Review of Models for Predicting
Performance of Narrow Tillage Tool
R. D. Grisso, John V. Perumpral
ASSOC. MEMBER
ASAE
ABSTRACT
OUR narrow tillage tool models are reviewed.
Assumptions, capabilities and limitations associated
with each model are discussed. Tillage tool performance
under two different soil conditions have been predicted
using the four models. Simulated results are compared
with the experimental results.
F
INTRODUCTION
During the last two decades several mathematical
models have been developed for predicting the
performance characteristics of tillage tools in soils. These
include two-dimensional models for wide tools and threedimensional models for narrow tillage tools. In general,
the validity of the individual models have been
established by comparing predictions with experimental
results. Despite exhaustive studies with individual
models, no attempts have been made to compare these
models. A comparison including assumptions,
capabilities, and limitations associated with each model
would be extremely valuable for all interested in tillage
mechanics. Therefore, the objective of this paper is to
compare selected models to predict the behavior of
narrow tillage tools.
Four models—developed by (a) Hettiaratchi and
Reece (1967), (b) Godwin and Spoor (1977), (c) McKyes
and Ali (1977), and (d) Perumpral et al. (1983) were
selected for this study. A brief discussion of each model
and a comparison of observed and simulated results from
the different models is included. The tool angles and
depths are limited to 0 to 90 deg with the horizontal and
0 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-dimensional
failure assumes that the failure configuration is
composed of forward and sideways failure regimes. The
former refers to the failure ahead of the soil-tool
interface, and the latter involves the horizontal sideways
movement of the soil away from the center of the
interface. The total force on the tool due to the threedimensional failure is the vector sum of the forces due to
forward failure (Pt), the sideways force (Ps), and the
adhesion force on the interface.
Article was submitted for publication in September, 1984; reviewed
and approved for publication by the Power and Machinery Div. of
ASAE 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., Virginia
Polytechnic Institute and State University, Blacksburg, VA.
1062
MEMBER
ASAE
The relationship developed for Pt utilizes the earlier
analysis of plane soil wedges in two-dimensional soil
failure by Hettiaratchi et al. (1966). Assuming that the
forward failure regime in front of a loaded interface
extends the full width and depth of the interface, the
expression for the force due to forward failure is
expressed as:
Pf = 7 D 2 B N 7 + c D B N c
+ AdDBNa + qDBNq*
[1]
The N-factors in this equation can be obtained from
graphs available in Hettiaratchi et al. (1966). The
forward failure force component (Pt) makes an angle d,
the soil-metal friction angle, with the normal to the
interface.
The sideways failure force Ps is composed of cohesive
and gravitational forces. This force can be expressed as:
Ps = 7(d + q/T)2 bNS7 + cdbNsc
[2]
where, d
= effective depth of failure wedge
b
= effective width of failure wedge
N sy andN sc = dimensionless factors due to
gravitation and cohesion,
respectively.
The N-factors in equation [2] depend on the roughness
of the interface. Separate relationships are given for
perfectly smooth (6 = 0) and perfectly rough (6 = <)>)
interfaces and are computed utilizing the charts in
Hettiaratchi et al. (1966). The N-factors computed are,
therefore, suitable for predicting sideways failure force
on vertical tools. For inclined interfaces, multiplication
of equation [2] by the inclination factor ( K J is
recommended. The inclination factor is given by the
expression:
t a n " 1 (sin a cot i//)
where,
i// = (45 + 0/2)
[3b]
The generalized relationship for the sideways force due
to a sideways failure is:
Ps = [7 (d + q/7) 2 bNS7 + cdbNsc] Ka
[4]
By combining equations [1] and [4] and including the
adhesion force on the interface, the draft and lift
•Notations common to all four models are defined in Table 1.
© 1985 American Society of Agricultural Engineers 0001-2351/85/2804-1062$02.00
TRANSACTIONS of the ASAE
TABLE 1. DEFINITION OF NOTATIONS COMMON TO
ALL FOUR MODELS.
Notations
assuming a constant radius (r) for the side crescents and
extending it to an angle rj, where:
Definitions and units
T? = cos" 1 | Dcota/r J
A^
c
0
8
7
B
D
a
q
r
0
Fx
Fz
P
N
Soil-metal adhesional factor, N/cm^
Cohesional factor, N/cm^
Internal soil friction angle
Soil-metal friction angle
Unit weight of soil, N/cm^
Tool width, cm
Tool depth, cm
Tool or rake angle from forward horizontal
Surcharge pressure on the soil-free surface, N/cra'
Rupture distance from tool to crescent, cm
Rupture angle from direction of travel
Horizontal or draft force, N
Vertical or lift force, N
Total tool force, N
Dimensionless earth pressure coefficients
[7]
To obtain a relationship for the force due to the side
crescents, a wedge-shaped elemental volume and forces
acting on it were considered. An equation for the passive
force due to the elemental volume was developed and
integrated to obtain the relationship for the total force by
one of the side crescents.
The relationships for the draft and lift forces on the
tool were developed by combining the relationships for
side crescents and the center wedge to give:
F x = [yD2Ny
+ cDN c + q D N J [B + rsinT?] sin(a+5)
+ A d BD[N a sin(o: + 6) + cos(a)]
[8]
(vertical) forces on a tillage tine can be expressed as:
F x = Pfsin(a + 5) + Pssina + AdDBcota
[5]
F z = Pfcos(a + 5)4- Pscosa + AdDB
[6]
F z = - [ 7 D 2 N 7 + cDN c + q D N J [B + rsinT?] cos(a
+ 5) - A d BD[N a cos(a + 5) - sin(a)]
Hettiaratchi and Reece (1967) conducted two separate
experimental investigations to examine the validity of the
model. Comparison of experimental and simulated
results for a vertical tine at low depth-to-width ratios
showed that the model over-predicted the tool forces.
However, the general shape of the predicted curve was
similar to that developed from the experimental results.
In the case of inclined tools, the model had a tendency to
under-predict the tool forces.
Godwin and Spoor (1977)
The Godwin-Spoor model was developed to predict
forces on narrow tillage tines with a wide range of depthto-width ratios. Two separate models were developed for
tools operating at depths less v or greater than critical
depth. 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 center
wedge with width the same as the tool width and two side
crescents. They considered the forces contributed by
each section to obtain the total force on the tool. The
relationship developed by Hettiaratchi et al. (1966) for
wide tools (equation [1]) was utilized to obtain the tool
force from the center position of the wedge. The equation
for the force due to the side crescents was developed by
[9]
The use of equations [8] and [9] requires prior
knowledge of the rupture distance (r). Godwin and Spoor
(1977) developed a graph using the information from
Payne (1956), Payne and Tanner (1959), and
Hettiaratchi and Reece (1967) to describe the
relationship between distance ratio (rupture
distance/depth) and tool angle.
Godwin and Spoor (1977) conducted low speed
verification tests in a sandy loam soil. The results of the
experimental tests were compared with the predicted
total force. The predicted draft force compared
reasonably well with the experimental results while poor
comparisons were obtained for the predicted vertical
force. Generally the vertical force was overpredicted.
However, as the depth to width ratio increased, the
agreement between the two improved.
McKyes and Ali (1977)
The three-dimensional model by McKyes and Ali
(1977) is similar to the Godwin-Spoor model. One major
difference is that the McKyes-Ali model does not require
prior knowledge of the rupture distance (r) for
computing the forces on the tool. The curved failure
surface from the tip of the tool was assumed to be
straight and makes an angle ft with the horizontal.
Considering an impending soil failure condition, a
relationship for the tool force was developed in terms of
the failure angle (p) and pertinent soil and tool
parameters. Through appropriate mathematical
operation the wedge which created the minimum passive
force was determined.
The rupture distance was assumed to be:
r = D | cotjS + cota |
[10]
where, p is the unknown failure angle.
Fig. 1—Three-dimensional failure wedge in front of
narrow tools for depths less than critical depth (Godwin
and Spoor, 1977).
Vol. 28(4):July-August, 1985
Knowing the rupture distance, equation [7] is used to
determine the crescent angle.
In order to develop a relationship for the total force on
the tool, as in the models described earlier, McKyes and
Ali (1977) also considered the forces contributed by the
center wedge and side crescents. Thus, the total draft
1063
force on the tool was expressed as:
F
= F
[11]
+ 2 F _
The final expression for equation [11] is similar to
equation [1], and the N-factors for the draft-force were
defined as follows:
N,7 X
NL
(r/2D) I 1 + 2r sin 7?/3B
i
cot(a + 5) + cot(j3 + 0)
) 1 + cotj3 cot(j3 + 0) \ \ 1 + r sin T?/B(
. . .[13]
cot(a + 5) + cot(]3 + 0)
(r/D)(l+ r sin 7?/B)
NL
[12]
cot(a + 8) + cot(j3 + 0)
[14]
center wedge were replaced by two sets of forces acting
on the sides of the center wedge. As in McKyes and Ali
(1977), the slip surface was assumed to be straight. The
center wedge and various forces considered for the
analysis are shown in Fig. 2.
The force vectors R, SF 2 , and CF 2 shown on faces
" a b c " and "def' are the forces replacing the side
crescents. The equation for earth pressure at rest was
utilized to estimate these forces. The rupture plane
"ebcf' was assumed to be straight, making an angle ft
with the horizontal. Plane " a b e d " is the interface, and
on this plane adhesional force and total tool forces exist.
Assuming that the wedge shown in Fig. 2 is in
equilibrium, the following equations for the draft and lift
forces were developed by summing the forces in the
horizontal and vertical directions:
F z = Pcos(a + 5')
= W w + 2SF2sinj3 + SF lS in|3
To determine the failure wedge, the angle p was
determined on a trial basis by minimizing the NyX
function of equation [12]. The ft angle obtained from this
process was used for the remaining computations.
McKyes and Ali (1977) compared the N-factors from
equations [12] through [14] with those for wide tools by
setting B = °°. The comparison showed very close
agreement for smooth tools (6 = 0). However, for tools
with a rough surface and for tool angles greater than 90
-<j>, the N-factors obtained from equations [12] through
[14] were much higher. To compensate for this poor
a g r e e m e n t , M c K y e s a n d Ali (1977) s u g g e s t an
interpolation procedure which is discussed in detail in
their paper.
McKyes and Ali (1977) also compared their simulated
results with other models and experiments. In general,
the results from the different methods agreed well. Even
though the McKyes-Ali model may not be as rigorous as
many other models already developed, it appears to be
straight forward and it could be easily used for threedimensional soil-cutting problems.
Perumpral, Grisso and Desai (1983)
The model by Perumpral et al. (1983) is similar to
models developed by McKyes and Ali (1977) and Godwin
and Spoor (1977). However, the side wedges flanking the
+ 2CF2sinj3 + CF^injS
+ ADFsina - Qcosj3 . . .
[15]
F x = Psin(a: + 5')
= 2SF2cosj3 + SF IC OSJ(3
+ 2CF2cosj3 + CF1cosj3
+ Qsin|3 - ADFcosa. . .
[16]
The 6' in equations [15] and [16] represents the soilmetal friction angle. In the models discussed previously a
constant friction angle 6 was used. In this model, the
friction angle is expressed as a function of tool angle.
The relationship for 6' is shown in Table 2. Combining
equations [15] and [16], a relationship for P can be
written as:
P=
sin(j3 + 8' + a + 0)
- ADF 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 terms of soil and tool parameters:
- [ A d BD(l + h/D)] cos(/3 + 0 + a)/sin a
+ yAx [2K Q z sin 0 + B sin(0 4- (l)]
+ c cos 0 [2 At + BD/sin |3]
sin(j3 + 0 + a + 8')
[18]
TABLE 2. SOIL PARAMETERS USED IN THE
COMPUTER SIMULATION.
SoU
Parameters
7
c
0
Fig. 2—The proposed failure wedge and the
forces (Perumpral et al., 1983).
1064
6'
Perumpral et al.
(1983)
McKyes a n d Ali
(1977)
0.0206 N / c m 3
0.015 N / c m 3
0.29 N / c m 2
0.023 N / c m 2
31°
35°
24° for 0 < 61°
23.3°
5 0 . 5 - .450 for (3 > 61°
TRANSACTIONS of the ASAE
m
m Hettiaratchi & Reece (1967)
o — — a Godwin & Spoor (1977)
<>_—o McKyes & Ali (1977)
•
• perumpral et al. (1983)
•
Observed Data (Perumpral et al.
Equation [17] can be expressed in the form of the
previous models as:
500
P = 7BD 2 N 7 + cBDN c + A d BDN a
[19]
250
where:
A
i
50
BD 2
N
=
7
J 2K n z sin 0 + B sin(0 + |8) (
'
; ——^-*sin(j3 + a + 0 + 5')
25
[20]
L
J
30
40
50
Tool Angle
N =
](h 2 ——+ - — - X
<*
BD sing I
sin (0 + CK + 0 + 6')
N
-(1 + h/D) cos(g + 0 + a)
cos
c
=
a
[21]
,
I1500
coefficient of earth pressure at rest,
expressed as: KQ = 1 - sin0
70
80
90
100
(Degrees)
Fig. 3—Draft force - tool angle relationship for 5 cm wide tool at a
depth of 5 cm.
sin(j3 + a + 0 + 5')sin(a)
where, K 0 =
60
.
Hettiaratchi & Reece (1967)
Godwin & Spoor (1977)
-' McKyes & Ali (1977)
(1983)
P e r u m p r a l e t al>
Observed Data (Perumpral et al., 1983)
•
1250
[23]
1000
750
h
z
= height of soil heave in front of the tool
at failure
= average depth at which the centroid of
the failure wedge is located from the soil
surface,
500
Tool Depth
[241
expressed as: z - (1/3)(D + h)
A, =
area of each side, " a b c " and
Fig. 4—Draft force - tool width relationship for a vertical tool at a
depth of 5 cm.
"def\
Hettiaratchi & Reece (1967)
Godwin & Spoor (1977)
McKyes & Ali (1977)
Perumpral et al. (1983)
Observed Data (Perumpral et al., 1983)^
expressed as:
A x - 0.5D 2 (1 + h/D) [(1 + h/D)cota + cotjS]
. . . [25]
In equation [18] all parameters except failure plane
angle (p) are known. According to Passive E a r t h
Pressure theory, passive failure takes place when the
resistance by the soil wedge is at a minimum. This theory
implies t h a t t h e wedge which c r e a t e s m i n i m u m
resistance must be identified to determine the rupture
angle. Mathematically, this wedge can be identified by
solving the equation:
5
Tool Width
dP/dj3 = 0
[26]
Since closed form differentiation of equation [26] is
rather complex, a numerical procedure was developed to
minimize the function and to determine the failure plane
angle (/?).
Perumpral et al. (1983) conducted verification tests
under laboratory conditions using artifical soil. The
simulation results compared well with experimental
data.
RESULTS AND DISCUSSION
The purpose of this study was not to select a best
model but to compare the basis for the models, the
assumptions involved, and the capabilities of the model.
All models discussed are based on the Passive Earth
Pressure theory. A majority of the assumptions involved
with the models are the same as those associated with the
Vol. 28(4):July-August, 1985
(cm)
7
5
(cm)
Fig. 5—Draft force - tool width relationship for a vertical tool at a
depth of 10 cm.
earth pressure theory. The models discussed neglect the
inertial forces and are suitable only for predicting the
forces on a narrow tine moving at extremely slow speeds.
Most models assume a constant soil-tool friction angle
(d). On the other hand, based on the observations made
during the preliminary laboratory tests, Perumpral et al.
(1983) assumed the soil-metal friction angle to be a
function of the tool angle. All models had one term to
account for the force due to adhesion at the interface,
with the exception of the McKyes-Ali model.
To compare the capabilities of the models, tool forces
for different tool geometry were predicted and the results
are compared with the experimental data. Tests results
obtained by Perumpral et al. (1983) and McKyes and Ali
(1977) under two different soil conditions were used. For
1065
Hettiaratchi & Reece (1967)
Godwin & Spoor (1977)
McKyes & Ali (1977)
Perumpral et al. (1983)
Observed Data (Perumpral et al. , 1983)
H e t t i a r a t c h i & Reece (1967)
Godwin & Spoor (1977)
o— -—o McKyes & A l i (1977)
Perumpral e t a l . (1983)
Observed Data (McKyes & A l i ,
m
a
•
•
•
a——a
60
51)
-
40
-
30
^
20
^
-
10
IrlT^—
1977)
^
^ " ^
^
^ , — - - " . ' ^ - • ^ . ' - *
—^-J^
'
i
'
60
Tool Angle
5
Tool Width
Fig. 9—Draft force - tool angle relationship for a 5 cm wide tool at a
depth of 5 cm.
(cm)
Fig. 6—Lift force - tool width relationship for a vertical tool at a depth
of 10 cm.
—•
—°
—o
"•
(Degrees)
7.5
Hettiaratchi & Reece (1967)
Godwin & Spoor (1977)
McKyes & All (1977)
Perumpral et al. (1983)
Observed Data (Perumpral et al., 1983)
Hettiaratchi & Reece (1967)
Godwin & Spoor (1977)
McKyes & Ali (1977)
Perumpral et al. (1983)
Observed Data (McKyes & Ali, 1977)
u
125
600
400
50
200
25
0
-200
40
50
Tool Angle
60
Tool Width
70
(cm)
Fig. 10—Draft force - tool width relationshp for a vertical tool at a
depth of 5 cm.
(Degrees)
Fig. 7—Lift force - tool angle relationship for a 10 cm wide tool at a
depth of 10 cm.
H e t t i a r a t c h i & Reece (1967)
m Godwin & Spoor (1977)
a
a
o——o McKyes & A l i (1977)
Perumpral e t a l . (1983)
Observed Data (Perumpral e t a l . ,
A
D
O
,
•
•
•
A
O
-
A
A
1983)
A
D
h
^
^
^ - ^
^ ^
O
ODD
L
o
O
O
8
^
D
•
1
o
H e t t i a r a t c h i & Reece (1967)
Godwin & Spo or (1977)
McKyes & AJ i (1977)
Perumpral et a l . (1983)
'"A
A ^ ^
\
h
-
o
§
^ y ^
D
8 o
B
I
Observed
r~
1
5
1
10
Tool Depth (cm)
I
15
1
Rupture Distance
1
(cm)
Fig. 11—A relationship between predicted and experimentally
observed rupture distance.
Fig. 8—Lift force - tool depth relationship for a vertical, 10 cm wide
tool.
each model, computer programs were developed to
simulate the tool performance. Using parameters given
in Table 2, tool combinations (five tool angles, three tool
depths, and three tool widths) under two soil conditions
were obtained from the computer simulation.
Comparisons of predicted results with experimental
data from Perumpral et al. (1983) are shown in Figs. 3
through 8. Similar comparisons utilizing the
experimental data from McKyes and Ali (1977) are
shown in Figs. 9 and 10. The graphs shown were selected
to indicate the effect of different tool parameters on the
draft and lift forces. Plots of draft force as a function of
1066
different tool parameters under both soil conditions
(Figs. 3-5, 9 and 10) clearly indicate that the
Hettiaratchi-Reece model consistently overpredicts the
draft force. In general, the simulated results using other
models agreed well with the experimental results. Similar
plots for lift forces (Figs. 6 to 8) show that the McKyesAli model did not agree well with the experimental
observations. Lift force obtained using the HettiaratchiReece model showed good correlation with experimental
data in all but one case (Fig. 7).
Fig. 11 shows a plot between predicted and
experimentally-observed rupture distance. In general,
reasonable agreement was obtained.
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 lift
fores on slow-moving narrow tools under different soil
conditions.
2. All models were found to be useful for predicting
the rupture distance.
3. All of the models except the Hettiaratchi-Reece
model gave reasonable predictions. The draft force
agreed well with experimental observations in three
cases. The Hettiaratchi-Reece model had a tendency to
over-predict the draft force.
4. All models except the McKyes-Ali model
predicted the lift force on the tool reasonably well.
Vol. 28(4):July-August, 1985
References
1. Godwin, 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 narrow
blades. J. Terramech. 14(2):43-58.
5. Payne, P. C. 1956. The relationship between the mechanical
properties of soil and the performance of simple cultivation
implements. J. Agric. Eng. Res. l(l):23-50.
6. Payne, P. C. and D. W. Tanner. 1959. The relationship between
rake angle and the performance of simple cultivation implements. J.
Agric. Eng. Res. 4(4):312-325.
7. Perumpral, J. V., R. D. Grisso, and C. S. Desai. 1983. A soiltool model based on limit equilibrium analysis. TRANSACTIONS of
the ASAE 26(4):991-995.
1067