Development and Further Refinement of a Semi- empirical Wheel Traction Model

Abstract

In this paper the theoretical basis, evolution and results of the field tests regarding the modelling of the agricultural tire-soil traction model are presented. The model is a reasonable compromise between the simpler empirical models, for which the range of applicability is limited to the cases having similar conditions to the ones from which the models were derived, and the analytical models, which require in-situ evaluation of a large number of soil properties. The model is based on the Mohr-Coulomb failure criteria, assuming that the maximum traction force is limited only by the soil shear strength. A computer program was developed in order to solve the system of equations introduced by the model, with the traction force and traction efficiency being evaluated. In the initial model the tire-soil contact patch was assumed to be an ellipse and no modifications of the tire cross-section were taken into account. Further developments took into account a super ellipse shape of the tire-ground contact surface, effect of tire slip over the contact patch area and deformation of the tire cross-section.

Keywords

• traction model
• Mohr-Coulomb failure criteria
• goodness-of-fit
• traction force
• traction efficiency

-

1 Introduction wheel-soil interaction models into empirical, analytical and numerical models.

Empirical methods are mainly based on soil properties (cone index, plate sinkage, shear strength) using similitude and dimensional analysis.

The semi-empirical (analytical) models represent a physical-based approach, which considers the mechanics of the wheel-soil interaction and are suitable for practical applications (Battiato&Diserens, 2017). In the semi-empirical models, the shear deformation of soil is considered; the models are based on soil parameters obtained by the means of a bevameter technique (penetration and shear tests), assuming that the vertical deformation of soil is similar to the deformation under a sinking plate, while the shear deformation of soil under a traction device is similar to the shear action of a torsion device (Tiwari et al., 2010). The parameters involved in the equations are determined experimentally.

This paper presents the evolution of a semi-empirical tire-ground interaction model; while the basic elements of the model remained the same, different assumptions regarding the shape of the tire-ground contact area and the tire deformation were used in time. 2 Tire-ground interaction model

2.1 Initial model

We have chosen to use a Bekker type model, assuming that the circumferential force limits the value of the wheel net traction force.

In order to evaluate the dimensions of the contact area, the model assumes that, under the vertical load (G, Fig. 1), the wheel sinks into the soil, reaching depth (zc) and the load induces tire deflection (zp) (Rosca et al., 2004). As a result, the radius of the contact patch becomes rd (rd >r0), and the circular length of the contact patch is: lc = 2×b×rd = 2×a×r0 . ( 1 ) 0

Using the Bekker equation (Bekker, 1969) and assuming the tire is perfectly elastic, we get:

2b 4 4 k × ò rdn+1 × [cos(b - j) - cos b]n × dj + × q p × a3 × r02 , 3 q p × b3 × rd2 =

3 z c = r0 - z p - r0 × cosb , z p = r0 × (1 - cos a) - rd × (1 - cosb) , where qp is the tire volume stiffness, zp is the tire deformation due to the vertical load G (G = qp·DVp) and zc is the soil deformation.

The tire change in volume due to deflection DVp was evaluated considering that the tire radius increases from r0 to rd as the tire flattens in the contact area, while the tire width was considered constant, as shown in Fig. 2 (Ghiulai&Vasiliu, 1975). ( 2 ) ( 3 ) ( 4 )

The shape of the tire-ground contact patch was assumed elliptical; the minor axis, lw was calculated using zp (Upadhyaya & Wulfsohn, 1990), while the major axis, lc, results when solving the system of equations ( 1, 2, 3, 4 ).

A computer program, based on an iterative process, was used in order to solve the system of equations ( 1, 2, 3, 4 ) and find lc, zp, zc and rd (Rosca et al., 2014).

The maximum traction force was assumed to be limited only by the maximum shear strength of the soil, given by the Mohr-Coulomb criterion, based on soil cohesion and the internal friction angle of the soil.

According to Wulfsohn & Upadhyaya (1992) and Lach (1996), the shear stress developed at the interface between the vehicle tire and the terrain is a function of shear displacement J:

ae - J ö t = tmax × çç1 - e K ÷ ,

÷ è ø where K is the soil shear deformation modulus and J is the shear displacement, given by the relation presented by El-Gawwad et al. (1999).

The net traction force and traction efficiency were calculated with the formulae given by the ASAE S296 standard. ( 5 )

2.2 Model development

Variable shear area. The first improvement of the model took into account the fact that, according to some authors (Komandi, 1993; Abd El-Gawwad et al., 1999) the shear area varies during the traction as a function of slip:

Ash = A t × [1 - (1 - s)× e-Y ], ( 6 ) where Y = c1 × lm1 × sm2 , with the values of the constants c1, m1 and m2 depending c upon the nature of the ground surface. Shape of the contact patch. Another improvement of the tire-ground interaction model is to consider the shape of the contact patch to be a super ellipse, based on the results presented by Keller (2005), who also considered the contact patch as a super ellipse and made measurements of the vertical stress below tires using compression cells. The value of the super ellipse exponent was calculated with the formula presented by Keller (2005):

n = 2.1× (b × d)2 + 2 where b is the tire width and d is the outer diameter. ( 7 ) Deformation of the tire cross-section. The next step consisted in approximating the shape of the tire cross-section with an ellipse (Koutný, 2007), as shown in Fig. 4a. Under the effect of vertical load (G, Fig. 1), the cross-section was deformed (Fig. 4b); thus, the minor semi-axis has decreased to h-zp, while the major axis has increased from b to lw.

a) b) Fig. 4. Tire cross-section deformation a) tire section parameters; b) tire section deformation under load; di – rim diameter; h – tire section height; b-tire width (undeformed); lw – tire width (under load); zp - tire deflection under vertical load

The major axis of the ellipse was calculated assuming that its perimeter remained unchanged: l w = b 2 + 2 × h × z p - z 2p . ( 8 ) 2.3

Experimental tests

In order to validate the theoretical results, field tests were developed, using the U650 tractor, equipped with the P2V plow; Table 1 presents the main features of the driving wheel and tire.

During the experiments, drive wheel slip and net traction force were measured directly, for wheel slips up to 30%.

2.4 Goodness-of-fit analysis

In order to evaluate the goodness-of-fit between model and experimental data the following criteria were considered (Schunn & Wallach, 2005): • percentage of points within 95% confidence interval of data (Pw95CI); • mean absolute deviation (MAD; • root mean squared deviation (RMSD); • mean scaled absolute deviation (MSAD); • Pearson correlation coefficient r2. 3 Results and discussion tire cross section was considered, due to the increased value of the contact surface area.

5 10 1%5 20 25 30 Fig. 5. Traction force (variable shear area) 5 10 15 20 25

30

The goodness-of-fit analysis showed that, compared to the previous model, the most significant differences were recorded for the traction efficiency: the Pearson correlation coefficient r2 increased from 0.186 to 0.216, the mean absolute deviation (MAD) decreased from 0.058 to 0.051, root mean squared deviation (RMSD) decreased from 0.0752 to 0.0686 and the mean scaled absolute deviation (MSAD) decreased from 5.225 to 4.557.

When referring to the values of the traction force, all the goodness-of-fit parameters recorded better values for the modified traction model.

Constant area 2.24 2.8 3.75 4.13 4.5 4.9 + 10,25 Constant area 0.6824 0.6990 0.6900 0.6740 0.6500 0.6170 + 3.1% model experiment b const. b elipse Experiment experiment model, b const. model, b elipse 0 5 10 25 30

35 15

20 slip [%] 5 10

The time evolution of a semi-empirical model for the prediction of traction performance of a tractor driving wheel is presented in this study.

The model was developed in several stages: a) constant tire-soil shear area, elliptical shape of the contact patch and no deformation of the tire cross section; b) variable tire-soil shear area (depending on wheel slip), elliptical shape of the contact patch and no deformation of the tire cross section; c) variable tire-soil shear area, super ellipse shape of the contact patch and no deformation of the tire cross section; d) variable tire-soil shear area, super ellipse shape of the contact patch and deformation of the tire cross section.

A goodness-of-fit analysis, based on several statistic criteria, was performed in order to validate the model; model predicted data and experimental data from ploughing tests were used in this analysis.

The successive development stages led to a better fit between theoretical and experimental data referring to traction force and traction efficiency. 0.8 0.7 0.7 y0.6 c n iec0.6 iff e ion0.5 t c a rT0.5 0.4 0.4 0.3 20. ***, ASAE D497.7, (1999), Agricultural Machinery Management Data. St. Joseph, Michigan, U.S.A

References

1. Abd El-Gawwad , K.A , Crolla , D.A. , Soliman , A.M.A. , El-Sayed , F.M. ( 1999 ) Offroad tyre modelling IV: extended treatment of tyre-terrain interaction for the multispoke model . Journal of Terramechanics , 36 , p. 77 - 90 .

2. Battiato , A. , Diserens , E. , ( 2017 ) Traction performance simulation on differently textured soils and validation: A basic study to make traction and energy requirement accessible to practice . Soil&Tillage Research , 166 , p. 18 - 32 .

3. Bekker , G.M. ( 1969 ) Introduction to terrain-vehicle systems . Univ. of Michigan Press, Ann Arbor.

4. Diserens , E. , Défossez , P. , Duboisset , A. , Alaoui , A. , ( 2011 ) Prediction of the contact area of agricultural traction tyres on firm soil . Biosystems engineering , 110 , p. 73 - 82 ;

5. Ghiulai , C. , Vasiliu , Ch. , ( 1975 ) Dynamics of ground vehicles (in Romanian), Didactics and Pedagogy Publishing House, Bucharest, Romania.

6. Hallonborg , U. ( 1996 ). Super ellipse as a tyre-ground contact area . Journal of Terramechanics 33 , p. 125 - 132 .

7. Hambleton , V. , Drescher , A. ( 2008 ) Development of improved test rolling methods for roadway embankment construction . Research 2008 - 08 , University of Minnesota, Minnesota Department of Transportation Research Services Section (available at http://www.lrrb.org/PDF/200808.pdf).

8. Janosi , Z. , Hanamoto , B. , ( 1961 ) The analytical determination of drawbar pull as a function of slip, for tracked vehicles in deformable soils . Proceedings of the 1st Intl. Conference Mech. Soil-Vehicle Systems. Turin , Italy.

9. Keller, T., ( 2005 ) A model for prediction of the contact area and the distribution of vertical stress below agricultural tyres from readily available tyre parameters . Biosystems Engineering , 92 ( 1 ), p. 85 - 96 .

10. Komandi , G. ( 1993 ) Reevaluation of the adhesive relationship between the tire and the soil . Journal of Terramechanics , 30 ( 2 ), p. 77 - 83 .

11. Koutný , F. , ( 2007 ) Geometry and mechanics of pneumatic tires . Zlin, CZE (available at: http://wanderlodgegurus.com/database/Theory/TireGeometry.pdf).

12. Lach , B. ( 1996 ) The influence of tire inflation pressure - a comparison between simulation and experiment . 1st East European Conference of the ISTVS, Wroclaw.

13. Rosca , R. Rakosi , E. , Manolache , Gh. ( 2004 ) Wheel Traction Prediction - A Comparison Between Models and Experimental Data . SAE Technical Paper Series 2004 - 01 -2707, SAE International, Warrendale, PA.

14. Roşca , R. , Cârlescu , P. , Ţenu , I. , ( 2014 ) A semi-empirical traction prediction model for an agricultural tyre, based on the super ellipse shape of the contact surface . Soil&Tillage Research , 141 , p. 10 - 18 .

15. Schmid , I. C. ( 1995 ) Interaction of the vehicle terrain results from 10 years research at IKK . Journal of Terramechanics , 32 ( 1 ), p. 3 - 27 .

16. Schunn , C.D. , Wallach , D. , ( 2005 ) Evaluating goodness-of-fit in comparison of models to data. Psychologie der Kognition: Reden and Vorträge anlässlich der Emeritierung , W. Tack (Ed.), University of Saarland Press, Saarbrueken, Germany, p. 115 - 154

17. Tiwari , V.K. , Pandey , K.P. , Pranav , P.K. ( 2010 ) A review on traction prediction equations . Journal of Terramechanics , 47 , p. 191 - 199 .

18. Upadhyaya , S.K. , Wulfson , Dvoralai ( 1990 ) Relationship between tire deflection characteristics and 2-D tire contact area . Transactions of the ASAE , 33 ( 1 ), p. 25 - 30 .

19. Wulfson , Dvoralai, Upadhyaya, S.K. ( 1992 ) Prediction of traction and soil compaction using three-dimensional soil-tyre contact profile . Journal of Terramechanics , 29 ( 6 ), p. 541 - 564 .