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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 06, June 2019, pp. 11-18, Article ID: IJMET_10_06_002
Available online at
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication
THE INFLUENCE OF DRIVING AXLE
LOCATION ON THE LATERAL FORCE OF
VEHICLE
Nguyen Khac Tuan
Thai Nguyen University of Technology, Thai Nguyen city, Vietnam
ABSTRACT
The lateral force acting on the tires produces a side slip angl

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le that affects the
directional stability of vehicle. This paper presents some research results of the
influence of driving axle location on the lateral force of vehicle with 4x2 wheel
formula when vehicle is turning. The single track dynamical models of FWD and RWD
vehicle while cornering are created. Based on these two dynamical models, the system
equations of motion built are enable to study the influence of driving axle location on
the lateral force. Some calculated simulation results are shown for illustration.
Key words: lateral force, driving axle, turning, transmission system
Cite this Article: Nguyen Khac Tuan, The Influence of Driving Axle Location on the
Lateral Force of Vehicle. International Journal of Mechanical Engineering and
Technology 10(6), 2019, pp. 11-18.
1. INTRODUCTION
The transmission system plays an important role in the overall performance of vehicle.
Especially, it has a significant influence on the dynamical characteristic, fuel consumption
and stability of the vehicle [14,16]. So far, there have been a number of publications related to
transmission system [13-16]. However, these works mainly focus on studying dynamic loads
appearing in transmission system [13,15,16] or investigating characteristics of acceleration
[1,2], the fuel consumption of vehicles with different powertrain structures [1,2,6,10].
Literature [1-16] showed very few published works that are concerned with the influence of
transmission layout on the stability of vehicles.
In actual working conditions, the lateral force acting on the tires produces a side slip
angle that affects the directional stability of vehicle [1,2,6,9,12]. Lateral forces can be caused
by many factors such as the impact of side wind, uphill, cornering... The main purpose of this
study is to investigate the influence of the location of driving axle on the lateral force acting
on the wheels of the vehicle with 4x2 wheel formula while cornering [4,5,9,11].
Nguyen Khac Tuan
12 editor@iaeme.com
2. MATERIAL AND METHOD
2.1. Dynamical model of vehicle while cornering
Consider the single track models shows in Figure 1. The parameters used in the vehicle’s
model are defined as below: θ – the steer angle of the front axle;
z
- the angular speed of the
vehicle about the vertical axis ; Fx, Fy: the external forces in the x and y diretions; VA, VD: the
velocity of the front and rear tires, respectively; Vx, Vy: the speed of vehicle in the x and y
diretions; Rx1i, Rx1i: the forces resulting from braking or tractive effort and/ or rolling
resistance, acting parallel to the plane of the wheel to the front and rear wheels, respectively;
Ry1i, Ry2i, : the lateral forces, acting to the front and rear wheels respectively; (i=r or f ), f, r:
the index for front wheel drive (FWD) and rear wheel drive (RWD), respectively; a,b: the
lengths from the mass center to the front and rear axles, respectively.
Rear wheel drive Front wheel drive
Figure 1. Single track models of vehicle during cornering
The equations of motion are writen by summing forces in the x and y directions and
summing moments about a vertical axis through the mass center. For the system shown in
figure, 1a can be written as:
2 1 1
1 2 1
1 1 2
cos sin
cos sin
cos sin
jx x r x x r y r
jy y r y r x r y
jz y r x r y r z
F R F R R
F R R R F
M R R a R b M
(1)
where Rxr1 is in the opposite direction of velocity VA and has the form as:
Rx1r=f.Z1
For the system shown in figure 1b, the equations of motion can be written as:
1 1 2
1 2 1
1 1 2. .
cos sin
cos sin
cos sin
jx x f y f x f
jy y f y f x f y
jz y f x f y f z
F R F R Rx
F R R R F
M R R a R b M
(2)
a
b
A
C
D
vD
vA
vy
Rx2r
vx
Rx1r
Ry1r
Ry2r
R
vC
Fjx
Fjy
O
Mz
Mjz
y
x
a
b
A
C
D
vD
vA
vy
Rx2f
vx
Rx1f
Ry1f
Ry2f
R
vc
O
Fjx
jy
Mz
Mjz
y
x
F
The Influence of Driving Axle Location on the Lateral Force of Vehicle
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In which, Rxr2 is in the opposite direction of velocity VD and has the form as:
Rx2f=f.Z2
In the equation (1) and (2): Fjx, Fjy are the centrifugal forces in the x and y diretions; Mz is
the moment of turning resistance; Mjz is the yaw moment. These can be expressed as:
jx x xP m j (3)
jy y yP m j (4)
2.jz z z z zM J m (5)
where: Jz is the moment of inertia of the vehicle about the vertical axis; m is the mass of
the vehicle.
Solving equation (1) can determine the forces Ry1r, Ry2r and Rx2r acting on the tires of
RWD vehicle:
1 1
2
1
2
tan
cos
tan
cos
jy y z jz
y r
jy y z jz
y r
jy y z jz
x r x jx
F F b M M
R fZ
L
F F a M M
R
L
F F b M MfZ
R F F
L
(6)
Following a similar approach, from quation (2), the forces Ry1f, Ry2f and Rx1f can be
calculated as:
1 2
2
1 2
)
( )
(
cos . sin
cos sin
jy y z jz
y f x jx
jy y z jz
y f
jy y z jz
x f jx x
M
F
F M
F M
F
F F b M
R F f Z
L
a F a M
R
L
F b M
R F fZ
L
(7)
2.2. Forces and moments acting on the vehicle while cornering
External forces in the x direction. The external force Fx in the direction x is [1,10]:
Fx=Fi + Fw =m.sin + K.F.v
2
x0 (8)
where: Fi is the climbing resistance; Fw is the aerodynamic drag force; is the the angle
of uphill gradient; Cd denotes the aerodynamic drag coefficient, ρ is the air density and A is
the cross sectional area.
Turning resistance moment Mz. During cornering of vehicle, the turning resistance
moment Mz is generated by the difference of torque at the left and right of driving wheels.
This value depends on the locking coefficient of the differential between the wheels, which is
determined by the equation [9]:
bt bn bt bn
d
bt bn k
M M M M
k
M M M
(9)
Mk – total torque, supplied to both wheels of driving axle.
Moment Mz can be determined as:
Nguyen Khac Tuan
14 editor@iaeme.com
. .bt bn k d
z
d d
B M M B M k
M
r r
(10)
Centrifugal forces and yaw moment. In order to determine the centifugutial forces Pjx, Pjy
and yaw moment Mjz we have to determine normal and tangential accelerations jx, jy and εz.
Figure 2. Scheme to determine the
acceleration of the vehicle
Figure 3. Simulink model for calculating accelerations jx and jy
Figure 2 shows the scheme for determining the acceleration of the vehicle jx, jy. We have:
( ) ( )
y
y N DO T CD x Z
dV
j j j V
dt
(11)
( ) ( )
x
x T DO N CD y Z
dV
j j j V
dt
(12)
where: JN(DO), JT(DO): normal and tangential acceleration of point D relative to point O,
respectively; JN(CD), JT(CD): normal and tangential acceleration of point C relative to point D,
respectively.
Angular speed is expressed as [6,10]:
1 2xx
z
VV
R L
(13)
Where: δ1, δ2 slip angles of the front- and rear wheels; Rδ - turning radius.
The relationship between velocities Vx and Vy has the form:
tany xV V (14)
ψ: the body slip angle
After a few transformations we get:
1 2x
y
V b a
V
L
(15)
1 2 1 2x x
y
dV
b a V b adV dt
dt L
(16)
Substituting equations (15), (16) into equations (11), (12) và (13) and then substituting
into (3), (4) and (5), yields:
O
CD j
T(DO)
j
N(CD)
j
T(CD)
j
N(DO)
The Influence of Driving Axle Location on the Lateral Force of Vehicle
15 editor@iaeme.com
2 1 2
. .
xx
jx x
V b adV
P m j m
dt R L
(17)
2 1 2 1 2
. .
x
x
x
jy y
dV
b a V b a
V dtP m j m
R L
(18)
1 2 1 2
2 2. . . . .
x
x
jz z z z z z
dV
V
dtM J m m
L
(19)
After detemining Fx, Fy, Fjx, Fjy and subsituting their values into equations (6) and (7) the values of
lateral forces Ry1f, Ry2f, and Ry1r, Ry2r of FWD and RWD vehicles with the aid of Matlab-simulink
software are obtained
3. RESULTS AND DISCUSSIONS
For illustration we show the results of calculating the influence of driving axle location on
lateral forces while vehicle cornering with the aid of Matlab-Simulink 2018 software.
Parameters for the model are chosen as: m=1290 (kg); a=1.19(m); b=1.37(m); B=1.5(m);
jz=1752 (kg/m
2
);kd=0.05;f=0.02.
The results of calculation of lateral forces generated in the front axle (Ry1f, Ry1r) and rear
axle (Ry2f, Ry2r) of FWD and RWD vehicles while cornering are presented in Figures 4 – 8.
Figure 4. The lateral force vs velocity for FWD and RWD vehicle with 4x2 wheel formula
From the graph in Figure 4 it can be seen that the lateral forces in the rear wheels are
relatively similar for both RWD and FWD layouts (Ry2f= Ry2r). However, the lateral force
generated in the front wheel is markedly different in two cases, in which the lateral force
acting on front wheel of FWD is smaller than that in RWD vehicle (Ry1f<Ry1r).
The analysis of results in Figures 5 and 6 also shows that the values of the lateral forces in
both layouts of driving axle depend strongly on the two factors: velocity and the steering
angle of front wheel. In the same cornering condition of vehicle, the greater the automobile
velocity grows, the greater the lateral force is generated. Also, the greater the steering angle θ,
the greater the lateral force acting on the wheels. This can be explained as follows: the lateral
forces value generated while vehicle cornering is strongly depend on the centrifugal forces
Fjx, Fjy and moment Mjz, especially the centrifugal forces Fjy in y direction; according to
equations (18) and (19), these forces and moment are directly proportional to the value of
Nguyen Khac Tuan
16 editor@iaeme.com
velocity Vx and rotation angle θ. Therefore, when the speed Vx or the steering angle θ of front
wheel of cornering vehicle increases, the value of lateral forces acting on the wheels grows.
Figure 5. The lateral force acting on rear wheels of RWD vs velocity with different of steering angles
Figure 6. The lateral force acting on front wheels of FWD vs steering angle with different of
velocities
Figure 7. The lateral forces vs velocity of RWD
vehicle with θ=0,2 rad
Figure 8.The lateral forces vs velocity of FWD
vehicle with θ=0,6 rad
In addition, when comparing the lateral force acting on the wheels of both case front
wheel drive and rear wheel drive vehicles (Figure 7 and 8), it is shown that the lateral force
acting on the front wheel is always greater than the lateral forces acting on the rear wheel.
The Influence of Driving Axle Location on the Lateral Force of Vehicle
17 editor@iaeme.com
This result is due to, beside the effect of centrifugal forces and yaw moment. Since the front
wheel also has functions of a steering wheel, it bears an additional lateral force generated by
steering angle θ.
4. CONCLUSIONS
This study presented a model for calculating the lateral forces acting at the wheels of FWD
and RWD vehicles with 4x2 wheel formula when cornering. The main conclusions are:
(i) The lateral force acting on the wheel depends on many factors, of which the most
important factors are the velocity of vehicle and the steering angle of the steering wheel;
(ii) For both FWD and RWD vehicles, the lateral forces acting on front wheel are always of
greater than those of rear wheel;
(iii) The value of lateral force acting on front wheel of FWD vehicle is smaller than the RWD
vehicle with 4x2 wheel formula. In other words, in the same cornering condition, FWD
vehicle has better directional stability than RWD vehicle.
ACKNOWLEDGEMENTS
The work described in this paper was supported by Thai Nguyen University of Technology
for a scientific project.
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18 editor@iaeme.com
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