- Email: [email protected]
Mechanics of wheels on soft soils— A method for presenting test results
Journal of Terramechanics, 1966,Vol. 3, No. 1, pp. 13 to 22. Pergamon Press Ltd.
Printed in Great Britain.
MECHANICS OF WHEELS ON SOFT SOILS-A METHOD FOR PRESENTING TEST RESULTS ETIENNE LEFLAIVE* INTRODUCTION RESEARCH in the field of vehicle mobility covers a wide range of subjects, from the physiological effects of vehicle vibrations on hmnan beings to the differentiation of various types of snow that vehicles may be required to cross. In their quest to find a vehicle that will go "anywhere." military mobility researchers have examined vehicles with oblate and hemispherical wheels, with tracks made of rubber-covered air chambers in many forms and with neither wheels nor tracks, i.e. walking vehicles, air-cushioned vehicles and vehicles propelled by Archimedean screws. Researchers in vehicle mobility for such civil purposes as transportation, earthmoving, agriculture and forestry are vitally concerned with the challenging economics aspects of the problem. Determination of the effects of the environment on the movement of vehicles leads to such absorbing problems as how the arrangement of trees in a forest affects the ability of a vehicle to move through them, or-how much resistance to vehicle movement is provided by dense stands of small trees that must be overriden. In contrast to this variety of interesting topics, the subject of this paper is the commonplace mechanics of a single, simple wheel on a soil condition prepared to be as uncomplicated as possible. The objective of the study of wheel mechanics in soft soil, besides the intellectual satisfaction of understanding a physical phenomenon, is to predict what will happen to a wheel in given conditions. Assuming that all the necessary characteristics of the wheel and the soil are known, the problem is to determine the relations among the load on the wheel, the torque, the pull that the wheel can develop, and the soil conditions. Two other quantities are of interest as well. Slip is important with respect to efficiency because for a given number of revolutions of the wheel, it reduces the distance over which the pull does work. Sinkage should be known, for it must remain smaller than the clearance of the vehicle. A first impulse is to attempt to solve the problem by determining the pull, the torque and possibly the slip and the sinkage by means of a theoretical analysis using the relations between stress and strain for the soil and for the wheel, the boundary conditions, and the conditions of equilibrium. While a theoretical study with very simple assumptions may indeed be helpful as a guide for the interpretation of test results, from our present knowledge an accurate description of the effects of variables, such as inflation pressure, tyre shape, tyre flexibility, treads, etc., cannot be expected because the analytical difficulties of a theoretical approach are considerable. Furthermore, many of the necessary physical laws for a correct analysis are not in hand. *Mobility and Environmental Division, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Mississipi. Communicated by D. R. Freitag.
13
14
ETIENNE LEFLA1VE
One of the main difficulties is that the differences in behaviour of various wheels or tyres probably are due to differences in the flow patterns of the soil under the moving wheels. As such a flow pattern is definitely three-dimensional and obviously has no axial symmetry, the problem is very complex. Another difficulty involves the complexity of the mechanical properties of the pneumatic tyres. However, even more important is the fact that stress-strain relations for soils are not known. Only stress displacement curves are available in given experimental conditions. This means that, generally, the stress displacement curves obtained in shear tests cannot be extrapolated or applied to other conditions because it is not known how the strains are distributed in the soil sample. In other words, it is known that stresses and strains are related, but not h o w - - i n spite of considerable research effort. This point is emphasized because such stress-strain relations are necessary to take into account the effect of slip in a theory of rolling motion. A given slip corresponds to a certain displacement between the wheel and the soil. The effect of this displacement, in terms of stresses, as compared with the result of a shear test, poses a question for which no sound answer exists. It could be assumed, as in civil engineering soil mechanics, that the displacement is large enough to mobilize the full friction of the soil and to allow application of Coulomb's law (which does not consider deformation quantitatively). However, this assumption would not provide a basis for description of the variation of pull and torque with slip. It would also be necessary to decide how to determine the angle of friction, and this again presents a problem. A final difficulty in the path of a theoretical approach is the extensive and rapid soil deformation that can occur under a moving wheel. Soil properties in such conditions may be different from those under slowly applied loads. Unexpected soil behaviour in the domain of large and rapid strain perhaps could explain some experimental results otherwise presently unexplainable. This discussion may be summarized by saying that not enough is known about mechanical properties of soils for a satisfactory theoretical treatment of the wheel problem. Knowledge of soil mechanics is a necessary guide for a physical understanding of the wheel behaviour, but it cannot lead to a direct theoretical solution. Obviously, the alternative to theorizing is to experiment with wheels in soft soils. Experiments may be conducted with actual vehicles or with single wheels, rigid or pneumatic tyred. Various kinds of soils may also be used--soils in their natural state, reprocessed soils, or artificial soils. Actual vehicles are used for practical reasons, but they are not convenient for close control and measurement of the variables. Rigid wheels and artificial soils are employed for simplification, but they must be used with care because all the consequences of these simplifications are not known. Their main role should be to investigate specific questions raised in the interpretation of tests conducted with pneumatic-tyred wheels on natural or reprocessed soils. The foregoing general statements contain nothing new. As a matter of fact, the Waterways Experiment Station (WES) fully recognized several years ago the necessity for tests with pneumatic tyres on natural soils. As no theory was assumed beforehand, the tests essentially were of an exploratory nature. A large number
MECHANICS OF WHEEL~ oN SOFT SOILS
15
of tests have been conducted, and a great quantity of carefully measured data is available [1, 2]. The point that this paper attempts to emphasize concerns the next step in research, following the achievement of such exploratory experiments; namely, the drawing of conclusions from the test results. Confusion arises from the number of parameters and variables playing a part in the phenomenon. This results in a great difficulty in obtaining a clear and general view of what the test results, in fact, are. In other words, the problem is to decide in which terms the results should be expressed to provide a good description of the experimental facts. If a clear view of what occurs can be obtained, then it becomes feasible to develop theories to explain the results. A good description of experimental results on rolling motion should be, as much as possible : (a) Complete; that is, it should include the various aspects (force, work, efficiency, etc.) and various phases (towed, self-propelled, maximum pull) of rolling motion (b) Concise (c) Easy to visualize (d) A good model of the physical processes (e) Capable of being expressed in a relatively simple analytical form. The last requirement suggests one of the consequences of establishing a framework for the observed facts; it naturally leads to an empirical system for dealing with the problem, such as graphical representations, empirical parameters, etc. This is useful and is not necessarily opposed to physical understanding and theoretical treatment, particularly if the fourth-mentioned requirement is fulfilled, because a good model contributes to the development of meaningful empirical parameters. It is the author's hope that this long introduction places the method of representation of test results given below within the framework of the wheel-soil mechanics research. BASIC CONCEPTS Graphical representations will be used to present the results of the tests and to describe the principles of the analysis technique. Slip has been used as the reference variable. This is customary, for slip is a primary physical process in rolling motion on soft soils. Sinkage, an alternative reference variable, is difficult to determine, and furthermore, a part of it is due to slip, while the remaining portion corresponds to bearing capacity. Since one value of sinkage may be composed of components from these two different sources in different proportions, sinkage is not appropriate as a basic reference variable. The quantities plotted against slip are energy quantities: torque energy, pull energy and dissipated energy. The torque energy is the work of the torque, either positive or negative; a positive work corresponds to a driving torque. The pull energy is the work done by the pull; it is positive when the wheel is developing a tractive effort and negative when the wheel is towed. The dissipated energy is the difference between the torque energy and the pull energy; it is always positive.
16
ETIENNE LEFLA1VE
The torque energy and the pull energy are determined from the test data. Then, the variations of the torque energy and the dissipated energy are studied in terms of the variables controlling the wheel behaviour. If the torque energy and the dissipated energy can be predicted through an understanding of the physical processes, the pull can be determined as their difference. One purpose of this approach was to differentiate the phenomena playing a part in rolling motion. Indeed, the reason for studying the energy input (torque) on one hand and the energy dissipation on the other hand is that it is useful to consider these two processes as independent, from the physical viewpoint. Evidently, both of these factors are influenced by the basic parameters controlling rolling motion and by secondary phenomena occurring when the wheel is in motion, such as the effect of sinkage on geometric conditions, variation of soil density due to shear, etc. However, it is proposed to regard them as two separate phenomena, and the production of pull as the result of their superposition. Another reason for considering energy quantities is their suitability for dealing with efficiency. Although the problem of efficiency for a wheel is not discussed herein, an efficiency parameter is defined. This quantity is a function of the energy quantities mentioned above and, likewise, has been plotted against slip. DEFINITIONS The parameters representing the tyre condition basically are those employed at the WES. Instead of inflation pressure, the rigidity of the tyre is expressed by the relative deflection of the loaded tyre on a hard surface. A rolling radius, R, at a given deflection, 8, and a given load, W, is defined as the distance the wheel travels in one revolution on a hard surface under these conditions, divided by 27r. Pull is denoted by the symbol F and the torque by M. Soil strength is represented by the cone index. It indicates whether the soil is dense or loose, firm or soft. Such an index is sufficient, at least, for the purpose of the present development, which is to build a framework to present test results, and not for an analysis of the soil behaviour under the wheel action. Two different coefficients to express slip are defined for use in this paper. One called normal slip is : s=
theoretical distance-actual distance travelled in a given time theoretical distance '
which is a conventional arrangement of the relevant parameters. The other coefficient has been termed differential slip and is defined as: g=
theoretical distance-actual distance actual distance
Either will be used in this paper as is most appropriate according to the circumstances. With co=rotational speed of the wheel, in rad/sec ,v = translation speed t----time
MECHANICS OF ~ L ~ I ~ N
SOFT SOILS
17
normal slip, s, and differential slip, g, are : Root - vt Root =1Root - ~ t
g = - • ,t
v
R"~' and
Roo
,o
1;
so that g g+l or $ g=i-s
The torque energy coefficient, "q, is defined as the energy supplied or withdrawn at the wheel axle, per unit of vertical load and per unit of distance travelled by the wheel.
~--- ~v--g- = w--~
=~-k -(l+g)
From the last expression it can be seen that the relation of torque energy coefficient to differential slip is a straight line for a constant torque, load and radius. The pull energy coefficient, ~, is the energy recovered or supplied as a pull, per unit of load and per unit of distance travelled. Fv Wv
F W
is also the ratio pull/load. The dissipated energy coefficient, p, is the energy that is not recovered mechanically, per unit of load and per unit of distance travelled. p=~-k=
Moo Wv
F W
p is the coefficient of rolling resistance, as proposed by Phillips [3]. The previously mentioned efficiency parameter is denoted traction efficiency (it is not within the scope of this paper to discuss this designation). It is defined: k where ~qa is the torque energy required for the wheel to simply propel itself in the soil. ~" is the ratio of the energy available as pull energy to the additional energy input required to produce a pull, beyond the point of self-propulsion. 7/, p, X and I" are dimensionless. REFERENCE CONDITION OF THE RIGID WHEEL ON A HARD SURFACE To help understand the meaning of the parameters just defined and to form a basis upon which rolling motion in general can be studied, it has been found useful
18
ETIENNE LEFLAIVE
to examine the simple case of a rigid wheel rolling and slipping on an unyielding surface. When dealing with actual conditions of pneumatic tyres on soft soil, the theoretical case of the rigid wheel on a hard surface will be taken as a reference to reveal more clearly by comparison the particular features of rolling motion on deformable soils. The case considered is strictly theoretical. A constant coefficient of friction, f, between the wheel and the surface is assumed, and W and R are taken as unity. Figure 1 shows the curves for torque, torque energy coefficient, ~7; pull energy
jj
~
+f ......................... ~M..
o
•
-f
W AND V
BRAKING
DRI VING WHEEL
OPPOSITE WAYS
NONROTA TING WHEEL
Fro. 1. Theoretical diagram for a rigid wheel on a rigid ground with a constant coefficient of friction, f. coefficient, h; and dissipated energy coefficient, p, vs. differential slip, g. The torque, M, has the limiting value + J for any positive differential slip and - f for any negative differential slip. For zero slip, M may have any value between + f and - f . The torque energy coefficient is "0=M (1 +g). Thus, for g :> 0, 77= + J ( l +g); for g = 0 , r j = M ; and for g < 0 , - 0 = - f ( l + g ) . The point g = - 1 corresponds to a non-rotating wheel (normal slip, s, is infinite); from g = 0 to g = - 1 , the wheel must be braked and energy is withdrawn from the wheel; for g ,< - 1, positive energy must be supplied to rotate the wheel in the direction opposite to its movement. The pull energy coefficient, h, is equal to the torque energy coefficient, ~9, when slip is zero, since at zero slip no energy is dissipated in friction. For positive differential slip, h = + f; for negative differential slip, h = - J . The dissipated energy coefficient, p, is equal to 77- h; it is zero for zero slip. For positive differential slip, p = + f ( l + g ) - f = f g ; for negative differential slip, p=-f(l+g)-(-])=-fg. Considering the variations of 7, /9 and h when slip
MECHANICS OF WHEELS ON SOFT SOILS
19
art
...
$
+1
E f f i c i e n c y ~./'~ f o r a rigid wheel on a rigid ground.
Fzo. 2.
increases, it is observed that the increase in torque energy coefficient is exactly absorbed by the corresponding increase in dissipated energy coefficient, and the pull remains constant. This result may appear obvious for the theoretical reference condition; however, for a wheel rolling on a yielding soil, the torque energy coefficient and the dissipated energy coefficient usually do not vary in such a parallel manner. Figure 2 represents the variation of traction efficiency, ~r, with respect to normal slip, s. As the torque energy coefficient for self-propulsion, ~7o, is zero for the theoretical reference condition, 'r=
~,
n-p
. ~q
.
~/
+-l(]+g)-(+/g) .
]
. . _+/(l+g)
1 --S
l+g
2C le IE
14 12
"r/ ,.0 08 0.6 h" / 04 Toward g=-I point 0-2
%
Load tb ~25
i
;)
~12 ,!.
n
1
,!6 ,!e 2:0 z.2
1
720 ,I
Cone index 57
57
J ,I
I
I+g FIG. 3.
1
2.4 2.6 2.e 3.0 3.2 3.4
Results of tests of smooth pneumatic tyres in dry sand.
20
ETIENNE LEFLAIVE TYPICAL
EXPERIMENTAL
RESULTS
The few results shown in Figs. 3, 4 and 5 refer to tests of smooth pneumatic tyres in dry sand. They are given as examples. The similarities between these curves derived from actual tests of pneumatic-tyred wheels on sand and the reference curves (Figs. 1 and 2) indicate the possibility of a comparison. This comparison is discussed below. DIFFERENCES BETWEEN THE REFERENCE CONDITION AND ACTUAL RESULTS Real wheels rolling in real soils will not behave in the ideal manner of Fig. 1 because (a) the real wheel sinks in the soil, (b) the soil must be deformed in the
14 12
j
~
D
.~
I0 P 0E ~ ' ~ . js 0.z 0.2
xi-x
0
FIG. 4.
,°~ " ~~e,~-~ x•f x ~ " ~ ~" ~xX
l
1
02
0'4
I
J
75 x 26 b,cycleft re 15% 15 Yo deflection c
"~"~"",
xx~
I
0-6 08
I
t0
i
l
I2 t4 g
R e s u l t s of tests of s m o o t h
I 2 3
Lood Ib Coneindex 225 25 iO0 24 I00 68
J
~6
I
I
18 20
pneumatictyres
I
22
I
24
in dry sand.
direction of the wheel movement to develop frictional resistance in this direction, and (c) actual conditions are three-dimensional. Since the real wheel sinks at zero slip, the dissipated energy coetficient, p, is not zero under actual conditions but has a certain value, p0. Therefore, the dissipated energy curve should be as shown in Fig. 6(a). Sinkage also results in a certain length of contact between the wheel and the soil, and torque may be affected through the resulting stress distribution.
16xl5 -6R. 2-PR. terro-tire 25% deflection 08 ~ ' N ~% ~ .
~
1"06
..-Z~
~
Lood Ib
Cone index
zz5
57
04 02
I
FIG..5.
I 02
1
I 04
l
i "~f~ 06 s
L 08
I ~1
10
1
I ~2
Results of tests of smooth pneumatic tyros in dry sand.
MECHANICS OF WHEELS ON,SOFT SOILS
21
The fact that the soil-wheel displacement in the direction of the wheel movement is needed to develop friction in its direction for the actual wheel leads to a torque versus slip curve with a finite slope in the neighbourhood of the origin. This and the resulting torque energy curve are shown in Fig. 6(£,). The pull energy curve that should result with the original p vs. g curve (p0 = 0) is shown in Fig. 6 (c). The pull energy curve that results when both sinkage and soil yielding under shear are considered is shown in Fig. 6(d). It is obtained by transqating downward the curve of Fig. 6(c) and shows that slip at the point of selfpropulsion (~, =0) is a function of both p0 and of the shape of the torque versus slip curve. The third main difference between real wheels and the reference condition is considered to be the effect of stress and deformations normal to the plane of the wheel. This probably is the reason why different torque energy curves are obtained for a given wheel on a given soil when the load is varied (see Fig. 3). Also, it is an important experimental fact that the pull usually does not remain constant in the large slip range, as Fig. 6(d) would indicate. This may be expressed in the proposed system by stating that the torque energy curve and the dissipated energy curve do not have the same slope. It is believed that the three-dimensional pattern of the flow of the soil under the moving wheel is responsible for this occurrence. However, discussion of this effect is too complex to be attempted here.
p
/ j
if 0
g
(a)
0
J
(b)
f
0
g
/
-
(c) (d) FIo. 6. (a) Dissipated energy curve, (b) Torque energy curve, (c) Pull energy curve, (d) Pull energy curve considering sinkage and soil yielding.
B
22
ETIENNE LEFLAIVE
Figure 4 (and other similar graphs not shown) yields an interesting practical result. The dissipated energy coefficient, or rolling resistance coefficient, curves are parallel for a given tyre on a given type of soil for various loads and soil strength conditions. Therefore, p can be expressed as the sum of p0 and a quantity proportional to slip, the coefficient of proportionality being the slope of the p vs. g curves. This coefficient may be taken as a characteristic of the tyre for a certain type of soil. Similarly, the parallelism of the curves for traction efficiency, T, versus normal slip, s, yields another coefficient, which is the slope of these curves (Fig. 5). This is the basis for an analytical formulation of rolling motion, which has been developed for wheels on dry sand, using the assumption of a straight-line relation for dissipated energy coefficient vs. differential slip and for traction efficiency vs. normal slip. The author hopes to have the opportunity to present it in another paper. SUMMARY This paper has emphasized the fact that a strictly theoretical solution of the problem of wheels rolling on soft soils is extremely difficult and, therefore, will probably not be achieved soon. This statement is made in spite of the considerable efforts that have been made in the direction of theoretical analyses and in spite of all the means now available to solve such complex problems, such as applications of theory of plasticity to soils, experimental techniques such as measurement of stresses, and the help of electronic computers. Therefore in view of the large amount of experimental data available and of the large number of parameters involved, a method of presenting the test results is proposed, which is intended to help the researcher gain a comprehensive picture of the physical phenomena involved. The method, which uses energy parameters, may be useful both in developing a simplified empirical system and in contributing toward more refined theoretical studies. The present approach has limitations. One is its inadequacy for dealing with high slip conditions, where most parameters become infinite. Another is the limited consideration given to sinkage. It would be of interest to evaluate to what extent the limitations are of importance for practical purposes. REFERENCES [!] ]. L. McRAE and S. J. KNIGHT. The Terrain-Vehicle Programmes of the U.S. Army Engineer Waterways Experiment Station. J. Terramechanics, 1, No. 1, 98 (1964). [2] W. ]. TURNBULLand D. R. FltE1TAG. The Behaviour of Sand under Pneumatic Tyres. Proc. 1st Int. Conj. Mechanics o t Soil-Vehicle Systems, pp. 486-505 (1961). [3] J. R. PmLL]PS. The Powered Vehicular Wheel Plane-Rolling in Equilibrium: A Consideration of Slip and Rolling Resistance. Proc. Ist Int. Conj. Mechanics of SoilVehicle Systems, pp. 541-554 (1961).
Printed in Great Britain.
MECHANICS OF WHEELS ON SOFT SOILS-A METHOD FOR PRESENTING TEST RESULTS ETIENNE LEFLAIVE* INTRODUCTION RESEARCH in the field of vehicle mobility covers a wide range of subjects, from the physiological effects of vehicle vibrations on hmnan beings to the differentiation of various types of snow that vehicles may be required to cross. In their quest to find a vehicle that will go "anywhere." military mobility researchers have examined vehicles with oblate and hemispherical wheels, with tracks made of rubber-covered air chambers in many forms and with neither wheels nor tracks, i.e. walking vehicles, air-cushioned vehicles and vehicles propelled by Archimedean screws. Researchers in vehicle mobility for such civil purposes as transportation, earthmoving, agriculture and forestry are vitally concerned with the challenging economics aspects of the problem. Determination of the effects of the environment on the movement of vehicles leads to such absorbing problems as how the arrangement of trees in a forest affects the ability of a vehicle to move through them, or-how much resistance to vehicle movement is provided by dense stands of small trees that must be overriden. In contrast to this variety of interesting topics, the subject of this paper is the commonplace mechanics of a single, simple wheel on a soil condition prepared to be as uncomplicated as possible. The objective of the study of wheel mechanics in soft soil, besides the intellectual satisfaction of understanding a physical phenomenon, is to predict what will happen to a wheel in given conditions. Assuming that all the necessary characteristics of the wheel and the soil are known, the problem is to determine the relations among the load on the wheel, the torque, the pull that the wheel can develop, and the soil conditions. Two other quantities are of interest as well. Slip is important with respect to efficiency because for a given number of revolutions of the wheel, it reduces the distance over which the pull does work. Sinkage should be known, for it must remain smaller than the clearance of the vehicle. A first impulse is to attempt to solve the problem by determining the pull, the torque and possibly the slip and the sinkage by means of a theoretical analysis using the relations between stress and strain for the soil and for the wheel, the boundary conditions, and the conditions of equilibrium. While a theoretical study with very simple assumptions may indeed be helpful as a guide for the interpretation of test results, from our present knowledge an accurate description of the effects of variables, such as inflation pressure, tyre shape, tyre flexibility, treads, etc., cannot be expected because the analytical difficulties of a theoretical approach are considerable. Furthermore, many of the necessary physical laws for a correct analysis are not in hand. *Mobility and Environmental Division, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Mississipi. Communicated by D. R. Freitag.
13
14
ETIENNE LEFLA1VE
One of the main difficulties is that the differences in behaviour of various wheels or tyres probably are due to differences in the flow patterns of the soil under the moving wheels. As such a flow pattern is definitely three-dimensional and obviously has no axial symmetry, the problem is very complex. Another difficulty involves the complexity of the mechanical properties of the pneumatic tyres. However, even more important is the fact that stress-strain relations for soils are not known. Only stress displacement curves are available in given experimental conditions. This means that, generally, the stress displacement curves obtained in shear tests cannot be extrapolated or applied to other conditions because it is not known how the strains are distributed in the soil sample. In other words, it is known that stresses and strains are related, but not h o w - - i n spite of considerable research effort. This point is emphasized because such stress-strain relations are necessary to take into account the effect of slip in a theory of rolling motion. A given slip corresponds to a certain displacement between the wheel and the soil. The effect of this displacement, in terms of stresses, as compared with the result of a shear test, poses a question for which no sound answer exists. It could be assumed, as in civil engineering soil mechanics, that the displacement is large enough to mobilize the full friction of the soil and to allow application of Coulomb's law (which does not consider deformation quantitatively). However, this assumption would not provide a basis for description of the variation of pull and torque with slip. It would also be necessary to decide how to determine the angle of friction, and this again presents a problem. A final difficulty in the path of a theoretical approach is the extensive and rapid soil deformation that can occur under a moving wheel. Soil properties in such conditions may be different from those under slowly applied loads. Unexpected soil behaviour in the domain of large and rapid strain perhaps could explain some experimental results otherwise presently unexplainable. This discussion may be summarized by saying that not enough is known about mechanical properties of soils for a satisfactory theoretical treatment of the wheel problem. Knowledge of soil mechanics is a necessary guide for a physical understanding of the wheel behaviour, but it cannot lead to a direct theoretical solution. Obviously, the alternative to theorizing is to experiment with wheels in soft soils. Experiments may be conducted with actual vehicles or with single wheels, rigid or pneumatic tyred. Various kinds of soils may also be used--soils in their natural state, reprocessed soils, or artificial soils. Actual vehicles are used for practical reasons, but they are not convenient for close control and measurement of the variables. Rigid wheels and artificial soils are employed for simplification, but they must be used with care because all the consequences of these simplifications are not known. Their main role should be to investigate specific questions raised in the interpretation of tests conducted with pneumatic-tyred wheels on natural or reprocessed soils. The foregoing general statements contain nothing new. As a matter of fact, the Waterways Experiment Station (WES) fully recognized several years ago the necessity for tests with pneumatic tyres on natural soils. As no theory was assumed beforehand, the tests essentially were of an exploratory nature. A large number
MECHANICS OF WHEEL~ oN SOFT SOILS
15
of tests have been conducted, and a great quantity of carefully measured data is available [1, 2]. The point that this paper attempts to emphasize concerns the next step in research, following the achievement of such exploratory experiments; namely, the drawing of conclusions from the test results. Confusion arises from the number of parameters and variables playing a part in the phenomenon. This results in a great difficulty in obtaining a clear and general view of what the test results, in fact, are. In other words, the problem is to decide in which terms the results should be expressed to provide a good description of the experimental facts. If a clear view of what occurs can be obtained, then it becomes feasible to develop theories to explain the results. A good description of experimental results on rolling motion should be, as much as possible : (a) Complete; that is, it should include the various aspects (force, work, efficiency, etc.) and various phases (towed, self-propelled, maximum pull) of rolling motion (b) Concise (c) Easy to visualize (d) A good model of the physical processes (e) Capable of being expressed in a relatively simple analytical form. The last requirement suggests one of the consequences of establishing a framework for the observed facts; it naturally leads to an empirical system for dealing with the problem, such as graphical representations, empirical parameters, etc. This is useful and is not necessarily opposed to physical understanding and theoretical treatment, particularly if the fourth-mentioned requirement is fulfilled, because a good model contributes to the development of meaningful empirical parameters. It is the author's hope that this long introduction places the method of representation of test results given below within the framework of the wheel-soil mechanics research. BASIC CONCEPTS Graphical representations will be used to present the results of the tests and to describe the principles of the analysis technique. Slip has been used as the reference variable. This is customary, for slip is a primary physical process in rolling motion on soft soils. Sinkage, an alternative reference variable, is difficult to determine, and furthermore, a part of it is due to slip, while the remaining portion corresponds to bearing capacity. Since one value of sinkage may be composed of components from these two different sources in different proportions, sinkage is not appropriate as a basic reference variable. The quantities plotted against slip are energy quantities: torque energy, pull energy and dissipated energy. The torque energy is the work of the torque, either positive or negative; a positive work corresponds to a driving torque. The pull energy is the work done by the pull; it is positive when the wheel is developing a tractive effort and negative when the wheel is towed. The dissipated energy is the difference between the torque energy and the pull energy; it is always positive.
16
ETIENNE LEFLA1VE
The torque energy and the pull energy are determined from the test data. Then, the variations of the torque energy and the dissipated energy are studied in terms of the variables controlling the wheel behaviour. If the torque energy and the dissipated energy can be predicted through an understanding of the physical processes, the pull can be determined as their difference. One purpose of this approach was to differentiate the phenomena playing a part in rolling motion. Indeed, the reason for studying the energy input (torque) on one hand and the energy dissipation on the other hand is that it is useful to consider these two processes as independent, from the physical viewpoint. Evidently, both of these factors are influenced by the basic parameters controlling rolling motion and by secondary phenomena occurring when the wheel is in motion, such as the effect of sinkage on geometric conditions, variation of soil density due to shear, etc. However, it is proposed to regard them as two separate phenomena, and the production of pull as the result of their superposition. Another reason for considering energy quantities is their suitability for dealing with efficiency. Although the problem of efficiency for a wheel is not discussed herein, an efficiency parameter is defined. This quantity is a function of the energy quantities mentioned above and, likewise, has been plotted against slip. DEFINITIONS The parameters representing the tyre condition basically are those employed at the WES. Instead of inflation pressure, the rigidity of the tyre is expressed by the relative deflection of the loaded tyre on a hard surface. A rolling radius, R, at a given deflection, 8, and a given load, W, is defined as the distance the wheel travels in one revolution on a hard surface under these conditions, divided by 27r. Pull is denoted by the symbol F and the torque by M. Soil strength is represented by the cone index. It indicates whether the soil is dense or loose, firm or soft. Such an index is sufficient, at least, for the purpose of the present development, which is to build a framework to present test results, and not for an analysis of the soil behaviour under the wheel action. Two different coefficients to express slip are defined for use in this paper. One called normal slip is : s=
theoretical distance-actual distance travelled in a given time theoretical distance '
which is a conventional arrangement of the relevant parameters. The other coefficient has been termed differential slip and is defined as: g=
theoretical distance-actual distance actual distance
Either will be used in this paper as is most appropriate according to the circumstances. With co=rotational speed of the wheel, in rad/sec ,v = translation speed t----time
MECHANICS OF ~ L ~ I ~ N
SOFT SOILS
17
normal slip, s, and differential slip, g, are : Root - vt Root =1Root - ~ t
g = - • ,t
v
R"~' and
Roo
,o
1;
so that g g+l or $ g=i-s
The torque energy coefficient, "q, is defined as the energy supplied or withdrawn at the wheel axle, per unit of vertical load and per unit of distance travelled by the wheel.
~--- ~v--g- = w--~
=~-k -(l+g)
From the last expression it can be seen that the relation of torque energy coefficient to differential slip is a straight line for a constant torque, load and radius. The pull energy coefficient, ~, is the energy recovered or supplied as a pull, per unit of load and per unit of distance travelled. Fv Wv
F W
is also the ratio pull/load. The dissipated energy coefficient, p, is the energy that is not recovered mechanically, per unit of load and per unit of distance travelled. p=~-k=
Moo Wv
F W
p is the coefficient of rolling resistance, as proposed by Phillips [3]. The previously mentioned efficiency parameter is denoted traction efficiency (it is not within the scope of this paper to discuss this designation). It is defined: k where ~qa is the torque energy required for the wheel to simply propel itself in the soil. ~" is the ratio of the energy available as pull energy to the additional energy input required to produce a pull, beyond the point of self-propulsion. 7/, p, X and I" are dimensionless. REFERENCE CONDITION OF THE RIGID WHEEL ON A HARD SURFACE To help understand the meaning of the parameters just defined and to form a basis upon which rolling motion in general can be studied, it has been found useful
18
ETIENNE LEFLAIVE
to examine the simple case of a rigid wheel rolling and slipping on an unyielding surface. When dealing with actual conditions of pneumatic tyres on soft soil, the theoretical case of the rigid wheel on a hard surface will be taken as a reference to reveal more clearly by comparison the particular features of rolling motion on deformable soils. The case considered is strictly theoretical. A constant coefficient of friction, f, between the wheel and the surface is assumed, and W and R are taken as unity. Figure 1 shows the curves for torque, torque energy coefficient, ~7; pull energy
jj
~
+f ......................... ~M..
o
•
-f
W AND V
BRAKING
DRI VING WHEEL
OPPOSITE WAYS
NONROTA TING WHEEL
Fro. 1. Theoretical diagram for a rigid wheel on a rigid ground with a constant coefficient of friction, f. coefficient, h; and dissipated energy coefficient, p, vs. differential slip, g. The torque, M, has the limiting value + J for any positive differential slip and - f for any negative differential slip. For zero slip, M may have any value between + f and - f . The torque energy coefficient is "0=M (1 +g). Thus, for g :> 0, 77= + J ( l +g); for g = 0 , r j = M ; and for g < 0 , - 0 = - f ( l + g ) . The point g = - 1 corresponds to a non-rotating wheel (normal slip, s, is infinite); from g = 0 to g = - 1 , the wheel must be braked and energy is withdrawn from the wheel; for g ,< - 1, positive energy must be supplied to rotate the wheel in the direction opposite to its movement. The pull energy coefficient, h, is equal to the torque energy coefficient, ~9, when slip is zero, since at zero slip no energy is dissipated in friction. For positive differential slip, h = + f; for negative differential slip, h = - J . The dissipated energy coefficient, p, is equal to 77- h; it is zero for zero slip. For positive differential slip, p = + f ( l + g ) - f = f g ; for negative differential slip, p=-f(l+g)-(-])=-fg. Considering the variations of 7, /9 and h when slip
MECHANICS OF WHEELS ON SOFT SOILS
19
art
...
$
+1
E f f i c i e n c y ~./'~ f o r a rigid wheel on a rigid ground.
Fzo. 2.
increases, it is observed that the increase in torque energy coefficient is exactly absorbed by the corresponding increase in dissipated energy coefficient, and the pull remains constant. This result may appear obvious for the theoretical reference condition; however, for a wheel rolling on a yielding soil, the torque energy coefficient and the dissipated energy coefficient usually do not vary in such a parallel manner. Figure 2 represents the variation of traction efficiency, ~r, with respect to normal slip, s. As the torque energy coefficient for self-propulsion, ~7o, is zero for the theoretical reference condition, 'r=
~,
n-p
. ~q
.
~/
+-l(]+g)-(+/g) .
]
. . _+/(l+g)
1 --S
l+g
2C le IE
14 12
"r/ ,.0 08 0.6 h" / 04 Toward g=-I point 0-2
%
Load tb ~25
i
;)
~12 ,!.
n
1
,!6 ,!e 2:0 z.2
1
720 ,I
Cone index 57
57
J ,I
I
I+g FIG. 3.
1
2.4 2.6 2.e 3.0 3.2 3.4
Results of tests of smooth pneumatic tyres in dry sand.
20
ETIENNE LEFLAIVE TYPICAL
EXPERIMENTAL
RESULTS
The few results shown in Figs. 3, 4 and 5 refer to tests of smooth pneumatic tyres in dry sand. They are given as examples. The similarities between these curves derived from actual tests of pneumatic-tyred wheels on sand and the reference curves (Figs. 1 and 2) indicate the possibility of a comparison. This comparison is discussed below. DIFFERENCES BETWEEN THE REFERENCE CONDITION AND ACTUAL RESULTS Real wheels rolling in real soils will not behave in the ideal manner of Fig. 1 because (a) the real wheel sinks in the soil, (b) the soil must be deformed in the
14 12
j
~
D
.~
I0 P 0E ~ ' ~ . js 0.z 0.2
xi-x
0
FIG. 4.
,°~ " ~~e,~-~ x•f x ~ " ~ ~" ~xX
l
1
02
0'4
I
J
75 x 26 b,cycleft re 15% 15 Yo deflection c
"~"~"",
xx~
I
0-6 08
I
t0
i
l
I2 t4 g
R e s u l t s of tests of s m o o t h
I 2 3
Lood Ib Coneindex 225 25 iO0 24 I00 68
J
~6
I
I
18 20
pneumatictyres
I
22
I
24
in dry sand.
direction of the wheel movement to develop frictional resistance in this direction, and (c) actual conditions are three-dimensional. Since the real wheel sinks at zero slip, the dissipated energy coetficient, p, is not zero under actual conditions but has a certain value, p0. Therefore, the dissipated energy curve should be as shown in Fig. 6(a). Sinkage also results in a certain length of contact between the wheel and the soil, and torque may be affected through the resulting stress distribution.
16xl5 -6R. 2-PR. terro-tire 25% deflection 08 ~ ' N ~% ~ .
~
1"06
..-Z~
~
Lood Ib
Cone index
zz5
57
04 02
I
FIG..5.
I 02
1
I 04
l
i "~f~ 06 s
L 08
I ~1
10
1
I ~2
Results of tests of smooth pneumatic tyros in dry sand.
MECHANICS OF WHEELS ON,SOFT SOILS
21
The fact that the soil-wheel displacement in the direction of the wheel movement is needed to develop friction in its direction for the actual wheel leads to a torque versus slip curve with a finite slope in the neighbourhood of the origin. This and the resulting torque energy curve are shown in Fig. 6(£,). The pull energy curve that should result with the original p vs. g curve (p0 = 0) is shown in Fig. 6 (c). The pull energy curve that results when both sinkage and soil yielding under shear are considered is shown in Fig. 6(d). It is obtained by transqating downward the curve of Fig. 6(c) and shows that slip at the point of selfpropulsion (~, =0) is a function of both p0 and of the shape of the torque versus slip curve. The third main difference between real wheels and the reference condition is considered to be the effect of stress and deformations normal to the plane of the wheel. This probably is the reason why different torque energy curves are obtained for a given wheel on a given soil when the load is varied (see Fig. 3). Also, it is an important experimental fact that the pull usually does not remain constant in the large slip range, as Fig. 6(d) would indicate. This may be expressed in the proposed system by stating that the torque energy curve and the dissipated energy curve do not have the same slope. It is believed that the three-dimensional pattern of the flow of the soil under the moving wheel is responsible for this occurrence. However, discussion of this effect is too complex to be attempted here.
p
/ j
if 0
g
(a)
0
J
(b)
f
0
g
/
-
(c) (d) FIo. 6. (a) Dissipated energy curve, (b) Torque energy curve, (c) Pull energy curve, (d) Pull energy curve considering sinkage and soil yielding.
B
22
ETIENNE LEFLAIVE
Figure 4 (and other similar graphs not shown) yields an interesting practical result. The dissipated energy coefficient, or rolling resistance coefficient, curves are parallel for a given tyre on a given type of soil for various loads and soil strength conditions. Therefore, p can be expressed as the sum of p0 and a quantity proportional to slip, the coefficient of proportionality being the slope of the p vs. g curves. This coefficient may be taken as a characteristic of the tyre for a certain type of soil. Similarly, the parallelism of the curves for traction efficiency, T, versus normal slip, s, yields another coefficient, which is the slope of these curves (Fig. 5). This is the basis for an analytical formulation of rolling motion, which has been developed for wheels on dry sand, using the assumption of a straight-line relation for dissipated energy coefficient vs. differential slip and for traction efficiency vs. normal slip. The author hopes to have the opportunity to present it in another paper. SUMMARY This paper has emphasized the fact that a strictly theoretical solution of the problem of wheels rolling on soft soils is extremely difficult and, therefore, will probably not be achieved soon. This statement is made in spite of the considerable efforts that have been made in the direction of theoretical analyses and in spite of all the means now available to solve such complex problems, such as applications of theory of plasticity to soils, experimental techniques such as measurement of stresses, and the help of electronic computers. Therefore in view of the large amount of experimental data available and of the large number of parameters involved, a method of presenting the test results is proposed, which is intended to help the researcher gain a comprehensive picture of the physical phenomena involved. The method, which uses energy parameters, may be useful both in developing a simplified empirical system and in contributing toward more refined theoretical studies. The present approach has limitations. One is its inadequacy for dealing with high slip conditions, where most parameters become infinite. Another is the limited consideration given to sinkage. It would be of interest to evaluate to what extent the limitations are of importance for practical purposes. REFERENCES [!] ]. L. McRAE and S. J. KNIGHT. The Terrain-Vehicle Programmes of the U.S. Army Engineer Waterways Experiment Station. J. Terramechanics, 1, No. 1, 98 (1964). [2] W. ]. TURNBULLand D. R. FltE1TAG. The Behaviour of Sand under Pneumatic Tyres. Proc. 1st Int. Conj. Mechanics o t Soil-Vehicle Systems, pp. 486-505 (1961). [3] J. R. PmLL]PS. The Powered Vehicular Wheel Plane-Rolling in Equilibrium: A Consideration of Slip and Rolling Resistance. Proc. Ist Int. Conj. Mechanics of SoilVehicle Systems, pp. 541-554 (1961).