-
Journal
www.elsevier.com/locate/jterra
Journal of Terramechanics 44 (2007) 133152
ofTerramechanics
Review
Digging and pushing lunar regolith: Classical soil mechanicsand
the forces needed for excavation and traction
Allen Wilkinson *, Alfred DeGennaro
Fluid Physics and Transport Branch, NASA Glenn Research Center,
M.S. 110-3, 21000 Brookpark Road, Cleveland, OH 44135, United
States
Received 11 January 2006; received in revised form 6 September
2006; accepted 7 September 2006Available online 13 November
2006
Abstract
There are many notional systems for excavating lunar regolith in
NASAs Exploration Vision. Quantitative system performance
com-parisons are scarce in the literature. This paper focuses on
the required forces for excavation and traction as quantitative
predictors ofsystem feasibility. The rich history of terrestrial
soil mechanics is adapted to extant lunar regolith parameters to
calculate the forces. Thesoil mechanics literature often
acknowledges the approximate results from the numerous excavation
force models in use. An intent of thispaper is to examine their
variations in the lunar context. Six excavation models and one
traction model are presented. The effects of soilproperties are
explored for each excavation model, for example, soil cohesion and
friction, toolsoil adhesion, and soil density. Excava-tion
operational parameters like digging depth, rake angle, gravity, and
surcharge are examined. For the traction model, soil, opera-tional,
and machine design parameters are varied to probe choices.
Mathematical anomalies are noted for several models. Oneconclusion
is that the excavation models yield such disparate results that
lunar-field testing is prudent. All the equations and graphs
pre-sented have been programmed for design use. Parameter ranges
and units are included.Published by Elsevier Ltd on behalf of
ISTVS.
1. Introduction
The parameters described in Fig. 1 are used throughoutSections 2
and 3 of this paper. It will become clear that thetool digging
depth, d, soil internal cohesion forces, c (soilsoil sliding along
the shear failure surface plane), rakeangle, b, and external
friction forces (soil sliding on theblade) related to the external
friction angle, d, and toolsoiladhesion, Ca and/or l, are the most
prominent variables.Gravity manifests itself in this paper only by
way of the soilweight. A likely overconsolidation ratio greater
than onefor virgin regolith due to meteorite impact, tidal and
ther-mal lunar quakes, and gravity settlement will induce mem-ory
effects on soil cohesion and toolsoil adhesion. Thevacuum and
plasma active lunar ambient environment alsowill induce particle
surface activity that affects adhesion
0022-4898/$20.00 Published by Elsevier Ltd on behalf of
ISTVS.
doi:10.1016/j.jterra.2006.09.001
* Corresponding author. Tel.: +1 216 433 2075; fax: +1 216 433
3793.E-mail addresses: [email protected],
[email protected]
(A. Wilkinson).
and cohesion. Some models include an inertial forcerequired to
give the soil its kinetic energy (v2) for movingwith the blade.
Some models include surcharge pressure,q, due to soil mounding
above the reference level of the ori-ginal soil surface. These
models allow the machine force, T,to be applied to the excavator
blade at an arbitrary angle,d, with horizontal, H, and vertical, V,
forces resolved. Allmodels here include soil density and some
cohesion effectsas a minimum. The contributions of each of the
physicalsources in the models, like cohesion, are shown
separatelyin the figures, except for the Osman model. Appendix
Agives the exact expressions used for each physical sourcein each
model plot.
Table 1 gives the parameter settings used in Sections 2and 3,
except when a parameter is varied in the plots ofFigs. 26, 8 and 9.
This parameter set represents Apollolunar soil parameters to the
extent they exist [1]. AppendixB provides more definition and
plausible ranges for theseparameters from terrestrial and lunar
sources. No singlemodel uses all of these parameters.
mailto:[email protected]:[email protected]
-
Table 1Parameter set (SI units) used for Balovnev, Gill, McKyes,
Osman, Swickand Viking models
Values Units
Tool width, w 1 mTool length, l 0.7 mTool depth, d 0.5 mSide
length, ls 0.56 mSide thickness, s 0.02 mBlunt edge angle, ab 40
degBlunt edge thickness, eb 0.05 mMoon gravity, gM 1.63 m/s
2
Earth gravity, gE 9.81 m/s2
Soil specific mass, c 1680 kg/m3
Surcharge mass, q 1 kg/m2
Rake angle, b 45 degShear plane failure angle, q 30 degTool
speed, v 0.1 m/sGills cut resistance index, K 1000 N/mCohesion, c
170 N/m2
Internal friction angle, / 35 degSoiltool adhesion, Ca 1930
N/m
2
Soiltool normal force, N0 100 NExternal friction angle, d 10
degOsman-d1 1Osman-d2 1Osman-d3 1Osman-d4 1Osman-d5 1Osman-d6
1Osman-d7 1Log radius, w 0 1Base log radius, r0 1 mCalculated log
radius, r1 3.17 mRankine passive depth, t 0.5 m
Lunar soil parameters are used where available.
Fig. 1. Definition of angles and parameters [2].
134 A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44
(2007) 133152
When using excavation and traction results here, onewould
balance H with the drawbar pull while V contrib-utes to the
effective weight of the vehicle, either heavieror lighter. Using
known soil parameters, one adjustsmachine parameters until H and
drawbar pull match,with some established drawbar pull over-design
factor.Given that we do not know the soil parameters precisely,one
has to do soil parametric studies to be sure thedesign is
tolerant.
Beyond the scope of this work is the customary fieldtesting
required to validate design with reality. Field testingis a very
expensive proposition for lunar exploration thatneeds careful
consideration in order to avoid robotic orhuman support
failures.
2. 2-D models of excavation
2.1. Osman [2,3]
This model uniquely allows for a curved soil shear fail-ure
surface. A curved failure surface is more realistic inexcavating
than the flat surface other models assume, butis determined by some
authors as not a significant enoughquantitative contribution to
warrant the mathematical dif-ficulty. The Osman model contains
surcharge and toolsoileffects, while it lacks the inertial forces
that bring the soilfrom rest up to the speed of the tool. This
model also lacksexplicit inclusion of the soil internal friction
angle.
The excavation force equation is
T w 0:5gct2ftan245 0:5qgd1 e2w
0 tan q 14 tan q
r20gcd2
gqtftan245 0:5qgd4
d13
w 0:5r21 r20c
tan q 2ctftan45 0:5qgd4
gqtsin45 0:5q d5 Cald7
d16
1
and the horizontal and vertical components of the totalforce, T,
are
H T sinb d;V T cosb d:
The authors here do not develop the use of this modelsince the
dis are indeterminate in the original Osmanpaper. To get the dis an
optimizing iteration is requiredon a parameter k, related to the
location of the center ofrotation of the log-radius shear surface,
that is not mathe-matically explicit in the formulation by either
Osman orBlouin. As a result no quantitative conclusions are
meritedhere. Fig. 2 shows the behavior of H for several
parameterswhile holding all other parameters constant as in Table
1.The dis used in this figure are arbitrary. In this model cer-tain
values of q will cause the tan(f(q)) and 1/tan(q) func-tions to
become divergent.
2.2. Gill and Vanden Berg [2,4]
This model, like Osmans, is of older origins. It adds aninertial
force contribution with the tool velocity term. Sur-charge is
missing from this model, but simplified toolsoil
-
0.2 0.4 0.6 0.8 1.0
5000
6000
7000
8000
Horizontal Force vsRankine Passive Depth
OsmanRankine Passive Depth, t (m)
For
ce (
N)
0 50 100 150
5400
5450
5500
5550
5600
5650
5700
Horizontal Force vsSurcharge
Surcharge, q (N/m2)
For
ce (
N)
1500 2000 2500 3000 3500
5000
6000
7000
8000
Horizontal Force vsSoil Density
Soil Mass Density, (kg/m ) 3
For
ce (
N)
0 1000 2000 3000 4000
5000
1500
025
000
3500
0
Horizontal Force vsCohesion Stress
Cohesion, c (N/m2)
For
ce (
N)
0.0 0.5 1.0 1.5
4000
5000
6000
7000
8000
Horizontal Force vsLog Spiral Angle
Log Spiral Angle, w (radians)
For
ce (
N)
1 2 3 4 5
1000
020
000
3000
040
000
5000
0
Horizontal Force vsInitial Log Spiral Radius
Initial Log Spiral Radius, r0 (m)
For
ce (
N)
Fig. 2. Osmans required drawbar force variation for significant
parameters.
A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44 (2007)
133152 135
friction, soilsoil cohesion, and inertial forces are included.In
the simplest form, the model for the horizontal force canbe written
as
H No sin b dNo cos b K wwith an ideal penetration cutting force
term contained inK w. Blouin re-writes the force equation to allow
leavingout that force as
H H K w No sin b dNo cos b:
Eq. (2) breaks out the full complexity of No
H gc sinb qsin q
l d cosb q2 sin q
d sinb q tan b2 sin q
csin qsin q / cos q
cv2 sin bsinb qsin q / cos q
wdsin b d cos bsin q / cos qsinq b1 /d cosq b/ d : 2
-
0.2 0.4 0.6 0.8 1.0
020
0040
0060
0080
0010
000
Horizontal Force vsTool Depth
GillDepth, d (m)
For
ce (
N)
total
cut
tool_soil
depth
cohesion
kinetic
20 40 60 80
020
0060
0010
000
1400
0
Horizontal Force vsRake Angle
Rake Angle, , (deg)
For
ce (
N)
20 25 30 35 40 45 50
010
0020
0030
0040
00
Horizontal Force vsSoil Internal Friction Angle
Internal Friction Angle, (deg)
For
ce (
N)
0 1000 2000 3000 4000
020
0040
0060
0080
00
Horizontal Force vsCohesion Stress
Cohesion, c (N/m2)
For
ce (
N)
2 4 6 8 10
050
0010
000
1500
020
000
Horizontal Force vsGravity
Acceleration of Gravity, g (m/s2)
For
ce (
N)
1500 2000 2500 3000 3500
020
0040
0060
00
Horizontal Force vsSoil Density
Soil Mass Density, (kg/m3)
For
ce (
N)
Fig. 3. Gills drawbar force variation for significant
parameters.
136 A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44
(2007) 133152
The penetration K w term is kept in results here as it
hassignificant magnitude. The total and vertical componentsof this
force are
T H cscb d;V H cotb d:From this point forward plot results will
show the individ-ual contributions of physical sources of the
forces in order
to gauge their relative contributions. Appendix A givesthe exact
expressions used for each physical source notedby the legends. See
Fig. 3 for the predicted drawbar forcerequirement under the
variation of several parameters whileholding all other parameters
constant as in Table 1. Thismodel indicates that depth and toolsoil
friction are thedominant contributors to the drawbar pull required
forexcavation. Depth, gravity, and rake angle, b, present the
-
0.2 0.4 0.6 0.8 1.0
010
0020
0030
0040
00
Horizontal Force vsTool Depth
SwickDepth, d (m)
For
ce (
N)
totalsurchargetool_soildepthcohesionkinetic
20 40 60 80
100
00
1000
2000
3000
Horizontal Force vsRake Angle
Rake Angle, (deg)
For
ce (
N)
20 25 30 35 40 45 50
050
010
0015
0020
00
Horizontal Force vsSoil Internal Friction Angle
Internal Friction Angle, (deg)
For
ce (
N)
0 1000 2000 3000 4000
010
0020
0030
0040
00
Horizontal Force vsCohesion Stress
Cohesion, c (N/m2)
For
ce (
N)
2 4 6 8 10
010
0020
0030
0040
0050
00
Horizontal Force vsGravity
Acceleration of Gravity, g (m/s2)
For
ce (
N)
1500 2000 2500 3000 3500
050
010
0015
0020
00
Horizontal Force vsSoil Density
Soil Mass Density, (kg/m3)
For
ce (
N)
Fig. 4. Swick and Perumpral drawbar force variation for the
dominant parameters.
A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44 (2007)
133152 137
strongest functional dependence. The lunar soil cohesionused
here makes cohesion a small contribution. Cohesionas a parameter
projects roughly a 100% increase over therange, while the internal
friction angle, /, projects a 34% in-crease. Over a plausible range
of soil density the requireddrawbar force increases by 120% in a
linear way, althoughno effect of density on K was allowed here.
This model pre-sents mathematical problems as the rake angle, b,
ap-
proaches 90, as it might for a backhoe, due to the tanbfactor in
the first line of Eq. (2).
2.3. Swick and Perumpral [2,5]
The Swick and Perumpral model includes a moresophisticated
toolsoil friction term with the adhesionparameter Ca. All the
physical effects of a simple blade
-
0.2 0.4 0.6 0.8 1.0
010
0020
0030
0040
00
McKyesDepth, d (m)
For
ce (
N)
totalsurchargetool_soildepthcohesionkinetic
20 40 60 8010
000
1000
2000
3000
Rake Angle, (deg)
For
ce (
N)
20 25 30 35 40 45 50
050
010
0015
0020
00
Horizontal Force vs
Horizontal Force vsTool Depth
Horizontal Force vsRake Angle
Soil Internal Friction Angle
Internal Friction Angle, (deg)
For
ce (
N)
0 1000 2000 3000 4000
010
0020
0030
0040
00
Horizontal Force vsCohesion Stress
Cohesion, c (N/m2)
For
ce (
N)
2 4 6 8 10
010
0020
0030
0040
0050
00
Horizontal Force vsGravity
Acceleration of Gravity, g (m/s2)
For
ce (
N)
1500 2000 2500 3000 3500
050
010
0015
0020
00
Horizontal Force vsSoil Density
Soil Mass Density, (kg/m3)
For
ce (
N)
Fig. 5. McKyes drawbar force variation for significant
parameters.
138 A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44
(2007) 133152
without end effects are included in this model. The magni-tude
of the total force is written as follows:
T Ca cosb / qsin b
gcd2cot b cot q sin/ q
gqcot b cot q sin/ q c cos /sin q
cv2 sin b cos /sinb q
wdsinb / q d
3
and the horizontal and vertical components of the totalforce, T,
are
H T sinb d;V T cosb d:
See Fig. 4 for the variation of H for several parameters.
Thismodel is problematic for certain angle ranges. If b + / +q + d
P 180, then the multiplier on the last line of Eq. (3)passes
through a singularity and jumps negative. If b + /+ q < 90, then
the toolsoil force is non-physical for smallrake angles, b, which
comes from the Ca term in Eq. (3).This paper considered using the
rule that q = 45 + //2.However, that produced the singular
multiplier using lunarsoil parameters at larger rake angles. As a
result a fixed valueof q is chosen in Table 1, that of a sand [6].
This still produced
-
0.2 0.4 0.6 0.8 1.0
020
0040
0060
0080
00
Horizontal Force vsTool Depth
MuffDepth, d (m)
For
ce (
N)
totalfrictioncohesionkinetic
20 40 60 80
010
0030
0050
0070
00
Horizontal Force vsRake Angle
Rake Angle, (deg)
For
ce (
N)
0.00 0.05 0.10 0.15 0.20 0.25 0.30
050
015
0025
0035
00
Horizontal Force vsTool Velocity
Tool Velocity, v (m/s)
For
ce (
N)
0 1000 2000 3000 4000
020
0040
0060
0080
00
Horizontal Force vsCohesion Stress
Cohesion, c (N/m2)
For
ce (
N)
2 4 6 8 10
050
0010
000
1500
020
000
Horizontal Force vsGravity
Acceleration of Gravity, g (m/s2)
For
ce (
N)
1500 2000 2500 3000 3500
010
0030
0050
0070
00
Horizontal Force vsSoil Density
Soil Mass Density, (kg/m3)
For
ce (
N)
Fig. 6. Viking models drawbar force variation for significant
parameters under lunar conditions.
A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44 (2007)
133152 139
the negative toolsoil force contribution as seen in the
hor-izontal force vs. rake angle plot in Fig. 4.
If one accepts that the failure plane angle in triaxialshear
cell tests is the same angle as q, then using the Mohrstress circle
defined by triaxial test results, one getsq = 45 + //2 [7, p.
1415]. Hoek also suggests that qdepends on b as well [8]. Some
forms of dependence of qon b could either eliminate or exacerbate
the singular mul-tiplier problem, depending on the form.
Overall, depth remains the largest contributing term as itwas
for Gill. Depth, gravity, and rake angle show strongeffects.
However, soilsoil cohesion has a strong depen-dence at values
larger than the lunar value here with
a 270% increase over the range plotted. Soil internal
frictionangle, /, projects a 150% increase over the range. Soil
den-sity shows a 90% increase over the range plotted.
Surchargeremains small as its chosen mass per area here is
small.
2.4. McKyes [2,7]
Like Swick and Perumpral, the McKyes model includesall the
physical effects of a two-dimensional blade withoutend effects.
This model is an enhancement of early work byReece [9]. In the
simplest form this model shows a heritagefrom the Terzaghi
coefficients, Nx [10]. The total forceequation is
-
140 A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44
(2007) 133152
T wcgd2N c cdN cCadN ca gqdN q cm2dN a;
T cgdcotb cotq2
gqcotb cotq c1 cotqcotq/Ca1 cotbcotq/
cv2tanq cotq/
1 tanqcotb
wdcosb d sinb dcotq/
4
and the horizontal and vertical components of the totalforce, T,
are
H T sinb d;V T cosb d:
A feature of the results in Fig. 5 is the
quantitativeequivalence to Swick and Perumpral, even with a
differ-ent mathematical description. However, the chief differ-ence
between Eqs. (3) and (4) is in the handling of thetrigonometric
factors. This model shows the same math-ematical problems for the
angle sums as noted in Section2.3; negative toolsoil adhesion force
for some rake an-gles depending on the choices of q and / and a
vanishingdivisor on the last line of Eq. (4) for other rake
angles.Together these cause concern for the generality of
thesemodels.
2.5. Lockheed-Martin/Viking [1113]
This model was used by Martin-Marietta Corp. in coop-eration
with the Colorado School of Mines for design ofthe Mars Viking
lander robotic excavation arm. The origi-nal forms come from Luth
and Wismer testing sands forfriction terms and clays for cohesive
terms [12,13]. Theseequations are also in active application for
characterizationof a bucket-wheel excavator for lunar and Martian
use.The model includes toolsoil friction, soilsoil shear
resis-tance due to cohesion, and inertial effects. Velocity hassome
non-quadratic contribution in the cohesion equa-tions; v0.121 in
Hcohesion and v
0.041 in Vcohesion. Surcharge,shear plane failure angle and
soiltool adhesion are not fac-tors in this model. The use of
assorted exponents and con-stant terms make it hard to reflect on
the physical details ofthe model.
H and V stand for horizontal and vertical force compo-nents
respectively in Eqs. (5)
H friction cgwl1:5b1:73ffiffiffidp d
l sin b
0:77
1:05 dw
1:1 1:26 v
2
gl 3:91
( );
V friction cgwl1:5ffiffiffidpf0:193 b 0:7142g d
l sin b
0:777
1:31 dw
0:966 1:43 v
2
gl 5:60
( );
H cohesion cgwl1:5b1:15ffiffiffidp d
l sin b
1:21
11:5ccgd
1:212v3w
0:1210:055
dw
0:78 0:065
!(
0:64 v2
gl
;
V cohesion cgwl1:5ffiffiffidpf0:48 b 0:703g d
l sin b
11:5ccgd
0:412v3w
0:0419:2
dw
0:225 5:0
!(
0:24 v2
gl
: 5
Examining Fig. 6, friction contributions far outweigh
othercontributions under low cohesion lunar conditions andmodest
excavation speed. Inertial (kinetic) contributionsremain small as
they have in other models in this paper.Depth, gravity, and rake
angle dependence remain strong.Cohesion effects have a weakly
non-linear character as dis-tinct from the other models and
projects a 140% increaseover the plotted range. This model, unique
from the othersin this paper, almost triples the required drawbar
pull overthe density range considered. The angles d, /, and q do
notfigure into this model, and the rake angle does not createthe
mathematical problems of the Gill, Swick and Perump-ral, and McKyes
models.
2.6. 2-D section summary
In this section the Gill, Swick and Perumpral, along withthe
McKyes and Viking models, predict in the plots factor-of-2-like
differences between the models in the expecteddrawbar forces
required under identical lunar conditions.Cohesion causes drawbar
pull requirements to vary morethan soil internal friction for the
range of each parameterconsidered plausible here. It is beyond the
scope of this paperto survey the comparison of these predictions
with terrestrialfield test data. The authors suggest that the level
of quantita-tive variance in these models requires field tests in
situ on the
lunar surface to validate their design dependability for
long-
term space hardware procurement. Pending validation theauthors
cannot recommend one model over another.
3. 3-D models for excavation
Most excavator buckets have sides that cut through soiland
confine material loss from the bucket. This sectionadds the cutting
and toolsoil adhesion effects from thesesides. Some Russian workers
laid the foundations for thisthinking [1416].
3.1. Hemami [2,17]
Hemami has partitioned the excavation forces withattention to
all the physical actions as identified below.
-
Table 2Balovnev data set: Cartesian forces for various
excavation depths
Depth (m) Horizontal force (N) Vertical force (N)
1 0.05 74.84 52.402 0.10 156.71 109.733 0.15 262.87 184.074 0.20
393.34 275.425 0.25 548.11 383.796 0.30 727.17 509.177 0.35 930.53
651.578 0.40 1158.19 810.98
A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44 (2007)
133152 141
Fig. 7 illustrates these forces. Eq. (6) adds up all the
hori-zontal (x) components
F x f1x f2x f3x f4x f5x f6x; 6where f1 is the weight of the
accumulating material in thebucket, f2 is the resistance from
compacting the material,f3 is all the friction forces of material
sliding on bucket sur-faces, f4 is the penetration or cutting
resistance, f5 is theinertial force from accelerating the material
to the toolvelocity much like in Section 2. Finally, f6 is the
inertialforce to move the empty bucket.
3.2. Balovnev [2,14]
The non-inertial components of Hemamis model can befurther
defined by associating the fi to the horizontal forcesaccording to
Balovnev:
f4x P 1 P 2 P 3 and f 3x P 4;where P1 is the cutting and surface
friction resistance of aflat trenching blade with a sharp edge; P2
is the additionalcutting resistance due to resistance from a blunt
edge; P3 isthe resistance offered by cutting from the two
confiningsides of the bucket; and P4 is the resistance due to
frictionon those sides. Interestingly f1, f2, f5, f6 are not
included inthis picture. The Russian literature considers these
second-ary and small [16,15].
The horizontal component of the total force is now writ-ten
as
H f4x f3x P 1 P 2 P 3 P 4
wd1 cot b cot dA1dgc2 c cot / gq BURIED
d l sin b gc 1 sin /1 sin /
web1 tan d cot ab
A2ebgc
2 c cot / gq dgc 1 sin /
1 sin /
2sdA3dgc2 c cot / gq BURIED
d ls sin b gc1 sin /1 sin /
4 tan dA4lsd
dgc2 c cot / gq BURIED d ls sin b
gc 1 sin /1 sin /
; 7
Fig. 7. Definition of forces used in this section [2].
where BURIED = TRUE or FALSE is a 1 or 0 dependingon whether the
whole bucket is below the soil surface ornot.
A1 Ab; A2 Aab; A3 A4 A p2
are geometricfactors depending on the angle of a surface with
respectto a reference plane. To calculate a particular Ai replaceb
with the appropriate argument in the following equation:
Ab 1 sin / cos2b1 sin /
if b < 0:5 sin1sin dsin /
d
;
cos d cos
dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin2
/ sin2 d
q 1 sin / e
2bpdsin1 sin d
sin /
tan /
if b P 0:5 sin1sin dsin /
d
:
The total and vertical components of this force are
T H cscb d;V H cotb d:
Table 2 provides horizontal and vertical forces as a func-tion
of cutting depth. These numbers are between thoseof the Gill and
McKyes models. As a compromise they willbe used in the discussions
of Section 4.
Fig. 8 shows the dependence on digging depth, soildensity,
surcharge, soil cohesion, and gravity. The dis-continuity in the
Depth plot is caused by the bucketentering the buried condition.
The drawbar force showslinear dependence on soil cohesion and
gravity like allthe other models except the Viking model.
Surcharge,as in all the other models, has little effect unless it
isgreatly increased. Soil density causes a 133% increaseof the
drawbar force over the range plotted. Cohesion
9 0.45 1410.15 987.4010 0.50 1776.48 1243.9111 0.55 2158.48
1511.3812 0.60 2577.94 1805.0913 0.65 3034.87 2125.0414 0.70
3529.27 2471.2215 0.75 4061.13 2843.6416 0.80 4630.47 3242.2917
0.85 5237.27 3667.1718 0.90 5881.54 4118.3019 0.95 6563.27
4595.6520 1.00 7282.48 5099.25
-
142 A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44
(2007) 133152
projects a 740% increase over the plotted range, muchgreater
than any other model. Depth remains the stron-gest contributor to
the drawbar pull, as it was in theother models.
Fig. 9 shows the dependence on various angles. The
dis-continuity in the rake angle plot is due to the bucket
exitingthe buried condition as the rake angle increases. The
shearplane failure angle, q, does not enter this model. The
rakeangle in various angle summations does not create theproblem
noted for the Gill, Swick and Perumpral, andMcKyes models. However,
if d is greater than /, thenA(b) misbehaves in this model. Overall
though, the qualita-tive shapes of the curves are similar to the
other models.
The Balovnev model shows that the blunt edge has littleeffect.
Soiltool adhesion caused by the sides is a prominentcontributor.
Side effects are ignored in all Section 2 models.The soil internal
friction angle, /, projects a 130% increasein horizontal force over
the plot range.
3.3. 3-D section summary
In all the excavation models of this paper soil propertiesare
considered homogeneous. Chapter 9 and Section 1.4 of
0.2 0.4 0.6 0.8 1.0
020
0040
0060
00
Horizontal Force vsTool Depth
Balovnev.1Depth, d (m)
For
ce (
N)
totalsurchargedepthcohesionsharp_bladeblunt_bladesides_cutsides_friction
0 1000 2000 3000 4000
020
0040
0060
0080
0010
000
Horizontal Force vsCohesion Stress
Cohesion, c (N/m2)
For
ce (
N)
Fig. 8. Balovnevs drawbar force variation for some significant
parameters. NoAlso the sum of sharp_blade (P1), blunt_blade (P2),
sides_cut (P3), and sides_intended to help comparison to previous
models as well as examine the new f
the Lunar Sourcebook gives estimated fits of density withdepth
[1]. The interdependence of parameters like (b,/,q)and (g,/,c),
which seem physically plausible, are neitherdealt with here nor
often in the literature. The Balovnevmodel includes some larger
contributors to excavationforces not seen in Section 2. Cohesion
causes requireddrawbar pull to vary more than it does with soil
internalfriction and by a wider margin than in the 2-D models.Table
3 provides a list of excavation depths for a 1 m widebucket that
requires a drawbar pull expected for the Apollorover as estimated
in Section 4 for the moon. Lunar-fieldtest data is needed to sort
which depth is valid. Pendingvalidation the authors cannot
recommend one model overanother.
4. Traction [1820]
Three notable methods of performing traction calcula-tions are:
(1) agricultural engineerings mobility index-based [21] method, (2)
the NATO Reference MobilityModel [22,23, p. 120] method, and (3)
the normal and shearstress-based method of Bekker [20]. The first
two use datafrom cone-penetrometer measurements as the single
source
1500 2000 2500 3000 3500
050
010
0015
0020
0025
0030
00
Horizontal Force vsSoil Density
Soil Mass Density, (kg/m3)
For
ce (
N)
2 4 6 8 10
020
0040
0060
0080
00
Horizontal Force vsGravity
Acceleration of Gravity, g (m/s2)
For
ce (
N)
te that the sum of surcharge, cohesion, and depth add up to the
total force.friction (P4) add up to the total force in these plots.
This break-out was
eatures of this model.
-
20 40 60 80
050
010
0015
0020
0025
00
Horizontal Force vsRake Angle
Balovnev.2Rake Angle, (deg)
For
ce (
N)
0 5 10 15 20 25 30 35
020
0040
0060
0080
00
Horizontal Force vsExternal Friction Angle
External Friction Angle, (deg)
For
ce (
N)
totalsurchargedepthcohesionsharp_bladeblunt_bladesides_cutsides_friction
15 20 25 30 35 40 45
050
010
0015
0020
0025
00
Horizontal Force vsSoil Internal Friction Angle
Internal Friction Angle, (deg)
For
ce (
N)
0.01 0.02 0.03 0.04 0.05
050
010
0015
00
Horizontal Force vsBlunt Edge Thickness
Blunt Edge Thickness, eb (m)
For
ce (
N)
Fig. 9. Balovnevs required drawbar force variation for
additional significant parameters.
Table 3Spread of digging depth for roughly the same drawbar pull
on the moon
Model Depth (m) Required drawbarpull (N)
Gill
-
D
z
l
soil surface
Fig. 10. Simple definition of geometrical parameters for
traction.
144 A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44
(2007) 133152
may well behave as brittle dry granular soils. A form forsoils
in general (including brittle soils) is [20, p. 8]
H H 0 exp K2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiK22 1
q K1S L
exp K2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiK22 1
q K1S L
: 10
Table 4Parameter set base: (SI units) used for traction
calculations
Values Units
Calculated slippage, S 8.30 %Calculated sinkage, z 0.00863 mTire
squat, e 0.0081 mSoil rupture angle, q 62.5 radSoil specific mass,
c 1680 kg/m3
Vehicle mass, W 698.5 kgGravity level, gM 1.63 m/s
2
Terrain slope angle, h 0 degInternal friction angle, / 35
degSoil cohesion, c 170 N/m2
Calculated wheel/track contact area, A 0.0400 m2
Wheel nominal width, B 0.27 mWheel diameter, D 0.81 mCalculated
wheel contact length, L 0.188 mNumber of wheels/tracks, n 4Contact
grousers per wheel/track, Ng 4Grouser height, h 0.0015875
mCalculated k 825,185 Pa=mn
0
kc 1400 Pa=mn01
k/ 820,000 Pa=mn0
Calculated Kc 48.6Nc 25Calculated Kc 0.537Nc 1.5Calculated l0
0.00234 mkt 50Degree of brittleness K1 5.56 m
1
Slip strength K2 2.5Coefficient of surface adhesion, x 0.000518
m2/NShear deformation slip modulus, j 0.018 mSoil deformation
exponent, n0 1Tracked OR not 0 1 = TRUE,
0 = FALSECalculated net drawbar pull, DP 239 N
In Fig. 11 all three of these forms for drawbar pull are
com-pared using values for K1 and K2 that are selected to trackEq.
(9) as well as possible. K1 and K2 are called slip coeffi-cients,
and values do not exist for lunar soils. Larger valuesof K1 cause
steeper initial slope and quicker fall-off at largerslip in plots
of H = f(S). K1 is the degree of brittleness, acohesion effect.
Larger values of K2 cause an increase inthe magnitude of H = f(S)
for all slippage values. K2 isthe degree of slip strength, a
friction effect. Today onewould measure the drawbar pull versus
slippage and fitthe curve for the best values of K1 and K2.
Appendix B gives a range for K1 and K2 that span fromloose sand
(low cohesion and high friction angle) to brittle/compact (high
cohesion and low friction angle) soils in ter-restrial experience
[19, p. 2667]. Notice K1 and K2 usedhere are of lower cohesion and
higher friction angle soilthan a slightly moist sandy loam that
Bekker describes.
To get H0, the ideal tractive thrust available, for a vehi-cle
with a soil contact area of A = bL per wheel, track, orleg, and W
is the total vehicle weight, one uses the equation[18]
H 0 AcWn
l Ac Wn
tan /:
For an n-wheeled or n-legged powered vehicle, total soilthrust
is given by
H 0 nbLc W tan /:Different weight loading per wheel, track, or
leg is not con-sidered here for simplicity. Likewise the effects of
pre-con-solidation or confining pressure from neighboring wheelsor
legs on improved soil shear strength and the resultant ef-fects on
H0 and slippage is not considered.
For tracked vehicles, the soil thrust increases due to
thegreatly increased ground contact area. For two tracks,
soilthrust is given by
H 0 2bLc W tan /:
4.2. Grousers
Grousers, or cleats, are an additional way to engage moresoil
(excavation-like) for greater traction. This paperincludes the
number of grousers interacting with the soil asthe wheel diameter
changes, given a fixed grouser spacing.However, it does not include
the soil shear strength decreasewhen the grousers are tall enough
and close enough to dis-rupt the soil of neighboring grousers. The
Apollo lunar roverhad 54 titanium alloy chevrons, 1.6 mm high,
covering 50%of the wheel surface [24]. This leads to a 2.4 cm
spacingbetween grousers. For a grousered, n-wheeled/legged
vehi-cle, the maximum soil thrust is given by [19, p. 2267]
H 0 nT 0 W tan /;where T 0 sN g hbcos qsin q h
2
cos q.
For a double tracked vehicle with Ng grousers of thick-ness, or
height h, in contact with the soil, the maximum soilthrust per
track is
-
2 4 6 8 10
050
010
0015
0020
0025
00
Draw Bar Pull vsGravity
Acceleration of Gravity, g (m/s2)
Net
Dra
w B
ar P
ull (
N)
TotalCompressiveBulldozing
0 20 40 60 80 100
020
040
060
080
0
Draw Bar Pull vsSlippage Alone
Slippage, s (%)
Net
Dra
w B
ar P
ull (
N)
Simple PlasticLSB PlasticGeneral & Brittle
0.00 0.04 0.08 0.1210
000
500
00
5000
1000
0
Draw Bar Pull vsSinkage Alone
Sinkage, z (m)
Net
Dra
w B
ar P
ull (
N)
TotalCompressiveBulldozing
0e+00 2e+05 4e+05 6e+05 8e+05
300
100
010
020
030
0
Draw Bar Pull vsModulii of Soil Deformation
k or 100 X kc
Net
Dra
w B
ar P
ull (
N)
kkc
0 1000 2000 3000 4000
050
100
150
200
250
Draw Bar Pull vsSoil Cohesion
Soil Cohesion, c (N/m2)
Net
Dra
w B
ar P
ull (
N)
TotalCompressiveBulldozing
20 25 30 35 40 45 50
010
020
030
040
0
Draw Bar Pull vsSoil Internal Friction Angle
Internal Friction Angle, (deg)
Net
Dra
w B
ar P
ull (
N)
TotalCompressiveBulldozing
Fig. 11. Bekkers predicted drawbar pull as a function of
gravity, slippage, sinkage, and soil strength.
A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44 (2007)
133152 145
H 0 2N gbLc 12hb
W tan / h
bcot1
hb
:
4.3. Motion resistances
There are several forces that subtract from the effectivedrawbar
pull after slippage. These forces are sinkage, bull-dozing, and
hill climbing, along with others relevant to aspecific context like
wheel flexure elastic losses or soil iner-tia with slippage of
grousers or transmission and runninggear losses. This paper allows
for the first three. Inertialosses may be small at low speeds as we
saw with the exca-
vation equations earlier. Sinkage and bulldozing depend onsoil
behavior.
4.3.1. Soil compaction due to sinkage
Energy is lost in packing down the soil by a wheel, leg,
ortrack. The modulus of soil deformation, k = kc/b + k/ [20, p.240,
31340 and 447], provides a cohesional, kc, and fric-tional, k/,
measure of the resistance to compaction due tosinkage. One should
find kc and K1, as well as k/ and K2,coupled by their cohesive and
frictional physical origins,respectively. The computer code
supporting this paper usesthe general form of sinkage for n wheels
or tracks as follows:
-
146 A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44
(2007) 133152
z Wn A k
1n0
Pk
1n0
; 11
where the total vehicle weight, W, is rationalized by the
totalground contact area, n A, for the graphs and tables here.
Slippage induced sinkage [20, p. 139] is not covered inthis
paper. Multiple passes in the same track by a wheelis not covered
by Eq. (11), unless one has new measure-ments of k and A.
Another form for sinkage of a balanced n-wheeled vehi-cle is
given by
z 3W =n3 n0k b
ffiffiffiffiDp
! 22n01
:
This is not a special case of Eq. (11), but rather an
empiricalform which Bekker wrote down that described three
differentexperimental results for a rigid wheel if one uses
different val-ues of n 0. See Ref. [20, p. 4379] for more
discussion.
Using z of Eq. (11), the soil compaction resistance for
ntracked, wheeled, and legged vehicles is determinedthrough the
following equation [20, p. 458 and 484]:
Rc nb k
n0 1
zn01: 12
Table 5Drawbar pull versus sinkage under lunar conditions
Sinkage (m) Drawbar pull (N) Compressiveresistance (N)
Bulldozingresistance (N)
1 0.0010 292.27 0.446 0.7242 0.0078 249.26 26.764 17.4203 0.0145
150.01 93.687 49.7434 0.0213 2.37 201.216 94.5965 0.0280 206.39
349.350 150.4756 0.0348 461.08 538.090 216.433
4.3.2. Bulldozing
Bulldozing comes from the pushing of soil in front of awheel by
the action of that wheel. For soft soils with largesinkage, this
term can dominate over compaction losses.
A form for the bulldozing resistance per wheel or trackis [20,
p. 453, Eqs. (2)(25)]
Rbbsina/2sinacos/
2zcKcgcz2Kc
pgcl3090/540
pcl20
180cl20 tan 45
/2
: 13
Bulldozing for wheeled vehicles involves all the terms ofEq.
(13). But for tracked vehicles, only the first term is rel-evant.
The key parameters of Eq. (13) are [10]
Kc nc tan / cos2 /;
Kc 2nc
tan / 1
cos2 / and
l0 z tan245 /=2:
7 0.0415 765.79 767.435 291.7938 0.0483 1119.97 1037.385 376.0269
0.0550 1523.20 1347.940 468.701
10 0.0618 1975.11 1699.101 569.45111 0.0685 2475.38 2090.867
677.95512 0.0753 3023.72 2523.238 793.92513 0.0820 3619.88 2996.214
917.10214 0.0888 4263.60 3509.796 1047.245
4.3.3. Gravitational
This force is just the simple projection of the vehicleweight
vector along an incline during ascent or descent, writ-ten here
as
Rg W sin h: 14
15 0.0955 4954.67 4063.983 1184.12816 0.1023 5692.88 4658.776
1327.54217 0.1090 6478.02 5294.174 1477.28618 0.1158 7309.90
5970.177 1633.16719 0.1225 8188.35 6686.785 1795.00220 0.1293
9113.17 7443.999 1962.612
4.4. Drawbar pull
With all the forces in hand the net drawbar pull is writ-ten,
assuming R is independent of slip, as follows:
DP H R H Rc Rb Rg Rother: 15Table 4 gives the parameters used
for the calculations ofTable 5 and Figs. 11 and 12, except where
they were specif-ically overridden to isolate a particular effect.
The tablevalues reflect literature values for the moon and the
4wheel-drive Apollo lunar rover [1,24].
Table 5 presents the data behind the sinkage plot inFig. 11.
This predicts that if the Apollo lunar rover were tosink about 2.1
cm into the regolith, then it would be stuck.Compressive resistance
increases more rapidly with sinkagethan bulldozing resistance for
this range of sinkage. Note,this paper does not develop the
consequences of soil param-eter changes with depth, like density,
as they are not certainenough in the literature. For example, the
lunar surface pow-der layer that sometimes is crusty is not
accounted for at all.
Fig. 11 maps the effects of gravity, slippage, sinkage,
andmodulus of sinkage deformation, k, on traction. Realize
thatslippage and sinkage, both proportional to vehicle weight
inthese calculations, are increasing with increasing gravity.Also
soil shear strength increases with gravity since the nor-mal stress
from the vehicle weight increases. Bulldozing resis-tance increases
more rapidly than compressive resistance asgravity increases. At
about two-thirds of Earths gravitydrawbar pull has a maxima for
this soil and vehicle.
Recall there are three slippage models above. One mightexpect
that the lunar soil could be described as brittle ratherthan
plastic with drawbar pull declining after failure ataround 60%
slippage in Fig. 11. The coincidence of theGeneral and Brittle
curve and the Lunar SourceBook(LSB) Plastic curve is arbitrary, as
there is no lunar datato fit for K1 and K2. The K1 and K2 used for
the plots arerelevant to a loose frictional sand. The Simple
Plasticand LSB Plastic curves are markedly different usingknown
lunar parameters. The LSB Plastic equation isused here in all
calculations.
-
0.1 0.2 0.3 0.4 0.5
050
100
150
200
250
300
Draw Bar Pull vsWheel Width
Wheel Width, B (m)
Net
Dra
w B
ar P
ull (
N)
1 2 3 4 5
050
100
150
200
250
300
Draw Bar Pull vsWheel Diameter
Wheel Diameter, D (m)
Net
Dra
w B
ar P
ull (
N)
1500 2000 2500 3000 3500
220
225
230
235
240
245
Draw Bar Pull vsSoil Density
Soil Mass Density, (kg/m3)
Net
Dra
w B
ar P
ull (
N)
0 2000 4000 6000 8000
050
010
0020
0030
00
Draw Bar Pull vsVehicle Mass
Vehicle Mass, W (kg)
Net
Dra
w B
ar P
ull (
N)
Fig. 12. Bekkers predicted drawbar pull for wheel width,
diameter, soil density, and vehicle mass.
A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44 (2007)
133152 147
The reader should be aware that in the k dependence plot,when
either kc (the cohesive contribution) or k/ (the fric-tional
contribution) are varied, then the other one is set toits Lunar
Sourcebook value. That is why the k/ curve inter-sects the kc curve
when k/ reaches its lunar value. k/ has thedominant dependence for
the soil strength. When k/ < 415kPa and z > 1.7 cm bulldozing
resistance is predicted todominate compressive resistance. If that
friction contribu-tion were less than roughly 225 kPa, then the
rover wouldbe stuck. The sinkage at this point is about 3 cm, or
approx-imately 4% of the wheel diameter. Perhaps something likethis
happened during Apollo when the lunar rover (LRV)drove in some soft
soil. Small k/ gives rise to large sinkage,that causes the approach
angle, a, to be greater than90; the wheel has sunk over its axel.
It is worth asking whatwould happen when driving on lose excavated
regolith pilesin the future.
The k/ dependence has a mathematical pathology forthe smallest
k/. The problem arises from the sin(a + /)/sina coefficient for the
bulldozing resistance in Eq. 13.When a + / > 180, then Rb is
negative, adding to drawbarpull unphysically. This can be seen in
the Draw Bar Pullvs. k plot. However, it is unrealistic for a or /
to everbe greater than 90, and this pathology should not be
apractical problem.
Cohesion shows about a 20% decrease in drawbar pullover the same
range considered for excavation forces ear-
lier. However, soil internal friction angle increases
drawbarpull something more than 300% over the range used in
theexcavation sections here.
Fig. 12 shows that traction is not strongly dependent onsoil
density (11% decrease), assuming other parameters likek are
constant. Not surprisingly narrow wheels get stuck.For a width less
than 3.8 cm the loaded lunar rover is stuckin this calculation. On
the other hand, changing the widthto greater than 3040 cm does not
improve traction much.Increasing the wheel diameter decreases
traction in thismodel. That is a consequence of decreased soil
shearstrength with wheel bearing pressure from increased con-tact
area and the 1/L2 dependence of slippage used here.The exponential
in Eq. (9) dominates the 1/(S L) factorleading to a decreasing H
with wheel diameter. If S L werea constant, then increased wheel
diameter would increasedrawbar pull as the increasing grouser
number in soil con-tact would overcome decreased wheel bearing
pressure, buta law of diminishing returns occurs. The number of
grous-ers per unit circumferential length was held
constant.Increasing vehicle mass shows the strongest effect in
thisfigure. At about 209 kg, the mass of the Apollo rover with-out
cargo, the drawbar pull is roughly 26 N. An optimummass for maximum
drawbar pull is seen. It may pay to self-load an excavator or rover
with regolith to greatly improveits traction without having to
bring mass from the Earth.However, overloading beyond the optimum
is detrimental.
-
148 A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44
(2007) 133152
4.5. Traction summary
Table 4 predicts that the Apollo lunar rover could pro-vide
about 239 N of drawbar pull fully loaded on a levelsurface.
Compressive resistance is the most significant lossfor traction,
given lunar soil parameters. However, whenk/ is small and sinkage
is large, or when cohesion or grav-ity are larger, then bulldozing
resistance dominates. Brittlelunar soil slippage effects are
unknown for lack of tractionmeasurements during Apollo. Sinkage
gets one stuck rathereasily on the moon. Increased soil density and
cohesionhave a modest decreasing effect on traction, while soil
inter-nal friction has a large effect. This is the opposite of
whatwas seen for cohesion and density effects on excavation.Not
surprisingly wheel width and diameter play an impor-tant role for
designers, and slippage constraints can makelarger a diameter
helpful or a handicap. A band of vehiclemass offers a maxima for
traction performance.
This work enables convenient calculations for a trackedvehicle
as well, but is not presented here.
5. Conclusions
Besides base-lining calculations using Apollo soil
sampleresults, this paper presents selected soil and machine
para-metric studies to highlight the most significant parametersfor
both the excavating bucket and the tractive vehicle driv-ing the
bucket. If all were as modeled, this work suggests theApollo rover
could excavate from roughly 1 to 15 cm depthdepending on which
excavation model is most accurate.However, the lunar rover could
get stuck if it sank nomi-nally more than 2.1 cm into virgin
regolith (ignoring theloose powder on the surface). There is no
available workon traction and sinkage into excavated lose lunar
regolith,and getting stuck on such soil will be more likely.
Soil cohesion has a large increasing effect on excavationforces
but a small decreasing effect on traction. Soil internalfriction
has a relatively small increasing effect on excavationforces but a
rather large increasing effect on traction. Soildensity produces a
moderate to strong increase in excavationforces, while producing a
small decrease in traction. Thecompeting effects of these soil
parameters when balancingexcavation forces against traction forces
suggests that versa-tility of equipment designs for diverse lunar
soils is hard toaccomplish.
Some mathematical anomalies of the excavation andtraction models
are noted.
A key capability of these results is that one could testmodels
and lunar parameters with small scale excavationand traction
devices on lunar robotic precursor missionsimmediately. All
equations in this paper have been com-puter programmed for future
calculations. These modelshave the better part of a century of
application to terrestrialsoils and machines. In all such
applications patient field test-ing was needed to establish the
soil parameters to match themodels. However, the predictability of
these equations forthe moon are unknown, without field testing,
across the
widely variable soils of the lunar highlands, maria, and fro-zen
poles. Excavation and traction tests during robotic lunarmissions
is strongly recommended by this work.
Modern excavating companies like Caterpillar and JohnDeere are
actively working on more granular physics-basedpredictive
algorithms, knowing that soil mechanics predic-tions use many
empirical parameters to fit observationsand that tight tolerance
machine designs are not possiblewithout more fundamental physics
understanding. As duringApollo, NASA could choose to advance the
state-of-the-artin terramechanics, starting with robotic lunar
missions.
Acknowledgements
This work was funded by the NASA Exploration Initia-tive
transitional support under In-Situ Resource Utiliza-tion. We thank
Chris Gallo and Dr. Juan Agui and Dr.Richard Rogers for editorial
improvements to this manu-script. Edward Katich, a summer intern,
assisted with liter-ature collection and database entry.
Appendix A. Expressions used to calculate curves for
eachphysical force of each model
Gill and Vanden Berg model:
Cutting: Kw,Toolsoil:
lgcsinb q
sin q wdsin b d cos bsin q / cos qsinq b1 /d cosq b/ d :
Depth:
gcsinb q
sin qd cosb q
2 sin q d sinb q tan b
2 sin q
wdsin b d cos bsin q / cos qsinq b 1 /d
cosq b/ d:
Cohesion:
csinqsinq/ cosq
wdsinb d cosbsinq/ cosqsinq b1/d cosq b/ d :
Kinetic:
cv2 sin bsinb qsin q / cos q
wdsin b d cos bsin q / cos qsinq b1 /d cosq b/ d :
Swick and Perumpral model: (Each term multiplied bysin(b + d)
for horizontal component)
Surcharge:
gqcot b cot q sin/ q wdsinb / q d :
-
A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44 (2007)
133152 149
Toolsoil:
Cacosb / q
sin b wdsinb / q d :
Depth:
gcd2cot b cot q sin/ q wd
sinb / q d :
Cohesion:
c cos /sin q
wdsinb / q d :
Kinetic:
cv2sin b cos /sinb q
wdsinb / q d :
McKyes model: (Each term multiplied by sin(b + d) forhorizontal
component)
Surcharge:
gqcot b cot q wdcosb d sinb d cotq / :
Toolsoil:
Ca1 cotb cotq/ wd
cosb d sinb d cotq/ :
Depth:
cgdcot b cot q
2 wdcosb d sinb d cotq / :
Cohesion:
c1 cot q cotq / wdcosb d sinb d cotq / :
Kinetic:
cv2tan q cotq /
1 tan q cot b wd
cosb d sinb d cotq / :
Lockheed-Martin/Viking model:
Friction:
cgwl1:5b1:73ffiffiffidp d
l sin b
0:771:05
dw
1:1 1:26 v
2
gl 3:91
( ):
Cohesion:
cgwl1:5b1:15ffiffiffidp d
l sin b
1:2111:5c
cgd1:21 2v
3w
0:121(
0:055 dw
0:78 0:065
! 0:64 v
2
gl
):
Kinetic:
cgwl1:5b1:73ffiffiffidp d
lsinb
0:771:26
v2
gl cgwl1:5b1:15
ffiffiffidp d
lsinb
1:21
11:5ccgd
1:212v3w
0:1210:055
dw
0:78 0:065
! 0:64 v
2
gl
( ):
Balovnev model:
Surcharge:
gqwd1 cot b cot dA1 web1 tan d cot abA2 2sdA3 4 tan dA4lsd:
Depth:
wd1 cotb cotdA1dgc2BURIED d l sinb
gc1 sin/1 sin/
web1 tand cotabA2dgc
1 sin/1 sin/
2sdA3dgc2BURIED d ls sinb gc
1 sin/1 sin/
4 tandA4lsddgc2BURIED d ls sinb
gc1 sin/1 sin/
:
Cohesion:
c cot /wd1 cot b cot dA1 web1 tan d cot abA2 2sdA3 4 tan
dA4lsd:
Sharp blade:
wd1 cot b cot dA1dgc2 c cot / gq
BURIED d l sin b gc 1 sin /1 sin /
:
Blunt blade:
web1 tand cotabA2ebgc
2 c cot/ gq dgc 1 sin/
1 sin/
:
Sides cut:
2sdA3dgc2 c cot / gq BURIED d ls sin b
gc 1 sin /1 sin /
:
Sides friction:
4 tan dA4lsddgc2 c cot / gq BURIED d ls sin b
gc 1 sin /1 sin /
-
Appendix B. Define notation
Model symbol Coding parameter Units {Range} description
Tooleb eb m {0.005:0.10} Blunt edge thicknessl l m {0.1:1} Tool
length (front to back)ls ls m {0.8 * l} Length of side plates s m
{0.01:0.1} Side plate thicknessw w m {0.3:3} Tool widthab alphab
deg {10:45} Blunt edge angle
Soil s = c + r(c) tan/c c N/m2 {68:4500} Cohesion, Bekker p. 332
and 340 [20], LSB
p. 529 [1]c gamma kg/m3 {1,200:3,500} Specific mass {[1] p. 494
and 536},
Blouin mixed this up with specific weight betweenmodels
/ phi deg {20:50} Internal friction angle
Toolsoil Tool shear = Ca + No(c) tandCa Ca N/m
2 {200:5000} Soiltool adhesiond delta deg {0:50} External
friction angle
Operation
d d m {0.05:1.0} Tool depthd1 d1 Dimensionless {unknown} Osmans
graphical distanced2 d2 Dimensionless {unknown} Graphical
distanced3 d3 Dimensionless {unknown} Graphical distanced4 d4
Dimensionless {unknown} Graphical distanced5 d5 Dimensionless
{unknown} Graphical distanced6 d6 Dimensionless {unknown} Graphical
distanced7 d7 Dimensionless {unknown} Graphical distanceK K N/m
{0:10} Cutting resistance indexq q kg/m2 {0.5:100} Surcharge * g =
N/m
2
r0 r0 m {0.5:5} Initial radius of log spiralr1 r1 m Calculated
radius of log spiralt t m {0.05:1} Depth of Rankine passive zonev v
m/s {0.01:0.3} Tool speedw 0 wprime deg {0:90} Polar angle in log
spiralb beta deg {5:90} Rake or cutting angleq rho deg q = 45 + //2
OR {20:55} [6] soil rupture angle
(traction) OR shear plane failure angle (excavation),w.r.t. soil
surface
r sigma N/m2 Normal stress WnA per wheel, for tractions tau N/m2
Soil shear strength = c + rtan/
Gravity
g gM 1.63 m/s2 Gravitational acceleration (MOON)g gE 9.81 m/s2
Gravitational acceleration (EARTH)
Miscellaneous
Na m Inertia coefficientNc m Cohesion coefficientNca m Adhesion
coefficientNo No N/m
2 {0:1,000} Load normal to bladeNq m Surcharge coefficientNc m
Weight coefficient
150 A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44
(2007) 133152
-
Appendix B (continued )
Model symbol Coding parameter Units {Range} description
Forces
H H N Horizontal component of T OR Soil Thrust intraction
T T N Resultant vector cutting forceV V N Vertical component of
T
Traction
A A m2 Ground contact area per wheel; leg; or track;p4 b L for
wheel OR B L for track and leg
b b m {0.025:0.5} Effective wheel ground contact width 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB
e e
p, NOT USED
B B m {0.025:1.0} Width of wheel/track/legD D m {0.175:5} Wheel
diameterDP DP N Net drawbar pullh h m {0.001:0.10} Grouser heightH0
H0 N Ideal soil thrust without slipk k Pa
mn01, Patel; Pa
mn0, Bekker Modulus of sinkage deformation = kc + b k/ per
Patel [18] (soil consistency); Bekker otherwisedefines k = kc/b
+ k/, {[20] p. 240, 31340 and 447}
kc kcPa
mn01{0:1593, Terrestrial range} modulus of cohesion ofsoil
deformation, {[20] p. 2403} & {[1] p. 529}
kt kt Dimensionless Tangential stiffness of wheel or track,
derived fromS L physical observation
k/ kphiPamn0
{0:160,000, Terrestrial range} modulus of friction ofsoil
deformation, {[20] p. 2403} & {[1] p. 529}
K1 K1 1/m {3.9:39} Degree of brittleness or compactness
orcoherence (cohesion effect) [19, p. 265 ff]
K2 K2 Dimensionless {3:1} Degree of slip strength (friction
effect)[19, p. 265 ff]
Kc Kc Dimensionless {[20] p. 453} For bulldozing resistance,
corrects Patel[18]
Kc Kgamma Dimensionless {[20] p. 453} For bulldozing resistance,
corrects Patel[18]
l0 l0 m {[20] p. 453} Distance of rupture for
bulldozingresistance, corrects Patel [18]
L L m {0.1:5} Ground contact length: wheel:L 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD ee
p, NOT USED wheel or track.
n 0 nprime Dimensionless {0:1.2} Soil deformation exponent,
{[20] p. 340}n n Dimensionless {1:6} Number of wheels/tracks/legsnc
nc Dimensionless {0:3} Coefficient of passive earth pressure,
{[20]
p. 139 and 142} & [10]Ng Ng Dimensionless {0:20} Number of
grousers touching soil per track/
wheel/legnc ngamma Dimensionless {15:35} Coefficient of passive
earth pressure, {[20]
p. 139} & [10]P P N/m2 Ground pressure = W/A = rR R N Total
resistive forceRb Rb N Bulldozing resistanceRc Rc N Soil compaction
resistanceRg Rg N Gravitational resistanceRr Rr N Rolling
resistance of elastic tire, NOT USEDRother Rother N Other resistive
forces
(continued on next page)
A. Wilkinson, A. DeGennaro / Journal of Terramechanics 44 (2007)
133152 151
-
Appendix B (continued )
Model symbol Coding parameter Units {Range} description
S S Dimensionless {(0.10:0.12)/L} Low speed slip, {[20] p. 138
and 447}say S L = 1012 cm, max. traction. S = x W/kt L
2 per Patel.T0 T0 N Available grousered wheel thrust s Ng hbcos
qsin q
h2
cos qW W N = kg m/s2 {50:8000} * g Vehicle weight, WLRV = 460
lbs
vehicle + 1080 lbs cargo = 1540 lbs onEarth = 698.5 kg = 6852 N
on Earth = 1138 N onthe moon
z z m Sinkage, not counting multiple passes in wheel
tracks.Bekker definition is used in this paper,z WnAk
1n0 Pk
1n0
a a deg Approach angle arccos 1 2 zeD
. e is oftennegligible
e E m {0.01:0.10} Wheel squat/deflection (