Traction prediction equations for radial ply tyres PDF

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Journal o f Terramechanics, Vol. 26, No. 2, pp. 149-175, 1989. 0022-4898/8953.00 + 0.00 Printed in Great Britain. Pergamon Press plc. © 1989 ISTVS TRACTION PREDICTION EQUATIONS FOR RADIAL PLY TYRES S. K. UPADHYAYA,* D. WULFSOHN* and G. JUBBALt Summary---Three radial tyres (16.9R38, 18.4R38 and 24.5R...

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Journal o f Terramechanics, Vol. 26, No. 2, pp. 149-175, 1989. Printed in Great Britain.

0022-4898/8953.00 + 0.00 Pergamon Press plc. © 1989 ISTVS

TRACTION PREDICTION EQUATIONS FOR RADIAL PLY TYRES S. K.

UPADHYAYA,* D. WULFSOHN* and G. JUBBALt

Summary---Three radial tyres (16.9R38, 18.4R38 and 24.5R32) wcre tested in two different soil types (Capay clay and Yolo loam), five different soil treatments (conditions) in each of the two soils, at three different axle loads and two different inflation pressures to obtain traction prediction equations for radial tyres. The field data were analysed to obtain traction prediction equations under varying soil and loading (vertical load, inflation pressure) conditions for radial ply tyres. Simulation studies were conducted to determine the effect of soil conditions, dynamic load on the axle and inflation pressure on the tractive performance of radial ply tyres. The results indicate that (i) changes in soil conditions influence tyre performance much more than changes in tyre loading and dimensions; (ii) in a given soil condition the 24.5R32 tyre appears to perform better than the other two tyres; (iii) the performance of the 18.4R38 tyre was similar to that of the 16.9R38 tyre, although at lower soil cone index values, the 18.4R38 tyre performed slightly better than the 16.9R38 tyre. NOTATION a,c,a',b',c' Traction coefficients, dimensionless A 3-Dimensional soil-tyre contact area (L 2) b Tyre section width (L) Cone index (FL -2) C Soil cohesion (FL -2) Cs GT Gross traction (F) Slip, dimensionless i i' Proportionality constant, dimensionless Soil deformation (L) J k Soil shear modulus (L) K. Vertical tyre stiffness (FL-l) Tangential tyre stiffness (FL -l) Kt l Contact length (L) l' Distance travelled (L) Exponent used in soil sinkage formula, dimensionless n NT Net traction (F) Tyre inflation pressure (FL -2) P r Roiling radius (L) T Torque (FI_,) Tractive efficiency, dimensionless TE Va Actual forward velocity (LT -l) W Dynamic load (F) Distance along soil-tyre contact area (L) Y Vertical tyre deformation (L) Bt Bz Vertical soil deformation (L) 2x Loss of forward motion due to soil and tyre deformation (L) Tangential tyre deformation (L) "qt Soil moisture content, dimensionless 0 Tyre rotation as it traverses the contact length, dimensionless O

*Associate Professor and Graduate Assistant, Agricultural Engineering Department and tGraduate Assistant, Mechanical Engineering Department, University of California, Davis, CA 95616, U.S.A. 149

15o (T 1"m~lx

+ !,

S. K. UPADHYAYA, D. WULFSOHN and G. JUBBAL normal stress (FL-2) Shear stress (FL-2) Maximum shear stress (FL-z) Soil angle of internal friction, dimensionless Angle between surface normal and the vertical at point on soil-tyre contact surface, dimensionless Wheel angular velocity (T ~)

INTRODUCTION AND REVIEW OF LITERATURE USE OF radial ply tyres is on the rise because of their superior tractive characteristics compared to their bias ply counterparts. Radial ply tyres develop higher drawbar pulls than bias ply tyres in the 0 to 30% slip range [1,2]. Tests conducted at the National Swedish Testing Institute for Agricultural Machinery [3] indicated that radial ply tyres develop higher pull compared to bias ply tyres in all the soil conditions tested except in very cohesive clay soil. Taylor et al. [4] reported that the radial ply tyre performs better on firm surfaces and that its advantages diminish on softer soils. Gee-Ctough et al. [5] found that radial ply tyres perform better than bias ply tyres in a variety of British soil conditions when the radial ply tyre was not too highly inflated. Burt et al. [6] reported that radial ply tyres perform better than bias ply tyres at an intermediate axle load and a low inflation pressure. On a drier, less dense, higher cone-index soil use of radial ply tyres resulted in higher tractive efficiencies than bias ply tyres. Wulfsohn et al. [7] found that in a tilled Yoio loam soil an 18.4R38 radial ply tyre performed better than an 18.4-38 bias ply tyre with similar tread design. Hausz [8] stated that the tractive advantages of radial ply tyres over bias ply tyres are usually due to a larger foot print for the same axle load, and more even ground pressure distribution over the contact area. The performance characteristics of different radial ply tyres in different soil conditions at different dynamic loads and inflation pressures are of interest in properly equipping power units to perform field tasks efficiently. Such information is also valuable for selecting power units to match implements (i.e. useful in machinery selection and management). The objective of this study is to obtain traction prediction equations for radial ply tyres under various soil and loading conditions. In order to accomplish this objective three different radial ply tyres (18.4R38, 16.9R38 and 24.5R32) were tested in two different soil types (Capay clay and Yolo loam) with five different soil conditions [undisturbed, stubble disced two times, stubble disced two times plus final disced two times, stubble disced two times plus final disced four times, and stubble disced two times plus final disced four times and irrigated (to form a crust)] in each of the two soils, at two different inflation pressures (83 kPa, 124 kPa) and three different axle loads (low, medium, and high) resulting in 180 tests consisting of over 1500 runs. EXPERIMENTAL TECHNIQUE Two fields with different soil types (Capay clay and Yolo loam) were selected for conducting field tests. These fields were located in the vicinity of the University of California, Davis. Five 30 × 180 m plots were marked off in each of the two fields. One of the five plots was left undisturbed to provide a firm soil condition for tyre tests. This plot will be referred to as "undisturbed" or " U D " . The second plot was disced two times with a heavy stubble disc to a depth of approximately 150 mm. This will be referred to as "SD2". The third plot was stubble disced two times with the same heavy stubble disc

TRACTION PREDICTION FOR RADIAL PLY TYRES

151

and then disced two more times using a finishing disc. This treatment will be referred to as " S D 2 + D 2 " . The fourth plot was stubble disced two times and then disced four more times using a finishing disc. This will be called " S D 2 + D 4 " . The last plot was subjected to the same treatment as the fourth plot and then it was flood irrigated to create a crust. This treatment will be referred to as " S D 2 + D 4 R " . Although the five treatments were similar in both Yolo loam and Capay clay fields, the Capay clay field was flood irrigated and allowed to dry before any field work was done on it, whereas, the Yolo loam field was not flood irrigated before tillage treatments. This resulted in distinctly different moisture conditions in these two fields. The depth of tillage was approximately 150 mm in the SD2, S D 2 + D 2 , S D 2 + D 4 , and S D 2 + D 4 R treatments in both fields. These five treatments in each of the two soils provided us with ten soil conditions for tyre tests. In each plot three radial ply tyres--16.9R38, 18.4R38 and 24.5R32 were tested using the UC Davis single wheel tester. Upadhyaya et al. [9] have described this machine in detail. Each tyre was tested at two different inflation pressures (83 and 124 kPa) and three different dynamic axle loads (approximately 13.3, 20 and 26.7 kN). Within each tyre the dynamic axle load treatments were completely randomised for a given inflation pressure. All three axle load tests at a given inflation pressure for a given tyre in a given soil condition were conducted in a sub-plot of 30 × 30 m. This test procedure resulted in 180 traction tests. The traction test procedure was similar to the one used by Upadhyaya [10] and Wulfsohn et al. [7]. For a given inflation pressure and dynamic axle load for a particular tyre in a specific soil condition the test procedure consisted of several runs (usually 8 to 10). The first run was conducted at approximately zero slip in a slip control mode. As the tyre moved down the test track, the input torque, net traction, dynamic axle load, wheel speed and forward travel speed were acquired using a CR21-XL Campbell Scientific data logger. The next few runs (usually three) were conducted at successively higher levels of slip in a slip control mode. Note that during a given run slip was held constant at a preselected value. The later runs were conducted in a draft control mode with each successive run at a higher draft setting. During a given run draft was held constant at a preselected value in the draft control mode. Between successive runs the tester was rotated approximately five degrees by pivoting about the rear support wheels. A given test was considered complete when either the wheel slipped excessively or when the system torque limitations were reached [10]. In each subplot 12 cone index values were obtained using an hydraulically operated standard cone penetrometer. Six of the 12 cone index measurements were made along the test track in a location in between the first and the second axle load tests and the other 6 cone index values were obtained in a location in between the second and the third axle load tests. In addition, a set of six bulk density and moisture content readings were obtained in each subplot in the Yolo loam soil using a neutron probe strata gage. The bulk density readings were obtained at 50, 100 and 150 mm depth at each of the six locations. This instrument gave only the surface moisture content. In the Capay clay soil we obtained a set of five bulk density and moisture content readings in each subplot.

RESULTS AND DISCUSSION

The experimental data for each of the 180 runs were analysed as suggested by Upadhyaya [10], and Wulfsohn et al. [7] to obtain the following type of equations:

152

S . K . U P A D H Y A Y A . D. WULFSOHN and G. JUBBAL

NT/W

= a(1-e -~)

(1)

and

r/rW-

GT _ a,(l_b,e_C,i) W

(2)

where N T = net traction, kN, G T = gross traction, kN; W -- dynamic load on the axle, kN; i = slip = ( 1 - V a/r~0) × 100;

(3)

r = rolling radius, m; ~o = wheel angular velocity, rad s-J; V~, = actual forward speed, m s-l, a,c,a',b',c' = traction equation coefficients. These equations assume that the zero condition used in these tests is zero net traction on the test surface when the wheel slip is zero (cf. equation 1). From equations (1), (2) and (3) it follows that: Tractive efficiency,

Power output

TE -

Power input

NTV, -

×

× 100

100

To)

_ (NT/W)(V~) (T/rW) ~ (NT/W) (T/rW) ( 1 - 0

x 100

x 100 .

(4)

Figures 1 through 3 are typical plots of experimental and predicted data over 0 to 30% slip range. Figure 1 corresponds to a test conducted in the " S D 2 + D 4 R " treatment in the Yolo loam soil at an inflation pressure of 83 kPa and a dynamic load on the axle of 20.59 kN using the 16.9R38 tyre. Figure 2 corresponds to a similar treatment for the 18.4R38 tyre. The dynamic axle load for this case was 21.15 kN. Figure 3 corresponds to the results of a test in a similar soil condition using a 24.5R32 tyre also at the same 83 kPa inflation pressure. The corresponding axle load for this case was 22 kN. It is clear from Figs 1 to 3 (and also from the other 177 tests) that the traction data fit equations (1) and (2) very well in all cases. The correlation coefficient, R 2, was generally 0.95 or higher. We have found this to be true in our other studies as well [7,10,11]. Equations (1) and (2) tell us the variation of NT/W and T/rW with slip. The influence of soil and tyre parameters on traction performance are not clear as these effects are hidden in the regression coefficients a, c, a', b' and c'. In order to predict the performance of a given tyre in a specific soil condition under a prescribed dynamic axle load at a given inflation pressure, it is important to relate these traction equation coefficients to soil and tyre parameters. In order to develop these relationships, we proceed as follows:

T R A C T I O N P R E D I C T I O N F O R R A D I A L PLY TYRES

x Expt. NT/W, • Pred. NTIW,

.7

Expt. T/rW, • Pred. T/rW,

;

2

!

, M .... 1!

153

+ Expt. TE, • Pred. TE,

. .6 ~- ,5

f

~ 4

10

FIG. 1.

15 Slip %

20

25

30

Tractive characteristics of a 16.9R38 tyre in the "SD2 + D4R" treatment in a Yolo loam soil at an inflation pressure of 83 kPa and dynamic axle load of 20.59 kN.

x Expt. NT/W, • Pred. NT/W,

Expt. T/rW, • Pred. T/rW,

+ Expt. TE, • Pred• TE,

.9

.s

/

.2

0

0

FIG. 2.

5

10

15 Slip %

20

25

30

Tractive characteristics of an 18.4R38 tyre in the "SD2 + D4R" treatment in a Yolo loam soil at an inflation pressure of 83 kPa and dynamic axle load of 21.15 kN.

Upadhyaya et al. [11] have stated that for a pneumatic tyre on a deformable soil surface, N T and W are given by

NT

= L ~coslp" dA - L o'sinlFdA

(5)

154

S.K. UPADHYAYA,

I

x Expt. NT/W, • Pred, NT/W,

D. W U L F S O H N and G. J U B B A L Expt. T/rW, • Pred. T/rW,

+ Expt. TE, • Pred, TE,

,9 ,S .7

~-.6

*

S/ 10

0

FIG. 3.

15 Slip %

20

25

30

Tractive characteristics of a 25.4R32 tyre in the "SD2 + D4R" treatment in a Yolo loam soil at an inflation pressure of 83 kPa and dynamic axle load of 22 kN.

where -r = shear stress = Tmax (1 - e-//t'); Tmax maximum shear stress = Cs + ¢r tan ~b; c~ = cohesion; cr = normal stress; + = angle of internal friction of soil; j = soil deformation; k = soil shear modulus; ~ -- angle between the surface normal and the vertical at any given point on the contact surface (Fig. 4); A = actual 3-D soil-tyre contact surface. The lack of knowledge of the true 3-D soil-tyre contact surface and the distribution of normal stress on this surface makes it impossible to evaluate equations (5) and (6). In this analysis we will develop mathematical expressions for traction equation coefficients for a simplified case which will be used as a guide to analyse these coefficients using a step-wise regression technique. Let us assume that the contact surface lies in a horizontal plane and that it is rectangular in shape with width b and length l. Let the soil deformation at a point y along this contact length be represented by j. Let the corresponding tyre deformation =

NT

O

FIG. 4.

Schematic view of soil-tyre interaction.

T R A C T I O N P R E D I C T I O N F O R R A D I A L PLY T Y R E S

155

be "qt. Let both the maximum soil and tyre deformations be proportional to the contact length. Then we have,

N T = K,r/,

(7)

where lit tyre deformation in the tangential direction, K t = tangential tyre stiffness. Assuming normal stress or to be constant over the foot print and approximately equal to inflation pressure we get, =

cr = p = W /bl

(9)

where b = tyre section width. Assuming soil deformation to be proportional to contact distance at any point under the tyre, referring to Fig. 5, we have j = i'y

(10)

where y = distance along soil-tyre interface as shown in Fig. 5; i' = proportionality constant. Substituting equations (9) and (10) in equation (8) and carrying out the integration we get,

NT

= "r~bl[l+-~l{e-i'l/k-l}l

(11)

When a given point on the tyre traverses the contact length of the tyre, let the axle move forward by l'. Then l' is given by 1' = I - A

(12)

where A is the loss of forward motion due to the deformation of both soil and tyre, i.e.

A = i'l+ 77, --PIJ ;:

(13)

v

b

i i i i i

' i ,

FIG. 5.

_2'

1

Contact geometry and soil deformation for the simplified case.

156

S.K. UPADHYAYA, D. WULFSOHN and G. JUBBAL

But the slip, i, is defined as

i =

l - ( l - A)

A

l

l

(14)

Therefore, from equations (7), (13) and (14) we get, NT

(15)

i=i'+ Kfl

As slip approaches 100%, i.e. lim i ~ 1, i' approaches NT

i' ---~ l

(16)

Ktl

From equations (11) and (16) we get,

NT

since

~- "rmaxbl[1

e --~- = e - l / k

NT

1

(17)

"TVT

= e_l/k e+N:r/kKt = 0

Since N T / K t l is small we can expand ( 1 - N T / K , I ) -1 ~- I + N T / K t l +

....

(18)

Using equation (18) in equation (17) we get,

NT

= ~.~bl

or N T

=

1--~

1+

"cm~,bl(1 - k / D 1 + "r~,,bk/Ktl

(19)

As slip approaches 100% our regression equation gives, NT W

-

a

.

(20)

TRACTION PREDICTION FOR RADIAL PLY TYRES

157

From equations (19) and (20) we get, a W = "G~,bl(1 - k / l )

(21)

1 + "~,,bk/K,l

Once again, expanding the denominator of equation (21) in a Taylor series, we obtain, a

bl

v.

= (~/'Vm.x(1-k/l)[1-

b

k

m'~(~-~/(7) +''']

"

Since p = W/bl from equation (9), we obtain upon neglecting higher order terms,

=

vm~

Iv)

('G,Jp)(1-k/l)-

(k/l)

+ ('c~,x/p)2(k/l)2(W/K,l)

W

.

Neglecting terms of the order (k/l) 2 we get, a = (vm.~,/p)(1-k/l)

-

(v~,./p)Z(k/l)(W/Ktl)

.

(22)

The value of "rma× depends on soil type and condition, and normal stress at the soil-tyre interface ['rma× = cs + (r tan ¢]. Soil shear modulus is expected to depend upon vertical load, soil contact area (i.e. b and l) and soil type and condition. If we assume cone index, C, and moisture content, 0, represent soil type and condition, and (r = p (equation 9) we have,

%~, = f~ (C, 0, p)

k = f 2 ( W , b, l, C, O)

where ~r~ has dimensions [FL2]; C, [FL'Z]; O, [ - ] ; p, [FLZ]; k,

[L]; W, [F]; b, [L]; l, [L] .

(23a)

(23b)

158

S . K . UPADHYAYA, D. WULFSOHN and G. JUBBAL

From the Buckingham II theorem we get,

~r~,x/P = f3 (C/p, O)

(24)

k/l : f, (W/Cbl, b/l, O) .

(25)

From equations (22), (24) and (25) we have,

a =

(b/l), (C/p)2(W/Cbl), 01

C/p,--C-bf ( C / p ) ( W / C b l ) ,

(26)

Equation (26) was used as a basis for a step-wise regression analysis. We found that neither (W/Cbl) nor (C/p) (W/Cbl) were significant by themselves, but that their product was highly significant. Table 1 shows the variables which affect coefficient a significantly. Figure 6 is a graphic...