vehicle modeling using bg

8/4/2019 Vehicle Modeling Using BG

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Vehicle dynamics simulation using bond graphs

Germn Filippini, Norberto Nigro and Sergio Junco

Facultad de Ciencias Exactas, Ingenieray Agrimensura

Universidad Nacional de Rosario. Av. Pellegrini 250, S2000EKE Rosario, Argentina.

Abstract This work addresses the construction of a four-

wheel, nonlinear vehicle dynamic bond graph model and itsimplementation in the 20sim modeling and simulation

environment. Nonlinear effects arising from the coupling of

vertical, longitudinal and lateral vehicle dynamics, as well asgeometric nonlinearities coming from the suspension system

are taken into account. Transmission and (a simplified) engine

models are also included. The modeling task is supported by amultibody representation where the parts are handled as rigid

bodies linked by joints. The first step is the 3D-modeling of

each, chassis, suspension units, tires and joints, as bond graph

elements equipped with power ports for physical

interconnection. This is done with the help of vector ormultibond graphs in order to exploit their compactness and

simplicity of representation. These 3D-units are later

programmed as 20sim bond graph subsystems whoseassembling through the power ports allows for an automated,

modular approach to the construction of the overall vehicle

model. Simulation experiments corresponding to standard

vehicle dynamics tests are presented in order to show theperformance of the model.

Index Terms Bond Graphs, Multibody systems, Vehicledynamics.

I.INTRODUCTIONodeling and simulation has an increasing importance in

the development of complex, large mechanical systems.

In areas like road vehicles [1, 12, 15], rail vehicles [9], high

speed mechanisms, industrial robots and machine tools [10,

11, 6], simulation is an inexpensive way to experiment with

the system and to design an appropriate control system.

The above indicated kind of mechanical systems belong for

a major part to the class of systems of rigid bodies or

multibody systems. Such systems consist of a finite number of

rigid bodies, interconnected by arbitrary joints. The latter may

exhibit properties of rotational or translational freedom,

Norberto Nigro whish to thanks CONICET for its support to this research.

N. Nigro is with CIMEC-INTEC-CONICET and with the School of

Mechanical Engineering, Facultad de Ingeniera (FCEIA), UniversidadNacional de Rosario (UNR), Argentina. (Corresponding author. Phone: 54-

342-4511594; fax: 54-342-4550944; e-mail: [email protected]) .

Germn Filippini, is fellow at CIMEC-INTEC and teaching assistant at the

School of Mechanical Engineering, FCEIA-UNR ([email protected]).Sergio J. Junco is with the Department of Electronics, FCEIA-UNR,

Argentina ([email protected]).

damping and compliance, and are the place of the attachment

of drives or external forces.

In classical mechanics several procedures exist through

which differential equations can be derived for a system of

rigid bodies. In the case of large systems these procedures are

labor-intensive and consequently error-prone, unless they are

computerized [4].

This work applies the multibody theory through the

multibond or vector bond graph technique [2, 3, 5, 6, 7] to the

modeling of a complex four-wheel vehicle system. Primarily, bond graphs (BG) represent elementary energy-related

phenomena (generation, storage, dissipation, power exchange)

using a small set of ideal elements that can be coupled together

through external ports representing power flow. Thus, they are

well-suited for a modular modeling approach based on

physical principles. Hierarchical modeling becomes possible

through coupling of component or subsystems models through

their connecting ports. Besides these physical features

capturing energy exchange phenomena, it is also possible to

code on the graph the mathematical structure of the physical

system, in the sense of showing the causal relationships (in a

computational sense) among its signals [2]. On the one side,

this allows connecting BG-models to signal flow graphs or

block diagrams, and -on the other side- it turns the algorithmic

derivation of mathematical and computational models from

BG into a highly formalized task [2]. The conjunction of all

these features make of BG a physically based, object-oriented

graphical language most suitable for dynamic modeling,

analysis and simulation of complex engineering systems

involving mixed physical and technical domains in their

constitution [7].

The vehicle model developed in this paper considers the

vertical, longitudinal and lateral vehicle dynamics, takes into

account the geometrical non-linearities associated to the

suspension system, and includes Pacejka models [14] for thebehavior of the pneumatic tires. Simple, adequate models for

the engine and the transmission are also included to take into

account the vehicle traction.

The rest of the paper is organized as follows. Section II

presents a brief general description of a four-wheel vehicle and

of its constituents seen as rigid bodies, and discusses the

modeling assumptions. Section III first deals with the standard

mathematical modeling of these components and then with

their bond graph modeling and discusses the construction of

the 20sim library [18]. Section IV presents the full vehicle

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model obtained by assembling the library components. Section

V presents the simulation results and, finally, Section VI

brings the conclusions.

II. GENERALITIES AND MODELING ASSUMPTIONSA.Vehicle Chassis

The vehicle chassis is modeled as a rigid body with a local

coordinate reference frame (x, y, z) attached to the center ofmass and aligned with the inertia principal axes as is shown in

figure 1. It has mass m, and the following principal inertia

moments: roll (Jr) respect to the body x-axis,pitch (Jp) respect

to the body y-axis andyaw (Jy) respect to the body z-axis.

Figure 1: Full car model.

As also the four suspension subsystems are modeled as

spatial multibody systems, joint models are necessary to link

them to the chassis. The joints are represented as flexible

instead of rigid using a pseudo spring-damper system withelastic and damping constants. The flow and efforts actuating

on the joints depend on the relative position and the relative

orientation among the bodies.

In order to link two rigid bodies at a given joint it is

necessary to do some transformations (translations and

rotations) between the reference frames associated to each

body. In this way, the state variables expressed at the center of

mass of each body are transformed to a local reference frame

attached to the joint.

B.Engine and TransmissionMost vehicles are propulsed by internal (spark or

compression ignited) combustion engines which -for ourmodeling purposes- may be modeled through a given static

curve relating the engine speed and the load with its torque

and its power. Usually, these curves are obtained through

testing the engine at partial and full load.

The transmission is composed by the mechanical members

connecting the engine crankshaft with the traction wheels, the

gearbox and the planetary gear train including the differential

[8]. The main phenomena taken into account are the speed and

effort transformation among them.

C.Pneumatic tiresAside from aerodynamic and gravitational forces, all other

major forces and moments affecting the motion of a ground

vehicle are applied through the running gear-ground contact.

An understanding of the basic characteristics of the interaction between the running gear and the ground is, therefore,

essential to the study of performance characteristics, ride

quality, and handling behavior of ground vehicles. However, a

detailed explanation about pneumatic tires is out of the scopeof this paper. The following figure shows a summary of the

main forces, moments and angles that play a major role on themodeling of the pneumatic tire. Each one is associated with a

corresponding local axis located at the center of the contact

patch of the wheel. Traction (Fx), lateral (Fy) and normal (Fz)

forces along X, Y and Z local axis respectively and overturning

(Mx), rolling resistance (My) and aligning (Mz) torques alongthe same axis respectively are modeled in terms of the slip and

the camber angles and the pneumatic characteristics.

For more details about the fundamentals and the modeling ofpneumatic tires see [14, 1, 12].

Figure 2: Tire axis reference system

D.SuspensionsSuspension is the term given to the system of springs, shock

absorbers and linkages that connects a vehicle to its wheels.

This assembly is used to support weight, absorb and dampen

road shock, and help maintain tire contact as well as proper

wheel-to-chassis relationship. Without being a restriction for

future extensions of the overall vehicle model, only static

suspension systems are considered in this work.

E.Aerodynamic forcesThe aerodynamic forces were very simplified in this

analysis through the usage of empirical aerodynamic

coefficients [1, 16, 17]. In this work only the drag force was

included, being the lift and the pitching effects very similar in

terms of their mathematical expressions. In the future not only

the coefficients, also their sensitivity may be included.

III. MATHEMATICAL AND BGMODELINGThis section presents the subsystem modeling according to

the hypothesis of Section II. Previously, the essential issues

concerning the bond graph modeling of multibody systems as

Fx

x

y

z

roll

yaw

pitch

Fy

Kt

Kt

Kt

Kt

Bs

Bs

Bs

Bs

Ks

Ks

Ks

Ks

(1)

(2)

(3)

(4)

1r

1h


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used in this paper are addressed.

To determine the spatial motion of the rigid body the well

known Euler equations are used [2], which appear in (1) and

(2) in both, their intrinsic way of representation and their

tensorial counterparts.

The first one represents the conservation of linear

momentum, written as:

kjijk

rel

ii

i

rel

pdt

dp

dt

dpF

pdt

pd

dt

pd

F

+==

+== (1)

where pF ,, represents the external forces, the angular

velocity vector and the linear momentum vector respectively;

d/dt, d/dt|rel, ijk represent the derivative respect to the inertialframe, the derivative respect to the body (vehicle) attached

frame and the Levi-Civita tensor used to express the cross

product in tensor notation.

The second of the Euler equations sets up the conservation

of angular momentum:

kjijk

rel

ii

i

rel

hdt

dh

dt

dhM

hdt

hd

dt

hdM

+==

+==(2)

where hM, represent the external torque and the angular

momentum vector.

Figure 3: BG representation of the spatial rigid body

dynamics.

The BG representation of the 3-dimensional motion of a

rigid body based on the Euler equations is shown in figure 3.

The port variables of the above model are defined with respect

to the system attached to the center of mass of the rigid body.

Referring these port variables to the coordinates of

interconnection to other bodies is a must in order to be able to

couple the corresponding models.

Figure 4 shows the bond graph implementation of the

equation system representing how the port variables of two

arbitrary points 'A' and 'B' of a given spatial body transform

each other. The equations relating the linear and rotational

efforts are the following:

kjijki rFM

rFM

=

= (3)

For the flow variables the equations are the following:

kjijki rv

rv

=

=(4)

Figure 4: Power variables transformation between two points

A, Bbelonging to a given 3-dimensional rigid body.

To transform the dynamic equations from those expressed in

the body attached frame of reference (roll, pitch and yaw axes)

to a spatially fixed frame of reference (X,Y,Z : inertial frame)

it is necessary to choose some parameterization for the

rotations. Among the multiple possibilities, Euler angles are

used in this work. To transform these rotations the following

equations are used:

= (5.a); = (5.b); =G (5.c)

vv = (5.d); vv = (5.e); vv G = (5.f)

where,

=

cossin0

sincos0

001

=

cos0sin

010

sin0cos

=

100

0cossin

0sincos


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=

z

y

x

=

z

y

x

=

z

y

x

=

Z

Y

X

G

=

z

y

x

v

v

v

v

=

z

y

x

v

v

v

v

=

z

y

x

v

v

v

v

=

Z

Y

X

G

v

v

v

v

Its bond graphs representation is observed in figure 5, where

the power variables are transformed from (x,y,z) axes to the

rotated ones (X,Y,Z) according to the angles , , respect to

each axis. It may be observed that while the flow variables are

rotated from (xyz) to (XYZ), the effort variables are

transformed back from (XYZ) to (xyz).

Next, the models used for the different subsystems

belonging to the whole vehicle model are presented [2, 12]. Figure 5: 3D-rotation equations expressed in terms of power

variables.

Figure 6: BG modeling of Vehicle Chassis

A.Vehicle ChassisFigure 6 shows the vehicle chassis model composed by a

rigid body model, a coordinate system transformation to the

global system where the vehicle weight is imposed and four

translational transformations ( ir) to the pivots of the

suspension at each wheel, each one with their corresponding

model Engine and Transmission

B.Engine and TransmissionEngine. In bond graphs representation the engine is

modeled as an effort source (the engine output torque) variablewith the engine velocity normally expressed in rpm

(revolutions per minute) and the position of the butterfly valve

(accelerator command) as written by equation (6).

)()1()(),,( rpppprp TATAATTT += (6)

where Ap is the accelerator position ( 10 pA , Tp and Trare the engine output torque and the resistant torque

respectively for a given engine speed ( ) .


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While the engine curve, the accelerator position and the

resistant torque are data for the model, the engine speed is

computed by the whole model. In the Engine Torque model

show in figure 7 the output torque is computed by equation (6)

and its result Tis used to modulate the source MSe.

Figure 7: BG modeling of Engine

Figure 8: BG modeling of Transmission

Transmission. Figure 8 shows the main components of the

transmission and its structure. The gearbox (figure 9) is

modeled using a modulated transformer (MTF) that relates the

input and output effort variables through a variable

transformation factor that depends on the gearbox ratios

supplied to the model in advance.

Figure 9: BG modeling of Gearbox

The differential (figure 10) is modeled by a transformer(TF) modulated by the planetary drive train (differential) ratio.

The 0 junction imposes the same torque to both traction

wheels.

Figure 10:BG modeling of Differential

C.Pneumatic TiresFor all forces and moments acting on the tire the Pacejka

model [14], inspired by a lot of experiments carried out using

different types of pneumatic tires is used.

( )

( )

2

1/

( ) 1

/

1.12 ; 0.625 ; 1

46 ; 5 ; 0.6

b

n

x sign A e c D

B K d

A C D

K d n

= +

=

= = =

= = =(7)


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This model is briefly described with the expressions (7),

with -1<


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Figure 12: Bond graphs modeling of Suspension

Figure 13: Bond graphs modeling of Revolution Joint

E.Aerodynamic ForcesEquation 9 represents the aerodynamic drag force where

is the air density, Cx is the (experimentally computed) drag

coefficient,Af is the vehicle frontal area and V is the relative

velocity between the vehicle and the wind.

2

2

1

VACF fxaero

x = (9)

This is represented in the Bond Graph formalism by a

resistance component applied to the coordinate system fixed

to the center of mass of the rigid body that models the chassis,

as it is shown in figure 14.

Figure 14: Bond graphs modeling of Rigid Body with

Aerodynamics Resistance

IV. FULL VEHICLE VECTOR BGMODEL AND 20SIMIMPLEMENTATION

In figure 15 the whole system model is shown, which isbuilt via assembling the previous submodels, pretty much in

the same way as a real vehicle is constructed. This is the

powerful modular or objected-oriented modeling propertyof the BG technique.


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Figure 15: Bond graphs modeling of a vehicle

V.SIMULATION RESULTSAfter the validation of the bond graphs multibody

toolbox developed in this work, a vehicle dynamics test,

extracted from the bibliography is performed [1].

The data, coming from a Renault Clio RL 1.1 car, are the

following:

Aerodynamics coefficient 0.33

Frontal area 1.86 m2

Distance between axes 2.472 mVehicle weight 8100 N

Centre of mass height 0.6 m

Front axis weight 5100 N

Rear axis weight 3000 N

Maximum engine torque 78.5 Nm at 2500 rpm

Maximum engine power 48 CV at 5250 rpm

Planetary drive train (differential) ratio 3.571

First gearbox ratio 3.731

Second gearbox ratio 2.049

Third gearbox ratio 1.321

Four gearbox ratio 0.967

Five gearbox ratio 0.795

Reverse gearbox ratio 3.571Tires, type and dimensions 145 70 R13 S

Wheelbase 1.650 m

Maximum speed 146 km/h

Acceleration 0-100 km/h in 17 s

Time spent to do 1000 meters 38 s

Distance from centre of mass to front axes 0.916 m

Distance from centre of mass to rear axes 1.556 m

Pneumatic tire radius (unloaded) 0.2666 m

Air density 1.225 kg/m3

Unsprung masses (at each wheel) 38.42 kg

Tire Vertical stiffness 150 000 N/m

Tire inertia 1.95 Kgm2

Damper coefficient 475 N s / m

Suspension stiffness 14 900 N/m

Sprung mass - Yaw Inertia 2345.53Kg m2

Sprung mass - Pitch Inertia 2443.26Kg m2

Sprung mass - Roll Inertia 637.26 Kg m2

The first test is a sudden motion starting from rest. With

the engine butterfly valve fully opened and once the clutch

is released the vehicle response may be assessed in terms of

the vehicle acceleration in a straight road. Figures 16 to 21

show the results for this case. Figure 16 shows the engine

speed (rpm) in time where it may be noted the times at

which the gearbox is used. Figure 17 shows the longitudinal

vehicle speed in time reaching 100 km/h in less than 20

seconds. Figure 18 plots the longitudinal sliding of one of

the traction wheels. The friction force may be evaluated

from this figure and the vertical load on the tire (see figure

19). In figure 20 the chassis pitching angle is shown.

During the first 10 seconds until the fourth gear is selected

the pitching is oscillating reaching a negative smaller value

after this time when five gear is selected showing an

acceleration behavior. Figure 21 shows the rear wheel load.The time spent to reach 100 km/h was 17 seconds, agreeing

very well with the road test published by the manufacturer.


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Figure 16: Time evolution of the engine speed.

Figure 17: Longitudinal vehicle velocity [m/s] in time.

Figure 18: Evolution of one of the traction wheels

sliding.

Figure 19: Evolution of the load [N] over a traction

wheel.

Figure 20: Chassis pitching as a function of time.

Figure 21: Load [N] over one of the rear wheels.

In the second test the vehicle is forced to follow a curved

road. The simulation starts with the vehicle in third gear at

65 km/h; the wheel drive in turned such that the front

wheels turn 1 degree in 10 sec following a time law as

shown in figure 22.

Figure 22: Evolution of the turning angle [rad] of the

front wheels.

Figure 23: Trajectory in x-y [m]


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Figure 24: Sliding angle of one of the front wheels as a

function of time.

Figure 25: Sliding angle of one of the rear wheels as a

function of time.

Figure 26: Yaw vehicle response as a function of time.

After 5 seconds the wheel drives return to the original

position (0 degree). In figure 24 the lateral slip angle of one

of the front wheels is shown and in figure 25 the same for

one of the rear wheels. Figure 13 plots the trajectory

followed by the vehicle and figure 26 the yaw response ofthe vehicle.

Figure 27: Front wheels slip angle [rad] as a function of

time

Figure 28: Trajectory x-y [m]

Figure 29: Slip angle of one of the front wheels [rad].

Figure 30: Slip angle of one of the rear wheels [rad].


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Figure 31: Yaw angle [rad] as a function of time.

In the third test a zigzag maneuver is assessed. The

vehicle start the simulation at 65 km/h in third gear the

driver turn the wheel drive following a time law as shown

in figure 27, the front wheels turn an angle of one degree in

10 seconds.

After this time the wheel drive is turned as to reach one

degree in the opposite direction in 10 seconds and then

return to the original position in the remaining 5 seconds.Figure 29 shows the lateral slip angle of one of the front

wheels and figure 30 does the same for one of the rear

wheels.

Figure 28 shows the vehicle trajectory and figure 31

shows the vehicle yaw angle.

The behavior of this case presents some similarity with

the second one and it may be observed that in both cases the

directional stability is a priori acceptable.

However to do a more rigorous analysis we should

analyze the eigenvalues of the directional response matrix

that may be computed making a sensitivity analysis with

simulations. This task is left for future work.

For the vehicle application it is observed that formaneuvering in straight road the response of the vehicle

with third, four and fifth gears were very smooth

concluding that the vehicle is well optimized to be used at

velocities greater than 50km/h. For curves the transient

response allows for evaluating the directional stability with

changes in the position of the wheel drive. According to the

results obtained it would be more desirable to have a less

oscillatory behavior in turns.

VI. CONCLUSIONSOne of the main goals of this paper was the extension of

this formalism to include large spatial (3-dimensional)rotations. Several elements oriented to multibody systems

were developed allowing working with different reference

frames, operating with them through the usage of

translations and general transformations. This toolbox

works acceptable in the vehicle dynamics prediction and it

was successfully applied to another project based on vehicle

fault diagnostics [13].

REFERENCES

[1] F. Aparicio Izquierdo, C. Vera Alvarez, V. Das Lpez, Teora de los

vehculos automviles. U. P. de Madrid[2] Ronald C. Rosenberg, Donald L. Margolis, Dean C. Karnopp.

Modeling and Simulation of Mechatronic Systems, A Wiley-Interscience

Publication, New York.

[3] Albert M. Bos. Modeling Multibody Systems in terms of MultibondGraphs, with application to a motorcycle, PhD thesis at Twente Univ. 1986

[4] Ahmed A. Shabana. Dynamics of Multibody systems. A Wiley-Interscience Publication.

[5] Jinhee Jang, Changsoo Han. Proposition of a Modeling Method forConstrained Mechanical Systems Based on the Vector Bond Graph.

Journal of the Franklin Institute. Vol. 335B, No 3, pp. 451 469, 1998.[6] T. Erial, J. Stein, L. Louca. A Bond Graphs Based Modular Modeling

Approach towards an Automated Modeling Environment for

Reconfigurable Machine Tools. IMAACA 2004

[7] F. Cellier, "Hierachical nonlinear bond graph: A unified methodologyfor modelling complex physical systems", Simulation, Vol 58, No. 4, pp.

230-248.[8] J. M. Mera, C. Vera, J. Flez. 2WD Power Train Modeling with Bond

Graph applied to Vehicular Dynamics . Universidad Politcnica de

Madrid, Spain.

[9] W. Kortuem , A. Urzt, Simulation of active suspensions in ground

transportation - Application to Maglev vehicles, 11th IMACS world

congress, Oslo, Norway, 5-9 Aug, 1985[10] B. Paul, Analytical dynamics of mechanisms - a computer oriented

overview, Mechanism and Machine Theory, vol 10, pp. 481-507, 1975.

[11] B. Paul, Computer oriented analytical dynamics of machinery,

Proceedings of the NATO advanced study institute on computer aidedanalysis and optimization of mechanical system dynamics, ed. E.J. Haug,

IOWA City, Aug 1983, Springer, pp. 41-87, 1984

[12] G. Filippini. Dinmica Vehicular mediante bond graphs. Proyecto

final de carrera de grado. Escuela de Ingeniera Mecnica, Universidad

Nacional de Rosario, 2004[13] D. Delarmelina, L. Silva, S. Junco. Fault Diagnosis in Vehicle based

on its Dinamic Model. Escuela de Ingeniera Mecnica y Escuela de

Ingeniera Electronica, Universidad Nacional de Rosario, 2005.

[14] H. Pacejka. Tyre Modeling for Use in Vehicle Dynamics Studies.SAE Paper, No. 870421.

[15] H.B. Pacejka, Principles of plane motion of automobiles, IUTAMsymposium on the dynamics of automobiles, Delft Univ. of Tech. pp. 33-

59, 1975

[16] W.H. Hucho, editor. Aerodynamics of road vehicles. SAE, (1998).

[17] R. H- Barnard, Road vehicle aerodynamic design, Second Edition,MechAero, England, (2001)

[18] Getting Started with 20-sim 3.6, Controllab Products B.V., Enschede,

Netherlands. Internet: www.20sim.com, 2005.

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