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A geometrical model to define the limits of accuracy of sugar beet topping
J. agric. Engng Res. (1986) 35, 55-66
Model to Define the Limits of Accuracy of Sugar Beet Topping
A Geometrical
M. J. O’DOGHERTY* A geometrical model of the sugar beet topping operation was developed using a beet crown shape based on field measurements. Calculations were made of the degree of over- and undertopping for different positions of the cutting plane below the top of the crown and relationships between these quantities established. The effect of variations in the position of the correct topping plane was examined, together with the influence of different crop size distributions. The effect of inaccurate sensing of the top of the crown by the topper feeler wheel
was also considered. The results of the model prediction were compared with field observations, and the limits of the accuracy of topping mechanisms defined in relation to the constraints imposed by beet crown geometry and variability.
1. Introduction
Many experiments have been conducted in which the over- and undertopping of sugar beet crowns have been measured in the field.’ There has not, however, been a means of determining the theoretical limits of accuracy which can be achieved if beet crowns are correctly sensed. In particular, it is necessary to establish a relationship between over- and undertopping for different positions of the cutting plane when using a fixed knife topping unit. The effect of crop size distribution and the position of the topping plane is also of interest in relation to the limits which they impose on accuracy of topping. It was decided to adopt the idealized shape of beet crown used in earlier work which is characterized by a hyperboloid superimposed on a basal cylinder. ‘,’ This form was used as the basis for calculations of over- and undertopping using simple geometrical methods and statistical data on beet and crop variability. The results cover a range of possibilities, based on measured data, so as to give an indication of the limits of topping accuracy to be accepted. The model was based on the assumption of geometrical similitude of beet crowns, irrespective of size. The effect of the model parameters were considered under the following headings: (1) the effect of topping errors for a fixed position of the correct topping plane;
(2) (3) (4) (5)
effect effect effect effect
of of of of
beet size; variation in position of correct topping plane; beet size distribution in a crop; errors in sensing of beet crowns by the topping mechanism.
2. Over- and undertopping errors for fixed topping plane position The idealized beet cross-section is shown in Fig. I. The below ground volume has a cylindrical portion (D, = Do) and a conical portion with a total beet volume, V, = 0.9700,3. The assumption of D, = DG does not affect the topping error calculations because the below
* Crop Research Group, Silsoe, Bedford Received
Field Machinery
Division,
National
Institute
of Agricultural
Engineering,
Wrest Park,
MK4J 4HS
22 March
1985; accepted
in revised form 29 October
1985
55 0021-8634/86/090055
+ 12 $03.0010
& 1986 The British
Society for Research
in Agricultural
Engineering
56
A GEOMETRICAL
MODEL
OF
SUGAR
BEET
TOPPING
Notation a b
h 00 V” x Ds D,
H P, I’, To T,
V, V, V, E dT (TV c
semi-transverse axis of hyperbolae of cross-section of idealized beet crowns semi-conjugate axis of hyperbolae of cross-section of idealized beet crowns height of correct topping plane of idealized beet crowns above soil level volume of overtopping for an incorrectly topped beet volume of undertopping for an incorrectly topped beet distance of cutting plane below peak of idealized beet crown diameter of beet at soil level maximum diameter of beet height of crown for idealized beet geometry percentage of roots in a diameter class interval percentage of roots with topping plane in a position class interval overtopping as a proportion of clean, correctly topped beet undertopping as a proportion of clean, correctly topped beet total beet volume for idealized geometry volume of correctly topped beet crown correctly topped beet volume for idealized geometry error in topping plane position standard deviation of position of correct topping plane standard deviation of sensing error of apex of beet crown by feeler wheel total standard deviation of position of topping plane
ground volume was chosen to equal that indicated by measurements on beet. To be correctly topped the beet must be cut in a horizontal plane located at a distance of H-h below the top of the crown’** (pl ane t-t in Fig. 1). If the beet is cut above this plane (in a plane a-a) it is undertopped and if topped below it in a plane b-b, it is overtopped. Over- and undertopping is expressed as the beet volumes between planes b-b and t-t and planes a-a and t-t, calculated as a percentage of the volume of the beet below the t-t plane. This corresponds to over- and undertopping in field experiments as a percentage of the weight of correctly topped clean beet. Percentage topping errors resulting from cutting the crown at different horizons were calculated for the idealized beet shape by obtaining the volume of the crown at distances x below its peak. The volume of the hyperboloidal portion of the crown (u,) was calculated from:’ v = nb2 x x2(x + 3~) c
3a2
’
(1)
where a = @74QDs and b = 0.3490,. For values of x > 0.5420s the beet is cut in the cylindrical portion (Fig. I) and the appropriate cylindrical volume is then added to the hyperboloid volume for x = 0.5420,. Crown volumes were calculated for different values of x and the error volume (volume of beet between a cutting plane and the correct topping plane, for which x = 0.4590,) obtained from the difference between the appropriate crown volumes. The correctly topped crown volume’ is equal to O.l32D,j. The error volumes were then calculated as a percentage of the correctly topped volume (V, = O-8380,3). The effect of a given error, E, in the position of the cutting plane for beet of different sizes was examined. The calculated values of the percentage of over- and undertopping are shown in Fig. 2 for values E up to 12.7 mm. If an
M.
J.
57
O’DOGHERTY
Fig. 1. Idealized beet geometry showing principal dimensions and their relationship to soil diameter, D,. (t-t. correct topping plane; a-a: cutting plane for undertopped crown; b-b: cutting plane for overtopped crown; S-S: soil level)
error of only 5.1 mm is made, overtopping can range from 3.0% for a large beet (D, = 130 mm) to 8.6% for a small beet (Ds = 50 mm). Corresponding values of undertopping range from 2.7% to 6.6% for large and small beet respectively. If it is required to limit both over- and undertopping to 5% for a beet of average size (Ds = 92 mm), then an error of no greater than 5.8 mm can be tolerated for overtopping and 6.9 mm for undertopping.’ It is of interest that an error of only 2.5 mm results in over and undertopping of 2.2% and 1.9’?, respectively. These deductions highlight the serious effect that relatively small errors in the position of the cutting plane can have on the balance of over- and undertopping. They indicate the accuracy of cutting which is required in practice if a specified standard of topping is to be achieved. 3. Variation in position of topping plane The discussion so far has assumed that for a given beet diameter the correct topping plane is located at a constant distance below the top of the crown. Measurements by Wayman however, have shown that for a particular size grade of beet the topping plane position varies over a range of values. Some of this arises because the measurements were made for size grades with a class interval of 25 mm for the maximum diameter (Do). This is equivalent to a range of 10.3 mm in the position of the topping level, and, if this is subtracted from the values obtained by Wayman, the range of the topping plane position below the beet crown was found to be equal to 0.4180,. For the purpose of the model it was assumed that the distribution was normal with a range of 0.4Ds, so that the standard deviation oT = 0.0670,.
58
A GEOMETRICAL
-151 0
/
,
,
/
20
40
60
80
MODEL
OF
/
,
,
100
120
140
SUGAR
BEET
TOPPING
Diameter of beet at solI level CD,), mm
Fig. 2. Variation in percentage overtopping (TO) or undertopping (T,) with beet soil diameter (D,) for a range of errors (E) in cutting plane position relative to the correct topping plane
Such a distribution means that for any position of the cutting plane within the range there is both under and overtopping for a given beet size. This is in contrast to the fixed topping plane concept for which all the beet are either undertopped, overtopped or correctly topped. 4. Beet size distribution The balance of under- and overtopping will depend on the manner in which the sizes of the beet are distributed. A study of published distributions of root diameters within crops shows that it is possible to represent the size distributions by a family of curves.’ The bounding curves of this family are shown in Fig. 3, where the cumulative frequency of the number of roots is plotted against maximum root diameter on a linear-probability basis. The mean diameter at soil level ranged from 66 to 96 mm and the standard deviation was constant with a value of 19 mm. 5. Effect of beet size distribution and variation in
topping plane position Calculations of over- and undertopping for a range of beet sizes were made for maximum diameters ranging from 45 to 165 mm, using a class interval of 10 mm. The method used is described in the Appendix. For a particular cutting plane position there was generally both under- and overtopping for a particular size of beet which can be determined from curves appropriate to a specified value of 0,/D,. Fig. 4 shows the relationships for distribution P between over- and undertopping for cr,/D, = 0, 0.067 and 0.095. These curves show clearly
M.
J.
59
O’DOGHERTY
99.9
-
99
-
95
-
90
-
ae
-
80
c g m;
e
40-
$ +$
zo-
2
IO-
I -
0.01 0
I
I
I
I
20
40
60
80
Beet
maximum
dmneter
I
I
I
100
120
140
(Do),mm
Fig. 3. Relationship between cumulative percentage frequency of occurrence of roots (n) and beet maximum diameter (Do). (P and Q are limits of distributions observed infield crops)
0
2
4
6 Undertopping
8
IO
12
14
CL),%
Fig. 4. Relationship between overtopping (TJ and undertopping (T,) for beet crop size distribution P. (Standard deviations of topping plane position, or, expressed as a ratio of beet diameter at soil level, D,)
that both tares increase with increasing variability of the topping plane position, as the bounding curve for the standard of topping progressively moves away from the axes. For balanced topping the tares increase from 2.2% when a,/D, = 0 to 3.676 when crT/Ds = 0.095 for root size distribution P. For distribution Q, the equivalent tares for balanced topping
60
A
GEOMETRICAL
MODEL
OF
SUGAR
BEET
TOPPING
20 t 8
15-
2 ; 8 t z 0
IO -
5-
0 3 2
-5-
c” B ,o -10
-
$ s -151 0
,
,
,
[
,
,
IO
20
30
40
50
60
70
DWmce of cuttmg (x), mm plane below crown apex
Fig. 5. Variation in overtopping (T,) and/or undertopping (T,) with distance of cutting plane (x) below the beet crown apex, for crop size distributions P and Q
were 2.9% and 4.2%, which indicates that crop size distribution does not have a large influence on topping tares if individual beet are accurately sensed. The curves of Fig. 4 show that there is a constraint on the degree of over- and undertopping which can be achieved. If it is required to reduce undertopping, for example, then an increase in overtopping must be accepted, which is determined by the form of the curves. Hence, changes in the level of the cutting plane below the crown top can only result in movement along the aTIDS curve for a particular crop size distribution. The curves show clearly that “perfect topping” cannot be realized because of physical constraints. The origin of co-ordinates corresponds to the concept of perfect topping (no under- or overtopping), but if the distance of the cutting plane below the crown peaks is fixed it is only possible to work on curves of the form shown. Fig. 5 shows the percentage of over- and undertopping plotted against the distance of the cutting plane below the crown top (x) for root size distributions P and Q and indicate the sensitivity of the balance of topping to the topping plane position. For example, for a,/D, = 0 an error of 6.4 mm can result in the balance of topping for distribution P being moved over a large proportion of its bounding curve from [T’, = 0.5x, T, = 6*3x] for + 6.4 mm to [T, = 5.4x, T, = 0.5x] for - 6.4 mm. Although size distribution does not have a large effect on topping tares, the correct cutting plane position to achieve a balance of topping differs markedly. The balanced cutting plane is at 45.7 mm below the peak for distribution P, but at 35.5 mm for distribution Q. Hence changes in crop size distribution can significantly affect the optimum knife setting. A point of practical interest is the effect of a mechanism which compensates for variations in beet size by changing the knife setting so that the cut is made at the correct topping plane for all beet. This is equivalent to always cutting at a distance x/D, = 0.459 below the crown. A simple compensating technique of this nature does not, however, eliminate topping tares because of variation in the position of the topping plane; tares range from 1.3% to 2.5% for overtopping and 1.8% to 3.4% for undertopping as a,/D, varies between 0.05 and 0.10.
M.
J.
61
O’DOGHERTY
0
0
2
4
6
UndertoppIng
8
IO
12
CT, ), %
Fig. 6. Relationship between overtopping (TO) and undertopping (T.) for a range of standard deviations, a, qf cutting plane position about the correct topping plane position, expressed as a ratio of beet diameter at soil level, D,. (Beet crop size distribution Q)
6. Errors in crown sensing The effect of errors in sensing of the top of the beet crown by the feeler wheel of the topping unit was assessed, assuming that they were random and followed a normal distribution. If the error variance in sensing by the feeler wheel is oi, the total variance g. For a given mean position of the about the mean topping position is given by 0’ = gp + CJ cutting plane, the under- and overtopping was calculated for a series of cutting planes on either side of the mean, in proportion to the frequency of occurrence of such a positional error with a normal distribution. The total standard deviations in topping the beet (a/D,) are 0.072, 0.084, 0.101 and 0.120 for values of a,/D, of 0.025, 0.05, 0.075 and 0.10 respectively,assuming that a,/DS = 0.067. Fig. 6showscurvesforcropdistributionQforvaluesof a/D, ranging from 0 to 0.12. The curve a/DS = 0.067 is the lower bound for beet for which aTIDS = 0.067 and that for a/OS = 0.12 represents large sensing errors (as/Ds = 0.10). Improvements in sensing accuracy will move the topping characteristic curves towards that for a/DS = 0.067. The difference between perfect and poor sensing represents an increase in balanced tares from 3.6% to 4.7%. The improvement is greater for topping away from the balanced condition. If overtopping was controlled at 2%, for example, improvement in sensing represents a potential reduction in undertopping from 7.4% to 52%. 7. Comparison with field measurements The predictions of the model have been validated in relatively few experiments in which the variation in the balance of topping was examined over a range of topper knife settings which was sufficient to establish the relationship between the tares. Experiments conducted by Wayman and Wayman and Maughan5 show a good measure of agreement with the model values, with balanced topping tares in a range of approximately 304 to 6%. Earlier work’-’ in 1965-67 exhibited rather higher tares, however, with balanced topping in the range 6.5% to 7.5%. Some work with an early type of feeler wheel topper lo
62
A GEOMETRICAL
0
2
4
6
UndertoppIng
8
MODEL
IO
12
OF SUGAR
BEET
TOPPING
14
CT,),%
Fig. 7. Typical characteristic curves of topper performance at 7.2 km/h relating percentage overtopping trailing arm (TJ and percentage undertopping (T,) from jield experiments of Wayman. (-, lightweight unit; - - - , lightweight unit mounted on parallel linkage)
gave results which compare well with more recent work and with the model. Fig. 7 shows some typical results given by Wayman” from other work which have the form of topping characteristic predicted by the model. 8. Discussion Calculations for single beet showed the degree of precision necessary when cutting off the crown. To achieve an accuracy of 5% over- and undertopping, the cut should be within 5.X to 6.9 mm of the correct topping plane (assuming the mean position) and an average size of beet (D, = 92 mm). Greater precision is necessary for small beet; for example, an error of only 2.5 mm can result in 4% overtopping and 3.5% undertopping on a beet of diameter D, = 50 mm. An important finding from the model was that for a fixed knife position it is not possible to attain perfect topping in a practical crop size distribution. Physical constraints result in a relationship between under- and overtopping and a decrease in one is necessarily accompanied by an increase in the other. The form of the relationship sets a bound for topping performance. Even if a mechanism is used which compensates for beet size, so that all beet are cut at their mean topping plane position, there will be both under- and overtopping because of the variation in topping plane position. Perfect topping can only be achieved for a variable topping plane position if the correct topping plane is sensed for individual beet. For disparate size distributions, if individual beet crowns are accurately sensed, the differences between topping tares are relatively small. The effect of errors in crown sensing is to increase the overall variance in the position of the cutting plane in relation to the correct topping plane, with an increase in the topping tares. The model shows the benefit to be obtained from improvements in the sensing accuracy. For example, the balanced tare for perfect sensing for size distribution Q and oJD, = 0.067 is 3*6%, compared with 4.7% for inaccurate sensing (o/D, = 0.10). The model emphasizes the important effect that beet variability has on the topping accuracy and
M.
J.
63
O’DOGHERTY
predicts that it is probably not possible to achieve a balanced tare of less than 3% to 4%, depending on individual beet variability and crop size distribution. The relationship between under- and overtopping obtained from the model is generally in good agreement with results from field measurements of topping tares. The measure of agreement lends support to the approach and indicates that although the model is a simple one, the physical concept is correct. In some experimental work the tares observed were higher than those predicted, which is to be expected in view of the simplifications made. Larger tares arise in practice because of gross errors in sensing because of geometrical limitations, particularly where a relatively small beet follows a large beet. In addition, there are errors due to departures of beet crown shape from the idealized shape and inclination of the correct topping plane to the horizontal. The topper may push the beet out of vertical alignment so that the cut is inaccurately made and, in some cases, the forces on the crown are such that it is not cut cleanly, resulting in random fracturing which increases one or other of the tares. The dynamic behaviour of the unit as it passes over the beet stand also affects the sensing of the crown and hence the position at which the knife enters it. 9. Conclusions A geometrical model was developed to study the relationship between over- and undertopping of sugar beet crowns. The model enabled studies to be made of the effect of root size distribution, variance in topping plane position and errors in crown sensing. The most important finding was that there is a functional relationship between under- and overtopping which is determined by the position of the cutting plane and is dependent on the size distribution of the crop and the variance of the position of the correct topping plane for a particular beet size. A balance between over- and undertopping must be accepted for a fixed knife setting, the particular values of which depend on crop variability. Perfect topping cannot be attained because of geometrical limitations, unless each root is sensed individually for its correct topping level and the information used to position a cutting element at the appropriate level. If a mechanism is used which compensates for beet size, the variance of topping plane position will result in both over- and undertopping. The model showed that for significant differences in crop size distribution the difference between balanced tares was only about 0.6x, but the knife setting required differed by 10 mm. The minimum balanced topping tare likely to be achieved is in the range 3% to 4%. The total error between the cutting plane and the correct topping plane is compounded of variation in topping plane position and errors in crown sensing. Balanced topping increases from 3.6% to 4.7% due to sensing errors from a particular size distribution. The effect on undertopping is, however, more significant; for 2% overtopping, the undertopping increased from 5.2% to 7.4%. The model highlights the accuracy required to achieve good topping. The cutting plane must be within approximately 6 to 7 mm of the correct topping plane to achieve an accuracy of 5% for an average size of beet. An error of only 2.5 mm can result in topping errors of about 2%. The geometrical model was simple in concept but showed good agreement with observations from field experiments. It enables useful deductions to be made in relation to the limits of topping which can be achieved in practice. In general, the values of over- and undertopping obtained in the field would be expected to be greater than those predicted by the model. References ’ O’Dogherty, M. J. The mechanics of a sugar beet topping mechanism. Ph.D. thesis, University of Reading, 1976
64
A GEOMETRICAL
MODEL
OF
SUGAR
BEET
TOPPING
M. J. The design of feeler wheel toppers for sugar beet: 1. Topper dynamics. Journal of Agricultural Engineering Research 1986, 34: 305-3 18 3 Wayman, J. A. An investigation into the variations in sugar beet root and crown dimensions with regard to the design of mechanical topping units, 1970. Departmental Note DN/R/l41/1600, National Institute of Agricultural Engineering, Silsoe, 1971 4 Wayman, J. A. The effects of topper unit modifications, plant distribution and harvester speed on sugar beet top tares, 1969. Departmental Note DN/R/028/1600. National Institute of Agricultural Engineering, Silsoe, 1970 5 Wayman, J. A.; Maughan, G. L. The effects of harvester speed and plant distribution on sugar beet topping, 1968. Departmental Note DN/256/RD, National Institute of Agricultural Engineering, Silsoe, 1969 6 Wayman, J. A.; Maughan, G. L. The effect of harvesting speed on the standard of topping of sugar beet, 1965-67. I.I.R.B. Journal 1969, 4(2): 104-l 11 ’ Wayman, J. A.; Maughan, G. L. An investigation into the effects of speed and plant distribution on mechanical topping of sugar beet, 1965. Departmental Note DN/214/CD, National Institute of Agricultural Engineering, Silsoe, 1966 s Wayman, J. A.; Maughan, G. L. An investigation into the effects of speed and plant distribution on mechanical topping of sugar beet, 1966. Departmental Note DN/233/CD, National Institute of Agricultural Engineering, Silsoe, 1967 g Wayman, J. A.; Maughan, G. L. Mechanical topping of sugar beet: The effects of harvester speed and plant distribution, 1965-67. Departmental Note DN/243/CD, National Institute of Agricultural Engineering, Silsoe, 1968 lo Dunn, J. S.; Maughan, G. L. Investigation into the topping of a mechanically thinned crop of sugar beet, Wrest Park, 1954. Departmental Note DN/85/ATD, National Institute of Agricultural Engineering, Silsoe, 1955 l1 Wayman, J. A. Field trials of design variations of the prototype NIAE lightweight sugar beet topper, 1974. Departmental Note DN/R/632/1600, National Institute of Agricultural Engineering, Silsoe, 1976 * O’hgherty,
Appendix Method
of calculation of over- and under-topping for normal distributions crop size and topping plane position
of
A.1. Crop size Over- and undertopping were calculated for a series of distances (x) of the topping plane below the top of the beet crown. The procedure used for a given size distribution of roots was first to obtain the percentage of roots for class intervals of the graded diameter, D,, of 10 mm, for classes ranging from 45 to 55 mm up to 155 to 165 mm. The soil diameter D,( = 0.8970,) was then found for each of the class interval means. The value of x/D, was calculated for a series of values of x in the range 20.6 to 65.9 mm, where the extremes of the range correspond to the correct topping planes for the smallest and largest class interval means, for which D, = 44.8 and 143.5 mm, respectively. The percentage topping errors T, and T, were then obtained graphically and the corresponding volumes u, and v, obtained from the correctly topped beet volumes for each size grade mean diameter, calculated from the expression V, = 0.8380:. These volumes are given by: (v,), = $$
x VT),
(Al)
(V”), = $$
x VT),
(A2)
and
where the subscript distribution.
r denotes
that the parameters
refer to a class interval of the size
M.
J.
65
O’DOGHERTY
These volumes were then multiplied by the percentage number of roots, P,, in each class interval to give the appropriate weighting for the crop size distribution under consideration. These volumes were then summed over the whole size distribution for the total number of class intervals, n, and the total volumes of under- and overtopped material obtained, given by:
fl ~ruxI/,)r
(A3)
(U”)W= ik ,tl ~,(T,),(W,.
(A4)
tdv = iii5 and
The subscript w denotes the total volume of material, weighted appropriately for crop size distribution. A similar procedure was followed for the correctly topped root volumes, which were multiplied by the percentage of roots in the class interval, and the volumes summed to obtain the weighted volume for the crop size distribution, given by:
The over- and undertopping T, and T, was then calculated for the particular value of by expressing the sum of the over- and undertopped volumes as a percentage of the sum of the correctly topped volumes, so that:
H -h( = 0.4590,)
i P,(T,),(I/,), (T,), = ‘= l”
(A6)
and
(A7)
A.2. Topping plane position Over- and undertopping were calculated on the basis of the idealized beet shape, assuming that the distance of the topping plane below the top of the crown was normally distributed about its mean. The calculations were made for a series of distances (x) of the cutting plane below the top of the crown. The distribution was divided into class intervals of x/D, = 0.05, and the percentage of roots which would have topping planes situated in this interval were found from the normal probability integral for values of a,/D, = O-067 and O-095. The volumes between the cutting plane, located at x below the top of the crown, and the midplanes of each of the class intervals of the topping plane position were calculated from differences of the appropriate crown volumes [see Eqn (Al)] denoted by: (u,), and (US, where the subscript s denotes that the parameters refer to a class interval of the topping plane distribution. The volumes were then multiplied by the percentage of roots by number, P,, which would have topping planes located in the appropriate class interval, and the result summed over the total number of class intervals, n, to give the total amount of over- and undertopped material. i.e.
66
A GEOMETRICAL
MODEL
OF
SUGAR
BEET
TOPPING
The correctly topped volumes (V,), were then calculated for the mean of each class interval of topping plane distribution, from the known geometry of the idealized beet, and multiplied by the percentage of roots in the class to obtain a weighted correctly topped volume. These volumes were then summed to give the total weighted correctly topped volume, i.e.
To obtain the over- and undertopping To and T, for the idealized shape, the total volumes of over- and undertopped material were expressed as a percentage of the total correctly topped volume, i.e.
and
where the subscript x denotes that the over- and undertopping positional distribution of the topping plane.
are weighted for the
Model to Define the Limits of Accuracy of Sugar Beet Topping
A Geometrical
M. J. O’DOGHERTY* A geometrical model of the sugar beet topping operation was developed using a beet crown shape based on field measurements. Calculations were made of the degree of over- and undertopping for different positions of the cutting plane below the top of the crown and relationships between these quantities established. The effect of variations in the position of the correct topping plane was examined, together with the influence of different crop size distributions. The effect of inaccurate sensing of the top of the crown by the topper feeler wheel
was also considered. The results of the model prediction were compared with field observations, and the limits of the accuracy of topping mechanisms defined in relation to the constraints imposed by beet crown geometry and variability.
1. Introduction
Many experiments have been conducted in which the over- and undertopping of sugar beet crowns have been measured in the field.’ There has not, however, been a means of determining the theoretical limits of accuracy which can be achieved if beet crowns are correctly sensed. In particular, it is necessary to establish a relationship between over- and undertopping for different positions of the cutting plane when using a fixed knife topping unit. The effect of crop size distribution and the position of the topping plane is also of interest in relation to the limits which they impose on accuracy of topping. It was decided to adopt the idealized shape of beet crown used in earlier work which is characterized by a hyperboloid superimposed on a basal cylinder. ‘,’ This form was used as the basis for calculations of over- and undertopping using simple geometrical methods and statistical data on beet and crop variability. The results cover a range of possibilities, based on measured data, so as to give an indication of the limits of topping accuracy to be accepted. The model was based on the assumption of geometrical similitude of beet crowns, irrespective of size. The effect of the model parameters were considered under the following headings: (1) the effect of topping errors for a fixed position of the correct topping plane;
(2) (3) (4) (5)
effect effect effect effect
of of of of
beet size; variation in position of correct topping plane; beet size distribution in a crop; errors in sensing of beet crowns by the topping mechanism.
2. Over- and undertopping errors for fixed topping plane position The idealized beet cross-section is shown in Fig. I. The below ground volume has a cylindrical portion (D, = Do) and a conical portion with a total beet volume, V, = 0.9700,3. The assumption of D, = DG does not affect the topping error calculations because the below
* Crop Research Group, Silsoe, Bedford Received
Field Machinery
Division,
National
Institute
of Agricultural
Engineering,
Wrest Park,
MK4J 4HS
22 March
1985; accepted
in revised form 29 October
1985
55 0021-8634/86/090055
+ 12 $03.0010
& 1986 The British
Society for Research
in Agricultural
Engineering
56
A GEOMETRICAL
MODEL
OF
SUGAR
BEET
TOPPING
Notation a b
h 00 V” x Ds D,
H P, I’, To T,
V, V, V, E dT (TV c
semi-transverse axis of hyperbolae of cross-section of idealized beet crowns semi-conjugate axis of hyperbolae of cross-section of idealized beet crowns height of correct topping plane of idealized beet crowns above soil level volume of overtopping for an incorrectly topped beet volume of undertopping for an incorrectly topped beet distance of cutting plane below peak of idealized beet crown diameter of beet at soil level maximum diameter of beet height of crown for idealized beet geometry percentage of roots in a diameter class interval percentage of roots with topping plane in a position class interval overtopping as a proportion of clean, correctly topped beet undertopping as a proportion of clean, correctly topped beet total beet volume for idealized geometry volume of correctly topped beet crown correctly topped beet volume for idealized geometry error in topping plane position standard deviation of position of correct topping plane standard deviation of sensing error of apex of beet crown by feeler wheel total standard deviation of position of topping plane
ground volume was chosen to equal that indicated by measurements on beet. To be correctly topped the beet must be cut in a horizontal plane located at a distance of H-h below the top of the crown’** (pl ane t-t in Fig. 1). If the beet is cut above this plane (in a plane a-a) it is undertopped and if topped below it in a plane b-b, it is overtopped. Over- and undertopping is expressed as the beet volumes between planes b-b and t-t and planes a-a and t-t, calculated as a percentage of the volume of the beet below the t-t plane. This corresponds to over- and undertopping in field experiments as a percentage of the weight of correctly topped clean beet. Percentage topping errors resulting from cutting the crown at different horizons were calculated for the idealized beet shape by obtaining the volume of the crown at distances x below its peak. The volume of the hyperboloidal portion of the crown (u,) was calculated from:’ v = nb2 x x2(x + 3~) c
3a2
’
(1)
where a = @74QDs and b = 0.3490,. For values of x > 0.5420s the beet is cut in the cylindrical portion (Fig. I) and the appropriate cylindrical volume is then added to the hyperboloid volume for x = 0.5420,. Crown volumes were calculated for different values of x and the error volume (volume of beet between a cutting plane and the correct topping plane, for which x = 0.4590,) obtained from the difference between the appropriate crown volumes. The correctly topped crown volume’ is equal to O.l32D,j. The error volumes were then calculated as a percentage of the correctly topped volume (V, = O-8380,3). The effect of a given error, E, in the position of the cutting plane for beet of different sizes was examined. The calculated values of the percentage of over- and undertopping are shown in Fig. 2 for values E up to 12.7 mm. If an
M.
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57
O’DOGHERTY
Fig. 1. Idealized beet geometry showing principal dimensions and their relationship to soil diameter, D,. (t-t. correct topping plane; a-a: cutting plane for undertopped crown; b-b: cutting plane for overtopped crown; S-S: soil level)
error of only 5.1 mm is made, overtopping can range from 3.0% for a large beet (D, = 130 mm) to 8.6% for a small beet (Ds = 50 mm). Corresponding values of undertopping range from 2.7% to 6.6% for large and small beet respectively. If it is required to limit both over- and undertopping to 5% for a beet of average size (Ds = 92 mm), then an error of no greater than 5.8 mm can be tolerated for overtopping and 6.9 mm for undertopping.’ It is of interest that an error of only 2.5 mm results in over and undertopping of 2.2% and 1.9’?, respectively. These deductions highlight the serious effect that relatively small errors in the position of the cutting plane can have on the balance of over- and undertopping. They indicate the accuracy of cutting which is required in practice if a specified standard of topping is to be achieved. 3. Variation in position of topping plane The discussion so far has assumed that for a given beet diameter the correct topping plane is located at a constant distance below the top of the crown. Measurements by Wayman however, have shown that for a particular size grade of beet the topping plane position varies over a range of values. Some of this arises because the measurements were made for size grades with a class interval of 25 mm for the maximum diameter (Do). This is equivalent to a range of 10.3 mm in the position of the topping level, and, if this is subtracted from the values obtained by Wayman, the range of the topping plane position below the beet crown was found to be equal to 0.4180,. For the purpose of the model it was assumed that the distribution was normal with a range of 0.4Ds, so that the standard deviation oT = 0.0670,.
58
A GEOMETRICAL
-151 0
/
,
,
/
20
40
60
80
MODEL
OF
/
,
,
100
120
140
SUGAR
BEET
TOPPING
Diameter of beet at solI level CD,), mm
Fig. 2. Variation in percentage overtopping (TO) or undertopping (T,) with beet soil diameter (D,) for a range of errors (E) in cutting plane position relative to the correct topping plane
Such a distribution means that for any position of the cutting plane within the range there is both under and overtopping for a given beet size. This is in contrast to the fixed topping plane concept for which all the beet are either undertopped, overtopped or correctly topped. 4. Beet size distribution The balance of under- and overtopping will depend on the manner in which the sizes of the beet are distributed. A study of published distributions of root diameters within crops shows that it is possible to represent the size distributions by a family of curves.’ The bounding curves of this family are shown in Fig. 3, where the cumulative frequency of the number of roots is plotted against maximum root diameter on a linear-probability basis. The mean diameter at soil level ranged from 66 to 96 mm and the standard deviation was constant with a value of 19 mm. 5. Effect of beet size distribution and variation in
topping plane position Calculations of over- and undertopping for a range of beet sizes were made for maximum diameters ranging from 45 to 165 mm, using a class interval of 10 mm. The method used is described in the Appendix. For a particular cutting plane position there was generally both under- and overtopping for a particular size of beet which can be determined from curves appropriate to a specified value of 0,/D,. Fig. 4 shows the relationships for distribution P between over- and undertopping for cr,/D, = 0, 0.067 and 0.095. These curves show clearly
M.
J.
59
O’DOGHERTY
99.9
-
99
-
95
-
90
-
ae
-
80
c g m;
e
40-
$ +$
zo-
2
IO-
I -
0.01 0
I
I
I
I
20
40
60
80
Beet
maximum
dmneter
I
I
I
100
120
140
(Do),mm
Fig. 3. Relationship between cumulative percentage frequency of occurrence of roots (n) and beet maximum diameter (Do). (P and Q are limits of distributions observed infield crops)
0
2
4
6 Undertopping
8
IO
12
14
CL),%
Fig. 4. Relationship between overtopping (TJ and undertopping (T,) for beet crop size distribution P. (Standard deviations of topping plane position, or, expressed as a ratio of beet diameter at soil level, D,)
that both tares increase with increasing variability of the topping plane position, as the bounding curve for the standard of topping progressively moves away from the axes. For balanced topping the tares increase from 2.2% when a,/D, = 0 to 3.676 when crT/Ds = 0.095 for root size distribution P. For distribution Q, the equivalent tares for balanced topping
60
A
GEOMETRICAL
MODEL
OF
SUGAR
BEET
TOPPING
20 t 8
15-
2 ; 8 t z 0
IO -
5-
0 3 2
-5-
c” B ,o -10
-
$ s -151 0
,
,
,
[
,
,
IO
20
30
40
50
60
70
DWmce of cuttmg (x), mm plane below crown apex
Fig. 5. Variation in overtopping (T,) and/or undertopping (T,) with distance of cutting plane (x) below the beet crown apex, for crop size distributions P and Q
were 2.9% and 4.2%, which indicates that crop size distribution does not have a large influence on topping tares if individual beet are accurately sensed. The curves of Fig. 4 show that there is a constraint on the degree of over- and undertopping which can be achieved. If it is required to reduce undertopping, for example, then an increase in overtopping must be accepted, which is determined by the form of the curves. Hence, changes in the level of the cutting plane below the crown top can only result in movement along the aTIDS curve for a particular crop size distribution. The curves show clearly that “perfect topping” cannot be realized because of physical constraints. The origin of co-ordinates corresponds to the concept of perfect topping (no under- or overtopping), but if the distance of the cutting plane below the crown peaks is fixed it is only possible to work on curves of the form shown. Fig. 5 shows the percentage of over- and undertopping plotted against the distance of the cutting plane below the crown top (x) for root size distributions P and Q and indicate the sensitivity of the balance of topping to the topping plane position. For example, for a,/D, = 0 an error of 6.4 mm can result in the balance of topping for distribution P being moved over a large proportion of its bounding curve from [T’, = 0.5x, T, = 6*3x] for + 6.4 mm to [T, = 5.4x, T, = 0.5x] for - 6.4 mm. Although size distribution does not have a large effect on topping tares, the correct cutting plane position to achieve a balance of topping differs markedly. The balanced cutting plane is at 45.7 mm below the peak for distribution P, but at 35.5 mm for distribution Q. Hence changes in crop size distribution can significantly affect the optimum knife setting. A point of practical interest is the effect of a mechanism which compensates for variations in beet size by changing the knife setting so that the cut is made at the correct topping plane for all beet. This is equivalent to always cutting at a distance x/D, = 0.459 below the crown. A simple compensating technique of this nature does not, however, eliminate topping tares because of variation in the position of the topping plane; tares range from 1.3% to 2.5% for overtopping and 1.8% to 3.4% for undertopping as a,/D, varies between 0.05 and 0.10.
M.
J.
61
O’DOGHERTY
0
0
2
4
6
UndertoppIng
8
IO
12
CT, ), %
Fig. 6. Relationship between overtopping (TO) and undertopping (T.) for a range of standard deviations, a, qf cutting plane position about the correct topping plane position, expressed as a ratio of beet diameter at soil level, D,. (Beet crop size distribution Q)
6. Errors in crown sensing The effect of errors in sensing of the top of the beet crown by the feeler wheel of the topping unit was assessed, assuming that they were random and followed a normal distribution. If the error variance in sensing by the feeler wheel is oi, the total variance g. For a given mean position of the about the mean topping position is given by 0’ = gp + CJ cutting plane, the under- and overtopping was calculated for a series of cutting planes on either side of the mean, in proportion to the frequency of occurrence of such a positional error with a normal distribution. The total standard deviations in topping the beet (a/D,) are 0.072, 0.084, 0.101 and 0.120 for values of a,/D, of 0.025, 0.05, 0.075 and 0.10 respectively,assuming that a,/DS = 0.067. Fig. 6showscurvesforcropdistributionQforvaluesof a/D, ranging from 0 to 0.12. The curve a/DS = 0.067 is the lower bound for beet for which aTIDS = 0.067 and that for a/OS = 0.12 represents large sensing errors (as/Ds = 0.10). Improvements in sensing accuracy will move the topping characteristic curves towards that for a/DS = 0.067. The difference between perfect and poor sensing represents an increase in balanced tares from 3.6% to 4.7%. The improvement is greater for topping away from the balanced condition. If overtopping was controlled at 2%, for example, improvement in sensing represents a potential reduction in undertopping from 7.4% to 52%. 7. Comparison with field measurements The predictions of the model have been validated in relatively few experiments in which the variation in the balance of topping was examined over a range of topper knife settings which was sufficient to establish the relationship between the tares. Experiments conducted by Wayman and Wayman and Maughan5 show a good measure of agreement with the model values, with balanced topping tares in a range of approximately 304 to 6%. Earlier work’-’ in 1965-67 exhibited rather higher tares, however, with balanced topping in the range 6.5% to 7.5%. Some work with an early type of feeler wheel topper lo
62
A GEOMETRICAL
0
2
4
6
UndertoppIng
8
MODEL
IO
12
OF SUGAR
BEET
TOPPING
14
CT,),%
Fig. 7. Typical characteristic curves of topper performance at 7.2 km/h relating percentage overtopping trailing arm (TJ and percentage undertopping (T,) from jield experiments of Wayman. (-, lightweight unit; - - - , lightweight unit mounted on parallel linkage)
gave results which compare well with more recent work and with the model. Fig. 7 shows some typical results given by Wayman” from other work which have the form of topping characteristic predicted by the model. 8. Discussion Calculations for single beet showed the degree of precision necessary when cutting off the crown. To achieve an accuracy of 5% over- and undertopping, the cut should be within 5.X to 6.9 mm of the correct topping plane (assuming the mean position) and an average size of beet (D, = 92 mm). Greater precision is necessary for small beet; for example, an error of only 2.5 mm can result in 4% overtopping and 3.5% undertopping on a beet of diameter D, = 50 mm. An important finding from the model was that for a fixed knife position it is not possible to attain perfect topping in a practical crop size distribution. Physical constraints result in a relationship between under- and overtopping and a decrease in one is necessarily accompanied by an increase in the other. The form of the relationship sets a bound for topping performance. Even if a mechanism is used which compensates for beet size, so that all beet are cut at their mean topping plane position, there will be both under- and overtopping because of the variation in topping plane position. Perfect topping can only be achieved for a variable topping plane position if the correct topping plane is sensed for individual beet. For disparate size distributions, if individual beet crowns are accurately sensed, the differences between topping tares are relatively small. The effect of errors in crown sensing is to increase the overall variance in the position of the cutting plane in relation to the correct topping plane, with an increase in the topping tares. The model shows the benefit to be obtained from improvements in the sensing accuracy. For example, the balanced tare for perfect sensing for size distribution Q and oJD, = 0.067 is 3*6%, compared with 4.7% for inaccurate sensing (o/D, = 0.10). The model emphasizes the important effect that beet variability has on the topping accuracy and
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63
O’DOGHERTY
predicts that it is probably not possible to achieve a balanced tare of less than 3% to 4%, depending on individual beet variability and crop size distribution. The relationship between under- and overtopping obtained from the model is generally in good agreement with results from field measurements of topping tares. The measure of agreement lends support to the approach and indicates that although the model is a simple one, the physical concept is correct. In some experimental work the tares observed were higher than those predicted, which is to be expected in view of the simplifications made. Larger tares arise in practice because of gross errors in sensing because of geometrical limitations, particularly where a relatively small beet follows a large beet. In addition, there are errors due to departures of beet crown shape from the idealized shape and inclination of the correct topping plane to the horizontal. The topper may push the beet out of vertical alignment so that the cut is inaccurately made and, in some cases, the forces on the crown are such that it is not cut cleanly, resulting in random fracturing which increases one or other of the tares. The dynamic behaviour of the unit as it passes over the beet stand also affects the sensing of the crown and hence the position at which the knife enters it. 9. Conclusions A geometrical model was developed to study the relationship between over- and undertopping of sugar beet crowns. The model enabled studies to be made of the effect of root size distribution, variance in topping plane position and errors in crown sensing. The most important finding was that there is a functional relationship between under- and overtopping which is determined by the position of the cutting plane and is dependent on the size distribution of the crop and the variance of the position of the correct topping plane for a particular beet size. A balance between over- and undertopping must be accepted for a fixed knife setting, the particular values of which depend on crop variability. Perfect topping cannot be attained because of geometrical limitations, unless each root is sensed individually for its correct topping level and the information used to position a cutting element at the appropriate level. If a mechanism is used which compensates for beet size, the variance of topping plane position will result in both over- and undertopping. The model showed that for significant differences in crop size distribution the difference between balanced tares was only about 0.6x, but the knife setting required differed by 10 mm. The minimum balanced topping tare likely to be achieved is in the range 3% to 4%. The total error between the cutting plane and the correct topping plane is compounded of variation in topping plane position and errors in crown sensing. Balanced topping increases from 3.6% to 4.7% due to sensing errors from a particular size distribution. The effect on undertopping is, however, more significant; for 2% overtopping, the undertopping increased from 5.2% to 7.4%. The model highlights the accuracy required to achieve good topping. The cutting plane must be within approximately 6 to 7 mm of the correct topping plane to achieve an accuracy of 5% for an average size of beet. An error of only 2.5 mm can result in topping errors of about 2%. The geometrical model was simple in concept but showed good agreement with observations from field experiments. It enables useful deductions to be made in relation to the limits of topping which can be achieved in practice. In general, the values of over- and undertopping obtained in the field would be expected to be greater than those predicted by the model. References ’ O’Dogherty, M. J. The mechanics of a sugar beet topping mechanism. Ph.D. thesis, University of Reading, 1976
64
A GEOMETRICAL
MODEL
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SUGAR
BEET
TOPPING
M. J. The design of feeler wheel toppers for sugar beet: 1. Topper dynamics. Journal of Agricultural Engineering Research 1986, 34: 305-3 18 3 Wayman, J. A. An investigation into the variations in sugar beet root and crown dimensions with regard to the design of mechanical topping units, 1970. Departmental Note DN/R/l41/1600, National Institute of Agricultural Engineering, Silsoe, 1971 4 Wayman, J. A. The effects of topper unit modifications, plant distribution and harvester speed on sugar beet top tares, 1969. Departmental Note DN/R/028/1600. National Institute of Agricultural Engineering, Silsoe, 1970 5 Wayman, J. A.; Maughan, G. L. The effects of harvester speed and plant distribution on sugar beet topping, 1968. Departmental Note DN/256/RD, National Institute of Agricultural Engineering, Silsoe, 1969 6 Wayman, J. A.; Maughan, G. L. The effect of harvesting speed on the standard of topping of sugar beet, 1965-67. I.I.R.B. Journal 1969, 4(2): 104-l 11 ’ Wayman, J. A.; Maughan, G. L. An investigation into the effects of speed and plant distribution on mechanical topping of sugar beet, 1965. Departmental Note DN/214/CD, National Institute of Agricultural Engineering, Silsoe, 1966 s Wayman, J. A.; Maughan, G. L. An investigation into the effects of speed and plant distribution on mechanical topping of sugar beet, 1966. Departmental Note DN/233/CD, National Institute of Agricultural Engineering, Silsoe, 1967 g Wayman, J. A.; Maughan, G. L. Mechanical topping of sugar beet: The effects of harvester speed and plant distribution, 1965-67. Departmental Note DN/243/CD, National Institute of Agricultural Engineering, Silsoe, 1968 lo Dunn, J. S.; Maughan, G. L. Investigation into the topping of a mechanically thinned crop of sugar beet, Wrest Park, 1954. Departmental Note DN/85/ATD, National Institute of Agricultural Engineering, Silsoe, 1955 l1 Wayman, J. A. Field trials of design variations of the prototype NIAE lightweight sugar beet topper, 1974. Departmental Note DN/R/632/1600, National Institute of Agricultural Engineering, Silsoe, 1976 * O’hgherty,
Appendix Method
of calculation of over- and under-topping for normal distributions crop size and topping plane position
of
A.1. Crop size Over- and undertopping were calculated for a series of distances (x) of the topping plane below the top of the beet crown. The procedure used for a given size distribution of roots was first to obtain the percentage of roots for class intervals of the graded diameter, D,, of 10 mm, for classes ranging from 45 to 55 mm up to 155 to 165 mm. The soil diameter D,( = 0.8970,) was then found for each of the class interval means. The value of x/D, was calculated for a series of values of x in the range 20.6 to 65.9 mm, where the extremes of the range correspond to the correct topping planes for the smallest and largest class interval means, for which D, = 44.8 and 143.5 mm, respectively. The percentage topping errors T, and T, were then obtained graphically and the corresponding volumes u, and v, obtained from the correctly topped beet volumes for each size grade mean diameter, calculated from the expression V, = 0.8380:. These volumes are given by: (v,), = $$
x VT),
(Al)
(V”), = $$
x VT),
(A2)
and
where the subscript distribution.
r denotes
that the parameters
refer to a class interval of the size
M.
J.
65
O’DOGHERTY
These volumes were then multiplied by the percentage number of roots, P,, in each class interval to give the appropriate weighting for the crop size distribution under consideration. These volumes were then summed over the whole size distribution for the total number of class intervals, n, and the total volumes of under- and overtopped material obtained, given by:
fl ~ruxI/,)r
(A3)
(U”)W= ik ,tl ~,(T,),(W,.
(A4)
tdv = iii5 and
The subscript w denotes the total volume of material, weighted appropriately for crop size distribution. A similar procedure was followed for the correctly topped root volumes, which were multiplied by the percentage of roots in the class interval, and the volumes summed to obtain the weighted volume for the crop size distribution, given by:
The over- and undertopping T, and T, was then calculated for the particular value of by expressing the sum of the over- and undertopped volumes as a percentage of the sum of the correctly topped volumes, so that:
H -h( = 0.4590,)
i P,(T,),(I/,), (T,), = ‘= l”
(A6)
and
(A7)
A.2. Topping plane position Over- and undertopping were calculated on the basis of the idealized beet shape, assuming that the distance of the topping plane below the top of the crown was normally distributed about its mean. The calculations were made for a series of distances (x) of the cutting plane below the top of the crown. The distribution was divided into class intervals of x/D, = 0.05, and the percentage of roots which would have topping planes situated in this interval were found from the normal probability integral for values of a,/D, = O-067 and O-095. The volumes between the cutting plane, located at x below the top of the crown, and the midplanes of each of the class intervals of the topping plane position were calculated from differences of the appropriate crown volumes [see Eqn (Al)] denoted by: (u,), and (US, where the subscript s denotes that the parameters refer to a class interval of the topping plane distribution. The volumes were then multiplied by the percentage of roots by number, P,, which would have topping planes located in the appropriate class interval, and the result summed over the total number of class intervals, n, to give the total amount of over- and undertopped material. i.e.
66
A GEOMETRICAL
MODEL
OF
SUGAR
BEET
TOPPING
The correctly topped volumes (V,), were then calculated for the mean of each class interval of topping plane distribution, from the known geometry of the idealized beet, and multiplied by the percentage of roots in the class to obtain a weighted correctly topped volume. These volumes were then summed to give the total weighted correctly topped volume, i.e.
To obtain the over- and undertopping To and T, for the idealized shape, the total volumes of over- and undertopped material were expressed as a percentage of the total correctly topped volume, i.e.
and
where the subscript x denotes that the over- and undertopping positional distribution of the topping plane.
are weighted for the