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Vehicle Dynamics

CEE 320Steve Muench

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Outline

1. Resistancea. Aerodynamicb. Rollingc. Grade

2. Tractive Effort3. Acceleration4. Braking Force5. Stopping Sight Distance (SSD)

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Main Concepts

• Resistance• Tractive effort• Vehicle acceleration• Braking• Stopping distance

grla RRRmaF

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Resistance

Resistance is defined as the force impeding vehicle motion1. What is this force?

2. Aerodynamic resistance

3. Rolling resistance

4. Grade resistance

grla RRRmaF

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Aerodynamic Resistance Ra

Composed of:1. Turbulent air flow around vehicle body (85%)

2. Friction of air over vehicle body (12%)

3. Vehicle component resistance, from radiators and air vents (3%)

2

2VACR fDa

3

2VACP fDRa

sec5501

lbfthp

from National Research Council Canada

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Rolling Resistance Rrl

Composed primarily of

1. Resistance from tire deformation (90%)

2. Tire penetration and surface compression ( 4%)

3. Tire slippage and air circulation around wheel ( 6%)

4. Wide range of factors affect total rolling resistance

5. Simplifying approximation:

WfR rlrl

147101.0

VfrlWVfP rlrlR

sec5501

lbfthp

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Grade Resistance Rg

Composed of – Gravitational force acting on the vehicle

gg WR sin

gg tansin

gg WR tanGg tan

WGRg

For small angles,

θg W

θg

Rg

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Available Tractive Effort

The minimum of:1. Force generated by the engine, Fe2. Maximum value that is a function of the

vehicle’s weight distribution and road-tire interaction, Fmax

max,mineffort tractiveAvailable FFe

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Tractive Effort Relationships

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Engine-Generated Tractive Effort

• Force

• Power

r

MF de

e

0

2

minsec

60

rpm engine

550

lbft torque

sec

lbft550 hp

Fe = Engine generated tractive effort reaching wheels (lb)

Me = Engine torque (ft-lb)

ε0 = Gear reduction ratio

ηd = Driveline efficiency

r = Wheel radius (ft)

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Vehicle Speed vs. Engine Speed

0

12

irn

V e

V = velocity (ft/s)

r = wheel radius (ft)

ne = crankshaft rps

i = driveline slippage

ε0 = gear reduction ratio

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Typical Torque-Power Curves

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Maximum Tractive Effort

• Front Wheel Drive Vehicle

• Rear Wheel Drive Vehicle

• What about 4WD?

LhL

hflW

F

rlf

1

max

LhL

hflW

F

rlr

1

max

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Diagram

Ra

Rrlf

Rrlr

ma

Wθ

g

Fbf

Fbr

h

h

lf

lr

L

θg

Wf

Wr

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Vehicle Acceleration

• Governing Equation

• Mass Factor (accounts for inertia of vehicle’s rotating parts)

maRF m

200025.004.1 m

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Example

A 1989 Ford 5.0L Mustang Convertible starts on a flat grade from a dead stop as fast as possible. What’s the maximum acceleration it can achieve before spinning its wheels? μ = 0.40 (wet, bad pavement)

1989 Ford 5.0L Mustang Convertible

Torque 300 @ 3200 rpm

Curb Weight 3640

Weight Distribution Front 57% Rear 43%

Wheelbase 100.5 in

Tire Size P225/60R15

Gear Reduction Ratio 3.8

Driveline efficiency 90%

Center of Gravity 20 inches high

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Braking Force

• Front axle

• Rear axle

L

fhlWF rlr

bf

max

L

fhlWF rlf

br

max

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Braking Force

• Ratio

• Efficiency

rear

front

fhl

fhlBFR

rlf

rlr

maxg

b

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Braking Distance

• Theoretical– ignoring air resistance

• Practical

• Perception

• Total

grlb

b

fg

VVS

sin2

22

21

Gga

g

VVd

2

22

21

pp tVd 1

ps ddd

a

VVd

2

22

21

For grade = 0

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Stopping Sight Distance (SSD)

• Worst-case conditions– Poor driver skills– Low braking efficiency– Wet pavement

• Perception-reaction time = 2.5 seconds• Equation

rtV

Gga

g

VSSD 1

21

2

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Stopping Sight Distance (SSD)

from ASSHTO A Policy on Geometric Design of Highways and Streets, 2001

Note: this table assumes level grade (G = 0)

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SSD – Quick and Dirty

a

VVV

V

Ggag

VVd

222

221

22

21 075.1

2.11075.1

2.11

1

2

47.1

02.322.112.322

047.1

2

1. Acceleration due to gravity, g = 32.2 ft/sec2

2. There are 1.47 ft/sec per mph

3. Assume G = 0 (flat grade)

ppp VttVd 47.147.1 1

V = V1 in mpha = deceleration, 11.2 ft/s2 in US customary unitstp = Conservative perception / reaction time = 2.5 seconds

ps Vta

Vd 47.1075.1

2

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Primary References

• Mannering, F.L.; Kilareski, W.P. and Washburn, S.S. (2005). Principles of Highway Engineering and Traffic Analysis, Third Edition). Chapter 2

• American Association of State Highway and Transportation Officals (AASHTO). (2001). A Policy on Geometric Design of Highways and Streets, Fourth Edition. Washington, D.C.