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Rolling Radii of a Pneumatic Tyre on Deformable Soil
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Biosystems Engineering (2003) 85 (2), 153–161 doi:10.1016/S1537-5110(03)00043-6 PM}Power and Machinery
Rolling Radii of a Pneumatic Tyre on Deformable Soil P!eter Kiss Department of Automotive and Thermal Technology, Szent Istv!an University, H-2103 Go. do. llo+ , Hungary; e-mail: [email protected] (Received 13 May 2002; received in revised form 27 February 2003; published online 29 April 2003)
Several rolling characteristics can be examined simultaneously by means of a suitable model representing the interaction of a pneumatic tyre rolling on deformable soil. The rolling radii can be distinguished, namely: the radius which develops due to wheelslip; the radius defined by the kinetics of the interaction; and the distance between the centre of the wheel and the bottom of the tyre. These three separate radii vary in time while rolling takes place. Owing to the varying nature of these values, they can only be obtained using dynamic measurements. This paper describes how these radii vary during rolling and discusses their interdependence. Tractor drawbar pull tests were conducted using 001 s data sampling intervals. The kinetic rolling resistance was defined by the peripheral force generated at the tyre-soil interface and the distance between the centre of the wheel and the bottom of the tyre, which is influenced by the wheel-load. The kinematic rolling radius was determined by measuring slip; and the kinetic rolling radius was established by analysing pull-test performance data, that is, the tractive force and the driving wheel-torque. The acceleration of the centre of the wheel was integrated twice with respect to time to obtain the path of the centre of the wheel as a function of time. The terrain profile, recorded after the tyre passed over it, and the path of the centre of the wheel were then plotted in the same coordinate system, making it possible to determine the distance to the bottom of the tyre. # 2003 Silsoe Research Institute. All rights reserved Published by Elsevier Science Ltd
1. Introduction A driving wheel imparts dual loads on the soil: the vertical wheelload generates compaction; and the driving torque causes shear loads. This is why early tyre–soil interaction research focused on two relationships: (1) sinkage due to soil compaction; and (2) the relationships between driving torque, tractive force and slip. There is an interactive relationship between the pneumatic tyre and the soil. The tyre acts on the soil and the soil reacts against the tyre. This is a classic case of action-reaction. The vertical load deforms the soil while the tyre flattens to a certain degree at the bottom. Early researchers concentrated on the load–sinkage relationship and on the distribution of soil stresses. The reasons for this were that: (1) soil compaction due to tyre load is detrimental to agricultural production; and (2) soil bearing capacity is an important influential factor concerning cross-country mobility. There have been several different approaches for the description of the phenomenon of vertical soil loading. 1537-5110/03/$30.00
Bernstein (1913), Goryatchkin (1936), Bekker (1956), Reece (1965) and Saakyan (1965) have developed different mathematical formulae to describe normal soil stresses occurring under a wheel. As shown by Bekker once the stresses are known these can be applied to determine the work exerted to compress the soil, and consequently the rolling resistance due to soil compaction. This equation is valid for small slip values. The relationship between slip and sinkage was unknown when Bekker developed his formulae in the 1950s. Later Onafeko (1969), and Kim and Shin (1986) investigated how slip affected rolling resistance. Okello (1992) researched the effect of the elasticity of the tyre beyond slip and sinkage. Plackett (1985) investigated the distribution of normal stresses under vehicles, while Dwyer et al. (1974) examined for different soil types important parameters which influence cross-country mobility. Computer techniques were applied for the purpose of determining soil stresses and deformation resulting in finite element method (FEM) and distinct element method (DEM) (Perumpral et al., 1971; Nem!enyi & Mouazen, 1999). 153
# 2003 Silsoe Research Institute. All rights reserved Published by Elsevier Science Ltd
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Notation a% Fg Fgh Ft Fv l Mh p Q2 rcb rk
vertical acceleration of the centre of the wheel, m s2 rolling resistance encountered by the front wheel, N rolling resistance acting against the rear wheel, N tractive force at the rear wheel, N drawbar pull exerted by the driving wheel, N distance covered during one wheel revolution, m driving torque acting on one wheel, Nm tyre inflation pressure, bar vertical load on the rear wheel, kN distance between the bottom of the tyre and the centre of the wheel, m kinetic rolling radius, m
Recent research into the normal and tangential stresses generated in the tyre–soil interface has led to an initial understanding of the process of soil compaction (Burt, 1987). Better understanding of soil compaction is the basis for developing methods and practices to reduce it. Agricultural tractors have grown in size and mass and, therefore, several researchers have focused on the analysis of stresses created in soil (Chancellor & Schmidt, 1962; Soane, 1980; Schwanghart, 1991). Wong (1991) investigated for various soil conditions and soil types the normal stresses occurring in the tyre–soil interface. The other main area of interest in tyre–soil research is how traction is generated under the tyre. The internal soil resistance, that is, the soil reaction against displacement, slip and changes in shape and volume, determines the tractive capability of the running gear. Internal soil resistance is characterised by the soil shear strength, which can be determined by methods well established in classical soil mechanics. Shear boxes and triaxial machines are used most frequently for this purpose (Boonsinsuk & Yong, 1984). Soils can be divided into two categories on the basis of shear tests: (1) those which possess internal cohesion; and (2) those which do not. Many researchers have created formulae to describe soil-shear diagrams and, from these they have derived equations for traction (Micklethwaite, 1944; Bekker, 1956; J!anosi & Hanamoto, 1964; Poncyliusz, 1986). In Hungary, Sitkei (1967) and Kom!andi (1975) developed methodologies to determine traction, based on the mechanical properties of soils.
rs rstat r% r%0 s t t0 v% v%0
rolling radius determined by slip, m distance between the centre of the wheel and the contact surface measured on concrete, m vertical position vector of the centre of the wheel, m initial vertical position vector of the centre of the wheel, m slip, % time, s initial time (time at the beginning of the testrun), s vertical velocity vector of the centre of the wheel, m s1 initial vertical velocity vector of the centre of the wheel, m s1
Tyre-soil inter-relationships, which were based on energetics, began to emerge in the 1970s. An ‘energetics model’ was introduced by Yong and Webb (1969), which evaluated the tractive performance of a rigid wheel moving on soft soil. The basic principle applied in the derivation of energetic models is the same as those employed for obtaining the energy balance in drawbar pull tests; that is, the useful output energy is equal to the input energy minus losses due to various resistances and slip. Among Hungarian researchers Kiss and Laib (1997) and Kiss et al. (1996) have conducted investigations into the dynamic processes involved in tyre–soil interaction by employing energetics. Complex processes take place when a pneumatic tyre rolls over off-road terrain. Tyre deformation and soil sinkage cause continuous changes in those physical parameters which influence the interaction of a tyre and the soil. Rolling over terrain and, hence, vehicle travel are basically non-stationary (dynamic) processes. The dynamic effect is caused primarily by the unevenness of the terrain profile and by the non-homogenous nature of the soil. Additional causes are, in case the vehicle exerts drawbar pull, the dynamic variations in resistance against the pulling force. Finally, slip can generate dynamic variations. Since these effects are stochastic the vibrations caused by them are also random. Vibration motions at the centre of the wheel may be studied by regarding them as motions in three different directions. When the vehicle moves in a straight-line vertical vibrations are caused by the terrain profile. Horizontal vibrations in the direction of travel are caused by the drawbar pull and/or vehicle acceleration. Lateral vibrations are the result of the presence of
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different profiles under the left and right sides of the vehicle. Also, differences in soil bearing capacity under the left and right wheels can generate lateral vibrations (Laib, 1995). It is important to note that these vibrations interact with each other. These vibrations cause energy losses and, meanwhile, they influence the interaction of the soil and the tyre (Kiss, 2001). The static load deforms the tyre and the soil. The rolling resistance and the drawbar pull cause further deformation. Additionally, constantly changing auxiliary load increase is present, generating ancillary tyre and soil deformation. The size of the contact surface varies, as well as its shape. The magnitude of the soil deformation and the position of the centre of the wheel change too. Thus, the vibrational acceleration influences the radius, while rolling takes place. As a result of the change in wheel load, the radius will become a time-dependent variable. The change in wheel load influences the adhesion between the soil and the tyre and, consequently, the slip varies continually. The radius, which prevails during rolling, is usually called the dynamic radius. Naming one particular radius dynamic radius is incorrect, because the name is used to denote several different radii. For example, the distance between the centre of the wheel and the bottom of the deformed tyre is often called dynamic radius (Ul’yanov & Mikhaylov, 1965), but often the radius defined by rolling with slip is denoted by the same term or ‘slip radius’ (Kom!andi, 1998). The same expression is also encountered in catalogues, published by tyre manufacturers, meaning the radius of the rolling circle. There are three radii which ‘exist’ during rolling:
This paper intends to present a series of tests conducted to determine these three radii as well as to analyse the factors, which influence them.
2. Test equipment and procedures To accomplish the objectives, field tests were performed with a John Deere 6600 tractor. The standard test methodologies were not followed strictly, because it was not the intention to determine the variation in pull and its maximum, but rather to examine the variation in soil–tyre interaction, and, in turn, to determine the rolling radii. The measured values were logged by means of a computerised test apparatus. Data were measured at a 001 s sampling rate. The tractor was operated in four different gears, with the tyres at five different inflation pressures. All tests were conducted in rearwheel-drive (Table 1). Braking was accomplished by using a John Deere Dyna-Cart, shown in Fig. 1. The terrain profile was measured at 20 cm intervals both before and after the vehicle passed over it, using a profilometer, which is based on the principle of communicating vessels. The profiles were measured at
(1) the distance between the bottom of the tyre and the centre of the wheel rcb; (2) kinematic rolling radius defined by motion (slip) rs; and (3) kinetic rolling radius defined by forces rk. Some of these are ‘real’ and can be measured. Others are ‘virtual’ and can only be calculated.
Fig. 1. The tractor with the dynamometer vehicle
Table 1 List of some preset and tested parameters for each test run Test segment no. 1 2 3 4 5 6 7 8 9
Tyre inflation pressure, bar
Vehicle velocity, km h1
Drawbar pull, kN
Slip, %
Gear
14 14 14 14 10 08 14 12 06
438 382 439 465 394 428 563 365 456
173 192 193 184 203 203 0 194 201
1840 3590 3620 4060 2630 1960 209 3120 1460
B2 C1 B3 C2 B2 B2 B2 B2 B2
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20 15
15
Profile height, cm
Profile height, cm
20
10
5
10 5 0 0
0 0 (a)
10
20
30
40
50
60
70
80
90 100 110
-5 10 20 30 40 50 60 70 80 90 100 110 Profile number (interval of 20 cm)
(b)
Profile number (interval of 20 cm)
20
Deformation, cm
15 10 5 0
0
10
20
30
40
50
60
70
80
90 100 110
-5
(c)
Profile number (interval of 20 cm)
Fig. 2. (a) Terrain profile in front of the tractor (‘virgin’ test segment); (b) terrain profile after the tractor (‘deformed’ test segment); and (c) vertical soil deformation in the rut formed by the right side of the tractor
the same points both on the left and right sides before and after the vehicle passed over the test lane. This was ensured by measuring them relative to a fixed base. Consequently, not only was the change in the profile determined but also the soil deformation, or sinkage. Each test was conducted on a ‘virgin’ terrain segment. The soil was sandy loam. The average moisture content was 8%. The field had been ploughed in the autumn and was settled by the time the tests took place. Dry soil density was 27 g cm3. Pore volume was 461%. Figure 2(a) shows the results from profiling the undeformed terrain. Figure 2(b) displays data obtained by profiling after the tractor passed over the segment. The curves show the variation of the terrain profile relative to the base. Figure 2(c) shows the magnitude of the vertical soil deformation under the front and rear wheels of the tractor. This is the difference between the curves in Figs 2(a) and (b). The computer stored the data via 15 channels. Fourteen were test channels and one was used for synchronisation (Table 2). The following instrumentation was used. Torque and pull were measured by strain gauges, rotational speed by an electronic tachometer, vehicle velocity by radar, acceleration by three dimensional (3D) piezoelectric
Table 2 List of measured data Test channel 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Measured value
Unit
Engine speed Engine torque Front wheel drive speed Front wheel drive torque Torque driving the left rear wheel Torque driving the right rear wheel Rear wheel drive speed Actual vehicle velocity Drawbar pull Longitudinal acceleration Lateral acceleration Vertical acceleration Exhaust gas temperature Fuel consumption
min1 Nm min1 Nm Nm Nm min1 km h1 kN m s2 m s2 m s2 8C kg h1
accelerometers, exhaust temperature by a NiCrNi thermocouple and fuel consumption by a flow-volume meter. The drawbar pull tests yielded a large number of data, with a data matrix for each test segment having 14 columns and 1300–2400 rows. In addition, profile measurements were stored in another file. The nine test
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60 Engine Performance, kW
Transmission losses
Wheel perf.
perf.
50 40 30 20 Traction perf.
10 0
0
2
Traction & slip perf.
4
Slip losses
Rolling resistance
6 8 10 Measuring time, s
12
14
16
Fig. 3. Towing test performance balance for test segment no. 9 for a two wheel drive tractor in gear B2 with a tyre inflation pressure of 06 bar, drawbar pull of 2014 kN and wheelslip of 146%; perf., performance
point from the centre rcb is measurable and it can also be computed. It is a ‘radius-like’ characteristic. It can be measured even while the tyre is rolling under varying load, albeit the required test method is not simple. To obtain this radius by computation requires knowledge of the static wheel radius, the vertical acceleration of the centre of the wheel, and the deformed soil profile under the same wheel measured synchronously. All three parameters are often measured during field tests. First, the motion-characteristics were determined using the vertical acceleration a% 5 a% ðtÞ which is measured continuously over the time period t at the centre of the wheel. Then the velocity of the wheel centre is given by Z t v% 5 ð1Þ a% ðtÞ dt þ v%0 5 v%ðtÞ t0
runs yielded nine basic data matrixes. As sampling was conducted at 001 s time intervals, a very large number of data was studied and, consequently, there was no need to fit curves to test points. By making use of the test data, energy performance parameters and the performance balance for a test run were determined. Engine performance is obtained from engine speed and torque; similarly, from driving wheel speed and driving torque the performance at the driving wheel axle can be calculated. The difference between these performance values is equal to the performance used up by the drive train. The slip was calculated by relating the average wheel speed measured when the tractor was pulling a load and when it moved with no drawbar load. Slip-performance (loss) is the product of the slip and the performance at the driving wheel. Drawbar performance is the product of the drawbar pull and vehicle velocity. The performance needed to overcome the rolling resistance is the driving wheel performance minus the sum of the slip performance and the drawbar performance. The calculated performance values were synchronous with the test data. (Fig. 3) This allows the detailed time history of the performance balance to be studied. The performance balance shows the instantaneous value of several performance components as a function of time. From the dynamic test data the radii can be determined as defined in this paper.
3. Results 3.1. The distance between the bottom of the wheel and its centre The bottom of the wheel is considered to be the instantaneous point at the wheel–soil interface, which is directly under of the wheel centre. The distance of this
where v%0 5 v%ðt0 Þ is the initial vertical velocity vector of the centre of the wheel. By integrating the velocity function the time–displacement function of the wheel centre is given by: Z t Z t r% 5 ð2Þ a% ðtÞ dt dt þ v%0 ðt t0 Þ þ r%0 t0
t0
where: r% is the vertical position vector of the centre of the wheel; and r%0 is the initial vertical position vector of the centre of the wheel. The mathematical operation was performed by means of MATLAB Simulink (Dynamic System Simulation Software) using initial conditions v%0 ; r% 5 0: The numerical integration employed the trapezoidal formula. The next step is to plot in the same coordinate system the profile of the wheel-rut and the synchronous displacement of the centre. Naturally, this is done for the same test segment, with these curves vertically displaced, so that the distance of the horizontal lines fitted to both curves is equal to the static tyre radius. Then the distance between the appropriate points of the two curves becomes the distance between the centre and the bottom of the wheel (Fig. 4). Table 3 shows the data obtained with respect to the distance between the bottom and the centre of the wheel rcb. The columns of table show the average load on the rear wheel Q2, the static radius rstat, inflation pressure p and the characteristic values for rcb, namely its maximum, minimum, average and the difference between the maximum and minimum. Figure 5 depicts rcb for a Michelin 650/65R 38X M 108 tyre at different inflation pressures, while operating under 20 kN load. It can be clearly seen that the drawbar load causes a smaller average rcb for the rolling tyre than the static radius rstat at 164 kN static wheel load. The diagram shows the maximum and minimum values of rcb. The range within which rcb varies in time
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can also be seen. As expected, increasing inflation pressure results in increasing rcb. It is evident from Fig. 5 that rcb values calculated from the acceleration of the wheel centre and the profile left behind the tyre are in consonance with the static radius: when the load increases the radius decreases which is in agreement with the load-deformation characteristic of the tyre.
3.2. The kinematic rolling radius The kinematic rolling radius rs is derived from the kinematics of the rolling process. It is a virtual radius, which can only be computed. Its magnitude can vary
Distance between the wheel centre and bottom of the tyre
rstat
rs 5
l 2p
ð3Þ
where l is the distance covered during one wheel revolution. Figure 6 depicts rs and rcb as a function of slip. It can be seen that rs decreases as the slip increases. However, rcb increases slightly with increasing slip. This is probably due to the increasing wheel vibrations caused by increasing slip. The two curves intersect at approximately zero slip. This agrees with theoretical
96 92 Distance, cm
Distance, cm
The path of the centre of the wheel
between zero to infinity. In the theoretical case of a rigid wheel rolling without slip, the kinematic radius is equal to the radius of the wheel, or geometric radius. When there is positive slip present (driving wheel, sometimes called driven wheel) the kinematic radius is smaller than the geometric radius and it is larger for negative slip (towed wheel). When the wheel spins in one place, rs is zero; and when the wheel is blocked by the brake and skids, rs is infinity. The kinematic radius, defined by slip, can be calculated by the following formula:
Terrian profile recorded after the tyre passed over it
Maximum
88
rstat
84 80 76
Average Minimum
72 0.6
Time, s
Fig. 4. Determination of the distance between the bottom and the centre of the wheel rcb from the displacement function of the centre and the profile left behind the tyre; rstat, distance between the centre of the wheel and the contact surface measured on concrete
0.8
1.0 1.2 Tyre inflation pressure, bar
1.4
Fig. 5. The distance between the bottom of the tyre and the centre of the wheel (rcb) as a function of inflation pressure; rstat, distance between the centre of the wheel and the contact surface measured on concrete
Table 3 The distance between the wheel centre and the bottom of the tyre Test segment no rsat, cm at 164 kN p, bar Q2, kN
1 2 3 4 5 6 7 8 9
902 902 902 902 858 836 902 880 814
14 14 14 14 10 08 14 12 06
1926 1981 1993 2005 1942 1920 1722 1980 1963
The distance between the centre and the bottom of the tyre (rcb), cm Minimum
Maximum
Average
Interval
782 780 820 790 770 755 830 780 745
921 920 910 930 885 860 920 910 835
868 871 872 880 832 809 879 852 791
139 140 90 140 115 105 90 130 90
rstat, distance between the centre of the wheel and the contact surface measured on concrete; p, tyre inflation pressure; Q2, vertical load on the rear wheel.
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120
90 Distance between the centre and the bottom
88
80
Distance between the wheel centre and bottom of tyre
86
60
Radius, cm
Radius, cm
100
Kinematic radius
40 20 0 0
10
20
30
40
50
84 82 Kinetic radius 80 78
Slip, %
Fig. 6. The kinematic radius as a function of slip (Michelin 650/ 65 R 38 X M 108)
76
0.6
0.8
1.0
1.2
1.4
Tyre inflation pressure, bar
considerations, since, when one neglects tyre and soil deformation, at zero slip the kinematic radius is equal to the distance between the centre of the wheel and its bottom. 3.3. The kinetic rolling radius The kinetic rolling radius rk acts in or near the interface surface of a rolling deformable tyre and a deformable soil. It is the distance between the centre of the wheel and the resultant of the elementary tractive forces acting along the ground contact surface. The magnitude of this radius determines the magnitude of the tractive force generated by the driving torque. Its value can only fall within a narrow range, unlike that of the kinematic rolling radius. The resultant of the elementary tractive forces does not lie necessarily in the contact surface. It may occur above it. The tyre is pressed deep into soft soil and, hence, the contact is a three-dimensional surface, not a plane. The elementary tractive forces act along the entire surface. They are not distributed evenly, but rather proportionally to the adhesion between the tyre and the soil. As a result the resultant tractive force acts above the bottom of the tyre. The kinetic rolling radius can be found by computation of the forces at the soil interface for both the front and rear wheels on one side of a rear-wheel-drive tractor: Ft 5 Fge þ Fgh þ Fv
Fig. 7. Comparison of the kinetic radius and the centre-bottom distance for different inflation pressure
where Mh and Ft are the driving moment and tractive force for the same driving wheel. Figure 7 shows rcb and rk as a function of inflation pressure. The tyre was the same as depicted in Fig. 6. The kinetic radius is smaller than the distance between the centre and the bottom of the tyre. This is in agreement with theoretical considerations as well as with practical experience. When the inflation pressure is low, the sidewall of the tyre flattens and the bottom of the tyre matches closely the shape of the bearing surface. In this case, the tractive forces fall within the plane of the bottom of the tyre or are close to it. For higher inflation pressure the sidewall deforms less, the cross-section of the tyre becomes more round and the contact surface becomes a deeper elliptical surface. Here, more elemental tractive forces act above the bottom and, hence, the resultant will act higher too. It follows from the above that rk and rcb differ more and more as the inflation pressure increases for a tyre rolling on deformable surface. Table 4 contains numerical values for rk and rcb and the last column depicts their difference as a percentage with respect to rcb. The percent values are negative since, as shown above, the kinematic radius is always smaller than the centre-bottom distance. Note that for a rigid plane surface the two radii are equal.
ð4Þ
where: Ft is the tractive force developed by one rear wheel; Fge and Fgh represent the rolling resistance acting against the front and rear wheels on the same side, respectively; and Fv is the drawbar pull for that side. This can be calculated by considering the entire drawbar pull and the ratio between the left and right-side driving torques. Once Ft is known, rk is given by Mh ð5Þ rk 5 Ft
4. Discussion It has been shown that the three radii discussed in this paper differ from each other. Only in special cases are they the same. On soft, deformable soil, the wheel centre-bottom distance is not equal to the other two radii. The physical process occurring at the tyre–soil interface is best characterised by the kinetic and kinematic radii. These two are equal only for a rigid wheel moving on a rigid surface at zero slip. (However,
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Table 4 Comparison of the kinetic radius and the wheel centre-bottom distance Test segment no 1 2 3 4 5 6 7 8 9
Tyre inflation pressure (p), bar
The distance between the centre and the bottom of the tyre (rcb), cm
Kinetic rolling radius(rk), cm
Difference %
14 14 14 14 10 08 14 12 06
868 871 872 880 832 809 879 852 791
840 830 858 848 829 800 877 838 789
32 47 16 36 10 11 02 16 03
they may be equal for an instant as they vary due to vibrations caused by uneven soil profile.) In case of a tyre moving on soft soil, there is always slip-deformation and, hence, slip is always present. Therefore the two radii will always differ from each other. Both characterise the tyre–soil interaction; rs characterises the interaction from the viewpoint of kinematics (slip), whereas rk is a characteristic of the force interplay between the tyre and the soil. The difference between these two radii is best exemplified at 100% slip, where rs is zero, but this is when the wheel exerts the highest tractive force, so rk must be more than zero.
When the wheel rotational speed is not constant, the driving torque must be reduced by the inertia needed to accelerate the wheel.
Acknowledgements The author wishes to express his thanks to John Deere Mannheim (Germany) for their help in the performance of our tests. Also I am grateful to the MTA (Hungarian Academy of Sciences) for their support of the research described herein.
References 5. Conclusions There are three different radii for a tyre rolling on offroad terrain: (1) the distance between the centre and the bottom of the tyre, which is influenced by loading conditions and inflation pressure; (2) the kinematic rolling radius determined by slip; and (3) the kinetic radius, which is the distance between the centre of the wheel, where the driving torque acts, and the tractive force generated in the tyre-soil interface. The kinematic rolling radius has long been derived from the distance covered in one revolution. The kinetic rolling radius is obtained from the driving torque and the tractive force, which is generated in the tyre–soil interface. When the tractor moves with uniform speed, the angular acceleration of the driving wheel is zero. In this case, the kinetic rolling radius is defined as the distance between the centre of the wheel and the resultant of the elementary tractive forces generated in the tyre–soil interface.
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Biosystems Engineering (2003) 85 (2), 153–161 doi:10.1016/S1537-5110(03)00043-6 PM}Power and Machinery
Rolling Radii of a Pneumatic Tyre on Deformable Soil P!eter Kiss Department of Automotive and Thermal Technology, Szent Istv!an University, H-2103 Go. do. llo+ , Hungary; e-mail: [email protected] (Received 13 May 2002; received in revised form 27 February 2003; published online 29 April 2003)
Several rolling characteristics can be examined simultaneously by means of a suitable model representing the interaction of a pneumatic tyre rolling on deformable soil. The rolling radii can be distinguished, namely: the radius which develops due to wheelslip; the radius defined by the kinetics of the interaction; and the distance between the centre of the wheel and the bottom of the tyre. These three separate radii vary in time while rolling takes place. Owing to the varying nature of these values, they can only be obtained using dynamic measurements. This paper describes how these radii vary during rolling and discusses their interdependence. Tractor drawbar pull tests were conducted using 001 s data sampling intervals. The kinetic rolling resistance was defined by the peripheral force generated at the tyre-soil interface and the distance between the centre of the wheel and the bottom of the tyre, which is influenced by the wheel-load. The kinematic rolling radius was determined by measuring slip; and the kinetic rolling radius was established by analysing pull-test performance data, that is, the tractive force and the driving wheel-torque. The acceleration of the centre of the wheel was integrated twice with respect to time to obtain the path of the centre of the wheel as a function of time. The terrain profile, recorded after the tyre passed over it, and the path of the centre of the wheel were then plotted in the same coordinate system, making it possible to determine the distance to the bottom of the tyre. # 2003 Silsoe Research Institute. All rights reserved Published by Elsevier Science Ltd
1. Introduction A driving wheel imparts dual loads on the soil: the vertical wheelload generates compaction; and the driving torque causes shear loads. This is why early tyre–soil interaction research focused on two relationships: (1) sinkage due to soil compaction; and (2) the relationships between driving torque, tractive force and slip. There is an interactive relationship between the pneumatic tyre and the soil. The tyre acts on the soil and the soil reacts against the tyre. This is a classic case of action-reaction. The vertical load deforms the soil while the tyre flattens to a certain degree at the bottom. Early researchers concentrated on the load–sinkage relationship and on the distribution of soil stresses. The reasons for this were that: (1) soil compaction due to tyre load is detrimental to agricultural production; and (2) soil bearing capacity is an important influential factor concerning cross-country mobility. There have been several different approaches for the description of the phenomenon of vertical soil loading. 1537-5110/03/$30.00
Bernstein (1913), Goryatchkin (1936), Bekker (1956), Reece (1965) and Saakyan (1965) have developed different mathematical formulae to describe normal soil stresses occurring under a wheel. As shown by Bekker once the stresses are known these can be applied to determine the work exerted to compress the soil, and consequently the rolling resistance due to soil compaction. This equation is valid for small slip values. The relationship between slip and sinkage was unknown when Bekker developed his formulae in the 1950s. Later Onafeko (1969), and Kim and Shin (1986) investigated how slip affected rolling resistance. Okello (1992) researched the effect of the elasticity of the tyre beyond slip and sinkage. Plackett (1985) investigated the distribution of normal stresses under vehicles, while Dwyer et al. (1974) examined for different soil types important parameters which influence cross-country mobility. Computer techniques were applied for the purpose of determining soil stresses and deformation resulting in finite element method (FEM) and distinct element method (DEM) (Perumpral et al., 1971; Nem!enyi & Mouazen, 1999). 153
# 2003 Silsoe Research Institute. All rights reserved Published by Elsevier Science Ltd
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Notation a% Fg Fgh Ft Fv l Mh p Q2 rcb rk
vertical acceleration of the centre of the wheel, m s2 rolling resistance encountered by the front wheel, N rolling resistance acting against the rear wheel, N tractive force at the rear wheel, N drawbar pull exerted by the driving wheel, N distance covered during one wheel revolution, m driving torque acting on one wheel, Nm tyre inflation pressure, bar vertical load on the rear wheel, kN distance between the bottom of the tyre and the centre of the wheel, m kinetic rolling radius, m
Recent research into the normal and tangential stresses generated in the tyre–soil interface has led to an initial understanding of the process of soil compaction (Burt, 1987). Better understanding of soil compaction is the basis for developing methods and practices to reduce it. Agricultural tractors have grown in size and mass and, therefore, several researchers have focused on the analysis of stresses created in soil (Chancellor & Schmidt, 1962; Soane, 1980; Schwanghart, 1991). Wong (1991) investigated for various soil conditions and soil types the normal stresses occurring in the tyre–soil interface. The other main area of interest in tyre–soil research is how traction is generated under the tyre. The internal soil resistance, that is, the soil reaction against displacement, slip and changes in shape and volume, determines the tractive capability of the running gear. Internal soil resistance is characterised by the soil shear strength, which can be determined by methods well established in classical soil mechanics. Shear boxes and triaxial machines are used most frequently for this purpose (Boonsinsuk & Yong, 1984). Soils can be divided into two categories on the basis of shear tests: (1) those which possess internal cohesion; and (2) those which do not. Many researchers have created formulae to describe soil-shear diagrams and, from these they have derived equations for traction (Micklethwaite, 1944; Bekker, 1956; J!anosi & Hanamoto, 1964; Poncyliusz, 1986). In Hungary, Sitkei (1967) and Kom!andi (1975) developed methodologies to determine traction, based on the mechanical properties of soils.
rs rstat r% r%0 s t t0 v% v%0
rolling radius determined by slip, m distance between the centre of the wheel and the contact surface measured on concrete, m vertical position vector of the centre of the wheel, m initial vertical position vector of the centre of the wheel, m slip, % time, s initial time (time at the beginning of the testrun), s vertical velocity vector of the centre of the wheel, m s1 initial vertical velocity vector of the centre of the wheel, m s1
Tyre-soil inter-relationships, which were based on energetics, began to emerge in the 1970s. An ‘energetics model’ was introduced by Yong and Webb (1969), which evaluated the tractive performance of a rigid wheel moving on soft soil. The basic principle applied in the derivation of energetic models is the same as those employed for obtaining the energy balance in drawbar pull tests; that is, the useful output energy is equal to the input energy minus losses due to various resistances and slip. Among Hungarian researchers Kiss and Laib (1997) and Kiss et al. (1996) have conducted investigations into the dynamic processes involved in tyre–soil interaction by employing energetics. Complex processes take place when a pneumatic tyre rolls over off-road terrain. Tyre deformation and soil sinkage cause continuous changes in those physical parameters which influence the interaction of a tyre and the soil. Rolling over terrain and, hence, vehicle travel are basically non-stationary (dynamic) processes. The dynamic effect is caused primarily by the unevenness of the terrain profile and by the non-homogenous nature of the soil. Additional causes are, in case the vehicle exerts drawbar pull, the dynamic variations in resistance against the pulling force. Finally, slip can generate dynamic variations. Since these effects are stochastic the vibrations caused by them are also random. Vibration motions at the centre of the wheel may be studied by regarding them as motions in three different directions. When the vehicle moves in a straight-line vertical vibrations are caused by the terrain profile. Horizontal vibrations in the direction of travel are caused by the drawbar pull and/or vehicle acceleration. Lateral vibrations are the result of the presence of
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different profiles under the left and right sides of the vehicle. Also, differences in soil bearing capacity under the left and right wheels can generate lateral vibrations (Laib, 1995). It is important to note that these vibrations interact with each other. These vibrations cause energy losses and, meanwhile, they influence the interaction of the soil and the tyre (Kiss, 2001). The static load deforms the tyre and the soil. The rolling resistance and the drawbar pull cause further deformation. Additionally, constantly changing auxiliary load increase is present, generating ancillary tyre and soil deformation. The size of the contact surface varies, as well as its shape. The magnitude of the soil deformation and the position of the centre of the wheel change too. Thus, the vibrational acceleration influences the radius, while rolling takes place. As a result of the change in wheel load, the radius will become a time-dependent variable. The change in wheel load influences the adhesion between the soil and the tyre and, consequently, the slip varies continually. The radius, which prevails during rolling, is usually called the dynamic radius. Naming one particular radius dynamic radius is incorrect, because the name is used to denote several different radii. For example, the distance between the centre of the wheel and the bottom of the deformed tyre is often called dynamic radius (Ul’yanov & Mikhaylov, 1965), but often the radius defined by rolling with slip is denoted by the same term or ‘slip radius’ (Kom!andi, 1998). The same expression is also encountered in catalogues, published by tyre manufacturers, meaning the radius of the rolling circle. There are three radii which ‘exist’ during rolling:
This paper intends to present a series of tests conducted to determine these three radii as well as to analyse the factors, which influence them.
2. Test equipment and procedures To accomplish the objectives, field tests were performed with a John Deere 6600 tractor. The standard test methodologies were not followed strictly, because it was not the intention to determine the variation in pull and its maximum, but rather to examine the variation in soil–tyre interaction, and, in turn, to determine the rolling radii. The measured values were logged by means of a computerised test apparatus. Data were measured at a 001 s sampling rate. The tractor was operated in four different gears, with the tyres at five different inflation pressures. All tests were conducted in rearwheel-drive (Table 1). Braking was accomplished by using a John Deere Dyna-Cart, shown in Fig. 1. The terrain profile was measured at 20 cm intervals both before and after the vehicle passed over it, using a profilometer, which is based on the principle of communicating vessels. The profiles were measured at
(1) the distance between the bottom of the tyre and the centre of the wheel rcb; (2) kinematic rolling radius defined by motion (slip) rs; and (3) kinetic rolling radius defined by forces rk. Some of these are ‘real’ and can be measured. Others are ‘virtual’ and can only be calculated.
Fig. 1. The tractor with the dynamometer vehicle
Table 1 List of some preset and tested parameters for each test run Test segment no. 1 2 3 4 5 6 7 8 9
Tyre inflation pressure, bar
Vehicle velocity, km h1
Drawbar pull, kN
Slip, %
Gear
14 14 14 14 10 08 14 12 06
438 382 439 465 394 428 563 365 456
173 192 193 184 203 203 0 194 201
1840 3590 3620 4060 2630 1960 209 3120 1460
B2 C1 B3 C2 B2 B2 B2 B2 B2
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20 15
15
Profile height, cm
Profile height, cm
20
10
5
10 5 0 0
0 0 (a)
10
20
30
40
50
60
70
80
90 100 110
-5 10 20 30 40 50 60 70 80 90 100 110 Profile number (interval of 20 cm)
(b)
Profile number (interval of 20 cm)
20
Deformation, cm
15 10 5 0
0
10
20
30
40
50
60
70
80
90 100 110
-5
(c)
Profile number (interval of 20 cm)
Fig. 2. (a) Terrain profile in front of the tractor (‘virgin’ test segment); (b) terrain profile after the tractor (‘deformed’ test segment); and (c) vertical soil deformation in the rut formed by the right side of the tractor
the same points both on the left and right sides before and after the vehicle passed over the test lane. This was ensured by measuring them relative to a fixed base. Consequently, not only was the change in the profile determined but also the soil deformation, or sinkage. Each test was conducted on a ‘virgin’ terrain segment. The soil was sandy loam. The average moisture content was 8%. The field had been ploughed in the autumn and was settled by the time the tests took place. Dry soil density was 27 g cm3. Pore volume was 461%. Figure 2(a) shows the results from profiling the undeformed terrain. Figure 2(b) displays data obtained by profiling after the tractor passed over the segment. The curves show the variation of the terrain profile relative to the base. Figure 2(c) shows the magnitude of the vertical soil deformation under the front and rear wheels of the tractor. This is the difference between the curves in Figs 2(a) and (b). The computer stored the data via 15 channels. Fourteen were test channels and one was used for synchronisation (Table 2). The following instrumentation was used. Torque and pull were measured by strain gauges, rotational speed by an electronic tachometer, vehicle velocity by radar, acceleration by three dimensional (3D) piezoelectric
Table 2 List of measured data Test channel 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Measured value
Unit
Engine speed Engine torque Front wheel drive speed Front wheel drive torque Torque driving the left rear wheel Torque driving the right rear wheel Rear wheel drive speed Actual vehicle velocity Drawbar pull Longitudinal acceleration Lateral acceleration Vertical acceleration Exhaust gas temperature Fuel consumption
min1 Nm min1 Nm Nm Nm min1 km h1 kN m s2 m s2 m s2 8C kg h1
accelerometers, exhaust temperature by a NiCrNi thermocouple and fuel consumption by a flow-volume meter. The drawbar pull tests yielded a large number of data, with a data matrix for each test segment having 14 columns and 1300–2400 rows. In addition, profile measurements were stored in another file. The nine test
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60 Engine Performance, kW
Transmission losses
Wheel perf.
perf.
50 40 30 20 Traction perf.
10 0
0
2
Traction & slip perf.
4
Slip losses
Rolling resistance
6 8 10 Measuring time, s
12
14
16
Fig. 3. Towing test performance balance for test segment no. 9 for a two wheel drive tractor in gear B2 with a tyre inflation pressure of 06 bar, drawbar pull of 2014 kN and wheelslip of 146%; perf., performance
point from the centre rcb is measurable and it can also be computed. It is a ‘radius-like’ characteristic. It can be measured even while the tyre is rolling under varying load, albeit the required test method is not simple. To obtain this radius by computation requires knowledge of the static wheel radius, the vertical acceleration of the centre of the wheel, and the deformed soil profile under the same wheel measured synchronously. All three parameters are often measured during field tests. First, the motion-characteristics were determined using the vertical acceleration a% 5 a% ðtÞ which is measured continuously over the time period t at the centre of the wheel. Then the velocity of the wheel centre is given by Z t v% 5 ð1Þ a% ðtÞ dt þ v%0 5 v%ðtÞ t0
runs yielded nine basic data matrixes. As sampling was conducted at 001 s time intervals, a very large number of data was studied and, consequently, there was no need to fit curves to test points. By making use of the test data, energy performance parameters and the performance balance for a test run were determined. Engine performance is obtained from engine speed and torque; similarly, from driving wheel speed and driving torque the performance at the driving wheel axle can be calculated. The difference between these performance values is equal to the performance used up by the drive train. The slip was calculated by relating the average wheel speed measured when the tractor was pulling a load and when it moved with no drawbar load. Slip-performance (loss) is the product of the slip and the performance at the driving wheel. Drawbar performance is the product of the drawbar pull and vehicle velocity. The performance needed to overcome the rolling resistance is the driving wheel performance minus the sum of the slip performance and the drawbar performance. The calculated performance values were synchronous with the test data. (Fig. 3) This allows the detailed time history of the performance balance to be studied. The performance balance shows the instantaneous value of several performance components as a function of time. From the dynamic test data the radii can be determined as defined in this paper.
3. Results 3.1. The distance between the bottom of the wheel and its centre The bottom of the wheel is considered to be the instantaneous point at the wheel–soil interface, which is directly under of the wheel centre. The distance of this
where v%0 5 v%ðt0 Þ is the initial vertical velocity vector of the centre of the wheel. By integrating the velocity function the time–displacement function of the wheel centre is given by: Z t Z t r% 5 ð2Þ a% ðtÞ dt dt þ v%0 ðt t0 Þ þ r%0 t0
t0
where: r% is the vertical position vector of the centre of the wheel; and r%0 is the initial vertical position vector of the centre of the wheel. The mathematical operation was performed by means of MATLAB Simulink (Dynamic System Simulation Software) using initial conditions v%0 ; r% 5 0: The numerical integration employed the trapezoidal formula. The next step is to plot in the same coordinate system the profile of the wheel-rut and the synchronous displacement of the centre. Naturally, this is done for the same test segment, with these curves vertically displaced, so that the distance of the horizontal lines fitted to both curves is equal to the static tyre radius. Then the distance between the appropriate points of the two curves becomes the distance between the centre and the bottom of the wheel (Fig. 4). Table 3 shows the data obtained with respect to the distance between the bottom and the centre of the wheel rcb. The columns of table show the average load on the rear wheel Q2, the static radius rstat, inflation pressure p and the characteristic values for rcb, namely its maximum, minimum, average and the difference between the maximum and minimum. Figure 5 depicts rcb for a Michelin 650/65R 38X M 108 tyre at different inflation pressures, while operating under 20 kN load. It can be clearly seen that the drawbar load causes a smaller average rcb for the rolling tyre than the static radius rstat at 164 kN static wheel load. The diagram shows the maximum and minimum values of rcb. The range within which rcb varies in time
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can also be seen. As expected, increasing inflation pressure results in increasing rcb. It is evident from Fig. 5 that rcb values calculated from the acceleration of the wheel centre and the profile left behind the tyre are in consonance with the static radius: when the load increases the radius decreases which is in agreement with the load-deformation characteristic of the tyre.
3.2. The kinematic rolling radius The kinematic rolling radius rs is derived from the kinematics of the rolling process. It is a virtual radius, which can only be computed. Its magnitude can vary
Distance between the wheel centre and bottom of the tyre
rstat
rs 5
l 2p
ð3Þ
where l is the distance covered during one wheel revolution. Figure 6 depicts rs and rcb as a function of slip. It can be seen that rs decreases as the slip increases. However, rcb increases slightly with increasing slip. This is probably due to the increasing wheel vibrations caused by increasing slip. The two curves intersect at approximately zero slip. This agrees with theoretical
96 92 Distance, cm
Distance, cm
The path of the centre of the wheel
between zero to infinity. In the theoretical case of a rigid wheel rolling without slip, the kinematic radius is equal to the radius of the wheel, or geometric radius. When there is positive slip present (driving wheel, sometimes called driven wheel) the kinematic radius is smaller than the geometric radius and it is larger for negative slip (towed wheel). When the wheel spins in one place, rs is zero; and when the wheel is blocked by the brake and skids, rs is infinity. The kinematic radius, defined by slip, can be calculated by the following formula:
Terrian profile recorded after the tyre passed over it
Maximum
88
rstat
84 80 76
Average Minimum
72 0.6
Time, s
Fig. 4. Determination of the distance between the bottom and the centre of the wheel rcb from the displacement function of the centre and the profile left behind the tyre; rstat, distance between the centre of the wheel and the contact surface measured on concrete
0.8
1.0 1.2 Tyre inflation pressure, bar
1.4
Fig. 5. The distance between the bottom of the tyre and the centre of the wheel (rcb) as a function of inflation pressure; rstat, distance between the centre of the wheel and the contact surface measured on concrete
Table 3 The distance between the wheel centre and the bottom of the tyre Test segment no rsat, cm at 164 kN p, bar Q2, kN
1 2 3 4 5 6 7 8 9
902 902 902 902 858 836 902 880 814
14 14 14 14 10 08 14 12 06
1926 1981 1993 2005 1942 1920 1722 1980 1963
The distance between the centre and the bottom of the tyre (rcb), cm Minimum
Maximum
Average
Interval
782 780 820 790 770 755 830 780 745
921 920 910 930 885 860 920 910 835
868 871 872 880 832 809 879 852 791
139 140 90 140 115 105 90 130 90
rstat, distance between the centre of the wheel and the contact surface measured on concrete; p, tyre inflation pressure; Q2, vertical load on the rear wheel.
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120
90 Distance between the centre and the bottom
88
80
Distance between the wheel centre and bottom of tyre
86
60
Radius, cm
Radius, cm
100
Kinematic radius
40 20 0 0
10
20
30
40
50
84 82 Kinetic radius 80 78
Slip, %
Fig. 6. The kinematic radius as a function of slip (Michelin 650/ 65 R 38 X M 108)
76
0.6
0.8
1.0
1.2
1.4
Tyre inflation pressure, bar
considerations, since, when one neglects tyre and soil deformation, at zero slip the kinematic radius is equal to the distance between the centre of the wheel and its bottom. 3.3. The kinetic rolling radius The kinetic rolling radius rk acts in or near the interface surface of a rolling deformable tyre and a deformable soil. It is the distance between the centre of the wheel and the resultant of the elementary tractive forces acting along the ground contact surface. The magnitude of this radius determines the magnitude of the tractive force generated by the driving torque. Its value can only fall within a narrow range, unlike that of the kinematic rolling radius. The resultant of the elementary tractive forces does not lie necessarily in the contact surface. It may occur above it. The tyre is pressed deep into soft soil and, hence, the contact is a three-dimensional surface, not a plane. The elementary tractive forces act along the entire surface. They are not distributed evenly, but rather proportionally to the adhesion between the tyre and the soil. As a result the resultant tractive force acts above the bottom of the tyre. The kinetic rolling radius can be found by computation of the forces at the soil interface for both the front and rear wheels on one side of a rear-wheel-drive tractor: Ft 5 Fge þ Fgh þ Fv
Fig. 7. Comparison of the kinetic radius and the centre-bottom distance for different inflation pressure
where Mh and Ft are the driving moment and tractive force for the same driving wheel. Figure 7 shows rcb and rk as a function of inflation pressure. The tyre was the same as depicted in Fig. 6. The kinetic radius is smaller than the distance between the centre and the bottom of the tyre. This is in agreement with theoretical considerations as well as with practical experience. When the inflation pressure is low, the sidewall of the tyre flattens and the bottom of the tyre matches closely the shape of the bearing surface. In this case, the tractive forces fall within the plane of the bottom of the tyre or are close to it. For higher inflation pressure the sidewall deforms less, the cross-section of the tyre becomes more round and the contact surface becomes a deeper elliptical surface. Here, more elemental tractive forces act above the bottom and, hence, the resultant will act higher too. It follows from the above that rk and rcb differ more and more as the inflation pressure increases for a tyre rolling on deformable surface. Table 4 contains numerical values for rk and rcb and the last column depicts their difference as a percentage with respect to rcb. The percent values are negative since, as shown above, the kinematic radius is always smaller than the centre-bottom distance. Note that for a rigid plane surface the two radii are equal.
ð4Þ
where: Ft is the tractive force developed by one rear wheel; Fge and Fgh represent the rolling resistance acting against the front and rear wheels on the same side, respectively; and Fv is the drawbar pull for that side. This can be calculated by considering the entire drawbar pull and the ratio between the left and right-side driving torques. Once Ft is known, rk is given by Mh ð5Þ rk 5 Ft
4. Discussion It has been shown that the three radii discussed in this paper differ from each other. Only in special cases are they the same. On soft, deformable soil, the wheel centre-bottom distance is not equal to the other two radii. The physical process occurring at the tyre–soil interface is best characterised by the kinetic and kinematic radii. These two are equal only for a rigid wheel moving on a rigid surface at zero slip. (However,
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Table 4 Comparison of the kinetic radius and the wheel centre-bottom distance Test segment no 1 2 3 4 5 6 7 8 9
Tyre inflation pressure (p), bar
The distance between the centre and the bottom of the tyre (rcb), cm
Kinetic rolling radius(rk), cm
Difference %
14 14 14 14 10 08 14 12 06
868 871 872 880 832 809 879 852 791
840 830 858 848 829 800 877 838 789
32 47 16 36 10 11 02 16 03
they may be equal for an instant as they vary due to vibrations caused by uneven soil profile.) In case of a tyre moving on soft soil, there is always slip-deformation and, hence, slip is always present. Therefore the two radii will always differ from each other. Both characterise the tyre–soil interaction; rs characterises the interaction from the viewpoint of kinematics (slip), whereas rk is a characteristic of the force interplay between the tyre and the soil. The difference between these two radii is best exemplified at 100% slip, where rs is zero, but this is when the wheel exerts the highest tractive force, so rk must be more than zero.
When the wheel rotational speed is not constant, the driving torque must be reduced by the inertia needed to accelerate the wheel.
Acknowledgements The author wishes to express his thanks to John Deere Mannheim (Germany) for their help in the performance of our tests. Also I am grateful to the MTA (Hungarian Academy of Sciences) for their support of the research described herein.
References 5. Conclusions There are three different radii for a tyre rolling on offroad terrain: (1) the distance between the centre and the bottom of the tyre, which is influenced by loading conditions and inflation pressure; (2) the kinematic rolling radius determined by slip; and (3) the kinetic radius, which is the distance between the centre of the wheel, where the driving torque acts, and the tractive force generated in the tyre-soil interface. The kinematic rolling radius has long been derived from the distance covered in one revolution. The kinetic rolling radius is obtained from the driving torque and the tractive force, which is generated in the tyre–soil interface. When the tractor moves with uniform speed, the angular acceleration of the driving wheel is zero. In this case, the kinetic rolling radius is defined as the distance between the centre of the wheel and the resultant of the elementary tractive forces generated in the tyre–soil interface.
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