Development of Analytical Model for Modular Tank Vehicle Carrying Liquid Cargo

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.


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 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][7][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.

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 equations were derived based on the principles of Newtonian mechanics and conservation of linear and angular momentums for a solid body.

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   , , x y z .The orientations of coordinate axes were ex- pressed 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   , , shows the coordinate systems for fixed unit and articulated vehicles.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.

Coordinate System
Yaw s  : 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: The transformation matrix, connecting the inertial system and the system fixed to the sprung mass, was obtained by combining the three matrices as follows: (10) where: (please see Equation ( 5) below) and:

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   , , 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.
Introducing the transformation matrices between the two systems, the relationship between the angular velocities can be calculated by the following equations: sin sin sin cos cos cos Therefore, the transformation matrix, that connects the sy  (12)   The angular velocity of the unsprung mass can fin s (13) By introducing the transformation th stem fixed to the unsprung mass and the inertial system, can be obtained by combining the above two transformations (( 10) and ( 11)).

 
       be deed by the following equation: matrices between e inertial system and the system fixed to the unsprung mass, the angular velocity was expressed in terms of Euler angles as follows: On the other hand, the road e co  (15)   xcitation forces are in ntact 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.

Sprung Mass Kinematics
For the derivation of equations of m is necessary to calculate the expression for the acceleration of an arbitrary point on the vehicle.translational velocity s v and an angular velocity s w .For a given vector q , following expression [10] was obtained: the 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: Therefore, substitution of Equation (18) in Equation ( 19) gives: The acceleration of the point can be calculated di p by fferentiating Equation ( 20) with respect to time: 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   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 ass with the sam p q r as that of the sprung mass of the vehicle as sh ure 5. own in Fig Copyright © 2013 SciRes.WJM Hence, the acceleration of the center of mass of the liquid can be obtained by replacing the expression   p L r r  the interactio odeled as a in Equation (21).Moreover, in this study n ween the vehicle and the liquid was m coordinates of the vector bet multi-body system using small time step t  .As the L r an were updated at each time step, the relative velocity d acceleration relative to the coordinate system fixed to the glected.sprung mass were ne-

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.
Equation ( 25) with respect to time: where: The acceleration was calculated by differentiating where suffixes   s and     T , suspension forces transmitass for each axis can be expressed as follows:  The suspension forces in the system attached to the sprung mass can be defined using the transformation matrix that con ects the unsprung mass and sprung mass (Equation e internal forces can be eliminated according to t e dynamic equations of motion for each axis , as illustrated in Figure 7.
o allows to consider a joint as a point.With this assumption, the number of degrees of freedom was reduced.Thus w expressions for velocity and acceleration of the trailer

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 f translational motion.This e can calculate the 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: The following relations can be obtained by introducing the expressions of Equation (32) in Equation (31).
The transformation matrix 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):

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 a 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.

Linear Motion
The application of Newton's laws eventually gives the equations of linear motion for the tractor and trailer.
The equations of translational motion can be obtained by the combination of the Equations (36), ( 22), ( 24), (29), and (30).These equations ere represented by second ial equations for sprung mass si: where: i : axle number.
: number of tires in each axle.for the other axles.

Angular Motion
It is important to model the rigid body as a system o material points with masses given by the following expression: Substituting Equation ( 21) in (41), the following expression can be obtained: The second and third terms of Equation ( 43) can be simplified [10]  r q I r I p I p q I r q I t t

M I w w I w q I p I r I p r I p r I t r I
p q I r q I p q I p I t t T he matrix of inertia si I was expressed in th tem e sys si as follows: Since the tractor body and the trailer body can be modeled as contained bodie atical expression s, all mathem s can be expressed by integrals    instead of a sums    .
The moments applied to the sprung mass due to the liquid load and the suspension forces expressed in axis sy lows: stem fixed to the sprung mass were calculated as fol- S s of omen 48) and the moments due to the fifth wheel constraints (35), the final equations of angular motion of the sprung mass (si) can be obtained as follows: k: axle number k = 3 for tractor and k = 2 for trailer.
The equation for rolling motion of the sprung mass of each axle can be given by the following where: : axle index.i j : number of tires in each axle.
for the front axle of the tractor and k : 2 k  4 k  for other axles.

t f
l and ansm ass th 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.

Suspension Model
The external forces acting on the vehicle are generated mainly due to the contac orces between whee ground.These forces are tr itted to the sprung m rough the suspension system of the vehicle.To simplify the model, the suspension system was represented static si uj uj uj uj where uj e is the suspension deflections and can be calculated based on the geometry of the vehicle.

Tire Model
The tire is an essential element in a vehicle.It represents the contact between wheel and ground.The forces and moments transmitted to the vehicle wheel-ground interaction are com These forces and moments depend primarily on normal forces, longitudinal and lateral load transfer, slip rate by the tires due to plex and nonlinear.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].
ith this model, it is possible to obtain lateral and lo ns ba as, braking/traction alone or direction case, as illustrated in Figure 11.
The calculation was based on the evaluation of friction W ngitudinal forces in the case of combined motio sed only on measured data for separate motions such x  and y  .The calculation of these coefficients de- s on the rate of longitudinal slip pend  and slip angle  .The rate of longitudinal slip of the tire ulated by the formula: and slip angle can be calc where w r is the radius of the wheel and w  the velocity of rotation of the wheel.V axe is the velocity of lateral translation of the axis and U p is the longitudinal velocity of the tire as shown in Figure 12.
The expressions can be evaluated from f m the velocity of center o ass of the vehicle.

Tire Vertical Load
In this study, the vertical load of the tire was modeled as inear sp ing.Therefore, the vertical force depended on the spring constant.
The tire deflections were calcula a l r ted from the geometry of the vehicle as follows: To calculate the combined forces, a dim sional vector of slip amplitude en  and direction  was defined [20] as: The coefficients of friction between the tire and the ground, in the case of the combined forces, took the forms below: Finally, longitudinal and lateral forces in the case of combined motion were calculated:

Vehicle Model Validation
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.Truck-Sim, the most popular software in this field, was used to represent and study the dynamics of vehicles in a computer environment.It is po le 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 Truck-Sim environment, the maneuvers are predefined paths, i.e., the excitation is repre a displacement vector.However, for the model oped in this study, the idered 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 ssib sented by devel 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 cons 70 km h v  were chosen to compare the two models.
he 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 tra led from the center of mass.T a good correla-    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).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.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 lation between the two models.

Conclusions
A complete nonlinear three-dimensional vehicle model 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 answer some quer bility of the heavy vehicles, in particular for the tanktruck.


Fixed to the Sprung MassThe three rotational motions of the sprung mass were expressed by the three Euler angles: yaw s (around axis x ) as shown by Figure2.

fr
: represents the position of center of sprung mass from the inertial system.position of the roll center relative to th r e system attached to the unsprung ity was calculated by differentiating Equation (24) with respect to time:
obtain the equation of angular tion.Accordin Newton's equation, angular momentum relative to the inertial system can be mo Copyright © 2013 SciRes.WJM of tractor.: sprung mass of trailer.


Figure 9. Forces and moments applied on the tire.

Figure 11 .
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].

Figure 14
Figure 14 shows t Figure 1 trajectory 4. (a) Simple lane change maneuver and the desired ; (b) Double lane change maneuver and the desired trajectory.Maneuvrer used for the com tween the model and TruckSim software.
Figure 16.(a)-(d) Double han ane vehicle (Solid: trucksim; Dashed: model).wasable t handle such situation in a better way.This difference can be e plaine ased o ass ption t at the fifth wheel w on gi f r m el.However, in TruckSim it was modeled by a

Table 2 . Geometric parameters of a ticulated vehicle.
1 CM and the axle number 3  m 4.522 -Distance between   1 CM and the axle number 4   1 CM and the axle number 5  