The design of a variable geometry linkage to improve depth control of tractor mounted implements

J. agric. Engng Res. (1988) 39, 85-97

The Design of a Variable Geometry Linkage to Improve Depth Control of Tractor Mounted Implements P. A. COWELL*; P. F. HERBERT? It has long been known that when long implements such as mouldboard ploughs are mounted on a tractor three-point linkage an excessive variation in working depth can occur when operating over an undulating ground profile. The principal cause of such variation has been shown to lie in the geometrical constraint imposed on the implement by the conventional three-point linkage. A variable geometry linkage is described in which the length of the top link is varied continuously as the tractor crosses an undulation. This permits a greater degree of articulation between the tractor and the implement and leads to improved control of depth.

1. Introduction Field tests on the depth control performance of mouldboard ploughs mounted on a tractor three-point linkage have indicated that large errors in working depth can occur when operating over undulating ground. ’ Experiments conducted over a single sinusoidal

undulation of 10m wavelength and 100mm amplitude showed that the errors produced were most pronounced when traversing the hollow of the undulation to such an extent that at times the plough came completely out of the ground. Control of the depth of the front furrow was generally worse than that of the rear furrow. Some variation in the depth of cultivation can be tolerated in practice provided it is not too excessive. However, if a field surface is such that the plough persistently runs shallow in the hollows there will be a tendency for the operator to set the plough to run deeper than might otherwise be necessary. It has been deduced’ that the reason for this unsatisfactory behaviour on undulating ground lies in the geometrical design of the three-point linkage. The implement has only one degree of freedom in the longitudinal-vertical plane relative to the tractor. As a consequence the depth of only one point on the implement can be directly controlled, the depth of every other point on the implement being predetermined by the pitch attitude of the tractor and the geometry of the linkage. Mouldboard ploughs in particular offer great resistance to the heel penetrating deeper than the share, so that if the plough is forced on to its heel, such as might be the case when the front wheels of the tractor rise over an undulation, further penetration is prevented and the front of the plough will rise towards the ground surface. This process is assisted by the draught control system which responds by reducing depth whenever an additional upward support force is induced on the plough heel. This problem has been recognized for some time, and Hesse’ and Mijller have developed a control system in which the implement is given two degrees of freedom relative to the tractor, both of which are independently controlled. Two independent measurements of working depth (using depth wheels) are made and the information so gathered is used to control implement height (vertical position of the lower links) and implement pitch (variable * Silsoe College, Shoe, Bedford MK45 4DT. UK t JCB Materials Handling Ltd., Rocester, Staffs. ST14 5JP, UK Received 3 November 1986; accepted in revised form 30 August 1987 002I-8634/88/020085+ 13 $03.00/O

85 c 1988The

British Society for Research in Agricultural

Engineering

VARIABLE

86

GEOMETRY

LINKAGE

FOR

TRACTORS

Notation a

an amplitude, mm b an amplitude, mm 1 horizontal distance from front share point to instantaneous centre of rotation by implement, m s forward distance travelled by implement, m time, s extension of displacer ram, mm Xl3 contraction of top link ram, mm Y depth of front share point, m Yo initial depth of front share point, m Y, equilibrium depth of implement, m

xi

piston area of displacer piston area of top link effective area of piston displacer ram, mm2 effective area of piston top link ram, mm2

ram, mm2 ram, mm2 rod side of

1

rod side of

lower link angle, deg. a phase angle, rad angular frequency of harmonic motion, rad/s slope of major axis of ellipse

length top link). This is a true variable geometry linkage in which the length of one link is actively controlled by an automatic control system. It suffers from the practical disadvantage that it requires the operator to attach two depth sensors to every implement he uses with the system. The purpose of the present research3 was to seek a means of controlling the linkage geometry in a passive way (i.e. no external power input through the hydraulic system) so as to produce a more nearly optimal result, preferably without the need for additional sensors. This requires a more detailed study of the kinematic and dynamic behaviour of the implement linkage combination. 2. Kinematic analysis Fig. 1 (top) illustrates a three-furrow plough mounted on a conventional three-point linkage, with the instantaneous centre of rotation at 0. The dotted figure illustrates the effect of raising the front of the tractor on the pitch attitude of the plough. Fig. I (bottom) shows the same plough mounted with its centre of rotation fixed at the centre of the rear axle. The dotted figure indicates that in this case the effect of raising the front of the tractor on the pitch attitude of the plough is largely eliminated. This initial illustration suggests that locating the centre of rotation at or near to the centre of the rear axle would permit the tractor to articulate better relative to the plough and so minimize interference from the pitch motion of the tractor. In order to extend this analysis to the case of a sinusoidal undulation a computer model of a tractor and plough operating over a single sinusoid was developed.3 For each point along the path the front furrow was set to the correct depth by locating the lower links at the appropriate angle. This was followed by setting the rear furrow to the correct depth and adjusting the top link to the appropriate length. The latter action usually affects the front furrow depth so an iterative procedure was followed until both front and rear depths were equal within acceptable limits. Allowance was made for the various stages of entry to the sinusoid, viz.:

(1) tractor front wheels enter sinusoid; (2) tractor rear wheels enter sinusoid; (3) front share enters sinusoid;

P.

A.

COWELL;

P.

F.

HERBERT

87

Fig. I. Illustrating ._ the effect of raising the front wheels of the tractor on the pitch attitude of the plouKh. ““Theheel of the pl&gh is maintained at a constant d&h

rear share enters sinusoid; tractor front wheels leave sinusoid; tractor rear wheels leave sinusoid; :;; front share leaves sinusoid; (8) rear share leaves sinusoid.

(4) (5)

As the tractor crosses the undulation both the angle of the lower links relative to the tractor and the length of the top link vary. Fig. 2 illustrates the result for a tractor with a wheelbase of 2.25 m carrying a three-furrow plough whose overall length from front to rear share is 1.42 m. The wavelength of the sinusoid is 10 m and the amplitude is 100 mm. The computer model suffers from some imperfections associated particularly with entry and exit to the sinusoid and an improved model was developed later’ after the system had been designed and gives a more accurate representation. The two models, however, give results which in all essentials are similar, so the latter is not illustrated here. Fig. 2 shows that there is no single ideal length for the top link. There are several preferred lengths according to the position of the combination along the sinusoid. If the calculation is continued to illustrate the effect of traversing a second successive sinusoid, the characteristic settles down to the form of an ellipse whose principal axis is inclined to the horizontal. Although a trigonometrical proof is rather tedious it may readily be inferred that the characteristic generated by plotting the angle of the lower links against the top link length when operating over a pure sinusoidal surface will be elliptical to a first-order approximation. The front and rear wheels of the tractor as well as the front and rear shares of the implement are required to move vertically all with the same amplitude and frequency, differing only in phase. Since the combination of two sine functions of the same frequency but differing in phase is itself a sine function with amplitude magnification (or attenuation) and phase shift, it follows that for small deflections all other parts of the tractor and plough will perform simple harmonic motion. The combination of two simple harmonic motions at right angles produces a Lissajous figure which, for inputs of the same frequency, is an ellipse.

VARIABLE

88

GEOMETRY

LINKAGE

FOR

TRACTORS

900

Fi;_____;&LIQ e

4 650

-

7 Lowering

Ralslng

3

i 600

I

_ -10

-8

I -6

I

I -4

-2

Bottom

link angle

I

I

I

,

0

2

4

6

(I, deg.

Fig. 2. Showing the relationship between top link length and bottom link angle requiredfor constant front and rear furrow depth when crossing a sinusoidal undulation. The dotted line shows the relationship derived by the instantaneous centre method. Wavelength, 10 m; tractor wheelbase, 2.25 m; amplitude, 100 mm; plough length (front share to rear share), 1.42 m

If the motion on the x and y axes respectively are given by y = b cos (ot+P)

x = a cos wt,

it can be shown that the major axis of the ellipse so produced has the slope

The characteristic of the conventional linkage drawn on Fig. 2 is a straight line parallel to the horizontal axis. This offers a poor compromise to the ideal requirement of top link length; a better fit would be provided by a characteristic corresponding to the major axis of the ellipse. This principle provides the basis for the design of the variable geometry linkage described in this paper. The results of Fig. 2 refer to a specific tractor and implement combination operating over a particular undulation of wavelength 10 m. If the wavelength of the surface profile is changed, the slope of the ideal characteristic changes. The complete analysis of the problem

?? -

1a

I 14

I

I61

6

I

I

8

IO Wavelength

I I? of undulation,

I

s

m

Fig. 3. Computed ideal ratio between top link length and bottom link angle (slope of the major axis of the ellipse). Tractor wheelbase, 2.22 m; plough length (front share to rear share), 3.12 m

P.

A.

COWELL;

P.

F.

HERBERT

89

E 020 E ; z

780

H 5

740-

_c

-

P i” 700 660 t

t 620 t I

-20

I

-16

I

I

-I2

I

I

-8

I.

I

Bottom

Fig. 4. Relationship

I

-4

I

I

0 llnk

angle

I

4

I

I

8

I

I

I2

I

I

I

16

a, deg

between top link length and bottom link angle to produce rotation the centre of the rear axle

ofimplement about

leads to some rather complex mathematical expressions which are beyond the scope of this paper. However, a computer simulation has been carried out4 on a different tractor and implement combination and the result is shown in Fig. 3. This indicates that as the wavelength increases greater proportional changes in the top link are required. As the wavelength increases, however, the amplitude of movement of the lower links and that of the ideal length of the top link decreases, so the ratio between them becomes less critical. The choice of the actual ratio between top link length and lower link angle should therefore preferably correspond to a characteristic at the lower end of the wavelength scale. A sensible practical bottom limit to the wavelength chosen might be based on the notion that the minimum wavelength it would be feasible to attempt to follow would be about five or six times the overall length of the implement. An alternative approach to deciding the appropriate relationship between top link length and bottom link angle is to arrange for the instantaneous centre of rotation of the implement relative to the tractor to occur at or near to the centre of the tractor rear axle. Fig. 4 shows the result so obtained for the tractor used in Fig. 2, and was produced by graphical analysis. The relationship yields a ratio of approximately 7 mm per degree which is a slightly less than that derived by the elliptical analysis, and was the ratio chosen for the design of the experimental system (see also Fig. 2). 3. Design

The analysis indicates that the length of the top link should shorten as the lower links are raised. Fig. 5 illustrates the system. A double acting displacer ram, (l), is connected between the end of one of the lift arms (C) and a point fixed relative to the frame of the tractor (D). A second double acting ram, (2) replaces the conventional top link and the two rams are hydraulically connected as shown, so that as one ram extends the other retracts. The net oil flow out of one cylinder must exactly match the oil admitted to the other. Thus

90

VARIABLE

GEOMETRY

LINKAGE

FOR

TRACTORS

External spool valve

Fig. 5. Variable geometry linkage as used in experimental work

and

A,X,

= A3XA

which yields the condition for correct functioning

A, = A, It follows that

A3

A,’

X* = A, XFI 4’ Each interconnecting line is coupled to an external spool valve on the tractor. Thus, when oil is pumped into the system the length of the top link can be adjusted to the correct length; this provides a convenient means of adjusting the pitch of the implement from the tractor seat. Since spool valves suffer from a degree of internal leakage it is necessary to provide a pilot pressure operated lock valve which locks a given quantity of oil into the system and prevents creep. It also helps to maintain correct register of the implement when it returns to work after having been raised. It is possible, due to general design constraints and the fact that a range of top link lengths are required for different implements and working depths, that the displacer ram or top link ram may reach the end of its travel before the hydraulic lift has reached the top of its stroke. It is advisable therefore to make the end mounts and the rams themselves strong enough to withstand the loads so exerted. Although this is not shown in the diagram, it may also be necessary in practice to provide overload pressure relief in the variable geometry linkage to enable it to withstand shock loads generated when carrying loads in the raised position. In a practical realization of the variable geometry linkage it was found that if equal sized rams were used and the displacer ram was mounted between points with approximately the same location as those used for the conventional assister ram, the correct displacement relationship was obtained. The analysis in Section 2 indicates the kinematic relationship necessary for equal front and rear depth of an implement, but it is not a sufficient condition. The path taken by a mounted

P.

A.

COWELL;

I’.

F.

HERBERT

91

implement when it is being raised on the linkage by the hydraulic lift is governed by the dynamic behaviour of the control system, more particularly the rate at which oil is being pumped into the lift cylinder. By contrast on the lowering part of the cycle, if the rate of oil flow is sufficiently great (which in practice it often is unless a deliberate restriction is placed in the line) the limitation to the rate of penetration is governed by the geometry of the hitch.lr5 The governing equation for penetration of mouldboard ploughs mounted on an unrestrained linkage is Y = Y,+(Y,-YP’, where s is the horizontal distance from the front share to the virtual hitch point. The rate of penetration is given by dY ye-y ds=-? The effective hitch length is much reduced when using the variable geometry linkage. By itself this would tend to produce a faster than normal rate of penetration. However, the equilibrium depth ye also affects the rate of penetration. If the top link is in compression during work this will induce a lift force in the displacer ram and consequently will reduce ye, thereby reducing penetration rate. By contrast a tensile force in the top link such as may be experienced with a heavy implement such as a reversible mounted plough would induce a downward force on the linkage by the displacer ram, an increase in ye and a subsequent increase in penetration rate. One may conclude that very heavy implements penetrate at a faster rate; with lighter implements there is a trade off between the effect of hitch length and equilibrium depth. The equilibrium depth attainable by lighter implements (those which generate a compressive force in the top link) will be less when using the variable geometry linkage. It should be understood that the actual forward distance travelled to reach equilibrium is unaffected by considerations of implement weight; it is only in the matter of how far the implement can penetrate that the weight plays a part. 4. Experimental

test procedure and results: Phase 1

An experimental plot of one hectare was deep ploughed, rotary cultivated, disced, levelled and finally consolidated by rolling. The undulation was excavated to give a single sinusoid of approximately 10 m wavelength and 100 mm amplitude. It was left fallow for two seasons and all surface growth was moved and carted from the site prior to the field trials taking place. A series of tests was conducted using a three-furrow Overurn mouldboard plough and a 49 kW MF575 tractor. The plough was adjusted to cut a 300 mm wide furrow at a depth of 150 mm. Although the plough had a depth wheel it was raised clear of the ground throughout the series of tests. Trials were carried out at speeds of 0.8 m/s, 1.1 m/s and 1.4 m/s. Each run was repeated four times. After each run the heights of the soil surface and the furrow bottom were measured at 0.5 m intervals using a level and staff. With this technique measurements could be made of both the surface profile and the furrow depth to an accuracy of +5 mm. The tractor was set to operate in draught control. In order to maintain the same depth of working throughout the series of tests, a slightly different draught control setting was required at each speed to allow for the increase in draught force as speed increases. The field results are summarized in Fig. 6, which represent half the total number of runs carried out. The worst and best result obtained in each category is illustrated, the selection being made on a visual assessment of how well (or otherwise) the plough was able to maintain an even depth overall.

;

5

IO

MD=l40mm

SD=l8mm

SD=22mm

MD= 149 mm SD=23 mm

MD=l49mm

linkage

MD=154mm

L

Variable geometry

MD=154mm SD=58mm

I-I m/s

linkage

m

SD=22 mm

SD=29mm

I

MD=147 mm SD=27mm

Fi

I

m

I

I

SD=34mm

1

MD= 145 mm SD=21 mm

k+=--=--I

I

MD=l60mm

l-4 m/s

MD=150 mm SD=56 mm

m

Fig. 6. Experimental tests using MF575 tractor with a three-furrow mouldboard plough. Curve A shows the surface profile, curve 13 shows the rear furrow profile, and curve C the depth of the rear furrow. In each category the worst result is shown on the left and the best result on the right. Approximate wavelength, 10 m; approximate amplitude, 100 mm. MD, mean depth; SD, standard deviation

SD=28mm

MD=l53mm

20

MD=l57mm

I5

0.8 m/s

MD=132 mm SD=27 mm

,5:vi

0

m

Conventional

P.

A.

COWELL;

I’.

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93

HERBERT Table 1

Averaged results of the depth of the rear furrow operating over a single sinusoid. Each result is the average of four repeated runs

Linkage

Forward velocity, m/s

Mean depth, mm

Standard deviation of depth, mm

0.8 1.1 1.4 0.8 1.1 1.4

148.4 155.9 150.8 148.6 141.6 148.5

26 1 436 41.0 23.8 23.5 24.1

Conventional

Variable

geometry

i

CoeJEcien t CJ/ variation sf depth, % 17.6 28.1 27.3 16.0 15.9 16.2

Table 2 Analysis of data on depth of rear furrow (Phase I)

Sample Conventional linkage Variable geometry

Size, n

Degrees of freedom

Mean CV, %

Sum of squares ojdeviations OfCV

Standard deviation of cv, s

12 12

11 11

24.31 16.07

697.86 40.43

1.91 1.92

Sum = 22

Diff. = 8.27

c x2 = 738.29

These results indicate the variation in rear furrow depth, and suggest that the variable geometry linkage promoted a considerable improvement in depth control performance at high speeds. This is confirmed in Table 1 which is a summary of all the tests carried out. The coefficient of variation of depth (CV) or (standard deviation)/(mean depth) for each individual test run is a convenient measure of the accuracy of depth control and is taken as the response variable for statistical analysis. It is here expressed as a percentage. From the experimental data Table 2 can be drawn up. The mean value of the coefficient of variation of depth of all tests was 24.3% for the conventional linkage and 16.1% for the variable geometry linkage. The significance of the 8.27% difference between the two mean coefficients of variation may be tested by setting up the hypothesis that the two means are the same and applying Student’s t-test. First it is necessary to carry out an F-test on the variances to check for inequality F = 697.86/12 = 17 26 40.43112 ’ ’ From a 5% level (two tail) F distribution table the critical value of F is 3.48 which indicates that the variances of the two populations are sufficiently different to require modification of the t-test.“’ For equal sized populations: Pooled mean square s2 = c n-l

= 67.12.

94

VARIABLE

GEOMETRY

LINKAGE

FOR TRACTORS

Pooled standard error of the difference in the two mean values

= 2.365. The observed value of 8.27 - __ = 3.50. ’ - 2.635 From t tables the value of t for 11 degrees of freedom is 3.1 at the 1% level, and 4.4 at the 0.1% level. From this series of tests it may be concluded that the improvement shown by the variable geometry linkage was highly significant.

5. Experimental tests and results: Phase 2 Although the experimental results of Phase 1 showed a positive benefit when using the variable geometry linkage the land on which the tests had been carried out was very well consolidated having been left fallow for two seasons. Also no measurements of front furrow depth had been made. A second series of tests was conducted on a light well drained soil on which a single sinusoid of the same dimensions as in Phase I was constructed. The soil was prepared in the same way, but was left undisturbed for approximately three months and consequently was much less consolidated. This is important because errors experienced by the conventional linkage are in large part generated by the heel of the rear body which cannot penetrate hard ground; in soft soil heel penetration is easier so the induced error may well be less. The tractor used was an MF590 carrying a three-furrow mouldboard plough (Ransomes TS90). The depth support wheel was not used. The depths of the front share and the rear share were recorded separately using instrumented depth sensing wheels running on the ground surface. Forward speeds and mean depths per test run are summarized in Table 3; Fig. 7 illustrates the test results. From the experimental data Tables 4 and 5 can be drawn up. The test variable as before is the coefficient of variation of depth (CV). In the case of the front furrow (Table 4) the mean value of the coefficient of variation of depth for the conventional linkage was 9.16(%) greater than that for the variable geometry linkage. Table 3 Showing the range of mean depths and forward speeds in tbe Phase 2 experiments

Linkage

Conventional Variable geometry

Rear Front Rear Front

Mean depth of lest run,

Forward speed,

mm

mls

148 to 155 to 141 to 150 to

167 178 170 181

1.67 to 2.21 1.65 to 2.15

P. A. COWELL; 40

0 407c

0

e is

0

c

zs

0

0

30-

0

0

8:

x

_

x “Q

zoX

0

9d 6% Et rz

X X

X

0c$ _ g30-

0

Pm

;g’ : ;z

95

P. F. HERBERT

-

0

Xx

X

00

x

x

0

SL 0%

“‘;,

0

0

XX

gi20-

x

$-’

0 X

0 ox xX

IO I.5

I

I

I

I.6

I.7

A.8

I

I

I

I

I.9

2.0

2.1

2.2

Forward

Fig.

7. Comparison

of depth variation 0, Conventional

I

I

I

I

I.6

I.7

I.8

I.9

IO 2.3

I.5 speed,

X

I

I

I

2.0

2.1

2.2

i

3

m/s

over sinusoid MF590 tractor with TS90 linkage; x , variable geometry linkage

three-furrow

plough.

The variance ratio F = 1.84, which from tables indicates that the normal t-test may be applied to test for the significance of the difference between means. The pooled estimate of population standard deviation S=

J-

289.68 ___ = 4.01. 18

Standard error of difference between means s

J

$+i=

1.79.

Table 4 Analysis of data on depth of front furrow (Phase 2)

Sample

Conventional linkage Variable geometry

Mean CV, %

Sum of squares of deviations OfCV

Standard deviation of cv. s

Size, n

Degrees of freedom

10

9

30.14

18776

4.51

10

9

21.58

101.92

3.31

Sum = 18

Diff. = 9.16

Sum = 289.68

Table 5 Analysis of data of depth of rear furrow (Phase 2)

Sample

Conventional linkage Variable geometry

Size, n

Degrees of freedom

Mean CV. %

Sum of squares qf deviations OfCV

Standard deviation gf-cv, s

10 10

9 9 Sum = 18

23.01 19.93 Diff. = 3.08

145.05 284.94 Sum = 429.99

4.01 5.63

96

VARIABLE

GEOMETRY

LINKAGE

FOR

TRACTORS

The observed value of 9.16 = 5.11, t = 1.79 -

which indicates a difference which is significant at the 0.1% level. In the case of the rear furrow the mean value of the coefficient of variation of depth for the conventional linkage was 3.08(%) greater than that for the variable geometry. In this case F = 1.96 (for greater mean square), S=

429.99 ___ = 4.89 18

J

S.E. = 2.19 and observed

3.08 ~ = 1.41. t = 2.19

t tables indicate that the difference between the two means in this case is significant only at the 20% level. 6. Effect of variable geometry on hydraulic lift capacity The effect of introducing the variable geometry linkage is to move the instantaneous centre of rotation of the implement to a point close to the rear axle. This decreases the velocity ratio and hence the mechanical advantage between the lift cylinder and the centre of gravity of the implement. Consequently the maximum lift capacity of the linkage is reduced. The reduction in lift capacity depends on the mast height of the implement and the distance behind the ends of the lower links at which the load is applied, but a typical reduction in lift capacity is around 30% as illustrated in Fig. 8. The change in instantaneous centre also results in a greater pitch angle rotation of the implement as it is raised. With long implements this can produce an excessively tail high attitude when in the fully raised position.

30

130

L x 125 f

28 26 24 -

120 .!J

22

II5

I IO 5 E ;:

‘4 12

5

IO t

-200

Vanable

-100

geometry

linkage hft capoclty

0 Vertical

diplacement

of hnkoge y, mm

Fig. 8. MF590 linkage lif capacity

\

P.

A.

COWELL;

P.

F.

97

HERBERT

None of the above problems is unsurmountable are beyond the scope of this paper to describe.

and solutions have been developed but

7. Discussion and conclusions In the phase 1 series of field tests the rear furrow depths only were measured. In phase 2 both front and rear furrow depths were measured. It has been shown previously’ that the depth of the front furrow can be calculated from the measured depth of the rear furrow, and that the errors in front furrow depth exceed those of the rear when crossing a sinusoidal undulation. This evidence is confirmed in the phase 2 tests. The phase 1 series of tests showed that where the ground was well consolidated there was a highly significant improvement in rear furrow depth control when using the variable geometry linkage. It may be inferred that there was a similar improvement in front furrow depth control. In the phase 2 series of tests on less well consolidated ground there was a very highly significant improvement in front furrow depth control when using the variable geometry linkage. An improvement in rear depth control was shown but it was not significant. It may be concluded that a passive variable geometry linkage of the type described reduces the errors in depth induced by the geometry of the hitch linkage when crossing undulating ground. The ability to alter the pitch angle of the implement from the tractor seat is a useful benefit. There are penalties associated with the use of a variable geometry linkage. These include reduced penetrability of light implements in firm soils, a reduction in the lift capacity of the tractor hydraulics and a tendency to raise the rear of the implement too high in the fully raised position. These problems can be eliminated but they require additional design features.

Acknowledgements The authors gratefully acknowledge the support of sponsoring members of the NIAE Industry Tractor Contract (ITC) and Department of Industry (DOI), and for their permission to reproduce Figures 3, 7 and 8. They would also like to acknowledge the help of C. D. Watt, M. J. de Flufy and D. R. White in producing the same figures.

References ’ Cowell, P. A.; Len, S. C. The field performance of tractor draught control systems. Journal of Agricultural Engineering Research 1967, 12(3): 205-221 * Hesse, H,; Miiller, R. Eine elektrohydraulische Zwei-Griissen-Tiefenregelung fur grosse Schlepperanbaupfliige (A twin monitor electrohydraulic depth regulator for large tractor mounted ploughs). Grundlagen der Landtechnik 1972, 22(3): 75-79 3 Herbert, P. F. Improvements to the kinematic design of tractor three point linkages. M.Sc. thesis (unpublished), Silsoe College, 198 1 4 Cowell, P. A.; Watt, C. D. An investigation of the performance of a variable geometry three point linkage. ITD Report No. 3, British Society for Research in Agricultural Engineering, July 1984 5 Cowell, P. A.; Sial, F. S. A theory for the dynamic behaviour of mouldboard ploughs during penetration. Journal of Agricultural Engineering Research 1976, 21: 3 13-323 6 Snedecor, G. W.; Cochran, W. G. Statistical Methods, 5th edition. Iowa State University Press, Section 4.9 ’ Ryan, B. F.; Joiner, B. L.; Ryan, T. A. Jr. Minitab Handbook, 2nd edition. Duxbury Press