Revisiting a Tire Wheel Traction Model Using a Modified Width
of the Tire-ground Contact Area
Radu Roșca 1, Petru-Marian Cârlescu 1, Ioan Țenu 1 and Virgil Vlahidis 1
1
    Iaşi University of Life Sciences (IULS), Sadoveanu St., no. 3, Iaşi, 700490, Romania


                 Abstract
                 Over the years a semi-empiric model describing the interaction between the agricultural tire
                 wheel and terrain was developed and improved. The model was used to predict the traction
                 force and traction efficiency of the driving wheel, based on the formulae given in the ASAE
                 D497.7 standard. The results provided by the model were validated by experimental data based
                 on a goodness-of-fit analysis. For all the previous models the goodness-of-fit analysis has
                 proved that the theoretical results were very well correlated with the experimental data for the
                 traction force (values of the Pearson coefficient r 2 exceeding 0.9), while less reliable results
                 were obtained for the traction efficiency (values of the Pearson coefficient r 2 comprised
                 between 0.20 and 0.65, depending on the geometry taken into account for the shape of the tire
                 cross-section). In order to improve the goodness-of-fit between the model data and the
                 experimental data in the present study three models for the tire-ground interaction were
                 considered: the initial one, developed earlier, which took into account a constant width of the
                 tire, and two modified models, based on an elliptical shape of the tire cross-section, with the
                 width of the tire-ground contact patch smaller than the tire cross-section width (major axis of
                 the ellipse which defines the shape of the cross-section). In the first of these two models the
                 sheared area in the tire-ground interface has varied with the travel reduction of the wheel, while
                 for the second one the shear area was considered constant. Based on the goodness-of-fit
                 analysis it was concluded that the constant shear area model provided the best results, with the
                 Pearson correlation coefficient significantly improved for the traction efficiency (r 2 = 0.838),
                 while preserving a high value for the traction force (r2 = 0.896). The model could provide
                 reliable results regarding the traction force and traction efficiency, in certain soil conditions,
                 thus removing the need for experimental tests.

                 Keywords 1
                 shear area, super-ellipse, traction force, traction efficiency

1. Introduction
    The use of tractor simulation and prediction models is an essentially low-cost approach for
evaluating the significance of different factors affecting the actual tractor operation. Under these
circumstances, traction prediction modelling has been driven by the fact that the tire-soil interface is
the primary cause of low traction efficiency (estimated to be on the order of 60% on farmland, for
transmission efficiency of nearly 90%) [1], without having to build physical prototypes or perform
numerous field tests.
    The tire-soil interaction has been studied by numerous authors in the attempt to develop traction
models for the agricultural tractor driving wheels. The basis for the traction models was established by
Bekker in 1956 [2, 3] by developing the basic theory of the wheel-soil interaction. Such models are
based on empirical, semi-empirical or analytical methods. Tiwari et al. [4] emphasized some of the
difficulties limiting the widespread use of analytical models, including the complex tire-soil interaction,

Proceedings of HAICTA 2022, September 22–25, 2022, Athens, Greece
EMAIL: rrosca@uaiasi.ro (A. 1); pcarlescu@yahoo.com (A. 2); itenu@uaiasi.ro (A. 3); rogrimex@gmail.com (A. 4)
ORCID: 0000-0003-4222-2165 (A. 1); 0000-0003-1039-0412 (A. 2); 0000-0001-5633-522X (A. 3); 0000-0001-9450-3749 (A. 4)
              ©️ 2022 Copyright for this paper by its authors.
              Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
              CEUR Workshop Proceedings (CEUR-WS.org)




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based on a large number of tire and soil parameters. Semi-empirical models are based on the vertical
deformation of the soil and on the shear deformation of the soil under a traction device. Empirical
models are simpler than analytical and semi-empirical models; however, their applicability is limited
to cases in which the service and experimental conditions used to develop the model are similar [5].
    Dimensional analysis has also been used in order to develop traction models [6].
    Semi-empirical models for wheel-soil interaction, based on Bekker’s theory [6], offer a valid
framework for modelling the traction performance of the tractor-soil system [7].
    The shape of the tire-soil contact area directly affects the traction performance of the driving wheel
and numerous algorithms for estimating the contact area of traction tires on agricultural ground were
developed over time. The geometry and mechanical properties of the tires should be considered when
developing models for the traction of agricultural vehicles [8], but not all the models take into account
the tire volume and tire stiffness [9].
    In this context, the present paper uses a previously developed traction model [10], further enhancing
it by taking into account the geometry and deformation of the tire cross-section in the tire-soil contact
area and also a modified width of the contact patch. The aim of this approach was to obtain a better
goodness-of-fit between the predicted values of the traction efficiency and the experimental ones.
    The paper contains the following sections: "Materials and methods", where the theoretical basis for
the models are based are presented, "Assessment of the models", where the comparative results between
models data and experimental data are presented, and the "Conclusions" section.

2. Materials and Methods

    The model for the tire-soil interaction was based on the schematics presented in Figure 1a [10],
assuming that, under the vertical load G, the radius of the tire, in the contact area, increases from r0 to
rd, while the tire sinks into the soil to the depth zc.
    The shape of the tire-soil contact patch is considered to be a super ellipse (Figure 1b) [11].




                               a)                                                     b)
 Figure 1: Schematics of the model [10]
 a) tire and soil deformation; b) contact patch
 zp-tire deflection under load; zc-tire sinkage into the soil; lc-length of the contact patch; l1w-width of
 the contact patch.



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    The length of the contact patch (major axis of the contact super ellipse, lc) is given as:
                                     lc = 2rdsin = 2 r0sin.                                                                   (1)
    Assuming that the tire is perfectly elastic we get:
                                          G = Z = q p  V ,                                                                        (2)
where qp is the volume stiffness of the tire and Vp is the variation of the tire volume in the tire-ground
contact zone.
   In order to evaluate the variation of the tire volume, the initial model [10] assumed that the width of
the tire footprint was equal to the width b of the tire (Figure 2). In the upgraded model it was assumed
that the transversal cross-section of the tire is an ellipse [12], as shown in Figure 3. With no vertical
load, the major axis of the cross-section is b (tire width) and the minor axis is h (section height); under
load, the minor axis of the section decreases to h-zp and the major axis increases to lw. The value of lw
was calculated assuming that the perimeter of elliptical cross-section remains the same for the initial
and final shape:
                                     l w = b 2 + 2  h  z p − z p2 .                                  (3)




 Figure 2: Tire deformation in the contact area                          Figure 3: Deformation of the tire cross-section
 (initial model)                                                         in the contact area (upgraded model)

    According to Figures 1b and 3 the width of the tire-soil contact area (minor axis of the super ellipse)
is l1w <lw:

                                            l1w = lw  1 −
                                                                (h − z p − 2  zc )2 .
                                                                    (h − z p )2
                                                                                                                                     (4)

  The tire volume change Vp was calculated as:
                          Vp = 0.5      r0  b  h −   rd  lw  (h − z p ) .                                              (5)
  The pressure-sinkage relationship [2]:
                                                p = k  zn                                                                           (6)
was applied for the case of the tire under load, finally leading to:
                                  2
                         G = k  rdn+1  cos( −  ) − cos  n  b( )  cos(  −  )  d ,
                                                                                                                                    (7)
                                   0
    The assumption that the tire is perfectly elastic (eq. 2, 5 and 7) led to the following equation:
     2

                                                                                                                              
  k  b( )  rdn + 1  cos( −  ) − cos  n  cos(  −  )  d == qp  0 ,5      r0  b  h −   rd  lw  (h − zp ) ,
                                                                                                                                    (8)
      0
where:




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                                       l c2  l 1kw − 2  l 1w  rd  sin( −  )k
                          b( ) = 2  y = k                                                        (9)
                                                               l ck
   The schematics shown in Figure 1a also led us to the following equations:
                                  z c = r0 − z p − r0  cos                                      (10)
                                  z p = r0  (1 − cos  ) − rd  (1 − cos  )                     (11)
  The system consisting of equations (1), (8), (10) and (11) was solved in an iterative process, using a
computer program based on the flowchart shown in Figure 4.




Figure 4: Flowchart of the computer program [10]

   The computer program was written in MS QuickBasic and Figure 5 presents a partial view of the
program listing and the results window provided by the program.
   The area of the tire-ground contact patch was calculated as (Figure 1b):




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                                                     lc / 2

                                          At = 2      b( )  dx
                                                       0
                                                                                                         (12)

where b() is given by equation (9).
   The traction force developed by the tire depends on the maximum shear stress which may be
achieved in the tire-soil interface, given by the Mohr-Coulomb equation [13]:
                                          max = c + p  tg                           (13)
where c is soil cohesion [kPa], p is the vertical pressure [kPa] and  is the internal friction angle.




 Figure 5: Program listing and results window

    The maximum shear stress max is not immediately available at the beginning of the contact area, but
is reached asymptotically, according to the Janosi and Hanamoto equation [10]:
                                                              − 
                                                                 J
                                            =  max   1 − e K                                   (14)
                                                                   
                                                                   
where K is the soil shear deformation modulus and J is the shear displacement.
    Shear displacement is obtained by integration of slip velocity along the contact patch [14]:
                                     J = rd  2   − (1 − s )  sin 2                           (15)
where s is the wheel slip (wheel travel reduction).
    Some authors [15] took into account the fact that the sheared area in the tire-ground interface varies
with the travel reduction of the wheel, s:
                                               
                                       Ash = At  1 − (1 − s )  e −t                              (16)
    This assumption was also investigated in our work; as a result, the maximum traction force provided
by the wheel, considered as the product of shear stress and shear area, was calculated as:
    •    Ft =   At , when the entire contact area was considered;
    •    Ft =   Ash , when the hypothesis of the variable shear area was considered.
    The net traction force and traction efficiency were calculated using the formulae given by the ASAE
D497.7 standard [16]:
                                                 FN = Ft − Rr                                       (17)
                                        tr = (1 − s )  (1 − Rr / Ft )                             (18)
where Rr is the rolling resistance of the wheel, calculated with the help of the wheel numeric, Bn, and
cone index, CI [10].




                                                              71
3. Assessment of the Models

    Field experiments were performed in order to assess the models. A ploughing equipment, composed
of the Romanian U-650 tractor and the variable width P2V plough was used for the tests.
    The tractor was equipped with the adequate implements for measuring the traction force and wheel
slip [10].
    Table 1 presents the main features of the driving wheel and tire; soil characteristics for the test field
are shown in Table 2.
    The maximum wheel slip during ploughing was 30% and different traction forces were achieved by
modifying the working width of the plough.
    The experimental data were compared with the data predicted by:
    •    the initial model [10];
    •    the upgraded model, based on the variable shear area (given by eq. 16);
    •    the upgraded model, based on the constant shear area (given by eq. 12).

Table 1.
Characteristics of the U-650 tractor and drive wheels
                        Item                                                 Value
         Load on the driving wheel [kN]                                      11.75
                     Type of tire                                          14.00 – 38
           Overall diameter of tire [m]                                       1.58
                   Tire width [m]                                            0.367
                   Lug width [m]                                              0.04
                   Lug length [m]                                             0.24
                   Lug height [m]                                            0.025
            Distance between lugs [m]                                        0.195
    Transversal radius of the under tread [m]                                0.300

Table 2
Characteristics of the test soil
                     Item                                                  Value
      Soil deformation modulus, K [m]                                      0.05
                                            k                                55
 Coefficients for the sinkage equation
                                            n                               1.3
           Soil cohesion, c [kPa]                                            25
       Angle of internal friction,  [0]                                     32
     Cone penetrometer index, CI [kPa]                                      970

    A goodness-of-fit analysis was performed in order to compare the predicted data with the
experimental ones; the analysis took into account the following criteria [17]:
    •    percentage of points within the 95% confidence interval of data (Pw95CI);
    •    mean absolute deviation (MAD);
    •    root mean squared deviation (RMSD);
    •    mean scaled absolute deviation (MSAD);
    •    Pearson correlation coefficient r2.
    The goodness-of-fit analysis was performed for both the traction force and the traction efficiency.
    As mentioned before, three models were evaluated: the initial one, the one assuming that the tire
cross-section is an ellipse and that the tire-soil shear area is affected by wheel slip and the one also
assuming an elliptical shape of the tire cross-section but which took into account a constant shear area.
    The results characterizing the tire-ground contact are summarized in Table 3. These data show that,
due to the fact that the width of the contact patch was considered to be equal with the tire width in the
initial model, that model has predicted a larger contact area compared with the other two models; as a


                                                      72
result, the maximum shear stress at the tire-ground interface was lower for the initial model. The
dynamic radius obtained for this model was greater and therefore led to a greater length of the contact
patch.

Table 3
Parameters for the tire-ground contact
                                                  Tire-ground model
     Item          constant width of tire      deformable cross-section,    deformable cross-section,
                      (Rosca, 2014)               variable shear area          constant shear area
     lc [m]                0.530                         0.534                        0.534
    lw [m1                 0.367                         0.384                        0.384
    l1w [m]                   -                          0.222                        0.222
     zc [m]               0.0271                        0.0275                       0.0275
     rd [m]                1.400                         1.257                        1.257
    At [m2]                0.145                        0.1086                       0.1086
   max [kPa]               53.3                          61.2                         61.2

   There were no differences between the upgraded models as far as the tire-ground contact geometry
was concerned because the effect of wheel slip on the shear area was taken into account at a later stage
(see Figure 4), for the prediction of traction force and traction efficiency.
   The results referring to the traction force are presented in Figure 6 and Table 4.
   The results depicted in Figure 6 prove that the initial model and the one based on the elliptic shape
of the tire cross-section and a constant shear area have provided quite similar predictions as long as
wheel slip did not exceed 25%, while the model based on the on the elliptic shape of the tire cross-
section and a variable shear area has provided significantly lower values for the traction force.




Figure 6: Predicted and experimental data for traction force




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   The results of the goodness-of-fit analysis, shown in Table 4, have confirmed that the model based
on the elliptical shape of the tire cross-section and variable shear area has provided the most inaccurate
results, as it predicted much lower values of the traction force than the ones obtained from the
experimental tests. Despite the achievement the highest value of the Pearson correlation coefficient, all
the other items were unfavorable: thus, the value of MSAD criterion means that, on average, the model
was 8.817 standard errors off from the experimental data; the percentage of model predictions that lie
within the 95% confidence interval of each corresponding experimental data point was only 11.1%; the
mean absolute deviation between each model point and the corresponding experimental point was
1.6408 (the highest value of all the models taken into account).
   On the other hand, for the model based on the constant shear area, although the value of r2 was lower
(but only by 2.7% in comparison with the initial model), the values of the all the other items taken into
account within the goodness-of-fit analysis (Pw96CI = 88.9%; MSAD = 1.184 etc.) led us to the
conclusion that this model was the most accurate for describing the tire-soil interaction.

Table 4
Results of the goodness-of-fit analysis for traction force
                                                  Tire-ground model
    Item        constant width of tire        deformable cross-section,      deformable cross-section,
                   (Rosca, 2014)                  variable shear area           constant shear area
       2
      r                 0.921                            0.981                        0.896
   MAD                 0.3353                           1.6408                        0.3981
   MSAD                 3.116                            8.817                        1.184
  Pw95CI                 66.7                             11.1                         88.9
   RMSD                0.4379                            1.685                        0.4952

    Figure 7 and Table 5 present the results referring to the traction efficiency. As it can be concluded
from Table 5, the main problem of the initial model was that it provided relatively poor results regarding
the traction efficiency (r2 = 0.203) and this was the driving force that steered us towards the
improvement of that specific model.
    The data presented in Figure 7 clearly show that the model based on the elliptical shape of the tire
cross-section and constant shear area was the most adequate for describing the tire-soil interaction
process in terms of traction efficiency, while the values predicted by the model based on variable shear
area were significantly lower than the corresponding experimental data.
    The results shown in Table 5 prove that the model based on a constant shear area has predicted more
reliable results than the initial one due to the following:
    •    a significantly higher value of the Pearson correlation coefficient in comparison with the initial
    one (0.838 vs. 0.203)
    •    all the predicted data points were within the 95% confidence interval of the corresponding
    experimental data (Pw95CI = 100);
    •    all the other items taken into account for the goodness-of-fit analysis had lower values than the
    ones recorded for both the initial model and the one using a variable shear area.

4. Conclusions

    Over the years a semi-empiric model describing the interaction between the agricultural tire wheel
and terrain was developed and improved. The model was used to predict the traction force and traction
efficiency of the driving wheel.
    In order to improve the goodness-of-fit between model and experimental a new geometry of the tire
cross-section was taken into account, assuming that the tire has an elliptical shape, which deforms under
vertical load while still preserving the elliptic shape.
    The data predicted by the original model were compared with the ones predicted by the modified
model and with experimental data, collected during ploughing.



                                                     74
   The best results were obtained when the constant tire-soil shear area was considered; the Pearson
correlation coefficient has significantly improved for the traction efficiency (r 2 = 0.838), while
preserving a high value for the traction force (r2 = 0.896).
   The percentage of model-predicted data points which fall within the 95% confidence interval of
experimental data has increased to 88.9% for the traction force and to 100% for the traction efficiency.




Figure 7: Predicted and experimental data for traction efficiency

Table 5
Results of the goodness-of-fit analysis for traction efficiency
                                                  Tire-ground model type
   Item          constant width of tire          deformable cross-section,      deformable cross-section,
                      (Rosca, 2014)                  variable shear area           constant shear area
     r2                    0.203                            0.726                         0.838
   MAD                    0.0569                           0.0953                         0.0198
  MSAD                     5.147                            1.966                         0.309
 Pw95CI                     55.6                             88.9                           100
  RMSD                     0.075                           0.1279                         0.0231
    It should be emphasized that, in this stage, the presented results are valid only for the specific soil
conditions from the test field. More experiments on terrains with different soil characteristics are needed
in order to extend the validity of the upgraded model, based on the constant shear area.
    One promising direction seems to be the use of a rheological model for the shear deformation of
soil, but this is still a difficult task as no reliable data regarding the rheological properties of different
soils are available [18].

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