8- MISCELLANEOUS AUTOMOTIVE TOPICS, Rev. A

Brian Paul Wiegand, B.M.E., P.E.
AUTOMOTIVE DYNAMICS and DESIGN
1
Bill Molzon 2008
John Melberg 2008

2
Brian Paul Wiegand, B.M.E., P.E
CAMBER
ANGLE
(ε)
CASTER
ANGLE
( )
CASTER
TRAIL
(t)
(Generally suspension is designed to
have a small amount of negative
camber to maximize tire-road contact
patch on the outside wheel in
turning [counters the effect of tread
“lift” due to lateral distortion and
possibly creates some beneficial
camber thrust].)
(Although a rolling tire provides a
certain amount of self-aligning
torque through the phenomenon of
pneumatic trail, the front suspension
is usually set with a certain amount
of positive caster. Positive caster
enhances directional stability, but
also increases steering wheel effort.)

3
TOE IN/OUT
(b-c)
TOE ANGLE
(δ)
Negative
Positive
(Road cars are designed to have a
small amount of “toe in” to
enhance directional stability; most
race cars are designed to have a
small amount of “toe out” to
enhance maneuverability.)
SCRUB, FROM SUSPENSION
MOVEMENT OR STEERING
(Suspension and Steering Designers
seek to minimize scrub but generally
can not eliminate it completely.
Designers therefore often balance toe,
scrub, and camber effects so as to
neutralize each other.)

4
KINGPIN
ANGLE
(σ)
SCRUB
RADIUS
(rσ)
POSITIVE
NEGATIVE
(Scrub radius
affects directional
stability, steering
effort, tire wear,
magnitude of bump
feedback, and
braking stability.
The result is that
there is always
some compromise,
and where that
compromise is
made depends on
the vehicle use and
type. KPI is usually
in the range of 11
to 15.3 deg; the
higher the more
stabile.)
(Note how the positive wheel
camber εW interacts with the
kingpin angle σ.)

5

6
Everything so far throughout this class has been predicated on
the , and not the actual rates of the springs
themselves. Quite simply, this is because a spring as installed in
the suspension system of a car generally works at a mechanical
disadvantage, and tends to be become “softer” at the wheel, as
per the following illustration:
Summation of the moments about “ ”
gives the relationship of force “ ” to force
“ ”, and similar triangles gives the
relationship of deflection “ ” to deflection
“ ”:

7
As can be seen from those relationships, an
as measured at the spring (“ ”) will result in a very
different “ ” as measured at the wheel
(“ ”). Since the force “ ” is less than “ ” and the
deflection “ ” is greater than “ ”, the
at the wheel “ ” will be a good deal than the
” at the spring. In fact, if a spring of known
spring rate “ ” is installed as shown in the figure, then the
at the wheel “ ” can be calculated as
per the following example:

8
The quantity “ ” could be called the “ ”
or the “ ”, but in either case is symbolized by
“ ” and is used to define the relationship between the
suspension spring rate and the suspension spring rate
at the wheel:
There is a further complication that occurs if the
suspension spring is not installed in an upright position
as shown in the previous figure, but in an
position…

9

10

11

12
Consider a vehicle that has a “double wishbone” IFS; the front suspension
spring rate (at the wheel) is 98.5 lb/in, with a corresponding “spring base” of
51.75 in. The vehicle also has a front anti-roll bar (ARB) of 84.7 lb-ft/deg
stiffness. This ARB roll stiffness is directly additive to the front axle roll
stiffness as calculated per the following formulae for an independent
suspension:
Where:
= Independent suspension roll
stiffness (lb-ft/degree).
= Combined stiffness of suspension
spring and tire at the wheel (lb/in).
= The “spring base” (track) width
at the axle line (in).

13
Where:
= Non-independent suspension
roll stiffness (lb-ft/deg).
= Effective suspension spring
stiffness at the mount to axle (lb/in).
= “Spring base” distance between
spring mounts along the axle (in).
= The spring stiffness of a tire
on the axle (lb/in).
= The “spring base” (track) width
at the axle (in).
The vehicle has a “live” (driven) beam suspension in the rear. The rear
suspension spring rate (at the wheel) is 128.8 lb/in (23.00 kg/cm), with
“spring bases” of 51.25 (132.7 cm) and 38.0 in (101.9 cm). The rear axle roll
stiffness is calculated per the following formulae for a non-independent
suspension:

14
Enough information has been given in the preceding vu-
graphs to be able to determine that the example vehicle
has front and rear axle roll resistances of 276.6 lb-ft/deg
(375.0 Nm/deg) and 167.8 lb-ft/deg (227.5 Nm/deg),
respectively. That is a total resistance to rotation about
the “ROLL AXIS” of 444.4 lb-ft/deg (602.5 Nm/deg). Now,
regarding that roll axis; how is its location determined
and what is the roll moment about that axis? The basic
relation between the roll moment, the roll stiffness, and
the roll angle is:

Each suspension type
requires a particular approach
to locating its static roll
center. Once the static roll
center has been determined
the full range of the roll
center’s movement through a
full cycle of suspension rise
and fall may be plotted and
the relationship determined
through regression analysis.
However, for an early
conceptual study the roll
centers are usually just taken
as being constant at the their
initial static positions.
15

16
The front and rear RC’s change in location dynamically, so the roll
axis orientation will also change, but the roll moment arm “h
cos(Φ)” can initially be taken as constant. The basic roll relation
can be replaced by the exact relation…

17

18
So, vehicle roll axis orientation affects the load transfer
front-to-rear balance in maneuver, and thereby the
directional stability. The key to determining the roll axis
location was the fact that it only takes two points to define a
straight line, in this conventional case the front and rear
. But, if more axles are added to the configuration then
things can become somewhat indeterminate if care has not
been taken to keep all the roll centers on the same line.
When Daimler Chrysler of Brazil added a third axle to an
existing bus design for increased load capacity the
prototype IBC-2036 exhibited “ ” during
lane change maneuver. To identify the cause and find a
solution to this problem Daimler Chrysler engaged the
services of Debis Humaitá IT Services Latin America Ltd.

19
IT Services found that the roll center of the third suspension was
14.06 in (371 mm) below the line between the first and second axle
roll centers. In its essence, the problem resulted from the fact that
the third axle addition had created a condition of two competing roll
axes, resulting in rather schizophrenic behavior.

20
Ideally the third axle should have been revised so its
would fall on the line, but practical
considerations of design modification complexity
and expense forced a compromise solution. That
compromise solution consisted of modifying the
rear suspension to get the third to fall 6.42 in
(165 mm) above the line; the compromise is
closer to the ideal position by 7.64 in (206 mm). This
modification to the prototype bus brought the yaw
“wiggle-wobble” during a lane change maneuver to
within acceptable limits for Daimler Chrysler
(although still not ideal).

21

22
In 1758 Erasmus Darwin (1731-1802), then future grandfather of
Charles Darwin (1809-1882), invented the so-called “Jeantaud-
Ackermann” steering system in order to rectify some of the
shortcomings of the “axle” steering system then in use for horse
drawn carriages, coaches, and wagons. The main shortcoming of the
axle steering system was that the steered wheels had to be
significantly smaller in radius than the non-steered wheels in order to
clear the body/chassis at full lock; this incurred significantly greater
road shock transmission as such shock varies inversely with the size
of the wheel. Another shortcoming was that axle steering reduced the
vehicle resistance to overturn when cornering. A third shortcoming,
which would not be readily apparent until the birth of the automobile,
was that the encountering of a bump in the road by one of the steered
wheels would generate a large impact moment about the axle pivot
point. The drivers of some of the earliest automobiles would
experience these moments as severe “kick-back” through the tiller.

23
At around 1816, Darwin’s steering innovation was apparently
independently reinvented by a German carriage manufacturer
named Georg Lankensperger. A certain Rudolf Ackermann
became acquainted with Lankensperger and his invention and,
acting ostensibly as Lankensperger’s agent, obtained an
English patent (number 4212) in his own name in 1818.
However, the geometry of the steering arrangement in
Ackermann’s patent illustration was both extreme (control arm
inclination with respect to the longitudinal axis was only about
9 to 13 degrees, while Darwin’s arrangement involved angles of
23 to 30 degrees) and difficult to discern. Consequently,
“Ackermann Steering” became associated with the vastly
inferior geometrical layout of zero degrees inclination of the
steering arms with regard to the longitudinal axis.

24
Rudolf Ackermann’s 1818 English patent number 4212 steering
geometry illustration, and Erasmus Darwin’s 1758 steering
geometry:
1818 1758

25
In 1878 French carriage maker Charles Jeantaud
corrected the “Ackermann Steering” back to the
original (Darwinian) concept. However, to this day the
steering geometry in question is still known as
“Ackermann Steering”. However the time was still not
right for a general acceptance; Rudolf Ackermann,
despite all his scheming perfidy, never profited from
his patent. Life is not only unfair, but often ironic.
The following diagram (four wheel) is of “Darwin
Steering” geometry in a low speed (no drift angles)
turn, which is the only kind of turn for which such
geometry is truly valid…

26
Low Speed
4 Wheel
Model
High Speed
2 Wheel
(“Bicycle”)
Model

27
The exact steering geometry, mainly a matter of the angle of inclination
of the control arms w.r.t. the longitudinal axis, is dependent on the
application. For conventional road cars of moderate performance the
angle has been traditionally determined by simple geometric
considerations (The Motor Vehicle, pg. 873). However, race cars, which
make turns at the very limit of lateral acceleration, often use an “anti-
Ackermann” steering because the inside wheel is essentially
completely unloaded vertically at such limiting acceleration; only the
steering angle of the dominant outer wheel is important.
Since for race and ultra-high performance cars the geometry must be
determined by dynamic considerations (max lateral acceleration),
which is very dependent on specific tire and vehicle characteristics, the
“slip angles” adopted by the tires at maximum lateral acceleration is
the prime determinant for steering geometry design (“Tech Tip:
Steering Geometry”, Optimum G1 article, pg. 1-5).
(1- OptimumG is an international vehicle dynamics consultant group located in Denver, CO, that works with automotive
companies and motorsports teams to enhance their understanding of vehicle dynamics through seminars, consulting, and
software development.)

28
Regarding control arm inclination, an engineer named Walter F. Korff
worked out a table that applies to beam axles with the control arms
behind the axle line. One could use Mr. Korff's table as a starting point
in conceptual design, then adjust the angles as the design is refined
using more sophisticated methodology (“X” angles seem to be based
on the trigonometric relation “tan-1(t/2lwb)” with 6 to 7 degrees added to
account for “slip”):

29
In 1976 F1 designer Robin Herd, perhaps inspired by the 1975 Tyrrell
Project 34 (P34) six-wheel F1 race car, constructed a six-wheel/three-
axle variation on the standard March 761 that he termed the March “2-
4-0”. Herd reduced the rear “wheel” size from the usual 26 in (66.0 cm)
diameter to 20 in (50.8 cm) diameter (equal to the usual front “wheel”
diameter) while doubling the number of rear wheels by having two rear
drive axles in tandem. Herd felt that this modification would improve
the vehicle aerodynamics and increase traction at the rear while
avoiding the problems associated with unusual tire sizes. While he
was probably quite correct in his assessment of the benefits of this
new configuration, he unwittingly created new problems.

30
One problem was that
Herd had created two
conflicting turn centers,
points “ ” and “ ” in
the figure shown, which
had to have an adverse
effect on handling.
The second problem
was that the two driven
rear axles would tend to
rotate at different rates,
yet there was no
differential “ ” between
the two rear axles to
accommodate this rate
variation.
MARCH 2-4-0
CONFIGURATION
CORRECTIVE
MEASURES

31
March Engineering at the time was financially strapped,
and the development of a proper transaxle for the
unusual dual drive axle configuration was not possible.
However, construction went ahead anyway as it was felt
that the publicity generated by such a novelty would
result in some financial success even if there was no
racing success. The 2-4-0 started testing in late 1976 and
immediately the transaxle casing revealed itself as
insufficiently stiff, allowing gears to slip out of mesh. The
gearbox was crudely strengthened (
), and the improved vehicle revealed itself
as having “incredible traction” in the wet, but further
development was halted as problems continued.

32
The improved traction that was demonstrated by the 2-4-0 concept
inspired British hill climb competitor Roy Lane to think that the
concept might still be valid for his very different type of competition,
where raw traction was much more significant. Lane bought a 2-4-0
rear end and mated it to a March 771 forward structure. In the wet the
car performed spectacularly, but in the dry it did not do so well.
Reportedly the car was but perhaps
even more significant was that the because
of the lack of a differential between the two rear drive axles. The shaft
between the two axles would “wind up” as the car was driven in a
turn, when the shaft
suddenly unwound. A proper transaxle incorporating a third between-
axles differential and making appropriate use of titanium was
indicated, but was beyond Lane’s financial capability, so the 2-4-0
concept died without the chance for vindication.

33
Other than “Conflicting Roll Centers” and “Conflicting Turn Centers”,
there are many other factors than can adversely affect the steering of
an automobile; “Scrub”, “Bump Steer”, “Roll Steer”, “Compliance
Steer”, “Camber Thrust”, “Conicity”, and “Ply Steer” constitute some
of the most common factors.
Scrub is when the kinematics of the suspension cause a lateral tire
movement during upward movement of the
suspension. The lateral stiffness of the tire times
the lateral scrub = a lateral disturbance force (in
the case shown it pulls to the left). Note that in
the illustration there is also an accompanying
camber change which in this case also results
in another force pulling to the left: the “Camber
Thrust”. In the situation shown these two forces
would be additive; a goal of the Suspension
Engineer is to try to get such forces to cancel
each other out.

34
Bump Steer is the tendency of a wheel on a car to steer as the wheel
moves upwards relative to the body. It is typically measured in units
of steer angle per unit of upwards motion (deg/foot, or rad/meter, etc.).
Typical values are from 2 to 10 degrees per meter for IFS wheels; solid
axles generally have zero Bump Steer (but may still have “Roll Steer”)
in most cases. If Bump Steer can be held to less than 0.20 degree of
toe out change per inch of suspension travel then it should not be a
problem, but no amount of bump steer is desirable. Excessive Bump
Steer increases tire wear and makes the vehicle more difficult to
handle on rough roads.
In a Bump Steer situation, both wheels may rise together, or just one
wheel may rise; note that if both wheels on an axle line move upwards
by the same amount there tends to be no net steering effect. The
linearity of the bump steer vs. lift plot is important. If it is not “linear”
then the length of the tie rod or some other parameter may need to be
adjusted. Bump or roll can result in inadvertent “toe out” steering due
to poor positioning of steering pivot points and/or poor angular
orientation of steering components, or due to improper length of outer
tie rods.

35
For instance, Bump Steer (and Roll Steer) can result from the outer
tie rod being of position and/or length such that for the same travel
upward it results in greater lateral movement than the lower
suspension arm. The following illustrates both good and bad outer tie
rod geometry:
(Figure {highly modified} from: Riley, Robert Q.; Automobile Ride, Handling, and Suspension Design, Robert Q. Riley
Enterprises LLC, Phoenix, AZ.)

36
“Roll Steer” is the term for the tendency on modern cars to have the
IFS wheels steer “outward” as the vehicle enters a turn and the sprung
mass rolls; this produces a condition of roll understeer. The rear
suspension is usually set up to minimize roll steer, where possible. Roll
Steer is an important part of the stability “budget” used in a “Bundorf
Analysis” to determine a vehicle's total understeering characteristic.
“Compliance Steer” is due to the use of “rubber” bushings in the
mounting of suspension components. When longitudinal and/or lateral
forces are generated at the ground contact section of a tire, those
forces cause deflections at the bushings. The deflections alter the
alignment, and may thereby produce in effect a “steer angle”. This
compliance “steer angle” may add to or subtract from the intended
steer angle created by the use of the steering wheel.
1 - “Bundorf Analysis” is a measure of the characteristics of a vehicle that govern its understeer balance. The
understeer characteristics are measured in units of degrees of yaw per “g” of lateral acceleration. This analysis
technique is named after GM automotive engineer R.Thomas Bundorf.

37
“Camber Thrust” refers to the force
generated perpendicular to the direction
of travel of a rolling tire due to
its camber. Thrust is generated whenever
a rotating tire goes into a cambered
position; points on the tread, which
would freely follow an elliptical path
projected onto the ground, are instead
forced to follow a straight path when
coming into contact with the ground due
to the traction force. The total point path
deviation towards the direction of the
lean is a deformation of the tire
tread/carcass caused by the total force in
the direction of the lean, i.e., the “Camber
Thrust”.

38
“Camber Thrust” is approximately linearly proportional to camber angle for
small angles, reaches its steady-state value nearly instantaneously after a
change in camber angle, and so does not have an associated “Relaxation
Length”1. “Camber Stiffness” is a parameter used to determine the Camber
Thrust generated by a cambered tire and is influenced by tire inflation
pressure and normal load. For small angles, the Camber Thrust is
approximated by the product of the Camber Stiffness and the Camber
Angle. The Camber Stiffness is the rate of change of camber force with
Camber Angle (∆FY/∆ε) at constant normal force and inflation pressure. The
net Camber Thrust is usually toward the front of the tire/ground contact
patch and so generates a Camber Torque such that it tends to steer a tire
towards the direction of lean (an alternate explanation for this torque is that
the two sides of the contact patch are at different radii from the axle and so
would travel forward at different rates unless constrained by traction with
the pavement, but that is more commonly termed “conicity”).
1 “Relaxation length” is a property of pneumatic tires that defines the delay between when a steer angle is introduced
and when the slip angle and cornering force reach steady-state values. It has also been defined as the distance that
a tire rolls before the lateral force builds up to 63% of its steady-state value. It can be calculated as the ratio
of cornering stiffness to lateral stiffness, where cornering stiffness is the ratio of cornering force to slip angle, and
lateral stiffness is the ratio of lateral force to lateral displacement.

39
(Figure from: Riley, Robert Q.; Automobile Ride, Handling, and Suspension Design, Robert Q. Riley Enterprises
LLC, Phoenix, AZ. It would have been helpful if the tire designation and inflation pressure had been given.)
Donald Bastow’s Car Suspension and Handling (1980) has essentially the
same explanation of Camber Thrust generation (pp. 74-75), and relates
how “old time” vehicles generally had much higher front suspension
static camber angles than are prevalent today, and in the interest of
decreased tire wear the suspensions were designed with a Toe-In setting
intended to negate the Camber Thrust.

40
(Figures from Bastow’s Automobile Ride, Handling, and Suspension Design, pg. 117 on left, with figure on right
from an unidentified source regarding motorcycle tire behavior. Motorcycles can corner with such a high degree of
camber and slip angles that the tire is in danger of “rolling off” the rim. Note that the camber thrust would seem to be
simply additive to the slip angle thrust per the motorcycle tire plot. For passenger car suspension design camber is
generally used to counter “curl up” of the tire contact patch under lateral load for improved traction.)
Bastow also provides an estimation formula for Camber Thrust (pg. 82) as
the normal load times the tangent of the camber angle “Fcamber = N ×
tan(ε)”, which for small angles can be further approximated as “Fcamber = N
× εdegrees /57.3”. Since the previous graph is for “N = 1000 lb” it would be
interesting to see how closely those approximations follow that
plot…Bastow provides example problems involving camber thrust (pp. 88
and 91) with an interesting plot on page 117 (shown below, left).

41
Bastow’s estimation formulae for Camber Thrust may be useful if there is
absolutely no other information given, but like all such formulae may
produce results that are far from acceptable in today’s demanding
environment. Taking the previous graph for “N = 1000 lb” and using the
approximation formulae provided by Bastow produces the following
results:

42
“Conicity” and “Ply Steer” refers to tire construction
features that cause lateral forces to develop in use; the
importance of these features has increased with the rise
of radial tire popularity. Generally, these features are
considered to be flaws in tire manufacture or design. Of
the two, “Conicity” seems to be the most troublesome
as “Ply Steer” effects generally are small and tend to go
unnoticed in normal applications. When extreme
enough, both their effects on vehicle performance are
manifested in vehicle pull and drift.
The explanation for “Conicity” force/torque is that the
two sides of the contact patch are at different radii from
the axle, and so would travel forward at different rates
as the tire rotates; therefore the large radius side goes
forward faster than the small radius side forcing a turn
toward the small radius side…

43
“Conicity” and “Ply Steer” refers to tire
construction features that cause lateral forces to
develop in use; the importance of these features
has increased with the rise of radial tire popularity.
Generally, these features are considered to be flaws
in tire manufacture or design. Of the two, “Conicity”
seems to be the most troublesome as “Ply Steer”
effects generally are small and tend to go unnoticed
in normal applications. When extreme enough, both
their effects on vehicle performance are manifested
in vehicle pull and drift.
The explanation for “Conicity” force/torque is
that the two sides of the contact patch are at
different radii from the axle, and so would travel
forward at different rates as the tire rotates;
therefore the large radius side goes forward faster
than the small radius side forcing a turn toward the
small radius side…

44
The presence of gears in a mechanical system can have important
implications with respect to the . In the system
depicted below the gear “ ” is the drive (or input) gear by virtue of being
directly connected to the motivating source, while gear “ ” is the driven
(or output) gear:

45
The significance of concretely identifying the “ ” and the
“ ” lies in the fact that the of a gear set is
determined in the form of an “ ”) ratio,
which is perhaps contrary to intuition. Thus the gear “ ”,
which is the ratio of the angular velocity of the drive gear (“ ”) to
the angular velocity of the driven gear (“ ”), is for the illustrative
example a matter of 1500 rpm to 1000 rpm (1500:1000), or 1.5
(1500/1000). If the angular velocities (or the angular accelerations:
“ ” and “ ”) are not known, then the gear may be
determined from the numbers of gear teeth (“ ”), or from the
gear “pitch” diameters (“ ”), or from the “pitch” radii (“
”) per the following relationships (note the reversal of input to
output at about the half-way point):

46
The “ ” is seemingly more rationally defined as the ratio of
the torque about the driven gear (“ ”) to the torque about the drive gear
(“ ”), and is for the illustrative figure a matter of 100 lb-ft (135.6 N-m) to 150
lb-ft (203.4 N-m), or 1.5 (150/100). Note that the and the
are (“ ”), but
:
Consequently, there is no need to make the distinction in notation between
the gear speed ratio “ ” and the gear torque ratio “ ”; one can just speak
of a gear ratio “ ” which is understood to require different (inverse)
handling depending upon whether one is looking for the or
the :

47
One purpose of this class segment is to show how an “ ” system
to the system of the previous figure can be determined.
Consider that the torque “ ” about gear “ ” necessary to cause an angular
acceleration “ ” of the gear and its rotating mass, which have a combined
rotational inertia “ ”, is:
Now, if the rotational inertia “ ” of the motivating source and gear “ ” can
be ignored for the moment (say that it is zero), then calculating backwards
from the above output torque about “ ” the necessary input torque about
“ ” is:
Of course, “ ” could not actually be zero, so the total torque required at “ ”
to angularly accelerate the entire system would be:

48
Remembering that “ ”, we can substitute “ ” for “ ” in the
previous equation, and obtain:
Then, by consolidating terms:
Thus the of the previous
figure has an about “ ” of “ ” or
(1281.5 kg-m2). There is one final caveat to all this, and that involves
the matter of (“ ”). If the “ ” where
to be included in the last equation, then the resulting formula for the
would be “ ”:

49
If the efficiency were, say, , then the for this example
system would be (1300.1 kg-m2), which makes sense as the energy loss
would make the system harder to accelerate rotationally, i.e., the system would seem to
have greater rotational inertia. Remember the efficiencies represent energy loss, not
increased stored kinetic energy, so whether to include the efficiency values or not in
the determination of an equivalent (effective) inertia is dependent upon the use to
which the result is to be put. With that in mind, the
of the previous figure may be depicted as:

50
If for some reason one wanted to find the of the geared
system about “ ”, then the process is very similar. Treating “ ” as if it were
the driver, the torque about “ ” to drive both “ ” and “ ” would be:
Again, utilizing the values from the geared system figure, the resulting
is now (2883.3 kg-m2). Again, if we were to
include the gear set efficiency “ ” then the formula would be:
The value for “ ” stays the same regardless of which direction the power
flow is regarded as being, so with “ ” pegged at the same 97% as before, the
value of the is (2930.5 kg-m2); the
corresponding equivalent ungeared system is as depicted in the following
figure…

51

52
The second purpose of this class segment is to show the relevance of the
“ ” and the “ ” has to the problem of
. For an automobile in acceleration the
, which when
adjusted to account for the resistance forces results in the
equation:
Where:
= Net accelerative force or thrust (lb, kg).
= Engine torque (lb-ft, kg-m).
= Transmission gear ratio (dimensionless).
= Transmission efficiency (dimensionless).
= Axle gear ratio (dimensionless).
= Axle efficiency (dimensionless).
= Dynamic rolling radius at the drive wheels (ft, m).
= Tire rolling resistance force (lb, kg).
= Aerodynamic resistance force (lb, kg).

53
The automotive upon which the acts
is determined per:
Where the symbolism unique to this equation is:
= Weight of the vehicle (lb, kg).
= Gravitational constant, which is required only when the weight and
inertias are expressed in force units and must be converted to mass
units. For example, when the units employed are in feet, pounds,
and seconds (English FPS system) then “ ” = 32.174 ft/s2 (standard
value).
= Rotational inertia about front axle line (lb-ft2, kg-m2).
= Rotational inertia about the crankshaft axis (lb-ft2, kg-m2).
= Rotational inertia about transmission 3rd motion axis, plus 1st and
2nd transmission motion axis inertias translated to the 3rd motion axis
(lb-ft2, kg-m2).
= Rotational inertia about rear axle line (lb-ft2, kg-m2).

54
This is set up to work in conjunction with the
as just presented previous, and those equations
have been used as the basis for a computer simulation of automotive
acceleration, with the results displaying a high degree of accuracy.
Alternatively, the equation for the automotive may
,
. For application to
automotive acceleration performance the efficiencies could be
incorporated as per the previous rotational acceleration examples, thus
the
:
However, then it is necessary to from the
corresponding version of the in order not to “double
count” the energy losses from gearing inefficiency.

55
When the application considered is longitudinal
deceleration, i.e. , then the effect of the gear set
efficiencies would seem appropriately accounted for by
inclusion in the determination as per:
Here the efficiencies are the geared
contributions to the ,
and thereby decreasing the amount of kinetic energy
requiring dissipation during braking. This is most
appropriate as the energy lost through gearing
inefficiency should result in less energy requiring
dissipation though the brakes.

56
POI (Products of Inertia) are the most neglected of all mass
property values because of their nature; they are not like other
mass properties. Compared to the other mass properties, the
use of, and hence the need for, POI may seem obscure. For
one thing, POI do not appear in a commonly encountered
dynamic relationship such as “ ” or “ ” so the
need for accurate POI values is less clear. Furthermore, POI
values can be difficult to intuitively assess; an aircraft or
automobile that is significantly larger than another such
vehicle is intuitively expected to possess greater weight and
MOI (Moment of Inertia), which is generally correct, but that is
not the case with POI. A small vehicle might possess a
significantly greater POI than a similar appearing but much
larger vehicle, which is contra-intuitive. Also, unlike weight
and MOI, POI can even be negative in value!

57
POI is a measure of the degree of asymmetrical distribution of mass in a
particular plane. The mathematical POI definition for each of the three
planes (XY, XZ, and YZ) is:
This has certain implications. If there is symmetry in mass distribution
about either of the axes that define the plane in question then the value of
that POI is zero. The sign of the POI value tends to depend upon which
quadrant of the considered plane has the most significant concentration of
mass; consider the following diagram of the XZ plane:
(Neg)
(Neg)
(Plus)
(Plus)

58
As was noted, the usefulness of POI values may seem
somewhat obscure due to the fact that the POI do not
constitute part of a common dynamic equation such as
“ ”. However, POI values are critically important
in the rotational transformation of MOI values from one
reference axis system to another reference axis system.
In the following limited case the “new” axis system ( ,
etc.), must share a common origin with the “old” system
( , etc.), but will be at some angle “ ” to it. In such a
case the transformation relation for the longitudinal MOI
( ) is:

59
When the first reference system is the Dynamic Reference Axis
System, and the second is the Principal Reference Axis System,
then the angle “ ” may be given special symbolism as “ ”.
These systems have a common origin at the vehicle c.g., in
contrast to the Design Reference Axis System whose origin is
located in an “arbitrary” fashion in that it is not at the c.g., but
placed according to some industrial or corporate standard:

60
The Principal Reference Axis System is a mass
property, like the weight or c.g. location, of the body
being considered. The angle “ ” which establishes the
angular orientation of that system’s X-axis with respect
to the design or dynamic X-axis may be determined per
the equation:
λ = Tan -1(2Pxz /[Iz – Ix]) / 2
Generally for a vehicle this represents a simple rotation
of the Principal XZ-Plane with respect to the Dynamic XZ
-Plane about a common Y-Axis, and there are no further
angular displacements (which is not the most general
case).

61
Another special case is when an inertia transformation
from the Design Axes System to the Roll Axes System
is required. Here we have not just an angular
transformation, but a translational one along a distance
“hr” as well:
Iroll = Ixs Cos2 - 2Pxzs Sin Cos + Izs Sin2 + Ws hr
2

62
The general case of mass properties transformation
from one Axes System to another Axes System is much
more complicated, requiring a reference axes
transformation program like “XFORM.BAS”. The input
data required for this program includes the nine possible
angular relationships between the two system axes, and
the two system coordinate sets for a common point:

63
Then the entire set of mass properties, excluding the
weight, has to be input in order to get the mass
properties as transformed into the new axes system:
Those mass properties, in the new reference system,
are then output:

64
Transformation of a
body’s mass
properties into some
new axes system is
only the secondary
function of
“XFORM.BAS”,
finding the Principal
Axes System of a
body whose mass
properties are known
in some other axes
system is the primary
function:

65
This test object is being spun
about the X-X axis at some
constant angular velocity “”.
Note that the object is
“statically balanced”; the spin
axis passes through the
object’s center of gravity.
However, the object is
“dynamically unbalanced” in
that the object has an
asymmetrical mass distribution
in the XY plane, which for the
sake of simplicity in exposition
is illustrated as being the result
of two component masses “m1” and “m2” positioned as shown. This dynamic
unbalance results in a set of rotating force vectors “F” at the upper and lower
bearings which can be measured, and from which the POI can be readily
calculated: Pxy = F L / 2

66
However spinning automobiles in such a fashion to find the Pxz is
not a practical proposition, and other empirical methods of
determination have traditionally proven less than accurate.
As a result automotive
Pxz data has been
scarce and unreliable,
but that has been
changing with time. The
chart to the right lists
data of both seemingly
good (in blue) and poor
(in red) quality; note
that it is the more
recent data that tends
to be the more accurate.

67

68
The matter of gyroscopic reactions is an area often neglected, yet very
important. Gyroscopic torque reactions result from the inevitable
changes in rolling axis orientation (precession) due to camber and
steering angle changes as the wheel moves with respect to the body due
to bumps, dips, and vehicle maneuver. Such unwanted reactions can
cause “shimmy” and “tramp”, and it is best to reduce those reactions as
much as possible in the interest of control and wear.
For a quantitative determination of gyroscopic effects , use the equation:
Where:
= Gyroscopic torque reaction due to precession (lb-ft).
= Rotational inertia of rolling mass (lb-ft-sec2).
= Angular velocity of rolling mass (radians/sec).
= Angular precession velocity due turn or camber change (rad/sec).

69
Let’s say a vehicle is traveling at a velocity of 60 mph (96.6 kph) when the
right front wheel encounters a 2 in (5.1 cm) high bump forcing the IFS to
go through 1.5 degrees of camber change (determined from the
suspension kinematics). The rolling inertia of 28.53 lb-ft2 (1.25 kg-m2)
corresponds to a rolling weight of approximately 52.1 lb (23.63 kg) with a
radius of gyration of 8.93 inches (0.74 feet, or 0.23 m). The suspension is
equipped with a 152/92R16 tire of about 12.63 in (1.05 ft, or 32.1 cm)
rolling radius; it is assumed that the time it takes for the tire to “engulf”
the bump/rise to bump height will be approximately equal to the time
necessary for the vehicle to travel a distance equal to the tire cross-
section height (aspect ratio × section width):
t = (((0.92 × 152 mm) / 25.4 mm/in) / 12 in/ft) /
((60 mph × 5280 ft/mile) / 3600 sec/hr))
= 0.0052 seconds

70
This allows for the determination of “ ” as camber angle change
per time (in radians per second):
= 1.5 degrees / 0.0052 seconds
= 288.5 deg/sec, or 45.9 rad/sec
As for the determination of “ ” it would be equal to the number of
tire rotations per time at speed (in radians per second):
= 60 mph × (5280 ft/m)/(3600 sec/hr)/(2 π × 1.05 ft)
= 13.3 cps, or 83.8 rad/sec
So finally we can determine the reaction torque (converting weight
units to mass units: mass = weight / 32.174 ft/sec2):
= (28.53 lb-ft2 / 32.174 ft/sec2) x 45.9 rad/sec × 83.8 rad/sec
= 3410.7 lb-ft, or 347.8 kg-m

71
Elimination of shimmy and tramp was one of the reasons for
adoption of independent front suspension. With a beam axle a
precession motion at one side was mirrored at the other side,
causing a double impact. This effect was to become especially
significant at the front axle as that was the steering axle. At the rear
axle the lack of steering complications helped slow the advance to
independent suspension; retention of the simpler and cheaper beam
axle continues to be a viable option up to the present day.

Early vehicles were satisfactory with a beam axle front suspension as
such vehicles tended to have lighter weight tire/wheel combinations
(though admittedly of somewhat larger overall diameter) and traveled at
slower speeds than later vehicles. Also, it wasn’t until around 1915 that
it became customary to have brake mechanisms included at the front
wheels (previously generally rear wheel only). As roads improved and
engines became more powerful, there was an increase in speed. With
heavier wheel/tire designs that included the mass of front brakes, the
use of beam axle front suspensions became less desirable. Also the
placement of the engine, although still in the front of the vehicle,
needed to be somewhat more forward in response to a need to modify
vehicle mass properties in an effort to control pitch motions; but then
the engine and the beam axle tended to vie for the same space. The
introduction of IFS incurred more expense, but meant the engine could
sit between the right and left front suspensions, which not only affected
a change in mass properties but tended to allow for shorter hood lines,
while the problems of shimmy and tramp, pitch motion, and rough ride
could all be alleviated. Brian Paul Wiegand, B.M.E., P.E.
72

73
There is something called the “Right Hand Rule” which allows for the
determination of the direction of gyroscopic torque reactions, of which the
following illustration is provided:

74
Fuel economy today is generally expressed in terms of distance traveled per
fuel consumed (ex.: mpg), and for two distinctly different operating
conditions: “City” and “Highway”. “City” operation tends to be at “low”
average speeds with many acceleration/deceleration cycles, while “Highway”
operation tends to be at “high” average speeds with few
acceleration/deceleration cycles. Since constant speed operation of internal
combustion engines tends to be their most efficient mode of operation, and
since internal combustion engine operation tends to be the largest single
source of energy loss for automobiles, the resultant “Highway” fuel
distance/consumption is always better than the “City” fuel
distance/consumption. To arrive at a set of “City” and “Highway” fuel
consumption figures for a vehicle the US EPA tests that vehicle by running it
through a series of driving routines, also called cycles or schedules, that
specify vehicle speed for each point in time during the test(s). For the 2007
and earlier model years, only the “City” and “Highway” cycles were used, but
starting with 2008 additional tests were used to adjust the “City” and
“Highway” results to account for higher speeds, air conditioning use, and
colder temperatures. Pre-2008 “City” and “Highway” figures have been
retro-adjusted dating back to 1985 (see http://www.fueleconomy.gov/mpg).

75
In 1972 a test cycle was invented by the newly created EPA to measure exhaust
emissions. This test cycle, which sought to mimic rush-hour traffic in downtown Los
Angeles with an average speed of 21.2 mph, is called the “City Cycle”, and is still in
use today. This dyno test is 11 miles long, takes just over 31 minutes to complete,
involves 23 stops, reaches a top speed of 56 mph, and has maximum acceleration
equivalent to a 18-second 0-to-60-mph run.
A second “Highway Cycle” was added in the late 1970’s as part of the introduction of
corporate average fuel-economy (CAFE) regulations. This 10.3-mile cycle, with an
average speed of a 48.4 mph and acceleration no more severe than in the city test, may
have been somewhat realistic in the days of the national 55-mph speed limit, but
doesn’t come close to approximating the highway behavior of today.
As a result of the lack of the test realism, mpg figures were adjusted downward, by
10% for the City and 22% for the Highway, starting in the early 1980s in an attempt to
produce better “Monroney Sticker” values, but the EPA still gave an optimistic
evaluation of fuel economy for decades.
In 1987 Congress allowed the states to increase highway speed limits from 55 to 65
mph, but it would take another 21 years for the EPA to adopt tests that provide more
realistic projections. To accomplish this, the EPA added three additional test cycles to
the original two for 2008 model-year cars.

76
US EPA altered the testing procedure effective MY2008 which adds three new
Supplemental Federal Test Procedures (SFTP) to include the influence of higher
driving speed, harder acceleration, colder temperature and air conditioning use:
SFTP US06 is a high speed/quick acceleration loop that lasts 10 minutes, covers 8
miles (13 km), averages 48 mph (77 km/h) and reaches a top speed of 80 mph
(130 km/h). Four stops are included, and brisk acceleration maximizes at a rate of
8.46 mph (13.62 km/h) per second. The engine begins warm and air conditioning is not
used. Ambient temperature varies between 68 °F (20 °C) to 86 °F (30 °C).
SFTO SC03 is the air conditioning test, which raises ambient temperatures to 95 °F
(35 °C), and puts the vehicle's climate control system to use. Lasting 9.9 minutes, the
3.6-mile (5.8 km) loop averages 22 mph (35 km/h) and maximizes at a rate of 54.8 mph
(88.2 km/h). Five stops are included, idling occurs 19 percent of the time and
acceleration of 5.1 mph/sec is achieved. Engine temperatures begin warm.
Lastly, a cold temperature cycle uses the same parameters as the current city loop,
except that ambient temperature is set to 20 °F (−7 °C).

77
In May 2011, the NHTSA and EPA issued a joint
final rule establishing new requirements for a
fuel economy and environment label that is
mandatory for all new passenger cars and
trucks starting with MY2013, and voluntary for
2012 models. The ruling includes labels for
alternative fuel and alternative propulsion
vehicles available in the US market. A common
fuel economy metric was adopted to allow the
comparison of alternative fuel and advanced
technology vehicles with conventional internal
combustion engine vehicles in terms of miles
per gallon of gasoline equivalent (MPGe). The
new labels also include for the first time an
estimate of how much fuel it takes to drive 100
miles (160 km) providing US consumers with
fuel consumption per distance traveled figures,
the metric commonly used in many other
countries.

78
By Federal regulation, an attached “Monroney Sticker” provides testimony as
to the fuel economy of every new car:
The combined EPA “City”-“Highway” fuel economy is calculated by a
“weighted” average:
0.55 × 18 mpg + 0.45 × 25 mpg = 21.15 ≈ 21 mpg

79
A quantitative estimation of the fuel economy of an
internal combustion engine (ICE) powered vehicle can
not be determined by the “plugging in” of a few
significant parameters into a relatively simple equation.
In practice, fuel economy is estimated through the use
of complex computer simulations which require the
input of a great many parameters, which ultimately is
empirically verified through a rigorous and involved
test program. However, if there is enough relevant
parametric information available regarding a vehicle of
known fuel economy, then the fuel economy of a
conceptual design can be reasonably estimated
through a series of parametric comparisons.

80
Per a Transportation Research Board (TRB) “Special Report 286”,
produced with funding by the DOT circa 2006, the energy usage for a late
model midsize conventional passenger car in the Highway and City
driving cycles is allocated:

81
A change in fuel economy due to aerodynamic drag is easy to
estimate: say the drag coefficient went from 0.42 for a
“baseline” vehicle to 0.36 for an “advanced concept” vehicle,
with an accompanying reduction in frontal area from 18.53 ft2
(1.72 m2) to 18.45 ft2 (1.71 m2). That results in a total drag
reduction of -14.66%, which per the previous table of fuel
consumption allocation results in a Highway/City driving cycle
improvement of -1.61%/-0.44% in fuel consumption:
-14.66% ᵡ 0.11H / 0.03C = -1.61%H / -0.44%C
With regard to rolling resistance the matter can be a little more
complex; say the static and dynamic resistance coefficients
have changed, but so have the normal loads due to weight,
weight distribution, and aerodynamic lift. The summary of all
the relevant data may be tabulated as per the following chart…

82
Note the use of EPA City and Highway Drive Cycle average speeds to
calculate the aerodynamic lift effect on the normal loads, leading to a
determination of the average total rolling resistance force reduction during
the Highway and City Drive Cycles of 66.21%/69.31% for the “advanced
concept” with respect to the “baseline”. Again referring to the fuel
consumption factors table, this indicates a further Highway/City reduction in
fuel consumption of:
-66.21% / -69.31% × 0.07H / 0.04C = -4.63%H / -2.77%C

83
Next, the energy requiring dissipation through braking is directly proportional
to the braking effective mass “me”. Say that for the “baseline” and the
“advanced concept” the varying effective mass due to gearing (same engine)
result in the following braking “me” values:
On this basis the “advanced concept” should dissipate 1.5% less energy
through braking than the “baseline”, which translates per the table of
Highway/City fuel consumption to decreases of -0.03%/-0.09%, respectively:
-1.5% ᵡ 0.02H / 0.06C = -0.03%H / -0.09%C

84
The varying effective mass due to gearing would result in different
acceleration “me” values as well, but the Highway/City “Engine Drive Mode”
factors of 69% Highway/62% City are each only partially attributable to
acceleration, probably on the order of 10% and 50%. So the factors to be
applied toward any reduction in acceleration fuel consumption are 6.9%
Highway and 31% City. Most of the Highway acceleration would be done in the
higher gears, resulting in this comparison of effective mass which indicates a
-0.14 % (0.069 x -2%) improvement:
3rd Gear: 3974Adv/4054Base = 0.980 4th Gear: 3891Adv/3972Base = 0.980
It is also reasonable to assume that most of the City acceleration would be
done in the lower gears, resulting in this comparison of effective mass which
indicates a -0.53% (0.31 x -1.7%) improvement:
1st Gear: 5216Adv/5296Base = 0.985 2nd Gear: 4204Adv/4284Base = 0.981

85
Therefore, for this example the total reduction in fuel consumption can be
summarized:
If the “baseline” vehicle had test EPA fuel economy mpg figures of 11.9 City /
18.3 Highway with a 14.8 Combined (0.55 × 11.9 + 0.45 × 18.3) mpg, then by this
comparison the “advanced concept” vehicle would have estimated EPA mpg
figures of 12.4 City / 19.5 Highway, for a 15.6 Combined (0.55 × 12.4 + 0.45 ×
19.5) mpg.

86
The conventional automotive
configuration of two axle lines and four
wheels, with each wheel located in a
corner of the automotive plan view, is
one of only a large number of possible
wheel/axle configurations, but has
prevailed for so long that this fact is
often forgotten. The choice of which
wheels steer and which wheels are
driven compounds the number of
configuration variations possible. When
other configuration variations are also
considered, such as varying the vehicle
tire/wheel size/type fore to aft or even
side to side (as on some circle track
racers), or whether the engine is to be
front, mid, or rear located, then the
number of all possible configurations
becomes infinite.

87
There is a tire phenomenon known as “standing wave”* which
has been addressed by various researchers such as McGivern
and Shirk:
*This is a misnomer; the “standing wave” is actually traveling around the tire circumference at a
velocity that is equal to the tire peripheral velocity so the appearance is one of a stationary
wave, but it really is a traveling wave.
“…at high speeds automobile tires develop…“tread
ripple” or “standing wave”. This phenomena
(sic)…is dangerous. It produces an intense heating
effect…with over-stressed fibers…tire failure.”
“…tests indicate that this…is…a function of
speed…tire size…tire pressure and make…weight
of tire…

88
McGivern and Shirk created a model of the tire tread as a
series of spring/mass systems; they determined that the
critical velocity “Vc” for catastrophic “standing wave” tire
failure may be estimated by:
Where:
Vc = The tire “standing wave” critical velocity (mph).
SN = The tire section width (in).
KZ = The tire vertical stiffness (lb/in).
tw = The tire tread width (in).
tt = The tire tread thickness (in).
δ = The tire tread material density (lb/in3).

89
Krylov and Gilbert created a more sophisticated model of the tire tread
wherein they came up with “Ccrit” as a combination of two different
modeling effects, “Ccr” (beam without tension) and “Cmem” (membrane
under tension). This involved a multi-step process beginning with:
Where:
λ = Standing wave phenomenon wavelength (m).
Ri = Tire inflated (no load) radius (m).
b = Tire belt width (m).
h = Tire belt thickness (m).
E = Young’s modulus (N/m2).
k = Tire foundation stiffness (N/m2).
ρ = Tire mass density (kg/m3).
Pi = Tire inflation pressure (Pa).
J = Tire belt second moment of area = h3b/12 (m4)
S = Tire tension = PiRib (N).
μ = Tire belt mass per unit length = ρhb (kg/m).

90
In a contemporary research paper by Jung-Chul An and Jin-Rae
Cho the following statement is made regarding their comparison
of such “simple” critical velocity models with detailed finite
element models (FEM), and with empirical test results:
“One interesting observation is that the critical speeds
predicted by the simple tire model are vastly
overestimated, which implies that the disregard of the
detailed tread grooves results in a…overestimation…”
Supporting this statement was the fact that, for a P205/60R15 tire
inflated to 137.9 kPa (20.0 psi) and under a 475 kg (1047 lb) load,
the “simple model” used by An and Cho returned a “Vc” value of
127 mph (205 kph), while their “detailed FEA model” resulted in
113 mph (182 kph); empirical testing showed the tire to have an
actual “Vc” of 118 mph (190 kph). However…

91
For this same P205/60R15 tire the McGivern/Shirk equation returned a
value of 149 mph (240 kph) when using “realistic” values for “δ”
(0.043352 lb/in3) and “tt” (0.6299 in). Using those same “realistic”
values, the critical velocity equations of researchers Krylov/Gilbert
returned 135 mph (217 kph) for the P205/60R15. All this would seem to
bear out An and Cho’s statement that “simple models” tend to
overestimate the standing wave critical velocity.
But, when using “conservative” values for “δ” (0.05 lb/in3) and “tt” (0.75
in), McGivern/Shirk produced 127 mph (204 kph) and Krylov/Gilbert
produced 119 mph (192 kph). Note that this Krylov/Gilbert result is
almost exactly equal to the empirical result of 118 mph (190 kph), and
that the McGivern/Shirk result is now much closer. This would seem to
indicate that, when lacking the resources for FEA modeling or empirical
testing, the use of “simple models” such as that developed by
McGivern/Shirk or Krylov/Gilbert is justified, as those models will give
reasonably accurate results when appropriately “conservative” values
for density and tread thickness are used.

92
A traditional determination of the critical velocity generally
results in reasonable critical velocity values although it
doesn’t take the water depth into account! The following
version of the Horne and Joyner “traditional” equation has the
tire-road contact pressure “Pc” substituted for the original
inflation pressure “Pi”, which makes the equation much more
informative with regard to actual individual tire cases:
Where:
Vh = The critical velocity for hydroplaning (mph).
Pc = The tire/road contact pressure (psi).

93
This “traditional” determination of the critical velocity at which
hydroplaning occurs produced results of 63mph (102 kph) and
61mph (98 kph) respectively for two very different cases.
Considering how this shows so little differentiation between such
different cases, the noted Czech researcher F. Koutný derived an
alternate way of calculating the “Vh”:
Where:
Vh = The critical velocity for hydroplaning (mm/sec).
Lc = The longitudinal dimension of the tire/road contact (mm).
ts = The “sink time” required for tire penetration to road
surface (sec).

94
That, however, is deceptively simple in appearance, as the sink
time “ts” must be determined by numerical iteration of the
following transcendental equation:
Where:
ht = The total “sink” distance before firm contact (mm).
h0 = The height of the water surface above road
roughness (mm).
ts = The “sink” time required for the tire to make firm
contact (sec).
a = One half the tire/road contact length (mm).
b = One half the tire/road contact width (mm).
Pc = The tire/road contact pressure (kN/mm2).
δ = The water density (kg/mm3).

95
Use of these Koutný equations to compute the “Vh” for the
same two example cases, as previously was used for the
traditional equation, results in 116 mph (186 kph) and 79 mph
(127 kph) respectively, for a water depth of 0.1 in (2.54 mm).
Since these values seem high, possibly the best values for “Vh”
that can be obtained would be via a “weighted” average of the
Koutný and traditional (Horne-Joyner) results, such as:
Case #1 Tire (N = 910 lb/ 413 kg, Pi = 40 psi/276 kPa):
Vh = (116 + 3×63)/4 = 76 mph, or 122 kph
Case #2 Tire (N = 455 lb/206 kg, Pi = 40 psi/276 kPa):
Vh = ( 79 + 3×61)/4 = 65 mph, or 105 kph
This averaging would seem to be a very reasonable
methodology judging from these results.

 AN ELECTRONIC COPY OF “MASS PROPERTIES and
ADVANCED AUTOMOTIVE DESIGN”, SAWE PAPER
3602, WILL BE PROVIDED EVERY INTERESTED
STUDENT.
 SUGGESTED FURTHER READING:
Fenton, John; Handbook of Vehicle Design Analysis,
Warrendale, PA; SAE R-191, 1999.
Milliken, William F., and Douglas L. Milliken; Race Car
Vehicle Dynamics, Warrendale, PA; SAE R-146, 1995.
Wong, Jo Jung; Theory of Ground Vehicles, Hoboken,
NJ; John Wiley & Sons Inc., 2008.
Reimpell, Jörnsen; Helmut Stoll, and Jürgen W.
Betzler; The Automotive Chassis; Jordan Hill, UK;
Butterworth-Heinemann, SAE R-300, 2001.
96

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