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Fundamentals of Vehicle Dynamics, T. D. Gillespie, ©2013 1

Chapter 1

Introduction


Course Objectives

• What do you need to know to understand vehicle

dynamics?– Important vehicle and system properties

– Mechanics by which each system functions

• Why do you need to know it?– How will a system’s properties affect performance?

– What are the design conflicts for achieving good performance in

different modes?

2


Chapter 1 Objectives

• Establish background information for understanding

vehicle dynamics

• I.e., how do we model vehicle dynamics?– Coordinate systems

– The role of equations

– Newton’s Second Law

• Dynamic loads on axles


Cugnot Vehicle (1769)

• The first motorized vehicle

• Also, the first accident!

P. 2

3


Sunbeam Mabley (1901)

• Development took some peculiar turns!


Speed Increase in Early Autos

• As speed increases, so do the dynamics

193019201910190018901880

0

20

40

60

80

100

Year

Sp

ee

d (

mp

h)

Mercer Raceabout

Olds Limited

30/98 Vauxhall

Panhard Levassor

Peugeot

Victoria

Winton

Daimler

Fiat

Peugeot

Fir

st c

oncr

ete

road

Fir

st t

raff

ic s

ignal

Pav

ed r

oad

New

York

to S

an F

ranci

sco

Sp

ee

d (

km

/h)

0

20

40

60

80

100

120

140

160

4


Forces on a Car

• To understand dynamics we need to know the forces

on the vehicle

• Gravity

• Aerodynamics

• Primary forces come from the tires

DLC Tire Forces.exeDLC Truck Tire Forces.exe


SAE Vehicle-fixed Coordinate System

• For many analyses the vehicle can be treated as a lumped

mass

• Need to define a coordinate system (directions relative to

the vehicle)

P. 8

Roll

Vertical

Yaw

Pitch

Z

Y

X

pq

r

CG

5


ISO Vehicle-fixed Coordinate System

• Some use a system from the International Standards

Organization (ISO)

• Both systems will be in the new SAE J670

Roll

Vertical

Yaw

Pitch

Z

Y

X

pq

r

CG


Earth-fixed Coordinate System

X

Y

Vehicle PathCourse Angle (Positive)

Heading Anglex Projected

Projection of Instantaneous Velocity

Sideslip Angle, β (Neg. angle shown)

Steer Angle

y Projected

ψ

ν

• Need to define motions relative to an inertial frame

P. 9

6


Engineering Models

• Objective is to model automotive vehicles and

systems

• Engineers use equations to define models

• Equations are only approximations of nature

• Consider the Perfect Gas Model


Perfect Gas Model

nRTPV =• The model expresses:

– What variables are important

• (P, V, n, R, and T)

– How they relate

• Suppose I am interested in pressure

• Pressure is:

– Proportional to n and T

– Inversely proportional to VV

nRTP =

7


Engineering Models

• Models that are simple, explicit equations teach us

how something works– The textbook focuses on these

• More comprehensive models (tires, suspension

systems) require other solutions– When needed we integrate them into simulation models like

• CarSim

• TruckSim

• BikeSim

• SuspensionSim


Newton’s Second Law

• All dynamics start with NSL

• Translational systems

xx MaF =∑ Fx = Force in the x-direction

M = Mass of the body

ax = Acceleration in x-direction

• Rotational system

xxxx IT α=∑Tx = Torque around the x-axis

Ixx = Moment of inertia about x-axis

αx = Acceleration about the x-axis

8


02

f x c A hx h hz h c

LW L Ma h Mg h sin L PM R h R d Mg c cosθ θ+ + + + + + − =

cos sin / 2c x c A hx h hz hf

Mg c Ma h Mg h L L PM R h R dW

L

θ θ− − − ⋅ − − −=

cos sin / 2 ( )c x c A hx h hz h

r

Mg b Ma h Mg h L L PM R h R L dW

L

θ θ+ + − ⋅ + + + +=

• Summing moments about point A

M g

Θ


Static Loads

• Sitting statically on a level surface:

L

cWW fs =

L

bWWrs = AB

M gc

Wf Wr L

b c

P. 13

9


Acceleration at Low Speed

• Acceleration on a level surface with no aerodynamic reactions

c

xfs

c

xf

g

a

L

hWW

g

a

L

h

L

cWW −=−= )(

c

xrs

c

xr

g

a

L

hWW

g

a

L

h

L

bWW +=+= )(

P. 13


Climbing a Grade

• No aerodynamic or acceleration effects

)sincos

( θθ

L

h

L

cWW f −

⋅=

• For small angles: cosθ = 1, sinθ = θ

θL

hWWW fsf −=

)sincos

( θθ

L

h

L

bWWr +

⋅=

θL

hWWW rsr +=

• θ = Grade angle (in radians)

P. 14

10


Road Grades

• Road grade is usually

expressed in % Run

RiseGrade 100(%) =

RunRise

θ1tan ( )Rise

Runθ −=

• Example – 5% grade 0.05RiseRun

=

1tan (0.05) 2.86deg 0.0499 rad 0.05 radθ −= = = ≅

(2.862 ) 0.999 1Cos = ≅� (2.862 ) 0.0499 0.05Sin = ≅�

• Good to about 20% (<2% error)


Composite Mass

• Longitudinal • Vertical

1

1

n

i i

icomposite n

i

i

m x

X

m

=

=

=∑

∑

1

1

n

i i

icomposite n

i

i

m z

Z

m

=

=

=∑

∑

Car

Passenger

Cargo

Z

Xx1 x2 x3

z1

z2

z3Composite

Finding the

Composite

CG Locationm1

m2

m3

11


Car-trailer Combinations

• Analyze trailer first to determine hitch forces

1.85 m

550 kg90 kg 3.0 m 1.1 m

908 kg 964 kg

0.35 m

1.85 m

550 kg90 kg

)( θ+==∑ xtrhxx aMFF

ctrtrhzz gMWFF =+=∑


The End