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Mathematical models for soil displacement under a rigid wheel
Journal of Terramechanics
Journal of Terramechanics 43 (2006) 287–301
www.elsevier.com/locate/jterra
Mathematical models for soil displacement under a rigid wheel Koichiro Fukami a,*, Masami Ueno b, Koichi Hashiguchi c, Takashi Okayasu c a
Department of Upland Farming Research, National Agricultural Research Center for Kyushu, Okinawa Region NARO, Miyazaki 885-0091, Japan b Faculty of Agriculture, University of the Ryukyus, Okinawa 903-0213, Japan c Faculty of Agriculture Graduate School, Kyushu University, Fukuoka 812-8581, Japan
Received 1 October 2004; received in revised form 28 February 2005; accepted 30 May 2005 Available online 2 November 2005
Abstract Soil–wheel interaction especially soil deformation caused by the wheel motion was investigated experimentally using a sophisticated soil bin test apparatus and an on-line measurement system for soil displacement. Based on these test results, characteristics of soil deformation were summarized focusing on the behavior and distribution of displacement increment vectors. Mathematical models were examined in order to describe the displacements of soil particles. Properties of the displacement loci are described. The magnitude of the displacement increment vector, its horizontal and vertical components are discussed, and characteristics of these distributions with respect to the relative horizontal distance from the vertical centerline of the wheel to the target point are clarified. Shapes of these distribution curves were closely similar to those of the derivatives of a Gaussian function. A distribution curve of the horizontal displacement increment had two peaks and that of the vertical one had three peaks. Based on the results, mathematical models for those displacement increments were proposed by employing a Gaussian function through multiplication of a linear function and a quadratic function. Predicted distributions and displacement loci of the models agreed with high
*
Corresponding author. Tel.: +81 0986 24 4277. E-mail address: kofukami@affrc.go.jp (K. Fukami).
0022-4898/$20.00 Ó 2005 ISTVS. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jterra.2005.05.005
288
K. Fukami et al. / Journal of Terramechanics 43 (2006) 287–301
accuracy to the measured values. The mathematical models were extended taking into consideration the wheel slip. The predicted distributions according to test conditions agreed very well to the measured results. Ó 2005 ISTVS. Published by Elsevier Ltd. All rights reserved. Keywords: Soil displacement; Rigid wheel; Displacement increment; Model; Gaussian function; Displacement locus; First derivative; Second derivative
1. Introduction Traveling performances of a wheel are determined by the deformation state and the stress distribution in soil near the wheel. Therefore, in order to improve the prediction accuracy of the traveling performance, it is necessary to clarify quantitatively the characteristics and mutual relations among traveling performances, soil deformation and stress distribution. Especially, soil deformation is a key item to carry out these soil–wheel interactions. Modeling and analysis of the ground contact stress distribution have been advanced using various sensors in the previous studies [1]. Although the results are not sufficient, many applications based on those are developed. On the other hand, quantitative analyses of soil deformation have not been developed sufficiently due to difficulty of measurement and the complexity of soil properties. There are some reports concerned with soil deformation under a wheel, but these are limited qualitative analyses. Thus, the modeling of soil deformation has not been initiated, except through application of the theory of elasticity. Recently, numerical methods such as the finite element method have been developed remarkably, and are expected to be useful in analyses of traveling performance. Some problems restrict the development of numerical methods. One problem is that there are no experimental data to verify the analytical results. Therefore, detailed analyses on the properties of soil deformation and its modeling were conducted in this study, based on the measurement of soil deformation near the wheel. An on-line measurement system for soil deformation was employed in the traveling tests. The locus of soil displacement was analyzed to clearly define its characteristics. The displacement increments of soil were separated into the horizontal and vertical components to allow the properties to be shown in detail. Modeling of these components was conducted using the Gaussian function and its derivatives.
2. Methods 2.1. Traveling test of a wheel A series of wheel traveling test was executed to measure the soil deformation under the wheel. A sophisticated wheel test apparatus developed by the authors in
K. Fukami et al. / Journal of Terramechanics 43 (2006) 287–301
289
previous studies [2] was employed for the traveling tests using a rigid wheel. The apparatus consisted of a soil bin, a rigid wheel mounted on a carriage, and the driving units and measurement systems as shown in Fig. 1. The apparatus was capable of developing a uniform soil condition in the soil bin and making precise measurements during the experiment. The soil bin was 414 mm wide, 1888 mm in length, and 617 mm deep. ‘‘Toyoura standard sand’’ with air-dried condition was used for the soil bin test. The soil bin was filled up by spraying from a hopper. Thickness of soil layer was set 600 mm and initial void ratio of soil layer was regulated about 0.61 in average. The test wheel was 300 mm diameter, 410 mm in width, and coated with a 5 mm thickness of rubber. Its weight was 217 N and its peripheral speed was 0.0833 mm/s. The forward velocity was, therefore, quite low. The soil bin and wheel had nearly the same width, so the plane strain condition was met. Movement of soil particles was limited to two-dimensions, vertical and longitudinal. The measured variables were the rotational angle of the wheel, travel distance, sinkage, dynamic load, torque on the wheel shaft, soil reaction and its application line, normal and tangential stresses applied to the contact surface. The data were converted to digital data and stored on a hard disk. During a test, the wheel was slowly lowered down to the soil surface, initially without any rotation. Rotation of wheel was then started and forward travel began. The former stage was called the initial sinkage stage and the latter, the traveling stage.
Fig. 1. System used for measurement of soil displacement.
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Fig. 2. Side view showing initial configuration of markers.
2.2. Analysis of soil deformation Soil displacement was measured using many 5 mm diameter markers placed on the inner plane of the soil bins sidewall. The markers were, therefore, set between the soil and the sidewall. The markers moved with the soil particles, so the displacement of a marker was considered to coincide with that of the soil. Configurations of markers were recorded by taking a series of photographs using a high-resolution camera, with one photograph taken for each one degree of wheel rotational angle. The coordinates of markers were then calculated using the image processing capability of the on-line measurement system [3]. The displacement increment of a marker was equivalent to the difference of the coordinates in two sequential photographs. The initial depth of the markers in the top row was 10 mm. The initial vertical distance between rows and the initial horizontal distance between columns of the configuration was 10 mm. Fig. 2 shows a sample of the initial positions of the markers and the principle of the soil deformation analysis. Strain increments in the soil were calculated using a displacement function and the finite element mesh component for strain analysis consisted of three markers.
3. Results and discussion 3.1. Characteristics of soil deformation 3.1.1. Locus of soil particles Displacement loci of soil particles at 3.9% and 21.9% slip are shown in Fig. 3. The loci of five markers that had an initial depth of 10 mm are shown in Fig. 3. Initial positions of these loci were set at the same point on the graph. Also, the markers were chosen to avoid the influence of the initial sinkage stage and to allow all movement following the wheel pass to be completed. The numbers in the legend
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Fig. 3. Displacement loci of soil particles that had initial depths of 10 mm.
of Fig. 3 designate the marker numbers. When a soil particle was forward of the wheel, it moved forward and upward as the wheel approached. The particle then turned to move forward and downward. The direction of movement then changed to rearward and downward, and the particle reached its lowest position just as the wheel was directly above the marker. After the wheel passed, the marker moved rearward and upward, and its motion finally stopped. This tendency was seen at high slips, and negative values too. Therefore, at any value of slip, the path of a soil particle as the wheel traveled was a spiral. Although shapes of soil particle loci presented in some previous studies [4,5] are similar to those in Fig. 3, those reported in
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the previous studies were qualitative, rough analyses. Quantitative measurements of soil particles were conducted in this study. At 3.9% slip, the loci of five markers agreed well with one another. At 21.9% slip, however, the loci had different shapes, especially at their final positions, and these differences were about 1 mm. Moreover, the rearward displacement at 21.9% slip was about 2 mm greater than at 3.9% slip. 3.1.2. Definition of displacement increment The phenomenon that path of a soil particle is a spiral has been confirmed by Wong [3] and Yong [4], so it may be regarded as a general phenomenon. In this study, displacement increments were calculated for each one degree of wheel rotation angle. Characteristics of displacement were then analyzed with respect to the relative horizontal distance between the target and the vertical centerline of the wheel. Definitions of relative horizontal distance, displacement increment, and its horizontal and vertical component are shown in Fig. 4. Dux and Duy are horizontal and vertical components of the displacement increment, respectively. The variables D, ux, and Dux are positive when the vectors coincide with the travel direction of the wheel. The variables uy, and Duy are negative when their vectors point downward. Therefore, when Dux > 0 and Duy < 0, then Du is in the forward and downward direction.
Fig. 4. Geometries between wheel position and displacement increments of the soil.
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3.1.3. Characteristics of the displacement increment of soil particles having initial depths of 10 mm Fig. 5 shows the terms |Du|, Dux and Duy with respect to the relative horizontal distance, D, at 3.9% slip. The solid lines in Fig. 5 show the arithmetic average of the values of five markers. As D decreased from D = 70, the magnitude of the displacement increment, |Du|, increased rapidly and reached its peak nearly directly beneath the wheel axle, i.e., D = 0. As D decreased further, |Du| decreased rapidly until D reached approximately 30. The peak |Du| occurred slightly forward of the wheel axle. The shape of the distribution curve is similar to that of a Gaussian distribution. The distribution domain of |Du| coincided with the range that the soil particle displaced with respect to D. The magnitude |Du| dose not provide information about the direction of the displacement increment, Du, so the distributions of the components Dux, and Duy are discussed here. As D decreased from D = 70, the horizontal displacement Dux increased rapidly until it reached its positive peak, and then decreased until it reached its negative peak at about D = 0. The value of D at Dux = 0 was D = 20 and was approximately midway between the values of D at the two peaks. As D decreased from where Dux was at
Directly beneath axle
0.2 | Δ u | (mm)
Averaged value Direction of travel
0.1
0.0
Δ u x (mm)
0.1
pf
0.0 xi ( Δ ux=0 )
–0.1
pb
–0.2
Δ u y (mm)
0.1
pu1
pu2
0.0 –0.1
xr1 (Δ uy=0)
–0.2 200
100
pd
0
xr2 (Δ uy=0)
–100
–200
Relative horizontal distance D (mm) Fig. 5. Distributions of |Du|, Dux and Duy with the relative horizontal distance for soil particles having initial depths of 10 mm.
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its negative peak, Dux increased and approached zero. Thus, the appearance of both a positive and a negative peak was a distinctive feature of this distribution curve for Dux. This indicates there were two domains; the soil particle was pushed forward when it was forward of the wheel and was kicked rearward when it was to the rear of the wheel. The negative peak of Dux was slightly forward of where D = 0. As D decreased, the vertical displacement increment, Duy, increased at first and a minor positive peak occurred at about D = 50. The Duy function then decreased until its negative peak occurred at about D = 20, and Duy then increased until its major peak occurred at D = 10. The Duy function then decreased. The positions, where Duy = 0 occurred at D = 0 and D = 40. Distinctive features of the Duy distribution curve were the occurrence of two positive peaks and one negative peak. 3.2. Modeling of the displacement increments 3.2.1. Basic function The phenomena that a soil particle drew a spiral and the typical shapes of distribution curves with respect to the relative horizontal distance were regarded as general characteristics of soil deformation under the traveling wheel. Thus, the modeling of the shapes of such distributions enabled us to evaluate the strain increment and strain at an arbitrary point in the soil. In addition, stress analysis would be capable of substituting the calculated strain increments into an elasto–plastic constitutive equation for the soil. Results of finite element analyses would then be verified, or the soil displacement would be substituted directly into the finite element method for the prediction of drawbar pull, torque and other terms relating the traveling performances. The shapes of distributions of |Du|, Dux and Duy had distinctive peaks with respect to the relative horizontal distance as mentioned above. Therefore, models were examined using the following Gaussian function as a basic function in order to express such distinctive shapes. ( ) 2 ðx xc Þ y ¼ Am exp ; ð1Þ 2w2 where Am was a peak value, xc was a peak position, x is a target position and w is a width of half value. A similar shape of Dux with positive and negative peaks could be derived by differentiating the equation with respect x as follows: ( ) 2 A ðx x Þ m c y 0 ¼ 2 ðx xc Þ exp . ð2Þ w 2w2 It was also obtained by multiplying the original Gaussian function and a linear function. Kawasaki [5] applied the function Eq. (2) to express a distribution of shear stress in the soil when a wheel traveled on the soil. Furthermore, the shape of the vertical displacement increment with two positive peaks and one negative peak was examined by the secondary differentiation of the Gaussian function in Eq. (1). The function in Eq. (2) was differentiated again with respect to x, and the following function was obtained:
K. Fukami et al. / Journal of Terramechanics 43 (2006) 287–301
( ) o Am n ðx xc Þ2 2 2 y ¼ 4 ðx xc Þ w exp . w 2w2 00
295
ð3Þ
Fig. 6 shows the shapes of these functions. The first derivative as a horizontal component drew a symmetry shape to the intersection, and the secondary derivative as a vertical one drew a symmetry shape to the principal line. However, measured curves
y Direction of travel w
xc
Am
Am / e
0.5
x - y’ w
xc
–Am /( w· e0.5 )
x Am /(w · e0.5 )
xc + w 3
y’’
xc – w 3
2 Am /(w 2 · e1.5. )
x –Am / w 2
w
xc
Fig. 6. Shapes of the Gaussian function and its derivatives.
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drew distorted shapes that were different from these functions as shown in Fig. 6. Therefore, the shapes should be adjusted to the measured shapes while maintaining the fundamental features of the curves. 3.2.2. Modeling of each displacement increment Modeling for the displacement increments Dux and Duy was carried out based on the Gaussian function. The function in Eq. (2) relating to Dux was obtained by multiplying the original Gaussian function and a linear function. The peak position of the former coincided to the intersection of the latter so that the shape of the function was symmetric about the y-axis. The measured curve was adjusted by parallel translation of the linear function. Fig. 7 showed the modified shape following the parallel translation. The function describing Dux was assumed by the modification as follows: ( ) 2 ðx xc Þ DU x ¼ Amx ðx xi Þ exp ; ð4Þ 2w2 where DUx = Dux/(Vo Æ Dt), x = D/R. (x ⎯ x p )2 ⎯ q ( x ⎯ xc )2 ⎯ q
x ⎯ xi
Δ Ux
x ⎯ xc w
w
xc
xi
Δ Uy
xc
A my
A mx
x
x xp
Δ Uy
Δ Ux ( pf x , pf y )
xi xc
( pu 1x, pu 1y )
2 q xp xc
x
x
( pbx , pby )
( pd x , pd y ) Fig. 7. Shapes of the model functions.
( pu2x , pu 2 y )
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Both the Dux and Duy components were divided by the circumference displacement increment VoDt for describing the dimensionless quantities. Further, the relative horizontal displacement D was converted to a non-dimensional quantity x using the radius R of the wheel. Vo is a circumference speed, xi is a coordinate of x as Dux= 0. xc and w were a peak position and a width of half value, respectively. The function in Eq. (3) was obtained by multiplying the original Gaussian function and a quadratic function. Both principal axes were coincident in this case, so the shape of the distribution curve was symmetric. The asymmetrical shape in the measured values could be expressed by the translation of the principal axis of the quadratic function. In other words, a quadratic function with a peak (xp,q) was multiplied by the original Gaussian function as follows.
Fig. 8. Fitting results of DUx, DUy and |DU| for soil particles having initial depths of 10 mm.
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DU y ¼ Amy
) ðx xc Þ2 ðx xp Þ q exp ; 2w2 n
2
o
(
ð5Þ
where Duy = Duy/(VoDt), xp and q were a peak position and value of the quadratic function. 3.2.3. Fitting to models Fig. 8 shows examples of application of Eqs. (4)–(6) to the measured values. A non-linear least square method was employed to the fitting based on the Gauss– Newton method. The values of parameters obtained by curve fitting and the degree of fitness, i.e., the coefficients of determination are shown in Table 1. The measured values of horizontal and vertical displacement increments were obtained by averaging the values of five markers, respectively. The coefficients of determination of DUx and DUy were greater than 0.96 and 0.93, respectively, for any value of slip. Therefore, the distributions of DUx and DUy were approximated with sufficient accuracy using the proposed models. The dimensionless magnitudes of the displacement increment were calculated by the following equation: jDU j ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DU 2x þ DU 2y .
ð6Þ
Fig. 9 shows the displacement loci predicted by the proposed models. These curves were obtained by the integration of values of Dux and Duy. The predicted curves agreed well with the measured curves. The displacement increment in the initial stage was calculated to be rather small, so this caused a slight difference to occur in the final stage for 3.9% slip. However, the fundamental tendency to draw the spiral with the travel of
Table 1 Parameter values for Gaussian functions and their derivatives, and coefficients of determination Parameter
Slip (%) 3.9
21.9
Amx Amy xc w xi xp q
0.623 2.944 0.113 0.104 0.150 0.143 0.014
0.647 1.814 0.136 0.126 0.215 0.213 0.023
Coefficient of determination R2(DUx) R2(DUy)
0.98 0.93
0.96 0.97
K. Fukami et al. / Journal of Terramechanics 43 (2006) 287–301
Σ Δ u x , Σ Δ u y (mm)
1.0
Direction of travel
3.9% 21.9%
299
Σ Δ ux
0.0
–1.0 Σ Δ u y –2.0
3.9% 21.9%
Measured value Predicted curve
–3.0 100
50
0
–50
–100
Relative horizontal distance D (mm) 0.5 21.9%
Vertical displacement (mm)
Slippage 3.9% Slip3.9%
0.0 –0.5 –1.0 –1.5 1.0
Measured value Predicted curve
0.0
–1.0
–2.0
–3.0
Horizontal displacement (mm) Fig. 9. Predicted loci for soil particles having initial depths of 10 mm.
wheel was expressed well. Thus, it is suggested that the proposed models for displacement increments will fit well to the shapes of distributions and the loci of displacement.
4. Extended models including slip The unknown coefficients in the models were approximated by the following equation in order to describe the displacement increment at arbitrary values of slip. Fig. 10 shows the predicted results. A ¼ A1 s2 þ A2 s þ C;
ð7Þ
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3.00
0.30
Amy 2.00
0.20
Amx
1.00
0.10
0.00 0.15
0.00 0.30
0.10
0.20
0.05
xi
0.10
xc
0.00 0.15
0.00 0.03
0.10
0.02
0.05
xp
0.01
q
w 0.00
0.00
0
5
10
15
20
25
0
Slip (%)
5
10 15 Slip (%)
20
25
Fig. 10. Relationships between unknown coefficients and slip at 10 mm depth.
Vertical displacement (mm)
0.5
10%
Direction of travel 4%
0.0
20%
–0.5 –1.00 1.0
0.5
0.0 –0.5 –1.00 –1.5 Horizontal displacement (mm)
–2.0
Fig. 11. Results of prediction at 10 mm depth for slip values from 4% to 20%.
–2. 5
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where A was an arbitrary unknown coefficient, A1, A2 and C were the constants in the quadratic function. Fig. 11 shows the predicted displacement loci with a slip range of 4–20% obtained by substituting A from Eq. (7) for Amx, xi, xc, and w in Eq. (4) and Amy, xp, and q in Eq. (5). The tendency that the final position of displacement locus shifted rearward as slip increased was expressed well as shown in the Figure. Thus, predictions of the displacement increments in the various conditions become possible by making clear the relations between the unknown coefficients and parameters in the models.
5. Conclusions Soil displacement increments under a rigid wheel were measured experimentally and the characteristics were summarized. The displacement locus of a soil particle was a spiral for various conditions including wheel slip values from 4% to 20%. The displacement increment was resolved into its horizontal and vertical components for detailed analysis. The distribution curves of these revealed distinctive features; the former and the latter had two peaks and three peaks, respectively. The distribution curves were modeled using a Gaussian function and its derivatives. The predicted results using the models agreed well with the measured values. The models were extended taking account of arbitrary wheel slip. Analyses of strain distribution and stress distribution should be conducted in order to carry out more detailed investigations of the traveling performances.
References [1] Nohse Y, Hashiguchi K, Ueno M, Shikanai T, Izumi H, Koyama F. A measurement of basic mechanical quantities of off-the-road traveling performance. J Terramechanics 1991;28(4):358–70. [2] Hashiguchi K, Okayasu T, Ueno M, Shikanai T. Image processing on-line measurement for soil displacement. J JSAM 1998;60(6):11–8. [3] Wong JY. Behavior of soil beneath rigid wheels. J Agr Eng Res 1967;12(4):257–69. [4] Yong RN, Fattah EA. Prediction of wheel–soil interaction and performance using the finite element method. J Terramechanics 1976;13(4):227–40. [5] Kawasaki H, Hirakawa D, Tatsuoka F. Experimental study on stress distribution in sand subject to roller compaction. In: The 37th Japan National Conference on Geotechnical Engineering; 2000. p. 1623–24.
Journal of Terramechanics 43 (2006) 287–301
www.elsevier.com/locate/jterra
Mathematical models for soil displacement under a rigid wheel Koichiro Fukami a,*, Masami Ueno b, Koichi Hashiguchi c, Takashi Okayasu c a
Department of Upland Farming Research, National Agricultural Research Center for Kyushu, Okinawa Region NARO, Miyazaki 885-0091, Japan b Faculty of Agriculture, University of the Ryukyus, Okinawa 903-0213, Japan c Faculty of Agriculture Graduate School, Kyushu University, Fukuoka 812-8581, Japan
Received 1 October 2004; received in revised form 28 February 2005; accepted 30 May 2005 Available online 2 November 2005
Abstract Soil–wheel interaction especially soil deformation caused by the wheel motion was investigated experimentally using a sophisticated soil bin test apparatus and an on-line measurement system for soil displacement. Based on these test results, characteristics of soil deformation were summarized focusing on the behavior and distribution of displacement increment vectors. Mathematical models were examined in order to describe the displacements of soil particles. Properties of the displacement loci are described. The magnitude of the displacement increment vector, its horizontal and vertical components are discussed, and characteristics of these distributions with respect to the relative horizontal distance from the vertical centerline of the wheel to the target point are clarified. Shapes of these distribution curves were closely similar to those of the derivatives of a Gaussian function. A distribution curve of the horizontal displacement increment had two peaks and that of the vertical one had three peaks. Based on the results, mathematical models for those displacement increments were proposed by employing a Gaussian function through multiplication of a linear function and a quadratic function. Predicted distributions and displacement loci of the models agreed with high
*
Corresponding author. Tel.: +81 0986 24 4277. E-mail address: kofukami@affrc.go.jp (K. Fukami).
0022-4898/$20.00 Ó 2005 ISTVS. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jterra.2005.05.005
288
K. Fukami et al. / Journal of Terramechanics 43 (2006) 287–301
accuracy to the measured values. The mathematical models were extended taking into consideration the wheel slip. The predicted distributions according to test conditions agreed very well to the measured results. Ó 2005 ISTVS. Published by Elsevier Ltd. All rights reserved. Keywords: Soil displacement; Rigid wheel; Displacement increment; Model; Gaussian function; Displacement locus; First derivative; Second derivative
1. Introduction Traveling performances of a wheel are determined by the deformation state and the stress distribution in soil near the wheel. Therefore, in order to improve the prediction accuracy of the traveling performance, it is necessary to clarify quantitatively the characteristics and mutual relations among traveling performances, soil deformation and stress distribution. Especially, soil deformation is a key item to carry out these soil–wheel interactions. Modeling and analysis of the ground contact stress distribution have been advanced using various sensors in the previous studies [1]. Although the results are not sufficient, many applications based on those are developed. On the other hand, quantitative analyses of soil deformation have not been developed sufficiently due to difficulty of measurement and the complexity of soil properties. There are some reports concerned with soil deformation under a wheel, but these are limited qualitative analyses. Thus, the modeling of soil deformation has not been initiated, except through application of the theory of elasticity. Recently, numerical methods such as the finite element method have been developed remarkably, and are expected to be useful in analyses of traveling performance. Some problems restrict the development of numerical methods. One problem is that there are no experimental data to verify the analytical results. Therefore, detailed analyses on the properties of soil deformation and its modeling were conducted in this study, based on the measurement of soil deformation near the wheel. An on-line measurement system for soil deformation was employed in the traveling tests. The locus of soil displacement was analyzed to clearly define its characteristics. The displacement increments of soil were separated into the horizontal and vertical components to allow the properties to be shown in detail. Modeling of these components was conducted using the Gaussian function and its derivatives.
2. Methods 2.1. Traveling test of a wheel A series of wheel traveling test was executed to measure the soil deformation under the wheel. A sophisticated wheel test apparatus developed by the authors in
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289
previous studies [2] was employed for the traveling tests using a rigid wheel. The apparatus consisted of a soil bin, a rigid wheel mounted on a carriage, and the driving units and measurement systems as shown in Fig. 1. The apparatus was capable of developing a uniform soil condition in the soil bin and making precise measurements during the experiment. The soil bin was 414 mm wide, 1888 mm in length, and 617 mm deep. ‘‘Toyoura standard sand’’ with air-dried condition was used for the soil bin test. The soil bin was filled up by spraying from a hopper. Thickness of soil layer was set 600 mm and initial void ratio of soil layer was regulated about 0.61 in average. The test wheel was 300 mm diameter, 410 mm in width, and coated with a 5 mm thickness of rubber. Its weight was 217 N and its peripheral speed was 0.0833 mm/s. The forward velocity was, therefore, quite low. The soil bin and wheel had nearly the same width, so the plane strain condition was met. Movement of soil particles was limited to two-dimensions, vertical and longitudinal. The measured variables were the rotational angle of the wheel, travel distance, sinkage, dynamic load, torque on the wheel shaft, soil reaction and its application line, normal and tangential stresses applied to the contact surface. The data were converted to digital data and stored on a hard disk. During a test, the wheel was slowly lowered down to the soil surface, initially without any rotation. Rotation of wheel was then started and forward travel began. The former stage was called the initial sinkage stage and the latter, the traveling stage.
Fig. 1. System used for measurement of soil displacement.
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Fig. 2. Side view showing initial configuration of markers.
2.2. Analysis of soil deformation Soil displacement was measured using many 5 mm diameter markers placed on the inner plane of the soil bins sidewall. The markers were, therefore, set between the soil and the sidewall. The markers moved with the soil particles, so the displacement of a marker was considered to coincide with that of the soil. Configurations of markers were recorded by taking a series of photographs using a high-resolution camera, with one photograph taken for each one degree of wheel rotational angle. The coordinates of markers were then calculated using the image processing capability of the on-line measurement system [3]. The displacement increment of a marker was equivalent to the difference of the coordinates in two sequential photographs. The initial depth of the markers in the top row was 10 mm. The initial vertical distance between rows and the initial horizontal distance between columns of the configuration was 10 mm. Fig. 2 shows a sample of the initial positions of the markers and the principle of the soil deformation analysis. Strain increments in the soil were calculated using a displacement function and the finite element mesh component for strain analysis consisted of three markers.
3. Results and discussion 3.1. Characteristics of soil deformation 3.1.1. Locus of soil particles Displacement loci of soil particles at 3.9% and 21.9% slip are shown in Fig. 3. The loci of five markers that had an initial depth of 10 mm are shown in Fig. 3. Initial positions of these loci were set at the same point on the graph. Also, the markers were chosen to avoid the influence of the initial sinkage stage and to allow all movement following the wheel pass to be completed. The numbers in the legend
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291
Fig. 3. Displacement loci of soil particles that had initial depths of 10 mm.
of Fig. 3 designate the marker numbers. When a soil particle was forward of the wheel, it moved forward and upward as the wheel approached. The particle then turned to move forward and downward. The direction of movement then changed to rearward and downward, and the particle reached its lowest position just as the wheel was directly above the marker. After the wheel passed, the marker moved rearward and upward, and its motion finally stopped. This tendency was seen at high slips, and negative values too. Therefore, at any value of slip, the path of a soil particle as the wheel traveled was a spiral. Although shapes of soil particle loci presented in some previous studies [4,5] are similar to those in Fig. 3, those reported in
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the previous studies were qualitative, rough analyses. Quantitative measurements of soil particles were conducted in this study. At 3.9% slip, the loci of five markers agreed well with one another. At 21.9% slip, however, the loci had different shapes, especially at their final positions, and these differences were about 1 mm. Moreover, the rearward displacement at 21.9% slip was about 2 mm greater than at 3.9% slip. 3.1.2. Definition of displacement increment The phenomenon that path of a soil particle is a spiral has been confirmed by Wong [3] and Yong [4], so it may be regarded as a general phenomenon. In this study, displacement increments were calculated for each one degree of wheel rotation angle. Characteristics of displacement were then analyzed with respect to the relative horizontal distance between the target and the vertical centerline of the wheel. Definitions of relative horizontal distance, displacement increment, and its horizontal and vertical component are shown in Fig. 4. Dux and Duy are horizontal and vertical components of the displacement increment, respectively. The variables D, ux, and Dux are positive when the vectors coincide with the travel direction of the wheel. The variables uy, and Duy are negative when their vectors point downward. Therefore, when Dux > 0 and Duy < 0, then Du is in the forward and downward direction.
Fig. 4. Geometries between wheel position and displacement increments of the soil.
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3.1.3. Characteristics of the displacement increment of soil particles having initial depths of 10 mm Fig. 5 shows the terms |Du|, Dux and Duy with respect to the relative horizontal distance, D, at 3.9% slip. The solid lines in Fig. 5 show the arithmetic average of the values of five markers. As D decreased from D = 70, the magnitude of the displacement increment, |Du|, increased rapidly and reached its peak nearly directly beneath the wheel axle, i.e., D = 0. As D decreased further, |Du| decreased rapidly until D reached approximately 30. The peak |Du| occurred slightly forward of the wheel axle. The shape of the distribution curve is similar to that of a Gaussian distribution. The distribution domain of |Du| coincided with the range that the soil particle displaced with respect to D. The magnitude |Du| dose not provide information about the direction of the displacement increment, Du, so the distributions of the components Dux, and Duy are discussed here. As D decreased from D = 70, the horizontal displacement Dux increased rapidly until it reached its positive peak, and then decreased until it reached its negative peak at about D = 0. The value of D at Dux = 0 was D = 20 and was approximately midway between the values of D at the two peaks. As D decreased from where Dux was at
Directly beneath axle
0.2 | Δ u | (mm)
Averaged value Direction of travel
0.1
0.0
Δ u x (mm)
0.1
pf
0.0 xi ( Δ ux=0 )
–0.1
pb
–0.2
Δ u y (mm)
0.1
pu1
pu2
0.0 –0.1
xr1 (Δ uy=0)
–0.2 200
100
pd
0
xr2 (Δ uy=0)
–100
–200
Relative horizontal distance D (mm) Fig. 5. Distributions of |Du|, Dux and Duy with the relative horizontal distance for soil particles having initial depths of 10 mm.
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its negative peak, Dux increased and approached zero. Thus, the appearance of both a positive and a negative peak was a distinctive feature of this distribution curve for Dux. This indicates there were two domains; the soil particle was pushed forward when it was forward of the wheel and was kicked rearward when it was to the rear of the wheel. The negative peak of Dux was slightly forward of where D = 0. As D decreased, the vertical displacement increment, Duy, increased at first and a minor positive peak occurred at about D = 50. The Duy function then decreased until its negative peak occurred at about D = 20, and Duy then increased until its major peak occurred at D = 10. The Duy function then decreased. The positions, where Duy = 0 occurred at D = 0 and D = 40. Distinctive features of the Duy distribution curve were the occurrence of two positive peaks and one negative peak. 3.2. Modeling of the displacement increments 3.2.1. Basic function The phenomena that a soil particle drew a spiral and the typical shapes of distribution curves with respect to the relative horizontal distance were regarded as general characteristics of soil deformation under the traveling wheel. Thus, the modeling of the shapes of such distributions enabled us to evaluate the strain increment and strain at an arbitrary point in the soil. In addition, stress analysis would be capable of substituting the calculated strain increments into an elasto–plastic constitutive equation for the soil. Results of finite element analyses would then be verified, or the soil displacement would be substituted directly into the finite element method for the prediction of drawbar pull, torque and other terms relating the traveling performances. The shapes of distributions of |Du|, Dux and Duy had distinctive peaks with respect to the relative horizontal distance as mentioned above. Therefore, models were examined using the following Gaussian function as a basic function in order to express such distinctive shapes. ( ) 2 ðx xc Þ y ¼ Am exp ; ð1Þ 2w2 where Am was a peak value, xc was a peak position, x is a target position and w is a width of half value. A similar shape of Dux with positive and negative peaks could be derived by differentiating the equation with respect x as follows: ( ) 2 A ðx x Þ m c y 0 ¼ 2 ðx xc Þ exp . ð2Þ w 2w2 It was also obtained by multiplying the original Gaussian function and a linear function. Kawasaki [5] applied the function Eq. (2) to express a distribution of shear stress in the soil when a wheel traveled on the soil. Furthermore, the shape of the vertical displacement increment with two positive peaks and one negative peak was examined by the secondary differentiation of the Gaussian function in Eq. (1). The function in Eq. (2) was differentiated again with respect to x, and the following function was obtained:
K. Fukami et al. / Journal of Terramechanics 43 (2006) 287–301
( ) o Am n ðx xc Þ2 2 2 y ¼ 4 ðx xc Þ w exp . w 2w2 00
295
ð3Þ
Fig. 6 shows the shapes of these functions. The first derivative as a horizontal component drew a symmetry shape to the intersection, and the secondary derivative as a vertical one drew a symmetry shape to the principal line. However, measured curves
y Direction of travel w
xc
Am
Am / e
0.5
x - y’ w
xc
–Am /( w· e0.5 )
x Am /(w · e0.5 )
xc + w 3
y’’
xc – w 3
2 Am /(w 2 · e1.5. )
x –Am / w 2
w
xc
Fig. 6. Shapes of the Gaussian function and its derivatives.
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drew distorted shapes that were different from these functions as shown in Fig. 6. Therefore, the shapes should be adjusted to the measured shapes while maintaining the fundamental features of the curves. 3.2.2. Modeling of each displacement increment Modeling for the displacement increments Dux and Duy was carried out based on the Gaussian function. The function in Eq. (2) relating to Dux was obtained by multiplying the original Gaussian function and a linear function. The peak position of the former coincided to the intersection of the latter so that the shape of the function was symmetric about the y-axis. The measured curve was adjusted by parallel translation of the linear function. Fig. 7 showed the modified shape following the parallel translation. The function describing Dux was assumed by the modification as follows: ( ) 2 ðx xc Þ DU x ¼ Amx ðx xi Þ exp ; ð4Þ 2w2 where DUx = Dux/(Vo Æ Dt), x = D/R. (x ⎯ x p )2 ⎯ q ( x ⎯ xc )2 ⎯ q
x ⎯ xi
Δ Ux
x ⎯ xc w
w
xc
xi
Δ Uy
xc
A my
A mx
x
x xp
Δ Uy
Δ Ux ( pf x , pf y )
xi xc
( pu 1x, pu 1y )
2 q xp xc
x
x
( pbx , pby )
( pd x , pd y ) Fig. 7. Shapes of the model functions.
( pu2x , pu 2 y )
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Both the Dux and Duy components were divided by the circumference displacement increment VoDt for describing the dimensionless quantities. Further, the relative horizontal displacement D was converted to a non-dimensional quantity x using the radius R of the wheel. Vo is a circumference speed, xi is a coordinate of x as Dux= 0. xc and w were a peak position and a width of half value, respectively. The function in Eq. (3) was obtained by multiplying the original Gaussian function and a quadratic function. Both principal axes were coincident in this case, so the shape of the distribution curve was symmetric. The asymmetrical shape in the measured values could be expressed by the translation of the principal axis of the quadratic function. In other words, a quadratic function with a peak (xp,q) was multiplied by the original Gaussian function as follows.
Fig. 8. Fitting results of DUx, DUy and |DU| for soil particles having initial depths of 10 mm.
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DU y ¼ Amy
) ðx xc Þ2 ðx xp Þ q exp ; 2w2 n
2
o
(
ð5Þ
where Duy = Duy/(VoDt), xp and q were a peak position and value of the quadratic function. 3.2.3. Fitting to models Fig. 8 shows examples of application of Eqs. (4)–(6) to the measured values. A non-linear least square method was employed to the fitting based on the Gauss– Newton method. The values of parameters obtained by curve fitting and the degree of fitness, i.e., the coefficients of determination are shown in Table 1. The measured values of horizontal and vertical displacement increments were obtained by averaging the values of five markers, respectively. The coefficients of determination of DUx and DUy were greater than 0.96 and 0.93, respectively, for any value of slip. Therefore, the distributions of DUx and DUy were approximated with sufficient accuracy using the proposed models. The dimensionless magnitudes of the displacement increment were calculated by the following equation: jDU j ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DU 2x þ DU 2y .
ð6Þ
Fig. 9 shows the displacement loci predicted by the proposed models. These curves were obtained by the integration of values of Dux and Duy. The predicted curves agreed well with the measured curves. The displacement increment in the initial stage was calculated to be rather small, so this caused a slight difference to occur in the final stage for 3.9% slip. However, the fundamental tendency to draw the spiral with the travel of
Table 1 Parameter values for Gaussian functions and their derivatives, and coefficients of determination Parameter
Slip (%) 3.9
21.9
Amx Amy xc w xi xp q
0.623 2.944 0.113 0.104 0.150 0.143 0.014
0.647 1.814 0.136 0.126 0.215 0.213 0.023
Coefficient of determination R2(DUx) R2(DUy)
0.98 0.93
0.96 0.97
K. Fukami et al. / Journal of Terramechanics 43 (2006) 287–301
Σ Δ u x , Σ Δ u y (mm)
1.0
Direction of travel
3.9% 21.9%
299
Σ Δ ux
0.0
–1.0 Σ Δ u y –2.0
3.9% 21.9%
Measured value Predicted curve
–3.0 100
50
0
–50
–100
Relative horizontal distance D (mm) 0.5 21.9%
Vertical displacement (mm)
Slippage 3.9% Slip3.9%
0.0 –0.5 –1.0 –1.5 1.0
Measured value Predicted curve
0.0
–1.0
–2.0
–3.0
Horizontal displacement (mm) Fig. 9. Predicted loci for soil particles having initial depths of 10 mm.
wheel was expressed well. Thus, it is suggested that the proposed models for displacement increments will fit well to the shapes of distributions and the loci of displacement.
4. Extended models including slip The unknown coefficients in the models were approximated by the following equation in order to describe the displacement increment at arbitrary values of slip. Fig. 10 shows the predicted results. A ¼ A1 s2 þ A2 s þ C;
ð7Þ
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3.00
0.30
Amy 2.00
0.20
Amx
1.00
0.10
0.00 0.15
0.00 0.30
0.10
0.20
0.05
xi
0.10
xc
0.00 0.15
0.00 0.03
0.10
0.02
0.05
xp
0.01
q
w 0.00
0.00
0
5
10
15
20
25
0
Slip (%)
5
10 15 Slip (%)
20
25
Fig. 10. Relationships between unknown coefficients and slip at 10 mm depth.
Vertical displacement (mm)
0.5
10%
Direction of travel 4%
0.0
20%
–0.5 –1.00 1.0
0.5
0.0 –0.5 –1.00 –1.5 Horizontal displacement (mm)
–2.0
Fig. 11. Results of prediction at 10 mm depth for slip values from 4% to 20%.
–2. 5
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where A was an arbitrary unknown coefficient, A1, A2 and C were the constants in the quadratic function. Fig. 11 shows the predicted displacement loci with a slip range of 4–20% obtained by substituting A from Eq. (7) for Amx, xi, xc, and w in Eq. (4) and Amy, xp, and q in Eq. (5). The tendency that the final position of displacement locus shifted rearward as slip increased was expressed well as shown in the Figure. Thus, predictions of the displacement increments in the various conditions become possible by making clear the relations between the unknown coefficients and parameters in the models.
5. Conclusions Soil displacement increments under a rigid wheel were measured experimentally and the characteristics were summarized. The displacement locus of a soil particle was a spiral for various conditions including wheel slip values from 4% to 20%. The displacement increment was resolved into its horizontal and vertical components for detailed analysis. The distribution curves of these revealed distinctive features; the former and the latter had two peaks and three peaks, respectively. The distribution curves were modeled using a Gaussian function and its derivatives. The predicted results using the models agreed well with the measured values. The models were extended taking account of arbitrary wheel slip. Analyses of strain distribution and stress distribution should be conducted in order to carry out more detailed investigations of the traveling performances.
References [1] Nohse Y, Hashiguchi K, Ueno M, Shikanai T, Izumi H, Koyama F. A measurement of basic mechanical quantities of off-the-road traveling performance. J Terramechanics 1991;28(4):358–70. [2] Hashiguchi K, Okayasu T, Ueno M, Shikanai T. Image processing on-line measurement for soil displacement. J JSAM 1998;60(6):11–8. [3] Wong JY. Behavior of soil beneath rigid wheels. J Agr Eng Res 1967;12(4):257–69. [4] Yong RN, Fattah EA. Prediction of wheel–soil interaction and performance using the finite element method. J Terramechanics 1976;13(4):227–40. [5] Kawasaki H, Hirakawa D, Tatsuoka F. Experimental study on stress distribution in sand subject to roller compaction. In: The 37th Japan National Conference on Geotechnical Engineering; 2000. p. 1623–24.