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Numerical simulation of a 4WD–4WS tractor turning in a rice field
Journal of Terramechanics 36 (1999) 91±115
Numerical simulation of a 4WD±4WS tractor turning in a rice ®eld H. Itoh a,*, A. Oida b, M. Yamazaki b a
Faculty of Biology Oriented Science and Technology, Kinki University, Naga-gun Wakayama 649-64, Japan b Faculty of Agriculture, Kyoto University, Sakyo-ku, Kyoto 606-01, Japan Received 5 August 1997; accepted 18 November 1998
Abstract For the steady-state circular turning of a 4WD±4WS (4 wheel driven±4 wheel steered) tractor in a rice ®eld, a numerical simulation was achieved. Equations of motion of this tractor were developed in a vehicle ®xed x±y coordinate system. By comparing the calculated and measured results of acting forces on the tractor tires, this simulation was evaluated. Then, the characteristic parameters of the turning vehicle, which are the side slip angle and the yaw angular velocity of the vehicle center of gravity, were simulated in several combinations of the steering wheel angle and the forward speed. Also the same simulation applied to a 4WD±2WS tractor which had the same body as the 4WD±4WS tractor. The simulated results showed a clear dierence of turnability between 4WS and 2WS. # 1999 ISTVS. All rights reserved.
Nomenclature b c F H i I
Width of a tire Cohesion of soil Tractive force Thrust or longitudinal force of a tire Slip of a tire Yaw moment of inertia
* Corresponding author. Tel.: +81-736-3888; fax: +81-736-77-4754; e-mail: [email protected] 0022-4898/98/$20.00 # 1999 ISTVS. All rights reserved. PII: S002 2-4898(98)0003 7-8
92
j K m r R RAD S V Z
H. Itoh et al. / Journal of Terramechanics 36 (1999) 91±115
Soil shear deformation Soil shear deformation modulus Mass of a vehicle Radius of a tire Rolling resistance of a tire Turning radius of the tractor center of gravity Lateral force on a tire Forward speed of the center of gravity of the tractor Sinkage of rut
Greek alphabet !
Side slip angle of the center of gravity of the tractor Angle Radial or normal stress Shear stress Internal friction angle of soil Yaw angular velocity of the center of gravity of the tractor
Subscripts j1 j2 jl jt Ss xc yc 1 2
1
2 0 1 2 !t
Tangential soil shear deformation at the front region at a negative slip Tangential soil shear deformation at the rear region at a negative slip Lateral soil shear deformation Tangential soil shear deformation at a positive slip Lateral shearing force of tire Longditudinal coordinate of the turning center Lateral coordinate of the turning center Steer angle of the left rear tire Steer angle of the right rear tire Steer angle of the left front tire Steer angle of the right front tire Angle of the transaction Entry angle Exit angle Angular velocity of a tire
1. Introduction This research focused on the turning motion of a farm tractor in a rice ®eld. There are many corners and obstacles on roads during movement from the farmer's house
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to the ®eld, so an operator of a farm tractor has to do various kinds of turning maneuver. Also, in o-road situations such as a rice ®eld, many turning maneuvers are required. In order to make the turning motions of the farm tractor eective and stable, it is necessary to analyze the fundamental kinematic dynamics of the farm tractor on-road and o-road and to investigate the interactions between the running devices of the tractor and the ®eld soil [1±3]. The kinematic dynamics on the even paved road has already been studied in automobile engineering. However, there was little research in relation to the similar analysis on o-road vehicles, especially on deformable soft soil [4±11]. The interactions between the running device and the soil have been investigated in terramechanic ®elds. As one investigation method, the stress distribution curves beneath a pneumatic tire or a rigid wheel have been analyzed [12±19]. The objective of this research is to formulate the turning motion of a four-wheel drive and four-wheel steering (4WD±4WS) agricultural tractor in a steady-state circular turn in rice ®elds. A 4WS tractor came onto the Japanese market. It is a wellknown fact that the four-wheel steering system provides a higher turnability on a paved road for a vehicle than the conventional two-wheel steering system. It is expected that the 4WD±4WS tractor will also show a high performance in a rice ®eld. That is the reason why the 4WS tractor was selected in this study. Fundamental turning characteristics are to be investigated during the analysis of the steady-state turn. By combining the previous analysis on the turning behavior of the 4WD±4WS tractor on a paved road [9,20] with the stress distribution analysis for tires or wheels, a mathematical model of the turning behavior of the tractor in a rice ®eld was conducted. Some main speci®cations of the tractor, some soil parameters, the steering wheel angle and the forward speed were input to the model. Then, the side slip angle and the yaw angular velocity of the tractor center of gravity were output after 200 iteration of solving program for dierential equations. At the ®nal stage of iteration, values of acting forces on the tires were also output. These simulated results were compared with the measured results [21]. This comparison showed some errors, but in general this model could express the actual turning behavior of the tractor. After evaluating this model, eects of the input parameters on the output parameters were simulated and analyzed. 2. Mathematical models The motion of the tractor was analyzed as two dimensional and a vehicle ®xed x±y coordinate system was used. The origin of the coordinate axes was settled at the center of gravity of the tested tractor. Three equations of motion were developed from a dynamic equilibrium analysis of the forces in the x and y directions and the momentum about the center of gravity. The tractive force and lateral force were considered as the input forces on tractor tires as shown in Fig. 1. To develop the equations of motion, the following assumptions applied.
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1. The tractive force acts in the same direction as the wheel plane. 2. The lateral force acts at right angle to the wheel plane and is de®ned as positive when the force acts from the right side to the wheel plane. 3. The side slip angle of a tire is de®ned as positive when the moving direction of the tire deviates leftward from the wheel plane, and the side slip angle of the center of gravity is de®ned as positive when the moving direction of the center of gravity deviates leftward from the longitudinal center line (the x axis). 4. The yaw angular velocity of the center of gravity in the counterclockwise direction is de®ned as positive.
Fig. 1. Tractive forces, lateral forces and side slip angles of the tires of a 4WD±4WS tractor in a left turn in the x±y plane. (In the above ®gure, F and S indicate the tractive force and lateral force, respectively. Subscripts ``1'' and ``2'' express left and right. ``f'' and ``r'' denote front and rear.)
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5. The rolling motion due to the sinking of tires in the soft ground can be neglected. 6. The eect of wind can be neglected. 7. The steer angles of the rear tires of the 4WS tractor conform to the Ackermann steering geometry. The accelerations of the center of gravity in the x and y directions are same as those used in the previous analysis [9]. Then, the equations of motion are given by Eqs. (1±3): m V_ cos ÿ V _ ! sin Ff1 cos 1 Sf1 sin 1 Ff 2 cos 2 Sf 2 sin 2 Fr1 cos 1 Sr1 sin 1
1
Fr2 cos 2 Sr2 sin 2 m V_ sin V _ ! cos ÿFf1 sin 1 Sf1 cos 1 ÿ Ff 2 sin 2 Sf 2 cos 2 ÿ Fr1 sin 1 Sr1 cos 1
2
ÿ Fr2 sin 2 Sr2 cos 2 I!_ ÿdfl Ff1 cos 1 ÿ dfl Sf1 sin 1 ÿ lfl Ff1 sin 1 lfl Sf1 cos 1 dfr Ff 2 cos 2 dfr Sf 2 sin 2 ÿ lfr Ff 2 sin 2 lfr Sf 2 cos 2 ÿ drl Fr1 cos 1 ÿ dr1 Sr1 sin 1
3
lrl Fr1 sin 1 ÿ lrl Sr1 cos 1 drr Fr2 cos 2 drr Sr2 sin 2 lrr Fr2 sin 2 ÿ lrr Sr2 cos 2 _ _ and !_ are dierential coecients of V, and !. where, V, The tractive force is the sum of the thrust H and rolling resistance R of the tire. The lateral force is the lateral shearing force of soil. These forces are shown in Fig. 2. Based on the results of the study on stress distribution under a tire or a rigid wheel, it is necessary to make the assumption that the pressure distribution is not uniform over the soil±tire contact area especially in the case of a soft, deformable ground. The stress distribution should be taken into account to predict the forces acting on vehicle tires. In this research, the model originated from Wong [18,19] was basically adopted.
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Fig. 2. Forces acting on a tire (H, R, Ss , Vt , Vl and t are thrust, rolling resistance, lateral shear force, forward velocity, lateral moving velocity and side slip angle of tire, respectively.)
By integrating the stress distribution curves shown in Fig. 3, the normal load, the thrust, the rolling resistance and the lateral shear force were obtained. The angle is zero at bottom dead center (B. D. C.) and is positive from B. D. C. to the entry angle 1 . The angle from B. D. C. to the exit angle 2 is negative. The following assumptions were adopted to calculate the forces acting on a tire. (a) (b) (c) (d)
The The The The
contact patch of a tire is rectangle. tire is rigid. stress distribution is uniform over the width of the tire. surcharge pressure can be neglected.
2.1. The equations of stress distributions The shear stress is assumed to be represented by Bekker's equation as shown in Eq. (4) [22]: j 4 c tan 1 ÿ exp ÿ K
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Fig. 3. Stress distribution curves under a rigid wheel. (In the ®gure, r, Z, 0 , 1 and 2 are radius of tire, sinkage, angular position of transition point, entry angle and exit angle, respectively.)
In this case the shear stress is determined by the internal friction angle , cohesion of soil c, soil deformation modulus K, and the radial stress . The soil parameters can be obtained by measurements [21]. In order to get the shear stress, the radial stress distribution curve has to be modeled ®rst. Reviewing the measured results of the radial stress distribution reported by Onafeko [16] and Oida [15], it was assumed that the radial stress distribution curve under a wheel can be represented by a parabola. Furthermore, it was assumed that the rut recovery, which is represented by a ratio, 1 ÿ 2 =21 , is to be 0.7, though the value increases with the slip of a tire [15]. The radial stress distribution curve is given by Eq. (5): a ÿ 1 ÿ 2
5
where, ``a'' in Eq. (5) is constant. Then, if the tangential and the lateral shear deformations of soil are calculated, equations of the tangential and the lateral stress distribution curves will be obtained. 2.2. Tangential shear deformation of soil at a positive slip The tangential shear deformation can be calculated by integrating the slip velocity Vj of a point on the wheel rim. The slip velocity is the dierence between the
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circumferential speed and the tangential component of the longitudinal forward velocity at the point on the rim. Wong conducted the tangential shear deformation jt at a positive slip i as Eq. (6) [18]. t 6 jt Vjdt r 1 ÿ ÿ 1 ÿ i sin 1 ÿ sin 0
2.3. Tangential shear deformation of soil at a negative slip The measured results of the stress distributions obtained by the above three researchers indicated that there was a negative tangential stress at the rear part of the soil±wheel contact region at the slightly positive or the negative slip. Wong analyzed the tangential stress distribution of a towed rigid wheel on sand. He showed the transition point where the direction of the tangential stress reversed. Uelmann [17] also indicated the negative tangential stress on a pure cohesive soil and it was described in his paper that there was upward soil ¯ow at the front part of the soil±wheel contact region for a towed wheel. Though the rice ®eld used for the test was the silty clay loam, Wong's model was adopted. The soil±tire contact area is to be divided into two regions at the transition point which is represented by the angle, 0 . The front region covering from 1 to 0 gives the positive tangential stress and in the rear region covering from 0 to 2 the tangential stress becomes negative. The soil shear deformation was calculated for each region. Wong suggested that at the transition point the direction of the absolute velocity of the rim coincides with one of the slip lines on the boundary of the soil mass. Furthermore, it is well known that the soil failure occurs at angle of (45 ÿ =2) from the principal stress plane. Then the angle of the absolute velocity of the rim was assumed to be equal to one of (45 ÿ =2) at the transition point. Then the 0 was obtained by Eq. (7): 1 cos 0 ÿ 1ÿi tan 45 ÿ sin 0 2 The tangential shear deformation j1 at the front region derived from Eq. (8). 1 1 r!t f1 1 ÿ i K ÿ cos d j1 !t r1 ÿ 1 1 ÿ iK ÿ 1 ÿ i sin 1 ÿ sin The constant K was expressed as follows: 1 1 ÿ i sin 1 ÿ sin 0 ÿ1 K 1ÿi 1 ÿ 0
7
8
9
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According to the measurement of the soil ¯ow by Wong, the tangential soil deformation j2 at the rear region became the same form as Eq. (6). t 10 j2 Vj dt r 0 ÿ ÿ 1 ÿ i sin 0 ÿ sin 0
2.4. Lateral shear deformation of soil The lateral shear deformation arises when the tire has the slip to lateral direction. The lateral shear deformation can be derived from the integral of the lateral moving velocity of a tire Vl as shown in Fig. 2. The lateral moving velocity is represented by Eq. (11): Vl Vf sin t
11
The lateral shear deformation of soil jl is given by the integral of the lateral moving velocity as shown in Eq. (12). t Vt sin t Vt cos t 12 1 ÿ ; !t jl Vl dt !t r 1 ÿ i 0 where !t is the angular velocity of a tire. 2.5. Equations of the distribution curves of tangential and lateral stresses Substituting the radial stress and the shear deformation of soil into Eq. (4), the equations of the distributions of tangential and lateral stresses are obtained. For the tangential stress distribution curve, three equations are represented as follows: jt p c tan 1 ÿ exp ÿ K j1 13 1 c tan 1 ÿ exp ÿ K jj2 j jj2 j j2 < 0 2 c tan 1 ÿ exp ÿ K j2 where p is the tangential stress at a positive slip and 1 and 2 are the tangential stresses at the front and rear regions at a negative slip, respectively. The lateral stress is denoted as negative when the side slip angle is positive: jjl j ÿjjl j 14 l c tan 1 ÿ exp ÿ K jl where l is the lateral stress.
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2.6. Forces acting on tractor tires The normal load on a tire can be calculated by integrating the vertical components of the radial and the tangential stresses over the soil±tire entire contact area. There are two equations to give the normal load. One is for a positive slip and another is for a negative slip, as shown in Eqs. (15) and (16). In the case of a positive slip: Wt rb
1 2
cos d
1 2
p sin d
15
where Wt is the normal load on a tire and b is the width of the tire. In the case of a negative slip: Wt rb
1 2
cos d
1 0
1 sin d
0 2
2 sin d
16
On the other hand, considering the momentum equilibrium about the tractor center of gravity, the normal load on each tire can be calculated another way as shown in a previous report [9]. This normal load should be equal to the right side of Eqs. (15) or (16). This relationship identi®ed the constant ``a'' in Eq. (5), if the calculated slip was substituted into the above equations. The successive iteration was achieved to calculate the constant ``a'' in terms of each tire. The thrust of a tire H is given by integration of the horizontal components of the radial and the tangential stresses over the soil±tire contact area. There are also two equations in relation to the slip of the tire. In the case of a positive slip: H rb
1 2
p cos d ÿ
0 2
sin d
17
In the case of a negative slip: H rb
1 0
1 cos d ÿ
0 2
sin d
18
The rolling resistance on a tire is calculated by integrating the backward horizontal components of the radial and tangential stresses over the soil±tire contact area. The rolling resistance is denoted as negative and is given by Eq. (19) or Eq. (20) at a positive or a negative slip, respectively. In the case of a positive slip: 1 sin d < 0 19 R ÿrb 0
where, R is the rolling resistance of the tire.
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In the case of a negative slip: R ÿrb
1 0
sin d ÿ
0 2
2 cos d < 0
20
The lateral shear force on a tire is calculated by integrating of the lateral stress over the entire contact area as shown in Eq. (21): Ss br
1 2
l d
21
2.7. Sinkage The entry angle 1 and the exit angle 2 are determined by the sinkage of a tire because the tire was assumed not to de¯ect and to be rigid. There was no measured data in terms of the relation between the sinkage and the slip of the tire of the tested tractor. Therefore, the static sinkage of the tire was predicted by using the pressure±sinkage relationship. The pressure±sinkage relationship, ``p f Z'' was examined in the rice ®eld and was approximated to 6th power polynomial Eq. (21). On the other hand, the relationship between the contact pressure p and the sinkage of the tire Z can be derived geometrically from Fig. 3 and the relation is expressed by Eq. (22): Wt 1 ÿ 2 ÿ ; aw 22 p 21 2aw br cosÿ1 rÿZ r where, aw is a ratio of actual contact length to static length with the same sinkage and Wt is the normal load on a tire. The contact pressures which are calculated by the two equations should be same. Therefore, Z should satisfy the next equation: f Z
Wt ÿ 2aw br cosÿ1 rÿZ r
23
2.8. Slip If the slip of the tire is calculated, the normal load, lateral shear force, thrust and the rolling resistance of the tire can be obtained. The thrusts of the left and right tires are equal by the action of the dierential gears which are furnished to the front and rear axles. So, it was assumed that the slip of the left tire gives a thrust which is equal to that of the right tire. As is well known, the slip is derived from the longitudinal forward velocity (Fig. 2) and the circumferential speed of a tire. The longitudinal forward velocity of each tire
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is calculated by the side slip angle, yaw angular velocity and the forward speed of the center of gravity of the tested tractor as follows: q Vffl V sin lfl !2 V cos ÿ dfl !2 cos f1 q Vffr V sin lfr !2 V cos dfr !2 cos f 2 24 q Vfrl V sin ÿ lrl !2 V cos ÿ drl !2 cos r1 q Vfrr V sin ÿ 1rr !2 V cos drr !2 cos r2 Vf indicates the longitudinal forward velocity of a tire and the subscripts ``¯'' and ``rr'' mean the ``front left'' and the ``rear right'', respectively. The angle indicates the side slip angle of the tire and its subscript is the same as that shown in Fig. 1. The circumferential speed is related to the reduction gear ratio and the dierential gear. If the forward speed, V in m/s is given, the number of revolutions per second of the engine and the front and rear axle shafts at a straight run are represented by Eq. (25): REVe
V ; REVf REVe RATf ; REVr REVe RATr RATr 2 rr 25
where, REVe , REVf and REVr are the number of revolutions per second of the engine and the front and rear axle shafts at a straight run. RATf and RATr are the reduction gear ratios of the front and rear drive shafts. rr is the radius of the rear tires. Furthermore the dierential gear con®nes the number of revolutions of the left and right tires as follows: 2REVf REVfl REVfr ; 2REVr REVrl REVrr
26
where, REVfl and REVfr are the number of the revolutions per second of the front left and right tires, respectively. REVrl and REVrr are the number of revolutions per second of the rear left and right tires, respectively. If the slip of a tire, for example, the slip of the front left tire, ifl is given, the circumferential speed can be calculated by the longitudinal forward velocity. Vcfl
Vffl 1 ÿ ifl
27
where Vcfl is the circumferential speed of the front left tire. Then the number of revolutions per second of the front right tire is derived from Eq. (28): REVfr 2REVf ÿ REVfl ; REVfl where, rf is the radius of the front tires.
Vcfl 2rf
28
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Then the circumferential speed of the front right tire, Vcfr is represented by Eq. (29). Vcfr 2rf REVfr
29
The slip of the front right tire, ifr can be calculated by using the Eqs. (29) and (24). ifr
Vcfr ÿ Vffr Vcfr
30
Substituting the slips of the front left and right tires into the Eqs. (17) or (18), the thrusts of the front left and right tires are calculated and they are compared each other. If the dierence between the two thrusts is smaller than a tolerance, the above-described slips are adopted, whereas if the dierence is greater than the allowed value, a new slip of the front left tire is attempted and so on. This process is also achieved for the rear left and right tires. 3. Solution of the motion equations 3.1. Initial values The equations of motion were solved by the Runge±Kutta±Gill method. The initial values of the side slip angle, yaw angular velocity and the forward speed of the center of gravity had to be given to the equations of motion at the ®rst stage. At the ®rst stage of the solution, it was assumed that the forward speed was zero and stood still. The step response was treated. In order to determine the initial values, moving direction of virtual wheel was considered. The virtual wheel was located on the center of the front and rear axles as shown in Fig. 4. The side slip angles of the front and rear virtual wheels should be zero as the initial condition. The moving directions of the front and rear virtual wheels, f , r were schematically shown in Fig. 4 and were expressed in Eq. (31).
f
V sin lf ! V sin ÿ lr ! ; r V cos V cos
31
The left sides of Eq. (31) could be assumed to be equal to the average angle of the left and right tires as shown in Eq. (32):
f
1 2 1 2 ; r 2 2
32
If f , r were given, the initial values of the side slip angle and the yaw angular velocity of the center of gravity, ini , !ini are represented by Eq. (33): lf tan r lr tan f V tan f cos ini ÿ sin ini 33 ; !ini ini tanÿ1 1f 1r lf
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Fig. 4. Moving directions of virtual wheels. ( f , r are the moving directions of front and rear virtual wheels.)
The initial value of the forward speed of the center of gravity is arbitrarily chosen. 3.2. Solution of the equations of motion Substituting the above equations [from Eqs. (4) to (33)] into Eqs. (1±3), Eq. (34) is obtained. 34 a11 V_ a12 _ b1 ; a21 V_ a22 _ b2 ; I!_ b3 Then the dierential coecients of the side slip angle, d =dt, yaw angular velocity, d!=dt and the forward speed, dV=dt of the center of gravity are represented by Eq. (35). d b1 a21 ÿ b2 a11 _ dV b1 ÿ a12 d! b3 ;V ; !_ _ I dt a12 a21 ÿ a11 a22 dt a11 dt
35
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The above equations were solved by the Runge±Kutta±Gill method and then the side slip angle, yaw angular velocity and the forward speed of the center of gravity were calculated. In the method the slip and the side slip angle of the tire were set constant during each step. The slip of the tire was derived from the related parameters which had been calculated before step 1 because the slip, sinkage and the forces acting on tires could not be calculated at the same time. As for the condition of the Runge±Kutta±Gill method, the interval between each step was set 0.01 s and 200 iteration was achieved by using a computer. 3.3. Eect of soil shear deformation modulus The soil shear deformation modulus was obtained from the measured results of the shear stress in relation to the soil deformation. The average normal pressure beneath a tire was estimated by the measured results of the normal load on a tire [21]. The estimated normal pressure was less than 39.2 kPa. So, the result of the shear stress-deformation curve at the normal pressure of 19.6 kPa was adopted to determine the soil deformation modulus. From the curve the soil deformation modulus was obtained as 0.3 cm. Substitution of the above deformation modulus into the equations of motion proved a failure in simulating the turning behavior of the tested tractor, because the deformation modulus was so small that the calculated tangential stress became so large at a small soil deformation [see Eq. (13)]. The eects of the deformation modulus on the calculated normal loads and thrusts of the right front and rear tires of the 4WS tractor in left turns were examined by computer simulation as shown in Fig. 5.
Fig. 5. Eect of soil shear deformation modulus on the normal loads and thrusts of the right front and rear tires. In the above notations, ``ÿ350/0.91/EXP'' means that experimental result at steering wheel angle of ÿ350 and the forward speed of 0.91 m/s and so on. ``SIM'' indicates the simulated result.
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The calculated normal loads on the tires were not in¯uenced by the shear deformation modulus and did not approximate the experimental results. On the other hand, there was a case that the calculated thrust at the soil deformation modulus of 3.5 cm approximated the measured one. Furthermore, the eects of the soil shear deformation modulus on other forces such as the lateral force and the rolling resistance were also investigated. The calculated rolling resistance closely resembled the measured one at the deformation modulus of 3.5 cm, but there were some cases that showed a large dierence between the simulated and the measured lateral forces at the shear deformation modulus. Although the soil shear deformation modulus of 3.5 cm was very much larger than the measured result, it seemed that this value was enough to oer adequate calculated results as a whole. Accordingly, the value, 3.5 cm was substituted as the soil shear deformation modulus into the equations of motion. 4. Evaluation of the simulation Using the equations of motion, the normal loads, lateral forces, rolling resistances, thrusts, slips, side slip angles of four driven tires, and the turning radius of the center of gravity were calculated. And then these simulated values in relation to right side tires and turning radius were compared with the measured results shown in a previous report [21]. A left steering wheel angle was de®ned as negative. Results corresponding to the negative steering wheel angle shown hereafter show data on the outside tires of a turn, whereas the data on the inside tires are shown in the positive steering wheel angle. As for the forward speed, third, ®fth and seventh gear shown in result graphs correspond to 0.41 m/s, 0.91 m/s and 1.92 m/s, respectively. 4.1. Tire normal load Fig. 6 shows the measured and the simulated normal loads on the right front and rear tires. The simulated normal load on the outside front tire (for negative steering wheel angle) agreed with that of the measured result. 4.2. Rolling resistance Fig. 7 shows the simulated and the measured rolling resistances of the right front and rear tires. The computed rolling resistance of the outside front tire agreed with the measured one. 4.3. Lateral force and side slip angle Fig. 8 shows the simulated and the measured lateral forces and side slip angles of the tires. Excepting some points, dierence between the simulated and measured lateral forces on the front tires was not large.
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Fig. 6. The measured and the simulated tire loads (``3/F/SIM'' means ``the simulation results of the front right tire in 3rd gear'' and so on. ``EXP'' indicates '' the experimental result'').
Fig. 7. The measured and the simulated rolling resistances of the tires.
Fig. 8. The measured and the simulated results of lateral forces and side slip angles of the tires.
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4.4. Thrust and slip Fig. 9 shows the simulated and the measured values of thrusts and slip of the right front and rear tires. Most of the calculated thrust of the outside rear tire neared the measured one. 4.5. Turning radius of center of gravity of the tractor The turning radius was derived from Fig. 4 and Eqs. (36) and (37).
f
V sin 1f ! V sin ÿ lr ! ; r V cos V cos
RAD
q x2c y2c ; xc ÿyc tan f lf ; yc
36
lf lr tan f ÿ tan r
37
Fig. 10 shows the simulated and the measured turning radii of the center of gravity in the left and right turns. In the case of 4WS, the dierence between the simulated and the measured turning radius was small in the right turn (for positive steering wheel angle). In the left turn it was smaller than that of measured results. In the case of 2WS the calculated turning radius was smaller than that of the measured result in the left and right turns.
Fig. 9. The measured and the simulated results of thrusts and slip of the tires.
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Fig. 10. Comparison of the measured with the simulated turning radii of the center of gravity.
4.6. Evaluations of the equations of motion and the simulation program The lateral shear force, rolling resistance and the thrust of the tire were the function of the deformation modulus [see Eqs. (13) and (14) and Eqs. (17±21)]. The slip was also aected by the deformation modulus because the slip was determined by the dierence between the calculated thrusts of the left and right tires. The small soil deformation modulus will give the large thrust at a small slip which gives a small soil deformation [Eqs. (6),(8) and (10)]. As shown in Fig. 5 the eect of the deformation modulus on the acting forces on tires was inspected and a comparison of the larger value with the measured one was used in this simulation. Though the 3.5 cm of the deformation modulus was estimated as the most suitable value for this simulation algorithm, some errors were observed in the comparison between the simulated and measured results. The sinkage of a tire increases with the slip [15]. This slip±sinkage relation was not taken into account in this calculation. If the slip±sinkage relationship was introduced to the calculation of the tire sinkage, the partition of the integral of each stress distribution would be modi®ed. Some other assumptions described in Section 2 also aected the calculation of the forces on the tires. Especially assumptions (b) and (c) seemed to introduce overestimated forces. From the above considerations, it was found that the large deformation modulus, which is far from the measured value, has to be substituted into the equations motion and the slip±sinkage relationship should be added to the calculation of the tire sinkage.
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5. Numerical simulation Using the equations of motion explained in the above section, the eects of the steering wheel angle, forward speed on the side slip angle and the yaw angular velocity of the center of gravity were simulated in relation to the 4WD±4WS and the 4WD±2WS tractor at steady-state circular turn. In addition to this simulation, the location of the center of gravity was also examined. The moving paths of the center of gravity and the end of PTO shaft were derived from the simulated results. 5.1. Eect of steering wheel angle During the turn, the steering wheel was ®xed to a constant angle. The angle was determined from ÿ50 to ÿ450 with an interval of 50 . The forward speeds of 1.0, 2.0 and 3.0 m/s were selected. As shown in Fig. 1, the side slip angle is de®ned as positive when the moving direction of the center of gravity deviates leftward from the longitudinal center line (the x axis). The yaw angular velocity in the counterclockwise direction is de®ned as positive. Fig. 11(a) shows the eect of the steering wheel angle on the side slip angle and the yaw angular velocity in the left turns. The side slip angle in 2WS increased with the increase of the absolute value of steering wheel angle. On the other hand the slope of the increase began to decrease at the steering wheel angle of about ÿ250 in 4WS and approached a certain value at the forward speed of 1.0 m/s. The side slip angle decreased under the ÿ250 of the steering wheel angle at the speeds of 2.0 and 3.0 m/s in 4WS. These results indicate that for the 4WS tractor, in the case of the steering wheel angle under ÿ200 , steering of the rear tires in the opposite direction to that of the front tires brings the decrease of the directional dierence between the longitudinal center line of the tractor and the tangent line of the turning circle. Furthermore the positive side slip angle of the center of gravity indicates that the moving direction of the center of gravity is the inside of the turn. Therefore it seems that the moving direction of the center of gravity changes to the outside of the turn with the increase of the absolute value of the steering wheel angle and the forward speed in 4WS. This tendency coincided with the result of the direction of resultant forces obtained by the experiment on the rice ®eld. The yaw angular velocity increased linearly with the absolute value of the steering wheel angle. The yaw angular velocity in 2WS was greater than that in 4WS at the steering wheel angle over ÿ150 and under the steering wheel angle this tendency reversed. These results also show the eect of the steering of the rear tires. The yaw angular velocity in 4WS decreased due to the steering of the rear tires in the same direction as the front tires and it increased by means of steering the rear tires in the opposite direction to that of the front tires. 5.2. Eect of forward speed The forward speed was set from 0.5 m/s to 3.0 m/s with an interval of 0.5 m/s. The steering wheel angles of ÿ150, ÿ250, ÿ350 and ÿ450 were selected. Fig. 11(b)
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shows the eect of the forward speed on the side slip angle and the yaw angular velocity in left turns. The side slip angle decreased with the increase of forward speed both in 4WS and 2WS. The direction of the longitudinal center line of the tractor approached the tangential direction of the turning circle as the forward speed increased. The side slip angle at steering wheel angle of ÿ150 in 4WS was larger than that in 2WS due to steering the rear tires in the same direction as that of the front tire. The yaw angular velocity increased linearly with the forward speed. This fact shows that the tested tractor has a neutral steer characteristic even on the rice ®eld
Fig. 11. Simulation results of the side slip angle and the yaw angular velocity of the center of gravity in left turns. (a) Eect of the steering wheel angle, (b) eect of the forward speed, (c) eect of the location of the center of gravity.
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below the forward speed of 3.0 m/s which is near the maximum forward speed of the tested tractor. Based on the simulated results shown in Fig. 11(b), it was predicted that the side slip angle in 4WS would become negative at forward speeds over 3.0 m/s and steering wheel angle of ÿ450 . Furthermore the side slip angle in 4WS was smaller than that in 2WS and was predicted to become negative at higher speeds than 3.0 m/s. Therefore the 4WS tractor is more likely to have oversteer characteristics than the 2WS tractor, though the results of the yaw angular velocity showed neutral steer characteristics. 5.3. Eect of location of the center of gravity The location of the center of gravity was de®ned by the ratio of the length from the front axle to the center of gravity to the wheelbase. So, if the center of gravity is located at the rear part of the tractor, the ratio approaches 1.0. The lateral transfer of the location was not considered. The forward speed was set to 1.5 m/s and the four levels of the steering wheel angle, ÿ150, ÿ250, ÿ350 and ÿ450 , were selected. Fig. 11(c) shows the eect of the location of the center of gravity on the side slip angle and the yaw angular velocity of the center of gravity in left turns. The side slip angle decreased linearly with the increase of the ratio. In the case of 4WS the side slip angle became negative at the steering wheel angles of ÿ350 and ÿ450 when the ratio was 0.7 and 0.8. This indicated that the moving direction of the center of gravity turns to the outside of the turn when the center of gravity is located at the rear part of the tractor. Furthermore the side slip angle increased with the steering wheel angle when the ratio was under 0.5 and this tendency reversed when the ratio was over 0.6. These results indicated that the oversteer characteristic became evident when the center of gravity was located at the rear part of the tractor and on the contrary the understeer characteristic was observed when the center of gravity located at the front part. The yaw angular velocity did not change with the increase of the ratio. 5.4. Moving paths of the center of gravity and the end of PTO shaft The locations of the center of gravity and the end of PTO shaft during the steadystate turns were calculated in the X±Y space ®xed coordinate. The locations can be calculated by the simulated side slip angle, yaw angular velocity and the forward speed of the center of gravity. The starting point of the center of gravity was the origin of the coordinate. And then the moving paths of the two points during a complete circular turn were simulated. Fig. 12(a) and (b) show the moving paths of the center of gravity and the end of PTO shaft at the steering wheel angles of ÿ100 or ÿ450 and the forward speed of 3.0 m/s. At the steering wheel angle of ÿ100 , the moving trajectory of the center of gravity in 4WS was located outside of that in 2WS. On the other hand, at the steering
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Fig. 12. Simulated moving paths at steering wheel angles of ÿ100 and ÿ450 . (a) Paths of the center of gravity, (b) Paths of the PTO shaft.
wheel angle of ÿ450 the trajectory of the center of gravity in 4WS was located in the inside of that in 2WS, because the rear tires were steered in the same direction as the front tires over ÿ200 of the steering wheel angle. On the other hand, the rear tires turn in the opposite direction to the front tires when the steering wheel angle is under ÿ200 . As for the path of the end of PTO shaft, the same tendency was observed, but at the steering wheel angle of ÿ450 the end of the PTO shaft passed the negative domain of the Y axis. These results show the eect of the steer angle of the rear tires of the 4WS tractor on turning behavior. This tendency was also seen in the computer simulation on the paved road. Through this simulation, it was predicted that more eective lane change at the steering wheel angle over ÿ200 and a sharper turn at the large steering wheel angle under ÿ200 could be supplied by the four wheel steering system, compared with the ordinary steering system.
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6. Conclusions The computer simulation model of the turning behavior of the 4WD±4WS and 4WD±2WS tractors at the steady-state turn on the rice ®eld was developed. The forces acting on the tires were calculated by the three dimensional stress distributions, using Wong's originated model. Then the turning behavior at the steady-state turn was simulated. The following was clari®ed: 1. By comparing the simulated results with the measured ones, it was found that some simulated values did not agree with the measured results and the eect of the soil deformation modulus was considered as the main cause of the disagreement. Then it was predicted that the large deformation modulus which was far from the measured value has to be given to the equations of motion. 2. The calculated tire sinkage derived from the static relation between the pressure and the sinkage was likely to give an underestimate. The addition of the slip±sinkage relationship to the calculation of the tire sinkage is expected to give a more correct result. 3. The simulation of the eect of the steering wheel angle on the behavior of the tractor indicated that the steering of the rear tire in the opposite direction to that of the front tire could reduce the dierence between the direction of longitudinal center line of the tractor and the tangential direction of the turning circle. 4. The simulation of the eect of forward speed on the tractors behaviors indicated that the oversteer characteristic in 4WS seemed to be more obvious than that in 2WS, though the neutral steer was ascertained in the simulated results of the yaw angular velocity of the center of gravity. 5. In the simulation of the eect of location of the center of gravity it was predicted that the oversteer characteristic would become conspicuous if the center of gravity was located at the rear part of the tractor in both 2WS and 4WS. 6. The simulated moving paths of the center of gravity and the end of PTO shaft indicated that the simulation program could show typical characteristics of turnability of the 4WS tractor, that is, the easy change in the tractor by steering of the rear tires in the same direction as the front tires and the sharp change by the opposite steering of the rear tire to that of the front tire. 7. It was thought that this study is only the ®rst stage in predicting the precise turning performance of wheeled vehicles, o-the-road. From this analysis, the core of the method of analysis of the turning behavior of the tractor o-theroad seems to be successfully constructed. References [1] Crolla DA, Hales FD. The lateral stability of tractor and trailer combinations. J Terramechanics 1979;16(1):1±22. [2] Crolla DA. An analysis of o-road vehicle steering behavior. Proc of 7th Int Conf ISTVS 1981;1265±90. [3] Crolla DA. O-road vehicle dynamics. Vehicle System Dynamics 1981;10:253±66.
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[4] Oida A. Turning behavior of articulated-frame-steering tractorsÐPart 2. Motion of tractors with drawbar pull. J Terramechanics 1987;24(1):57±73. [5] Kitano M, Watanabe K, Ooi K. Turning motion of four-track steering vehicles. Proc of 2nd AsiaPaci®c Conf ISTVS, Bangkok 1988;377±88. [6] Oida A, Itoh H. Study on turning behavior of a 4wd±4ws farm tractor. Proc of 2nd Asia-Paci®c Conf ISTVS, Bangkok 1988;397±411. [7] Liljedahl JB, Turnquist TP, Smith DW, Hoki M. Tractors and Their Power Units, 4th ed., New York: Van Nostrand Reinhold 1989;289±307. [8] Kitano M, Watanabe K, Saito T. Turning motion of four-track steering vehicles, Proc of 10th Int Conf ISTVS, Kobe 1990;3:613±24. [9] Itoh H, Oida A. Dynamic analysis of turning performance of 4wd±4ws tractor on paved road. J Terramechanics 1990;27(2):125±43. [10] Itoh H, Oida I, Yamazaki M. Turning behavior of a 4WD±4WS farm tractor in a ®eld. Proc of 11th Int Conf ISTVS, Lake Tahoe 1993;1:286±95. [11] Wong JY. Theory of Ground Vehicles, 2nd ed., New York: Wiley-Interscience, 1993, pp. 262±303. [12] Krick G. Radial and shear stress distribution under rigid wheels and pneumatic tires operating on yielding soils with consideration of tire deformation. J Terramechanics 1969;6(3):73±98. [13] Oida A, Koppes H. Study on a small three-axial force transducer. Bull Fac Agric, Niigata University 1986;38:27±37. [14] Oida A, Satoh A, Itoh H. Measurement and analysis of normal, longitudinal and lateral stresses in wheel-soil contact area. Proc of 2nd Asia-Paci®c Conf ISTVS, Bangkok 1988;233±243. [15] Oida A, Satoh A, Itoh H, Triratanasirichai K. Three dimensional stress distributions on a tire-sand contact surface. J Terramechanics 1991;28(4):319±30. [16] Onafeko O, Reece AR. Stresses and deformations beneath rigid wheels. J Terramechanics 1967;4(1):59±80. [17] Uelmann FL. The performance of rigid cylindrical wheels on clay soil, Proc of 1st Int Conf ISTVS, Turin 1961. [18] Wong JY, Reece AR. Prediction of rigid wheel performance based on the analysis of soil±wheel stresses Part I. Performance of driven rigid wheels. J Terramechanics 1967;4(1):81±98. [19] Wong JY, Reece AR. Prediction of rigid wheel performance based on the analysis of soil±wheel stresses Part II. Performance of towed rigid wheels. J Terramechanics 1967;4(2):7±25. [20] Itoh H, Oida A, Yamazaki M. Measurement of forces acting on 4wd±4ws tractor tires during steadystate circular turning on a paved road. J Terramechanics 1994;31(5):285±312. [21] Itoh H, Oida A, Yamazaki M. Measurement of forces acting on 4wd±4ws tractor tires during steadystate circular turning in a rice ®eld. J Terramechanics 1995;32(5):263±83. [22] Bekker MG. Introduction to Terrain-Vehicle Systems. Ann Arbor: The University of Michigan Press, 1969.
Numerical simulation of a 4WD±4WS tractor turning in a rice ®eld H. Itoh a,*, A. Oida b, M. Yamazaki b a
Faculty of Biology Oriented Science and Technology, Kinki University, Naga-gun Wakayama 649-64, Japan b Faculty of Agriculture, Kyoto University, Sakyo-ku, Kyoto 606-01, Japan Received 5 August 1997; accepted 18 November 1998
Abstract For the steady-state circular turning of a 4WD±4WS (4 wheel driven±4 wheel steered) tractor in a rice ®eld, a numerical simulation was achieved. Equations of motion of this tractor were developed in a vehicle ®xed x±y coordinate system. By comparing the calculated and measured results of acting forces on the tractor tires, this simulation was evaluated. Then, the characteristic parameters of the turning vehicle, which are the side slip angle and the yaw angular velocity of the vehicle center of gravity, were simulated in several combinations of the steering wheel angle and the forward speed. Also the same simulation applied to a 4WD±2WS tractor which had the same body as the 4WD±4WS tractor. The simulated results showed a clear dierence of turnability between 4WS and 2WS. # 1999 ISTVS. All rights reserved.
Nomenclature b c F H i I
Width of a tire Cohesion of soil Tractive force Thrust or longitudinal force of a tire Slip of a tire Yaw moment of inertia
* Corresponding author. Tel.: +81-736-3888; fax: +81-736-77-4754; e-mail: [email protected] 0022-4898/98/$20.00 # 1999 ISTVS. All rights reserved. PII: S002 2-4898(98)0003 7-8
92
j K m r R RAD S V Z
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Soil shear deformation Soil shear deformation modulus Mass of a vehicle Radius of a tire Rolling resistance of a tire Turning radius of the tractor center of gravity Lateral force on a tire Forward speed of the center of gravity of the tractor Sinkage of rut
Greek alphabet !
Side slip angle of the center of gravity of the tractor Angle Radial or normal stress Shear stress Internal friction angle of soil Yaw angular velocity of the center of gravity of the tractor
Subscripts j1 j2 jl jt Ss xc yc 1 2
1
2 0 1 2 !t
Tangential soil shear deformation at the front region at a negative slip Tangential soil shear deformation at the rear region at a negative slip Lateral soil shear deformation Tangential soil shear deformation at a positive slip Lateral shearing force of tire Longditudinal coordinate of the turning center Lateral coordinate of the turning center Steer angle of the left rear tire Steer angle of the right rear tire Steer angle of the left front tire Steer angle of the right front tire Angle of the transaction Entry angle Exit angle Angular velocity of a tire
1. Introduction This research focused on the turning motion of a farm tractor in a rice ®eld. There are many corners and obstacles on roads during movement from the farmer's house
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to the ®eld, so an operator of a farm tractor has to do various kinds of turning maneuver. Also, in o-road situations such as a rice ®eld, many turning maneuvers are required. In order to make the turning motions of the farm tractor eective and stable, it is necessary to analyze the fundamental kinematic dynamics of the farm tractor on-road and o-road and to investigate the interactions between the running devices of the tractor and the ®eld soil [1±3]. The kinematic dynamics on the even paved road has already been studied in automobile engineering. However, there was little research in relation to the similar analysis on o-road vehicles, especially on deformable soft soil [4±11]. The interactions between the running device and the soil have been investigated in terramechanic ®elds. As one investigation method, the stress distribution curves beneath a pneumatic tire or a rigid wheel have been analyzed [12±19]. The objective of this research is to formulate the turning motion of a four-wheel drive and four-wheel steering (4WD±4WS) agricultural tractor in a steady-state circular turn in rice ®elds. A 4WS tractor came onto the Japanese market. It is a wellknown fact that the four-wheel steering system provides a higher turnability on a paved road for a vehicle than the conventional two-wheel steering system. It is expected that the 4WD±4WS tractor will also show a high performance in a rice ®eld. That is the reason why the 4WS tractor was selected in this study. Fundamental turning characteristics are to be investigated during the analysis of the steady-state turn. By combining the previous analysis on the turning behavior of the 4WD±4WS tractor on a paved road [9,20] with the stress distribution analysis for tires or wheels, a mathematical model of the turning behavior of the tractor in a rice ®eld was conducted. Some main speci®cations of the tractor, some soil parameters, the steering wheel angle and the forward speed were input to the model. Then, the side slip angle and the yaw angular velocity of the tractor center of gravity were output after 200 iteration of solving program for dierential equations. At the ®nal stage of iteration, values of acting forces on the tires were also output. These simulated results were compared with the measured results [21]. This comparison showed some errors, but in general this model could express the actual turning behavior of the tractor. After evaluating this model, eects of the input parameters on the output parameters were simulated and analyzed. 2. Mathematical models The motion of the tractor was analyzed as two dimensional and a vehicle ®xed x±y coordinate system was used. The origin of the coordinate axes was settled at the center of gravity of the tested tractor. Three equations of motion were developed from a dynamic equilibrium analysis of the forces in the x and y directions and the momentum about the center of gravity. The tractive force and lateral force were considered as the input forces on tractor tires as shown in Fig. 1. To develop the equations of motion, the following assumptions applied.
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1. The tractive force acts in the same direction as the wheel plane. 2. The lateral force acts at right angle to the wheel plane and is de®ned as positive when the force acts from the right side to the wheel plane. 3. The side slip angle of a tire is de®ned as positive when the moving direction of the tire deviates leftward from the wheel plane, and the side slip angle of the center of gravity is de®ned as positive when the moving direction of the center of gravity deviates leftward from the longitudinal center line (the x axis). 4. The yaw angular velocity of the center of gravity in the counterclockwise direction is de®ned as positive.
Fig. 1. Tractive forces, lateral forces and side slip angles of the tires of a 4WD±4WS tractor in a left turn in the x±y plane. (In the above ®gure, F and S indicate the tractive force and lateral force, respectively. Subscripts ``1'' and ``2'' express left and right. ``f'' and ``r'' denote front and rear.)
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5. The rolling motion due to the sinking of tires in the soft ground can be neglected. 6. The eect of wind can be neglected. 7. The steer angles of the rear tires of the 4WS tractor conform to the Ackermann steering geometry. The accelerations of the center of gravity in the x and y directions are same as those used in the previous analysis [9]. Then, the equations of motion are given by Eqs. (1±3): m V_ cos ÿ V _ ! sin Ff1 cos 1 Sf1 sin 1 Ff 2 cos 2 Sf 2 sin 2 Fr1 cos 1 Sr1 sin 1
1
Fr2 cos 2 Sr2 sin 2 m V_ sin V _ ! cos ÿFf1 sin 1 Sf1 cos 1 ÿ Ff 2 sin 2 Sf 2 cos 2 ÿ Fr1 sin 1 Sr1 cos 1
2
ÿ Fr2 sin 2 Sr2 cos 2 I!_ ÿdfl Ff1 cos 1 ÿ dfl Sf1 sin 1 ÿ lfl Ff1 sin 1 lfl Sf1 cos 1 dfr Ff 2 cos 2 dfr Sf 2 sin 2 ÿ lfr Ff 2 sin 2 lfr Sf 2 cos 2 ÿ drl Fr1 cos 1 ÿ dr1 Sr1 sin 1
3
lrl Fr1 sin 1 ÿ lrl Sr1 cos 1 drr Fr2 cos 2 drr Sr2 sin 2 lrr Fr2 sin 2 ÿ lrr Sr2 cos 2 _ _ and !_ are dierential coecients of V, and !. where, V, The tractive force is the sum of the thrust H and rolling resistance R of the tire. The lateral force is the lateral shearing force of soil. These forces are shown in Fig. 2. Based on the results of the study on stress distribution under a tire or a rigid wheel, it is necessary to make the assumption that the pressure distribution is not uniform over the soil±tire contact area especially in the case of a soft, deformable ground. The stress distribution should be taken into account to predict the forces acting on vehicle tires. In this research, the model originated from Wong [18,19] was basically adopted.
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Fig. 2. Forces acting on a tire (H, R, Ss , Vt , Vl and t are thrust, rolling resistance, lateral shear force, forward velocity, lateral moving velocity and side slip angle of tire, respectively.)
By integrating the stress distribution curves shown in Fig. 3, the normal load, the thrust, the rolling resistance and the lateral shear force were obtained. The angle is zero at bottom dead center (B. D. C.) and is positive from B. D. C. to the entry angle 1 . The angle from B. D. C. to the exit angle 2 is negative. The following assumptions were adopted to calculate the forces acting on a tire. (a) (b) (c) (d)
The The The The
contact patch of a tire is rectangle. tire is rigid. stress distribution is uniform over the width of the tire. surcharge pressure can be neglected.
2.1. The equations of stress distributions The shear stress is assumed to be represented by Bekker's equation as shown in Eq. (4) [22]: j 4 c tan 1 ÿ exp ÿ K
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Fig. 3. Stress distribution curves under a rigid wheel. (In the ®gure, r, Z, 0 , 1 and 2 are radius of tire, sinkage, angular position of transition point, entry angle and exit angle, respectively.)
In this case the shear stress is determined by the internal friction angle , cohesion of soil c, soil deformation modulus K, and the radial stress . The soil parameters can be obtained by measurements [21]. In order to get the shear stress, the radial stress distribution curve has to be modeled ®rst. Reviewing the measured results of the radial stress distribution reported by Onafeko [16] and Oida [15], it was assumed that the radial stress distribution curve under a wheel can be represented by a parabola. Furthermore, it was assumed that the rut recovery, which is represented by a ratio, 1 ÿ 2 =21 , is to be 0.7, though the value increases with the slip of a tire [15]. The radial stress distribution curve is given by Eq. (5): a ÿ 1 ÿ 2
5
where, ``a'' in Eq. (5) is constant. Then, if the tangential and the lateral shear deformations of soil are calculated, equations of the tangential and the lateral stress distribution curves will be obtained. 2.2. Tangential shear deformation of soil at a positive slip The tangential shear deformation can be calculated by integrating the slip velocity Vj of a point on the wheel rim. The slip velocity is the dierence between the
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circumferential speed and the tangential component of the longitudinal forward velocity at the point on the rim. Wong conducted the tangential shear deformation jt at a positive slip i as Eq. (6) [18]. t 6 jt Vjdt r 1 ÿ ÿ 1 ÿ i sin 1 ÿ sin 0
2.3. Tangential shear deformation of soil at a negative slip The measured results of the stress distributions obtained by the above three researchers indicated that there was a negative tangential stress at the rear part of the soil±wheel contact region at the slightly positive or the negative slip. Wong analyzed the tangential stress distribution of a towed rigid wheel on sand. He showed the transition point where the direction of the tangential stress reversed. Uelmann [17] also indicated the negative tangential stress on a pure cohesive soil and it was described in his paper that there was upward soil ¯ow at the front part of the soil±wheel contact region for a towed wheel. Though the rice ®eld used for the test was the silty clay loam, Wong's model was adopted. The soil±tire contact area is to be divided into two regions at the transition point which is represented by the angle, 0 . The front region covering from 1 to 0 gives the positive tangential stress and in the rear region covering from 0 to 2 the tangential stress becomes negative. The soil shear deformation was calculated for each region. Wong suggested that at the transition point the direction of the absolute velocity of the rim coincides with one of the slip lines on the boundary of the soil mass. Furthermore, it is well known that the soil failure occurs at angle of (45 ÿ =2) from the principal stress plane. Then the angle of the absolute velocity of the rim was assumed to be equal to one of (45 ÿ =2) at the transition point. Then the 0 was obtained by Eq. (7): 1 cos 0 ÿ 1ÿi tan 45 ÿ sin 0 2 The tangential shear deformation j1 at the front region derived from Eq. (8). 1 1 r!t f1 1 ÿ i K ÿ cos d j1 !t r1 ÿ 1 1 ÿ iK ÿ 1 ÿ i sin 1 ÿ sin The constant K was expressed as follows: 1 1 ÿ i sin 1 ÿ sin 0 ÿ1 K 1ÿi 1 ÿ 0
7
8
9
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According to the measurement of the soil ¯ow by Wong, the tangential soil deformation j2 at the rear region became the same form as Eq. (6). t 10 j2 Vj dt r 0 ÿ ÿ 1 ÿ i sin 0 ÿ sin 0
2.4. Lateral shear deformation of soil The lateral shear deformation arises when the tire has the slip to lateral direction. The lateral shear deformation can be derived from the integral of the lateral moving velocity of a tire Vl as shown in Fig. 2. The lateral moving velocity is represented by Eq. (11): Vl Vf sin t
11
The lateral shear deformation of soil jl is given by the integral of the lateral moving velocity as shown in Eq. (12). t Vt sin t Vt cos t 12 1 ÿ ; !t jl Vl dt !t r 1 ÿ i 0 where !t is the angular velocity of a tire. 2.5. Equations of the distribution curves of tangential and lateral stresses Substituting the radial stress and the shear deformation of soil into Eq. (4), the equations of the distributions of tangential and lateral stresses are obtained. For the tangential stress distribution curve, three equations are represented as follows: jt p c tan 1 ÿ exp ÿ K j1 13 1 c tan 1 ÿ exp ÿ K jj2 j jj2 j j2 < 0 2 c tan 1 ÿ exp ÿ K j2 where p is the tangential stress at a positive slip and 1 and 2 are the tangential stresses at the front and rear regions at a negative slip, respectively. The lateral stress is denoted as negative when the side slip angle is positive: jjl j ÿjjl j 14 l c tan 1 ÿ exp ÿ K jl where l is the lateral stress.
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2.6. Forces acting on tractor tires The normal load on a tire can be calculated by integrating the vertical components of the radial and the tangential stresses over the soil±tire entire contact area. There are two equations to give the normal load. One is for a positive slip and another is for a negative slip, as shown in Eqs. (15) and (16). In the case of a positive slip: Wt rb
1 2
cos d
1 2
p sin d
15
where Wt is the normal load on a tire and b is the width of the tire. In the case of a negative slip: Wt rb
1 2
cos d
1 0
1 sin d
0 2
2 sin d
16
On the other hand, considering the momentum equilibrium about the tractor center of gravity, the normal load on each tire can be calculated another way as shown in a previous report [9]. This normal load should be equal to the right side of Eqs. (15) or (16). This relationship identi®ed the constant ``a'' in Eq. (5), if the calculated slip was substituted into the above equations. The successive iteration was achieved to calculate the constant ``a'' in terms of each tire. The thrust of a tire H is given by integration of the horizontal components of the radial and the tangential stresses over the soil±tire contact area. There are also two equations in relation to the slip of the tire. In the case of a positive slip: H rb
1 2
p cos d ÿ
0 2
sin d
17
In the case of a negative slip: H rb
1 0
1 cos d ÿ
0 2
sin d
18
The rolling resistance on a tire is calculated by integrating the backward horizontal components of the radial and tangential stresses over the soil±tire contact area. The rolling resistance is denoted as negative and is given by Eq. (19) or Eq. (20) at a positive or a negative slip, respectively. In the case of a positive slip: 1 sin d < 0 19 R ÿrb 0
where, R is the rolling resistance of the tire.
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In the case of a negative slip: R ÿrb
1 0
sin d ÿ
0 2
2 cos d < 0
20
The lateral shear force on a tire is calculated by integrating of the lateral stress over the entire contact area as shown in Eq. (21): Ss br
1 2
l d
21
2.7. Sinkage The entry angle 1 and the exit angle 2 are determined by the sinkage of a tire because the tire was assumed not to de¯ect and to be rigid. There was no measured data in terms of the relation between the sinkage and the slip of the tire of the tested tractor. Therefore, the static sinkage of the tire was predicted by using the pressure±sinkage relationship. The pressure±sinkage relationship, ``p f Z'' was examined in the rice ®eld and was approximated to 6th power polynomial Eq. (21). On the other hand, the relationship between the contact pressure p and the sinkage of the tire Z can be derived geometrically from Fig. 3 and the relation is expressed by Eq. (22): Wt 1 ÿ 2 ÿ ; aw 22 p 21 2aw br cosÿ1 rÿZ r where, aw is a ratio of actual contact length to static length with the same sinkage and Wt is the normal load on a tire. The contact pressures which are calculated by the two equations should be same. Therefore, Z should satisfy the next equation: f Z
Wt ÿ 2aw br cosÿ1 rÿZ r
23
2.8. Slip If the slip of the tire is calculated, the normal load, lateral shear force, thrust and the rolling resistance of the tire can be obtained. The thrusts of the left and right tires are equal by the action of the dierential gears which are furnished to the front and rear axles. So, it was assumed that the slip of the left tire gives a thrust which is equal to that of the right tire. As is well known, the slip is derived from the longitudinal forward velocity (Fig. 2) and the circumferential speed of a tire. The longitudinal forward velocity of each tire
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is calculated by the side slip angle, yaw angular velocity and the forward speed of the center of gravity of the tested tractor as follows: q Vffl V sin lfl !2 V cos ÿ dfl !2 cos f1 q Vffr V sin lfr !2 V cos dfr !2 cos f 2 24 q Vfrl V sin ÿ lrl !2 V cos ÿ drl !2 cos r1 q Vfrr V sin ÿ 1rr !2 V cos drr !2 cos r2 Vf indicates the longitudinal forward velocity of a tire and the subscripts ``¯'' and ``rr'' mean the ``front left'' and the ``rear right'', respectively. The angle indicates the side slip angle of the tire and its subscript is the same as that shown in Fig. 1. The circumferential speed is related to the reduction gear ratio and the dierential gear. If the forward speed, V in m/s is given, the number of revolutions per second of the engine and the front and rear axle shafts at a straight run are represented by Eq. (25): REVe
V ; REVf REVe RATf ; REVr REVe RATr RATr 2 rr 25
where, REVe , REVf and REVr are the number of revolutions per second of the engine and the front and rear axle shafts at a straight run. RATf and RATr are the reduction gear ratios of the front and rear drive shafts. rr is the radius of the rear tires. Furthermore the dierential gear con®nes the number of revolutions of the left and right tires as follows: 2REVf REVfl REVfr ; 2REVr REVrl REVrr
26
where, REVfl and REVfr are the number of the revolutions per second of the front left and right tires, respectively. REVrl and REVrr are the number of revolutions per second of the rear left and right tires, respectively. If the slip of a tire, for example, the slip of the front left tire, ifl is given, the circumferential speed can be calculated by the longitudinal forward velocity. Vcfl
Vffl 1 ÿ ifl
27
where Vcfl is the circumferential speed of the front left tire. Then the number of revolutions per second of the front right tire is derived from Eq. (28): REVfr 2REVf ÿ REVfl ; REVfl where, rf is the radius of the front tires.
Vcfl 2rf
28
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Then the circumferential speed of the front right tire, Vcfr is represented by Eq. (29). Vcfr 2rf REVfr
29
The slip of the front right tire, ifr can be calculated by using the Eqs. (29) and (24). ifr
Vcfr ÿ Vffr Vcfr
30
Substituting the slips of the front left and right tires into the Eqs. (17) or (18), the thrusts of the front left and right tires are calculated and they are compared each other. If the dierence between the two thrusts is smaller than a tolerance, the above-described slips are adopted, whereas if the dierence is greater than the allowed value, a new slip of the front left tire is attempted and so on. This process is also achieved for the rear left and right tires. 3. Solution of the motion equations 3.1. Initial values The equations of motion were solved by the Runge±Kutta±Gill method. The initial values of the side slip angle, yaw angular velocity and the forward speed of the center of gravity had to be given to the equations of motion at the ®rst stage. At the ®rst stage of the solution, it was assumed that the forward speed was zero and stood still. The step response was treated. In order to determine the initial values, moving direction of virtual wheel was considered. The virtual wheel was located on the center of the front and rear axles as shown in Fig. 4. The side slip angles of the front and rear virtual wheels should be zero as the initial condition. The moving directions of the front and rear virtual wheels, f , r were schematically shown in Fig. 4 and were expressed in Eq. (31).
f
V sin lf ! V sin ÿ lr ! ; r V cos V cos
31
The left sides of Eq. (31) could be assumed to be equal to the average angle of the left and right tires as shown in Eq. (32):
f
1 2 1 2 ; r 2 2
32
If f , r were given, the initial values of the side slip angle and the yaw angular velocity of the center of gravity, ini , !ini are represented by Eq. (33): lf tan r lr tan f V tan f cos ini ÿ sin ini 33 ; !ini ini tanÿ1 1f 1r lf
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Fig. 4. Moving directions of virtual wheels. ( f , r are the moving directions of front and rear virtual wheels.)
The initial value of the forward speed of the center of gravity is arbitrarily chosen. 3.2. Solution of the equations of motion Substituting the above equations [from Eqs. (4) to (33)] into Eqs. (1±3), Eq. (34) is obtained. 34 a11 V_ a12 _ b1 ; a21 V_ a22 _ b2 ; I!_ b3 Then the dierential coecients of the side slip angle, d =dt, yaw angular velocity, d!=dt and the forward speed, dV=dt of the center of gravity are represented by Eq. (35). d b1 a21 ÿ b2 a11 _ dV b1 ÿ a12 d! b3 ;V ; !_ _ I dt a12 a21 ÿ a11 a22 dt a11 dt
35
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The above equations were solved by the Runge±Kutta±Gill method and then the side slip angle, yaw angular velocity and the forward speed of the center of gravity were calculated. In the method the slip and the side slip angle of the tire were set constant during each step. The slip of the tire was derived from the related parameters which had been calculated before step 1 because the slip, sinkage and the forces acting on tires could not be calculated at the same time. As for the condition of the Runge±Kutta±Gill method, the interval between each step was set 0.01 s and 200 iteration was achieved by using a computer. 3.3. Eect of soil shear deformation modulus The soil shear deformation modulus was obtained from the measured results of the shear stress in relation to the soil deformation. The average normal pressure beneath a tire was estimated by the measured results of the normal load on a tire [21]. The estimated normal pressure was less than 39.2 kPa. So, the result of the shear stress-deformation curve at the normal pressure of 19.6 kPa was adopted to determine the soil deformation modulus. From the curve the soil deformation modulus was obtained as 0.3 cm. Substitution of the above deformation modulus into the equations of motion proved a failure in simulating the turning behavior of the tested tractor, because the deformation modulus was so small that the calculated tangential stress became so large at a small soil deformation [see Eq. (13)]. The eects of the deformation modulus on the calculated normal loads and thrusts of the right front and rear tires of the 4WS tractor in left turns were examined by computer simulation as shown in Fig. 5.
Fig. 5. Eect of soil shear deformation modulus on the normal loads and thrusts of the right front and rear tires. In the above notations, ``ÿ350/0.91/EXP'' means that experimental result at steering wheel angle of ÿ350 and the forward speed of 0.91 m/s and so on. ``SIM'' indicates the simulated result.
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The calculated normal loads on the tires were not in¯uenced by the shear deformation modulus and did not approximate the experimental results. On the other hand, there was a case that the calculated thrust at the soil deformation modulus of 3.5 cm approximated the measured one. Furthermore, the eects of the soil shear deformation modulus on other forces such as the lateral force and the rolling resistance were also investigated. The calculated rolling resistance closely resembled the measured one at the deformation modulus of 3.5 cm, but there were some cases that showed a large dierence between the simulated and the measured lateral forces at the shear deformation modulus. Although the soil shear deformation modulus of 3.5 cm was very much larger than the measured result, it seemed that this value was enough to oer adequate calculated results as a whole. Accordingly, the value, 3.5 cm was substituted as the soil shear deformation modulus into the equations of motion. 4. Evaluation of the simulation Using the equations of motion, the normal loads, lateral forces, rolling resistances, thrusts, slips, side slip angles of four driven tires, and the turning radius of the center of gravity were calculated. And then these simulated values in relation to right side tires and turning radius were compared with the measured results shown in a previous report [21]. A left steering wheel angle was de®ned as negative. Results corresponding to the negative steering wheel angle shown hereafter show data on the outside tires of a turn, whereas the data on the inside tires are shown in the positive steering wheel angle. As for the forward speed, third, ®fth and seventh gear shown in result graphs correspond to 0.41 m/s, 0.91 m/s and 1.92 m/s, respectively. 4.1. Tire normal load Fig. 6 shows the measured and the simulated normal loads on the right front and rear tires. The simulated normal load on the outside front tire (for negative steering wheel angle) agreed with that of the measured result. 4.2. Rolling resistance Fig. 7 shows the simulated and the measured rolling resistances of the right front and rear tires. The computed rolling resistance of the outside front tire agreed with the measured one. 4.3. Lateral force and side slip angle Fig. 8 shows the simulated and the measured lateral forces and side slip angles of the tires. Excepting some points, dierence between the simulated and measured lateral forces on the front tires was not large.
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Fig. 6. The measured and the simulated tire loads (``3/F/SIM'' means ``the simulation results of the front right tire in 3rd gear'' and so on. ``EXP'' indicates '' the experimental result'').
Fig. 7. The measured and the simulated rolling resistances of the tires.
Fig. 8. The measured and the simulated results of lateral forces and side slip angles of the tires.
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4.4. Thrust and slip Fig. 9 shows the simulated and the measured values of thrusts and slip of the right front and rear tires. Most of the calculated thrust of the outside rear tire neared the measured one. 4.5. Turning radius of center of gravity of the tractor The turning radius was derived from Fig. 4 and Eqs. (36) and (37).
f
V sin 1f ! V sin ÿ lr ! ; r V cos V cos
RAD
q x2c y2c ; xc ÿyc tan f lf ; yc
36
lf lr tan f ÿ tan r
37
Fig. 10 shows the simulated and the measured turning radii of the center of gravity in the left and right turns. In the case of 4WS, the dierence between the simulated and the measured turning radius was small in the right turn (for positive steering wheel angle). In the left turn it was smaller than that of measured results. In the case of 2WS the calculated turning radius was smaller than that of the measured result in the left and right turns.
Fig. 9. The measured and the simulated results of thrusts and slip of the tires.
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Fig. 10. Comparison of the measured with the simulated turning radii of the center of gravity.
4.6. Evaluations of the equations of motion and the simulation program The lateral shear force, rolling resistance and the thrust of the tire were the function of the deformation modulus [see Eqs. (13) and (14) and Eqs. (17±21)]. The slip was also aected by the deformation modulus because the slip was determined by the dierence between the calculated thrusts of the left and right tires. The small soil deformation modulus will give the large thrust at a small slip which gives a small soil deformation [Eqs. (6),(8) and (10)]. As shown in Fig. 5 the eect of the deformation modulus on the acting forces on tires was inspected and a comparison of the larger value with the measured one was used in this simulation. Though the 3.5 cm of the deformation modulus was estimated as the most suitable value for this simulation algorithm, some errors were observed in the comparison between the simulated and measured results. The sinkage of a tire increases with the slip [15]. This slip±sinkage relation was not taken into account in this calculation. If the slip±sinkage relationship was introduced to the calculation of the tire sinkage, the partition of the integral of each stress distribution would be modi®ed. Some other assumptions described in Section 2 also aected the calculation of the forces on the tires. Especially assumptions (b) and (c) seemed to introduce overestimated forces. From the above considerations, it was found that the large deformation modulus, which is far from the measured value, has to be substituted into the equations motion and the slip±sinkage relationship should be added to the calculation of the tire sinkage.
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5. Numerical simulation Using the equations of motion explained in the above section, the eects of the steering wheel angle, forward speed on the side slip angle and the yaw angular velocity of the center of gravity were simulated in relation to the 4WD±4WS and the 4WD±2WS tractor at steady-state circular turn. In addition to this simulation, the location of the center of gravity was also examined. The moving paths of the center of gravity and the end of PTO shaft were derived from the simulated results. 5.1. Eect of steering wheel angle During the turn, the steering wheel was ®xed to a constant angle. The angle was determined from ÿ50 to ÿ450 with an interval of 50 . The forward speeds of 1.0, 2.0 and 3.0 m/s were selected. As shown in Fig. 1, the side slip angle is de®ned as positive when the moving direction of the center of gravity deviates leftward from the longitudinal center line (the x axis). The yaw angular velocity in the counterclockwise direction is de®ned as positive. Fig. 11(a) shows the eect of the steering wheel angle on the side slip angle and the yaw angular velocity in the left turns. The side slip angle in 2WS increased with the increase of the absolute value of steering wheel angle. On the other hand the slope of the increase began to decrease at the steering wheel angle of about ÿ250 in 4WS and approached a certain value at the forward speed of 1.0 m/s. The side slip angle decreased under the ÿ250 of the steering wheel angle at the speeds of 2.0 and 3.0 m/s in 4WS. These results indicate that for the 4WS tractor, in the case of the steering wheel angle under ÿ200 , steering of the rear tires in the opposite direction to that of the front tires brings the decrease of the directional dierence between the longitudinal center line of the tractor and the tangent line of the turning circle. Furthermore the positive side slip angle of the center of gravity indicates that the moving direction of the center of gravity is the inside of the turn. Therefore it seems that the moving direction of the center of gravity changes to the outside of the turn with the increase of the absolute value of the steering wheel angle and the forward speed in 4WS. This tendency coincided with the result of the direction of resultant forces obtained by the experiment on the rice ®eld. The yaw angular velocity increased linearly with the absolute value of the steering wheel angle. The yaw angular velocity in 2WS was greater than that in 4WS at the steering wheel angle over ÿ150 and under the steering wheel angle this tendency reversed. These results also show the eect of the steering of the rear tires. The yaw angular velocity in 4WS decreased due to the steering of the rear tires in the same direction as the front tires and it increased by means of steering the rear tires in the opposite direction to that of the front tires. 5.2. Eect of forward speed The forward speed was set from 0.5 m/s to 3.0 m/s with an interval of 0.5 m/s. The steering wheel angles of ÿ150, ÿ250, ÿ350 and ÿ450 were selected. Fig. 11(b)
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shows the eect of the forward speed on the side slip angle and the yaw angular velocity in left turns. The side slip angle decreased with the increase of forward speed both in 4WS and 2WS. The direction of the longitudinal center line of the tractor approached the tangential direction of the turning circle as the forward speed increased. The side slip angle at steering wheel angle of ÿ150 in 4WS was larger than that in 2WS due to steering the rear tires in the same direction as that of the front tire. The yaw angular velocity increased linearly with the forward speed. This fact shows that the tested tractor has a neutral steer characteristic even on the rice ®eld
Fig. 11. Simulation results of the side slip angle and the yaw angular velocity of the center of gravity in left turns. (a) Eect of the steering wheel angle, (b) eect of the forward speed, (c) eect of the location of the center of gravity.
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below the forward speed of 3.0 m/s which is near the maximum forward speed of the tested tractor. Based on the simulated results shown in Fig. 11(b), it was predicted that the side slip angle in 4WS would become negative at forward speeds over 3.0 m/s and steering wheel angle of ÿ450 . Furthermore the side slip angle in 4WS was smaller than that in 2WS and was predicted to become negative at higher speeds than 3.0 m/s. Therefore the 4WS tractor is more likely to have oversteer characteristics than the 2WS tractor, though the results of the yaw angular velocity showed neutral steer characteristics. 5.3. Eect of location of the center of gravity The location of the center of gravity was de®ned by the ratio of the length from the front axle to the center of gravity to the wheelbase. So, if the center of gravity is located at the rear part of the tractor, the ratio approaches 1.0. The lateral transfer of the location was not considered. The forward speed was set to 1.5 m/s and the four levels of the steering wheel angle, ÿ150, ÿ250, ÿ350 and ÿ450 , were selected. Fig. 11(c) shows the eect of the location of the center of gravity on the side slip angle and the yaw angular velocity of the center of gravity in left turns. The side slip angle decreased linearly with the increase of the ratio. In the case of 4WS the side slip angle became negative at the steering wheel angles of ÿ350 and ÿ450 when the ratio was 0.7 and 0.8. This indicated that the moving direction of the center of gravity turns to the outside of the turn when the center of gravity is located at the rear part of the tractor. Furthermore the side slip angle increased with the steering wheel angle when the ratio was under 0.5 and this tendency reversed when the ratio was over 0.6. These results indicated that the oversteer characteristic became evident when the center of gravity was located at the rear part of the tractor and on the contrary the understeer characteristic was observed when the center of gravity located at the front part. The yaw angular velocity did not change with the increase of the ratio. 5.4. Moving paths of the center of gravity and the end of PTO shaft The locations of the center of gravity and the end of PTO shaft during the steadystate turns were calculated in the X±Y space ®xed coordinate. The locations can be calculated by the simulated side slip angle, yaw angular velocity and the forward speed of the center of gravity. The starting point of the center of gravity was the origin of the coordinate. And then the moving paths of the two points during a complete circular turn were simulated. Fig. 12(a) and (b) show the moving paths of the center of gravity and the end of PTO shaft at the steering wheel angles of ÿ100 or ÿ450 and the forward speed of 3.0 m/s. At the steering wheel angle of ÿ100 , the moving trajectory of the center of gravity in 4WS was located outside of that in 2WS. On the other hand, at the steering
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Fig. 12. Simulated moving paths at steering wheel angles of ÿ100 and ÿ450 . (a) Paths of the center of gravity, (b) Paths of the PTO shaft.
wheel angle of ÿ450 the trajectory of the center of gravity in 4WS was located in the inside of that in 2WS, because the rear tires were steered in the same direction as the front tires over ÿ200 of the steering wheel angle. On the other hand, the rear tires turn in the opposite direction to the front tires when the steering wheel angle is under ÿ200 . As for the path of the end of PTO shaft, the same tendency was observed, but at the steering wheel angle of ÿ450 the end of the PTO shaft passed the negative domain of the Y axis. These results show the eect of the steer angle of the rear tires of the 4WS tractor on turning behavior. This tendency was also seen in the computer simulation on the paved road. Through this simulation, it was predicted that more eective lane change at the steering wheel angle over ÿ200 and a sharper turn at the large steering wheel angle under ÿ200 could be supplied by the four wheel steering system, compared with the ordinary steering system.
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6. Conclusions The computer simulation model of the turning behavior of the 4WD±4WS and 4WD±2WS tractors at the steady-state turn on the rice ®eld was developed. The forces acting on the tires were calculated by the three dimensional stress distributions, using Wong's originated model. Then the turning behavior at the steady-state turn was simulated. The following was clari®ed: 1. By comparing the simulated results with the measured ones, it was found that some simulated values did not agree with the measured results and the eect of the soil deformation modulus was considered as the main cause of the disagreement. Then it was predicted that the large deformation modulus which was far from the measured value has to be given to the equations of motion. 2. The calculated tire sinkage derived from the static relation between the pressure and the sinkage was likely to give an underestimate. The addition of the slip±sinkage relationship to the calculation of the tire sinkage is expected to give a more correct result. 3. The simulation of the eect of the steering wheel angle on the behavior of the tractor indicated that the steering of the rear tire in the opposite direction to that of the front tire could reduce the dierence between the direction of longitudinal center line of the tractor and the tangential direction of the turning circle. 4. The simulation of the eect of forward speed on the tractors behaviors indicated that the oversteer characteristic in 4WS seemed to be more obvious than that in 2WS, though the neutral steer was ascertained in the simulated results of the yaw angular velocity of the center of gravity. 5. In the simulation of the eect of location of the center of gravity it was predicted that the oversteer characteristic would become conspicuous if the center of gravity was located at the rear part of the tractor in both 2WS and 4WS. 6. The simulated moving paths of the center of gravity and the end of PTO shaft indicated that the simulation program could show typical characteristics of turnability of the 4WS tractor, that is, the easy change in the tractor by steering of the rear tires in the same direction as the front tires and the sharp change by the opposite steering of the rear tire to that of the front tire. 7. It was thought that this study is only the ®rst stage in predicting the precise turning performance of wheeled vehicles, o-the-road. From this analysis, the core of the method of analysis of the turning behavior of the tractor o-theroad seems to be successfully constructed. References [1] Crolla DA, Hales FD. The lateral stability of tractor and trailer combinations. J Terramechanics 1979;16(1):1±22. [2] Crolla DA. An analysis of o-road vehicle steering behavior. Proc of 7th Int Conf ISTVS 1981;1265±90. [3] Crolla DA. O-road vehicle dynamics. Vehicle System Dynamics 1981;10:253±66.
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[4] Oida A. Turning behavior of articulated-frame-steering tractorsÐPart 2. Motion of tractors with drawbar pull. J Terramechanics 1987;24(1):57±73. [5] Kitano M, Watanabe K, Ooi K. Turning motion of four-track steering vehicles. Proc of 2nd AsiaPaci®c Conf ISTVS, Bangkok 1988;377±88. [6] Oida A, Itoh H. Study on turning behavior of a 4wd±4ws farm tractor. Proc of 2nd Asia-Paci®c Conf ISTVS, Bangkok 1988;397±411. [7] Liljedahl JB, Turnquist TP, Smith DW, Hoki M. Tractors and Their Power Units, 4th ed., New York: Van Nostrand Reinhold 1989;289±307. [8] Kitano M, Watanabe K, Saito T. Turning motion of four-track steering vehicles, Proc of 10th Int Conf ISTVS, Kobe 1990;3:613±24. [9] Itoh H, Oida A. Dynamic analysis of turning performance of 4wd±4ws tractor on paved road. J Terramechanics 1990;27(2):125±43. [10] Itoh H, Oida I, Yamazaki M. Turning behavior of a 4WD±4WS farm tractor in a ®eld. Proc of 11th Int Conf ISTVS, Lake Tahoe 1993;1:286±95. [11] Wong JY. Theory of Ground Vehicles, 2nd ed., New York: Wiley-Interscience, 1993, pp. 262±303. [12] Krick G. Radial and shear stress distribution under rigid wheels and pneumatic tires operating on yielding soils with consideration of tire deformation. J Terramechanics 1969;6(3):73±98. [13] Oida A, Koppes H. Study on a small three-axial force transducer. Bull Fac Agric, Niigata University 1986;38:27±37. [14] Oida A, Satoh A, Itoh H. Measurement and analysis of normal, longitudinal and lateral stresses in wheel-soil contact area. Proc of 2nd Asia-Paci®c Conf ISTVS, Bangkok 1988;233±243. [15] Oida A, Satoh A, Itoh H, Triratanasirichai K. Three dimensional stress distributions on a tire-sand contact surface. J Terramechanics 1991;28(4):319±30. [16] Onafeko O, Reece AR. Stresses and deformations beneath rigid wheels. J Terramechanics 1967;4(1):59±80. [17] Uelmann FL. The performance of rigid cylindrical wheels on clay soil, Proc of 1st Int Conf ISTVS, Turin 1961. [18] Wong JY, Reece AR. Prediction of rigid wheel performance based on the analysis of soil±wheel stresses Part I. Performance of driven rigid wheels. J Terramechanics 1967;4(1):81±98. [19] Wong JY, Reece AR. Prediction of rigid wheel performance based on the analysis of soil±wheel stresses Part II. Performance of towed rigid wheels. J Terramechanics 1967;4(2):7±25. [20] Itoh H, Oida A, Yamazaki M. Measurement of forces acting on 4wd±4ws tractor tires during steadystate circular turning on a paved road. J Terramechanics 1994;31(5):285±312. [21] Itoh H, Oida A, Yamazaki M. Measurement of forces acting on 4wd±4ws tractor tires during steadystate circular turning in a rice ®eld. J Terramechanics 1995;32(5):263±83. [22] Bekker MG. Introduction to Terrain-Vehicle Systems. Ann Arbor: The University of Michigan Press, 1969.