An Empirical Model for Tractive Performance of Rubber-Tracks

7/28/2019 An Empirical Model for Tractive Performance of Rubber-Tracks

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Journal of Terramechanics JT05-010

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An Empirical Model for Tractive Performance of Rubber-tracksin Agricultural Soils

Robert Grisso, J ohn Perumpral and Frank ZozProfessor, William Cross Jr. Professor and Head Emeritus, Biological Systems Engineering,

Virginia Tech, Blacksburg, VA and Retired Engineer, John Deere Product Engineering Center,Waterloo, IA, USA

Corresponding Author: Dr. Robert Grisso, Biological Systems Engineering, 200 Seitz Hall(0303), Virginia Tech, Blacksburg, VA 24061-0303, 540-231-6538, FAX: 540-231-3199,[email protected]

Abstract: Mathematical models capable of describing the interaction between the traction

devices and soils have been effective in predicting the performance of off-road vehicles. Such a

model capable of predicting the performance of bias-ply tires in agricultural soils was first

developed by Brixius [1]. When the soil and vehicle parameters are known, this model uses an

iterative procedure to predict the tractive performance of a vehicle including pull, tractive

efficiency, and motion resistance. Al-Hamad et al. [2] modified the Brixius equations to predict

the performance of radial tires. Zoz and Grisso [3] have demonstrated that the use of spreadsheet

templates is more efficient than the original iterative procedure used to predict the performance

of 2WD and 4WD/MFWD tractors. As tractors equipped with rubber-tracks are becoming

popular, it is important that we have the capability to predict the performance for off-road

vehicles equipped with rubber-tracks during agricultural operations. This paper discusses the

development of an empirical model to accomplish this goal and its validity by comparing the

predicted results with published experimental results.

Keywords: Rubber-tracks, Traction Mechanics; Traction Prediction, Traction Model


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INTRODUCTION

Through out the world, farm tractors are used extensively to carry out different agricultural

operations. During the last five decades tractors used in agricultural operations have undergone

many changes. Selected examples of these changes include increased horsepower ratings, tractor

configuration (2WD and 4WD/ MFWD), and use of different types of tractive devices such as

(bias-ply tires to radial tires to rubber-tracks). Production capabilities of these tractors depend

heavily on tractor configuration, type of tractive devices used and terrain conditions. Ability to

predict and optimize the performance of these tractors during field operations has been of great

interest to scientists, manufacturers, and users.

In an effort to meet this need, Zoz [4] developed a set of graphs based on field tests

conducted in three types of soils: firm, tilled and soft or sandy, and on concrete with 2WD

tractors. He demonstrated that the set of graphs developed could be used to predict the drawbar

pull, travel speed, drawbar horsepower and travel reduction of 2WD tractors under different soil

conditions.

Wismer and Luth [5] studied the single wheel behavior in an indoor soil-bin facility. Using

dimensional analysis and the results of carefully planned tests, they developed equations to

predict the pull and tractive efficiency of tractors under different slip when certain conditions are

satisfied.

Similar sets of equations were developed by Zoz and Brixius [6] to predict the performance

of tractors on concrete. Nebraska tractor test results were used to develop these relationships.

Based on these equations, they have also developed a computer program to predict the vehicle

performance on concrete.


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In 1987, Brixius revised the relationships originally developed by Wisner and Luth [5].

Using the data from approximately 2,500 field tests involving 121 soil-tire combinations and

improved curve fitting techniques, Brixius [1] came up with a revised set of traction equations.

In addition to providing better predictions, these equations developed for bias-ply tires, extended

the range of applications. Al-Hamad et al. [2] modified the relationships developed by Brixius to

predict the performance of vehicle equipped with radial tires.

A review of literature has revealed that a great deal of experimental studies have been

conducted to assess the tractive performance of rubber-tracks in different soils and to compare it

with the performance of other types of tractive devices ([8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,

19, 20, 21, 22, 23, and 24]). However, to our knowledge, only limited studies have dealt with the

development of mathematical models to predict the performance of rubber tracks in agricultural

soils. Upadyaya et al. [8] and Zoz [9] have tested rubber-tracks and developed regression

equations to predict net-traction, motion resistance, and tractive efficiency as a function of travel

reduction or slip. They used regression analysis to minimize data scatter and developed useful

relationships for specific test conditions. The limitation of these relationships, however, is that

they may be useful only for the field and vehicle conditions that existed during the collection of

experimental data. Therefore, the overall objective of this study was to develop an empirical

model specifically to predict the tractive performance of rubber-tracks in a variety of agricultural

soils and to establish its validity by comparing the predicted results with the experimental.

Rubber-Track Mechanics

In many respects, the mechanics of the rubber-tracks and wheel systems are very similar, and

a brief discussion of mechanics of rubber-tracks is included in this section. A more detailed

review of the same is available in Zoz and Grisso [3]. Figure 1 shows the forces on a rubber-


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tracks system. The torque input (T) to the axle develops a gross thrust (GT). Part of the gross

thrust is used to overcome the motion resistance (MR). The remainder is the net traction (NT) or

pull available for useful work.

Though there are similarities between tires and rubber-tracks, the dynamic load distribution

on rubber-tracks is significantly different. For example, the location of the dynamic load

resultant (eh) depends on the static weight distribution, the design of the suspension system

supporting the bogie wheels, and the vehicle weight transfer characteristics [7].

To maximize the tractive performance and to minimize the soil disturbance, ideally the

pressure distribution on a rubber-track should be uniform and, the dynamic weight distribution in

the front and rear should be equal. The dynamic weight distribution on rubber-tracks depends on

factors such as static weight, tractor dimensions, location of center of gravity, angle and the

magnitude of pull. Unlike in the case of tires, both the magnitude and uniformity in dynamic load

distribution are important during the testing of rubber-track systems.

Traction Equations for Rubber Tracks

Since our goal was to develop a traction model with the capability to predict the rubber-track

performance in a variety of agricultural soils and for different track systems, we decided to

modify the following original equations developed for tires by Brixius[1]. Brixius expressed

GTR (Gross Traction Ratio) and MRR (Motion Resistance Ratio) as a function of mobility

number (Bn) and wheel slip (s). He determined the dimensionless numbers in the equations

using a curve-fitting technique and the following are the generalized equations he developed:

( )( )

+

+

=

db

2K1

h

1K1

W

dbCI

nB (1)


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4C

s3

Ce1n

B2

Ce1

1C

Wr

TGTR +

=

= (2)

nB

s6

C

4C

nB

5C

W

MMRR

++== (3)

MRRGTRW

NTNTR == (4)

Where,Bn Mobility numbers CI Cone Indexb unloaded tire section width d unloaded tire diameterr tire rolling radius tire deflectionh tire section height W Dynamic load on the tractive devicess Wheel slip M motion resistanceNT Net traction or pull NTR Net traction ratio

T Axle Torque

Equations 1-3 include six coefficients (C1-C6) and two tire constants (K1 & K2). These

constants and coefficients may change depending on the type of tractive devices. For bias-ply

tires, values of C1, C2, C3, C4, C5, C6, K1, and K2 are 0.88, 0.1, 7.5, 0.04, 1.00, 0.5, 5 and 3,

respectively [1].

Zoz [25] created a Lotus 1-2-3template for Brixius equations. This template helped the

users to predict the performance of tractors or different configurations equipped with bias-ply or

radial tires in different agricultural soils.

As radial tires became popular, there was interest in models capable of predicting

performance of tractors equipped with radial tires. Al-Hamed et al. [2] modified the Brixius

equations to meet this need. Using experimental data and curve fitting techniques, a new set of

coefficients C1 thru C6 and K1 & K2 to represent the radial tires was generated. They are 0.88,

0.08, 9.5, 0.032, 0.90, 0.5, 5 and 3, respectively.


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When spreadsheet use, became more common, Zoz and Grisso [3] employed the spreadsheet

for predicting the tractive performance of tractors. This spreadsheet has the capability to handle

tractor configuration, bias-ply, radial tires, and different agricultural soil conditions.

Recognizing the advantages of pneumatic tires and tracks, more and more farm tractors are

now being equipped with rubber-tracks. Even though field studies have been conducted to

compare the performance of rubber-tracks and MFWD tractors in different soils [11 and 12], to

date very little has been done to develop a mathematical model to predict the tractive

performance of rubber-tracks.

In order to develop a generalized model for rubber-tracks, first we used a trial and error

procedure to determine the values of the coefficients (C1 C6) and constants (K1 and K2) for

rubber-tracks. Using the test data collected with rubber-tracks in different soils, we determined

the coefficients and constants that provided the best fit and developed the following

relationships:

a) Gross-traction-slip,

b) Motion resistance-slip, and

c)Tractive efficiency-slip

For comparison purposes, the values of the constants and coefficients for bias-ply tires, radial

tires, and rubber-tracks are included in Table 1.

The following are the modified relationships for predicting the tractive performance of

rubber-tracks:

+

=

TL

TW61

5

CI/0.698-e-1W

TLTWCI

nB (5)


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( )DWI0.03s17e1n

B0.025e11.10GTR +

= (6)

( ) ( )n

B

s0.5

DWI

0.03

DWI0.7

n

B

1.75MRR

++

= (7)

( )sGTR

NTRTE

= 1 (8)

Where, TW and TL are track width and track length respectively.

Since the dynamic weight ratio (DWR), the ratio between dynamic loads on the rear and

front, play a significant role in the overall performance of rubber tracks, it is necessary to express

the coefficients C4 and C5 in terms of dynamic weight index, DWI, and,

( )( )

+

=

1DWR

1DWR0.7ABS1DWI (9)

The tractive efficiency is its maximum when the DWI reaches it maximum value of one.

DWI is maximum when the weight distribution is equal in the front and the rear (DWR =1).

The values for C4 and C5 shown in table 1 are assuming equal weight distribution in the front and

rear.

Validation of the Model

The validity of the model developed was examined by comparing the predicted and

experimental results. Net Traction Ratio (NTR), and tractive efficiency (TE)-slip relationships

were developed for 44 cases based on field tests [9, 13, 14, and 15] conducted in sandy loam, silt

loam, clay and clay loam soils under tilled and untilled conditions and in subsoiled sandy loam

with four different track widths (406, 457, 635, and 813 mm) and compared against predicted. In

order to assess the closeness between the two, the Pearson Correlation Coefficients and the

Average absolute differences at 30 different track slips in the range of 1-30% were determined


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and the average for each of the 44 cases considered is presented in table 2. High correlation

coefficient and low absolute difference values indicate good agreement between the predicted

and experimental results except in clay soils.

In order to further illustrate the agreement between the predicted and experimental results,

the NTR and TE were plotted as a function of track slip for four different cases (Fig. 2). Curves

for two different track widths 813 mm (case 44), and 406 mm (case 40) are shown in Fig. 2a.

As expected wider track widths provided better performance in terms of net traction developed

and tractive efficiency. Even though case 44 provided high correlation coefficient and low

absolute difference values for both NTR and TE, the model seems to under predict the NTR at

higher track slips.

Figure 2b compares the performance of 406 mm rubber-track in untilled (case 9) and tilled

(case 5) sandy loam. In general, there is good agreement between the predicted and experimental

results. As expected the track performance in untilled soil with higher CI is slightly better than in

tilled soil with lower CI value. The maximum tractive efficiencies (TEmax) in both cases

occurred at slips between 6-7 percent. The predicted TEmax values are 0.831 and 0.815 for

untilled and tilled soils, respectively.

To further illustrate the validity of the model, we determined the maximum tractive

efficiencies from predicted and experimental results and the corresponding net traction and track

slip values at TEmax and plotted these ratios against each other as shown in Fig. 3 for each of the

44 cases in Table 2. The fact that most points for all three ratios clustered around 1:1 line, once

again illustrates very good agreement between predicted and experimental results.


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Model Application

The model developed can be used effectively for a number of different applications. Figure 4

is included to demonstrate one such use. The horizontal and vertical axes of this figure represent

Net Traction Ratio and traction performance ratios such as TE and track slip, respectively. Plots

include predicted TE, and slip curves for different mobility numbers. For a given mobility

number (Bn), the figure provides the information on the maximum TE possible and the

corresponding Net Traction ratio and track slips at which the vehicle has to operate to obtain

these ratios. For example, for Bn=40, to attain a maximum TE and a Net Traction ratio of 0.43

the vehicle must operate at a track slip of approximately 7.1%. In the same soil (which provided

a Bn value of 40), if higher TE and NTR are desired, one could select wider and or longer track

to provide a higher Bn number. This model together with the spreadsheet [3] will provide the

user the flexibility to determine the influence of different parameters on Bn values and develop

similar performance curves quickly for different Bn values. This model can also be used

effectively to compare the performance of vehicle with rubber-tracks or tires and for conducting

parametric studies as illustrated in Zoz and Grisso [3].

Conclusion

An empirical model to predict the tractive performance of vehicles equipped with rubber-

tracks has been developed. Comparison of predicted and experimental results shows that the

model developed is effective in predicting the performance of rubber-tracks during agricultural

operations.


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References

[1] Brixius, WW. Traction prediction equations for bias-ply tires. ASAE Paper No. 871622. St. Joseph,MI: ASAE, 1987.

[2] Al-Hamad, SA, Grisso RD, Zoz FM, Von Bargen K. Tractor performance spreadsheet for radial tires.Computers and Electronics in Agric, 1994:10(1):45-62.

[3] Zoz, FM, Grisso RD. Traction and Tractor Performance. ASAE Distinguished Lecture Series #27. St.Joseph, MI: ASAE, 2003.

[4] Zoz, FM. Predicting tractor field performance. Trans. ASAE, 1972:15:249-255.

[5] Wismer, RD, Luth HJ . Off-road traction prediction of wheeled vehicles. ASAE Paper No. 72619. St.Joseph, MI: ASAE, 1972.

[6] Zoz, FM, Brixius WW. Traction prediction for agricultural tires on concrete. ASAE Paper 79-1046,1979.

[7] Corcoran, PT, Gove DS. Understanding the mechanics of track traction. In Proc. Int'l Conference onSoil Dynamics, 4:664-678, 17-19 June. Auburn, AL: Auburn University, Office of ContinuingEducation, 1985.

[8] Upadhyaya, SK, Chancellor WJ , Wulfsohn D, Glancey JL. Sources of variability in traction data.J.Terramechanics, 1988:25(4):249-272.

[9] Zoz, FM. Rubber and tire tractive performance. SAE Technical Paper Series 972731. Warrendale,PA: SAE, 1997.

[10] Culshaw, D. Rubber tracks for traction.J. Terramechanics, 1988:25(1): 69-80.

[11] Shell, LR, Zoz FM, Turner RL. Field performance of rubber rubber and MFWD tractors in Texassoils. In Rubber and Tire Traction in Agricultural Vehicles, 65-73. SAE SP-1291. Warrendale, PA:SAE, 1997.

[12] Turner, RJ, Shell LR, Zoz FM. Field performance of rubber rubber and MFWD tractors in southernAlberta soils. In Rubber and Tire Traction in Agricultural Vehicles, 75-85. SAE SP-1291.Warrendale, PA: SAE, 1997.

[13] Bashford, LL, Kocher MF. Rubbers vs tires, rubbers vs rubbers, tires vs tires. Applied Engng in Ag,1999:15(3):175-181.

[14] Esch, JH, Bashford LL, Von Bargen K, Ekstrom RE. Tractive performance comparisons between arubber rubber track and four-wheel-drive tractor. Trans. ASAE, 1990:33(4):1109-1115.

[15] Upadhyaya, SK, Rosa UA, Josiah MN, Koller M. Effects of rubber width and grouser wear on thetractive characteristics of rubber-tracked vehicles. Trans. ASAE, 2001:17(3):267-271.

[16] Okello, JA, Dwyer, M.J , Cottrell, FB. The tractive performance of rubber tracks and a tractor drivingwheel tyre as influenced by design parameters.J. agric. Engng Res. 1994:59(1):33-43.


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[17] Okello, JA, M Watany, DA Crolla. A theoretical and experimental investigation of rubber trackperformance models.J. agric. Engng Res. 1998:69(1):15-24.

[18] Okello, JA. Prediction and experimental validation of the field tractive performance of a rubbertrack. J. agric. Engng Res. 194:59(3):163-171.

[19] Marsili, A, Servadio, P, Pagliai, M, Vignozzi, N. Changes of some physical properties of a clay soilfollowing passage of rubber- and metal-tracked tractors. Soil and Tillage Research, 1998:49(2):185-199.

[20] Blunden, BG, McBride, RA, Daniel, H., Blackwell, PS. Compaction of an earthy sand by rubbertracked and tyred vehicles.Australian Journal of Soil Research 1994:32:1095-1108.

[21] Dwyer, MJ; Okello, JA; Scarlett, AJ. Theoretical and experimental investigation of rubber tracks foragriculture.J. Terramechanics, 1993:30(4):285-298.

[22] Ma ZD, NC Perkins. Modeling of track-wheel-terrain interaction for dynamic simulation of trackedvehicle systems. Proceedings of the 1999 ASME Design Engineering Technical Conferences

September 12-15, 1999, Paper DETC99/VIB-8200, Las Vegas, Nevada

[23] Rahman, A, Yahya, Mohd. Zohadie, Wan Ishak and Desa Ahmad. Design parameters optimizationsimulation of a prototype segmented rubber track vehicle for Sepang peat in Malaysia.AmericanJournal of Applied Sciences 2005:2(3): 655-671.

[24] Sandu, C, Freeman, JS. Connectivity algorithm for an extended rubber-band track model.HeavyVehicle Systems, A Series of the Int. J. of Vehicle Design, 2002:9(4):334355.

[25] Zoz, FM. Predicting tractor field performance (updated). ASAE Paper No. 871623. St. J oseph, MI:ASAE, 1987.


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Captions:

Table 1. Comparison of constants and coefficients in the generalized traction model for bias-plytires, radial tires, and rubber-tracks.

Table 2. Comparison of predicted and experimental results from 44 cases with a range of soil and trackwidth conditions.

Figure 1. Rubber-tracks drive nomenclature and mechanics.

Figure 2. Comparison of predicted (lines) and experimental (symbols) Net Traction Ratio and TE Sliprelationships. (a) Effect of track width on track performance in wet untilled loam soil (Solid &Diamonds - 813 mm ; Dash & Square - 457 mm.) (b) Effect of soil condition on theperformance 406 mm rubber track (Solid & Diamond - untilled soil with CI =1.31 MPa; Dash& Square - tilled soil with CI =1.10 MPa)

Figure 3. Predicted and experimental performance ratios plotted against each other.

Figure 4. Tractive Efficiency and Slip curves for three Mobility Numbers as a function of Net TractionRatio.


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Table 1. Comparison of constants and coefficients in the generalized traction model for bias-plytires, radial tires, and rubber-tracks.

Coefficients &Constants

Bias-Ply TiesBrixius [1]

Radial TiresAl-Hamad et al. [2]

Rubber-tracks

K1 5 5 5

K2 3 3 6C1 0.88 0.88 1.10C2 0.10 0.08 0.025C3 7.5 7.0 17.0C4 0.04 0.03 0.03

1C5 1.0 1.20 1.75

1C6 0.5 0.5 0.5

1DWR is assumed to be one.


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Table 2. Comparison of predicted and experimental results from 44 cases with a range of soil and trackwidth conditions.

Case Bn TW TL CI

mm mm MPa TE NTR TE NTR

1 Sandy Loam Subsoi led 46.3 406 2261 1.00 0.997346 0.999905 0.0472663 0.02014116

2 Sandy Loam Subsoi led 56.1 635 2261 1.00 0.996561 0 .99798 0 .0282443 0.04510789



5 San dy Lo am Till ed 49.0 406 2261 1.10 0.998363 0.996981 0.0129518 0.03853785



8 San dy Lo am Till ed 49.0 406 2261 1.10 0 .975066 0.99996 0.009508 0.04693027

9 Sandy Loam Un til led 54.7 406 2261 1.31 0.990594 0.996663 0.0063653 0.02009918

10 San dy Lo am Un til led 66.1 635 2261 1.31 0.978403 0.9994 0.0097118 0.01749651



13 Silty L oam Till ed 35.9 457 2261 0.34 0.951008 0.942627 0.0218614 0.16603619




17 Si lt y Loam Un til led 37.5 457 2261 0.41 0.945927 0.947026 0.0316202 0.1753590618 Silty L oam Un til led 36.1 635 2261 0.43 0.905658 0.95606 0.0223477 0.13643306

19 Si lt y Loam Un til led 39.4 635 2261 0.57 0.872817 0.985759 0.0372502 0.02778831

20 Si lt y Loam Un til led 46.2 813 2261 0.41 0.958382 0.973264 0.0101703 0.09209565

21 Clay Tilled 53.7 457 2261 1.01 0.820267 0.792466 0.0151746 0.15879987

22 Clay Tilled 39.1 457 2261 0.48 0.60586 0.754862 0.0277168 0.0991268

23 Clay Tilled 51.1 635 2261 0.69 0.797113 0.85587 0.0307369 0.12043289

24 Clay Tilled 44.2 635 2261 0.46 0.828185 0.939957 0.0296629 0.05800577

25 Clay Tilled 68.2 813 2261 1.04 0.936452 0.853057 0.0190983 0.197039

26 Clay Tilled 42.9 813 2261 0.28 0.726106 0.811423 0.0393161 0.07126382

27 Clay Untilled 54.5 457 2261 1.03 0.905086 0.996594 0.0263013 0.03134426

28 Clay Untilled 41.5 457 2261 0.57 0.672038 0.84052 0.0450066 0.11782215

29 Clay Untilled 62.4 635 2261 1.03 0.901489 0.914425 0.0124913 0.11090144

30 Clay Untilled 52.0 635 2261 0.72 0.850327 0.926415 0.0208519 0.06013871

31 Clay Untilled 67.9 813 2261 1.03 0.967229 0.995113 0.0069084 0.09847739

32 Clay Untilled 55.2 813 2261 0.68 0.917268 0.946129 0.0092776 0.05634723

33 Loam Tilled 44.7 457 2261 0.69 0.740461 0.832527 0.0262387 0.07627296

34 Loam Tilled 43.5 457 2261 0.65 0.854858 0.981217 0.0384508 0.06026457

35 Loam Tilled 57.5 635 2261 0.89 0.915002 0.923621 0.010294 0.11262731

36 Loam Tilled 51.1 635 2261 0.69 0.785959 0.758413 0.0169448 0.08446186

37 Loam Tilled 51.3 813 2261 0.56 0.881195 0.930607 0.0165314 0.07695967

38 Loam Tilled 57.5 813 2261 0.74 0.896175 0.905043 0.0106538 0.07261524

39 Loam Untilled 54.5 457 2261 1.03 0.923889 0.997448 0.0232292 0.02364217

40 Loam Untilled 48.7 457 2261 0.83 0.980841 0.999865 0.0255197 0.02022895

41 Loam Untilled 62.4 635 2261 1.03 0.94936 0.988319 0.0088979 0.05442887

42 Loam Untilled 60.3 635 2261 0.97 0.9512 0.99077 0.0095515 0.03012802

43 Loam Untilled 67.9 813 2261 1.03 0.926902 0.981142 0.0176778 0.09050577

44 Loam Untilled 73.1 813 2261 1.17 0.994844 0.998042 0.0068254 0.05062752

Brixius parameters

Soil Conditions

Pearson

Correlation

Average Absolu te

Difference

Experimental cases [9, 13, 14, and 15].


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MR

slrrr

NT T

W1

Va

Wd

GT

W2 W3 W4

W5

Ground LineDh

rt

Vt =Velocity, theoreticalVa =Velocity, actualT =Axle torqueGT =Gross traction (theoretical pull)NT =Net traction (actual pull)

MR =Motion resistance

W =Weight, staticWd =Weight, dynamicslr =Loaded radius, staticrr =Rolling radiusrt =Torque radius

Vt

Va

T

GT

NT

MR

W

Wd

slr

rr

rt

eh

Figure 1. Rubber-tracks drive nomenclature and mechanics.


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(a) Effect of track width on track performance in wet untilled loam soil (Solid & Diamonds - 813

mm ; Dash & Square - 457 mm.)

(b)Effect of soil condition on the performance 406 mm rubber-track (Solid & Diamond - untilledsoil with CI =1.31 MPa; Dash & Square - tilled soil with CI =1.10 MPa)

Figure 2. Comparison of predicted (lines) and experimental (symbols) Net Traction Ratio and TE Sliprelationships.


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Figure 3. Predicted and experimental performance ratios plotted against each other.

Figure 4. Tractive Efficiency and Slip curves for three Mobility Numbers as a function of Net TractionRatio.

Documents

An Empirical Model for Tractive Performance of Rubber-Tracks