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The effect of wheel speed on rolling resistance
Jottrnal o[ Terramechan'cs, 1971, Vol. 8, No. 1, pp. 51 to 58. Pergamon Press
Printed in Great Britain.
THE EFFECT OF WHEEL SPEED ON ROLLING RESISTANCE R. G. POPE*
1. INTRODUCTION IT HAS been suggested in a previous paper [1] that the rolling resistance of a wheel on a clay soil may be dependent upon wheel speed. The intention of this paper is to present some results which verify this suggestion and confirm the influence of wheel speed on the rolling resistance as predicted by plate penetration tests. 2. WHEEL TESTS The tests were conducted using two rigid steel wheels (Fig. 1) both 10 in. dia. and 2½ in. rim width. Wheel A had solid side plates and wheel B was cut away
FIG. 1.
T e s t wheels.
*Department of Civil Engineering Royal Military College of Science, Shrivenham, Swindon, Wilts, England. Comunicated by A. R. Reece.
51
52
R.G. POPE
at the rim. The reason for using two wheels was to investigate the side cohesion drag on wheel A. The soil used in the mobility test rig was an artificial clay made up in the proportions 26-8% glycerine, 5.6% water, and 67.7% powered blue clay by weight. The justification for the use of this artificial clay is described elsewhere [1]. Before packing the soil in the test bed it was thoroughly mixed and passed through a pug mill. A kango h a m m e r was used to pack the clay and it was packed high and left to equalise for several days. The surface was then skimmed and levelled with a smooth steel roller (Fig. 2). The test bed was 24 in. wide by 9 in. deep by 20 ft
FIG. 2. Steel roller. long. Prior to each test the bed was rolled and levelled--eight passes. Two runs were made in each prepared bed as preliminary tests had shown there was no appreciable interference. Due to limitations of the carriage speed the wheels were tested over a range of ground speeds from 2 ft/min, to 15 ft/min. It would have been more appropriate to test the wheels at higher speeds and an attempt is being made to do this but the results are expected to be similar. The wheels were freely towed under dead loads and measurements of rolling resistance was measured through a I00 lb. load cell and sinkage with a 2 in. travel linear transducer, both directly recorded on a u.v. recorder. For each wheel load, a set sequence of tests were carried out so that the effects of soil variation and remoulding due to previous passes would be minimised. Each test sequence was carried out in one day and is shown opposite:
THE EFFECT OF WHEEL SPEED ON ROLLING RESISTANCE
53
FIG. 3. Wheel B under test. Test bed preparation
1
2
3
4
Test speed in left hand side of bed
2 ft/min
8 ft/min
15 ft/rnin
4 ft/min
Test speed in right hand side of bed
4 ft/min
15 ft/min
8 ft/min
2 ft/min
3. PLATE PENETRATION TESTS T h e pressure sinkage r e l a t i o n s h i p was a s s u m e d to fit the e x p r e s s i o n :
p=(ck~ + ~bk~)
(u t,,, "
F o r the soil used in the tests k~ was k n o w n to be a p p r o x i m a t e l y zero a n d was a s s u m e d to be zero for the analysis. T h e expression then simplifies to :
T h e values of ckc, n, and m were f o u n d f r o m sinkage tests with a 1 in. × 5 in. rigid steel footing at sinkage rates f r o m 0"02 to 20 i n . / m i n . Because of the difficulty o f testing the soil in the bins the plate p e n e t r a t i o n tests were m a d e in a s a m p l e of the s a m e soil p a c k e d into a rigid b o x at the same C o n e I n d e x V a l u e of 60 [2] as the soil in the test bins.
54
R.G. POPE
Figure 4 shows the effect of sinkage rate and m was found from the slope of the graph to be 0"15. This value is high compared with results obtained from circular footings in the same soil, possibly due to the width, 1 in., of the footing used or a shape effect. The values of (25.0) and n (0'31) were found from Fig. 5 at a. sinkage speed u0 of 0"2 in./min.
ck,,
i -7
-~
G
%/ o
5
~a o. ,J.
2_ 501
/'.~01
£', 3017
Log { s i n k o g e r o t e } ,
1397 in/rain
FIG. 4. Log (pressure) against log (sinkage rate) for 1 in. X 5 in. footing. 4. PREDICTED AND EXPERIMENTAL ROLLING RESISTANCE The expression used for predicting rolling resistance has been previously derived [1] and is as follows: l
R
(n+ 1)
[
3W
(bk~)1~'¢' ~1""'~j (3 -n)VD
] 'c-''~~+''°/(~'-~L+'')] (2)
where k.,~ (.ck,,+ ybk,;, ) (. V'" ) -
b"
D~'"u,/"
"
The experimental results were taken as the mean of the measured results from three controlled zonal lengths, each 1 ft long, of the test bin where the soil had been found most consistent in the preliminary tests. The results are plotted, together with the predicted results from the plate penetration tests, in Figs. 6 and 7. The
THE EFFECT OF WHEEL SPEED ON ROLLING RESISTANCE
55
agreement between predicted and experimental results is good particularly for the heavier load which is to be expected since the wheels are nearer the relative plate sinkage. For this heavier toad the predicted difference in rolling resistance between
15]4
-
. o rb```
J 12 .......
t
i
I
~
i
;
1.9
]-5
T6
T7
T.8
T9
oo
({)
Fla. 5. Log (pressure) against log (Z/B) for 1 in. x 5 in. footing at sinkage rate 0"2 in./min. Wheel A o Wheel B
Predicted 28
24
e~
'S
20
16
FIG. 6.
[
2
;
4
_
~
8
i
i5
Wheel speed, ftlmin Rolling resistance against wheel speed for wheel load ]24 lb.
56
R.G. POPE A Wheel A o [J
Wheel B
Predicted
6C
¢3 tn
5o or-
4o
I
i.
2
4
i
8
Wheel speed, F I G . 7.
15 =
ft/min
Rolling resistance against wheel speed for wheel load 204 lb.
the extreme wheel speeds was 8"1 lb. and the measured values were 6"5 lb. 7.3 lb. for wheels A and B respectively. The side plate drag on wheel A did not significantly affect the results. anomalous result at the lower sinkage caused by the lighter load is probably to variation in the soil surface state between testing the two wheels. For
\ 08
~ Wheel A o Wheel B ~:1 Predicted
\
07
"--CA 06
05
~ 04 09
03
02
0
2
4
8
Wheel speed,
15
ft/min
FIG. 8. Sinkage against wheel speed for wheel load 204 lb.
and The due the
THE EFFECT OF WHEEL SPEED ON ROLLING RESISTANCE approximately ½ in. sinkage at the heavier load the drag area 2-8 in s and a drag of about 12 lb. could have been expected based cohesion of the soil. This has obviously not occurred probably that the drag (area) is considerably less than 2"8 in 2 because the soil around the wheel and the rut was not cut clearly (Fig. 9). "
~/
/
/
57
would be about on the estimated due to the fact was pulled down
I
Wheel oxle
z ~/ /
J / /
JJ 2 __or,~t_ -soil
J/
surface J/~'
FIG. 9. Rut Formation.
EXPERIMENTAL SINKAGE The sinkage results for the heavier wheel load are shown in Fig. 8. Results for the lighter load are not presented since they were incomplete and inaccurate. The expression used for predicted sinkage has been previously derived [1] and is as follows : 5.
PREDICTED AND
z=
(3 - n) bk,~~ D
(3)
It can be seen from Fig. 8 that there is considerable discrepancy between the predicted and experimental results. This is accentuated by the fact that the predicted rolling resistance was less than the experimental and accordingly one would expect the predicted sinkage to be less. The sinkage discrepancy is probably caused by neglecting the rear part of the contact area which is likely to have more effect on the sinkage than on the rolling resistance. Schuring [3] suggests that at small sinkages, z / b ~ 10 per cent, the rear contact area may be as much as 40 per cent of the front contact area. As expected the sinkage of wheel A was slightly less than that of wheel B since wheel A was given some support from the adhesion on the side plates. 6. CONCLUSIONS The theory is adequate to predict the effect of wheel speed on rolling resistance though there is error in the sinkage prediction at the low sinkage investigated. Providing plate penetration tests are performed at a rate of deformation comparable to the vehicle situation then the error involved in neglecting the rate effect will be negligible compared to the non-homogeneity of the soils.
58
R.G.
POPE
NOTATION
b D kc, k~ p u,, W c k~ m, n R V z
width, in. wheel diameter, in. sinkage moduli pressure, lb. in -~ plate sinkage velocity ft/sec. wheel load, lb. cohesion, lb. in -~ constant sinkage exponents rolling resistance, lb. wheel ground speed, ft/sec. sinkage, in.
Acknowledgement --The author wishes to thank members of the Staff of the Royal Military College of Science who have aided him in his study and in particular A / P r o f e s s o r J. M. Hawkes who instigated the Soil-Vehicle facility at the College.
[l] [2] [3]
REFERENCES R. co. POPE. The effect of sinkage rate on pressure, sinkage relationships and rolling resistance in real and artificial clays. J. Terramechanics 6 (4), 31-38 (1969). M.E.X.E. Prediction of soil strength. Report No. 1187, February 1970, Christchurch, England. O. SCHURING. The energy loss of a wheel. Proc, 2nd lnt, Confi Soil Vehicle Mechanics, Quebec (1966).
Printed in Great Britain.
THE EFFECT OF WHEEL SPEED ON ROLLING RESISTANCE R. G. POPE*
1. INTRODUCTION IT HAS been suggested in a previous paper [1] that the rolling resistance of a wheel on a clay soil may be dependent upon wheel speed. The intention of this paper is to present some results which verify this suggestion and confirm the influence of wheel speed on the rolling resistance as predicted by plate penetration tests. 2. WHEEL TESTS The tests were conducted using two rigid steel wheels (Fig. 1) both 10 in. dia. and 2½ in. rim width. Wheel A had solid side plates and wheel B was cut away
FIG. 1.
T e s t wheels.
*Department of Civil Engineering Royal Military College of Science, Shrivenham, Swindon, Wilts, England. Comunicated by A. R. Reece.
51
52
R.G. POPE
at the rim. The reason for using two wheels was to investigate the side cohesion drag on wheel A. The soil used in the mobility test rig was an artificial clay made up in the proportions 26-8% glycerine, 5.6% water, and 67.7% powered blue clay by weight. The justification for the use of this artificial clay is described elsewhere [1]. Before packing the soil in the test bed it was thoroughly mixed and passed through a pug mill. A kango h a m m e r was used to pack the clay and it was packed high and left to equalise for several days. The surface was then skimmed and levelled with a smooth steel roller (Fig. 2). The test bed was 24 in. wide by 9 in. deep by 20 ft
FIG. 2. Steel roller. long. Prior to each test the bed was rolled and levelled--eight passes. Two runs were made in each prepared bed as preliminary tests had shown there was no appreciable interference. Due to limitations of the carriage speed the wheels were tested over a range of ground speeds from 2 ft/min, to 15 ft/min. It would have been more appropriate to test the wheels at higher speeds and an attempt is being made to do this but the results are expected to be similar. The wheels were freely towed under dead loads and measurements of rolling resistance was measured through a I00 lb. load cell and sinkage with a 2 in. travel linear transducer, both directly recorded on a u.v. recorder. For each wheel load, a set sequence of tests were carried out so that the effects of soil variation and remoulding due to previous passes would be minimised. Each test sequence was carried out in one day and is shown opposite:
THE EFFECT OF WHEEL SPEED ON ROLLING RESISTANCE
53
FIG. 3. Wheel B under test. Test bed preparation
1
2
3
4
Test speed in left hand side of bed
2 ft/min
8 ft/min
15 ft/rnin
4 ft/min
Test speed in right hand side of bed
4 ft/min
15 ft/min
8 ft/min
2 ft/min
3. PLATE PENETRATION TESTS T h e pressure sinkage r e l a t i o n s h i p was a s s u m e d to fit the e x p r e s s i o n :
p=(ck~ + ~bk~)
(u t,,, "
F o r the soil used in the tests k~ was k n o w n to be a p p r o x i m a t e l y zero a n d was a s s u m e d to be zero for the analysis. T h e expression then simplifies to :
T h e values of ckc, n, and m were f o u n d f r o m sinkage tests with a 1 in. × 5 in. rigid steel footing at sinkage rates f r o m 0"02 to 20 i n . / m i n . Because of the difficulty o f testing the soil in the bins the plate p e n e t r a t i o n tests were m a d e in a s a m p l e of the s a m e soil p a c k e d into a rigid b o x at the same C o n e I n d e x V a l u e of 60 [2] as the soil in the test bins.
54
R.G. POPE
Figure 4 shows the effect of sinkage rate and m was found from the slope of the graph to be 0"15. This value is high compared with results obtained from circular footings in the same soil, possibly due to the width, 1 in., of the footing used or a shape effect. The values of (25.0) and n (0'31) were found from Fig. 5 at a. sinkage speed u0 of 0"2 in./min.
ck,,
i -7
-~
G
%/ o
5
~a o. ,J.
2_ 501
/'.~01
£', 3017
Log { s i n k o g e r o t e } ,
1397 in/rain
FIG. 4. Log (pressure) against log (sinkage rate) for 1 in. X 5 in. footing. 4. PREDICTED AND EXPERIMENTAL ROLLING RESISTANCE The expression used for predicting rolling resistance has been previously derived [1] and is as follows: l
R
(n+ 1)
[
3W
(bk~)1~'¢' ~1""'~j (3 -n)VD
] 'c-''~~+''°/(~'-~L+'')] (2)
where k.,~ (.ck,,+ ybk,;, ) (. V'" ) -
b"
D~'"u,/"
"
The experimental results were taken as the mean of the measured results from three controlled zonal lengths, each 1 ft long, of the test bin where the soil had been found most consistent in the preliminary tests. The results are plotted, together with the predicted results from the plate penetration tests, in Figs. 6 and 7. The
THE EFFECT OF WHEEL SPEED ON ROLLING RESISTANCE
55
agreement between predicted and experimental results is good particularly for the heavier load which is to be expected since the wheels are nearer the relative plate sinkage. For this heavier toad the predicted difference in rolling resistance between
15]4
-
. o rb```
J 12 .......
t
i
I
~
i
;
1.9
]-5
T6
T7
T.8
T9
oo
({)
Fla. 5. Log (pressure) against log (Z/B) for 1 in. x 5 in. footing at sinkage rate 0"2 in./min. Wheel A o Wheel B
Predicted 28
24
e~
'S
20
16
FIG. 6.
[
2
;
4
_
~
8
i
i5
Wheel speed, ftlmin Rolling resistance against wheel speed for wheel load ]24 lb.
56
R.G. POPE A Wheel A o [J
Wheel B
Predicted
6C
¢3 tn
5o or-
4o
I
i.
2
4
i
8
Wheel speed, F I G . 7.
15 =
ft/min
Rolling resistance against wheel speed for wheel load 204 lb.
the extreme wheel speeds was 8"1 lb. and the measured values were 6"5 lb. 7.3 lb. for wheels A and B respectively. The side plate drag on wheel A did not significantly affect the results. anomalous result at the lower sinkage caused by the lighter load is probably to variation in the soil surface state between testing the two wheels. For
\ 08
~ Wheel A o Wheel B ~:1 Predicted
\
07
"--CA 06
05
~ 04 09
03
02
0
2
4
8
Wheel speed,
15
ft/min
FIG. 8. Sinkage against wheel speed for wheel load 204 lb.
and The due the
THE EFFECT OF WHEEL SPEED ON ROLLING RESISTANCE approximately ½ in. sinkage at the heavier load the drag area 2-8 in s and a drag of about 12 lb. could have been expected based cohesion of the soil. This has obviously not occurred probably that the drag (area) is considerably less than 2"8 in 2 because the soil around the wheel and the rut was not cut clearly (Fig. 9). "
~/
/
/
57
would be about on the estimated due to the fact was pulled down
I
Wheel oxle
z ~/ /
J / /
JJ 2 __or,~t_ -soil
J/
surface J/~'
FIG. 9. Rut Formation.
EXPERIMENTAL SINKAGE The sinkage results for the heavier wheel load are shown in Fig. 8. Results for the lighter load are not presented since they were incomplete and inaccurate. The expression used for predicted sinkage has been previously derived [1] and is as follows : 5.
PREDICTED AND
z=
(3 - n) bk,~~ D
(3)
It can be seen from Fig. 8 that there is considerable discrepancy between the predicted and experimental results. This is accentuated by the fact that the predicted rolling resistance was less than the experimental and accordingly one would expect the predicted sinkage to be less. The sinkage discrepancy is probably caused by neglecting the rear part of the contact area which is likely to have more effect on the sinkage than on the rolling resistance. Schuring [3] suggests that at small sinkages, z / b ~ 10 per cent, the rear contact area may be as much as 40 per cent of the front contact area. As expected the sinkage of wheel A was slightly less than that of wheel B since wheel A was given some support from the adhesion on the side plates. 6. CONCLUSIONS The theory is adequate to predict the effect of wheel speed on rolling resistance though there is error in the sinkage prediction at the low sinkage investigated. Providing plate penetration tests are performed at a rate of deformation comparable to the vehicle situation then the error involved in neglecting the rate effect will be negligible compared to the non-homogeneity of the soils.
58
R.G.
POPE
NOTATION
b D kc, k~ p u,, W c k~ m, n R V z
width, in. wheel diameter, in. sinkage moduli pressure, lb. in -~ plate sinkage velocity ft/sec. wheel load, lb. cohesion, lb. in -~ constant sinkage exponents rolling resistance, lb. wheel ground speed, ft/sec. sinkage, in.
Acknowledgement --The author wishes to thank members of the Staff of the Royal Military College of Science who have aided him in his study and in particular A / P r o f e s s o r J. M. Hawkes who instigated the Soil-Vehicle facility at the College.
[l] [2] [3]
REFERENCES R. co. POPE. The effect of sinkage rate on pressure, sinkage relationships and rolling resistance in real and artificial clays. J. Terramechanics 6 (4), 31-38 (1969). M.E.X.E. Prediction of soil strength. Report No. 1187, February 1970, Christchurch, England. O. SCHURING. The energy loss of a wheel. Proc, 2nd lnt, Confi Soil Vehicle Mechanics, Quebec (1966).