2. 352 T. MURO
SOIL-WHEEL INTERACTION DURING DRIVING ACTION
Amount of slip
Figure 1 shows the distributions of the slip velocity and the amount of slip around
the peripherical contact part of a rigid wheel during driving action. The tangential slip
velocity Vs at an arbitrary point X and central angle 0 on the peripheral contact part
of the rigid wheel is determined for the radius of rigid wheel R, the angular velocity
o) (= -dO~dO, the wheel speed V in the direction of ground surface as follows:
V~ = Rco- VcosO. 1)
Here, 0 is the angle between OM and radius vector OX, and is positive for
counterclockwise direction. The slip ratio id during driving action is defined for
RoJ > V as follows:
V
id = 1 ---- (2)
R {o
Therefore, the slip velocity V, can be calculated as,
V~ = R~o{1 - (1 - id)COS0}. (3)
In general, Vs decreases gradually to the minimum value R~o- V at the bottom-
dead-center M, after taking the maximum value at the point A and the entry angle
0 = Of, and then increases slightly toward the point E and the exit angle 0 = -0r.
The amount of slip ]d(0) is shown as the integration of V, from the initial time
t = 0 and 0 = Of at the beginning of contact with soil to an arbitrary time t = t and
0 = 0. That is,
£ t
jd(O) = RJ0co{1 - (1 - id)COS 0} dt
0t
l"
= RJ {1 - (1 - id)cos0}d0
= R{(0f - 0) - (1 - id)(sin Of- sin 0)} (4)
is obtained, jd(0) takes a positive value for the whole range of the contact part AE,
and it increases parabolically from 0 at the point A to the maximum value at the
point E.
. Ro = V),
FIG. 1. Distributions of slip velocitv V, and amount of slip jd (during driving action , >
3. DRIVEN RIGID WHEEL ON SOFTGROUND 353
Rolling locus and soil deformation
Figure 2 shows the general flow pattern beneath a driven rigid wheel in sandy soil
and the position of an instantaneous center. The forward and backward flow zone
NCA and NDE is bounded by a logarithmic spiral for the radial shear zone and then
a straight line which meets the ground surface at the angle 1r/4- ~p/2 at point C and
D for the passive failure zone, respectively [4]. ~p is the angle of internal friction of
the soil. As the instantaneous c_,enter of the driven wheel I is situated above the
bottom-dead-center M, section NA of the rim has a generally forward and downward
movement and moves the soil forward, and section NE of the rim has a generally
backward and downward movement and drives the soil directly backward by means of
the tangential shear stress. Two slip lines sl and s2 occur forward and backward
beneath the point N and intersect each other at the angle + (rr/4- ~p/2) to the
direction of maximum principal stress Ol and at the angle +_(rr/4 + ~p/2) to the
direction of minimum principal stress 03, as shown in Fig. 3. In this figure, the
direction of slip line Sl at the point N agrees with that of the resultant velocity vector
between vehicle speed V and peripherical speed Rto which intersects at a right angle
with the radius vector IN. The position of point N and the direction of slip lines Sa, s2
depend on the slip ratio id which is determined from V and Rto. Wong and Reece [5]
executed an experiment for a sandy soil and verified that the normal stress o
developed on the peripheral surface of the rigid wheel takes the maximum value on
the point N, and the relation between the location of N, i.e. the central angle ON and
the slip ratio id, can be expressed for the entry angle Of as,
ON = (a + bid)Of. (5)
The angle r/between the slip line sl and the tangential direction of point N can be
expressed as,
V sin 0
tan ~ =
R to - V cos 0
(1 - id) sin 0
1 - (1 -- id)COS0
= cot (~ + 6) (6)
ol 2
FIG.2. Soildeformationbeneatha drivenrigidwheel.
4. 354 T. MURO
I
O~ .w ~V
o'1
Fich3. Directions of slip line s~, se and maximum principal stress ~J, o,
where ~ is the angle between the direction of slip line sl and that of resultant applied
stress p which is calculated from normal stress o and shear resistance r, and 6 is the
angle between the direction of p and the radial direction at point N. The angle
should be equal to qJ, when the combination of the normal stress pcos_~ and the
shear resistance _+psin5 on the surface of slip line s~ and st satisfies the failure
criterion of the sandy soil. Therefore, the central angle #~< could be calculated
theoretically.
The resultant velocity vector of all the soil particles on the peripheral contact part
of the rigid wheel is always rotating around the instantaneous center I. The position
of I locates above the bottom-dead-center M for the distance OI, which could be
calculated as follows (Fig. 2):
OI = RcosO + Rsin0tan7
= RcosO + _1 (V - R~ocosO)
{t)
V
St)
= R(1 - ia). (7)
From the above equation, the instantaneous center I is located at point M and
OI = R for id = 0% at V = Rco, and it is located at the point O, i.e. the axis of the
rigid wheel, and OI = 0 for id = 100% at V = 0. The position of instantaneous center
depends on the slip ratio, or the wheel speed and the angular velocity of the wheel,
and was verified experimentally by Wong [6].
As is well known, the angle fi between the resultant velocity vector and radial
direction at point X takes a minimum value at 4_OIX = ~r/2, and 0 = cos -I (l - i(0.
Next, the trajectories, e.g. the rolling locus of a point on the peripheral contact
5. DRIVEN RIGID WHEEL ON SOFT GROUND 355
part of the rigid wheel, during driving action should be considered. Figure 4 shows
the rolling locus of an arbitrary point F on the peripheral surface of a rigid wheel for
the slip ratio id and Rco = V during driving action. When the relation between the
central angle ol of the point F and the time t can be expressed as 0L-- cot, the position
of the axis of the rigid wheel O moves to O' for the distance Vt = Vol/co.
Therefore, the coordinates (X, Y) can be calculated as follows:
Vc~
X- + Rsin0~
co
= R{c~(1 - id) + sin ol} (8)
Y = R(1 + cos c~). (9)
The above equations represent a trochoid curve. The coordinates of point P at
ol = zr/2 are calculated as X = R{(1 - id)rr/2 + 1) and Y = R, those of point Q at
ol = zr are calculated as X = zrR(1 - id) and Y = 0, those of point S at c~= 3rr/2 are
calculated as X = R{(1 - id)3zr/2 -- 1) and Y = R, and those of point T at ol = 27r are
calculated as X = 2~rR(1 - id) and Y = 2R. The gradient of the tangent to the rolling
locus dY/dX is equal to the direction of the resultant velocity vector of an arbitrary
point on the peripherical contact surface of the rigid wheel and it can be expressed as
follows:
dY _ sin cr (10)
dX 1- io + cosa
Therefore, the gradient of the tangent to the rolling locus at points P, Q, S and T is
calculated as -1/(1- id), 0, +1/(1- id) and 0, respectively. In general, the rolling
locus of the point on the peripheral contact part of the rigid wheel draws a loop line,
in accordance with the amount of sinkage s. During one revolution of the rigid wheel,
there is a loss 2zrRid in the wheel advance and the moving distance is expressed as
2rrR(a - id) [7].
Driving force
Figure 5 shows the contact pressure distributions applied on the peripheral contact
surface of a rigid wheel during driving action. The normal stress a and the shear
resistance r have positive values for the entire contact portion of the rigid wheel, and
¥
I
T
Y W , s I
(~ ~R ~' 2~R(1--id) 2~R
R (I -id)
LX
FIG.4. Trochoid curveof a point on peripheral surface of rigid wheel during driving action (Rco=>V).
6. 356 1". MURO
=- V
Fu,. 5. Contact pressure distributions applied on pcripherical contact part of a rigid wheel during driving
action.
the angle between the resultant applied stress p and the radial direction of wheel
surface 6 is shown as 6 = tan-I(r/o). The amount of sinkage z at an arbitrary point X
on the peripheral surface, the amount of sinkage so at the bottom-dead-center M and
the amount of rebound u0 at the point E can be expressed as follows, using the
central angle 0, the entry angle 0f and the exit angle 0,.
z = R(cos0 - cos0f)
s'~ = R(I - cos Of)
ul~ = R(1 - cos Or).
(11)
(12)
(13)
The length of the rolling locus l = l(cr) can be derived from equations (8) and (9).
Substituting the relation 0 = 7r - a' into l(cr), the length of trajectory l = l(O) is given
by integrating the minute locus from 0 = 0 to 0 = Ofas follows:
= RJ0{(1 - id)2 -- 2(1 -- i,0COS0 + 1}l/ed0. (14)
l(0)
As shown in Fig. 6, the direction of the resultant force between the effective
driving force Td and the axle load W is given as the angle ~ = tan-l(Td/W) to the
vertical axis. In this case, the relation between contact pressure q(O) and soil
deformation d(O) in the direction of the resultant force agrees well with the plate
loading, unloading and sinkage relationship [8]. Here, q(O) is the component of the
resultant applied stress p(O) to the direction of the angle ~ to vertical axis. d(O) is the
component of the rolling locus l(O) to the same direction of the resultant force. As
XM is the element of the rolling locus in the same direction of the resultant velocity
vector, d(O) should be calculated as the integration of XH from 0 = 0 to 0 = 0~.
7. DRIVEN RIGID WHEEL ON SOFT GROUND 357
/w
0
Td /,
V
d
=tan-1 (Td/W)
/ H XM=~/--(~/2- 8- ~)
Ad(8)= X H = XM eos(Z H XM)
= Ag.(e) sin(e + g +@)
q = p c0s(~ -t- 0 -- ~)
Fro. 6. Component of a rolling locus Ad(0) in the direction of applied stress q during driving action.
which is the component of XM in the direction of the angle ~ to vertical axis, as
follows:
Of
t"
d(O) = RJ0{(1 - id)2 -- 2(1 -- ij)cos0 + 1}1/2
xsin{O+ ~+tan_ 1 (1-- id) Sin O }dO. (15)
1 _7 - cosO
The velocity vector Vp in the direction of the stress q(O) on the peripheral surface
of a rigid wheel can be calculated as the component of the resultant velocity vector
between Ro) and V, as follows:
Vp = {(R~o- VcosO)e + (VsinO)2} 1/2
×sin(0+ ~ +tan-1 (1-- id) sin 0 /
1 ._7(-1 £ -i~cosO]" (16)
Here, the plate loading and unloading test should be executed in consideration of
the "size effect" [9] of the contact length of wheel b = R(sin Of+ sin Or) and the
"velocity effect" [10] of the loading speed Vp. Therefore, the relations between the
8. 358 T. MURO
resultant applied stress p(O), the stress component q(O) and the soil deformation
d(O) could be determined as follows:
For 0m:~×=< 0 _-<0j
For -O r ~ 0 < 0max
p(O) =
p(o) =
q(O)
cos (c~ + o - `5)
k~_~{a(0)}"'
cos(~ + 0 - ,5)
q(O)
cos(~+ 0-6)
kl~{d(Omax) }.... k2{d(Omax) - d(0)}';:
= (17)
cos(~+ 0- 6)
1 +
1 + ;~v~
kc, kc,
k~ - + ke,, k2 - - + k¢,~
b b
where 0max is the central angle 0 in correspondence with the maximum stress
q = q(0max). The coefficients kc,, k,p, and kc~, k~, and the indices nl and n2 should
be determined from the quasi-static plate loading and unloading test at the loading
speed V0. The another coefficient ~ and the index K should be determined from the
dynamic plate loading test at the loading speed Vp. Therefore, the distribution of
normal stress a(O) can be calculated as follows:
o(o) = p(O) cos 6. (18)
Also, the distribution of shear resistance r(0) can be calculated by substituting the
amount of slip jd(0) into the Janosi-Hanamoto equation [11], as follows:
r(0) = {c~ + o(0)tanq~}[[1 - exp[- a{jd(O)}B
= {c~, + o(0)tanq~}
x [[1 - exp[- aR{(O~- O) - (l - id)(sin 0f- sin 0)}~]. (19)
The above equation could be applied for loose sandy soil or weak clayey soft
ground, but another equation by Oida [12] could be applied for hard compacted
sandy ground.
Afterwards, the angle ,5(0) between the resultant applied stress p(O) and the radial
direction can be calculated as follows:
{r(0) 1 (20)
,5(0) = tan -l o(0) J"
Next, the axle load W and the driving torque Qd can be related to the contact
pressure distribution a(O) and r(0), as
.O t
W = BRI {a(O)cosO + r(O)sinO}dO (21l
3- O:
9. DRIVEN RIGID WHEEL ON SOFF GROUND 359
Of
f
= BR2IoT(O dO
Qd
and then the driving force can be calculated as the value of Q JR.
(22)
Compaction resistance
In general, a considerable amount of slip sinkage under the driven rigid wheel will
occur due to the dilatancy phenomenon during shear action of soil on the peripheral
interface. The amount of slip sinkage Ss [8] could be expressed as the function of
contact pressure p and amount of slip Is as
ss = copC~jc2 (23)
where the coefficient Co and the indices cl, c2 are the terrain-wheel system constants
and they should be determined experimentally for the given steel plate and the
terrain.
Figure 7 shows the rolling motion of a rigid wheel during driving action for the
point X on the ground surface. To calculate the amount of slip sinkage Ss at point X
on the ground surface, it is necessary to determine the amount of slip Js on the point
X. An arbitrary point F on the peripheral surface of the rigid wheel meets the point
A on the ground surface at the time t = 0 and the central angle 0 is the angle of
radius vector O'F to vertical axis at an arbitrary time t. Then, the rotation angle tot
of the wheel is expressed as
tot = Of- 0 (24)
where Of is the entry angle. And the moving distance OO' of the wheel during the
time t is given as follows:
00' = Vt
= Rtot(1 - id)
= R(1 - id)(0f -- 0). (25)
IR (sin 8f+si n 8r)H
~Vt
X
F~G.7. Rolling motion of a driven rigid wheel for the point X on ground surface.
10. 360 T. MURO
The transit time to. in which the rigid wheel has passed on the point X. is given as the
time when the distance OO' reaches the contact length of wheel OO" and the central
angle of the radius vector O"F becomes 0 = 0~ as follows:
OO" = R(sin 0f + sin 0,.)
sin Of + sin 0~
0~ = 0f - (26)
1 - i d
Substituting equation (24) into the above equation.
sin Of + sin 0,.
t~ = (27)
co(1 - id)
is obtained. As the amount of slip of the rigid wheel during one revolution is 2gRi d,
the amount of slip of soil at the point X can be calculated as the ratio of te to 2~/~o as
follows:
re(sin Of + sin Or)
j~ = 2~Rid
2~r(0(1 -- id)
id
= R(sin Ot + sin 0r) - - . (28)
1 - id
Considering the small interval of central angle (Of + O0/N and the small interval of
amount of slip jJN for the tiny time interval te/N, the vertical component of
resultant applied stress p can be calculated as p(On)cos (0,, - On) for the nth interval
of central angle 0n. Then, substituting into equation (23), the amount of slip sinkage
st is given as the summation of the minute amount of slip sinkage as follows:
= E ,
s~ co {p(On)cos(On -- a,,)}'~ X L L (29)
11 = [
where
0n -- n (Of + Or).
N
Then, the total sinkage, i.e. the rut depth of rigid wheel s is given from the
amounts of the static sinkage so and the slip sinkage s~, and the amount of rebound
Uo as follows:
s = So - u0 + s~. (30)
The product of the compaction resistance Lcd applied in front of the rigid wheel
and the moving distance 27rR(1 - id) during one revolution of wheel should be equal
to the rut making work which is calculated as the integration of the contact pressure
p acting on a plate of length 27rR(1 - id) and width B from the depth z = 0 to the
total amount of sinkage z = s, as follows:
F"5
2#R(1 - id)Lcd = 27rR(1 - id)BI..p dz.
at/
11. DRIVEN RIGID WHEEL ON SOFTGROUND 361
Then, the compaction resistance Lcd could be determined as follows, considering
the vertical velocity effect:
t" s
Lcd = kl~B|z "~dz (31)
J0
where
1 + AV~
1 + AV'~
z-- sin(cos-l( z)).
Effective drivingforce
As shown in Fig. 8, the effective driving force To and the axle load W act on the
point of central axis O of the rigid wheel, the driving force Qd/R acts on the
bottom-dead-center M, and the vertical reaction force N and the compaction
resistance Lcd act on the point G at the amount of eccentricity ed and the vertical
distance l d from the point O.
The amount of eccentricity ed0 for the no-slip sinkage state is given as follows,
considering the moment equilibrium of vertical stress applied on the peripheral
contact surface around the central axis O:
BR2[of
ed0 -------~-j_o~p cos (0 - 6) sin 0 cos 0 dO. (32)
Then, the real eccentricity ed of the vertical reaction force N can be modified as
(sin 0~ + sin O~)(Rsin Or + ed0)
ed = -- R sin Or (33)
sin Of + sin Or
Qd
Qd/R
I L~d
Qd
FIG.8. Variousforces appliedon a rigidwheel duringdrivingaction.
12. 362 T.MURO
where
0~=COS l(1--~) and 0~=Or.
Then the vertical and horizontal force balances and the moment balance are equated
as follows:
W = N (34)
Td = ((2~)h --Led= Qd
COSOed-R L~.~, (35)
Qd
Qj - -- R + Lc~tl~l - Ned = 0. (36)
R
Therefore, the effective driving force Td can be calculated as the difference
between the horizontal component of the driving force (QJR)h and the compaction
resistance Led, as shown in equation (35).
Since the relation between ed and Id is expressed from equation (36) as
Ne~t
l~l - (37)
Lcd
the position of point G, that is, ed and l,~ can be determined.
Energy equilibrium
The effective input energy El supplied by the driving torque Qd is the sum of the
output energies which are the sinkage deformation energy E2 required to make a rut
under the rigid wheel, the slip energy E3 developed at the peripheral contact part and
the effective drawbar pull energy E4 required to develop an effective driving force, as
follows:
E 1 = E 2 + E 3 + E 4. (38)
Each unit energy value per second is given as follows:
El = Qd m
QdV
- (39)
R(1 - id)
E2 = LcdR~O(1 - id)
= LcdV (40)
E3 = Qd~Oid
- QdidV (41)
R(1 - id)
E4 = TdRm(1 - id)
= T~V. (42)
The optimum effective driving force Tdoptis defined as the effective driving force at
13. DRIVEN RIGID WHEEL ON SOFT GROUND 363
the optimum slip ratio idopt which the effective drawbar pull energy E 4 takes the
maximum value. And the tractive power efficiency E d is defined as follows:
TdR(1 -- id)
g d = (43)
Qd
SIMULATION ANALYSIS
Flow chart
Figure 9 shows the flow chart to calculate the tractive performance of a driven rigid
wheel running on a flat soft ground. First of all, the wheel dimensions such as the
axle load W, the radius R and the width B of the rigid wheel, and the peripheral
W, B , R, Ro~ I
½
I ko t. k ~t. ]] i. k~z. k ~,~. IIz. Vu. ~ . K
c.. tan¢ , a , o.. ol . o z
At rest . ~
l,
Iw-w(o)l<. 1
I e~-u. 0 .... p.(O), a,,(O), ru(O). 80(0). s,. I
At driving state:[~___l
i
[ p,(0), a,(o), r,(o), a,(o) ~-q
l IPr(O)--pr,(O)l<,~
I P~(O)" a~(O), r~(O), Sr(O)
#
I o r,,,- o r,,, j I < *
Iv
,w-w<0.0r,,<. I
0n,., 0rm. pu,(O), am(0)
r,.,(O). $.,(0). s.,. u.,
I ,s, = s,,,+s., I
{,
[o,,, o..... (0). ~,<o)...... I
t
q O.,, Td,,L .... e..,.E..,.E,,.Ez,.Ea,.E., I
FIG. 9. Flow chart to simulate the tractive performance of a rigid wheel during driving action.
14. 364 T. MURO
velocity R~,, are given as the input data. Also the terrain-wheel system constants such
as the coefficients k,,. k¢,, and k,:. k¢,: and indices hi, he, and the loading rate ~"~
measured from the quasi-static plate loading and unloading test. the coefficient Z and
the index k measured from the dynamic plate loading test, the soil constants c~,. tan q~
and a measured from the plate traction test, and the coefficient ~~. the indices ~'~. ~'~
measured from the slip sinkage test should be read as the othcr input data.
At rest. the distribt, tions of the normal stress o(O). the shear resistance r(t:l), the
resultant applied stress p(tt) and the friction angle h(O) arc calculated for a given
entry angle Of,~,and a given exit anglc #m. Then. to determine the real values of th0
and #m, the distributions of o(O), V(#), F(O) ant] h(O) are repeatedly calculated by
means of the two-division method until the vertical equilibrium cquation (21) can bc
satisfied precisely.
In the driving state for a given slip ratio i,i. the Rflh)wing two calculations should be
donc for thc %n.ntryangle Or< which is .just determined. For the forward periphcral
contact part AM. lhc distributions of of(O) calculated from equation (18;). r~(fl)
calculated from equation (19). ?h(O) calculated from equation (20). which can be
calculated from l;,t(O) given in equation (17) arc calculated repeatedly until the real
distributions of lh(~)) are determined. After thai. for the backward peripheral contact
..-...
part ME, the distributions of o,(fl), T,([)). (3,.(t)) calculated from p,(tl) arc calculated
until thc exit angle tim, is determined.
Next. those calculations should be done for the given entr angle tt>i until the
vertical equilibrium equation (21) is satisfied precisely for the given slip ratio i,t.
Then. the final valucs of ttt< and (t~.<. and the final distributions of th(tt) and o~(t~),
rt(O ), hl(# ), and l.,,(trJ) and o,(H), r,(#), h~(#), and the final amount of sinkage .~,)/
and amount of rebound uq~ can be determined. Furthermorc. thc total amount o|
sinkage .si calculated from equation (30) can be determined lrom the amount of slip
sinkage .s~i givcn m equation (29). Then. the driving torque O~li calculated from
equation (22). thc driving force Odi/R, the compaction resistancc L,I i calculated from
equation (31). the amount of eccentricity cdi calculated from equation (33). the
tractive powcr efficiency Edi calculated from equation (43). and several energy values
El;, E2i. E3i and f54, calculated from equations (39)-(42) can be dctcrmined for all
the slip ratios i~l. Also. lhe optimum slip ratio /d,,pt and the optimum effective driving
force Tdopt can be determined. Finally. those relations between (.)d/R- i,i. -l'~l- i,t-
.s- id. ill. t~,.- id. e,t - i~1. F~l - i,i and E t . E-,. t5~. E., - id. and the distributions of
the normal stress o(fl) and the shear resistance r(#) can be graphically expressed bx
use of a microcomputer.
I~Jxperimental ver~licatio~t
As an example, the tractive performance of the driven rigid wheel with the axle
load W= 1.52kN, radius R= 16cm, width B=9.5cm running on a weak soft soil
ground at the peripherical speed R¢o = 7.07 cm/s has been simulated. The analytical
simulation results have been verified from the experimental test data. All the
terrain-wheel system constants are shown in Table 1.
The sandy soil was dried in air and the water content w- 2.38%, the specific
density G~ = 2.66 Mg/m -~, the average grain size Ds~ = 0.78 ram, the coefficient of
uniformity U~,= 12.(I. The size of soil bin is 120 cm length, 10 cm width and 35 cm
depth. The sandy soil was filled uniformly in the two dimensional soil bin, bv means
of the free falling method from a 35 cm height. The initial density of the sandy soil
16. 366 T. MURO
10
$
(cm)
g
15
0 20 40
i T
6O
I t
i d !IX)
Fits. ll. Relations betwen total amount of sinkage s and slip ratio id during driving action.
15
ed
(c,,)
10
/
/
//
/
i ~ (Z)
L ~ i i I I
0 20 40 60
Fro. 12. Relation between amount of eccentricity e,~of vertical reaction force and slip ratio i~ during driving
action.
Figure 13 shows the relations between the entry angle Of, the exit angle Or and the
slip ratio ia. Of increases parabolically with the increment of id after taking the
minimum value 0fmi, = 0.629 rad at id "-70%, but 0r increases gradually with id from
0r = 0.175 tad at ia- 0% and decreases rapidly after reaching the maximum value
0rmax = 0.244 rad. Figure 14 shows the relations between the several energy values
El, E2, E3, E 4 and the slip ratio id. The effective input energy Ej increases
gradually with the increment of io and increases rapidly from id ~. 55%. The sinkage
deformation energy E2 increases parabolically with id from the initial point
0.934 kNcm/s at id- 0%. The slip energy E 3 increases almost linearly with id and
increases rapidly from id -- 55%. The effective drawbar pull energy E 4 increases with
id and reaches the maximum value E4max = 2.359 kNcm/s at id = 10%. After that, E 4
decreases almost parabolically to the negative values.
Figure 15 shows the relation between the tractive power efficiency Ed and the slip
ratio id. Ed decreases almost hyperbolically with id from the maximum value
Edmax ----68.1% at id - 0% and turns to the negative values from ia = 43%. Figure 16
17. DRIVEN RIGID WHEEL ON SOFT GROUND 367
9O
8,
8.
6O
(deg)
30
J
J
8r
I I I
20 40 60
FIG. 13. Relations between entry angle Of, exit angle 0r and slip ratio id during driving action.
10
E~, E2
E3,E4
5
(kNcm
/sec)
0
-5
EL
I ' 2'0 40"k~6O
F~o. 14. Relations between energy values El, E2, E3, E4 and slip ratio id during driving action.
100
Ed
(Z)
50
-50
-100
Fio. 15. Relation between tractive power efficiency Ed and slip ratio ia.
18. 368 T. MURO
shows the distributions of the normal stress o(01 and the shear resistance r(0) at
idopt ----- 10%. The shapes of stress distribution agree well with the experimental test
data as shown in the paper published by Onafeko et al. [13]. In this case, the
maximum value of the normal stress Om~,,,= 156.2 kPa is obtained at Ox = 0.300 tad
and 0N/0f = 0.450.
Figure 17 shows the relation between the angle ratio 0N/0 r to obtain the maximum
normal stress and the slip ratio i d, The ratio 0:</0f increases slightly with the
increment of id, and the constants in equation (5), a = 0.391 and b = 1.82 × 10-3 arc
obtained. In this case, it is clarified that the position showing the maximum normal
stress shifts forward 39.1% of the entry angle and it increases slightly with the
increment of slip ratio.
CONCLUSIONS
TO predict the tractive performance of a driven rigid wheel on soft ground, not only
the effects of slip ratio on slip sinkage, motion resistance and driving torque but also
the effect of rolling locus in the direction of external resultant force between the
effective driving force and the axle load have been taken into account. It is shown
that the following new analytical methods of soil-wheel interaction can give better
prediction than existing theory.
Q,~ 0 @ , ~ V
~. 8,', -~ i a = 10 Z
> "->~//LY/::-
.)~ // ,l
..---'4o
200 L 5;--"
_20 o ~ i ~ 20'
0 o
F~G.16. Distributions of contact pressure at ij = 1(1% during driving action: (a) normal stress o(0):
(b) shear resistance r(0),
ON 1.0
Of 0.8
0.6
0.4
0.2
ON/Of = 1.82 x 10-3 i d + 0.391
D | •
I I m I I
0 10 20 30 40 50
FiG. 17. Relation between angle ratio 0y/0f and slip ratio id during driving action.
19. DRIVEN RIGID WHEEL ON SOFT GROUND 369
(l) The resultant stress between the normal stress and the shear resistance applied
around the peripheral contact surface of the rigid wheel should be predicted by use of
the dynamic pressure-sinkage curve of soil, considering the rolling locus of the wheel
in the direction of the external resultant force of the effective driving force and the
axle load.
(2) As the terrain-wheel constants, the coefficients k,,, k~, k,~ and k@, the indices
rtl, rt2, the loading rate V0 from the quasi-static plate loading and unloading test, the
coefficient ~, and the index K from the dynamic plate loading test, the soil constants
ca, tan q~ and a from the plate traction test, and the coefficient c0, the indices c~, c2
from the slip sinkage test should be determined precisely, considering the size effect
of the plate and the velocity effect of the loading rate.
(3) The effective driving force can be predicted as the difference of the driving
force calculated as the integration of shear resistance around the peripheral contact
surface and the locomotion resistance of wheel. The locomotion resistance can be
calculated as the compaction resistance of soil from the total amount of sinkage of
wheel.
(4) The tractive performances of the driven rigid wheel of the axle load 1.52 kN,
the radius 16 cm, the width 9.5 cm running on the weak sandy soil terrain, i.e. the
analytical relations between the driving force, the effective driving force and the slip
ratio, the total amount of sinkage and the slip ratio have been verified experiment-
ally. As a result, it is clarified that the optimum effective driving force is 0.097 kN at
the optimum slip ratio 10%, and the maximum one is 0.127 kN at the slip ratio 21%.
(5) The position showing the maximum normal stress around the peripheral contact
surface of the rigid wheel shifts forward 39.1% of entry angle and it increases slightly
with the increment of slip ratio.
REFERENCES
[1] J. Y. WONGand A. R. REECE, Soil failure beneath rigid wheels. Proc. 2nd Int. Conf. ISTVS, Quebec,
pp. 425-445 (1966).
[2] R. N. YON• and E. J. WINDISH,Determination of wheel contact stresses for measured instantaneous
soil deformation. J. Terramechanics 7 (3/4), 57-67 (1970).
[3] R. N. YONG and E. A. FATrAH, Prediction of wheel soil interaction and performance using the finite
element method. J. Terrameehanics 13 (4), 227-240 (1976).
[4] K. TERZAC~HJ,Theoretical Soil Mechanics, pp. 367-415. John Wiley, New York, (1943).
[5] J. Y. WONt3 and A. R. RLECE, Prediction of rigid wheel performance based on the analysis of
soil-wheel stresses, Part 1. Performance of driven rigid wheel. J. Terramechanics 4 (1), 81-98 (1967).
[6] J. Y. WON~, Behaviour of soil beneath rigid wheels. J. agric. Engng Res. 12 (4), 257-269 (1967).
[7] R. N. YONG, E. A. FA'r'rA, and N. SKIADAS, Vehicle Traction Mechanics, pp. 157-193 Elsevier,
Amsterdam (1984).
[8] T. Mu,o, Grouser shape effect on tractive performance of a flexible tracked vehicle carrying up slope
terrain. Proc. 4th North American Regional Meeting, ISTVS, Sacramento, pp. 9-16 (1992).
[9] T. MuRo and K. YOKOTA,Size effect on tractive performance of a bulldozer running weak terrain.
Proc. JSCE 445, 101-108 (1992).
[101 M. GRAHN, Prediction of sinkage and rolling resistance for off-the-road vehicles considering penetra-
tion velocity. J. Terramechanics 28 (4), 339-347 (1991).
[11] Z. JANOSIand B. HANAMOTO,The analytical determination of drawbar-pull as a function of slip for
tracked vehicles. Proc. 1st Int. Conf. Mech. SoiI-Veh. Systems', Torino, pp. 707-736 (1961).
[12] A. OIoA, Study on equation of shear stress-displacement curves on Kacigin-Guskov equation. Proc.
JSAM 37, 20-25 (1975).
[13] O. ONAFEKOand A. R. REEC'E,Soil stresses and deformations beneath rigid wheels. J. Terramechan-
ics 4 (1), 59-80 (1967).
![Journal of Terramechanics,Vol. 30, No. 5, pp. 351-369, 1993.
Printed in Great Britain.
0022-4898/93 $6.00+0.00
Pergamon Press Ltd
© 1993 ISTVS
TRACTIVE PERFORMANCE OF A DRIVEN RIGID WHEEL
ON SOFT GROUND BASED ON THE ANALYSIS OF
SOIL-WHEEL INTERACTION
TATSURO MURO*
Summary--A new analytical method has been presented to predict the tractive performance
of a rigid wheel running on soft ground. The resultant stress of the normal stress and the
shear resistance applied around the peripherical contact part of the rigid wheel should be
calculated by use of the dynamic pressure-sinkage curve measured from the plate loading and
unloading test, considering the rolling locus of the wheel in the direction of the external
resultant force of the effective driving force and the axle load. The effective driving force
could be calculated as the difference of the driving force, i.e. the integration of shear
resistance and the locomotion resistance calculated from the total amount of sinkage. As a
result, the analytical relations between the driving force, the effective driving force and the
slip ratio, the amount of sinkage and the slip ratio, the amount of eccentricity of resultant
force and the slip ratio, and the entry angle, the exit angle and the slip ratio could be verified
experimentally.
INTRODUCTION
VARIOUS kinds of compaction machine mounted on a rigid steel roller are used for
compacting soil, asphalt and roller-compacted dam concrete, etc. Wong and Reece [1]
confirmed experimentally that the angle showing the maximum normal stress around
the peripherical contact part of the rigid wheel running on a sandy soil could be
expressed as a linear function of the slip ratio. They also showed that the distribution
of shear resistance could be calculated from the normal stress and the amount of slip,
assuming a symmetrical distribution of the normal stress. Yong et al. [2, 3] analysed
the contact pressure distribution from the behavior of soil, i.e. the stress-strain
velocity relationship under the rolling wheel, by means of a visco-plastic theory and a
finite element method, and they also verified the tractive performances of a rigid
wheel by use of an energy equilibrium theory.
Here, the tractive performance of a rigid wheel running on a soft sandy soil ground
is analysed by use of the pressure-sinkage curve measured from the dynamic plate
loading and unloading test, considering the rolling locus of the wheel in the direction
of the external resultant force of the effective driving force and the axle load. The
contact distributions of normal stress and shear resistance, and the relations between
the driving force, the effective driving force and the slip ratio, the amount of sinkage
and the slip ratio, the amount of eccentricity of resultant force and the slip ratio, and
the entry angle, the exit angle and the slip ratio are simulated and verified
experimentally.
*Professor, Department of Civil and Ocean Engineering, Faculty of Engineering, Ehime University, 3
Bunkyo-cho, Matsuyama, 790, Japan.
351](https://image.slidesharecdn.com/muro1993-desbloqueado-220326014728/85/muro1993-desbloqueado-pdf-1-320.jpg)
![352 T. MURO
SOIL-WHEEL INTERACTION DURING DRIVING ACTION
Amount of slip
Figure 1 shows the distributions of the slip velocity and the amount of slip around
the peripherical contact part of a rigid wheel during driving action. The tangential slip
velocity Vs at an arbitrary point X and central angle 0 on the peripheral contact part
of the rigid wheel is determined for the radius of rigid wheel R, the angular velocity
o) (= -dO~dO, the wheel speed V in the direction of ground surface as follows:
V~ = Rco- VcosO. 1)
Here, 0 is the angle between OM and radius vector OX, and is positive for
counterclockwise direction. The slip ratio id during driving action is defined for
RoJ > V as follows:
V
id = 1 ---- (2)
R {o
Therefore, the slip velocity V, can be calculated as,
V~ = R~o{1 - (1 - id)COS0}. (3)
In general, Vs decreases gradually to the minimum value R~o- V at the bottom-
dead-center M, after taking the maximum value at the point A and the entry angle
0 = Of, and then increases slightly toward the point E and the exit angle 0 = -0r.
The amount of slip ]d(0) is shown as the integration of V, from the initial time
t = 0 and 0 = Of at the beginning of contact with soil to an arbitrary time t = t and
0 = 0. That is,
£ t
jd(O) = RJ0co{1 - (1 - id)COS 0} dt
0t
l"
= RJ {1 - (1 - id)cos0}d0
= R{(0f - 0) - (1 - id)(sin Of- sin 0)} (4)
is obtained, jd(0) takes a positive value for the whole range of the contact part AE,
and it increases parabolically from 0 at the point A to the maximum value at the
point E.
. Ro = V),
FIG. 1. Distributions of slip velocitv V, and amount of slip jd (during driving action , >](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)