classical soil mechanics
DESCRIPTION
SoilTRANSCRIPT
-
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
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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
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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
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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
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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
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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
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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 1347.940 468.70110 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 1962.6124.4. 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
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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 (