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pulled drive wheel
Journal of Terramechanics 40 (2003) 33–50 www.elsevier.com/locate/jterra
Traction performance of a pushed/pulled drive wheel I. Shmulevicha,*, A. Osetinskya a
Technion—Israel Institute of Technology, Agricultural Engineering, Haifa, 32000, Israel
Abstract A comprehensive method for prediction of off-road driven wheel performance is presented, assuming a parabolic wheel–soil contact surface. The traction performance of a driven wheel is predicted for both driving and braking modes. Simulations show significant non-symmetry of the traction performance of the driving and braking wheels. The braking force is significantly greater than the traction force reached in the driving mode. In order to apply the suggested model for prediction of the traction performance of a 4WD vehicle, the load transfer effect was considered. Simulated traction performances of front and rear driven wheels differ significantly, due to the load transfer. In the driving mode, the rear driven wheel develops a net traction force greater than that of the front wheel. On the other hand, in the braking mode the front driven wheel develops a braking force significantly greater than that of the rear driven wheel due to a pushed/pulled force affected by the load transfer. The suggested model was successfully verified by the data reported in literature and by full-scale field experiments with a special wheel-testing device. The developed approach may improve the prediction of off-road multi-drive vehicle traction performance. # 2003 ISTVS. Published by Elsevier Ltd. All rights reserved. Keywords: Net traction; Gross traction; Off-road; Load transfer; Soil–tire interaction
1. Introduction For many years numerous researchers have reported predictions of off-road wheel performances. Some dealt in analytical approaches, others in semi-empirical or empirical approaches. The diversity in the approaches of research on off-road tire performance points to the complexity of the issue. Each of the above-mentioned approaches has some limitations: the analytical approaches are difficult to use; empirical equations are limited to the tested cases and most semi-empirical methods * Corresponding author. Tel.: +972-4-829-2626; fax: +972-4-822-1529. E-mail address: [email protected] (I. Shmulevich). 0022-4898/$20.00 # 2003 ISTVS. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jterra.2003.09.001
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Nomenclature a b c dn dv e0 GT HBR j K Kc Kp kc kv kf MRc n in R Rr S si Va Vaj , and Voj Vj Vt,Va W W x, z x0 z0 vo ! , v
parameter (m1) wheel width (m) soil cohesion (kPa) elementary horizontal force (kN) elementary vertical load (kN) eccentricity of the center of the wheel in relation to the rear point of the contact surface 0 (m) gross traction force (kN) force required to pull the fully braked wheel (kN) shear displacement of the soil at point B (m) shear deformation modulus (m) carcass stiffness of the tire (kN/m) inflation pressure dependence of the tire (kN/m/bar) cohesive modulus of deformation (kN/m(n+1)) coefficient of vertical stiffness of the wheel (kN/m/m) frictional modulus of deformation (kN/m(n+2)) motion resistance due to soil compaction (kN) exponent of soil deformation inflation pressure of the tire (bar) unloaded radius of the wheel (m) rolling radius (m) slip (dec) slip index (m) magnitude of the actual longitudinal velocity of the wheel (m/s) tangential components of the actual longitudinal velocity and velocity of point B in relation to the center of the wheel, respectively (m/s) slip velocity of point B (m/s) theoretical and actual velocity (m/s) vertical load applied to the wheel (kN) dynamic variation of the vertical load on the wheel (kN) coordinates of the contact surface (m) length of the contact surface (m) rut depth (m) angle between the radius-vector of point B and vertical axle (rad) vertical displacement of the elementary contact area of the wheel (m) deflection of the wheel on a hard surface (m) angle of internal friction (rad) angular velocity of the wheel (s1): !=d (vo)/dt normal and shear stresses acting on the elementary contact area (kPa) vertical component of surface traction (kPa): v=dpv/(bdx)
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focus on predicting separate performances in driving and braking (pulled/pushed) modes. An external implement or an internal interaction between driven axles of a 4WD vehicle could, due to load transfer, significantly affect the performance of each driven wheel in the vehicle. Moreover, it could also be changed due to variations in terrain. Wheel properties and soil conditions affect traction performance differently, and cause the wheel to behave differently in driving and braking modes. In order to optimize the traction performance of the vehicle, the model for wheel traction performance should also be extended to the braking mode and modified to consider the load transfer effect. There are various approaches to analyze the traction performance of a vehicle with respect to load transfer. Some of them are based on the multi-body dynamic analysis of a moving vehicle or modeling with commercial CAD programs. These programs require large amounts of computing resources and have difficulties in evaluating the soil and the tire dynamic parameters. Based on a semi-empirical model proposed by Bekker [1] or based on empirical approaches, several researchers have developed models to predict vehicle traction performance with respect to the load transfer effect. These models predict the traction performance only for the driving mode and do not cover the braking mode. Other researchers studied the traction performance on the scaled models of 4WD tractors or on full-sized tractors in order to experimentally derive the influence of load transfer. It is difficult to establish general conclusions from the above-mentioned studies based on specific empirical data. The papers by Osetinsky and Shmulevich [2,3] present a comprehensive development of the wheel–soil interaction model considering the influence of load transfer on the traction performance of the wheel. The present research focuses on applying a semi-empirical model for wheel–soil interaction based on the assumption of a parabolic wheel–soil contact area in order to predict the traction performance of a driven wheel affected by load transfer. Verification of the suggested model was accomplished by full-scale field experiments.
2. Traction model development Prediction of traction performance of a driving wheel considers wheel–soil interaction in two interrelated aspects—force equilibrium and kinematics. In order to model the wheel–soil interaction, the form of the contact surface should be determined. In this work a simple analytical approach to the wheel–soil contact surface was suggested. The prediction of the form of the contact surface is based on modeling the interrelated wheel and soil deformations under various operating conditions. Several assumptions were made for the suggested model: the soil serves as a plastic non-linear medium, the wheel moves in steady state condition at a low velocity, its deformations are assumed to be linear elastic and the wheel–soil interaction is two-dimensional (a plane strain problem).
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2.1. Wheel–soil contact surface One of the main assumptions of the present work is that the wheel–soil contact surface can be represented in parabolic form in the longitudinal direction with the apex at the rear point of contact 0 (Fig. 1a) as follows: z ¼ z0 ax2
ð1Þ
Using the vertical soil pressure-sinkage relationship suggested by Bekker [1] the vertical load on the wheel–soil contact surface versus rut depth is derived as follows: rffiffiffiffiffi kc þ bkf z0 n W¼ ð2Þ z ð 3 nÞ 3 a 0 In order to adjust the form of the contact surface to the deflection of a moving pneumatic wheel, its state of equilibrium in the vertical direction is analyzed (Fig. 1b). The elementary vertical load pv acting on the contact area, with length dx, is caused by the vertical wheel stiffness kv and the deflection of a moving wheel: dpv ¼ kv dx
ð3Þ
The force equilibrium of a moving wheel in the vertical direction is found after integrating the elementary vertical load pv along the wheel–soil contact surface, resulting in the following analytical solution: 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 ax0 e0 ðx0 e0 Þ R 2 x0 e20 þ R 2 e20 x0 W ¼ kv þ 2 3 2 2 x e e 0 0 0 arcsin þ arcsin ð4Þ R R
Fig. 1. Driven wheel–soil interaction: (a) force equilibrium; (b) distribution of the elementary vertical load.
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The eccentricity e0 of the center of the wheel is found geometrically according to Fig. 1b as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !! x0 4R 2 2 e0 ¼ 1a x0 ð5Þ 2 1 þ ðax0 Þ2 The wheel properties can be presented by the coefficient of vertical stiffness kv, carcass stiffness Kc and the inflation pressure dependence Kp of the tire. It can be evaluated on the basis of the tire load-displacement relationship for a certain value of inflation pressure Pin as: kv ¼ h
R 2 arcsin
W i x 0 x0 ðR DÞ R
ð6Þ
The parameter (a) of the parabolic contact surface and the rut depth z0 under the moving wheel are derived from a simultaneous solution of Eqs. (2) and (4). A more detailed development of the above-mentioned model and its verification through experimental data is presented by Osetinsky and Shmulevich [2]. 2.2. Kinematics of the moving wheel and soil shear displacements The slip velocity of any peripheral point B on the contact surface (Fig. 2), can be calculated as: Vj ¼ Vaj þ Voj
ð7Þ
For the parabolic contact surface described by expression (1) the soil shear displacement at point B is found by Osetinsky and Shmulevich [2] after integrating the slip velocity over the time required to move from x0 to x:
Fig. 2. Illustration of slip velocity.
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ð x0 axðx 2e0 Þ þ R 2 e20 axðx 2e0 Þ þ R 2 e20 ! dx j¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 x 2 2 2 2 2 ax R e0 þðx e0 Þ 1 þ ð2axÞ
ð8Þ
2.3. Slip and slip index The well-known definition of wheel slip is: S¼
Vt Va Va ¼1 Vt Rr !
ð9Þ
The slip concept is used with high accuracy for studying rigid wheel kinematics, and can also be used for studying pneumatic tires after proper definition of the rolling radius. Its value is affected by operating conditions: the soil and tire properties, vertical load on the wheel, etc. It cannot be accurately predicted analytically and should be measured, according to the ASAE S296.2 standard, as the distance traveled per revolution of the towed wheel (zero slip), divided by 2. The measuring procedure is complicated and sometimes yields unreliable results. The soil shear displacement according to expression (8) is presented as a function of the ratio of the actual longitudinal and angular velocities of the wheel. This ratio is defined as the slip index, si: si ¼ Va =!
ð10Þ
The advantage of the slip index term lies in processing and presenting experimental data, because both velocities can be measured simply and with high reliability. For given work conditions the relationship between the slip and slip index is assumed to be constant: S¼1
si Rr
ð11Þ
The slip index is a dimensional parameter expressed in units of length and its physical meaning is clearly seen from expression (11). It is equal to the rolling radius of the towed wheel while slip is zero. This approach to slip problems was introduced for the first time by Phillips [4], but the above mentioned ratio [Eq. (10)] was not defined explicitly as a system parameter and that was probably the reason why this concept was not developed any further. 2.4. Wheel traction forces and motion resistance The gross traction force is a result of the surface traction in the horizontal direction and can be derived after analyzing the forces acting on the elementary area ds (Fig. 3).
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Fig. 3. Elementary forces applied to the soil contact area ds.
The elementary vertical load dpv is found according to [Eq. (3)] (detailed in paper [2] and the shear force assumed from the shear stress–displacement relationship according to Janosi and Hanamoto [5]: ¼ ðc þ tanÞ 1 ej=K ð12Þ The gross traction force is calculated by integrating dph along the wheel–soil contact area: ð x0 ð x0 v 2ax tan 1 ej=K c 1 ej=K ð2axÞ2 þ1 bdx dph ¼ GT ¼ 1 þ 2axð1 ej=K Þtan 0 0 ð13Þ The distance from the center of the wheel RGT, where the gross traction force is applied (Fig. 1a), is found on the basis of force equilibrium analysis and the respective torque M on the wheel. Assuming that the motion resistance is due to soil compaction [1], the integration of the horizontal component of the soil reaction Rs along the contact surface yields the motion resistance MRc: kc þ bkf ðnþ1Þ z ð14Þ MRc ¼ ð n þ 1Þ 0 It can be seen that expression (14) is identical to the one derived by Bekker [1] for a rigid wheel since it is non-dependent on the form of the contact surface. The net traction force is calculated according to: H ¼ GT MRc
ð15Þ
2.5. Traction performance of pushed/pulled wheel The parabolic model approach was verified by the experimental data reported for a rigid wheel [6] and a 9.5-16R-1 tire [7] operating in the driving mode, using the
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mathematical expressions derived above. The tire stiffness properties used by the model (Kp, Kc) are reported by Lines and Murphy [8]. A special procedure was developed, using Matlab software, to calculate wheel performance as a function of soil and wheel properties versus wheel slip and slip index. For given operating conditions, a force equilibrium in vertical direction results in a predictable parabolic contact surface between the wheel and soil. Based on this predicted surface, shear soil displacement and related stresses are calculated for various slip values, resulting in a prediction of motion resistance and gross and net traction forces. The predicted net traction force and the experimental data are shown versus slip in Fig. 4. The suggested method shows reasonable agreement between the predicted and reported results for the tire and for the rigid wheel: R2 is equal to 0.9687 and 0.8656, respectively. The suggested method was developed for the prediction of traction performance for a single wheel regardless of the operating modes (driving or braking). It can be seen that the predicted traction performance agrees with the experimental data in the driving mode. Thus, an assumption is made that the same method can be applied to predict the traction performance in the braking mode of the pushed or pulled wheel. In order to demonstrate the prediction of traction performance in the braking mode, the model was applied to the above mentioned tire. The results of the predicted traction performance versus slip and versus slip index are shown in Fig. 5. The net traction performance H plotted versus slip (Fig. 5a) runs from minus infinity (braking mode) to the value of one (full slip of the driving wheel). The same graph plotted versus slip index (Fig. 5b) looks like the mirror image of the former one. It runs from zero, indicating full slip of the driving wheel, to plus infinity—the braking mode of the wheel. It can be seen that the traction performance of a driven wheel operating in the driving and braking modes is not symmetrical under the same conditions. The force required to pull or push the tire in braking mode is greater than the traction force that is developed by the tire in driving mode. Under the specified operating conditions, the net traction force achieved by the tire is about 1.25 kN in
Fig. 4. Comparison between predicted and measured net traction force of a driven wheel.
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Fig. 5. Predicted net traction of the 9.5-16R-1 tire: (a) versus slip; (b) versus slip index.
the driving mode (similar to the measured value). However, a force of about 2.2 kN predicted to pull a braking tire, which is 75% more than the net traction force, is achieved in the driving mode. 2.6. Load transfer effect on the traction performance of a wheel The vertical load on the wheel of a moving vehicle changes dynamically due to load transfer. Increasing the vertical load on the wheel increases the traction performance of the wheel. Since the vertical load acting on the moving wheel affects the traction performance, an increase of the load on the wheel magnifies the gross traction force and the torque. On the other hand, reduction of the vertical load decreases the performance values. The gross traction of the tire discussed above, caused by different constant vertical loads, is shown in Fig. 6 as a cluster of dashed curves versus the respective values of slip and slip index. The vertical load applied to the driven tire varies between 1.23 and 9.08 kN. The predicted gross traction of the tire operating under a constant vertical load of 4.0 kN is shown in Fig. 6 by a solid line. Fig. 6b also depicts the maximum and minimum values of the rolling radius Rr max and Rr min of the tire. In order to predict the traction performance of the specified tire, affected by load transfer, the variations of the vertical load on the wheel should be considered. The
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Fig. 6. Predicted gross traction of the driven 9.5-16R-l tire operating as a front and rear wheel: (a) versus slip; (b) versus slip index.
dynamic load can be found from the equation of the equilibrium of the moving wheel [Eq. (13). The actual curve of the traction performance crosses the cluster of curves. Osetinsky and Shmulevich [3] present a comprehensive analysis of the influence of load transfer in four possible combinations of driven wheel operation in a 4WD vehicle: front and rear wheel in driving and braking modes. The load transfer changes the vertical load on the front and the rear wheels in different ways. While operating in the driving mode, the vertical load increases on the rear wheels and at the same time decreases on the front wheels. In the braking mode, the load transfer affects the traction performance of the wheel in the opposite way: the vertical load increases on the front wheel and decreases on the rear one. In addition, the geometrical dimensions of the vehicle influence the value of the dynamic vertical load, caused by load transfer. The predicted curves of traction performance of the same tire affected by load transfer are presented in Fig. 6. The curve representing the gross traction of the tire
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operating as a rear wheel (depicted by ‘*’) lies above the line representing the performance under a constant load (shown as a solid line), while the curve predicting the gross traction of the tire operating as a front wheel lies below it (depicted by ‘+’). This means, that for the same slip, the rear driving wheel develops a higher gross traction force than the front one. On the other hand, the force required to push the front braking wheel is higher than the force required to pull the rear braking wheel. There is only one common point for the three curves—the point depicting the towed wheel while the gross traction force is zero. 2.7. Prediction of the traction performance of a driven wheel Simulation of the traction performance of a wheel is schematically shown in Fig. 7. The wheel and soil properties and the working conditions serve as input parameters. At first, the rut depth z0 and the tire deflection 0 are calculated on the basis of the steady-state equilibrium of the wheel [Eqs. (2) and (4)]. The motion resistance MRc is evaluated according to expression (14). The next step is to determine the trafficability of the wheel which is limited by the maximum permissible deflection of the tire 0max according to the requirements of the manufacturer. On the other band, trafficability
Fig. 7. Flow-chart of the traction performance simulation for a driven wheel.
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of the wheel is limited on weak soils by the ‘flotation’ requirement—the maximum permissible value of the soil sinkage z0max. If it is possible to move the wheel, the vertical and horizontal components of the surface traction v and h can be calculated. Following this calculation, the gross traction force GT, the distance of its application RGT, the net traction force H and the rolling radius Rr are calculated. The simulations of traction performance of the driven wheel with different inflation pressures Pin, operating under various constant vertical loads W, are the results of this stage. The last phase of simulations involves the application of the load transfer effect and presentation of the results.
3. Method and materials In order to verify the proposed model, full-scale field experiments were carried out. A single-axle hydraulically driven semi-trailer was attached to a loader tractor, which also served as a hydraulic power supply. The testing device was built to apply static vertical load Ws and torque M. In addition, it measured the dynamic variations of the vertical load on the wheel W, the net traction force H, and the longitudinal velocity Va and angular velocity ! of the tested wheel. The experiments were carried out on sand and on tilled soil. The soil properties as well as the tire stiffness parameters were measured and are shown with respective operating conditions in Table 1. The testing device was equipped with transducers that measured wheel performance and included a data acquisition unit developed on the basis of the LabTech package with an A/D-converter DAS-16 board. Fig. 8 shows the main features of the testing device. The sampling in the driving and braking modes was performed successively during the same operational cycle. The tested wheel began its motion in driving mode, while moving the tractor, and during its movement, the drive of the tractor was turned on and its velocity gradually increased. This affected the variations of the slip of the tested wheel. The latter reached braking mode (negative slip), when the linear velocity of the loader tractor exceeded the limited velocity of the tested wheel, i.e., it was pushed/pulled. Recorded data continuously covered the overall range of variations of slip of the tested wheel. In order to simulate the operation of the tested wheel as a rear and front driven wheel, the wheel tester moved forwards and backwards.
Table 1 Soil properties and wheel specifications that were tested Type of terrain
Soil properties
Michelin 12.5-20XL-R specifications
kf n kc (kN/m(n+1)) (kN/m(n+2)) (–) Sand 251 Tilled soil 642
2497 2001
c K Ws Pin Kp Kc b/D (kPa) ( ) (mm) (kN) (bar) (kN/m/bar) (kN/m) (m)
0.761 0.0 1.028 9.5
30 25 18 40
9.02 0.8 3.53 1.4
40.8
17.4
0.3/1.02
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Fig. 8. Testing device: (a) schematic view; (b) rear view.
4. Results and discussion Net traction force H, torque M on the wheel, dynamic variation of the vertical load on the wheel W, were measured. The measured traction performance is presented as a function of the slip index, calculated according to expression (10). Figs. 9 and 11 show the plots of the respective traction performance including the data of all the repetitions that were carried out under a certain operating condition. The experimental results are plotted versus slip index and contain the following data: the family of stippled lines show the predicted traction performances of the tested wheel under the constant vertical loads, i.e., without load transfer; the force required to pull the fully braked rear wheel (with zero angular velocity while the respective braking torque is negative), shown by the dashed lines, designated as HBR. The left-hand side of each plot (positive traction) presents the driving mode. The right-hand side (negative traction) presents the braking mode of the tested wheel— pulled for the rear wheel and pushed for the front one (Fig. 9). The zero value of the slip index denotes the full slip of the rotating wheel (100% of slip). The section of conversion of the moving wheel from the driving to the braking mode occurs in the slip range close to zero.
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Fig. 9. Measured net traction force versus slip index on concrete: (a) front wheel; (b) rear wheel.
The measured net traction performance of the driven wheel (wheel properties shown in Table 1) inflated to 0.8 bar under a static vertical load of 5.1 kN, while operating as the front and the rear wheel on concrete is presented in Fig. 9a,b. Another important parameter of the traction performance is the torque M acting on the driving wheel. In general the wheel torque behaves in the same way as the respective presentation of the net traction force. The significant spread of measured data is caused by the dynamic nature of the testing system and non-homogeneity of the soil and consequently, the variability of its parameters. These factors, combined with the complex feature of the traction performances, make it difficult to reach a quantitative estimation of the dispersion of the measured data. In order to estimate the correctness of the measurements, torque M versus net traction force H was plotted for both the front and rear wheels on concrete (see Fig. 10). The measured data of traction performance as well as the analytically predicted data are plotted. Both sets are fitted by linear regressions with
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a high accuracy: R2=0.9997 for predicted data and R2=0.987 and 0.998 for the measured data of the front and rear wheel, respectively. The bold dashed line presents the linear regression of the experimental points, whereas the solid line indicates the regression of the analytically predicted data. The regression lines of the measured and predicted data shown in Fig. 10 are almost parallel—their angular coefficients differ by 1 and 8%. The deviations of the same coefficients calculated for the other tested modes also remain between those limits. These values serve as an indicator for the correctness of the predicted performances and the variance of the model from the measured data. The offset between the measured and predicted data is due to the moment of internal resistance of the drive between the torque-meter and the testing wheel. It also includes the resistance required for the deformations of the tire while moving. The simulated and measured net traction forces versus slip index are presented in Fig. 11, according to the operating conditions shown in Table 1. Fig. 11a,b refers to
Fig. 10. Measured and predicted torque on the driven wheel versus net traction force on concrete. Ws=5.1/kN; Pin=0.8 bar; (a) front wheel; (b) rear wheel.
48 I. Shmulevich, A. Osetinsky / Journal of Terramechanics 40 (2003) 33–50 Fig. 11. Measured and predicted net traction force versus slip index: (a) front wheel on tilled soil; (b) rear wheel on tilled soil; (c) front wheel on sand; (d) rear wheel on sand.
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tilled soil and Fig. 11c,d to sand. The experimental data is signified by empty squares, whereas a solid line refers to the predicted respective performance, considering the load transfer effect. In addition, the clusters of stippled curves depict the predicted net traction forces for different vertical loads, similar to Fig. 6. Both sets of figures present the net traction forces developed by the front and rear wheel. One can see that there is a significant influence of the load transfer on the measured net traction force and that it affects the performance on a front and rear wheel differently. When operating as a front wheel, the developed net traction force in the driving mode is substantially smaller than the resistance force in the braking mode. This can be seen on both soils through the value of the net traction force, which is about half that of the resistance force (Fig. 11a,c). On the other hand, when operating as a rear wheel, the net traction force in the driving mode is larger than the resistance force in the braking mode. For example, on sand the net traction force is about twice as large as the resistance force (Fig. 11b,d). Finally, it should be emphasized that on sand, the influence of the load transfer can be seen clearly, due to the heavier static vertical load and the higher trafficability of the wheel in these operating conditions. The measured net traction force developed by the tested wheel, operating as a front wheel, is dispersed more densely in the driving mode than in the braking mode (Fig. 11a,c). This is due to the higher stability of motion of the front wheel in the driving mode. In contrast, the measured data seems to imply that in the braking mode the motion of the rear wheel is more stable (Fig. 11b,d). It can be seen, that the simulated net traction performance in both driving and braking modes behaves in the same way as the measured data. The values of the predicted forces are similar to the measured values. The deviance of the measured data from the predicted force is caused by the non-homogeneity of the soil and consequently, the variability of its parameters. In addition, other reasons for this deviance include the dynamic nature of the testing system, as opposed to the static approach of the model.
5. Conclusions A semi-empirical model for the prediction of the traction performance of a wheel in both driving and braking modes is presented assuming the wheel–soil contact surface is parabolic. The main conclusions from this research can be summarized as follows. The proposed model demonstrates significant non-symmetry of traction wheel performance for driving and braking modes. The traction force developed by the wheel in driving mode is significantly smaller than the force required to pull (or push) the wheel in braking mode. This model also considers the load transfer effect on a driven wheel, and shows the difference in performance of the front and rear wheel as a result. In order to validate the model, full-scale experiments were carried out on a testing device, which was built for that purpose. The proposed method was successfully verified qualitatively by the data reported in literature and by the results of the field experiments with a
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single wheel tester. The results show that the proposed method may improve prediction of off-road multi-drive vehicle traction performance.
Acknowledgements We dedicate the paper in memorial to Prof. Dan Wolf, 1936–2001, who was the founder of this research work. The authors wish to thank the Technion Research and Development Foundation Ltd. and the Agricultural Engineering Department at the Technion—Israel Institute of Technology for sponsoring this research.
References [1] Bekker MG. Off-the-road locomotion. Ann Arbor: The University of Michigan Press; 1960. [2] Osetinsky A, Shmulevich I. Modeling of a pushed/pulled driven wheel. In: Proceedings of the Automation Technology for Off-Road Equipment (ATOE 2002) Conference, Chicago, IL, 2002a. p. 137–47. [3] Osetinsky A, Shmulevich I. Traction performance simulation of a pushed/pulled driven wheel. In: Proceedings of the Automation Technology for Off-Road Equipment (ATOE 2002) Conference, Chicago, IL, 2002b: p. 148–57. [4] Phillips JR. The powered vehicular wheel plane-rolling in equilibrium: a consideration of slip and rolling resistance. Proceedings of the 1st International Conference on the Mechanics of Soil-Vehicle Systems, Torino, Italy, 12–16 June 1961. [5] Janosi Z, Hanamoto B. The analytical determination of drawbar pull as a function of slip for tracked vehicles in deformable soils. In: Proceedings of the 1st International Conference on the Mechanics of Soil-Vehicle Systems, Torino, Italy, 12–16 June 1961. [6] Muro T. Tractive performance of a driven rigid wheel on soft ground based on the analysis of soil– wheel interaction. Journal of Terramechanics 1993;30(5):351–69. [7] Thangavadivelu S, Taylor R, Clark S, Slocombe J. Measuring soil properties to predict inictive performance of an agricultural drive tire. Journal of Terramechanics 1994;31(4):215–25. [8] Lines JA, Murphy K. The stiffness of agricultural tractor tires. Journal of Terramechanics 1991; 28(1):49–64.
Traction performance of a pushed/pulled drive wheel I. Shmulevicha,*, A. Osetinskya a
Technion—Israel Institute of Technology, Agricultural Engineering, Haifa, 32000, Israel
Abstract A comprehensive method for prediction of off-road driven wheel performance is presented, assuming a parabolic wheel–soil contact surface. The traction performance of a driven wheel is predicted for both driving and braking modes. Simulations show significant non-symmetry of the traction performance of the driving and braking wheels. The braking force is significantly greater than the traction force reached in the driving mode. In order to apply the suggested model for prediction of the traction performance of a 4WD vehicle, the load transfer effect was considered. Simulated traction performances of front and rear driven wheels differ significantly, due to the load transfer. In the driving mode, the rear driven wheel develops a net traction force greater than that of the front wheel. On the other hand, in the braking mode the front driven wheel develops a braking force significantly greater than that of the rear driven wheel due to a pushed/pulled force affected by the load transfer. The suggested model was successfully verified by the data reported in literature and by full-scale field experiments with a special wheel-testing device. The developed approach may improve the prediction of off-road multi-drive vehicle traction performance. # 2003 ISTVS. Published by Elsevier Ltd. All rights reserved. Keywords: Net traction; Gross traction; Off-road; Load transfer; Soil–tire interaction
1. Introduction For many years numerous researchers have reported predictions of off-road wheel performances. Some dealt in analytical approaches, others in semi-empirical or empirical approaches. The diversity in the approaches of research on off-road tire performance points to the complexity of the issue. Each of the above-mentioned approaches has some limitations: the analytical approaches are difficult to use; empirical equations are limited to the tested cases and most semi-empirical methods * Corresponding author. Tel.: +972-4-829-2626; fax: +972-4-822-1529. E-mail address: [email protected] (I. Shmulevich). 0022-4898/$20.00 # 2003 ISTVS. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jterra.2003.09.001
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Nomenclature a b c dn dv e0 GT HBR j K Kc Kp kc kv kf MRc n in R Rr S si Va Vaj , and Voj Vj Vt,Va W W x, z x0 z0 vo ! , v
parameter (m1) wheel width (m) soil cohesion (kPa) elementary horizontal force (kN) elementary vertical load (kN) eccentricity of the center of the wheel in relation to the rear point of the contact surface 0 (m) gross traction force (kN) force required to pull the fully braked wheel (kN) shear displacement of the soil at point B (m) shear deformation modulus (m) carcass stiffness of the tire (kN/m) inflation pressure dependence of the tire (kN/m/bar) cohesive modulus of deformation (kN/m(n+1)) coefficient of vertical stiffness of the wheel (kN/m/m) frictional modulus of deformation (kN/m(n+2)) motion resistance due to soil compaction (kN) exponent of soil deformation inflation pressure of the tire (bar) unloaded radius of the wheel (m) rolling radius (m) slip (dec) slip index (m) magnitude of the actual longitudinal velocity of the wheel (m/s) tangential components of the actual longitudinal velocity and velocity of point B in relation to the center of the wheel, respectively (m/s) slip velocity of point B (m/s) theoretical and actual velocity (m/s) vertical load applied to the wheel (kN) dynamic variation of the vertical load on the wheel (kN) coordinates of the contact surface (m) length of the contact surface (m) rut depth (m) angle between the radius-vector of point B and vertical axle (rad) vertical displacement of the elementary contact area of the wheel (m) deflection of the wheel on a hard surface (m) angle of internal friction (rad) angular velocity of the wheel (s1): !=d (vo)/dt normal and shear stresses acting on the elementary contact area (kPa) vertical component of surface traction (kPa): v=dpv/(bdx)
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focus on predicting separate performances in driving and braking (pulled/pushed) modes. An external implement or an internal interaction between driven axles of a 4WD vehicle could, due to load transfer, significantly affect the performance of each driven wheel in the vehicle. Moreover, it could also be changed due to variations in terrain. Wheel properties and soil conditions affect traction performance differently, and cause the wheel to behave differently in driving and braking modes. In order to optimize the traction performance of the vehicle, the model for wheel traction performance should also be extended to the braking mode and modified to consider the load transfer effect. There are various approaches to analyze the traction performance of a vehicle with respect to load transfer. Some of them are based on the multi-body dynamic analysis of a moving vehicle or modeling with commercial CAD programs. These programs require large amounts of computing resources and have difficulties in evaluating the soil and the tire dynamic parameters. Based on a semi-empirical model proposed by Bekker [1] or based on empirical approaches, several researchers have developed models to predict vehicle traction performance with respect to the load transfer effect. These models predict the traction performance only for the driving mode and do not cover the braking mode. Other researchers studied the traction performance on the scaled models of 4WD tractors or on full-sized tractors in order to experimentally derive the influence of load transfer. It is difficult to establish general conclusions from the above-mentioned studies based on specific empirical data. The papers by Osetinsky and Shmulevich [2,3] present a comprehensive development of the wheel–soil interaction model considering the influence of load transfer on the traction performance of the wheel. The present research focuses on applying a semi-empirical model for wheel–soil interaction based on the assumption of a parabolic wheel–soil contact area in order to predict the traction performance of a driven wheel affected by load transfer. Verification of the suggested model was accomplished by full-scale field experiments.
2. Traction model development Prediction of traction performance of a driving wheel considers wheel–soil interaction in two interrelated aspects—force equilibrium and kinematics. In order to model the wheel–soil interaction, the form of the contact surface should be determined. In this work a simple analytical approach to the wheel–soil contact surface was suggested. The prediction of the form of the contact surface is based on modeling the interrelated wheel and soil deformations under various operating conditions. Several assumptions were made for the suggested model: the soil serves as a plastic non-linear medium, the wheel moves in steady state condition at a low velocity, its deformations are assumed to be linear elastic and the wheel–soil interaction is two-dimensional (a plane strain problem).
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2.1. Wheel–soil contact surface One of the main assumptions of the present work is that the wheel–soil contact surface can be represented in parabolic form in the longitudinal direction with the apex at the rear point of contact 0 (Fig. 1a) as follows: z ¼ z0 ax2
ð1Þ
Using the vertical soil pressure-sinkage relationship suggested by Bekker [1] the vertical load on the wheel–soil contact surface versus rut depth is derived as follows: rffiffiffiffiffi kc þ bkf z0 n W¼ ð2Þ z ð 3 nÞ 3 a 0 In order to adjust the form of the contact surface to the deflection of a moving pneumatic wheel, its state of equilibrium in the vertical direction is analyzed (Fig. 1b). The elementary vertical load pv acting on the contact area, with length dx, is caused by the vertical wheel stiffness kv and the deflection of a moving wheel: dpv ¼ kv dx
ð3Þ
The force equilibrium of a moving wheel in the vertical direction is found after integrating the elementary vertical load pv along the wheel–soil contact surface, resulting in the following analytical solution: 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 ax0 e0 ðx0 e0 Þ R 2 x0 e20 þ R 2 e20 x0 W ¼ kv þ 2 3 2 2 x e e 0 0 0 arcsin þ arcsin ð4Þ R R
Fig. 1. Driven wheel–soil interaction: (a) force equilibrium; (b) distribution of the elementary vertical load.
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The eccentricity e0 of the center of the wheel is found geometrically according to Fig. 1b as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !! x0 4R 2 2 e0 ¼ 1a x0 ð5Þ 2 1 þ ðax0 Þ2 The wheel properties can be presented by the coefficient of vertical stiffness kv, carcass stiffness Kc and the inflation pressure dependence Kp of the tire. It can be evaluated on the basis of the tire load-displacement relationship for a certain value of inflation pressure Pin as: kv ¼ h
R 2 arcsin
W i x 0 x0 ðR DÞ R
ð6Þ
The parameter (a) of the parabolic contact surface and the rut depth z0 under the moving wheel are derived from a simultaneous solution of Eqs. (2) and (4). A more detailed development of the above-mentioned model and its verification through experimental data is presented by Osetinsky and Shmulevich [2]. 2.2. Kinematics of the moving wheel and soil shear displacements The slip velocity of any peripheral point B on the contact surface (Fig. 2), can be calculated as: Vj ¼ Vaj þ Voj
ð7Þ
For the parabolic contact surface described by expression (1) the soil shear displacement at point B is found by Osetinsky and Shmulevich [2] after integrating the slip velocity over the time required to move from x0 to x:
Fig. 2. Illustration of slip velocity.
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ð x0 axðx 2e0 Þ þ R 2 e20 axðx 2e0 Þ þ R 2 e20 ! dx j¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 x 2 2 2 2 2 ax R e0 þðx e0 Þ 1 þ ð2axÞ
ð8Þ
2.3. Slip and slip index The well-known definition of wheel slip is: S¼
Vt Va Va ¼1 Vt Rr !
ð9Þ
The slip concept is used with high accuracy for studying rigid wheel kinematics, and can also be used for studying pneumatic tires after proper definition of the rolling radius. Its value is affected by operating conditions: the soil and tire properties, vertical load on the wheel, etc. It cannot be accurately predicted analytically and should be measured, according to the ASAE S296.2 standard, as the distance traveled per revolution of the towed wheel (zero slip), divided by 2. The measuring procedure is complicated and sometimes yields unreliable results. The soil shear displacement according to expression (8) is presented as a function of the ratio of the actual longitudinal and angular velocities of the wheel. This ratio is defined as the slip index, si: si ¼ Va =!
ð10Þ
The advantage of the slip index term lies in processing and presenting experimental data, because both velocities can be measured simply and with high reliability. For given work conditions the relationship between the slip and slip index is assumed to be constant: S¼1
si Rr
ð11Þ
The slip index is a dimensional parameter expressed in units of length and its physical meaning is clearly seen from expression (11). It is equal to the rolling radius of the towed wheel while slip is zero. This approach to slip problems was introduced for the first time by Phillips [4], but the above mentioned ratio [Eq. (10)] was not defined explicitly as a system parameter and that was probably the reason why this concept was not developed any further. 2.4. Wheel traction forces and motion resistance The gross traction force is a result of the surface traction in the horizontal direction and can be derived after analyzing the forces acting on the elementary area ds (Fig. 3).
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Fig. 3. Elementary forces applied to the soil contact area ds.
The elementary vertical load dpv is found according to [Eq. (3)] (detailed in paper [2] and the shear force assumed from the shear stress–displacement relationship according to Janosi and Hanamoto [5]: ¼ ðc þ tanÞ 1 ej=K ð12Þ The gross traction force is calculated by integrating dph along the wheel–soil contact area: ð x0 ð x0 v 2ax tan 1 ej=K c 1 ej=K ð2axÞ2 þ1 bdx dph ¼ GT ¼ 1 þ 2axð1 ej=K Þtan 0 0 ð13Þ The distance from the center of the wheel RGT, where the gross traction force is applied (Fig. 1a), is found on the basis of force equilibrium analysis and the respective torque M on the wheel. Assuming that the motion resistance is due to soil compaction [1], the integration of the horizontal component of the soil reaction Rs along the contact surface yields the motion resistance MRc: kc þ bkf ðnþ1Þ z ð14Þ MRc ¼ ð n þ 1Þ 0 It can be seen that expression (14) is identical to the one derived by Bekker [1] for a rigid wheel since it is non-dependent on the form of the contact surface. The net traction force is calculated according to: H ¼ GT MRc
ð15Þ
2.5. Traction performance of pushed/pulled wheel The parabolic model approach was verified by the experimental data reported for a rigid wheel [6] and a 9.5-16R-1 tire [7] operating in the driving mode, using the
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mathematical expressions derived above. The tire stiffness properties used by the model (Kp, Kc) are reported by Lines and Murphy [8]. A special procedure was developed, using Matlab software, to calculate wheel performance as a function of soil and wheel properties versus wheel slip and slip index. For given operating conditions, a force equilibrium in vertical direction results in a predictable parabolic contact surface between the wheel and soil. Based on this predicted surface, shear soil displacement and related stresses are calculated for various slip values, resulting in a prediction of motion resistance and gross and net traction forces. The predicted net traction force and the experimental data are shown versus slip in Fig. 4. The suggested method shows reasonable agreement between the predicted and reported results for the tire and for the rigid wheel: R2 is equal to 0.9687 and 0.8656, respectively. The suggested method was developed for the prediction of traction performance for a single wheel regardless of the operating modes (driving or braking). It can be seen that the predicted traction performance agrees with the experimental data in the driving mode. Thus, an assumption is made that the same method can be applied to predict the traction performance in the braking mode of the pushed or pulled wheel. In order to demonstrate the prediction of traction performance in the braking mode, the model was applied to the above mentioned tire. The results of the predicted traction performance versus slip and versus slip index are shown in Fig. 5. The net traction performance H plotted versus slip (Fig. 5a) runs from minus infinity (braking mode) to the value of one (full slip of the driving wheel). The same graph plotted versus slip index (Fig. 5b) looks like the mirror image of the former one. It runs from zero, indicating full slip of the driving wheel, to plus infinity—the braking mode of the wheel. It can be seen that the traction performance of a driven wheel operating in the driving and braking modes is not symmetrical under the same conditions. The force required to pull or push the tire in braking mode is greater than the traction force that is developed by the tire in driving mode. Under the specified operating conditions, the net traction force achieved by the tire is about 1.25 kN in
Fig. 4. Comparison between predicted and measured net traction force of a driven wheel.
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Fig. 5. Predicted net traction of the 9.5-16R-1 tire: (a) versus slip; (b) versus slip index.
the driving mode (similar to the measured value). However, a force of about 2.2 kN predicted to pull a braking tire, which is 75% more than the net traction force, is achieved in the driving mode. 2.6. Load transfer effect on the traction performance of a wheel The vertical load on the wheel of a moving vehicle changes dynamically due to load transfer. Increasing the vertical load on the wheel increases the traction performance of the wheel. Since the vertical load acting on the moving wheel affects the traction performance, an increase of the load on the wheel magnifies the gross traction force and the torque. On the other hand, reduction of the vertical load decreases the performance values. The gross traction of the tire discussed above, caused by different constant vertical loads, is shown in Fig. 6 as a cluster of dashed curves versus the respective values of slip and slip index. The vertical load applied to the driven tire varies between 1.23 and 9.08 kN. The predicted gross traction of the tire operating under a constant vertical load of 4.0 kN is shown in Fig. 6 by a solid line. Fig. 6b also depicts the maximum and minimum values of the rolling radius Rr max and Rr min of the tire. In order to predict the traction performance of the specified tire, affected by load transfer, the variations of the vertical load on the wheel should be considered. The
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Fig. 6. Predicted gross traction of the driven 9.5-16R-l tire operating as a front and rear wheel: (a) versus slip; (b) versus slip index.
dynamic load can be found from the equation of the equilibrium of the moving wheel [Eq. (13). The actual curve of the traction performance crosses the cluster of curves. Osetinsky and Shmulevich [3] present a comprehensive analysis of the influence of load transfer in four possible combinations of driven wheel operation in a 4WD vehicle: front and rear wheel in driving and braking modes. The load transfer changes the vertical load on the front and the rear wheels in different ways. While operating in the driving mode, the vertical load increases on the rear wheels and at the same time decreases on the front wheels. In the braking mode, the load transfer affects the traction performance of the wheel in the opposite way: the vertical load increases on the front wheel and decreases on the rear one. In addition, the geometrical dimensions of the vehicle influence the value of the dynamic vertical load, caused by load transfer. The predicted curves of traction performance of the same tire affected by load transfer are presented in Fig. 6. The curve representing the gross traction of the tire
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operating as a rear wheel (depicted by ‘*’) lies above the line representing the performance under a constant load (shown as a solid line), while the curve predicting the gross traction of the tire operating as a front wheel lies below it (depicted by ‘+’). This means, that for the same slip, the rear driving wheel develops a higher gross traction force than the front one. On the other hand, the force required to push the front braking wheel is higher than the force required to pull the rear braking wheel. There is only one common point for the three curves—the point depicting the towed wheel while the gross traction force is zero. 2.7. Prediction of the traction performance of a driven wheel Simulation of the traction performance of a wheel is schematically shown in Fig. 7. The wheel and soil properties and the working conditions serve as input parameters. At first, the rut depth z0 and the tire deflection 0 are calculated on the basis of the steady-state equilibrium of the wheel [Eqs. (2) and (4)]. The motion resistance MRc is evaluated according to expression (14). The next step is to determine the trafficability of the wheel which is limited by the maximum permissible deflection of the tire 0max according to the requirements of the manufacturer. On the other band, trafficability
Fig. 7. Flow-chart of the traction performance simulation for a driven wheel.
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of the wheel is limited on weak soils by the ‘flotation’ requirement—the maximum permissible value of the soil sinkage z0max. If it is possible to move the wheel, the vertical and horizontal components of the surface traction v and h can be calculated. Following this calculation, the gross traction force GT, the distance of its application RGT, the net traction force H and the rolling radius Rr are calculated. The simulations of traction performance of the driven wheel with different inflation pressures Pin, operating under various constant vertical loads W, are the results of this stage. The last phase of simulations involves the application of the load transfer effect and presentation of the results.
3. Method and materials In order to verify the proposed model, full-scale field experiments were carried out. A single-axle hydraulically driven semi-trailer was attached to a loader tractor, which also served as a hydraulic power supply. The testing device was built to apply static vertical load Ws and torque M. In addition, it measured the dynamic variations of the vertical load on the wheel W, the net traction force H, and the longitudinal velocity Va and angular velocity ! of the tested wheel. The experiments were carried out on sand and on tilled soil. The soil properties as well as the tire stiffness parameters were measured and are shown with respective operating conditions in Table 1. The testing device was equipped with transducers that measured wheel performance and included a data acquisition unit developed on the basis of the LabTech package with an A/D-converter DAS-16 board. Fig. 8 shows the main features of the testing device. The sampling in the driving and braking modes was performed successively during the same operational cycle. The tested wheel began its motion in driving mode, while moving the tractor, and during its movement, the drive of the tractor was turned on and its velocity gradually increased. This affected the variations of the slip of the tested wheel. The latter reached braking mode (negative slip), when the linear velocity of the loader tractor exceeded the limited velocity of the tested wheel, i.e., it was pushed/pulled. Recorded data continuously covered the overall range of variations of slip of the tested wheel. In order to simulate the operation of the tested wheel as a rear and front driven wheel, the wheel tester moved forwards and backwards.
Table 1 Soil properties and wheel specifications that were tested Type of terrain
Soil properties
Michelin 12.5-20XL-R specifications
kf n kc (kN/m(n+1)) (kN/m(n+2)) (–) Sand 251 Tilled soil 642
2497 2001
c K Ws Pin Kp Kc b/D (kPa) ( ) (mm) (kN) (bar) (kN/m/bar) (kN/m) (m)
0.761 0.0 1.028 9.5
30 25 18 40
9.02 0.8 3.53 1.4
40.8
17.4
0.3/1.02
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Fig. 8. Testing device: (a) schematic view; (b) rear view.
4. Results and discussion Net traction force H, torque M on the wheel, dynamic variation of the vertical load on the wheel W, were measured. The measured traction performance is presented as a function of the slip index, calculated according to expression (10). Figs. 9 and 11 show the plots of the respective traction performance including the data of all the repetitions that were carried out under a certain operating condition. The experimental results are plotted versus slip index and contain the following data: the family of stippled lines show the predicted traction performances of the tested wheel under the constant vertical loads, i.e., without load transfer; the force required to pull the fully braked rear wheel (with zero angular velocity while the respective braking torque is negative), shown by the dashed lines, designated as HBR. The left-hand side of each plot (positive traction) presents the driving mode. The right-hand side (negative traction) presents the braking mode of the tested wheel— pulled for the rear wheel and pushed for the front one (Fig. 9). The zero value of the slip index denotes the full slip of the rotating wheel (100% of slip). The section of conversion of the moving wheel from the driving to the braking mode occurs in the slip range close to zero.
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Fig. 9. Measured net traction force versus slip index on concrete: (a) front wheel; (b) rear wheel.
The measured net traction performance of the driven wheel (wheel properties shown in Table 1) inflated to 0.8 bar under a static vertical load of 5.1 kN, while operating as the front and the rear wheel on concrete is presented in Fig. 9a,b. Another important parameter of the traction performance is the torque M acting on the driving wheel. In general the wheel torque behaves in the same way as the respective presentation of the net traction force. The significant spread of measured data is caused by the dynamic nature of the testing system and non-homogeneity of the soil and consequently, the variability of its parameters. These factors, combined with the complex feature of the traction performances, make it difficult to reach a quantitative estimation of the dispersion of the measured data. In order to estimate the correctness of the measurements, torque M versus net traction force H was plotted for both the front and rear wheels on concrete (see Fig. 10). The measured data of traction performance as well as the analytically predicted data are plotted. Both sets are fitted by linear regressions with
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a high accuracy: R2=0.9997 for predicted data and R2=0.987 and 0.998 for the measured data of the front and rear wheel, respectively. The bold dashed line presents the linear regression of the experimental points, whereas the solid line indicates the regression of the analytically predicted data. The regression lines of the measured and predicted data shown in Fig. 10 are almost parallel—their angular coefficients differ by 1 and 8%. The deviations of the same coefficients calculated for the other tested modes also remain between those limits. These values serve as an indicator for the correctness of the predicted performances and the variance of the model from the measured data. The offset between the measured and predicted data is due to the moment of internal resistance of the drive between the torque-meter and the testing wheel. It also includes the resistance required for the deformations of the tire while moving. The simulated and measured net traction forces versus slip index are presented in Fig. 11, according to the operating conditions shown in Table 1. Fig. 11a,b refers to
Fig. 10. Measured and predicted torque on the driven wheel versus net traction force on concrete. Ws=5.1/kN; Pin=0.8 bar; (a) front wheel; (b) rear wheel.
48 I. Shmulevich, A. Osetinsky / Journal of Terramechanics 40 (2003) 33–50 Fig. 11. Measured and predicted net traction force versus slip index: (a) front wheel on tilled soil; (b) rear wheel on tilled soil; (c) front wheel on sand; (d) rear wheel on sand.
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tilled soil and Fig. 11c,d to sand. The experimental data is signified by empty squares, whereas a solid line refers to the predicted respective performance, considering the load transfer effect. In addition, the clusters of stippled curves depict the predicted net traction forces for different vertical loads, similar to Fig. 6. Both sets of figures present the net traction forces developed by the front and rear wheel. One can see that there is a significant influence of the load transfer on the measured net traction force and that it affects the performance on a front and rear wheel differently. When operating as a front wheel, the developed net traction force in the driving mode is substantially smaller than the resistance force in the braking mode. This can be seen on both soils through the value of the net traction force, which is about half that of the resistance force (Fig. 11a,c). On the other hand, when operating as a rear wheel, the net traction force in the driving mode is larger than the resistance force in the braking mode. For example, on sand the net traction force is about twice as large as the resistance force (Fig. 11b,d). Finally, it should be emphasized that on sand, the influence of the load transfer can be seen clearly, due to the heavier static vertical load and the higher trafficability of the wheel in these operating conditions. The measured net traction force developed by the tested wheel, operating as a front wheel, is dispersed more densely in the driving mode than in the braking mode (Fig. 11a,c). This is due to the higher stability of motion of the front wheel in the driving mode. In contrast, the measured data seems to imply that in the braking mode the motion of the rear wheel is more stable (Fig. 11b,d). It can be seen, that the simulated net traction performance in both driving and braking modes behaves in the same way as the measured data. The values of the predicted forces are similar to the measured values. The deviance of the measured data from the predicted force is caused by the non-homogeneity of the soil and consequently, the variability of its parameters. In addition, other reasons for this deviance include the dynamic nature of the testing system, as opposed to the static approach of the model.
5. Conclusions A semi-empirical model for the prediction of the traction performance of a wheel in both driving and braking modes is presented assuming the wheel–soil contact surface is parabolic. The main conclusions from this research can be summarized as follows. The proposed model demonstrates significant non-symmetry of traction wheel performance for driving and braking modes. The traction force developed by the wheel in driving mode is significantly smaller than the force required to pull (or push) the wheel in braking mode. This model also considers the load transfer effect on a driven wheel, and shows the difference in performance of the front and rear wheel as a result. In order to validate the model, full-scale experiments were carried out on a testing device, which was built for that purpose. The proposed method was successfully verified qualitatively by the data reported in literature and by the results of the field experiments with a
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single wheel tester. The results show that the proposed method may improve prediction of off-road multi-drive vehicle traction performance.
Acknowledgements We dedicate the paper in memorial to Prof. Dan Wolf, 1936–2001, who was the founder of this research work. The authors wish to thank the Technion Research and Development Foundation Ltd. and the Agricultural Engineering Department at the Technion—Israel Institute of Technology for sponsoring this research.
References [1] Bekker MG. Off-the-road locomotion. Ann Arbor: The University of Michigan Press; 1960. [2] Osetinsky A, Shmulevich I. Modeling of a pushed/pulled driven wheel. In: Proceedings of the Automation Technology for Off-Road Equipment (ATOE 2002) Conference, Chicago, IL, 2002a. p. 137–47. [3] Osetinsky A, Shmulevich I. Traction performance simulation of a pushed/pulled driven wheel. In: Proceedings of the Automation Technology for Off-Road Equipment (ATOE 2002) Conference, Chicago, IL, 2002b: p. 148–57. [4] Phillips JR. The powered vehicular wheel plane-rolling in equilibrium: a consideration of slip and rolling resistance. Proceedings of the 1st International Conference on the Mechanics of Soil-Vehicle Systems, Torino, Italy, 12–16 June 1961. [5] Janosi Z, Hanamoto B. The analytical determination of drawbar pull as a function of slip for tracked vehicles in deformable soils. In: Proceedings of the 1st International Conference on the Mechanics of Soil-Vehicle Systems, Torino, Italy, 12–16 June 1961. [6] Muro T. Tractive performance of a driven rigid wheel on soft ground based on the analysis of soil– wheel interaction. Journal of Terramechanics 1993;30(5):351–69. [7] Thangavadivelu S, Taylor R, Clark S, Slocombe J. Measuring soil properties to predict inictive performance of an agricultural drive tire. Journal of Terramechanics 1994;31(4):215–25. [8] Lines JA, Murphy K. The stiffness of agricultural tractor tires. Journal of Terramechanics 1991; 28(1):49–64.