CEE 320 Winter 2006 Vehicle Dynamics CEE 320 Steve Muench
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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
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 =
+=
147101.0
VfrlWVfP rlrlR =
sec5501
lbfthp
⋅=
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Grade Resistance Rg
Composed of – Gravitational force acting on the vehicle
gg WR θsin=
gg θθ tansin ≈
gg WR θtan=Gg =θtan
WGRg =
For small angles,
θg W
θg
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
r
MF dee
ηε 0=
( ) π2
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
επ 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
µ
µ
−
−
=1
max
( )
LhLhfl
WF
rlr
µ
µ
+
+
=1
max
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Diagram
Ra
Rrlf
Rrlr
ma
Wθg
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 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 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 rlrbf
++= µµmax
( )[ ]L
fhlWF rlfbr
+−=
µµmax
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Braking Force
• Ratio
• Efficiency
( )( ) rear
front
fhl
fhlBFR
rlf
rlr =+−++
=µµ
µη maxgb =
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Braking Distance
• Theoretical– ignoring air resistance
• Practical
• Perception
• Total
( )( )grlb
b
fg
VVS
θµηγ
sin2
22
21
±+−=
±
−=G
ga
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.
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