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The Bekker theory of rolling resistance amended to take account of skid and deep sinkage
Journal of Terramechanics, 1978, Vol 15, No 2, pp 107 to 110
0022--4898/78/0601-0107$02.00/0
© Pergamon Press Ltd. Printed in Great Britain.
DISCUSSION T H E B E K K E R T H E O R Y O F R O L L I N G RESISTANCE A M E N D E D TO T A K E A C C O U N T O F SKID A N D DEEP S I N K A G E D. GEE°CLOUGH(WoE 13, pp. 87-105, 1976). A U T H O R ' S REPLY I WlSH to thank Karafiath and Nowatzki for their comments on my paper. They state that the rolling resistance of rigid wheels is basically a problem of soil bearing capacity which may be solved by plasticity methods. This is perfectly true and, using these methods, one would expect quite accurate solutions to the problem. Unfortunately these methods do not allow closed form solutions to the problem [2] and numerical techniques using digital computers have to be employed. This makes the plasticity method virtually useless from a practical point of view. As an example: if we are designing a new off-road wheeled vehicle and we wish to make fundamental decisions on the size and configuration of the wheels, we have a very large number of options open to us. Couple this to the widely differing soil conditions in which most off-road vehicles are expected to operate and it is immediately obvious that the "oneoff" computer solutions using plasticity methods are impractical. Methods which give closed form solutions are still the only practical way of approaching the problem, even though they may give answers which are less accurate than those found using formal plasticity methods. They claim that no definition of the parameters k and n as used in the expression (~)~orm*l = k N n is given. This is because I did not think one was necessary. However, if it will clarify matters, I have used a model which postulates that soil, in compression, will behave in a similar manner to a non-linear spring with a spring constant k and an exponent of displacement n. How these parameters are to be measured is still a matter for debate, the method usually adopted being to measure them from experiments in which plates of the same width as the wheel are pushed vertically into the soil. This was the method employed by Wills et al. [3] in the experiments which I used to check my theory and it will be remembered that my theory gave good agreement with their measured results. The necessity for using plates of the same width as the wheel is because the constant k is found to be a function of width, one which unfortunately is not properly described by the Bekker equation k .... k J b + ko4. If we had to measure a separate value of k for every wheel width we wished to investigate in any given soil this would make my method impractical. However, it has become apparent from exp.~riments which I have recently concluded, the results of which I hope to publish shortly, that it may be possible to measure k and n from experiments with a narrow rigid wheel. The forces calculated from these k and n values may then be corrected for any given wheel width in a very simple manner. *L. L. Karafiath and E. A. Nowatzki, 'Dis:us;ion of "The Bekker theory of rolling resistance amended to t a k e account of skid an:l d.-::3 sin~ag~" "J Terramechanics 14, 183-187 (1977). 107
108
DISCUSSION
Karafiath and Nowatzki state that the cumulative normal (N) and tangential (T) displacements calculated from my equations (6) and (7) have no physical meaning since the displacements are not in a straight line. This I find very puzzling since, of course, the statement is quite untrue. N and T are total displacements, at any angle of rotation of the wheel, of points on the periphery of the wheel along two specific, curved paths [1] (see Fig. 5 in Gee-Clough and Chancellor [5]). The overall displacement of a point on the periphery of the wheel may be found by adding vectorially the total normal and tangential displacements. I hesitate to state their case for them but I believe the point they were trying to make is the following. Consider a point on the periphery of the wheel, in contact with the soil, at an angular displacement 0-d0. Rotate the wheel a further small angular displacement dO and the normal force will change by an amount AFnl acting in a direction along AB in Fig. 1. Rotate the wheel another increment dO and the normal force
c FIG. 1. Vector addition of normal force.
changes by AFn2 acting along BC. In my calculation of normal forces I have used the arithmetic sum of AB and BC to calculate the change of normal force. In fact it should be the vector sum, i.e. the length AC. This is a valid criticism. The approximation I have used causes only small errors for small rotations. However as the angle of rotation increases and approaches 90 ° the errors become larger. This approximation is in fact the reason why the stresses calculated using my equations (8) and (9) have the form shown in Fig. 3B of Karafiath and Nowatzki's discussion paper. The reason why I have used this approximation is that I have not yet found a simple way of allowing for this vector rotation in my analysis. When I find one I shall incorporate it. The effect of allowing for this vector rotation will, in fact, increase still further the prediction accuracy of my method. As an example of the accuracy of prediction of their method, Karafiath and Nowatzki use data by Onafeko and Reece [6] for a 49 in. dia. x 12 in. rigid wheel on loose sand (Karafiath and Nowatzki, Fig. 2). It seems appropriate therefore to use the same data to examine the accuracy of my own method. Using equation (21) to calculate the sinkage for the given wheel load of 2085 lb, it will be found that a sinkage of 6.4 in. is predicted. This may be compared to the measured value of 6.2 in. The coefficient of rolling resistance calculated using my method is 0.388 and that measured by Onafeko and Reece is 0.417.
DISCUSSION
109
I am surprised that Karaiiath and Nowatzki argue against the proposition that if there is no applied moment about the axle of a towed wheel then there can be no moment due to soil stresses about the axle. This of course is an obvious proposition and, if proof were needed, can be shown to be true by measured values of the shear stresses at the interface of towed rigid wheels (e.g. Onafeko and Reece [6] from which Karafiath and Nowatzki's Fig. 2 is taken). The summation of internal and external forces and moments acting on the system must be zero and since there is no external moment about the axle there can be no moment due to internal forces. There is no inertia force acting on the wheel if it is travelling at constant velocity and if soil inertia is taken into account this will merely change the stress pattern but will not remove the requirement that the moment about the axle must be zero. The comment about wheel bearing friction being responsible for an external moment about the axle is completely misleading since there is no evidence that they have taken this into account in calculating their soil stresses. In any case wheel bearing friction is usually so small that it can be neglected. The computed shear stress distribution shown in their Fig. 2 cannot possibly apply to a free rolling towed wheel since there is obviously a decelerating moment about the axle caused by the unbalanced shear stress distribution. The skid values predicted by my equations (26)-(30) are found by calculating the moment about the axle due to peripheral shear stresses and equating it to zero as it must be. For the two extreme cases for which integration of the equations is possible it so happens that skid can be predicted knowing only the eagagement angle 0e. My use of the Janosi--Hanomoto equation in calculating these shear stresses is fundamentally the same as the method used by Wong and Reece [7]. Finally there is the question as to why I did not use the skid values given in Wills et al. [3]. The reason is that, when I was searching for data to check theoretical ×
/
0"3
./
m
0"Z
r,
/,
0'l
/
/
- -
Neasurtd
/ l o.i
l
0-2 Htosured
FIG. 2.
skid
I 0'3
velu¢
Measured and calculated values of skid.
-
CelGeloted
110
DISCUSSION
predictions against, I did not wish to use inaccurate values scaled from graphs in a Journal. I therefore obtained a copy of the M.Sc. thesis o f F . M. Barrett [8] from which the above mentioned paper was written and in which the measured forces and sinkages are tabulated. I was surprised to find no record of any direct measurements of skid. Rather than use the values given in Wills et al., in which skid is merely indicated along an irregular horizontal axis and whose origin I was unsure of, I left the question as to the accuracy of skid prediction open. Figure 2 shows measured values against calculated values of skid from experiments which I have recently concluded with rigid wheels in dry sand. I wish once more to thank Karafiath and Nowatzki for their comments and hope that, in some small way, this exchange of views can help re-stimulate a healthy debate on terramechanics which, it seems to me, has been missing for the last few years DAVID GEE-CLOUGH Principal Scientific Officer National Institute of Agricultural Engineering Wrest Park Silsoe Bedford England REFERENCES [1] D. GEE-CLOUGH,The Bckker theory of rolling resistance amended to take account of skid arm deep sinkage. J. Terramechanics 13, 87 (1976). [2] E. A. NOWATZKIand L. L. KARAFIATH,General yield conditions in a plasticity analysis of soil wheel interaction. J. Terramechanics 11, 29 (1974). [3] B. M. D. WILLS,F. M. BARRETTand G. J. SHAW, An investigation into rolling resistance theories for towed rigid wheels. J. Terramechanics 2, 24 0965). [4] M. G. BEKKER,Off-the-Road Locomotion. University of Michigan Press, Ann Arbor (1960). [5] D. GEE-CLOUGH and W. J. CHANCELLOR,Pull and lift characteristics of single lugs on rigid wheels in wet rice soils. Trans. A S A E 19, 433 (1976). [6] O. ONAFEKOand A. R. REECE,Soil stress and deformations I~eneath rigid wheels. J. Terramechanics 4, 59 (1967). [7] Jo-YUNG WONG and A. R. REECE, Prediction of rigid wheel performance based on the analysis of soil-wheel stresses. J. Terramechanics 4, 7 (1967). [8l F. M. BARRETT, An Investigation into Theories of Rolling Resistance of Towed Rigid Wheels. M.Sc. Thesis, Univ. Durham (England), 1964 (unoublished).
0022--4898/78/0601-0107$02.00/0
© Pergamon Press Ltd. Printed in Great Britain.
DISCUSSION T H E B E K K E R T H E O R Y O F R O L L I N G RESISTANCE A M E N D E D TO T A K E A C C O U N T O F SKID A N D DEEP S I N K A G E D. GEE°CLOUGH(WoE 13, pp. 87-105, 1976). A U T H O R ' S REPLY I WlSH to thank Karafiath and Nowatzki for their comments on my paper. They state that the rolling resistance of rigid wheels is basically a problem of soil bearing capacity which may be solved by plasticity methods. This is perfectly true and, using these methods, one would expect quite accurate solutions to the problem. Unfortunately these methods do not allow closed form solutions to the problem [2] and numerical techniques using digital computers have to be employed. This makes the plasticity method virtually useless from a practical point of view. As an example: if we are designing a new off-road wheeled vehicle and we wish to make fundamental decisions on the size and configuration of the wheels, we have a very large number of options open to us. Couple this to the widely differing soil conditions in which most off-road vehicles are expected to operate and it is immediately obvious that the "oneoff" computer solutions using plasticity methods are impractical. Methods which give closed form solutions are still the only practical way of approaching the problem, even though they may give answers which are less accurate than those found using formal plasticity methods. They claim that no definition of the parameters k and n as used in the expression (~)~orm*l = k N n is given. This is because I did not think one was necessary. However, if it will clarify matters, I have used a model which postulates that soil, in compression, will behave in a similar manner to a non-linear spring with a spring constant k and an exponent of displacement n. How these parameters are to be measured is still a matter for debate, the method usually adopted being to measure them from experiments in which plates of the same width as the wheel are pushed vertically into the soil. This was the method employed by Wills et al. [3] in the experiments which I used to check my theory and it will be remembered that my theory gave good agreement with their measured results. The necessity for using plates of the same width as the wheel is because the constant k is found to be a function of width, one which unfortunately is not properly described by the Bekker equation k .... k J b + ko4. If we had to measure a separate value of k for every wheel width we wished to investigate in any given soil this would make my method impractical. However, it has become apparent from exp.~riments which I have recently concluded, the results of which I hope to publish shortly, that it may be possible to measure k and n from experiments with a narrow rigid wheel. The forces calculated from these k and n values may then be corrected for any given wheel width in a very simple manner. *L. L. Karafiath and E. A. Nowatzki, 'Dis:us;ion of "The Bekker theory of rolling resistance amended to t a k e account of skid an:l d.-::3 sin~ag~" "J Terramechanics 14, 183-187 (1977). 107
108
DISCUSSION
Karafiath and Nowatzki state that the cumulative normal (N) and tangential (T) displacements calculated from my equations (6) and (7) have no physical meaning since the displacements are not in a straight line. This I find very puzzling since, of course, the statement is quite untrue. N and T are total displacements, at any angle of rotation of the wheel, of points on the periphery of the wheel along two specific, curved paths [1] (see Fig. 5 in Gee-Clough and Chancellor [5]). The overall displacement of a point on the periphery of the wheel may be found by adding vectorially the total normal and tangential displacements. I hesitate to state their case for them but I believe the point they were trying to make is the following. Consider a point on the periphery of the wheel, in contact with the soil, at an angular displacement 0-d0. Rotate the wheel a further small angular displacement dO and the normal force will change by an amount AFnl acting in a direction along AB in Fig. 1. Rotate the wheel another increment dO and the normal force
c FIG. 1. Vector addition of normal force.
changes by AFn2 acting along BC. In my calculation of normal forces I have used the arithmetic sum of AB and BC to calculate the change of normal force. In fact it should be the vector sum, i.e. the length AC. This is a valid criticism. The approximation I have used causes only small errors for small rotations. However as the angle of rotation increases and approaches 90 ° the errors become larger. This approximation is in fact the reason why the stresses calculated using my equations (8) and (9) have the form shown in Fig. 3B of Karafiath and Nowatzki's discussion paper. The reason why I have used this approximation is that I have not yet found a simple way of allowing for this vector rotation in my analysis. When I find one I shall incorporate it. The effect of allowing for this vector rotation will, in fact, increase still further the prediction accuracy of my method. As an example of the accuracy of prediction of their method, Karafiath and Nowatzki use data by Onafeko and Reece [6] for a 49 in. dia. x 12 in. rigid wheel on loose sand (Karafiath and Nowatzki, Fig. 2). It seems appropriate therefore to use the same data to examine the accuracy of my own method. Using equation (21) to calculate the sinkage for the given wheel load of 2085 lb, it will be found that a sinkage of 6.4 in. is predicted. This may be compared to the measured value of 6.2 in. The coefficient of rolling resistance calculated using my method is 0.388 and that measured by Onafeko and Reece is 0.417.
DISCUSSION
109
I am surprised that Karaiiath and Nowatzki argue against the proposition that if there is no applied moment about the axle of a towed wheel then there can be no moment due to soil stresses about the axle. This of course is an obvious proposition and, if proof were needed, can be shown to be true by measured values of the shear stresses at the interface of towed rigid wheels (e.g. Onafeko and Reece [6] from which Karafiath and Nowatzki's Fig. 2 is taken). The summation of internal and external forces and moments acting on the system must be zero and since there is no external moment about the axle there can be no moment due to internal forces. There is no inertia force acting on the wheel if it is travelling at constant velocity and if soil inertia is taken into account this will merely change the stress pattern but will not remove the requirement that the moment about the axle must be zero. The comment about wheel bearing friction being responsible for an external moment about the axle is completely misleading since there is no evidence that they have taken this into account in calculating their soil stresses. In any case wheel bearing friction is usually so small that it can be neglected. The computed shear stress distribution shown in their Fig. 2 cannot possibly apply to a free rolling towed wheel since there is obviously a decelerating moment about the axle caused by the unbalanced shear stress distribution. The skid values predicted by my equations (26)-(30) are found by calculating the moment about the axle due to peripheral shear stresses and equating it to zero as it must be. For the two extreme cases for which integration of the equations is possible it so happens that skid can be predicted knowing only the eagagement angle 0e. My use of the Janosi--Hanomoto equation in calculating these shear stresses is fundamentally the same as the method used by Wong and Reece [7]. Finally there is the question as to why I did not use the skid values given in Wills et al. [3]. The reason is that, when I was searching for data to check theoretical ×
/
0"3
./
m
0"Z
r,
/,
0'l
/
/
- -
Neasurtd
/ l o.i
l
0-2 Htosured
FIG. 2.
skid
I 0'3
velu¢
Measured and calculated values of skid.
-
CelGeloted
110
DISCUSSION
predictions against, I did not wish to use inaccurate values scaled from graphs in a Journal. I therefore obtained a copy of the M.Sc. thesis o f F . M. Barrett [8] from which the above mentioned paper was written and in which the measured forces and sinkages are tabulated. I was surprised to find no record of any direct measurements of skid. Rather than use the values given in Wills et al., in which skid is merely indicated along an irregular horizontal axis and whose origin I was unsure of, I left the question as to the accuracy of skid prediction open. Figure 2 shows measured values against calculated values of skid from experiments which I have recently concluded with rigid wheels in dry sand. I wish once more to thank Karafiath and Nowatzki for their comments and hope that, in some small way, this exchange of views can help re-stimulate a healthy debate on terramechanics which, it seems to me, has been missing for the last few years DAVID GEE-CLOUGH Principal Scientific Officer National Institute of Agricultural Engineering Wrest Park Silsoe Bedford England REFERENCES [1] D. GEE-CLOUGH,The Bckker theory of rolling resistance amended to take account of skid arm deep sinkage. J. Terramechanics 13, 87 (1976). [2] E. A. NOWATZKIand L. L. KARAFIATH,General yield conditions in a plasticity analysis of soil wheel interaction. J. Terramechanics 11, 29 (1974). [3] B. M. D. WILLS,F. M. BARRETTand G. J. SHAW, An investigation into rolling resistance theories for towed rigid wheels. J. Terramechanics 2, 24 0965). [4] M. G. BEKKER,Off-the-Road Locomotion. University of Michigan Press, Ann Arbor (1960). [5] D. GEE-CLOUGH and W. J. CHANCELLOR,Pull and lift characteristics of single lugs on rigid wheels in wet rice soils. Trans. A S A E 19, 433 (1976). [6] O. ONAFEKOand A. R. REECE,Soil stress and deformations I~eneath rigid wheels. J. Terramechanics 4, 59 (1967). [7] Jo-YUNG WONG and A. R. REECE, Prediction of rigid wheel performance based on the analysis of soil-wheel stresses. J. Terramechanics 4, 7 (1967). [8l F. M. BARRETT, An Investigation into Theories of Rolling Resistance of Towed Rigid Wheels. M.Sc. Thesis, Univ. Durham (England), 1964 (unoublished).