Narrow Till Age Tool Models

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Review of Models for PredictingPerformance of Narrow Tillage Tool

R . D . Gr i s s o , J ohn V . P e rum pra lASSOC. MEMBE R MEMBER

ASAE ASAE

ABSTRACT

FOUR narrow tillage tool models are reviewed.Assumptions, capabilities and limitations associated

with each model are discussed. Tillage tool performanceunder two different soil conditions have been predictedusing the four models. Simulated results are comparedwith the experimental results.

INTR ODUC TION

During the last two decades several mathematicalmodels have been developed for predicting theperformance characteristics of tillage tools in soils. Theseinclude two-dimensional models for wide tools and three-dimensional models for narrow tillage tools. In general,the validity of the individual models have beenestablished by comparing predictions with experimentalresults. Despite exhaustive studies with individualmodels, no attempts have been made to compare thesem ode l s . A com par i s on i nc l ud i ng a s s um pt i ons ,capabilities, and limitations associated with each modelwould be extremely valuable for all interested in tillagemechanics. Therefore, the objective of this paper is tocompare selected models to predict the behavior ofnarrow tillage tools.

Four models—developed by (a) Hettiaratchi andReece (1967), (b) Godwin and Spoor (1977), (c) McKyesand Ali (1977), and (d) Perumpral et al. (1983) were

selected for this study. A brief discussion of each modeland a comparison of observed and sim ulated results fromthe different models is included. The tool angles anddepths are limited to 0 to 90 deg with the horizontal and0 to critical depth (6 x tool width), respectively.

SOIL-TOOL INTERACTION MODELS

Hettiaratchi and Reece (1967)The Hettiaratchi-Reece model is based on the Passive

Earth Pressure theory. The analysis of three-dimensionalfailure assumes that the failure configuration iscomposed of forward and sideways failure regimes. Theformer refers to the failure ahead of the soil-toolinterface, and the latter involves the horizontal sideways

movement of the soil away from the center of theinterface. The total force on the tool due to the three-dimensional failure is the vector sum of the forces due toforward failure (P t), the sideways force (P s), and theadhesion force on the interface.

Article was submitted for p ublication in S eptember, 1984; reviewedand approved for publication by the Power and Machinery Div. ofASAE in March, 1985. Presented as ASAE Paper No. 81-1535.

The authors are: R. D. GRISSO, Graduate Research Associate,Agricultural Engineering Dept., Auburn University, AL; and JOHN V.PERUMPRAL, Professor, Agricultural Engineering Dept., VirginiaPolytechnic Institute and State University, Blacksburg, VA.

The relationship developed for P t utilizes the earlieranalysis of plane soil wedges in two-dimensional soilfailure by Hettiaratchi et al. (1966). Assuming that theforward failure regime in front of a loaded interfaceextends the full width and depth of the interface, theexpression for the force due to forward failure isexpressed as:

P f = 7 D 2 B N 7 + c D B N c + A d D B N a + q D B N q *

[ 1 ]

The N-factors in this equation can be obtained fromgraphs available in Hettiaratchi et al. (1966). Theforward failure force component (P t) makes an angle d,

the soil-metal friction angle, with the normal to theinterface.

The sideways failure force P s is composed of cohesiveand gravitational forces. This force can be expressed as:

P s = 7(d + q/T)2 bN S7 + cdbN sc [2]

whe re, d = effective dep th of failure wedgeb = effective width of failure wedgeN s yandN s c = dimensionless factors due to

gravitation and cohesion,respectively.

The N-factors in equation [2] depend on the roughnessof the interface. Separate relationships are given forperfectly smooth (6 = 0) and perfectly rough (6 = <)>)interfaces and are computed utilizing the charts inHettiaratchi et al. (1966). The N-factors computed are,therefore, suitable for predicting sideways failure forceon vertical tools. For inclined interfaces, multiplicationof equat ion [2] by the incl inat ion factor (KJ isrecommended. The inclination factor is given by theexpression:

t a n " 1 (s in a cot i//)

where,

i// = (45 + 0/2) [3b]

The gene ralized relationship for the sideways force du eto a sideways failure is:

Ps = [7 (d + q/7) 2 bN S7 + cdbN sc ] K a [4 ]

By combining equations [1] and [4] and including theadhesion force on the interface, the draft and lift

•Notations common to all four models are defined in Table 1.

1062 © 1985 American Society of Agricultural Engineers 0001-2351/85/2804-1062$02.00 TRA NSA CTION S of the ASAE



TABLE 1. DEFINITION OF NOTATIONS COMMON TOALL FOUR MODELS.

Notations Definitions and units

A^ Soil-metal adhesional factor , N/cm ^c Cohesional factor , N/c m^0 Inter nal soil friction angle8 Soil-metal friction angle7 Unit weight of soil, N/ cm ^B Tool width , cmD Tool depth , cma Tool or rake angle from forward h orizontalq Surcharge pressure on the soil-free surface, N/cr a'

r Ruptu re d is tance from tool to crescent , cm0 Ru ptu re angle from direct ion of travelF x Horizontal or draft force, NF z Vertical or lift force, NP Total tool force , NN Dimensionless earth pressure coefficients

(vertical) forces on a tillage tine can be expressed as:

F x = P fsin(a + 5) + P ssina + A dDBcota [5]

F z = P fcos(a + 5)4- P scosa + A dDB [6]

Hettiaratchi and Reece (1967) conducted two separateexperim ental investigations to examine the validity of the

model . Comparison of experimental and simulatedresults for a vertical tine at low depth-to-width ratiosshowed that the model over-predicted the tool forces.However, the general shape of the predicted curve wassimilar to that developed from the experimental results.In the case of inclined tools, the model had a tendency tounder-predict the tool forces.

Godwin and Spoor (1977)The Godwin-Spoor model was developed to predict

forces on narrow tillage tines with a wide rang e of depth-to-width ratios. Two separate models were developed fortools operating at depths less v or greater than criticaldepth. For this review we chose only the model for tool

depths less than the critical depth.The failure wedge considered for the analysis is shown

in Fig. 1. The total wedge was divided into a centerwedge with width the same as the tool width an d two sidecrescents. They considered the forces contributed byeach section to obtain the total force on the tool. Therelationship developed by Hettiaratchi et al. (1966) forwide tools (equation [1]) was utilized to obtain the toolforce from th e center position of the wedge. The eq uationfor the force due to the side crescents was developed by

Fig. 1— Three-dimensional failure wedge in front ofnarrow tools for depths less than critical depth (Godwinand Spoor, 1977).

assuming a constant radius (r) for the side crescents andextending it to an angle rj, where:

T? = co s" 1 | Dco ta / r J [7]

To obtain a relationship for the force due to the sidecrescents, a wedge-shaped elemental volume and forcesacting on it were considered. An e quation for the passiveforce due to the elemental volume was developed andintegrated to obtain the re lationship for the total force byone of the side crescents.

The relationships for the draft and lift forces on the

tool were developed by combining the relationships forside crescents and the center wedge to give:

F x = [yD2Ny + c D N c + qD N J [B + rsinT?] sin( a+5)

+ A dB D [ N asin(o: + 6) + cos(a)] [8]

F z = - [ 7D 2 N 7 + c D N c + q D N J [B + rsinT?] cos(a

+ 5) - A d B D [ N a cos (a + 5) - s in(a )] [9]

The use of equations [8] and [9] requires priorknowledge of the rupture distance (r). Godwin and Spoor(1977) developed a graph using the information from

Payne (1956), Payne and Tanner (1959), andHett iaratchi and Reece (1967) to describe ther e l a t i o n s h i p b e t w e e n d i s t a n c e r a t i o ( r u p t u r edistance/depth) and tool angle.

Godwin and Spoor (1977) conducted low speedverification tests in a sandy loam soil. The results of theexperimental tests were compared with the predictedtotal force. The predicted draft force comparedreasonably well with the experimental results while poorcomparisons were obtained for the predicted verticalforce. Generally the vertical force was overpredicted.However, as the depth to width ratio increased, theagreement between the two improved.

McKyes and Ali (1977)The three-dimensional model by McKyes and Ali

(1977) is similar to the Godw in-Spoor mode l. One majordifference is that the McKyes-Ali model does not requireprior knowledge of the rupture distance (r) forcomputing the forces on the tool. The curved failuresurface from the tip of the tool was assumed to bestraight and makes an angle ft with the horizontal.

Considering an impending soil failure condition, arelationship for the tool force was developed in terms ofthe failure angle (p ) and pertinent soil and toolpa ram et e r s . Th rough app rop r i a t e m a t hem at i ca loperation the wedge which created the minimum passiveforce was determined.

The rupture distance was assumed to be:

r = D | cotjS + co ta | [10]

where, p is the unknown failure angle.

Knowing the rupture distance, equation [7] is used todetermine the crescent angle.

In order to develop a relationship for the total force onthe tool, as in the models described earlier, McKyes andAli (1977) also considered the forces contributed by thecenter wedge and side crescents. Thus, the total draft

Vol. 28(4):July-August, 1985 1063



force on the tool was expressed as:

F = F + 2 F _ [11 ]

The f ina l expression for equa t ion [11] i s s imi la r to

equ a t io n [1] , an d the N-fac tors for the draf t - force w ere

def ined as fo l lows:

N,(r/2D) I 1 + 2r sin 7?/3B

7 X cot(a + 5) + cot(j3 + 0)

i [12]

NL

NL

) 1 + cotj3 cot(j3 + 0) \ \ 1 + r sin T?/B(

cot(a + 5) + cot(]3 + 0)

(r/D)(l+ r sin 7?/B)

c o t (a + 8) + cot(j3 + 0)

. . . [13 ]

[14 ]

To de t e rmine the f a i l u re w e dge , t he a ng le p w a s

de te rmine d on a t r i a l ba s i s by min imiz ing the N yX

func t ion o f e qua t ion [12 ] . Th e ft a ng le ob ta ine d f rom th i s

p roc e ss w a s u se d fo r t he r e ma in ing c ompu ta t ions .

Mc K ye s a nd A l i (1977 ) c om pa re d the N - fa c to r s f rom

equat ions [12] through [14] wi th those for wide tools by

se t t i ng B = ° ° . Th e c ompa r i son show e d ve ry c lo sea g re e me n t fo r smoo th too l s (6 = 0). However , for tools

wi th a rough surface and for tool angles grea te r than 90

-<j>, the N-fa c tors obta ined from equ at io ns [12] th ro ug h

[14 ] w e re muc h h ighe r . To c ompe nsa t e fo r t h i s poo r

a g r e e m e n t , M c K y e s a n d A li ( 1 9 7 7 ) s u g g e s t a n

in t e rpo la t ion p roc e du re w h ic h i s d i scusse d in de t a i l i n

t h e i r p a p e r .

Mc K ye s a nd A l i (1977 ) a l so c om pa re d the i r s im u la t e d

re su l t s w i th o the r mode l s a nd e xpe r ime n t s . In ge ne ra l ,

t he r e su l t s f rom the d i f f e re n t me thods a g re e d w e l l . Eve n

though the Mc K ye s-A l i mode l ma y no t be a s r i go rous a s

ma ny o the r mode l s a l r e a dy de ve lope d , i t a ppe a r s t o be

st ra ight forward and i t could be easi ly used for three-

d ime ns iona l so i l - c u t t i ng p rob le ms .

Perumpral , Grisso and Desai (1983)

The mode l by Pe rumpra l e t a l . (1983 ) i s s imi l a r t o

mod e l s de ve lope d by Mc K ye s a n d A l i (1977 ) a nd G od w ina nd Spoor (1977 ) . H ow e ve r , t he s ide w e dge s f l a nk ing the

center wedge were replaced by two se ts of forces ac t ing

on the sides of the cente r wedge . As in McKyes and Al i

(1977 ) , t he s l i p su r fa c e w a s a ssume d to be s t r a igh t . The

center wedge and var ious forces considered for the

ana lysis a re shown in Fig . 2 .

The force vec tors R , SF 2 , a nd C F 2 shown on faces

" a b c " a nd " d e f ' a re t he fo rc es r e p l a c ing the s ide

c re sc e n t s . The e qua t ion fo r e a r th p re ssu re a t r e s t w a s

u t i l i z e d to e s t ima te t he se fo rc e s . The rup tu re p l a ne

"ebcf' w a s a ssume d to be s t r a igh t , ma k in g a n a ng le ft

w i th the ho r i z on ta l . P l a ne " a be d" i s t he in t e r fa c e , a nd

on th is p lane adhesiona l force and to ta l tool forces exist .

A ssuming tha t t he w e dge show n in F ig . 2 i s i nequi l ibr ium, the fo l lowing equa t ions for the draf t and l i f t

forces were deve loped by summing the forces in the

ho r i z on ta l a nd ve r t i c a l d i r e c t ions :

F z = Pcos(a + 5')

= W w + 2SF2sinj3 + SF lS in |3

+ 2CF2sinj3 + CF^injS

+ AD Fs ina - Qcosj3 . . . [15]

F x = Psin(a: + 5')

= 2SF2cosj3 + SF IC OSJ(3

+ 2CF2cosj3 + CF1cosj3

+ Qsin|3 - ADFcosa. . . [16]

T h e 6' in equa t ions [15] and [16] represents the so i l -

meta l f r ic t ion angle . In the models d iscussed previously a

c ons t a n t f r i c t i on a ng le 6 w a s u se d . In t h i s mode l , t he

fr ic t ion ang le i s expre ssed as a func t ion of tool ang le .

The re l a t i onsh ip fo r 6' i s show n in Ta b le 2 . C ombin ing

equat ions [15] and [16] , a re la t ionship for P can be

w r i t t e n a s :

P = sin(j3 + 8' + a + 0)

- AD F cos(j3 + 0 + a)

+ 2 SF 2 cos 0 + W w sin(0 + 0) j+ 2 CF 2 cos 0 + CF 1 cos(0)

[17 ]

In t e rms o f so i l a nd too l pa ra me te r s :

sin(j3 + 0 + a + 8')

- [ A d B D (l + h/D) ] cos(/3 + 0 + a)/sin a

+ yA x [2 K Qz sin 0 + B sin(0 4- (l)]

+ c cos 0 [2 A t + BD/sin |3]

[18]

Fig. 2—The proposed failure wedge and the

forces (Perumpral et al., 1983).

T A B L E 2 . S O I L P A R A M E T E R S U S E D IN T H E

C O M P U T E R S I M U L A T I O N .

So UP a r a m e t e r s

7c

06 '

Pe r u m p r a l e t a l . M cK y es an d A l i( 1 9 8 3 ) ( 1 9 7 7 )

0 . 0 2 0 6 N / c m 3 0 . 0 1 5 N / c m 3

0 . 2 9 N / c m 2 0 . 0 2 3 N / c m 2

3 1 ° 3 5 °24° for 0 < 61° 23 . 3°5 0 .5 - . 4 5 0 f o r (3 > 6 1 °

1064 TRANSACTIONS of the ASAE



Equa t ion [17 ] c a n be e xp re sse d in t he fo rm o f t he

p re v ious mode l s a s :

P = 7 B D 2 N 7 + c B D N c + A d B D N a [19 ]

w h e r e :

A i

B D 2 J 2 K n z sin 0 + B sin(0 + |8) (N = ' - ; — — ^ - * - [20 ]

7 sin(j3 + a + 0 + 5') L J

c o s ](h 2 ——+ - — - X

N = <* BD si ng I [21]c sin (0 + CK + 0 + 6')

N = -(1 + h/D) cos(g + 0 + a)a sin(j3 + a + 0 + 5')sin(a)

w he re , K 0 = coeff ic ient of ear th pressu re a t rest ,

expressed as: KQ = 1 - sin0 [23]

h = hei gh t of soil hea ve in fro nt of th e tool

a t fa i lure

z = averag e de pth a t which the centr o id of

the fa i lure wedge is loca ted from the soi l

su r fa c e ,

expressed as: z - (1/3)(D + h) [241

A , = a re a o f e a c h s ide , " a bc " a nd " d e f \

expressed as:

A x - 0 .5D 2 (1 + h /D) [ (1 + h /D )cot a + cotjS] . . . [25 ]

In e qua t ion [18 ] a l l pa ra me te r s e xc e p t f a i l u re p l a ne

a n g l e (p ) a r e k n o w n . A c c o r d i n g t o P a s s i v e E a r t hPre ssu re t he o ry , pa ss ive f a i l u re t a ke s p l a c e w he n the

re s i s t a nc e by the soi l w e dge i s a t a m in i mu m. Th i s t he o ry

i m p l i e s t h a t t h e w e d g e w h i c h c r e a t e s m i n i m u m

re s i s t a nc e mus t be i de n t i f i e d to de t e rmine the rup tu re

a ng le . Ma the ma t i c a l ly , t h i s w e dge c a n be ide n t i f i e d by

so lv ing the e qua t ion :

dP/dj3 = 0 [26]

Since c losed form diffe ren t ia t ion of equ a t io n [26] i s

r a the r c omple x , a nume r i c a l p roc e du re w a s de ve lope d to

min imiz e the func t ion a nd to de t e rmine the f a i l u re p l a ne

angle (/?).

Pe rumpra l e t a l . (1983 ) c onduc te d ve r i f i c a t ion t e s t sunde r l a bo ra to ry c ond i t i ons u s ing a r t i f i c a l so i l . The

s imu la t ion r e su l t s c om pa re d w e l l w i th e xpe r ime n ta l

d a t a .

R E S U L T S A N D D I S C U S S I O N

The purpose of th is s tudy was not to se lec t a best

mode l bu t t o c ompa re t he ba s i s fo r t he mode l s , t hea ssu mp t ions i nvo lve d , a nd the c a pa b i l i t i e s o f t he m ode l .

A l l mode l s d i sc usse d a re ba se d on the Pa ss ive Ea r th

Pre ssu re t he o ry . A ma jo r it y o f t he a ssum pt ions i nvo lve d

w i th t he mode l s a re t he sa m e a s t hose a ssoc i a t e d w i th t he

500

250

50

25

m m Hettiaratchi & Reece (1967)

o——a Godwin & Spoor (1977)

<> _— o McKyes & Ali (1977)

• • perumpral et al. (1983)

• Observed Data (Perumpral et al.

30 40 50 60 70 80 90 100

Tool Angle (Degrees)

Fig. 3—Draft force - tool angle relationship for 5 cm wide tool at adepth of 5 cm.

Hettiaratchi & Reece (1967)

Godwin & Spoor (1977)

, -' McKyes & Ali (1977)

I- . P e r u m p r a l e t a l > (1983)

Observed Data (Perumpral et al., 1983)1500 •

1250

1000

750

500

Tool Depth (cm)

Fig. 4—Draft force - tool width relationship for a vertical tool at adepth of 5 cm.



McKyes & Ali (1977)

Perumpral et al. (1983)

Observed Data (Perumpral et al., 1983)^

575

Tool Width (cm)

Fig. 5—Draft force - tool width relationship for a vertical tool at adepth of 10 cm.

e a r th p re ssu re t he o ry . The mode l s d i sc usse d ne g le c t t he

iner t ia l forces and a re su i table only for predic t ing the

forces on a narrow t ine moving a t ext remely slow speeds.

Mo s t mode l s a ssu me a c ons t a n t so il - too l f r i c ti on a ng le(d) . O n the o the r ha nd , ba se d on the obse rva t ions ma de

dur ing the p re l imina ry l a bo ra to ry t e s t s , Pe rumpra l e t a l .

(1983) assumed the so i l -meta l f r ic t ion angle to be a

func t ion of the tool angle . Al l models had one te rm to

account for the force due to adhesion a t the in te rface ,

w i th t he e xc e p t ion o f t he Mc K ye s-A l i mode l .

To c ompa re t he c a pa b i l i t i e s o f t he mode l s , t oo l fo rc e s

for d i f fe rent tool geometry were predic ted and the resul ts

a re c ompa re d w i th t he e xpe r ime n ta l da t a . Te s t s r e su l t s

ob ta ine d by Pe rumpra l e t a l . (1983 ) a nd Mc K ye s a nd A l i

(1977) under two di ffe rent so i l condi t ions were used. For

Vol. 28(4):July-August, 1985 1065





McKyes & Ali (1977)


Observed Data (Perumpral et al. , 1983)

5 7.5

Tool Width (cm)

Fig. 6—Lift force - tool width relationship for a vertical tool at a dep thof 10 cm.

600

400

200

0

-200

— • Hettiaratchi & Reece (1967)

— ° Godwin & Spoor (1977)

—o McKyes & All (1977)

" • Perumpral et al. (1983)

Observed Data (Perumpral et al. , 1983)

40 50 60 70

Tool Angle (Degrees)

Fig. 7—Lift force - tool angle relationship for a 10 cm wide tool at a

depth of 10 cm.

1

L

h

h

r~

, ma ao——o

• ••

1

H e t t i a r a t c h i & R e ec e ( 1 9 67 )Godwin & Spoo r (197 7)

McKyes & Al i (1977 )P e r u m p r a l e t a l . ( 1 9 8 3 )O b s e r v e d D a t a ( P e r u m p r a l e t a l . , 1 9 8 3 )

\

1 I5 10 15

Tool Depth (cm)

Fig. 8—Lift force - tool depth relationship for a vertical, 10 cm wide

tool.

6 0

51)

4 0

3 0

2 0

10

-

-

-

-

m aa——a

o— -—o

• ••

IrlT^—

i

H e t t i a r a t c h i & R e ec e ( 1 9 6 7 )Godwin & Spoor (197 7)McKyes & Al i (197 7)P e r u m p r a l e t a l . ( 1 9 8 3 )O bs e r ve d D a t a ( Mc K yes & A l i , 1977 )

^ ^ ^ " ^^ ^ ^ , — - - " . ' ^ - • ^ . ' - *

—^-J^' '

60

T oo l A ng l e ( D e g r e e s )

Fig. 9—Draft force - tool angle relationship for a 5 cm wide tool at adepth of 5 cm.



McKyes & Ali (1977)


u Observed Data (McKyes & Ali, 1977)

125

50

25

Tool Width (cm)

Fig. 10—Draft force - tool width relationshp for a vertical tool at adepth of 5 cm.

-

-

-

ADO

o

H e t t i a r a t c h iGodwin & SpoMcKyes & AJ iP e r um pr a l e t

A

^ y ^ D

& Reece (1967)or (1977)

(1977)a l . ( 1983 )

A ^ ^

8 oB

'" A

o§

8

D ^ ^

I

A

A

OO•

ODD

1

O

A ^

D ^ - ^

o ^ ^

O

1

Observed Rupture Distance (cm)

Fig. 11—A relationship between predicted and experimentally

observed rupture distance.

each model, computer programs were developed tosimulate the tool performance. Using parameters givenin Table 2, tool combinations (five tool angles, three tooldepths, and three tool widths) under two soil conditionswere obtained from the computer simulation.

Comparisons of predicted results with experimentaldata from Perumpral et al. (1983) are shown in Figs. 3through 8 . S imi lar compar i sons u t i l i z ing theexperimental data from McKyes and Ali (1977) areshown in Figs. 9 and 10. The graph s shown were selectedto indicate the effect of different tool parameters on thedraft and lift forces. Plots of draft force as a function of

different tool parameters under both soil conditions

(Figs. 3-5, 9 and 10) clearly indicate that theHettiaratchi-Reece model consistently overpredicts thedraft force. In general, the simulated results using othermodels agreed well with the experimen tal results. Similarplots for lift forces (Figs. 6 to 8) show that the McKyes-Ali model did not agree well with the experimentalobservations. Lift force obtained using the Hettiaratchi-Reece model showed good correlation with experimentaldata in all but one case (Fig. 7).

Fig. 11 shows a plot between predicted andexperimentally-observed rupture distance. In general,reasonable agreement was obtained.

1066 TRANSACTIONS of the ASAE



CONCLUSIONS

The conclusions of this study are as follows.1. All models discussed in this paper can be

conveniently programmed to obtain the draft and liftfores on slow-moving narrow tools under different soilconditions.

2. All models were found to be useful for predictingthe rupture distance.

3. All of the models except the Hettiaratchi-Reecemodel gave reasonable predictions. The draft forceagreed well with experimental observations in three

cases. The Hettiaratchi-Reece model had a tendency toover-predict the draft force.4. All models except the McKyes-Ali model

predicted the lift force on the tool reasonably well.

References1. Godw in, R. J. and G. Spoor. 1977. Soil failure with narrow

tines. J. Agric. Eng. Res. 22(4):213-228.2. Hettiaratchi, D. R. P. and A. R. Reece. 1967. Symmetrical

three-dimensional soil failure. J. Terramech. 4(3):45-67.3. Hettiaratchi, D. R. P., B. D. Witney and A. R. Reece. 1966.

The calculation of passive pressure in two dimensional soil failure. J.Agric. Eng. Res. 11(2):89-107.

4 . McKyes, E. and O. S. Ali. 1977. The cutting of soil by narrowblades. J. Terramech. 14(2):43-58.

5. Payne, P. C. 1956. The relationship between the mechanicalproperties of soil and the performance of simple cultivationimplements. J. Agric. Eng. Res. l(l):23-50.

6. Payne, P. C. and D. W. Tanne r. 1959. The relationship between

rake angle and the performance of simple cultivation implements. J.Agric. Eng. Res. 4(4):312-325.7. Peru mpra l, J. V., R. D. Grisso, and C. S. Desai. 1983. A soil-

tool model based on limit equilibrium analysis. TRANSACTIONS ofthe ASAE 26(4):991-995.

Vol. 28(4):July-August, 1985 1067

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Narrow Till Age Tool Models