption has one degree of freedom. viz the
linear displacement of the wheel axle on the guiderail.
With regard to potential self-induced oscillations of this
system it is interesting to study the various damping
terns in that equation of motion.
DC Damping coefficient of the guiderail spring
DT Damping coefficient of tyre radial spring
W Wheel moment of inertia
VG Ground speed
XB Axle displacement relative to guiderail
RR Distance axle to ground
p Inclination of guiderail (positive front end up)
the complete damping term reads
-
IW. sin p .cosp .(VG + XB .cosp)
(RR+XB.sinp)'
The damping term resulting from the "Cog-Wheel"
assumption is negative (destabilizing). Since it increases
approximately proportional to ground speed, the
equation indicates that there might exist a critical ground
speed at which overall damping becomes negative.
This critical speed can he increased by a "hardware"
modification to the model by reducing inclination of the
guiderail. reality this woul
landing gear leg forward rake.
3-9
nean a reduction of
Improvement can also be achieved by a quasi "software"
modification. i.e. by pumping up the tyre to increase RR.
AZ. The "Slip Ratio" Assumption
As already discussed in present paper, circumferential
force on an almost freely rolling tyre is approximately
proportional to slip ratio SR.
If this assumption is introduced to the model then linear
displacement of the wheel axle and rotation of the wheel
are two separate degrees of freedom which are coupled
by the circumferential force.
This system is not as readily analyzed as the "Cog-
Wheel" system. However, from the main part of this
paper it is concluded, that one and the same axle linear
velocity relative to the guiderail will effect different slip
ratio rates, slip ratio rates becoming smaller with
increasing ground speed. That means that the ratio of
circumferential force oscillation amplitude to axle
displacement amplitude reduces with increasing ground
speed.
Apan from any phase shift effects it is therefore expected
that an eventual destabilizing effect of tyre
circumferential force on axle foreiaft motion will cease
with speed.
A3 Conclusion
Comparation of the "Cog-Wheel" model and the "Slip
Ratio" model perception lead to the conclusion that
sustained foreiaft bending oscillations may well occur
under unfavourable landing gear design parameters.
However. this type of self-induced oscillation should be
limited to the low to medium aircraft speed range.
In a realistic analysis of the phenomenon, care should be
taken that all important effects are included in the model.
For instance, effective "guiderail" inclination can be
influenced by fuselage bending and / or by stroking
motion of the oleo strut.