Vehicle Dynamics - rizayana · 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

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Vehicle Dynamics

CEE 320 Steve Muench

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Outline

1. Resistance a. Aerodynamic b. Rolling c. Grade

2. Tractive Effort 3. Acceleration 4. Braking Force 5. 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 motion 1. 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

ρ=

3

2VACP fDRa

ρ=

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 =

⎟⎠

⎞⎜⎝

⎛ +=147

101.0 VfrlWVfP rlrlR =

sec5501 lbfthp ⋅

=

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Grade Resistance Rg

gg WR θsin=gg θθ tansin ≈

gg WR θtan=Gg =θtan

WGRg =

For small angles,

θg Wθg

Rg

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Available Tractive Effort

The minimum of: 1. Force generated by the engine, Fe

2. 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

rMF de

eηε 0=

( )π2

minsec60

rpm engine550

lbft torquesec

lbft550 hp ×

⎟⎠

⎞⎜⎝

⎛×

⋅=⎟

⎠

⎞⎜⎝

⎛ ⋅

Fe = Engine generated tractive effort reaching wheels (lb)

Me = Engine torque (ft-lb)ε0 = Gear reduction ratioηd = Driveline efficiency

r = Wheel radius (ft)

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Vehicle Speed vs. Engine Speed

( )0

12ε

π irnV e −=

V = velocity (ft/s)r = wheel radius (ft)

ne = crankshaft rpsi = driveline slippageε0 = 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?

( )

LhLhfl

WF

rlf

µ

µ

−

−

=1

max

( )

LhLhflW

Frlr

µ

µ

+

+

=1

max

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Diagram

Ra

Rrlf

Rrlr

ma

Wθg

Fbf

Fbr

h

h

lf

lrL

θg

Wf

Wr

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Vehicle Acceleration

• Governing Equation

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

maRF mγ=−∑

200025.004.1 εγ +=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 rpmCurb 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 rlrbf

++=

µµmax

( )[ ]L

fhlWF rlfbr

+−=

µµmax

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Braking Force

• Ratio

• Efficiency

( )( ) rear

frontfhlfhlBFRrlf

rlr =+−

++=

µ

µ

µη maxgb =

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Braking Distance

• Theoretical – ignoring air resistance

• Practical

• Perception

• Total

( )( )grlb

b

fgVVS

θµη

γ

sin2

22

21

±+

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛±

−=

Ggag

VVd2

22

21

pp tVd 1=

ps ddd +=

aVVd

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

rtVG

gag

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

( )( ) ( )

( ) aVVVV

GgagVVd

222

221

22

21 075.1

2.11075.1

2.111

247.1

02.322.112.322047.1

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 mph a = deceleration, 11.2 ft/s2 in US customary units tp = Conservative perception / reaction time = 2.5 seconds

ps VtaVd 47.1075.12

+=

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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.