E579 – Mechatronic Modeling and Simulation
Bond Graph Simulation of
Bicycle Model
Instructor: Dr. Shuvra Das
By: Vishnu Vijayakumar
Contents
 Introduction
Bicycle Model
 Bond-graph Modeling
 Results and Discussion
 Future Work
 References

E579 - Term Project - Bicycle Model
2
Introduction

Types of Cornering

Slow-speed (parking lot maneuvers)
No Lateral Forces
 Therefore center of turn must lie on the
projection of the rear axle


High-speed
E579 - Term Project - Bicycle Model
3
Low-Speed Cornering
E579 - Term Project - Bicycle Model
4
High- Speed Cornering
Turning equations differ because lateral
acceleration will be present
 Tires must develop lateral forces
 Slip Angles will be present at each wheel
 For purpose of analysis it is convenient
to represent the vehicle by a bicycle
model

E579 - Term Project - Bicycle Model
5
Introduction
 Bicycle Model
 Bond-graph Modeling
 Results and Discussion
 Future Work
 References

E579 - Term Project - Bicycle Model
6
Bicycle Model
Bicycle model [1]
E579 - Term Project - Bicycle Model
7
Parameters










L = Wheel Base = 100.6 in = 8.38ft
R = Radius of turn = 200 ft
V = Forward Speed
g = Gravitational Acceleration = 32.2ft/s2
Wf = Load on front axle = 1901 lb
Wr = Load on rear axle = 1552 lb
Cαf = Cornering Stiffness of front tires = 464 lb/deg
Cαr = Cornering Stiffness of rear tires = 390 lb/deg
Tire Friction coefficient = 0.7 (Assumed)
Yaw Mass moment of Inertia = 600 lb-ft2 [4]
Example Problem [2]
E579 - Term Project - Bicycle Model
8
Equations
  57 . 3
L

R

f

r 
W f .V
f
r
2
C  f . g .R
W r .V
2
C  r . g .R
Equations for steering angles and
slip angles [2]
E579 - Term Project - Bicycle Model
9
Introduction
 Bicycle Model
 Bond-graph Modeling
 Results and Discussion
 Future Work
 References

E579 - Term Project - Bicycle Model
10
Bond Graph Representation
MGY
MGY1
I
I
Mass_y
I
Mass_x
1
OneJunction3
Moment_of_Inertia
mass
1
d/dt
1
OneJunction4
Differentiate1
OneJunction5
MSf
SignalGenerator3
MSf1
TF
c
MTF
MTF
sine_delta
0
0
ZeroJunction2
Submodel3
ZeroJunction1
1
1
OneJunction2
OneJunction1
R
R
Rear_Tire_friction
MTF
Submodel2
b_cos_delta
Front_tire_friction
Submodel4
inverse_cos_delta
delta_calc
SignalGenerator2
E579 - Term Project - Bicycle Model
11
Introduction
 Bicycle Model
 Bond-graph Modeling
 Results and Discussion
 Future Work
 References

E579 - Term Project - Bicycle Model
12
Steer Angle with Velocity
4
Steer angle
3.5
3
Understeer
2.5
2
0
5
10
15
20
Velocity
25
E579 - Term Project - Bicycle Model
30
35
13
Change of Steer angle with time
Steer Angle Vs Time
3.2
Steer Angle
3
2.8
2.6
2.4
2.2
0
5
10
time {s}
E579 - Term Project - Bicycle Model
15
20
14
Steer Angle Vs Lateral Acceleration
3.2
Delta
3
2.8
2.6
2.4
2.2
0
0.5
1
1.5
2
2.5
Lateral Acceleration
3
E579 - Term Project - Bicycle Model
3.5
4
4.5
15
Validation

Measurement of Understeer Gradient Using
Constant Radius Method
 Understeer can be measured by operating
the vehicle around a constant radius turn
and observing steering angle and lateral
acceleration
 Vehicle speed is increased in steps that will
produce lateral accelerations at reasonable
increments
E579 - Term Project - Bicycle Model
16
E579 - Term Project - Bicycle Model
17
Validation



At 60 mph velocity the
lateral acceleration gain
was calculated using the
formula
Lateral Acceleration wasa
calculated using the
formula
From graph Lateral
Acceleration gain =
0.407g/deg
V
ay

2
57 . 3 Lg

1
KV
2
 0 . 475 g / deg
57 . 3 Lg
y

V
2
R
ay 
V
2
Rg
E579 - Term Project - Bicycle Model

88
2
200  32 . 2
 1 .2 g
18
Introduction
 Bicycle Model
 Bond-graph Modeling
 Results and Discussion
 Future Work
 References

E579 - Term Project - Bicycle Model
19
Future Work
Enhance the model
 Load Transfer (Longitudinal)

E579 - Term Project - Bicycle Model
20
Introduction
 Bicycle Model
 Bond-graph Modeling
 Results and Discussion
 Future Work
 References

E579 - Term Project - Bicycle Model
21
References
1.
2.
3.
4.
Karnopp, Margolis, Rosenberg, “System
Dynamics”, Third Edition, 2000
Thomas Gillespie, “Fundamentals of Vehicle
Dynamics”, 1992
J.Y.Wong, “Theory of Ground Vehicles”,
1993
Divesh Mittal, “Characterization of Vehicle
Parameters affecting dynamic roll-over
propensity”, SAE2006-01-1951
E579 - Term Project - Bicycle Model
22
Questions?