Bond Graph Simulation of Bicycle Model

Instructor: Dr. Shuvra Das

By: Vishnu Vijayakumar

E579 – Mechatronic Modeling and Simulation

E579 - Term Project - Bicycle Model 2

Contents

IntroductionBicycle ModelBond-graph ModelingResults and DiscussionFuture WorkReferences

E579 - Term Project - Bicycle Model 3

Introduction

Types of CorneringSlow-speed (parking lot maneuvers)

No Lateral ForcesTherefore center of turn must lie on the

projection of the rear axleHigh-speed

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Low-Speed Cornering

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High- Speed Cornering

Turning equations differ because lateral acceleration will be present

Tires must develop lateral forces Slip Angles will be present at each wheelFor purpose of analysis it is convenient

to represent the vehicle by a bicycle model

E579 - Term Project - Bicycle Model 6

IntroductionBicycle ModelBond-graph ModelingResults and DiscussionFuture WorkReferences

E579 - Term Project - Bicycle Model 7

Bicycle Model

Bicycle model [1]

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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 9

Equations

RgC

VW

RgC

VW

R

L

r

rr

f

ff

rf

..

.

..

.

3.57

2

2

Equations for steering angles and slip angles [2]

E579 - Term Project - Bicycle Model 10

IntroductionBicycle ModelBond-graph ModelingResults and DiscussionFuture WorkReferences

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Bond Graph Representation

RFront_tire_friction

RRear_Tire_friction

1OneJunction1

1OneJunction2

0ZeroJunction1

0ZeroJunction2

MTFb_cos_delta

MTFsine_delta

MTFinverse_cos_delta

TFc

MGYMGY1

IMass_x

IMoment_of_Inertia

IMass_y

1OneJunction3

1OneJunction4

1OneJunction5

Submodel2

Submodel3

Submodel4

mass

SignalGenerator3

MSfMSf1

d/dt

Differentiate1

SignalGenerator2

delta_calc

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IntroductionBicycle ModelBond-graph ModelingResults and DiscussionFuture WorkReferences

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0 5 10 15 20 25 30 35Velocity

2

2.5

3

3.5

4Steer angle

Understeer

Steer Angle with Velocity

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Change of Steer angle with time

Steer Angle Vs Time

0 5 10 15 20time {s}

2.2

2.4

2.6

2.8

3

3.2Steer Angle

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Steer Angle Vs Lateral Acceleration

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5Lateral Acceleration

2.2

2.4

2.6

2.8

3

3.2Delta

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Validation

Measurement of Understeer Gradient Using Constant Radius MethodUndersteer 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

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Validation

At 60 mph velocity the lateral acceleration gain was calculated using the formula

Lateral Acceleration was calculated using the formula

From graph Lateral Acceleration gain = 0.407g/deg

deg/475.0

3.571

3.572

2

g

LgKVLg

Vay

gRg

Vay 2.1

2.32200

8822

R

Vay

2

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IntroductionBicycle ModelBond-graph ModelingResults and DiscussionFuture WorkReferences

E579 - Term Project - Bicycle Model 20

Future Work

Enhance the model Load Transfer (Longitudinal)

E579 - Term Project - Bicycle Model 21

IntroductionBicycle ModelBond-graph ModelingResults and DiscussionFuture WorkReferences

E579 - Term Project - Bicycle Model 22

References

1. Karnopp, Margolis, Rosenberg, “System Dynamics”, Third Edition, 2000

2. Thomas Gillespie, “Fundamentals of Vehicle Dynamics”, 1992

3. J.Y.Wong, “Theory of Ground Vehicles”, 1993

4. Divesh Mittal, “Characterization of Vehicle Parameters affecting dynamic roll-over propensity”, SAE2006-01-1951

Questions?