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 r2 exceeding 0.9), while less reliable results were obtained for the traction efficiency (values of the Pearson coefficient r2 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 (r2 = 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. shear area, super-ellipse, traction force, traction efficiency
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,
2022 Copyright for this paper by its authors. 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 [
In this context, the present paper uses a previously developed traction model [
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.
The model for the tire-soil interaction was based on the schematics presented in Figure 1a [
The shape of the tire-soil contact patch is considered to be a super ellipse (Figure 1b) [11]. The length of the contact patch (major axis of the contact super ellipse, lc) is given as: lc = 2rdsin = 2 r0sin.
Assuming that the tire is perfectly elastic we get:
G = Z = qp V , 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 [
Vp = 0.5 r0 b h − rd lw (h − zp ) .
p = k z n was applied for the case of the tire under load, finally leading to:
2 G = k rdn+1 cos( − )− cos n b( ) cos( − ) d , b( ) = 2 y = k lc2 l1kw − 2l1w rd sin( − )k
lck The schematics shown in Figure 1a also led us to the following equations: zc = r0 − zp − r0 cos (9) (10) zp = r0 (1− cos )− rd (1− cos ) 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 where c is soil cohesion [kPa], p is the vertical pressure [kPa] and is the internal friction angle. (13) (17)
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 [
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]:
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 [
FN = Ft − Rr
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 [
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 [
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 crosssection and a variable shear area has provided significantly lower values for the traction force.
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.
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.
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 (r2 = 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.
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]. 5. References [1] H. Catalan, P. Linares, V. Mendez. "Tractor_PT: A traction prediction software for agricultural