World Journal of Mechanics, 2013, 3, 122-138
doi:10.4236/wjm.2013.32010 Published Online April 2013 (http://www.scirp.org/journal/wjm)
Development of Analytical Model for Modular Tank
Vehicle Carrying Liquid Cargo
Messaoud Toumi1, Mohamed Bouazara1*, Marc J. Richard2
1
Department of Applied Sciences, University of Quebec at Chicoutimi, Saguenay, Canada
2
Department of Mechanical Engineering, Laval University, Quebec, Canada
Email: *mbouazar@uqac.ca
Received February 27, 2013; revised April 8, 2013; accepted April 15, 2013
Copyright © 2013 Messaoud Toumi et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The study of dynamics of tank vehicles carrying liquid fuel cargo is complex. The forces and moments due to liquid
sloshing create serious problems related to the instability of tank vehicles. In this paper, a complete analytical model of
a modular tank vehicle has been developed. The model included all the vehicle systems and subsystems. Simulation
results obtained using this model was compared with those obtained using the popular TruckSim software. The comparison proved the validity of the assumptions used in the analytical model and showed a good correlation under single
or double lane change and turning manoeuvers.
Keywords: Analytical Model; Tank Vehicle; Stability; Dynamic Behavior; Suspension
1. Introduction
In general, numerical models are developed to understand the liquid sloshing phenomenon coupled with tank
structure. They are able to determine the coupling behavior, only under specific conditions, such as periodic
accelerations. The effects of suspension system, tire and
road excitation on a moving vehicle have not been taken
into consideration. Regarding the vehicle itself, different
simple models for tractors and trailers have been described in literatures to study the dynamic behavior of
heavy vehicles during various maneuvers. Ellis [1] developed a simple model for tractor-trailer type bicycle
with four degrees of freedom where the load transfer was
modeled using an additional degree of freedom (rolling
motion). Hyun [2] adopted a model for vehicle with four
degrees of freedom for the active control of roll-over of
heavy vehicles. While various solid-liquid models have
been developed to determine the dynamic behavior of
vehicles carrying liquids, few models have been developed to reflect the effects of vehicle systems and subsystems, such as suspension and tire components. The models adopted for the vehicle systems are all based on simplified assumptions.
It is necessary to develop a comprehensive model because a vehicle is composed of various subsystems and
the effects of those need to be considered. AutoSim
*
Corresponding author.
Copyright © 2013 SciRes.
package, one of most popular software for modeling of
the behavior of a vehicle, was developed at the University of Michigan [3,4]. Three software applications were
created based on the AutoSim package [5]. These software applications are CarSim, TruckSim and BikeSim
for cars, heavy vehicles and motorcycles respectively.
However, the TruckSim software does not include the
effects of motion of a moving load [6-8]. They are easy
to use for conventional vehicles only. However, they
offer some models for unconventional designs and the
models find applications in some specific research projects. Another drawback with these tools is that they
work in a closed environment. Therefore the present
work focussed on development of custom made models.
2. Vehicle Kinematic
To develop the model of the vehicle, there are several
methods that could be exploited to derive the equations
of motion such as Lagrange, Newton and virtual work
methods. The popular alternative approach for dynamic
modelling of vehicles is to use of simple models having a
reasonable excution time. In this study, a new model was
developed based on the simplified Ervin model [4]. This
model was solved without any mathematical approximation and it took care of the complexity of liquid motion
inside the tank. The solutions of the equations were obtained using the mathematical software Maple [9]. The
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equations were derived based on the principles of Newtonian mechanics and conservation of linear and angular
momentums for a solid body.
2.1. Coordinate System
The large number of degrees of freedom for translation
and rotational motion, required to represent an articulated
vehicle, excludes the use of a single coordinate system.
In fact, the equations of motion can be written more easily if several coordinate systems are employed. The purpose of this section is to identify the orientation of the
various coordinate systems, and specify the variables
required to connect the processing unit vectors in the
various systems. The inertial coordinate system, the body
coordinate system fixed to the sprung mass and the coordinate system fixed to the unsprung mass were used to
describe the system. Newton’s laws are valid only for a
finite acceleration in an inertial coordinate system
xn , yn , zn . The orientations of coordinate axes were expressed in accordance with the Society of Automotive
Engineers’ standard (SAE), where the positive x axis
points anterior, the positive y axis is oriented to the right
and the positive z axis points downward. In our model,
each sprung mass was represented as a rigid body with
six degrees of freedom namely, longitudinal, lateral, vertical, roll, pitch and yaw. For the unsprung mass, there
were assigned two degrees of freedom namely, the roll
and vertical motions relative to the point of attachment of
the sprung mass. The equations were formulated such
that there was no limit to the number of sprung and unsprung masses. All the equations were solved, without
any mathematical simplification, using the symbolic
computational software Maple [9].
Three coordinate systems were used to develop the
equations of motion. The first one was attached to the
inertial system xn , yn , zn , the second one was attached
to each sprung mass xs , ys , zs and the third one was
attached to each unsprung mass xu , yu , zu . Figure 1
123
ET AL.
shows the coordinate systems for fixed unit and articulated vehicles.
2.1.1. Coordinate System Fixed to the Sprung
Mass
The three rotational motions of the sprung mass were
expressed by the three Euler angles: yaw s (around
axis z ), pitch s (around axis y ) and rolling motion
s (around axis x ) as shown by Figure 2.
The transformation matrix between inertial system and
the system fixed to the sprung mass was defined separately for the three successive rotations: yaw, pitch and
roll.
Yaw s :
in
kn s i1
T
jn
cos s
s sin s
j1 k1
sin s
cos s
0
0
T
0
0
1
(1)
Pitch s :
i1
j1 k1 s i2
T
j2
k2
T
in
j1
s
s
i1
jn
kn , k1
(a)
i2
i1
j1
k1
k2
(b)
i2
i1
j1
k1
k2
(c)
Figure 2. (a) Yaw: w sz s kn ; (b) Pitch: w sy s j1 ; (c) Roll:
Figure 1. Fixed unit and articulated tank vehicles.
Copyright © 2013 SciRes.
w sx s is . Sprung mass orientation defined by euler angles.
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124
Roll
cos s
s 0
sin
s
s :
i2
j2
0 sin s
1
0
0 cos s
k2 s is
T
0
1
s 0 cos s
0 sin
s
js
ks
(2)
T
sin s
cos s
0
(3)
The transformation matrix, connecting the inertial system and the system fixed to the sprung mass, was obtained by combining the three matrices as follows:
in
jn
k n Rsn is
T
js
ks
js
ks Rsn
T
1
in
2.1.3. Coordinate System Fixed to the Unsprung Mass
As mentioned earlier, two motions were assigned to each
unsprung mass, namely, the roll motion and vertical motion relative to the sprung mass. It may be noted that the
pitching motion of the unsprung mass, representing the
axle of vehicle, is infinitely small and can be neglected
[3]. The orientation of the sprung mass relative to the
inertial coordinate system was defined by two rotational
motions namely, yaw motion s and roll motion u
as illustrated in Figure 3.
The transformation matrix between the system fixed to
the unsprung mass and the inertial system can be expressed as:
Yaw s :
in
T
Rsn s s s
where: (please see Equation (5) below)
and:
is
ET AL.
kn
T
(6)
iu
x
y
z Rsn U s Vs Ws
s
C s C s
R S s C s
S s
Copyright © 2013 SciRes.
(10)
ju
ku u i1
T
j1 k1
T
sin u
cos u
0
(11)
s
i1
jn
kn , k1
(a)
(8)
i1 , iu
j1
ju
k1
(9)
n
s
cos s
0
s
j1
sin s sin s
sin s cos s
qs
rs
cos s
cos s
sin s
cos s
qs
rs
cos s
cos s
0
0
1
in
Introducing the transformation matrices between the
two systems, the relationship between the angular velocities can be calculated by the following equations:
s cos s qs sin s rs
T
(7)
ps is qs js rs k s s is s j2 s kn
s ps
j1 k1
sin s
0
1
s 0 cos u
0 sin
u
2.1.2. Linear and Angular Velocities of the Sprung
Mass
The equations of motion of each sprung mass were developed and written for the system fixed to the sprung
mass in terms of linear velocity U s , Vs , Ws and angular velocity ps , qs , rs of the center of mass of the
sprung mass. In order to calculate the velocity and Euler
angles, expressions connecting the linear and angular
velocities for both the systems were developed.
T
T
Roll u :
with index ( C cos , S sin ).
T
kn s i1
cos s
s sin s
0
(4)
jn
jn
ku
(b)
Figure 3. (a) Yaw; (b) Roll u . Unsprung mass orientation
defined by Euler angles.
S s Cs C s S s Ss
C s Cs S s S s Ss
C s Ss
S s Ss C s S s Cs
C s Ss S s S s Cs
C s Cs
(5)
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Therefore, the transformation matrix, that connects the
system fixed to the unsprung mass and the inertial system,
can be obtained by combining the above two transformations ((10) and (11)).
iu
ku u s in
T
ju
jn
kn
T
By introducing the transformation matrices between
the inertial system and the system fixed to the unsprung
mass, the angular velocity was expressed in terms of
Euler angles as follows:
p
(14)
u
On the other hand, the road excitation forces are in
contact with the unsprung mass. These forces are transferred to the sprung mass through the suspension system.
Therefore, the transformation matrix between the two
systems fixed to the sprung and unsprung masses needs
to be calculated.
iu
ju
ku u s s is
T
js
ks
T
Rsu u s s
where: (please see Equation (17) below).
translational velocity vs and an angular velocity ws .
For a given vector q , the following expression [10] was
obtained:
q q ws q
t f
t b
(12)
The angular velocity of the unsprung mass can be defined by the following equation:
wu pu iu rs k s
(13)
u
125
ET AL.
(15)
(16)
(18)
The indices f and b indicate that the derivative was
calculated with respect to the inertial system and system
of the body concerned respectively.
The velocity of point p in the vehicle, relative to inertial system, can be calculated by the following expression:
v p vs rp
t f
(19)
Therefore, substitution of Equation (18) in Equation
(19) gives:
v p vs rp ws rp vs rp ws rp (20)
t b
The acceleration of the point p can be calculated by
differentiating Equation (20) with respect to time:
ap vp
t f
vs rp ws rp ws vs rp ws rp
t b
(21)
2.2. Sprung Mass Kinematics
rp
vs ws vs w s rp ws ws rp 2 ws rp
For the derivation of equations of motion of the vehicle it
is necessary to calculate the expression for the acceleration of an arbitrary point on the vehicle. Figure 4 shows
O f as the coordinate system fixed to the road (inertial)
and Ob as the system of the body coordinate with a
Since the center of mass of the sprung mass coincides
with the origin of the coordinate system attached to the
sprung mass, acceleration of the center of mass of the
sprung mass was obtained by replacing rp 0 in
Equation (21): as vs ws vs .
p
rp
d
dt U qW rV
asx
d
W
a
V
rU
p
sy dt
a
sz
d W pV qU
dt
Vp
ap
b
xb
yb
zb
rb
r
In this study, it was assumed that the load of the liquid,
represented by the center of mass, can move as a material
point and can be represented by a remote vector
T
rL xL , yL , z L from the center of mass of the sprung
T
mass with the same angular velocity ps , qs , rs as that
of the sprung mass of the vehicle as shown in Figure 5.
x
y
f
z
Figure 4. Coordinate systems.
C s
R S s Su
Cu S s
u
s
Copyright © 2013 SciRes.
(22)
S s Ss
Cs Cu Ss Su C s
Su Cs Cu Ss C s
S s Cs
Cu Ss Su Cs C s
Su Ss Cu Cs C s
(17)
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126
Hence, the acceleration of the center of mass of the liquid
can be obtained by replacing the expression rp rL in
Equation (21). Moreover, in this study the interaction
between the vehicle and the liquid was modeled as a
multi-body system using small time step t . As the
ET AL.
coordinates of the vector rL were updated at each time
step, the relative velocity and acceleration relative to the
coordinate system fixed to the sprung mass were neglected.
aL vs ws vs w s rL ws ws rL (23)
d
d
d
dt U dt q z L dt r yL q W pyL qxL r V rxL pz L
aLx
d
d
d
aLy V r xL p zL r U qzL ryL p W pyL qxL
dt
dt
a dt
Lz d
d
d
W p yL q xL p V rxL pz L q U qzL ryL
dt
dt
dt
Equation (25) with respect to time:
2.3. Unsprung Mass Kinematics
The position of the unsprung mass is located in relation
to the point where the sprung mass is attached as shown
in Figures 5 and 6.
(24)
ru f rf rr rru
au as w s rr ws ws rr s
where:
rf : represents the position of center of sprung mass
from the inertial system.
rr xr , 0, zr s : represents the position of the roll
where suffixes s and u indicate systems fixed to
the sprung mass and unsprung mass respectively.
center relative to the system fixed to the sprung mass.
rru 0, 0, zu u : represents the position of the roll
center relative to the system attached to the unsprung
mass. The velocity was calculated by differentiating
Equation (24) with respect to time:
Vu Vs ws rr rr wu rru rru
(25)
Vs ws rr s wu rru rru u
w u rru wu wu rru
rru u
2 wu rru w u rru
ws ps , qs , rs
T
(26)
is the angular velocity of the sprung
mass and wu pu , 0, rs
T
is the angular velocity of the
unsprung mass.
As described in Figure 7, suspension forces transmitted to the sprung mass for each axis can be expressed as
follows:
The acceleration was calculated by differentiating
Fsxi1 Fsxi 2
Fsupi
Fsyi
F F
szi 2 ui
szi1
(27)
Figure 5. Vehicle mathematical model.
Copyright © 2013 SciRes.
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ET AL.
sprung mass Si can be defined using the transformation
matrix that connects the unsprung mass and sprung mass
(Equation (17)).The internal forces can be eliminated
according to the dynamic equations of motion for each
axis i , as illustrated in Figure 7.
Fsxi1 Fsxi 2
Fsupi Ruisi
Fsyi
(28)
F F
szi 2 ui si
szi1
4
Fysi mui aui jui Fwyi cos ui
j 1
4
Fwzi sin ui mui g sin ui
j 1
Figure 6. Unsprung mass kinematics.
4
Fxsi1 Fxsi 2 mui aui iui Fwxi
(29)
(30)
j 1
where:
Fszi1 and Fszi 2 are the vertical suspension forces at
left and right sides respectively.
Fsyi is the internal lateral force applied to the roll
center of each axis. This force is the result of the lateral
forces applied to the tires.
Fsxi1 and Fsxi 2 are the longitudinal suspension forces
at the right and left sides respectively.
The suspension forces in the system attached to the
2.4. Fifth Wheel Kinematics
The motion of the sprung mass of tractor and trailer are
coupled via the fifth wheel joint. Several studies suggested to consider joint connection as a rigid one in the
case of translational motion. This allows to consider a
joint as a point. With this assumption, the number of degrees of freedom was reduced. Thus we can calculate the
expressions for velocity and acceleration of the trailer
Figure 7. Vehicle model (front view).
Copyright © 2013 SciRes.
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128
depending on the velocity and acceleration of the tractor
[4]. If the harness is not rigid enough, it can be modeled
as an assembly of a spring and a damper in parallel [3].
However, torsional component of the fifth wheel acts in
the case of rolling motion. From Figure 8, velocity and
acceleration of point C were calculated with respect to
the two systems fixed to the sprung masses of tractor and
trailer as follows:
Vc Vs1 Vc / s1 Vs 2 Vc / s 2
ET AL.
ac / s1 w s1 rc / s1 ws1 ws1 rc / s1
ac / s 2 w s 2 rc / s 2 ws 2 ws 2 rc / s 2
(32)
The following relations can be obtained by introducing
the expressions of Equation (32) in Equation (31).
U1 qzc1
Vc Rss12 V1 r1 xc1 p1 zc1
W1 q1 xc1 s1 s 2
U 2 qzc 2
V2 r2 xc 2 p2 zc 2
W q x
2
2 c2
s 2
(31)
ac as1 ac / s1 as 2 ac / s 2
with:
rc / s1 xc1is1 zc1k s1 and rc / s 2 xc 2is 2 zc 2 k s 2
(33)
The transformation matrix Rss12 between the system
attached to the sprung mass of trailer s 2 and the system fixed to the sprung mass of tractor s1 can be calculated through the inertial system as follows:
where:
Vc / s1 ws1 rc / s1
Vc / s 2 ws 2 rc / s 2
Rss12 Rns 2 Rsn1 .
and
ts2
Rts1
Tractor
qs1 , V1
Ys
s1
s2
qs2 , U2
rs2 , W2
Xs2
ps1 , U1
Xs
Trailer
Ys2
qs2 , V2
rs1 , W1
Zs
rc1
rc2
c
Fifth wheel
n
Rts1
Zs2
n
Rts2
Yn
f
Xn
Zn
Figure 8. Fifth wheel kinematic.
d
d
U1 q1 zc1 q1 W1 q1 xc1 r1 V1 r1 xc1 p1 zc1
dt
dt
s2 d
d
d
ac Rs1 V1 r1 xc1 p1 zc1 r1 U1 q1 zc1 p1 W1 q1 xc1
dt
dt
dt
d
d
W1 q1 xc1 p1 V1 r1 xc1 p1 zc1 q1 U1 q1 zc1
dt
dt
s1 s 2
d
d
U 2 q2 zc 2 q2 W2 q2 xc 2 r2 V2 r2 xc 2 p2 zc 2
t
t
d
d
d
d
d
V2 r2 xc 2 p2 zc 2 r2 U 2 q2 zc 2 p2 W2 q2 xc 2
dt
dt
dt
d
d
W2 q2 xc 2 p2 V2 r2 xc 2 p2 zc 2 q2 U 2 q2 zc 2
dt
dt
s 2
Copyright © 2013 SciRes.
(34)
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The simultaneous solution of Equations (33) and (34)
gave the final expressions for the velocity and acceleration of the trailer as a function of the velocity and acceleration of the tractor.
The sweep on the roll angle between the tractor and
trailer was useful to calculate the constraint of the fifth
wheel for the roll motion (roll moment):
M cs1 k xc s1 s 2 is1
M cs 2 Rss12 k xc s s 2 is1
129
ET AL.
case of a unit vehicle. The free body diagram shown in
Figure 7 shows the external and internal forces and moments applied to each subsystem of the vehicle. To obtain the equations of linear and angular motions, it is
important to model the rigid body as a set of material
points.
3.1. Linear Motion
The application of Newton’s laws eventually gives the
equations of linear motion for the tractor and trailer.
(35)
Fi mi ai
3. Vehicle Kinetics
This section is devoted to the definition of variables with
some algebraic manipulations chosen for the equations of
motion. All kinetic parameters were developed for an
articulated vehicle. The same settings were applied in the
msi asi mLi aLi isi
(36)
The equations of translational motion can be obtained
by the combination of the Equations (36), (22), (24), (29),
and (30). These equations were represented by second
order differential equations for sprung mass si:
cos si Fsxj1 Fsxj 2 sin uj sin si Fsyj
k
k
j 1
j 1
cos uj sin si Fszj1 Fszj 2 sin si msi mLi g Constraint forces isi
k
(37)
j 1
msi asi mLi aLi jsi cos uj cos si sin uj cos si sin si Fsyj sin uj cos si Fszj1 Fszj 2
k
k
j 1
j 1
cos uj cos si sin si Fszj1 Fszj 2 sin si sin si Fsxj1 Fsxj 2
k
k
j 1
j 1
(38)
cos si sin si msi mLi g Constraint forces jsi
msi asi mLi aLi ksi cos uj sin si sin uj cos si cos si Fsyj sin uj sin si Fszj1 Fszj 2
k
k
j 1
j 1
cos uj cos si cos si Fszj1 Fszj 2 sin si cos si Fsxj1 Fsxj 2
k
k
j 1
j 1
(39)
cos si cos si msi mLi g Constraint forces ksi
where
i 1 : tractor
i 2 : trailer.
j : axle number.
k : axle number. k 3 for tractor and k 2 for
trailer.
L : liquid.
si : sprung mass i .
In this study the constraint forces due to the fifth
wheel were eliminated by using the kinematic Equations
(33) and (34) developed earlier. It should be noted that
all these equations of motion were programmed in the
Maple software in a systematic way. Therefore, to obtain
the equations of motion in the case of a unit vehicle, only
change of the indices i 1 , k 3 was needed.
The equation of vertical motion of the sprung mass for
each axis i was given by the folowing expression:
Copyright © 2013 SciRes.
mui aui kui
k
k
Fwzij cos ui Fwyij sin ui
j 1
j 1
mui g cos ui Fszi1 Fszi 2
(40)
where:
i : axle number.
j : number of tires in each axle.
k : k 2 fortractor front axle and k 4 for the
other axles.
3.2. Angular Motion
It is important to model the rigid body as a system of
material points p with masses m p to obtain the equation of angular motion. According to Newton’s equation,
angular momentum relative to the inertial system can be
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ET AL.
m p rp vs ws vs ms rs vs ws vs
given by the following expression:
rp
M s M p rp m p
p
(41)
p
The second and third terms of Equation (43) can also
be simplified [10] as:
p
Substituting Equation (21) in (41), the following expression can be obtained:
m p rp w s rp I s w s
p
vs ws vs w s rp
M s m p rp
p
ws ws rp
m p rp ws ws rp ws I s ws
(42)
p
Therefore Equation (42) takes the following form:
(44)
M s ms rs vs ws vs I s w s ws I s ws
M s m p rp vs ws vs
p
m p rp w s rp m p rp ws ws rp
p
(43)
Since the sprung mass center coincides with the origin
of the body axis system rs o , the expression of angular motion (44) can be formulated as follows:
p
The first term of Equation (43) can be simplified as:
d
d
rs qs I ys rs I xzs ps I xs ps qs I xzs rs qs I zs
d
d
t
t
d
2
2
M s I s w s ws I s ws qs I ys ps I xzs rs I xzs ps rs I zs ps rs I xs
dt
d
d
rs I zs ps qs I ys rs qs I xzs ps qs I xs ps I xzs
dt
dt
(45)
The matrix of inertia I si was expressed in the system si as follows:
2
2
y p z p mp
si
m p x p z p m p
si
z x m
p
p p
s
i
I si r 1 r r
2
p
T
p p
si
instead of a sums
si
x 2p z 2p m p
si
z p y p m p
si
x p z p m p
si
y p z p m p
si
2
2
x
y
m
p p p
si
(46)
system fixed to the sprung mass were calculated as follows:
xLi mLi aLi isi
0
1
M Li yLi mLi aLi jsi si si 0 (47)
z m a k
m g
si
Li Li Li
Li
Since the tractor body and the trailer body can be modeled as contained bodies, all mathematical expressions can
be expressed by integrals
x p y p m p
.
The moments applied to the sprung mass due to the
liquid load and the suspension forces expressed in axis
xuj
xuj
Fsxj1 Fsxj 2
0
k
si
si
0
suj Ruj
0 Ruj Fsyj ki si ui
j 1
z
F F j 1 z
szj 2
0
szj1
rj
rj
k
M suspi
Substituting in Equation (45), the terms of the moments due to the liquid charge (47), the moments of the
suspension (48) and the moments due to the fifth wheel
I
xsi
constraints (35), the final equations of angular motion of
the sprung mass (si) can be obtained as follows:
d
d
I xLi ps rs qs I zsi I zLi I ysi I yLi ps qs rs I xzsi I xzLi
dt
dt
yLi cos si cos si z Li cos si sin si mLi g y Li aLi z z Li aLi y
(48)
(49)
k i si uj M csi M supi isi
k
j 1
Copyright © 2013 SciRes.
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ysi
131
ET AL.
d
I yLi qs ps2 rs2 ( I xzsi I xzLi ) M csi M supi jsi ps rs I zsi I zLi I xsi I xLi
dt
z Li sin si xLi cos si cos si mLi g z Li aLi x xLi aLi z
(50)
k
d
d
I zsi I zLi I zuj rs ps qs I ysi I yLi I xsi I xLi rs qs ps I xzsi I xzLi
dt
j 1
dt
yLi sin si xLi cos si sin si mLi g xLi aLi y yLi aLi x
(51)
M Tj M csi M supi k si
k
j 1
k: axle number k = 3 for tractor and k = 2 for trailer.
The equation for rolling motion of the sprung mass of each axle i can be given by the following expression:
k
I xxui pui Fszi1 Fszi1 sui zui Fsyi Fwyij
j 1
Fwzi1 Fszi 3 Ti Fwzi 2 Fszi 4
where:
i : axle index.
j : number of tires in each axle.
k : k 2 for the front axle of the tractor and k 4
for other axles.
3.3. Suspension Model
The external forces acting on the vehicle are generated
mainly due to the contact forces between wheel and
ground. These forces are transmitted to the sprung mass
through the suspension system of the vehicle. To simplify the model, the suspension system was represented
with a linear spring and a damper assembled in parallel.
The vertical force applied on the vehicle through the
suspension system was assumed to be equal to the sum of
the static equilibrium force and the excitation forces.
Fsi K uj euj Cuj euj Fstatic
(53)
h
ri
cos ui zui
T d k
i
i
i
(52)
si ui
the contact between wheel and ground. The forces and
moments transmitted to the vehicle by the tires due to
wheel-ground interaction are complex and nonlinear.
These forces and moments depend primarily on normal
forces, longitudinal and lateral load transfer, slip rate
and slip angles as illustrated in Figure 9.
There are several models available for tires. Most
studies have used linear model or models based on tables
from experimental tests. The forces and moments were
characterized according to vehicle velocity, normal force,
longitudinal slip ratio and slip angle [4,11-14]. These
models usually have a better prediction capability for the
traction force of contact. However, their data are specific
for the type of tire which reduces their universal use.
There are other numerical models which use different
analytical approaches [15-17]. The choice of model of
where euj is the suspension deflections and can be calculated based on the geometry of the vehicle.
euj zs sin uj cos si cos uj cos si sin si suj
cos uj sin si xuj
euj
d
euj
dt
(54)
where:
i 1 : sprung mass of tractor.
i 2 : sprung mass of trailer.
j : axle number ( j 1, 2, 3 for tractor and j 4, 5
for trailer).
3.4. Tire Model
The tire is an essential element in a vehicle. It represents
Copyright © 2013 SciRes.
Figure 9. Forces and moments applied on the tire.
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132
the tire affects the calculation of the efforts at the wheelsoil interface. The data from these models are important
when one wants to make a dynamic model of a vehicle.
In this study the efforts of the tires were studied with
the model called slip circle [11,18]. The model is closely
related to the model of friction ellipse shown in Figure
10 [11].
With this model, it is possible to obtain lateral and
longitudinal forces in the case of combined motions
based only on measured data for separate motions such
as, braking/traction alone or direction case, as illustrated
in Figure 11.
The calculation was based on the evaluation of friction
x and y . The calculation of these coefficients depends on the rate of longitudinal slip and slip angle
. The rate of longitudinal slip and slip angle of the tire
can be calculated by the formula:
rww U p
Vaxe
Up
(b)
rww U p
(56)
rww
tan 1
(a)
(55)
Up
tan 1
ET AL.
Vaxe
Up
where rw is the radius of the wheel and w the velocity of rotation of the wheel. Vaxe is the velocity of lateral
translation of the axis and Up is the longitudinal velocity
of the tire as shown in Figure 12.
The expressions can be evaluated from the velocity of
center of mass of the vehicle.
Vaxej cos si Vsi rsi xuj psi zrj cos uj puj H uj (57)
U pj1 U si T j rsi
U pj 2 U si T j d j 2 rsi
(58a)
(c)
Figure 11. (a) Lateral tire-road contact force; (b) Longitudinal tire-road contact force; (c) Aligning moment generated
at tire-road contact. Experimental forces and moments
generated at tire-road contact for several vertical load
charge [19].
U pj 3 U si T j rsi
Fy max
Fy
Fx
Fx max
Braking
Traction
Figure 10. Friction ellipse concept.
Copyright © 2013 SciRes.
Figure 12. Tire model.
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M. TOUMI
U pj 4 U si T j d j 2 rsi
(58b)
3.4.1. Tire Vertical Load
In this study, the vertical load of the tire was modeled as
a linear spring. Therefore, the vertical force depended on
the spring constant.
(59)
Fwzij Ktij ij
The tire deflections were calculated from the geometry
of the vehicle as follows:
j1 01 zsi zuj0 zuj 1 cos uj
(60)
sin uj T j zuj cos uj
j 3 j1 T j cos uj
(61)
j 4 j1 T j d j cos uj
To calculate the combined forces, a dimensional vector of slip amplitude and direction was defined
[20] as:
2
(62)
sin
The coefficients of friction between the tire and the
ground, in the case of the combined forces, took the
forms below:
, x cos y ) sin
2
x , , cos
2
(63)
y , , sin
Finally, longitudinal and lateral forces in the case of
combined motion were calculated:
Fx , x , Fz
(64)
Fy , y , Fz
3.4.2. Braking Force
The braking force was calculated by taking into account
forces and moments developed in the wheel due to rotation (spin) of the wheel as shown in Figure 13. Acceleration of rotation (spin) of the wheels wi were calculated from the rotational motion of the wheels as follows
[21]:
I wiwi Td Tbi M ri Fxi rw
Td Tediff trans
Copyright © 2013 SciRes.
ww
T -T -Mri
d bi
rw
Figure 13. Wheel dynamics.
4. Vehicle Model Validation
j 2 j1 d j cos uj
tan
Iwi
Fxi
zrj sin si xuj cos si cos si zrj
2 sin
133
ET AL.
(65)
The TruckSim software, developed by the transportation
center of the University of Michigan (UMTRI), is specialized in the simulation of heavy vehicles [15,19,22].
The center also developed software applications: CarSim
for tourist vehicles and BikeSim for motorcycles. TruckSim, the most popular software in this field, was used to
represent and study the dynamics of vehicles in a computer environment. It is possible to analyze a large number of vehicle configurations, since the software has a
library of existing models in the transportation industry.
However, TruckSim can only add a load that is considered to be fixed on the semi-trailer. This feature does not
allow to study the dynamic behavior of liquid sloshing in
tanker trucks. The TruckSim library also has a number of
predefined trajectories and maneuvers that can enable
researchers to validate the behavior of the vehicle model
developed, for difficult maneuvers such as motion in a
curve, change of single and double lanes. In the TruckSim environment, the maneuvers are predefined paths,
i.e., the excitation is represented by a displacement vector. However, for the model developed in this study, the
excitation was defined by the angle of steering or braking
torque as an input parameter. From the output vector, the
response of the steering angle was recorded. This response was the input parameter for our model that considered the same configuration for a unit or articulated
vehicle as defined by the Tables 1 and 2 in the annexure
[22]. Two lane change maneuvers (single and double)
with a constant speed v 70 km h were chosen to compare the two models. Figure 14 shows the trajectory and
the steer angle during the two maneuvers [22].
Figures 15 and 16 represent the comparison between
model simulations for the unit vehicle and the TruckSim
software respectively. The vehicle directional responses
were evaluated using two difficult motions, such as
change of single and double lanes. This comparison was
characterized in terms of roll angle, lateral acceleration
of the center of mass and trajectory traveled from the
center of mass. The simulation showed a good correlaWJM
M. TOUMI
134
ET AL.
Table 1. Geometric parameters of unit vehicle.
Table 2. Geometric parameters of articulated vehicle.
(a)
(a)
Parameters
Symbols
Values
Spring mass (kg)
ms
4457 kg
Roll inertia moment (kg·m2)
I xs
2287 kg·m2
Pitch inertia moment (kg·m2)
I ys
35,408 kg·m2
Yaw inertia moment (kg·m2)
I zs
34,823 kg·m2
Height of mass center of
gravity of spring (m)
zcg
1.173 m
Distance between the center of
mass and the front axis (m)
xu1
1.135 m
Distance between the center of
mass and the rear axis 1 (m)
xu 2
3.252 m
Distance between the center of
mass and the rear axis 2 (m)
xu 3
4.522 m
Tractor
Parameters
Spring mass (kg)
Unspring mass.
Parameters
Axle 1
Axle 2
Axle 3
msu (kg)
527
1007
973
2
Isu (kg·m )
612
579
584
Izu (kg·m2)
612
579
584
K (kg·m2/rad)
432
3389.54
3389.54
2Ti m
2.022
2.06
2.06
2 Si m
0.828
1.029
1.031
di m
0
0.31
0.31
H ri m
0.533
0.686
0.704
s2
Trailer
6308
2800
Roll inertia moment (kg·m )
6879
2400
Pitch inertia moment (kg·m2)
21,711
40,000
19,665
40,000
130
-
1.02
1.7
4.601
-
-
5.5
1.384
-
3.242
-
4.522
-
-
3.9
-
5.2
2
2
Yaw inertia moment (kg·m )
Mixte inertia product
Ixz (kg·m2)
Height of mass (CM) center of
gravity of spring m
CM
m
Distance between CM
the fifth wheel m
Distance between CM
the axle number 1 m
Distance between CM
the axle number 2 m
Distance between CM
the axle number 3 m
Distance between CM
the axle number 4 m
Distance between CM
the axle number 5 m
Distance between
1
and
the fifth wheel
2
(b)
s1
and
1
and
1
and
1
and
1
and
1
and
(b)
Steer angle (deg)
2
Unsprung mass.
1
Parameters
Axle 1
Axle 2
Axle 3
Axle 4
Axle 5
-1
mus kg
527
1007
973
735
735
-2
I xu kg m 2
612
579
584
586
593
I zu kg m
612
579
584
586
593
K kg m deg
1186.3
1581.8
119.8
1468.2
1468.2
2Ti m
2.022
2.06
2.06
2.06
2.06
2 Si m
0.828
1.029
1.031
1.118
1.118
di m
0
0.31
0.31
0.31
0.31
H ri m
0.553
0.686
0.704
0.717
0.676
0
0
1
2
3
4
5
6
7
8
9
Time (s)
Tractor steer angle (deg)
(a)
5
4
3
2
1
0
-1
-2
-3
-4
-5
0
2
4
6
8
Time (s)
(b)
Figure 14. (a) Simple lane change maneuver and the desired
trajectory; (b) Double lane change maneuver and the desired trajectory. Maneuvrer used for the comparaison between the model and TruckSim software.
Copyright © 2013 SciRes.
2
2
tion between the two models. A small difference was
noted for the trajectory of the center of mass. This difference may be due to the steering angle error of the output vector obtained using TruckSim. In addition, excitement used for the TruckSim software was in closed loop
(predefined trajectory).
WJM
M. TOUMI
135
ET AL.
(a)
(a)
10
Trucksim
8
(Model)
Yaw angle (deg)
6
4
2
0
-2
-4
-6
-8
(b)
-10
0
2
4
6
8
10
Time (s)
5
Trucksim
(Model)
(b)
4
Y (m )
3
2
1
0
0
50
100
150
X (m )
200
(c)
7
Trucksim
Yaw angle (deg)
6
(Model)
5
(c)
4
3
2
1
0
-1
-2
0
2
4
6
8
10
Time (s)
(d)
Figure 15. (a)-(d) Single lane change maneuver for a unit
vehicle (Solid: trucksim; Dashed: model).
(d)
In case of articulated vehicles, the analysis was more
complicated. This difficulty was due to addition of the
hinge point where there were additional forces and moments acting between the tractor and trailer. Figures 17
and 18 represent the comparison between the two models
for the same difficult issues such as motions during
change of single and double lanes. A good overall correlation was observed. However, our model underestimated
the response to some extent. This difference in response
was observed due to yaw motion. TruckSim software
Copyright © 2013 SciRes.
Figure 16. (a)-(d) Double lane change maneuver for a unit
vehicle (Solid: trucksim; Dashed: model).
was able to handle such situation in a better way. This
difference can be explained based on the assumption that
the fifth wheel was considered as a rigid body for our
model. However, in TruckSim it was modeled by a
spring-damper combination, which can represent multibody connections between subsystems in a better way.
The increase in the response of yaw motion influenced
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M. TOUMI
136
(Model)
3
2
1
0
0
50
100
150
(Model)
Tractor
Trailer
1
0
-1
-2
200
0
1
2
3
X (m)
4
5
6
7
8
9
Time (s)
(a)
(b)
Trucksim
8
Yaw Angle (deg)
(c)
6
(Model)
Trucksim
6
4
Trailer
Yaw rate (deg/s)
Y (m)
4
Trucksim
2
-2
Trucksim
Lateral acceleration (m.s )
5
ET AL.
Tractor
4
2
0
Tractor
(Model)
Trailer
2
0
-2
-4
-2
-6
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
Time (s)
(d)
5
6
7
8
9
Time (s)
(e)
(f)
Figure 17. (a)-(f) Single lane change maneuver for an articulated vehicle (Solid: trucksim; Dashed: model).
6
Trucksim
5
Trucksim
16
(Model)
12
Yaw rate (deg/s)
4
Y (m)
3
2
1
0
-1
Tractor
(Model)
Trailer
8
4
0
-4
-8
-12
-16
-2
0
50
100
150
200
0
250
1
2
3
4
X (m)
(a)
(Model)
-2
Lateral acceleration (m.s )
Tractor
8
Yaw angle (deg)
(b)
Trucksim
12
Trailer
4
0
-4
-8
-12
0
1
2
3
4
5
5
6
7
8
9 10 11
Time (s)
6
7
8
9
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
(c)
Trucksim
(Model)
Tractor
0
1
2
Trailer
3
Time (s)
(d)
4
5
6
7
8
9
Time (s)
(e)
(f)
Figure 18. (a)-(f) Double lane change maneuver for an articulated vehicle (Solid: trucksim; Dashed: model).
the trajectory of the vehicle as illustrated by Figures 17
and 18. This may be due to the error of steering angle
recorded from the TruckSim output vector. Still, this
difference did not practically affect the very good corre
Copyright © 2013 SciRes.
lation between the two models.
5. Conclusions
A complete nonlinear three-dimensional vehicle model
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M. TOUMI
was developed and validated by Trucksim software. Both
unit vehicle and articulated vehicle combination systems
were considered in this study. The model gave realistic
results in simulation of handling maneuvers near and
beyond the adhesion limits.
The load-transfer for mobile charge due to the liquid
was accurately modeled and integrated into the vehicle
model as a multibody system. The dynamic responses of
tank vehicles were further investigated in view of variations in vehicle maneuvers, fill volume, road condition,
and tank configuration [8].
This research, can help better understanding of this
kind of complex problem. This will make it possible to
answer some queries in the field of safety and the stability of the heavy vehicles, in particular for the tanktruck.
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M. TOUMI
Definitions of Symbols
U si : Longitudinal velocity of sprung mass si ;
Vsi : Lateral velocity of the sprung mass si ;
rsi : Yaw rate of the sprung mass si ;
psi : Roll rate of the sprung mass si ;
d j : Distance between the two tires;
T j : Distance between the center of mass of the axle
j and the center of (one tire or dual tires);
xuj : Longitudinal distance between the center of mass
of the axle j and the center of the sprung mass si ;
zrj : Vertical distance between the roll center of the
axle j and the center of the sprung mass si ;
H uj : Vertical distance between the roll center roll of
the axle j and the road;
01 : Initial deflection due to static deflection;
zsi : Sprung mass vertical displacement si ;
zuj : The vertical distance between the roll center and
center of the axle j ;
Copyright © 2013 SciRes.
ET AL.
zuj0 : Initial vertical distance between roll center and
center of the axle j ;
zrj : The vertical distance between the roll center and
center of mass of the sprung mass si ;
M csi : Moment due to constraints of the fifth wheel;
M supi : Moment due to the suspension forces;
Td : Wheel drive troque;
Te : Engine torque;
Tbi : Wheel braking torque;
Twi : Wheel moment of inertia;
trans : Transmission ratio;
diff : Différentiel ratio;
: Torque function (the fraction of engine torque applied to the specific wheel);
Fxi : Longitudinal tire-road contact force;
M ri : Resistance moment of the tire.
WJM