Belt Tracks JTER

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1

2 An empirical model for tractive performance

3 of rubber-tracks in agricultural soils

4 Robert Grisso a,*, John Perumpral a, Frank Zoz b

5 a Biological Systems Engineering, 200 Seitz Hall (0303), Virginia Tech, Blacksburg, VA 24061-0303, USA6 b John Deere Product Engineering Center, Waterloo, IA, USA

7

8 Abstract

9 Mathematical models capable of describing the interaction between traction devices and10 soils have been effective in predicting the performance of off-road vehicles. Such a model capa-

11 ble of predicting the performance of bias-ply tires in agricultural soils was first developed by12 Brixius [Brixius WW. Traction prediction equations for bias-ply tires. ASAE Paper No.13 871622. St. Joseph, MI: ASAE; 1987]. When the soil and vehicle parameters are known, this14 model uses an iterative procedure to predict the tractive performance of a vehicle including15 pull, tractive efficiency, and motion resistance. Al-Hamad et al. [Al-Hamad SA, Grisso RD,16 Zoz FM, Von Bargen K. Tractor performance spreadsheet for radial tires. Comput Electron17 Agr 1994:10(1):4562] modified the Brixius equations to predict the performance of radial18 tires. Zoz and Grisso [Zoz, FM, Grisso RD. Traction and tractor performance. ASAE Distin-19 guished Lecture Series #27. St. Joseph, MI: ASAE; 2003] have demonstrated that the use of20 spreadsheet templates is more efficient than the original iterative procedure used to predict21 the performance of 2WD and 4WD/MFWD tractors. As tractors equipped with rubber-tracks22 are becoming popular, it is important that we have the capability to predict the performance23 for off-road vehicles equipped with rubber-tracks during agricultural operations. This paper24 discusses the development of an empirical model to accomplish this goal and its validity by25 comparing the predicted results with published experimental results.26 2005 Published by Elsevier Ltd on behalf of ISTVS.

27 Keywords: Rubber-tracks; Traction mechanics; Traction prediction; Traction model28

0022-4898/$20.00 2005 Published by Elsevier Ltd on behalf of ISTVS.

doi:10.1016/j.jterra.2005.12.002

* Corresponding author. Tel.: +1 540 231 6538; fax: +1 540 231 3199.E-mail address: [email protected](R. Grisso).

Journal of Terramechanics xxx (2005) xxxxxx

www.elsevier.com/locate/jterra

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Terramechanics

TER 308 No. of Pages 12

21 December 2005; Disk Used ARTICLE IN PRESS
mailto:[email protected]:[email protected]


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29 1. Introduction

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

31 agricultural operations. During the last five decades tractors used in agricultural32 operations have undergone many changes. Selected examples of these changes33 include increased horsepower ratings, tractor configuration (2WD and 4WD/34 MFWD), and use of different types of tractive devices such as (bias-ply tires to radial35 tires to rubber-tracks). Production capabilities of these tractors depend heavily on36 tractor configuration, type of tractive devices used and terrain conditions. Ability37 to predict and optimize the performance of these tractors during field operations38 has been of great interest to scientists, manufacturers, and users.39 In an effort to meet this need, Zoz[4]developed a set of graphs based on field tests40 conducted in three types of soils: firm, tilled and soft or sandy, and on concrete with

41 2WD tractors. He demonstrated that the set of graphs developed could be used to42 predict the drawbar pull, travel speed, drawbar horsepower and travel reduction43 of 2WD tractors under different soil conditions.44 Wismer and Luth[5] studied the single wheel behavior in an indoor soil-bin facil-45 ity. Using dimensional analysis and the results of carefully planned tests, they devel-46 oped equations to predict the pull and tractive efficiency of tractors under different47 slip when certain conditions are satisfied.48 Similar sets of equations were developed by Zoz and Brixius[6]to predict the per-49 formance of tractors on concrete. Nebraska tractor test results were used to develop50 these relationships. Based on these equations, they have also developed a computer51 program to predict the vehicle performance on concrete.52 In 1987, Brixius revised the relationships originally developed by Wisner and Luth53 [5].Using the data from approximately 2500 field tests involving 121 soiltire com-54 binations and improved curve-fitting techniques, Brixius[1]came up with a revised55 set of traction equations. In addition to providing better predictions, these equations56 developed for bias-ply tires, extended the range of applications. Al-Hamad et al.[2]57 modified the relationships developed by Brixius to predict the performance of vehi-58 cles equipped with radial tires.59 A review of literature has revealed that a great deal of experimental studies have

60 been conducted to assess the tractive performance of rubber-tracks in different soils61 and to compare it with the performance of other types of tractive devices [824].62 However, to our knowledge, only limited studies have dealt with the development63 of mathematical models to predict the performance of rubber-tracks in agricultural64 soils. Upadyaya et al.[8]and Zoz[9]have tested rubber-tracks and developed regres-65 sion equations to predict net traction, motion resistance, and tractive efficiency as a66 function of travel reduction or slip. They used regression analysis to minimize data67 scatter and developed useful relationships for specific test conditions. The limitation68 of these relationships, however, is that they may be useful only for the field and vehi-69 cle conditions that existed during the collection of experimental data. Therefore, the

70 overall objective of this study was to develop an empirical model specifically to pre-71 dict the tractive performance of rubber-tracks in a variety of agricultural soils and to72 establish its validity by comparing the predicted results with the experimental.

2 R. Grisso et al. / Journal of Terramechanics xxx (2005) xxxxxx




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73 2. Rubber-track mechanics

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

75 similar, and a brief discussion of mechanics of rubber-tracks is included in this sec-76 tion. A more detailed review of the same is available in Zoz and Grisso[3]. Fig. 177 shows the forces on a rubber-tracks system. The torque input (T) to the axle develops78 a gross thrust (GT). Part of the gross thrust is used to overcome the motion resis-79 tance (MR). The remainder is the net traction (NT) or pull available for useful work.80 Though there are similarities between tires and rubber-tracks, the dynamic load81 distribution on rubber-tracks is significantly different. For example, the location of82 the dynamic load resultant (eh) depends on the static weight distribution, the design83 of the suspension system supporting the bogie wheels, and the vehicle weight transfer84 characteristics[7].

85 To maximize the tractive performance and to minimize the soil disturbance, ide-86 ally the pressure distribution on a rubber-track should be uniform and, the dynamic87 weight distribution in the front and rear should be equal. The dynamic weight dis-88 tribution on rubber-tracks depends on factors such as static weight, tractor dimen-89 sions, location of center of gravity, angle and the magnitude of pull. Unlike in the90 case of tires, both the magnitude and uniformity in dynamic load distribution are91 important during the testing of rubber-track systems.

92 3. Traction equations for rubber-tracks

93 Since our goal was to develop a traction model with the capability to predict the94 rubber-track performance in a variety of agricultural soils and for different track95 systems, we decided to modify the following original equations developed for tires96 by Brixius [1]. Brixius expressed Gross Traction Ratio (GTR) and Motion Resis-97 tance Ratio (MRR) as a function of mobility number (Bn) and wheel slip (s). He

MR

slrrr

NT T

W1

Va

Wd

GT

W2 W3 W4

W5

Ground LineDh

rt

Vt = Velocity, theoreticalVa = Velocity, actualT = Axle torque

GT = Gross traction (theoretical pull)NT = Net traction (actual pull)

MR = Motion resistance

W = Weight, staticWd = Weight, dynamicslr = Loaded radius, static

rr = Rolling radiusrt = Torque radius

Vt

Va

T

GTNT

MR

W

Wd

slr

rrrt

eh

Fig. 1. Rubber-tracks drive nomenclature and mechanics.

R. Grisso et al. / Journal of Terramechanics xxx (2005) xxxxxx 3




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98 determined the dimensionless numbers in the equations using a curve-fitting tech-99 nique and the following are the generalized equations he developed:

100

Bn CI b dW

1 K1 d=h1 K2 b=d

1GTR T

r W C1 1 eC2Bn 1 eC3s C4 2

MRR MW

C5Bn

C4 C6 sffiffiffiffiffiBn

p 3

NTR NTW

GTR MRR 4102102

103 whereBnis the mobility numbers;b the unloaded tire section width;r the tire rolling

104 radius;h the tire section height;s the wheel slip; NT is the net traction or pull;Tthe105 axle torque; CI is the cone index;dthe unloaded tire diameter; dthe tire deflection;106 Wthe dynamic load on the tractive devices;Mthe motion resistance and NTR is the107 net traction ratio.108 Eqs.(1)(3) include six coefficients (C1C6) and two tire constants (K1 and K2).109 These constants and coefficients may change depending on the type of tractive110 devices. For bias-ply tires, values of C1, C2, C3, C4, C5, C6, K1, and K2 are 0.88,111 0.1, 7.5, 0.04, 1.00, 0.5, 5 and 3, respectively[1].112 Zoz [25] created a Lotus 1-2-3 template for Brixius equations. This template113 helped the users to predict the performance of tractors or different configurations114 equipped with bias-ply or radial tires in different agricultural soils.115 As radial tires became popular, there was interest in models capable of predict-116 ing performance of tractors equipped with radial tires. Al-Hamed et al. [2]117 modified the Brixius equations to meet this need. Using experimental data and118 curve-fitting techniques, a new set of coefficients C1C6 and K1 and K2 to repre-119 sent the radial tires was generated. They are 0.88, 0.08, 9.5, 0.032, 0.90, 0.5, 5 and120 3, respectively.121 When spreadsheet use, became more common, Zoz and Grisso[3]employed the122 spreadsheet for predicting the tractive performance of tractors. This spreadsheet has

123 the capability to handle tractor configuration, bias-ply, radial tires, and different124 agricultural soil conditions.125 Recognizing the advantages of pneumatic tires and tracks, more and more126 farm tractors are now being equipped with rubber-tracks. Even though field stud-127 ies have been conducted to compare the performance of rubber-tracks and128 MFWD tractors in different soils [11,12], to date very little has been done to129 develop a mathematical model to predict the tractive performance of rubber-130 tracks.131 In order to develop a generalized model for rubber-tracks, first we used a trial and132 error procedure to determine the values of the coefficients (C1C6) and constants (K1

133 and K2) for rubber-tracks. Using the test data collected with rubber-tracks in differ-134 ent soils, we determined the coefficients and constants that provided the best fit and135 developed the following relationships:





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136 (a) Gross-traction-slip.137 (b) Motion resistance-slip.138 (c) Tractive efficiency-slip.

139140 For comparison purposes, the values of the constants and coefficients for bias-ply141 tires, radial tires, and rubber-tracks are included inTable 1.142 The following are the modified relationships for predicting the tractive perfor-143 mance of rubber-tracks:

Bn CI TW TLW 1 eCI=0:698

5

1 6 TWTL

!

5

GTR

1:10

1

e0:025Bn

1

e17s

0:03

DWI 6

MRR 1:75

Bn 0:7 DWI 0:03

DWI 0:5 sffiffiffiffiffiBn

p 7

TE NTRGTR

1 s 8

145145

146 where TW and TL are track width and track length, respectively.147 Since the dynamic weight ratio (DWR), the ratio between dynamic loads on the148 rear and front, play a significant role in the overall performance of rubber-tracks,149 it is necessary to express the coefficientsC4and C5in terms of dynamic weight index,

150 DWI, and,

DWI 1 ABS 0:7 DWR 1DWR 1

9152152

153 The tractive efficiency is its maximum when the DWI reaches it maximum value of154 one. DWI is maximum when the weight distribution is equal in the front and the rear155 (DWR = 1). The values forC4 and C5shown inTable 1are assuming equal weight156 distribution in the front and rear.

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

Coefficients and constants Bias-ply ties Brixius[1] Radial tires Al-Hamad et al.[2] Rubber-tracks

K1 5 5 5K2 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

a

C5 1.0 1.20 1.75aC6 0.5 0.5 0.5

a DWR is assumed to be one.





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157 4. Validation of the model

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

159 and experimental results. Net traction ratio (NTR), and tractive efficiency (TE)slip160 relationships were developed for 44 cases based on field tests[9,1315]conducted in161 sandy loam, silt loam, clay and clay loam soils under tilled and untilled conditions162 and in subsoiled sandy loam with four different track widths (406, 457, 635, and163 813 mm) and compared against predicted. In order to assess the closeness between164 the two, the Pearson Correlation Coefficients and the Average absolute differences165 at 30 different track slips in the range of 130% were determined and the average166 for each of the 44 cases considered is presented inTable 2. High correlation coeffi-167 cient and low absolute difference values indicate good agreement between the pre-168 dicted and experimental results except in clay soils.

169 In order to further illustrate the agreement between the predicted and experimen-170 tal results, the NTR and TE were plotted as a function of track slip for four different171 cases (Fig. 2). Curves for two different track widths 813 mm (case 44), and 406 mm172 (case 40) are shown inFig. 2(a). As expected wider track widths provided better per-173 formance in terms of net traction developed and tractive efficiency. Even though case174 44 provided high correlation coefficient and low absolute difference values for both175 NTR and TE, the model seems to under predict the NTR at higher track slips.176 Fig. 2(b) compares the performance of 406 mm rubber-track in untilled (case 9)177 and tilled (case 5) sandy loam. In general, there is good agreement between the pre-178 dicted and experimental results. As expected the track performance in untilled soil179 with higher CI is slightly better than in tilled soil with lower CI value. The maximum180 tractive efficiencies (TEmax) in both cases occurred at slips between 6% and 7%. The181 predicted TEmaxvalues are 0.831 and 0.815 for untilled and tilled soils, respectively.182 To further illustrate the validity of the model, we determined the maximum trac-183 tive efficiencies from predicted and experimental results and the corresponding net184 traction and track slip values at TEmax and plotted these ratios against each other185 as shown inFig. 3 for each of the 44 cases in Table 2. The fact that most points186 for all three ratios clustered around 1:1 line, once again illustrates very good agree-187 ment between predicted and experimental results.

188 5. Model application

189 The model developed can be used effectively for a number of different applica-190 tions.Fig. 4 is included to demonstrate one such use. The horizontal and vertical191 axes of this figure represent net traction ratio and traction performance ratios such192 as TE and track slip, respectively. Plots include predicted TE, and slip curves for dif-193 ferent mobility numbers. For a given mobility number (Bn), the figure provides the194 information on the maximum TE possible and the corresponding net traction ratio

195 and track slips at which the vehicle has to operate to obtain these ratios. For exam-196 ple, forBn= 40, to attain a maximum TE and a Net Traction ratio of 0.43 the vehi-197 cle must operate at a track slip of approximately 7.1%. In the same soil (which





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

Case Soil conditions Brixius parameters Pearson correlati

Bn TW (mm) TL (mm) Cl (MPa) TE NT

1 Sandy loam Subsoiled 46.3 406 2261 1.00 0.997346 0.92 Sandy loam Subsoiled 56.1 635 2261 1.00 0.996561 0.93 Sandy loam Subsoiled 61.0 813 2261 1.00 0.994104 0.94 Sandy loam Subsoiled 46.3 406 2261 1.00 0.997324 0.95 Sandy loam Tilled 49.0 406 2261 1.10 0.998363 0.96 Sandy loam Tilled 59.3 635 2261 1.10 0.950108 0.97 Sandy loam Tilled 64.6 813 2261 1.10 0.999352 0.98 Sandy loam Tilled 49.0 406 2261 1.10 0.975066 0.99 Sandy loam Untilled 54.7 406 2261 1.31 0.990594 0.9

10 Sandy loam Untilled 66.1 635 2261 1.31 0.978403 0.911 Sandy loam Untilled 72.0 813 2261 1.31 0.994029 0.9

12 Sandy loam Untilled 54.7 406 2261 1.31 0.998466 0.913 Silty loam Tilled 35.9 457 2261 0.34 0.951008 0.914 Silty loam Tilled 36.6 635 2261 0.45 0.78922 0.915 Silty loam Tilled 35.5 635 2261 0.40 0.913543 0.916 Silty loam Tilled 44.1 813 2261 0.34 0.682614 0.917 Silty loam Untilled 37.5 457 2261 0.41 0.945927 0.918 Silty loam Untilled 36.1 635 2261 0.43 0.905658 0.919 Silty loam Untilled 39.4 635 2261 0.57 0.872817 0.920 Silty loam Untilled 46.2 813 2261 0.41 0.958382 0.921 Clay Tilled 53.7 457 2261 1.01 0.820267 0.722 Clay Tilled 39.1 457 2261 0.48 0.60586 0.723 Clay Tilled 51.1 635 2261 0.69 0.797113 0.8

24 Clay Tilled 44.2 635 2261 0.46 0.828185 0.925 Clay Tilled 68.2 813 2261 1.04 0.936452 0.826 Clay Tilled 42.9 813 2261 0.28 0.726106 0.827 Clay Untilled 54.5 457 2261 1.03 0.905086 0.928 Clay Untilled 41.5 457 2261 0.57 0.672038 0.829 Clay Untilled 62.4 635 2261 1.03 0.901489 0.9


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Table 2 (continued)

Case Soil conditions Brixius parameters Pearson correlation

Bn TW (mm) TL (mm) Cl (MPa) TE NTR

30 Clay Untilled 52.0 635 2261 0.72 0.850327 0.9231 Clay Untilled 67.9 813 2261 1.03 0.967229 0.9932 Clay Untilled 55.2 813 2261 0.68 0.917268 0.9433 Loam Tilled 44.7 457 2261 0.69 0.740461 0.8334 Loam Tilled 43.5 457 2261 0.65 0.854858 0.9835 Loam Tilled 57.5 635 2261 0.89 0.915002 0.92

36 Loam Tilled 51.1 635 2261 0.69 0.785959 0.7537 Loam Tilled 51.3 813 2261 0.56 0.881195 0.9338 Loam Tilled 57.5 813 2261 0.74 0.896175 0.9039 Loam Untilled 54.5 457 2261 1.03 0.923889 0.9940 Loam Untilled 48.7 457 2261 0.83 0.980841 0.9941 Loam Untilled 62.4 635 2261 1.03 0.94936 0.9842 Loam Untilled 60.3 635 2261 0.97 0.9512 0.9943 Loam Untilled 67.9 813 2261 1.03 0.926902 0.9844 Loam Untilled 73.1 813 2261 1.17 0.994844 0.99

Experimental cases[9,1315].


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198 provided aBnvalue of 40), if higher TE and NTR are desired, one could select wider199 and or longer track to provide a higher Bn number. This model together with the200 spreadsheet[3]will provide the user the flexibility to determine the influence of dif-201 ferent parameters on Bn values and develop similar performance curves quickly for

202 different Bn values. This model can also be used effectively to compare the perfor-203 mance of vehicle with rubber-tracks or tires and for conducting parametric studies204 as illustrated in Zoz and Grisso[3].

Fig. 2. Comparison of predicted (lines) and experimental (symbols) net traction ratio and TE slip

relationships: (a) effect of track width on track performance in wet untilled loam soil (solid and diamonds,813 mm; dash and square,457 mm); (b) effect of soil condition on the performance 406 mm rubber-track (solid and diamond, untilled soil with CI = 1.31 MPa; dash and square, tilled soil withCI = 1.10 MPa).





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205 6. Conclusion

206 An empirical model to predict the tractive performance of vehicles equipped with207 rubber-tracks has been developed. Comparison of predicted and experimental results

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

Fig. 4. Tractive efficiency and slip curves for three mobility numbers as a function of net traction ratio.





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257 [23] Rahman A, Yahya A, Zohadie M, Ishak W, Ahmad D. Design parameters optimization simulation of258 a prototype segmented rubber track vehicle for Sepang peat in Malaysia. Am J Appl Sci259 2005;2(3):65571.260 [24] Sandu C, Freeman JS. Heavy vehicle systems, connectivity algorithm for an extended rubber-band261 track model. A series of the Int J Vehicle Desig 2002;9(4):33455.262 [25] Zoz FM. Predicting tractor field performance (updated). ASAE Paper No. 871623. St. Joseph, MI:263 ASAE; 1987.264




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