Dynamics: Vehicle Braking Principle
Critical in the determination of stopping sight distance, roadway surface design, and accident avoidance systems
Outline
Principles of Braking
Braking Forces and Distribution Antilock Braking Systems
Braking distances
Safe braking distance
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Time-space diagram
dx = v (t ) dt 2 d x dv = = a (t ) 2 dt dt 3 d x da = = j (t ) 3 dt dt
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Integral form
x (t ) = v ( ) d
0
t
v (t ) = a ( ) d
o
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t
a (t ) = j ( ) d
0
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t
Special Cases: fixed location
xo
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Special Cases: constant speed
x(t)
vo
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time
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Special Cases: constant acceleration
Acceleration
x(t)
Deceleration
time
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Stopping Time
v ( 0 ) = v o , v ( t stop ) = 0
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Stopping Distance
Recall:
x(t ) = x0 + vo t + 1 2 aot 2
t stop = v0 a0
x stop = x t stop - x 0
vo 1 vo = vo - a0 ao 2 a0
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vo = 2 ao
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Braking Principles
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g
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Braking Force
Wr =
Wf =
Wl f - h(ma - Ra W sin g ) L
Wlr + h(ma - Ra W sin g ) L
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. . .
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Braking Force
Front axle
Fbf
Rear axle
max
=
W [lr + h( + f rl )]
L
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Fbr max =
W [l f - h( + f rl )]
L
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Braking Force Coefficient of road adhesion
Pavement Coefficient of road adhesion Maximum Good, dry Good, wet Poor, dry Poor, wet
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Slide 0.80 0.60 0.55 0.30 0.10
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1.00 0.90 0.80 0.60 0.25
Packed snow or ice
Antilock Braking System
What is Antilock Braking System (ABS)? How does it work? The ABS is a four-wheel system that prevents wheel lock-up by automatically modulating the brake pressure during an emergency stop. The anti-lock brake system consists of the following components: Hydraulic Control Unit (HCU) Anti-lock brake control module. Front anti-lock brake sensors Rear anti-lock brake sensors.
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Braking Force Ratio
A major brake design objective is to achieve the maximum attainable vehicle deceleration g (g=32.2 ft/s2 or 9.81 m/s2) To approach this maximum vehicle deceleration, vehicle braking systems must correctly distribute braking forces between the front and rear brakes. Brake Force Ratio (BFR), front force over rear force:
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Braking Force
Percentage of braking force that braking the system should allocate to the front axle for maximum braking is:
Percentage of braking force that the braking system should allocate to the rear axle for maximum braking is:
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Compromised Design
Braking efficiency:
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The maximum value for gmax is ?
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Compromised Design
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Light Truck
Passenger Car
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Braking Distance
Theoretical
Mass factor (1.04 for automobiles)
W + K aV12 + f rlW W sin bW S= ln 2 2 gK a W + K aV2 + f rlW W sin
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Ka =
2
CD Af
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Braking Distance
Theoretical
Considering braking efficiency:
b W + K aV12 + f rlW W sin bW S= ln 2 2 gK a b W + K aV2 + f rlW W sin
If V2=0:
2 bW K aV1 S= ln 1 + 2 gK a b W + f rlW W sin
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Braking Distance
Theoretical
ignoring air resistance
b (V12 - V22 ) S= 2 g (b + f rl sin g )
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Braking Distance
Theoretical
Practical: Worst-case conditions
Poor driver skills Low braking efficiency Weather conditions
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Braking Distance
Practical
V12 - V22 d= a 2g G g
For grade = 0
V12 - V22 d= 2a
Reaction
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Total
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Stopping Sight Distance (SSD)
Equation:
V12 SSD = + V1t r a 2g G g
Perception-reaction time Deceleration rate
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Stopping Sight Distance (SSD)
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from ASSHTO A Policy on Geometric Design of Highways and Streets, 2001
Note: this table assumes level grade (G = 0)
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EXAMPLE 1
A small truck is to be driven down a 4% grade at 75 mph. The coefficient of road adhesion is 0.95, and it is known that the braking efficiency is 80% when the truck is empty and it decreases by one percentage point for every 100 lb of cargo added. Ignoring aerodynamic resistance, if the driver wants the truck to be able to achieve a minimum theoretical stopping distance of 300 ft from the point of brake application, what is the maximum amount of cargo that can be carried?
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EXAMPLE 2
An engineering student is driving on a level roadway and sees a construction sign 600ft ahead in the middle of the roadway. The student strikes the sign at a speed of 35 mph. If the student was travelling at 55 mph when the sign was first spotted, what was the student's associated reaction time?
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