1. Vehicle dynamics simulation using bond graphs
Germán Filippini, Norberto Nigro and Sergio Junco
Facultad de Ciencias Exactas, Ingeniería y Agrimensura Universidad Nacional de Rosario. Av. Pellegrini 250, S2000EKE Rosario, Argentina

2. Generalities and Modeling Assumptions
Vehicle Chassis
Engine and Transmission
Pneumatic tires
Suspensions
Aerodynamic forces Fx ω x y z
roll
yaw
pitch Fy δ Kt Bs Ks
(1)
(2)
(3)
(4) h1 r1

3. Mathematical and BG Modeling
Rigid Body
Euler equations
Conservation of linear momentum
Conservation of angular momentum

4. Mathematical and BG Modeling
Power variables transformation between two points ‘A’, ‘B’
Equations relating the linear
and rotational efforts
Equations relating the linear
and rotational flows

5. Mathematical and BG Modeling
3-dimensional rotation equations expressed in terms
of power variables
Euler angles
Where,

6. Mathematical and BG Modeling
Vehicle Chassis

7. Mathematical and BG Modeling
Engine and Transmission ,
Ap accelerator position
Tp engine output torque
Tr engine resistant torque

8. Mathematical and BG Modeling
Gearbox and Differential ,

9. Mathematical and BG Modeling
Pneumatic Tire
Pacejka model
Adherence coefficient
Lateral force ,
with -1<  <1 the longitudinal slip

10. Mathematical and BG Modeling
Pneumatic Tire ,

11. Mathematical and BG Modeling
Suspension ,

12. Mathematical and BG Modeling
Aerodynamic forces ,
ρ air density
Cx drag coefficient
Af vehicle frontal area
V relative velocity between the vehicle and the wind

13. Mathematical and BG Modeling
Full Vehicle vectorBG Model ,

14. Simulation Data Renault Clio RL 1.1 Aerodynamics coefficient 0.33
Frontal area m2
Distance between axes m
Vehicle weight N
Centre of mass height m
Front axis weight N
Rear axis weight 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
Second gearbox ratio
Third gearbox ratio
Four gearbox ratio
Five gearbox ratio
Reverse gearbox ratio
Tires, type and dimensions R13 S
Wheelbase m
Maximum speed km/h
Acceleration km/h in 17 s
Time spent to do 1000 meters 38 s
Distance from centre of mass to front axes m
Distance from centre of mass to rear axes m
Pneumatic tire radius (unloaded) m
Air density kg/m3
Unsprung masses (at each wheel) kg
Tire Vertical stiffness N/m
Tire inertia Kgm2
Damper coefficient N s / m
Suspension stiffness N/m
Sprung mass - Yaw Inertia Kg m2
Sprung mass - Pitch Inertia Kg m2
Sprung mass - Roll Inertia Kg m2 ,

15. Time evolution of the engine speed.
Simulation Results
First Test
Time evolution of the engine speed.
Evolution of the load [N] over a traction wheel. ,
Longitudinal vehicle velocity [m/s] in time.
Chassis pitching as a function of time.
Evolution of one of the traction wheels sliding.
Load [N] over one of the rear wheels.

16. Simulation Results Second Test .
Sliding angle of one of the front wheels. ,
Evolution of the turning angle [rad] of the front wheels. .
Sliding angle of one of the rear wheels as a function of time.
Trajectory in x-y [m]
Yaw vehicle response as a function of time.

17. Simulation Results Third Test .
Slip angle of one of the front wheels [rad] ,
Evolution of the turning angle [rad] of the front wheels. .
Slip angle of one of the rear wheels [rad].
Trajectory x-y [m]
Yaw angle [rad] as a function of time.

18. Conclusions
One 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 work 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.