Mass Properties and Automotive Braking, Rev b

Society of Allied Weight Engineers, Inc.
Aerospace • Marine • Offshore • Land • Allied Industries
SAWE
Mass Properties and Automotive
Braking
SAWE Paper 3766
2022 SAWE 81st International Conference, Savannah GA
Brian Paul Wiegand, PE
Northrop Grumman Corporation, Retired

SAWE
Background
Have written papers concerning various automotive performance aspects
and the significance of Mass Properties to that performance:
• Acceleration (0-60 MPH, Quarter-Mile ET, Top Speed).
• Lateral acceleration (Max Lat G’s, Roll Stiffness, Roll Gain).
• Road Shock and Vibration (Transmissibility/Gain, Gyroscopic Reaction, Road Contact).
• Ride (Pitch and Bounce, Frequency of Motions, “Flat Ride”).
• Crash Survival (Peak Deceleration, Head Injury Criterion, Neck Injury Criterion).
• Directional Stability (Understeer Gradient)
Now a paper on yet another aspect of automotive performance and the
Mass Properties significance to that performance aspect:
• Braking (60 – 0 MPH, Etc.)
2
Society of Allied Weight Engineers, Inc. proprietary - Document contains no export controlled technical data

SAWE
Mass Properties and Automotive Braking
3
Apparently the driver was able to “lock up” the brakes; that is, there was no rotation of the
wheels at all, causing the coefficient of traction to drop to the dynamic level and skidding the car
to a stop leaving a good deal of rubber on the road. This is hard on the tires and not the fastest
way to a stop, to say nothing of the possible loss of directional control. The fastest, safest way to a
stop is with the tires rotating under just enough brake torque on the verge of “lock up” so that the
static coefficient of traction is still being utilized to develop the maximum braking force possible.

SAWE
4
Braking is the reverse of acceleration; just as effective mass was very
significant with regard to acceleration, it also is important with regard to braking. The
simplest (2-dimensional) depiction of a braking vehicle of weight “Wt” initially
moving with a velocity “V”, but decelerating to a stop in a distance “d”, is:

SAWE
5
The work done by the total resistance force “F”
between Points 1 and 2 is “F × d”. If the vehicle comes
to a complete stop (“V = 0”) at Point 2, then the work
done by that point has completely dissipated (equaled)
the vehicle’s kinetic energy as it existed at Point 1,
where the braking effort was initialized:

SAWE
6
Braking to a stop requires dissipating all the
kinetic energy of the vehicle, not just the energy
associated with the translationally moving mass, but
the rotationally moving mass as well. Substituting for
“m” the effective mass “me” (but dropping the “I2”
term for a “clutch disengaged” condition) brings the
model a little closer to reality:

SAWE
7
Where the symbolism is:
F = All resistance forces in parallel but opposite to velocity vector (lb, N).
d = The braking distance (ft, m).
Wt = The total weight of the vehicle (lb, kg).
g = The gravitational constant, required for English units (32.174 ft/s2).
I1 = Rotational inertia about front axle line (lb-ft2, kg-m2).
I3 = Rotational inertia about transmission 3rd motion axis, plus the
1st and 2nd transmission motion axis inertias translated to the
3rd motion axis (lb-ft2, kg-m2).
I4 = Rotational inertia about rear axle line (lb-ft2, kg-m2).
AR = Axle (differential) torque ratio (dimensionless).
AE = Axle (differential) energy efficiency (dimensionless).
RD = Dynamic rolling radius at drive wheels (ft, m)
V = The initial velocity when braking commences (ft/sec, m/sec).

SAWE
8
Another step toward reality involves “F”; there is
no single force acting on the vehicle to slow it down.
There are the traction forces “Ff, Fr” at the tire/ground
contact points (which result from the brake torques “Tf”
and “Tr”), and then there are rolling resistance forces
“ff, fr”, plus the aerodynamic drag force “D”:

SAWE
9
The aero drag “D” and lift “L” forces may be calculated with “V” in
feet-per-second, the areas “Af” “Ap” are in square-feet, and the resulting
“D” “L” forces are in pounds, per the equations:
The rolling resistance at the front axle “ff” and the rolling resistance
at the rear axle “fr” are expressed by the equations:
Here the velocity “V” is in units of mph, while the axle normal forces
“Nf” “Nr” and the resulting axle rolling resistance forces “ff” “fr” are all in
terms of pounds.

SAWE
10
The axle braking traction forces “Ff” and “Fr” are also dependent on
the axle normal loads “Nf” and “Nr” :
However, the traction coefficients “μf” and “μr” are themselves a
function of normal load or, more precisely, functions of contact area
pressure “Pc”:
And since “Pc = N/Ac” (contact pressure equals normal load divided
by contact area):

SAWE
11
Of course, the contact areas also vary in accord with the normal
loads. The front and rear tire contact areas (multiply by ”2” for per axle)
vary in accord with normal loads:
Where:
Ac = Tire to ground plane gross
contact area (in2)
Lc = Tire to ground contact area
length (in).
tw = Tire tread width, assumed
constant with load (in).
Ri= Tire no-load inflated radius
(in).
d = Tire vertical deflection under
load (in).

SAWE
12
To use the area equations requires calculation of the tire deflection
“d” under normal load “N” (lb)…
…and also requires the tire tread width “tw” which (when lacking a
measured value) can be approximated per the Michelin formula:
Where for these last two equations:
KZ = Tire vertical stiffness (lb/in).
tw = Tire tread width, assumed constant with load (in).
= Tire section aspect ratio (dimensionless).
SN = Tire nominal section width (mm).
N = The normal load on the tire (lb).
d0 = Tire deflection function “y-intercept” value (in).

SAWE
13
The tire vertical spring rate “Kz” can be approximated by the
Rhyne Equation:
Where:
KZ = Tire vertical stiffness (kg/mm).
Pi = Tire inflation pressure (kPa).
= Tire section aspect ratio (dimensionless).
SN = Tire nominal section width (mm).
DR = Wheel rim nominal diameter (mm).

SAWE
14
The tire deflection function “y-intercept” value (“d0” ) can be
approximated by one of the following equations depending on tire type:
Bias Tire Deflection Intercept Estimation Equation:
δ0 = -0.0000000989 Kz
2 - 0.0002913191 Kz+ 0.7597928883
Radial Tire Deflection Intercept Estimation Equation:
δ0 = 0.0000001014 Kz
2 - 0.0008292180 Kz + 0.8434083686
For determination of
the tire-to-ground net
contact area a “Land:Sea
Ratio” table is needed

SAWE
15
The brakes on any modern car can easily “lock” the wheels, but that’s not
max deceleration. For max deceleration the braking force must be exactly equal,
not greater than, the max static traction force “N × μ” that the tires are capable of.
This means that the torque “T” produced by the brakes must be equal to (not
more than) “F × R”:
Traditionally this meant that a lot of time and effort was spent on
developing brake proportioning systems, and later Anti-skid Braking Systems
(ABS). Modern ABS (and skilled drivers) can be so good that the above equations of
perfect brake balance may be assumed to hold. It is this assumption of perfect
brake proportioning which makes the functioning of the braking computer
simulation program “MAXDLONG.BAS” possible.

SAWE
16
Just as lateral traction force
causes a tire deformation known as
“slip angle”, longitudinal traction
force causes a tire deformation
known as “slip”. This phenomenon
should be accounted for if braking
distance is to be accurately
determined.
Per the SAE definition of “slip”
the apparent vehicle velocity “Rω” is
related to actual vehicle velocity “V”,
so that at full wheelspin “%S = ∞%”,
at free rolling “%S = 0%”, and at
locked brakes “%S = -100%”.

SAWE
17
The longitudinal traction force-to-slip
relationship changes with normal load which
makes it difficult to find a simple equation
relating slip to force. However, if the traction
force is “normalized” through division by
normal load the resulting family of traction
coefficient-to-slip relations cluster tightly
together:
This makes it relatively easy to
derive a simple approximate coefficient-
to-slip relation:

SAWE
18
fμ = -0.000000222 V3 + 0.000085521 V2 - 0.010726832 V + 0.997410713
To further complicate matters, the coefficient of
traction also varies with velocity, which is continuously
decreasing during braking. There is no established
general formulation to account accurately for the variation
with velocity, and many researchers take it as being
negligible, while some resort to rough “rules of thumb”.
This figure illustrates the variation in peak traction
coefficient with velocity, and also illustrates the transition
from “static” (rolling) to “dynamic” (sliding) traction.
Fortunately, a 1969 paper by researchers Harned,
Johnston, and Scharpf presents some actual data
regarding the variation in tire traction coefficient with
speed. If that data is subjected to a regression analysis
then an equation for the fraction of tire traction coefficient
“fμ” left available at speed (in mph) can be obtained:

SAWE
19
Now that all the relations for a braking simulation have been covered, the
normal loads would seem to be key. These loads result from the static
longitudinal weight distribution as modified by dynamic “weight transfer” and
aero drag and lift:
Where:
Wt = Total vehicle weight (lb).
Wb = Vehicle wheelbase (in).
LCG = Longitudinal distance of CG from front axle (in).
me = Vehicle “effective mass” (lb).
a = Vehicle deceleration (in/sec2).
VCG = Vertical distance of CG from ground plane (in).
D = Aerodynamic drag force (lb).
VCP = Vertical distance of CP from ground plane (in).
L = Aerodynamic lift force (lb).
LCP = Longitudinal distance of CP from front axle (in).

SAWE
20
THE MAXDLONG.BAS FLOW CHART

SAWE
21
THE MAXDLONG.BAS “VALIDATION”

SAWE
22
MAXDLONG.BAS OUTPUT FILE (SHORT FORM)

SAWE
23
The weight effect on braking is
exactly as anticipated; the greater the
vehicle weight then the greater the braking
distance required. When the
weight/braking distances are plotted in the
traditional manner the result is:
In an effort to wring more insight from
the weight/braking distance data a
different plotting format is tried:

SAWE
24
The longitudinal center of gravity
(LCG) effect on braking distances is also in
accord with anticipation as was that of
weight. Plotted in the traditional manner the
LCG effect is illustrated:
It was expected that as the LCG moved
aft the braking distances would start to
decrease, as the increasingly aft LCG position
would begin to compensate for the dynamic
“weight transfer” forward. To see if this was
so another change in plotting format was
required:

SAWE
25
Now the vertical center of gravity
(VCG) effect on braking will be presented:
The traditional plot has a high degree
of overlap; a revised plot is certainly in
order. Replotting the VCG data as per
previous figures presents a new plot as
follows:

SAWE
26
The plot of the braking curves with
rotational weight variation is as
anticipated; the braking distances
increase with increasing rotational
weight:
Again, more interesting information
is to be obtained through a reformatted
plot of the data:

SAWE
27
CONCLUSIONS
Decreasing weight decreased braking distance from all velocities
with an average of 0.02061 ft/lb, whereas decreasing rotational weight
decreased braking distance with an average of 0.06663 ft/lb. Therefore,
reducing the weight of rotating vehicle components is 3.2 times more
effective than reducing the weight of vehicle components that move in
translation only. Increasing weight and rotational weight increased
braking distance with an average of +0.01866 ft/lb and +0.06372 ft/lb
respectively, indicating that increasing rotational weight is about 3.4
times more detrimental to braking performance than increasing weight.
This illustrates that the effects of decreasing or increasing weight and
rotational weight about the nominal values are not symmetrical, but the
respective R2 values of 0.9888 and 0.9998 indicate that the effects are
near linear.

SAWE
28
CONCLUSIONS
The variations in LCG and VCG also produced results that were exactly
as anticipated, but very different from each other. The variation in LCG was
highly nonlinear in effect with R2 = 0.3369, while the VCG effect was much more
straight line at R2 = 0.8719. Increasing (moving aft) the LCG decreased
stopping distances up to a point (LCG = 71.56 in, or 81.76 cm, for the subject
vehicle) during which the movement tended to even out the dynamic fore/aft
tire-road contact pressures, and after which point the movement tended to
uneven out those pressures. As the static fore/aft tire-road contact pressures of
the subject vehicle were near even, any decrease in VCG, which is so
instrumental in the dynamic “weight transfer”, improved braking performance.
The bottom line of all of this is an emphasis on the importance of
light weight, aft bias LCG, and minimum VCG, for optimum braking
performance.

SAWE
29
Q&A (5 min)

SAWE
30
Back-up Slides

SAWE
31
There have long existed a number of methods for estimating the braking distances of a vehicle, some
graphical, and some like the following example from J.Y. Wong’s Theory of Ground Vehicles, 4th Ed.
(2008). The braking situation is illustrated:
Sb = braking distance (ft).
Wt = weight (lb).
V = initial vehicle speed (ft/s)
μ = coefficient of traction (dimensionless)
θs = slope angle (radians or degrees)
Cr = coefficient rolling resistance (dimensionless)
ηb = braking efficiency (dimensionless).
g = gravity (32.174 ft/s2)
Cae = “coefficient aero efficiency” (lb/ft),
which is:
ρ = atmospheric density (0.07528 lb/ft3 @ STP)
CD = drag coefficient (dimensionless)
Af = frontal area (ft2)

SAWE
32
Tire parameter determination begins with the tire designation. Starting around
1976 in the US tire designations have conformed to the “P-Metric” system, which meant
that passenger car tire designations would look like:
P152/92B16
In the US the “P”, for “passenger car”, is generally dropped, but the nominal
section width in millimeters (“152”), the aspect ratio (“92”), the tire type (“B”, for bias),
and the wheel diameter in inches (“16”) remain. We can find the inflated no-load radius
“Ri” directly from this designation. When a tire is measured for its nominal dimensions it
is placed on a wheel of appropriate width and inflated to a specific pressure. For the
example tire that means a 5 in wide rim and 40 psi inflation; from the resultant
designation the inflated no-load radius may be determined:
Ri = DR/2 + 0.03937 SH

SAWE
33
Ri = Tire no-load inflated radius (in).
DR = Wheel rim nominal diameter (in).
SH = Tire nominal shoulder height (SN × /100) (mm).
For the given example tire designation this means that at the measurement
conditions the inflated no load radius is:
Ri = 16/2 + 0.03937 (92/100)152 = 13.52 in
Adjustments may be made to this inflated radius to account for the small variations
resulting from the use of inflation pressures other than the measurement pressure by use
of a delta-strain equation.

SAWE
34
Subject Vehicle; Automobile Model:
1958 Jaguar XK150S
Nominal Parameter Values:
Wt =3639.80 lb
Wf = 1748.03 lb Wr = 1891.76 lb
Ws = 3117.93 lb
Wusf = 214.03 lb Wusr = 307.85 lb
LCG = 53.01 in VCG = 22.91 in
LCGs = 51.82 in VCGs = 24.62 in
husf = 12.62 in husr = 12.68 in
RG = 16.10 deg/g
Krollf = 276.59 lb-ft/deg
Krollr = 167.78 lb-ft/deg
Lwb = 102.00 in
tf = 51.75 in tr = 51.25 in
hrcf = -7.8 in hrcr = 14.4 in
hr = 20.66 in hra = 3.48 in

SAWE
35
1958 Jaguar XK150S Mass Properties

SAWE
36
1958,JAGUAR,XK150S_FHC
102,4.09,0.97
3640,53.01,22.91,85.96,3.53,86.27
0.42,18.53,0.20,36.48,45.4,30.1
DUNLOP RS4 6.00x16 W/TUBE ON 5-IN RIM
23,13.51,3.97,0.70,848,0.44
1.47,112.8,0.001195
1.2948,0.008094,0.006044
0.65,0.05,450000
DUNLOP RS4 6.00x16 W/TUBE ON 5-IN RIM
26,13.51,3.97,0.70,933,0.40
1.42,117.5,0.001370
1.2107,0.007933,0.006076
0.65,0.05,450000
5.66,5.66
INPUT FILE: JAGDECEL.DAT

SAWE
37
For heavy trucks, such as the tractor-trailer combos known as
“semis” or “eighteen wheelers”, the use of engine braking to reduce the
work load on the brakes is very common. To enhance the engine braking
effect, numerous systems have been developed such as the Berna,
Saurer, Williams, Oetiker, Clayton, Fowa, and Jacobs. The Jacobs
system has been very popular, and is commonly referred to as the “Jake
Brake”. Essentially all these systems enhance the energy absorption of
the engine by providing for more compression strokes (two-stroke
operation vs. four-stroke) and/or a higher back pressure (a valve shuts
the exhaust manifold off from atmospheric drain).
ENGINE BRAKING

SAWE
38
FIN

Mass Properties and Automotive Braking, Rev b

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Mass Properties and Automotive Braking, Rev b

Similar to Mass Properties and Automotive Braking, Rev b (20)

More from Brian Wiegand

More from Brian Wiegand (11)

Recently uploaded

Recently uploaded (20)

Mass Properties and Automotive Braking, Rev b

Editor's Notes