Comparison of the traffic performance of a two-axle four wheel drive (4WD), rear wheel drive (RWD), and front wheel drive (FWD) vehicle on loose sandy sloped terrain

Pergamon

Journal of Terramechanics, Vol. 34, No. 1, pp. 37-55, 1997 Elsevier Science Ltd © 1997 ISTVS. All fights reserved Printed in Great Britain 0022--4898/97 $17.00 + 0.00 PII: S0022-4898(97)00016-5

COMPARISON OF THE TRAFFIC PERFORMANCE OF A TWO-AXLE FOUR WHEEL DRIVE (4WD), REAR WHEEL DRIVE (RWD), AND FRONT WHEEL DRIVE (FWD) VEHICLE ON LOOSE SANDY SLOPED TERRAIN TATSURO MURO* Summm-y--The trat~c performances during driving and braking of a 5.88 kN weight wheeled vehicle with two-axle four wheel drive, rear wheel drive, and front wheel drive running up and down a loose sandy sloped terrain were compared by means of a simulation. For the given dimensions of the vehicle and the given terrain-wheel system constants, the relationship between the effective tractive and braking effort of the vehicle, the amount of sinkage of the front and rear wheels, the total amount of sinkage of the vehicle, and the slip ratio were calculated to estimate the optimum height of force of application and the optimum eccentricity of the center of gravity of the vehicle. It was observed that, during driving action, the maximum effective tractive effort of the four wheel drive vehicle (4WD) was larger than that of the rear wheel drive vehicle (RWD), which in turn was greater than that of the front wheel drive vehicle (FWD). During the braking action, the effective braking effort at skid -20% of the four wheel vehicle (4WB) was larger than that of the front wheel brake vehicle (FWB), in turn greater than that of the rear wheel brake vehicle (RWB), when the two-axle four wheel vehicle is moving up or down the loose sandy sloped terrain. The maximum terrain slope angle up which the two-axle wheeled vehicle is able to move during driving action was found to be about 0.067rr rad for the 4WD vehicle, about 0.03Dr rad for the RWD vehicle, and about 0.017zr rad for the FWD vehicle. The effective braking effort at skid-20% of 4WB, FWB and RWB was found to decrease with slope angle. © 1997 ISTVS.

NOMENCLATURE Br D e eD eft(b) erd(b)

G H hg ifd(b) ird(b)

Jd~b)(0)

£fd(b) ~rd(b)

L

Lfcd(b)

width of front wheel (cm) width of rear wheel (cm) wheel base from front to rear axle (cm) eccentricity eccentricity of vehicle's center of gravity (cm) eccentricity of normal ground reaction acting on front driving (or braking) wheel (cm) eccentricity of normal ground reaction acting on rear driving (or braking) wheel (cm) center of gravity of vehicle height of application force (cm) height of centre of gravity slip ratio of front driving wheel or skid of front braking wheel slip ratio of rear driving wheel or skid of rear braking wheel slippage of driving (or braking) wheel at an arbitrary point of central angle 0 (cm) height between front axle Or and application point of normal ground reaction acting on front driving (or braking) wheel (cm) height between rear axle Or and application point of normal ground reaction acting on rear driving (or braking) wheel (cm) distance between central axis of vehicle and application point land locomotion resistance acting on front driving (or braking) wheel (kN)

"Department of Civil and Ocean Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama 790-77, Japan. 37

38

T. Muro

land locomotion resistance acting on rear driving (or braking) wheel (kN) terrain-wheel system constant obtained from plate (kPa) my terrain-wheel system constant obtained from plate traction test bottom-dead-center of front wheel Mf bottom-dead-center of rear wheel Mr n,nO,n 1 terrain-wheel system constant obtained from plate loading and traction test front axle Of rear axle Or driving torque (Qfd~>0) or braking torque (Qfd~<0) acting around front axle Of (kNcm) Qfd(b) driving torque (Qrd ~>0) or braking torque (Qrd~<0) acting around rear axle Or (kNcm) Qrd(b) Qrd~b)/Rf driving (or braking) force acting on front wheel (kN) Q,dcb)/Rr driving (or braking) force acting on rear wheel (kN) radius of front wheel RS radius of rear wheel R~ total sinkage of vehicle (cm) s sinkage of front wheel (cm) $f sinkage of rear wheel (cm) Sr effective driving force (Tfd/> 0) or braking force (Trb~<0) acting on front axle Of (kN) Tfd~b~ effective driving force (Trd/>0) or braking force (Trb ~<0) acting on rear axle Or (kN) Trd(b) recovery of terrain at the front wheel (cm) Uf recovery of terrain at the rear wheel (cm) Ur v vehicle speed (cm s -I) vehicle weight (kN) W wr front axle load (kN) Wr rear axle load (kN) Lred(b)

mc

o of O fed(b) Orcd(b)

Ot

,~(o) r(o) 09f Wr

slope angle of terrain (rad): ~>0 during driving action; fl < 0 during braking action central angle of wheel (rad) entry angle of wheel (rad) entry angle of front driving (or braking) wheel (rad) entry angle of rear driving (or braking) wheel (rad) angle of inclination of vehicle (rad) normal stress acting on wheel at an arbitrary point of central angle 0 (kPa) shear resistance acting on wheel at an arbitrary point of central angle 0 (kPa) angular velocity of front wheel (rad s-~) angular velocity of rear wheel (rad s-~)

INTRODUCTION F o r excavating a sandy soil, and for pulling or pushing other vehicles, a n u m b e r o f wheeled machines are used at m a n y construction sites. The position o f the center o f gravity o f the wheeled vehicle and the position o f the height o f application o f the pulling or pushing force need to be controlled to obtain the largest value o f the m a x i m u m effective tractive effort [1]. In this paper, the traffic performance during driving and braking o f a 5.88 k N weight two-axle four wheel drive vehicle (4WD), rear wheel drive vehicle ( R W D ) , and front wheel drive vehicle ( F W D ) running up and d o w n loose sandy sloped terrain was analyzed theoretically. F o r the given dimensions o f the vehicle and the given terrain-wheel system constants, the relationship between the effective tractive and braking effort o f the vehicle, the a m o u n t o f sinkage o f the front and rear wheel, the total a m o u n t o f sinkage o f the vehicle, and the slip ratio were calculated to find out the o p t i m u m height o f the force o f application and the o p t i m u m eccentricity o f the center o f gravity o f the vehicle for use in c o m p a r i n g the traffic p e r f o r m a n c e o f these vehicles. The m a x i m u m effective tractive effort TDmax and the effective braking effort at s k i d - 2 0 % Ts-2o o f the vehicle were varied with the position o f the center o f gravity o f the

Comparison o f the traffic performance on sloped terrain

39

vehicle, but the effect of the height of application force was found to be insensitive [2]. The optimum eccentricity of the center of gravity to obtain the largest values of TDma~ and Tn-2o can be determined from the slope angle fl = +~r/36 rad of the terrain, in the case of 4WD, 4WB, RWD, RWB, FWD, and FWB. The maximum slope angle of the loose sandy terrain up which the two-axle four wheel vehicle is able to move during driving action and down during braking action was analyzed by simulation, with the four wheel vehicle operating at optimum eccentricity of the center of gravity and optimum height of the application force.

VEHICLE DIMENSIONS AND TERRAIN PROPERTIES The vehicle dimensions are shown in Table 1, and the terrain-wheel system constants at the site of front and rear wheel are shown in Table 2. The average line pressure of the wheel is 0.147 kN cm -l. To determine the terrain-wheel system constants [3] at the front wheel and the rear wheel, the plate loading and unloading test and the plate traction and sinkage tests were conducted for the air-dried loose decomposed granite soil and for the compacted soil after one pass of a roller having the same line pressure. SIMULATION ANALYTICAL METHOD Four wheel drive vehicle The vehicle dimensions and the forces acting on the four wheel drive (or brake) 4WD(B) vehicle moving up or down a loose sandy sloped terrain of angle ¢1during driving or braking action are shown in Figs 1 and 2. The 4WD vehicle has the same driving system as that of rigid 4TD [4]. The vehicle weight W acts vertically on the center of gravity G of the vehicle, the front axle load Wf and the rear axle load Wr act normally to the terrain surface, and the slope resistance Wftan [3 and Wrtan 13act on the front axle Of and on the rear axle Or in parallel to the terrain surface. Table 1. Terrain-whcei system constants Constants for Bekker's J a n o s i - H a n a m o t o ' s and M u r o ' s equation Plate loading and reloading test

Plate traction test

Plate slip sinkage test

kc k~ n kcr k¢¢ nr r Vo m¢ mf a ko no co ct c2

Front wheel

Rear wheel

Unit

85.9 9.85 0.554 180 49.3 0.174

87.3 9.89 0.551 179 49.6 0.174

N cm-(n+ 1) N c m -<"+2)

0.4 1.6 0.035 0 0.434 2.66 2.08 1.04 3.58 0.773 1.446

0 0.434 2.68 2.08 1.04 3.51 0.772 1.443

N c m -(~r+l) N cln -(nr + 2)

eras -1 kPa 1cm- i N crn -~n°+ 2) x 10 -3

40

T. Muro Table 2. Vehicle dimensions o f two-axle four wheel vehicle

Dimension Vehicle weight Radius o f front wheel Radius of rear wheel Width o f front wheel Width of rear wheel Eccentricity o f center of gravity G Height of center of gravity G Distance between central axis of vehicle and point F of acting effective tractive or braking effort T Height o f application o f effective tractive or braking effort T Peripheral speed o f rear wheel Wheel base from front to rear wheel axle Line pressure o f front wheel Line pressure of rear wheel

Symbol

Unit

W Rf Rr Bf Br e hg

5.88 kN 15 cm 15cm 10cm 10cm -0.30q).30 20 cm

L

50 cm

H

0-60 cm

Rtor D

7.07 cm s- 1 50 cm 14.7kNm -I 14.7kNm -]

W/4Bf W/4Br

The position of the center of gravity G of the wheeled vehicle is located according to the amount of eccentricity eD from the central axis of the vehicle and to the height hg perpendicular to the line OfOr. D is the wheel base from the front axle to the rear axle. Rf and Rr are the radius of front and rear wheels, respectively. The simulation program of the tractive (or braking) performance of a driven (or towed) rigid wheel on a soft ground had previously been verified experimentally by use of a rigid wheel of radius of 16 cm, width of 9.5 cm, and with axle load of 1.52 kN on the same loose air-dried sandy soil of initial density of 1.52 M gm -3 [5, 6]. Here, the traffic performance of a four wheel drive (or brake) vehicle was assumed to result from the combination of the effective driving (or braking) force of the front wheel

Front Wheel Width Bt

~-(G¢~) ~ ~ ~ e ~

el ; i ~

Fig. 1. Vehicle dimensions and forces acting on a four wheel drive vehicle (4WD) running on sloped terrain.

Comparison of the tratfic performance on sloped terrain

I

D e.D

41

.

Front Wheel Width Bf Fig. 2. Vehicle dimensions and forces acting on a four wheel brake vehicle (4WB) running on sloped terrain.

and that of the rear wheel during driving (or braking) action. The driving (or braking) torque Qfd(b) acts around the front axle Of and another driving (or braking) torque Qrd(b) acts around the rear axle Or. The effective driving (or braking) force Trd(b) acts on the front axle Of and another effective driving (or braking) force Trd(b) acts on the rear axle Or in parallel to the terrain surface. On the forward contact part of the front wheel, the parallel land locomotion resistance Lfcd(b) t o the terrain surface, the tangential driving (or braking) force Qfd(b)/Rf and the normal ground reaction W f - Qfd(b) sinOfed(b)/Rf act at a distance of the amount of eccentricity efd(b) = Rf sin 0fed(b) and £fd(b) = Rf cos 0fed(b). On the forward contact part of the rear wheel, the parallel land locomotion resistance Lrcd(b) t o the terrain surface, the tangential driving force Qra(b)/Rr, and the normal ground reaction W r - Qrd(b) sinOred(b)/Rr act at a distance of the amount of eccentricity erd(b)= Rrsin0red(b) and ~rd(b) = Rr COS0red(b). The position of the application point F of the effective tractive (or braking) effort TD(B) is located according to the distance L from the central axis of the vehicle and on the application height H from the line OfOr as shown in Figs 1 and 2. The angle of inclination of the vehicle Ot is defined as the angle between the line OfOr and the terrain surface. For the amount of sinkage sf of the front wheel measured at the bottomdead-center Mf and for the amount of sinkage Sr of the rear wheel measured at the bottom-dead-center Mr, the angle of inclination of the vehicle 0t is given as follows: 0t = sin-I Rf

-

uf + sf

O

- -

Rr

(1)

where uf is the amount of recovery of the terrain at the front wheel. The total amount of sinkage s of the vehicle, i.e. the rut depth, can be calculated as S = Sf -- Uf "q- Sr - - Ur

where Ur is the amount of recovery of the terrain at the rear wheel.

(2)

42

T. Muro

At vehicle speed V, the angular velocity (.of and Wr of the front and rear wheels, the slip ratio im (or skid if b) o f the front wheel and the slip ratio ird (or skid irb) o f the rear wheel are expressed as follows: V

R f (.of

ifd = 1 -- Rfw---f'

ieo -- T

V

1

(3)

Rrogr

ird ----- 1 --Rrco----~'

irb =

V

--

(4)

I

F r o m the force equilibrium in parallel and normal directions to the terrain surface, ,~ Trd(b) = ~Qfd(b) co n~ ~red(b) -- Lr~tb) -- Wf tan fl

Trd(b) ----Qrd(b) COS 0rcd(b) -- Lred(b) -- Wr tan Rr

(5)

fl

TD(B) : Tfd(b) -F- Trd(b)

=

Qfd(b)

Rf COS0fcd(b)+

(6) Qrd(b) COS 0red(b)

Rr

(7)

--(Lfcd(b) + Lrcd(b)) -- W s i n / ~

Wcos/~ = Wf + Wr.

(8)

The equilibrium o f m o m e n t s a r o u n d the rear axle Or can be expressed by equation (9).

WfD cos 0t - Tfd(b)D sin 0t

D

-- WCOS fl{~- -

(eD + hg tan 0t)} cos 0t

O " + HTD(B) cos0t - (L - ~-)TD(B) sm0t + Wsin D

{2 - (eD + hg t a n

fl[hg^ +

(9)

COS Ut

0t) } sin 0t] = 0

TO(n) can be calculated from the summation o f Tfd(b) acting on the front wheel and Trd(b) acting on the rear wheel. During driving action o f the wheel, the a m o u n t o f slippagejd(0) at an arbitrary point o f the circumference at the central angle 0 for a wheel o f radius R and width B at slip ratio id is calculated for the entry angle 0f as follows: jd(0) = R{(0f - 0) - (1 - i d ) ( s i n 0 f -- sin 0)}

(10)

The shear resistance rd(0) acting along the circumferential contact part of the driving wheel can be calculated as rd(0) = {me + mr tr(0)}[1 -where a(0) is the normal contact pressure.

exp{--ajd(O}]

(11)

Comparison of the trafficperformanceon sloped terrain

43

Then, the torque Od is given as follows: 0t

Qd = 2 B R 2 J rd(O)dO

(12)

--Or

During braking action of the wheel, the amount of slippage jb(O) at an arbitrary point of the circumference at the central angle 0 for a wheel of radius R and width B at slip ratio ib is calculated for the entry angle Or as follows: .~(0) = R{(Of - O) - ~ .

1

. (sin0f - sin0)}

(13)

1 -t- /b

The shear resistance rb(0) acting along the circumferential contact part of the braking wheel can be calculated as follows. In the case of cos Of - 1 ~
(14)

with nontraction atjq < A(O) ~
(15)

with reciprocal traction at jb(O)~
(16)

In the case of ib < COSOf - 1: rb(O) = --{mc + mf o(0)}[I -- exp{ajb(O)}]

(17)

where rp and jp are the shear resistance of the soil and the amount of slippage, respectively, at the beginning of the unloading state, and jq is the amount of slippage at the beginning of the reverse reloading state. Then, the torque Qb is given as follows: Of

Qb = 2 B R 2 J rb(O)dO

The compaction resistance

(18)

Lfed(b) of the front and rear wheels can be calculated as follows: sf

(19) 0

44

T. Muro

The relationship between TD(B) and ia(b)(= i~d(b) = ira(b)), Tra(b), and id(b), and Trd(b) and id(b), the relationship between Qfd(b), and ia(b), and Qrd(b) and id(b), and the relationship between sf, Sr, s, and ia(b), were determined.

Rear wheel drive vehicle Figures 3 and 4 show the vehicle dimensions and forces acting on the RWD(B) vehicle moving up or down loose sandy sloped terrain of angle ft. The traffic performance of the RWD(B) vehicle was assumed to result from the combination of the effective braking force of the front wheel during the pure rolling state and the effective driving (or braking) force of the rear wheel during driving (or braking) action. The driving (or braking) torque Qrd(b) acts around the rear axle Or. The effective braking force Tfb acts on the front axle Or in parallel to the terrain surface, and the effective driving (or braking) force Trd(b) acts on the rear axle Or in parallel to the terrain surface. On the forward contact part of the front wheel, the pure rolling resistance, i.e. the parallel land locomotion resistance Lfcb to the terrain surface and the normal ground reaction Wf act at a distance of the amount of eccentricity efb = Rfsin0reb and erb = Rfcos0feb. On the forward contact part of the rear wheel, the parallel land locomotion resistance to the terrain surface L~caCb), the tangential driving force Qra(b)/Rr, and the normal ground reaction W~- Qrd(b)SinO~ed(b)/Rr act at a distance of the amount of eccentricity erd(b) : Rr sin 0red(b) and £rd(b) = R r c o s 0red(b). At vehicle speed V, the angular velocity wf and tOr of the front and the rear wheel, the skid iro of the front wheel and the slip ratio ird (skid irb) of the rear wheel can be expressed as follows: Rfo)f ifb - - - -

'

-'.J e., Front

Wheel Width

'

V

W--

1

(20)

Ot.

I

e

7b

r"

Bf ~-(Q"¢h'Os/a~,~L RearV~rheelWidth~ °

Fig. 3. Vehicle dimensions and forces acting on a rear wheel drive vehicle (RWD) running on sloped terrain.

Comparison of the traffic performance on sloped terrain

45

TBcosOt

~0<0

~

i

_. _,o,aR,)~O~ i

Front Wheel Width Bf Fig. 4. Vehicle dimensions and forces acting on a rear wheel brake vehicle (RWB) running on sloped terrain.

V ird = 1 -- Rr-----~'

irb --

Rra}r V

1

(21)

F r o m the force equilibrium in parallel and normal directions to the terrain surface, the following relationships can be obtained: (22)

Tfb : --Lfcb -- Wf tan/~

Trd(b) = Qrd(b) COSOred(b) -- Lred(b) -- Wr tan Rr

TD(B)=

Tfb"b

(23)

Trd(b) (24)

---- QrRib)cos 0red(b)- L r e d ( b ) - L f c b - W s i n / ~

Wcos/3 = Wf + Wr

(25)

The equilibrium o f m o m e n t s a r o u n d the rear axle Or can be expressed as: D

WtD cos 0t + LfcbD sin 0t - Wcos/~{~ - (eD + hg tan 0t)} cos 0t D

hg

D

+HD(B) COS0t -- (L -- ~)TD(B) sin 0t + Wsin/3[c--~s 0t + {~-

-(eD + hg tan Or)} sin 0t]

= 0

(26)

46

T. Muro

The effective tractive (or braking) effort TD(s) can be calculated from the summation of the land locomotion resistance Lfcb, the effective driving (or braking) force Trd(b), and the slope resistance. For the pure rolling state of the front wheel, the skid should be determined by means of a numerical program which can calculate repeatedly until the braking torque Qb in equation (18) becomes zero. The relationship between TD(B) and ird(b), Tfb and ird(b), and Trd(b ) and ird(b), the relationship between Qrd(b) and ird(b), and the relationship between sf, Sr, s, and ird were determined.

Front wheel drive vehicle Figures 5 and 6 show the vehicle dimensions and forces acting on the FWD(B) vehicle moving up or down a loose sandy sloped terrain of angle/3. The traffic performance of the FWD(B) vehicle is assumed to result from a combination of the effective driving (or braking) force of the front wheel during driving (or braking) action and the effective braking force of the rear wheel during the pure rolling state. The driving (or braking) torque Qrd(b) acts around the front axle Of. The effective driving (or braking) force Trd(b) acts on the front axle Of in parallel to the terrain surface, and the effective braking force Trb acts on the rear axle Or in parallel to the terrain surface. On the forward contact part of the front wheel, the parallel land locomotion resistance Lfcd(b) , the tangential driving (or braking) force Qrd(b)/Rf, and the normal ground reaction Wf--Qfd(b) sinOfed(b)/Rf act at a distance of the amount of eccentricity efd(b) ---- Rf sin 0fed(b) and £fd(b) = Rf cos 0fed(b). On the forward contact part of the rear wheel, the pure rolling resistance, i.e. the parallel land locomotion resistance Lrcb to the terrain surface, and the normal ground reaction Wr act at a distance of the amount of eccentricity erb -----Rf sin 0=b and £rb ~---Rf COS Oreb. At vehicle speed V, the angular velocity ogf and O~rof the front and rear wheels, the slip ratio ifd (or skid ifb) of the front wheel, and the skid irb of the rear wheel can be expressed as

£

' Front Wlleel Width Bf

i

U

_,..7 '' / _/

W

"

~

p>u

~/e~, RearWheel Width B~ Fig. 5. Vehicledimensionsand forcesactingon a front wheeldrivevehicle(FWD)runningon slopedterrain.

Comparison of the traffic performance on sloped terrain

47

,,<0 Front Wheel Width Bf Fig. 6. Vehicle dimensions and forces acting on a front wheel brake vehicle (FWB) running on sloped terrain. V

ifd = 1

Rfmf

Rfcof'

ifb-- ~ Rrwr

irb =

V

-

1

1

(27)

(28)

From the force equilibrium in parallel and normal directions to the terrain surface, Qfd(b) Tfd(b) = - ~ f COS0fed(b) -- Lfcd(b) -- Wf tan/~

(29)

Trb = --Lrcb -- Wr t a n

(30)

TD(B) = Tfd(b) + Trb

(31)

--- QRIb) COS0fed(b)- Lfcd(b)- Lr~a - Wsinfl Wcos ~ = Wr + Wr.

(32)

From the equilibrium of moments around the rear axle Or, an equation the same as equation (9) was obtained. The effective tractive (or braking) effort Tt~(a) can be calculated from the summation of the effective driving (or braking) force Tfo(b) acting on the front wheel and the land locomotion resistance L~cb and the slope resistance. For the pure rolling state of the rear wheel, the skid should be determined by means of a numerical program that can calculate repeatedly until the braking torque Qb in equation (18) becomes zero.

48

T. Muro

The relationship between TD(B) and ifd(b), Trd(b) and ifd(b), and Trb and ird(b), the relationship between Qfd(b) and ifd(b), and the relationship between st, st, s and ifd(b) were determined.

N U M E R I C A L RESULTS

During driving action The effects of the position of the center of gravity of the vehicle eD on the effective tractive effort TD were analyzed with the two-axle 4WD, RWD, and F W D vehicle operating on loose sandy sloped terrain of an angle fl = zr/36 rad to pull another construction machine. The optimum amount of eccentricity eopt of the center of gravity to obtain the largest value of the maximum effective tractive effort should be determined within the range of the stability of the vehicle operation, i.e. - 1/6 ~
1.0 T T ~ ,

4WD

(kN)

RWD ~ FWD ~

....

~ =rd36rad H=35 cm

°f

-0.3 I..

-0.1 I I 0.3 /"

e

/ - 1/6

- ! .0

1/6

Fig. 7. Relationship between maximum effectivetractive effort Tomax of 4WD, RWD, and FWD and eccentricity of center of gravity e.

Comparisonof the trafficperformanceon slopedterrain

49

up another vehicle, the RWD vehicle can hardly run up the slope, and the FWD vehicle cannot climb the slope. Figure 9 shows the relationship between the amount of sinkage of the front wheel sf of the 4WD, RWD, and FWD vehicle and the slip ratio i. The amount of sinkage of the front driving wheel of FWD and 4WD increase with slip ratio, depending on the increasing axle load of the front wheel due to the optimum eccentricity. On the other hand, st of RWD does not increase with i because the front wheel is in the pure rolling state. Figure 10 shows the relationship between the amount of sinkage of the rear wheel Sr of the 4WD, RWD, and FWD vehicle and the slip ratio i. The amount of sinkage of the rear driving wheel of RWD and 4WD increases with slip ratio. Sr for RWD depends also on the increasing axle load of the rear wheel due to the optimum eccentricity. Sr for FWD does not change with i because the rear wheel is in the pure rolling state. Figure 11 shows the relationship between the total amount of sinkage of the vehicle s of the 4WD, RWD, and FWD vehicle and the slip ratio i. The total amounts of sinkage of the 4WD, RWD, and FWD vehicle increase with slip ratio up to 20%, but further increase in i results in significant increase in s. They take almost the same value due to the same vehicle weight.

During braking action The effects of the position of the center of gravity of the vehicle eD on the effective braking effort Ts were analyzed when the two-axle 4WB, RWB, and FWB vehicles are operating on the loose sandy sloped terrain at an angle ~ = -rr/36 rad, pushed by another construction machine. The optimum amount of eccentricity eopt of the center of gravity to obtain the largest value of the maximum absolute value of the effective braking effort should be determined within the range of the stability of the vehicle operation, i.e. -l/6~
TD

(kN) 30

40

-1

-2

-3 Fig. 8. Relationshipbetweeneffectivetractiveeffort TDof 4WD, RWD, and FWD, and slip ratio i.

50

T. Muro i 0

(%)

10

20

30

40

I

I

I

I

]

3

sf (cm)

4

5

- ----.....

4WD RWD FWD

,

~

13=rd36 rad H=35cm

",

e = eopt

Fig. 9. Relationship between amount of sinkage o f front wheel sr o f 4WD, RWD, and FWD and slip ratio i.

i 0

(%)

10

2O

30

40

I

I

1

I

3 4 Sr

(cm) 5

13=~/36rad H=35cm e = eopt

\

Fig. 10. Relationship between amount o f sinkage of rear wheel Sr o f 4WD, RWD, and F W D and slip ratio i.

i (%) 0

10

20

30

40

I

I

I

I

l

-

2

"~---~ ~

----.....

s 4 (¢m)

1~=~36ra d H = 35 cm

5

-

4WD RWD FWD

e = eopt

Fig. 11. Relationship between total amount o f sinkage of vehicle s of 4WD, RWD, and F W D and slip ratio i

Comparison of the traffic performance on sloped terrain

51

Figure 12 shows the relationship between the effective braking effort at skid-20%, Ts-20 of the 4WB, RWB, and FWB vehicle and the eccentricity of the center of gravity e of each vehicle. These simulation results are obtained for the slope angle/3 = -7r/36 rad and the application height H = 3 5 c m . The results clarify that eopt = - 1 / 6 for 4WB, eopt = 1/6 for RWB, and eopt = -1/6 for FWB. This means that the position of the center of gravity should be set backward for RWB and forward for 4WB and FWB to obtain the maximum effective braking effort. Simulation results were also obtained for the slope angle /3 = - ~ r / 3 6 rad and the application height H = 35 cm and for the optimum eccentricity of the center of gravity of the vehicle eopt= -1/6 for 4WB, eopt = 1/6 for RWB, and eopt = - 1 / 6 for FWB. Figure 13 shows the relationship between the effective braking effort TB of the 4WB, RWB, and FWB vehicle and the skid i. In this case, the RWB vehicle develops less effective braking effort than that of 4WB or FWB because the front axle load is larger than the rear one on the terrain sloped at/~ = -~r/36 rad. Figure 14 shows the relationship between the amount of sinkage of the front wheel sf of the 4WB, RWB, and FWB vehicle and the skid i. The amount of sinkage of the front wheel of 4WB is almost the same as that of FWB. But the amount of sinkage of the front wheel of RWB is smaller than that of 4WB or FWB because the front axle load of RWB is less than the rear axle load and also the front wheel of RWB is in the pure rolling state. It also showed that slip ratio i did not affect the amount of sinkage sf for RWB. Figure 15 shows the relationship between the amount of sinkage of the rear wheel Sr of the 4WB, RWB, and FWB vehicle and the skid i. The amount of sinkage of the rear wheel of 4WB is almost the same as that of FWB. But the amount of sinkage of the rear wheel of RWB becomes larger than that of 4WB or FWB because the rear wheel of RWB is in the braking state while the front axle load is larger than the rear one. Figure 16 shows the relationship between the total amount of sinkage of the vehicle s of the 4WB, RWB, and FWB vehicle and the skid i. The total amounts of sinkage for 4WB and FWB increased with skid when the skid decreased beyond -20%, while that of RWB did not vary with skid. e

-0.3 I

-0.1 I

0

0.1

I

0.3

L

I

I

[~= -rd36rad /-/= 35 cm

-1 ~4WB ~ - ~ RWB .....

FWB

! /

,

-2 /

f

t:

-1)6

• -3

1/6

Fig. 12. Relationship between effective braking effort Ta-20 at s k i d - 2 0 % of 4WB, RWB, and F W B and eccentricity o f center o f gravity, e.

52

T. Muro 13= -n/36 rad H=35cm e = eopt -40 !

-30 I

-20 i

- 10 )

(%)

./ -1

-2 To (kN)

FWB /

-3 Fig. 13. Relationship between effectivebraking effort TB of 4WB, RWB, and FWB and skid i. SLOPE A N G L E

During driving action The m a x i m u m effective tractive effort of the 4 W D vehicle was found to be larger than that of the R W D or F W D vehicle, when a two-axle four wheel vehicle was moving up loose sandy sloped terrain. The m a x i m u m slope angle of the terrain on which the 4WD, R W D , or F W D vehicle is able to move up the loose sandy sloped terrain was analyzed when the vehicle was moving up the sloped terrain at each optimum eccentricity of the center of gravity eopt = - 0 . 1 0 , 1/6, and - 1 / 6 , respectively, and at the application height H = 35 cm. Figure 17 shows the relationship between the m a x i m u m effective tractive effort TDmax and the slope angle 13 of the terrain for the 4WD, R W D , and F W D vehicle. The m a x i m u m -40

i (%)

-30

-20

-10

I

I

I

I

0

1

2 f

~

~= -~36 tad H=35cm e = eo~

4WB - - - - - RWB . . . . . FWB

3 4 Sf

5

(cm)

Fig. 14. Relationship between amount of sinkage of front wheel sf of 4WB, RWB, and FWB and skid i.

Comparison of the traffic performance on sloped terrain

-40

i (%) -30

-20

-10

53

0 1 2

4WB ~ - ~

RWB

.....

FWB

3 4

I~= -r,./36 rad H=35 cm

Sr

(cm)

e =eopt

5

Fig. 15. Relationship between amount of sinkage of rear wheel Sr of 4WB, RWB, and FWB and skid i.

effective tractive effort TDmax decreases with the increment of the slope angle ft. In this case, it is clarified that the maximum terrain slope angle ~max for the 4WD, RWD, and FWD vehicle to carry up another construction machine is about 0.067~r, 0.031zr, and 0.017zr rad, respectively.

During braking action The effective braking effort at skid-20% TB-20 for the 4WB vehicle was larger than that for the RWB or FWB vehicle when a two-axle four wheel vehicle is moving down loose sandy sloped terrain. For 4WB, RWB, and FWB, Ta-20 was analyzed by the simulation, with those vehicles moving down the sloped terrain at each optimum eccentricity of the center of gravity eopt = -1/6, 1/6, and -1/6, respectively, and at the application height H = 35 cm. Figure 18 shows the relationship between the effective braking effort at skid - 2 0 % TB-20 and the slope angle fl of the terrain for the 4WB, RWB, and FWB vehicle, respectively. As shown in this figure, the effective braking effort I Ta-20 I decreases generally with the increment of the slope angle I/~ 1. -40

i (%) -30

I

-20

I

-10

I

f f

~ ----..... 15= -a/36rad H= 35 cm e

=

eo~

4WB RWB FWB 4 S

5

(cm)

Fig. 16. Relationship between total amount of sinkage of vehicle s of 4WB, RWB, and FWB and skid i.

54

T. Muro

1 -

Tomax (kN)

13 (rad) ! ~r]36 ~x/18 : ~12 : i : : : iH=35c m ':

0

-1

.....

FWD i

Fig. 17. Relationship between maximum effective tractive effort Toma~and slope angle/~.

-n/12 !

13 (rad) -~J18 -~36 : i

0

-1 - - - - - RWB . . . . . FWB

:

-2 TB-2o

-3 Fig. 18. Relationship between effective braking effort Ta-2o at skid-20% and slope angle ft. It is clarified that [ TB-2O [ for the 4WB, FWB, and R W B vehicle decreases with the slope angle ] ill. CONCLUSIONS F r o m the results obtained in this investigation, the following conclusions can be drawn: 1. The driving (or braking) torque o f a wheel can be calculated by the integration o f the shear resistance developed on the part in contact with the terrain. The shear resistance is given as a function o f n o r m a l stress and a m o u n t o f slippage. D u r i n g driving (or braking) action, the a m o u n t o f slippage increases (or decreases) steadily from zero at the entry angle. 2. During the pure rolling state o f a wheel, the torque becomes zero due to the integration o f positive and negative distributions o f the shear resistance. As the a m o u n t o f slippage distribution has a positive peak value, the shear resistance should be calculated as the c o m b i n a t i o n o f the three states o f traction, untraction, and reciorocal reverse traction.

Comparison of the trafficperformanceon sloped terrain

55

3. For the slope angle fl = 7r/36 rad and the application height H = 35 cm, the optimum eccentricity eopt to obtain the largest value of the maximum effective tractive effort is found to be - 0 . 1 0 for 4WD, eopt = 1/6 for RWD, and eopt = - 1 / 6 for FWD. 4. For the slope angle fl = -7r/36 rad and the application height H = 35 cm, the optimum eccentricity e o p t tO obtain the largest value of the effective braking effort at s k i d - 2 0 % is found to be - 1 / 6 for 4WB, eopt ----- 1/6 for RWB, and eopt : - 1 / 6 for FWB vehicle. 5. During driving action, the maximum effective tractive effort of the 4WD vehicle is larger than that of the R W D vehicle, followed by the F W D vehicle. The maximum effective tractive effort of the two-axle wheeled vehicle varies with the position of the center of gravity of the vehicle and the slope angle of the terrain. 6. During braking action, the effective braking effort at skid - 2 0 % of the 4WB vehicle is larger than that of the FWB vehicle, in turn larger than that of the RWB vehicle. The maximum effective braking effort at skid - 2 0 % of the two-axle wheeled vehicle varies with the position of the center of gravity of the vehicle and the slope angle of the terrain. 7. During driving action, the total amount of sinkage of the 4WD, RWD, and F W D vehicle increases with slip ratio and the values are almost the same due to the same vehicle weight. 8. During braking action, the total amount of sinkage of the 4WB and FWB vehicle increase with skid, while that of the RWB does not vary with skid. 9. The maximum slope angle/5max of the terrain on which the two-axle wheeled vehicle is able to move up the terrain during driving action is about 0.067~r rad for the 4WD vehicle, 0.03Dr tad for the R W D vehicle, and 0.0177r rad for the F W D vehicle. 10.The effective braking effort at s k i d - 2 0 % [ Ta-20 I of 4WB, FWB, and RWB decrease with the slope angle I/5 I. REFERENCES 1. Muro, T., Tractive performanceof a four-wheeldrive vehiclemovingup a sloped weak terrain. Proceedings of the 6th European ISTVS Conference,Vienna, 1994,pp. 806-822. 2. Muro, T. and Fukagawa, R., Application height control system of a wheeled vehicle running on a loose sandy soil. Proceedingsof the 1lth International Symposiumon Automation and Robotics in Construction, ISARC, Brighton, 1994,pp. 495-502. 3. Muro, T, Arai, K., Fukagawa, R. and Tateyama, K., Recent Theory of Construction Engineering- Robot# zation and Systematization. Asakura Press, 1994, pp. 84-117. 4. Watanabe, K., Kitano, M. and Fujishima, A., Handling and stability evaluation of four-track steering vehicles considering traction force distribution control. Proceedings of the 5th North American ISTVS/ Workshop, Saskatoon, Canada, 1995,pp. 413-422. 5. Muro, T., Tractive performanceof a driven rigid wheel on soft ground based on the analysis of soil-wheel interaction. Journal of Terramechanics, 1993,30(5), 351-369. 6. Muro, T., Braking performancesof a towed rigid wheel on a soft ground based on the analysis of soil-compaction. Soils and Foundations, 1993, 33(2), 91-104.