J. agric. Engng Res. (1987) 37, 8 1-91
Ride Vibration
Transfer Functions
of Agricultural
Tractors:
between the Ground and the
Tractor Body
J. A. LINES*
Two experiments are described in which measurements were made of the transfer function
between the ground profile and the motion of an unsprung agricultural tractor. The results of
these experiments indicate that significant differences exist between the characteristics of rolling
and non-rolling tyres. The stiffness of tyres was found to decrease at the onset of rolling and
thereafter remain relatively constant for further increases in rolling speed. The damping
characteristic of the tyres was found according to circumstances to increase or decrease
substantially with changing rolling speed. The dynamic characteristics of rolling tyres were found
to be relatively unaffected by changes in vibration amplitude. The motion of the tractor was
found to be coherent with the profile of the ground only at frequencies lower than half the
passing frequency of the tyre lugs.
1. Introduction
There are numerous published records of attempts to model the ride vibration of
unsuspended vehicles such as agricultural tractors. Unfortunately the results are seldom
supported by experimental evidence. In a previous paper’ a simulation method has been
described and shown to be unable to predict the dynamic behaviour of a tractor with any
real reliability. In this simulation each tyre was modelled as three mutually perpendicular
springs orientated axially connecting the bottom of the tyre to the ground surface. A viscous
damper was coupled in parallel with each spring. The tractor was described as a rigid
massive body with a pivoted front axle. In order to establish the reason for the failure of this
simulation, the response of a tractor was measured in the laboratory by using a hydraulic
ram to vibrate a plate on which one of the tractor tyres was standing. The acceleration of the
plate and the consequent acceleration of the tractor in various directions was measured and
transfer functions calculated from these measurements. When these transfer functions were
inserted into the simulation in place of the predicted transfer functions, the simulation was
still not able to predict correctly the ride vibration behaviour of the tractor. Two further
experiments have now been carried out in order to enable these results to be understood and
to establish how a more accurate model of a driving tractor might be produced. In both of
these the tractor wheels were rolling, but while one was conducted in the laboratory, the
other was made using a modified tractor driven over a test track.
2. Background
It is now well established that the dynamic characteristics of tyres vary with rolling
speed. *- ” In so far as the tractor can be represented by idealized linear components, i.e.
masses, springs and viscous damping, the radial spring stiffness of tyres is found to decrease
at the onset of rolling and recover again slowly as rolling speed increases. However reports
about the amount by which the stiffness initially decreases, and the point at which it starts to
* Agricultural Vehicles Division, NIAE, Wrest Park, Silsoe, Bedford MK45 4HS, UK
Received 9 September 1985; accepted in revised form 29 June 1986
81
002 l-8634/87/07008
I+ II
%03.00/O
0
1987 The British Society for Research
in Agricultural
Engineermg
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recover differ widely. Laib* indicates that on tractor rear tyres he found only a 5% decrease
in tyre stiffness and a minimum stiffness occurring at about 1 km/h. This contrasts with the
results of Hooker3 and others whose work on road vehicle tyres indicates a minimum
stiffness occurring at 10 km/h which is 30% below the non-rolling tyre stiffness. The stiffness
of non-rolling tyres is reported to decrease substantially as the amplitude of deflection
increases.3*4 The same authors report that the stiffness of rolling tyres is relatively unaffected
by variation in amplitude.
The damping coefficient of rolling tyres is reported as decreasing with increasing rolling
speed. *M Laib* reports that at a frequency of 4 Hz the damping coefficient of a tyre
decreases tenfold between 0 and 10 km/h. Giihlich et d4 indicate that the damping
coefficient of a tractor tyre increases at the onset of rolling by 20% reaching a peak at about
0.3 km/h then decreasing in an exponential way to 40% of its non-rolling value by the time
the speed reaches 10 km/h.
The axial (lateral) and tangential (longitudinal) characteristics of agricultural tyres have
not been subjected to much investigation. Stayner and Boldero” have made free vibration
measurements
on a non-rolling tyre and related these to stiffness and damping
characteristics of a spring and viscous damper in parallel. Other authors13 have reported
force deflection curves. In contrast road vehicles have received much more attention both in
the development of relevant theory and the measurement of tyre characteristics. There are
however important differences between road and agricultural tyres which limits the
usefulness of much road vehicle research.
One such difference is the ratio between the tyre contact patch length (or more strictly the
tyre relaxation length) and the rolling speed. A road vehicle travelling at 20 m/s with a tyre
contact patch length of O-1 m completely changes the rubber in contact with the road surface
200 times a second. An agricultural tractor travelling at 2 m/s with a tyre contact patch
length of 0.4 m changes the rubber in contact with the road surface only five times a second.
Consequently the lateral and longitudinal response of these tyres to, say, a 3 Hz vibration is
likely to be quite different.
At zero or very low speed when the rate of change of the contact patch is low compared
with the frequency of vibration, the contact patch will not move with respect to the road
surface in response to the vibration. All the restraining force and energy dissipation takes
place in the tyre carcass. The simplest way of representing the longitudinal and lateral
characteristics of the tyre under these circumstances is with a spring and damper in parallel
in the longitudinal direction, and another such pair in the lateral direction connecting the
wheel rim to the ground.
At high rolling speeds a rolling tyre responds to side force with side slip (and similarly to
longitudinal force with tangential slip). The relationship between the slip angle (a) and force
(F) can be represented as
F = LA(1 -e-ka)
where F is the side force, L the radial load on the tyre, a the angle which the path of the tyre
contact patch makes with the plane of the tyre (in radians) and A and k are constants. Even
at low speeds and high levels of vibration the angle a is small, so the expression above can be
simplified to
F = LC,a
where Ak = C, which is the coefficient of slip. If the rolling speed of the wheel is I/, and the
lateral velocity of the tyre contact patch is &, then the above relationship can be represented
as
F=LC,$
J.
A.
83
LINES
The side slip of the contact patch on the road therefore responds with a velocity is which is
proportional to the side load applied. The tyre road interface therefore behaves as a viscous
damper in series with the characteristics of the tyre carcass and with a damping coefficient
given by
McAllister’4 has measured the side force developed on a variety of surfaces by agricultural
tyres. Values of C, obtained in the experiment are typically 4 rad-‘. Hence the apparent
damping at the tyre ground interface due to side slip for a medium size tractor with say
10 kN load on each rear tyre and driving at 4 m/s is 10000 Nsm-‘. This is substantially
larger than the tyre carcass damping which is typically 2000 Nsm- ’ (Stayner and Boldero).”
In the same way, the longitudinal forces are developed by tangential slip. For small slip
values a similar relationship to that given above is valid:
where j, is the difference between the no slip speed and the actual speed, C,_is the coefficient
of longitudinal slip and FL is the longitudinal force. A typical value” for C, is 85 which for
the same conditions as before implies a viscous damping at the tyre ground interface of
20 000 Nsm- ‘. Again this is substantially larger than the damping coefficient of a non-rolling
tyre which is around 2500 Nsm- ‘.
Agricultural tyres are operated in a regime between the low and high speed cases given.
The longitudinal deflection characteristics of tractor tyres are dominated by bending of the
individual tyre lugs” so the relaxation length is short. Lateral deflection of tractor tyres is
dominated by shearing of the tyre carcass so the relaxation length for this type of deflection
is longer. It is therefore to be expected that the longitudinal deflection characteristic of the
tyre will more nearly be described by the high rolling speed model than will the lateral
characteristics.
It was a purpose of the experiment to observe the effects on tractor vibrational changes in
tyre characteristic due to rolling speed.
In order to predict the vibration of a vehicle as it is driven over a track of a known profile
it is necessary that all the sources of excitation are included. It is also necessary that the
vibration caused by each known source adds linearly to the vibration caused by the other
sources and that the characteristics of the tractor do not change with changing excitation
amplitude. Measurements have not yet been reported which show to what extent these
conditions are met. The work presented here is also aimed at clarifying these points.
3. Experiments
In order to measure the transfer functions of the tractor between the ground contact patch
at one of its wheels and the vehicle body response, it was necessary to excite one wheel with
a signal which was quite independent of any signals at the other wheels. This situation does
not occur during normal tractor driving because on each side of the tractor the front and
rear wheels pass over virtually the same ground profile. Reported here are two different
methods which were used to create this situation.
3.1. Laboratory method
An agricultural tractor was placed on an hydraulic test stand” which enabled each of its
wheels to be independently vibrated in the vertical direction by an hydraulic actuator
84
RIDE
VIBRATION
OF TRACTORS
(Fig. I). The surface on which each of the wheels rested was covered by small diameter
(50 mm) rollers so that the wheels could be rotated. On this stand the tractor wheels were
driven under tractor power at various speeds. The tractor was excited by vibration under
one or more wheels. The vertical acceleration of the pads on which the wheels rested was
measured, as was the acceleration of the tractor body, measured on the cab floor under the
operator’s seat in the vertical, longitudinal, lateral, roll, pitch and yaw directions. In this
way the transfer functions of the tractor through each wheel could be measured at various
tyre rolling speeds and vibration amplitudes with and without the presence of uncorrelated
vibration input at the other wheels. The wheel for which the transfer function was being
calculated was excited with random excitation which had a reasonably flat acceleration
power spectrum from 0.5 to 12 Hz. The other three wheels were either not excited or excited
with vibration, approximately equal in acceleration rms to the first wheel. This excitation
was similar to that which would be received travelling along a rough track. The three inputs
were correlated accordingly.
The advantages of this method of exciting the tractor were that vibration amplitude and
driving speed could be independently controlled, that the filtering effect of the tyre
enveloping the track17 did not confuse comparisons made between different driving speeds,
and that long statistically reliable signal lengths could be used and easily repeated. The
method produced repeatable and well correlated measurements.
The disadvantage of this method was that the excitation used was somewhat removed
from reality. Because the rollers under the tyres were free to rotate in response to the tyres
rather than being constrained to rotate at a given velocity by the momentum of the tractor,
any longitudinal deflection of the tyre is likely to have resulted in acceleration of the rollers
rather than the tractor. The longitudinal vibration of the tractor was also affected by a
horizontal steel hawser which restrained it from driving off the front of the test stand. Since
the longitudinal vibration of the vehicle is linked by the inertia of the tractor to the pitch and
vertical vibration, measurements in these other two directions too may not be fully
representative of the real situation. A further drawback to this test method is that in real life
a change of ground level under a wheel is always accompanied by a change of gradient
whereas on the test stand the surfaces remained level. The response of a tractor wheel to
such a change of gradient is not yet fully understood, but one would expect it to affect the
longitudinal, and hence also the pitch and vertical response.
Fig. 1. The four ram hydraulic test stand
J.
A.
LINES
85
3.2. Test track method
A tractor was driven over a standard rough track” with one wheel on the rough profile
and the other three on relatively smooth concrete. This was achieved by changing the wheel
track width asymmetrically (Fig. 2). In this way the main vibration input was through only
one wheel, and this was not correlated with any input from the other three wheels. The
transfer function was again measured between the second differential of the ground elevation
under the tyre and the acceleration in six directions measured on the cab floor under the
driver’s seat. The ground elevation was measured simultaneously with the tractor
acceleration, using an optical non-contacting displacement transducer.lg A similar tractor
had been excited by a vertical input at one wheel using a hydraulic actuator as mentioned in
the introduction. In this way comparison could be made between the response of the tractor
when the tyres were rolling, and when not rolling. Further details of this experiment are
given elsewhere.”
The advantages of this method are that it very closely represents the actual situation of a
tractor driving on a rough surface. The disadvantages are that only one rather short (100 m)
track could be used which resulted in noisy and somewhat unreliable transfer function
measurements. It was not possible to alter the profile or amplitude of either the rough profile
for the measured wheel or the smooth profile for the other three wheels. In order to interpret
the measurements made at different speeds it was necessary to model the spatial filtering
effect of the tyre as it enveloped the ground profile. The filtering model used, smoothly
decreased the levels of the ground excitation as its frequency increased, so that a 50%
decrease in input level was reached as the ground input wavelength approached a length
twice that of the tyre contact patch. Details of the filter are given in reference 20. To assess
the effect which the wheel track asymmetry had on the tractor performance, it was driven
with both standard and asymmetric track widths over a track and the measured ride
accelerations were compared. These indicated that no significant difference in the vehicle
behaviour resulted from the asymmetric track widths.
Fig. 2. Track method used to excite tractor with (left) a rear wheel input and (right) a front wheel input
RIDE
86
VIBRATION
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TRACTORS
4. Results
4.1. General description
Transfer functions of the stationary tractor show a highly resonant system which is similar
for the two tractors used. Pitch and longitudinal transfer functions show a common
resonance frequency with the vertical and have in addition one other resonance. Roll and
lateral transfer functions have both of their resonance frequencies in common. The yaw
transfer function is less distinct but appears to share resonance frequencies with roll and
lateral. In this way the transfer functions are consistent with linear analysis which shows that
the system can be split into two independent sub-systems, one involving vertical,
longitudinal and pitch motion, the other involving roll, lateral and yaw motion.
4.2. Changes with amplitude
When the wheels are not rolling, changes in the amplitude of the input to the wheels cause
significant changes in the transfer functions (Fig. 3). In general the natural frequencies of the
system decrease-typically
by 10% for a 1 : 5 increase in r.m.s. level, and the effective level of
damping increases. The increase in damping varies considerably from one transfer function
to another. In most cases transfer functions for the front wheel inputs are more sensitive to
amplitude than those for the rear. Since the chan&e in input amplitude represents a
proportionally greater deflection of the smaller front tyres than the rear tyres, this is perhaps
understandable.
The effect of changes in excitation level on rolling wheels is much less than on stationary
wheels. An increase in the excitation level does not have an observable effect on the natural
frequencies but in some cases it increases the level of damping slightly. This effect seems to
be greater for the front wheels and for low driving speeds.
In the laboratory tests all the transfer functions were measured both with the other three
tractor wheels receiving no input, and with them receiving approximately equal amounts of
excitation to that of the measured wheel. This excitation was similar to that which might be
received travelling along a rough track, and the three inputs were correlated accordingly.
There is no evidence that the addition of this extra “noise” in the system changes the transfer
functions in any way.
4.3. Changes with rolling speed
4.3.1. Laboratory experiment
There are large differences between transfer functions measured with wheels rolling and
not rolling (Fig. 4). At a tyre rolling speed of 6 km/h the natural frequencies are found to be
typically 25% lower than for non-rolling tyres. No significant change can be observed
between the natural frequencies at 6 km/h and 11 km/h. This observation is consistent with
those of previous authors that tyre stiffness drops sharply at the onset of rolling and
thereafter increases only gradually. It appears from the transfer functions that the tyre
stiffness in all three axial directions (radial, lateral and longitudinal) could behave in this
way.
The measurements show large changes in damping as speed is varied between 0 km/h and
11 km/h. Damping of the lateral mode increases from about 20% to around 100% of the
critical level. Roll and yaw transfer functions show similar effects. The vertical transfer
function shows only a small change in damping. Longitudinal and pitch transfer functions
develop very much stronger resonances indicating a decrease in the damping level.
The transfer functions measured at 6 km/h show that the increase in damping in the
lateral, roll and yaw is gradual and proportional to rolling speed. However the apparent
J.
A.
LINES
87
Frequency,
Hz
Fig. 3. Transfer jiinctions showing vertical response to high (--)
and low (- -)
excitation for [left) stationary and (right) rolling wheels
1
Rear wheel
Vertm
levels qf rear wheel
Rear wheel
Lateral
8
3
Rear wheel
Longitudinal
Rear
wheel
-z
BL
0
8
0
0
Front wheel
Vertical
Front wheel
Lateral
Frequency,
Hz
Fig. 4. Laboratory experiment. Magnitudes of transfer functions, calculated between input at wheel and
tractor response at driver’s seat mounting point at 0 km/h (--),
6 km/h ( - - ) and I1 km/h (---,
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RIDE
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decrease in damping in vertical, longitudinal and pitch appears to be closely linked to the
change in tyre stiffness, varying as it does only slowly with rolling speed after the initial
change at the onset of rolling. It would seem then, that these two effects originate from two
different sources.
4.3.2. Test track experiment
Measurements of the transfer functions made on the test track show results that are
consistent with reduction of tyre stiffness at the onset of rolling and show only slow change
thereafter (Fig. 5). The decrease in -the apparent damping which was observed in the
laboratory
experiment for longitudinal
and pitch motion is not shown in these
measurements. The results show a progressive decrease in damping with increasing rolling
speed. The transfer function measured on the stationary tractor shows a much stronger
second resonance than any of the rolling measurements. The increase in damping observed
for roll, lateral and yaw movements is only apparent in the field experiment results for the
front wheel of the tractor. The rear wheel transfer function damping does not appear to
change substantially.
31
I
61:
:
Rear wheel
Vertical
o-
0
8
Rear wheel
‘Longltudlnal
Rear wheel
Roll
Front wheel
Vertical
Front wheel
Lateral
8
0
Frequency, Hz
Fig. 5. Field experiment. Magnitude of transfer function, calculated between input at wheel and response
),IIkm/h
(--)
and15kmlh
(....)
at driverS seat mounting point at 0 km/h (--
8
J.
A.
89
LINES
4.4. Additional excitation
The coherence associated with the measurements made in the laboratory is generally good
(Fig. 6). However at the wheel rotation frequencies (both front and rear), and at harmonics
of these, the coherence drops, indicating that the tractor is being excited by wheel rotation at
these frequencies. At half the tyre lug passing frequency and at the tyre lug passing frequency
of each wheel the coherence once again drops. Because of the power of excitation at these
frequencies and small variations in the driving speed, these bands of low coherence are wide.
Above half of the lug passing frequency the coherence remains much poorer than it was at
lower frequencies.
The coherence between excitation and response signals measured in the track experiment
was much poorer than for the laboratory experiment. In this case it decreases to a very low
value at a frequency which appears to be proportional to driving speed. This frequency
however was not obviously related to the tyre lug passing frequency as it was in the
laboratory experiment and was lower than the track slat passing frequency. The most likely
cause of this low value of coherence is the precision with which the profile of the track was
measured. Since the power in the ground surface displacement profile decreases rapidly with
increasing spatial frequency, the upper frequency limit of accurate frequency response
functions depends on the precision with which the track profiles have been surveyed.
5. Discussion
It is immediately apparent that for ride vibration simulations of unsprung vehicles to
produce accurate results the characteristics of rolling tyres must be used. Not only must the
tyre characteristics be measured on a rolling tyre, but the way in which the apparent
damping changes with rolling speed must also be understood and incorporated.
It has been previously noted’ that the predicted longitudinal ride vibration for a tractor is
closer to the measured vibration if the longitudinal damping of the tyres is increased to a
near critical level. It is therefore encouraging to see that the longitudinal damping in the field
experiment appears to be much higher for rolling than non-rolling cases. It is not clear at the
moment why the transfer functions measured across the rear wheel of the tractor on the test
track should vary from those of the front wheel and the results measured in the laboratory.
However, one common factor is that only the rear wheels of the tractor on the test track are
subjected to any significant amount of tangential deflection. In the other three cases either
because the wheels are not driven, or because of the test stand arrangement, tangential forces
between the wheel and the surface are very rapidly dissipated.
The amplitude non-linearity of the rolling tyre transfer functions is small. An important
01
0
1
12
0
Frequency,
Fig. 6. Coherence between
laboratory experiments
Hz
rear wheel input and tractor vertical response in (left) field
). 6km/h (- -), II km/h (+-)
and ISkmlh
at 0 km/h (--
and (right)
(....I
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VIBRATION
OF
TRACTORS
consequence of this is that the principle of superposition is valid, so that the vibration due to
the input at each wheel can be computed separately and summed to give the motion of the
tractor subjected to all four inputs simultaneously. Because of this it may well be possible to
make use of frequency domain methods for simulating ride vibrations, rather than the
comparatively slow and complex time domain methods.
6. Conclusions
The characteristics of rolling tyres are significantly different from those of non-rolling
tyres. The tyre stiffness characteristics appear to decrease at the onset of rolling and
thereafter to remain relatively constant with further increases in rolling speed. The
vibrational energy dissipated by the tyres may however increase proportionally with rolling
speed.
Amplitude non-linearities appear to be negligible for rolling tyres and the principle of
superposition therefore is applicable.
Wheel eccentricity and tyre lugs excite the tractor at specific frequencies. It appears that at
frequencies above half the tyre lug passing frequency, tractor vibration is only poorly
correlated with ground excitation.
Acknowledgement
The laboratory experiment described in this paper was carried out in the technical university of
Berlin, Institut fiir Landtechnik und Baumaschinen. I am grateful to Professor Dr Ing. H. Giihlich and
Dipl. Ing. A. Kising for their assistance during the experiment and for the use of their hydraulic test
stand. I would also like to thank Dr Koch of the Institut fur Wasserbau for the loan of analysis
equipment.
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