5- MASS PROPERTIES ANALYSIS and CONTROL

AUTOMOTIVE DYNAMICS:
MASS PROPERTIES
ANALYSIS
& CONTROL
Brian PaulWiegand, B.M.E., P.E.
1AUTOMOTIVE DYNAMICS and DESIGN

MASS PROPERTIES ANALYSIS
& CONTROL: THE SOCIETY
 The Society of Allied Weight Engineers is
an international organization whose
primary purpose is to promote the
recognition of “Mass Properties
Engineering” as a specialized branch of
engineering. The Society is organized into
22 chapters with approximately 800
individual and 20 corporate members
from across the United States, Europe,
United Kingdom, and Canada.

 The Society of Aeronautical Weight
Engineers was organized in 1939 in Los
Angeles, California, and was incorporated
as a nonprofit organization April 2, 1941.
As membership grew to include engineers
associated with shipbuilding, land
transportation, and other allied industries
and technologies, the Society name was
changed on January 1, 1973 to the Society
of Allied Weight Engineers, Inc.

 Much of the following material in this
course segment is derived from
publications of the society:
 Weight Engineers Handbook, SAWE; Los
Angeles, CA; Revised May 2002.
 Introduction to Aircraft Weight Engineering,
SAWE; Los Angeles, CA; 1996.
 Mass Properties Control System for Wheeled
and Tracked Vehicles, SAWE RP5 A; Los
Angeles, CA; 26 May 2007.

& CONTROL: MASS PROPERTIES

& CONTROL: UNCERTAINTIES

& CONTROL: STANDARD DEVIATION
Say that for a particular project a radar system is
purchased from a subcontractor. The specification
control drawing states only that the purchased item(s)
must come in under some maximum weight. If a
statistically significant number (“N”) of such units
have already been accepted by Q/C and are available,
then a scale weighing of each of these nominally
identical items will yield the desired information (“σ”)
when utilized as input to the algorithm:

Consider a hypothetical assembly of ten aluminum
cubes. The cubes are to be machined from stock to
have a side dimension of 4.6414 ±0.0155 inches as
per drawing specification (all others, if any, are
assumed rejected by Q/C). This results in a
maximum cube weight, assuming a material
density 0.1 lb per cubic inch, of 10.1 lb, and in a
minimum cube weight of 9.9 lb. Since this
represents the total component weight variation
possible, the range of 0.2 lb (10.1 lb – 9.9 lb)
represents a 6σ uncertainty; the nominal weight
±3σ is 10.0 lb ±0.1 lb per cube.

The 3σ uncertainty of a 10-cube (“n” =10) assembly
(100 lb nom) is found:
Consider what happens when the illustrative example
of a 10-cube assembly is increased to a 100-cube
assembly (1000 lb nom):
Then increased to a 1000-cube assembly (10,000 lb
nom):

The reason that uncertainty inputs are expressed in terms
of standard deviations is that, without some uniform
known measure of variation being used for input, the
output of an uncertainty analysis would be an unknown
quantity. An “answer” would be obtained, but it would be
useless; the situation would be like weighing an aircraft,
but without an inventory. If the individual component
mass property uncertainty inputs to the previous
uncertainty equations are all expressed as “3σ” (“3
sigma”, or 3 standard deviation) values, then the resultant
assembly uncertainty will also be a three sigma value (and
“1σ” inputs produces a “1σ” output, etc.). The utility of
using standard deviation values can be readily illustrated
when considering the normal distribution of random
variations...

& CONTROL: NORMAL DISTRIBUTION
(CUMULATIVE)

& CONTROL: NORMAL DISTRIBUTION
1σ
2σ
3σ
C
U
M
U
L
A
T
I
V
E

& CONTROL: WEIGHT ESTIMATION
Reasonably accurate aircraft initial weight estimates
have been made routinely for more than half a century,
and cars, trains, and ships long before that. If that were
not the case, history would not have borne witness to as
much progress as it has. The initial estimation of vehicle
weight is the first step in the weight control process; an
unrealistic estimate may doom a project to failure
regardless of the rigor of the subsequent weight control
effort. Likewise, a good initial estimate may not
guarantee success if subsequent control is lax. It is for
good reason that the discipline involved is called “Mass
Properties Analysis and Control”.

At the very onset the weight estimate may be no more
than an educated guess, but that soon gives way to an
estimate based on derived weights and “givens”
(engine, mission load, etc.). In the evolution toward the
contract weight estimate, there is a progression from
weights for only the most major functional groupings
onward through weights for sub-groups in ever
increasing detail. For military aircraft this process
culminates in a level of estimate that corresponds in
detail to that of a Mil-Std-1374 Group Weight Statement.
This consideration of the weight estimate in increasing
detail directly results in a decreasing total uncertainty,
for reasons just previously discussed.
.

Traditionally, the methodology used to predict the
weight of the various functional groups was empirically
based. That is to say, weight estimating relationships
(WERs) between functional group weights and certain
design parameters were determined by “regression
analysis” utilizing a normalized database of known
(existing) cases. “Comparison” of weights from the
resultant WER to the known weights of the generating
cases provided the means to quantify the uncertainty
inherent in the use of each WER. From the WER weights
and associate uncertainties (plus the “given” weights
and their associate uncertainties) a weight and
uncertainty for the entire project may be determined via
the appropriate “weight accounting” methodology.

& CONTROL: AUTOMOTIVE ESTIMATION
16
AUTOMOTIVE DYNAMICS and DESIGN
HOWEVER, when it comes to automobiles there is a
scarcity of published information of sufficient accuracy
and/or completeness so as to constitute a methodology
equal that used in the aerospace and marine industries.
Published automotive mass property estimation methods
seem to be available only in a non-comprehensive
fashion through a variety of scattered sources. This is
why this instructor wrote a paper with the intent to
systematize the information drawn from published
sources and, with the employment of techniques based
on those used in the aerospace industry, to augment and
improve upon the published information so as to develop
a basis for a comprehensive automotive mass properties
estimation methodology.

What is presented in this class is intended to provide
a possible overall framework for, and an initial “first
cut” at, the development of a comprehensive
methodology. Automotive design practitioners
working within the established industry may have a
far more potent methodology available to them, but in
the form of proprietary techniques that they are not at
liberty to divulge. It is the independent designer or
researcher that is most likely to find this class
segment to be of great value, and it is the purpose of
this class to aid such independent efforts through
promoting the development of a publicly accessible
methodology.

This class will present a preliminary “top-down”
methodology which requires as input only those
most basic and common overall parameters as
would be available in the earliest of design stages
or, for existing designs, from the commonly
available literature. This includes such parameters
as vehicle dimensions, applicable general legal
specification or regulation, general vehicle
configuration and category, type of suspension,
and level of technology (which is generally time
dependent).

& CONTROL: AUTO ESTIMATION OUTPUT
The desired estimated output will consist of
the curb weight and its attendant c.g.
coordinates and inertias, the unsprung weight
and its attendant c.g. coordinates and inertias,
the sprung weight and its attendant c.g.
coordinates and inertias, and the sprung
weight roll moment of inertia (i.e., a rotational
inertia about an essentially longitudinal axis,
the location of which is determined by the
suspension geometry).

& CONTROL: ESTIMATION OF TOTAL
VEHICLE WEIGHT, DATABASE TO WER
WER development begins with obtaining a suitable database for
analysis. There are two approaches that may be used to create such
a database. One approach, which is that used in this class, is to
“normalize the database” by striving for a consistency of data. For
instance, for the development of a WER to estimate the fuselage
group weight of transport aircraft, all fighter aircraft fuselage weight
data may be excluded. The rationale for this exclusion stems from the
fact that transport and fighter fuselages are designed to vastly
different criteria; a transport fuselage is mainly a large empty
pressure vessel to be subjected to relatively low maneuver loads,
while a fighter fuselage is just the opposite. A simple combining of
the data from both types can result in a decrease in accuracy of the
resultant WER; such diverse types are best represented by two
separate relations.

VEHICLE WEIGHT, DATABASE TO WER
Whatever the method, after an appropriate database is
assembled, the results of initial attempts at WER
formulation can provide direction for further database
refinement. WERs are established by “regression
analysis” wherein a “best-fit” curve is matched to the
data points; the overlay of derived function with the
data point scatter can be revealing with regard to
anomalous points. If sufficient cause can be assigned
to account for such “outliers” they can then be
normalized so as to be consistent with the mainstream
data, or they can be excluded from the database
altogether.

& CONTROL: WER BY REGRESSION
ANALYSIS METHOD OF LEAST SQUARES
“Method of Least Squares” determines “best fit” by
finding the line for which the sum of the square of the
“errors” (a.k.a. “offsets”, “residuals”) is a minimum:

& CONTROL: WER BY REGRESSION
ANALYSIS METHOD OF LEAST SQUARES
“Outliers” may have a disproportionate, and possibly
misleading, effect on the regression analysis result…
…because of the
effect of the large
outlier “(Yi – Yesti)2”
datum
in the “Method of
Least Squares”
as commonly
used for
Regression
Analysis.

MASS PROPERTIES ANALYSIS &
CONTROL:METHOD OF LEAST SQUARES
The sum of the square of the errors is used instead of the
sum of the errors because as “N” grows in size the sum
of the errors generally tends towards zero without any
improvement in fit; this is by virtue of negative errors
cancelling out positive errors. With the errors being
squared before summation this problem of self-
cancelation by opposition in sign is avoided. However,
the magnitude of the sum of the squared errors is still
dependent not only on the size of the errors themselves
but also on the size (“N”) of the sample. What is needed
is a relative value that would allow conclusions to be
drawn independent of the size of the sample, provided
that the size is sufficiently large to be representative of
the population. For this purpose the “Coefficient of
Determination” was formulated…

CONTROL: LEAST SQUARES R2
The “Coefficient of Determination” is formulated
as:
Note that the size of the sum of the errors squared
is now made relative to (divided by) the size of the
quantity:
This is the sum of the errors with respect to the line
representing the average value of the data sample;
this line represents worst possible fit to the data…

CONTROL: LEAST SQUARES R2
The average value of the data sample represents
the worst possible fit to the data…

VEHICLE WT, NORMALIZED DATABASE

VEHICLE WEIGHT: CORRELATION
This data has already been normalized as far as
intended for this usage, so the next logical step is to
see how the various parameters correlate with the
curb weight values:

VEHICLE WEIGHT: CORRELATION COEFF
There are various formulations for “Correlation Coefficient” as
symbolized by the lower case letter “r”, but in general they all
return numbers between “-1” (perfect inverse correlation) to “+1”
(perfect direct correlation), with a return value of “0” indicating no
correlation (since correlation coefficients are based on a
presumed linear relationship a “0” indication does not rule out a
curvilinear relationship). One equation for the “Sample
Correlation Coefficient” is:

VEHICLE WT: LOG MULTI-REGRESSION
The identification of parameters with a significant
correlation to the variable of interest completes
the second step in developing a WER. The third
step is to conduct a logarithmic multi-regression
analysis between the variable of interest and the
parameters identified as significant. The reason
for the employment of this particular type of
regression analysis is two-fold: 1) there are a
number of independent parameters (hence
“multi”), and 2) WERs tend to be nonlinear and of
the form: W = a X1
m1 X2
m2 X3
m3….XN
mn + c

Applications such as MS Excel have built-in regression analysis
routines, but they are usually limited in their capability. Excel will
perform a multi-regression analysis, but only for a linear relationship…
W = aX1 + bX2 + cX3 ….+zXN + B
…but we are dealing with a power relation…
W = a X1
m1 X2
m2 X3
m3….XN
mn + c
…so we must first do a logarithmic transformation of the data so that…
ln W = a m1 + b m2 + c m3….d mN + ln c
…and then invoke an Excel (or equivalent) regression analysis of the
transformed data…

Wt = 24.24544005 Loa
2.592751191 × Woa
-3.18666414 × Hoa
1.661373742
× Wb
0.463385968 × Tf
-0.079500729 × Tr
2.154947878
Wt = 308.2292681 Loa
0.042192705 × Woa
-0.26662407 × Hoa
-4.25391807
× Wb
2.162601987 × Tf
-1.90043119 × Tr
4.319906019
…giving us results for which an anti-logarithmic
transformation results in for the ’85 to’95 FWD cars…
…and for the ’76 to’88 RWD cars…
(R2 = 0.94082787, DOF = 3, “Satisfactory”)
(R2 = 0.970504, DOF = 3, “Satisfactory”)
…and as for the “R2”,“DOF”, and “Satisfactory”…

& CONTROL: EST OF TOTAL VEHICLE
WT: COEFFICIENT OF DETERMINATION
The “Coefficient of Determination”, “R2”, is a measure of the degree
to which the estimated variable values match up with the observed
variable values. When “R2” is zero the WER in question is no better a
“predictor” of the observed values than a simple average of those
values; when “R2” is close to 1.00 then the WER is a good
“predictor” of the observed data, and presumably will do well in
estimating values not part of the observed data but within the range
of observation. The formula for the “Coefficient of Determination” is:
The previous multi-regression analysis by way of logarithmic
transformation provides us with an excellent illustration of why it is
still important to know such fundamental equations despite the use
of computerized analysis …

& CONTROL: EST OF TOTAL VEHICLE
WT: COEFFICIENT OF DETERMINATION

VEHICLE WT: DEGREES OF FREEDOM
A high (or low) correlation coefficient value resulting from
comparison of the fitted line to the originating data is only a
partial picture. A line might be a very good fit to the observed
data, but a very poor predictor of future data values, because
there were insufficient observations, given the complexity of
the relationship, to discover the true nature of that
relationship. The number of observed data points “N” and the
complexity of the proposed relation, as indicated by the
number of parameters “n”, indicate what is called the
“Degrees of Freedom” of the analysis:

VEHICLE WIEGHT: WER R2 AND DOF

& CONTROL: ESTIMATION OF THE
UNSPRUNG WEIGHT
Total vehicle weight WERs as developed in the
preceding chapter need only be resorted to if the total
weight of an existing vehicle cannot be obtained any
other way, or as a rough check on the structural
efficiency of an existing design, or as a top level
estimate for a conceptual design. Regardless of why
and how a total vehicle weight (“Wt”) is obtained,
division of that weight into unsprung and sprung
amounts is the next step. To accomplish this, it is
easier to estimate the smaller, more determinate
unsprung weight (“Wus”) first, and then simply call
the remaining vehicle weight the sprung weight
(“Ws”).

UNSPRUNG WEIGHT
Here a less rigorous approach than developing a WER will be
used. This approach is touched upon in The Automotive
Chassis but is most fully developed in The Sports Car and New
Directions in Suspension Design. This methodology is cruder
than the use of a WER; in this cruder method just three factors
are considered as influencing the amount of unsprung weight.
One factor is the total weight of the vehicle; the greater the
vehicle weight borne by the suspension then the greater the
unsprung weight tends to be. Another factor is the type of
suspension, which includes such things as whether the
suspension must transmit driving and/or steering forces, and
whether the brakes are inboard or outboard. Lastly, a factor
may be utilized to adjust for the level of technology utilized in
the suspension; a highly optimized and/or exotic material
design will tend to reduce the unsprung weight.

UNSPRUNG WEIGHT, WT FACTORS

UNSPRUNG WEIGHT, LOT FACTORS
The previous factors represent a general Level of Technology
(LOT) typical of about 1969 to 1981. Those unsprung weight
factors are most likely to be applicable to a vehicle of that period.
However, a specific vehicle might be very unrepresentative of its
era; sometimes a vehicle is said to have been “ahead of its time”
(or sometimes the opposite). If called for, a different technology
level may be accounted for by the judicious use of auxiliary
adjustment factors such as those presented in this table:

UNSPRUNG WEIGHT, EXAMPLE CASE
In the paper “Developments in Vehicle Center of Gravity and
Inertial Estimation and Measurement”, Bixel et al presented a
means by which the total vehicle unsprung weight may be
empirically determined while the vehicle is on a VIMF (Vehicle
Inertia Measurement Facility) test device. They applied their
method to one vehicle, a 1994 Ford Taurus of 3237.3 lb curb
weight, with a vertical c.g. at 23.067 in above the VIMF platform,
and with axle heights of 11.8 in above the platform (most
methods assume the unsprung weight to be at axle height). In
two tests utilizing their method, they got values of 331.8 lb and
391.1 lb for the unsprung weight. Then the Transportation
Research Center in East Liberty, Ohio, tore the Taurus apart and
weighed the unsprung mass (which would require a good deal of
judgment as some components are considered only partially
unsprung!) resulting in a unsprung weight of 392.4 lb.

To provide a little more assurance as to what the
best value for the Taurus unsprung weight might
be, a detailed weight calculation of the 1994
Taurus unsprung weights was carried out by this
instructor. That investigation resulted in a front
axle unsprung weight of 197.1 lb and a rear axle
unsprung weight of 195.1 lb, for a total unsprung
weight of 392.2 lb. Note that this means the front
and rear suspension weights are nearly equal
despite the fact that the 1994 Taurus c.g. was
heavily biased toward the front in its curb weight
condition!

Although the 1994 Taurus has a curb weight distribution of 64/36
(2071.9 lb on the front axle and 1165.4 lb on the rear axle) the
calculated unsprung weights seem to fly in the face of the natural
assumption that the suspension weights would be directly
proportional to the curb weight axle loads. That seems
reasonable, until one realizes that the curb weight is just one of
many possible load conditions that a suspension must be
designed to accommodate, right up to the GVWR (Gross Vehicle
Weight Rating). The 1994 Taurus had a GVWR of 4722 lb; it could
accommodate up to six passengers (rated at about 190 lb each)
plus their luggage (rated at about 57.5 lb each). With such a
loading the weight distribution would move from the curb weight
condition of 64/36 to something closer to 50/50; the Campbell
method takes this into account, but does so implicitly by still
utilizing the curb weight for calculation as that is the weight most
commonly available.

To apply the Campbell method a little further information is
required: the front suspension was a McPherson Strut (coil spring
over) with lower longitudinal and lateral links; since the Taurus was
FWD this was a driven suspension. The rear suspension was also a
McPherson Strut (coil spring over) with lower longitudinal and lateral
links. The disk brakes were outboard located both at the front and the
rear, and no special technological efforts to reduce weight were
notable (so use of adjustment factors not required). Utilizing the
above information and the appropriate unsprung weight coefficients,
the fore and aft unsprung weights are estimated as:
Front : Wusf = 3237.3 x 0.0675 = 218.5 lb / axle
Rear : Wusr = 3237.3 x 0.0530 = 171.6 lb / axle
The distribution seems off, but the total unsprung weight of 390.1 lb
seems as accurate as one can possibly expect for an estimate.

However, with regard to the accuracy of the unsprung weight
distribution, the Campbell method would seem to require some
fine tuning. Possible opportunity for such “tuning” arises from the
fact that the method assumes the unsprung weight can vary in a
smooth continuous function as the vehicle weight varies. That is
certainly untrue in the case of wheels and tires, and for certain
other vehicle suspension components (e.g.: dampers) as well. The
wheels and tires are one of the largest contributing factors to the
unsprung weight, so the fact that such are built only in certain
discreet sizes, causing their weight function to move in a series of
“hops”, and not continuously, is not without consequence. This
consideration gives rise to the “Modified Campbell” method for
estimating the unsprung weight…

Therefore, factor out the wheel and tire weight as accounted for in
the Campbell method (use 0.59) to calculate the remaining
unsprung weight, and then use a source such as certain tire and
wheel weight tables to determine a more realistic weight for those
items, the final result should see some improvement. The Taurus in
question had General Tire Ameri-Tech G4S P205/70R14 tires (21.0
lb/ea) on 6x14 steel rims (17.25 lb/ea ). With plastic wheel cover, lug
nuts, and miscellaneous the total weight per wheel comes to about
40.37 lb, so the “Modified Campbell Method” results in:
Frt: Wusf = (3237.3 x 0.0675 x 0.59)+(2 x 40.37) = 209.7 lb/axle
Rr: Wusr = (3237.3 x 0.0530 x 0.59)+(2 x 40.37) = 182.0 lb/axle
The distribution is better, and the total unsprung weight of 391.7 lb
also seems good.

SPRUNG WEIGHT, EXAMPLE CASE
As noted at the start of this sprung/unsprung
weight estimation segment, estimate the
unsprung weight (“Wus”) first, and then simply
call the remaining vehicle weight the sprung
weight (“Ws”):

TOTAL VEHICLE C.G. COORDINATES
The crux of the estimation problem for the longitudinal and vertical
c.g. coordinates*, and indeed for the inertia values as well, is that
such quantities are very dependent upon the arrangement of the
internal masses within the vehicle, and no mere consideration of
only vehicle weight and external dimensions can be very precise in
divining such mass property values. However, if an estimation
procedure must be had, then obtaining “WERs” (if that acronym
can be used in a general way to refer to estimation routines derived
for entities other than weight) using vehicle databases normalized
by vehicle configuration, type, etc. (thus taking arrangement of the
internal masses into account in a very general way) offers the best
hope of obtaining a reasonable methodology for top-down c.g. (and
inertia) estimation.
*The lateral center of gravity, whether for an existing or a conceptual vehicle, can
almost always (certain circle track race cars being one exception) be initially estimated
as “0.0”, indicating a laterally balanced mass condition.

The longitudinal center of gravity for an existing
vehicle is almost always available in the published
literature, usually given in terms of percent weight
front versus percent weight rear, although it is
sometimes given as the weight on the front wheels
and the weight on the rear wheels. Regardless of how
the longitudinal weight distribution is given, the
longitudinal axle-to-c.g. lengths (often symbolized as
“a” and “b” in the technical literature) will need to be
determined as these will be necessary for various
common calculations later on:

a = [Nr / (Nf + Nr)] ᵡ Wb
b = [Nf / (Nf + Nr)] ᵡ Wb
Where:
a, b = the front axle to c.g. & rear axle to
c.g. lengths.
Nf, Nr = the front and rear normal loads.
Wb = the wheelbase.

However, if the longitudinal weight distribution for an
existing vehicle is not available, or if the vehicle in
question is not existing, but only a concept in the very
earliest of design stages, then the matter of estimating a
center of gravity location, without recourse to some
tedious “bottom-up” weight accounting approach, must
again be had by obtaining a “WER” (if that acronym can
be used in a general way to refer to estimation routines
derived for quantities other than weight) using vehicle
databases normalized by vehicle configuration, type,
and LOT (thus taking arrangement of the internal
masses into account, albeit in a very general way).

Attempts to base a longitudinal c.g. estimation
routine on just the wheelbase or LOA are fairly
common, and unsatisfactory. Using the same
database as was used for total vehicle weight, a
correlation analysis revealed that all the major
vehicle dimensions have some relevance to the
problem:

FWD (Frt. Eng.):
Lcg = 0.068329376 Wb + 2.407178089 Tf – 1.415494942 Tr –
0.340171358 Loa – 0.750920075 Hoa + 1.500133451 Woa – 0.694654904
RWD (Frt. Eng.):
Lcg = 0.73545172 Wb + 0.72866578 Tf – 1.06615718 Tr +
0.0431037 Loa – 0.2255865 Hoa – 0.4656507 Woa + 0.63206632
The “R2” values for these equations are 0.934072123
and 0.975930444, respectively. Furthermore, both the FWD
and RWD relations are now rated “satisfactory” per the
SAWE criteria (“r” = 0.96647407 and 0.987891919, DOF =
3). Provided that all the dimensional data is available, use
of these equations would appear to be the best way to
estimate the longitudinal c.g.

Estimation of the vertical center of gravity is yet another
challenge, but unlike the longitudinal center of gravity a number
of methods to overcome this challenge have been presented
over time in the technical literature. For instance, Taborek
presents the following estimation formula:
Hcg = .25 Wb
However, relating the vehicle vertical c.g. coordinate to just the
vehicle horizontal dimension of wheelbase seems questionable
on the face of it. Allen, Rosenthal, and Szostak have an
alternative suggestion which they determined by means of a
regression analysis to be “accurate within ±5%”:

However, this regression analysis was carried out using an
extremely diverse database that varied in technology level (1956
to 1980!), type (economy car to motor home!), and configuration
(FWD, RWD, and AWD!) Moreover, out of 18 vehicles in their
database (constituting at least four distinct type groups), only 5
were passenger vehicles of the sort this class is concerned with.
If they had concentrated on just the passenger vehicles
(admittedly, only 5 data points still varying in type and LOT), and
let the intercept “float”, they would have obtained something
more like this (units in inches):

To provide a more
relevant analysis,
the motley
database of Allen,
Rosenthal, and
Szostak, which
seemed to include
everything short of
the QE II, was
junked and a much
more reasonable
one(s) constructed:

The resulting estimating routines obtained via a linear regression
analysis of this database information were (units in meters):
FWD (Frt. Eng.): Hcg = 0.131506849 Hoa + 0.361472603
RWD (Frt. Eng.): Hcg = 0.532328767 Hoa – 0.195421918
These equations were called the “Wiegand” equations. Lastly a
truly appropriate database was constructed that reflected all the
major vehicle dimensions and a correlation analysis was run:

Again it seems that all the dimensions may contain some clues
as to the location of a c.g. coordinate, this time the vertical.
Utilizing all the dimensions, a linear multi-regression analysis
resulted in the following relationships (units in meters):

The question
now arises
as to how do
all these VCG
estimation
routines
compare.
Well, for the
FWD (front
engine):

And as to
how all these
VCG
estimation
routines
compare for
the RWD
(front
engine):

Note that equations such as the last two are only useful if all six
input parameters are available; what if only the overall length
(“Loa”) and the front track (“Tf”) values for a particular vehicle is
available? Well, it would be a tedious and not very productive
task to develop, in advance of any need, an estimation routine
for every possible combination of input parameters, especially
since many combinations are unlikely to be encountered.
Therefore, when none of the routines presented in this class are
applicable for the estimation of vehicle mass properties (and not
just the c.g.), then an appropriate database (drawn from a source
such as the NHTSA VIPMD) must be constructed and a
regression analysis conducted to produce a WER for the
particular combination of input parameters available.

UNSPRUNG WEIGHT C.G. COORDINATES
The c.g. estimating routines presented so far have
been for the total vehicle weight; it is now appropriate
to address the obtaining of c.g. coordinates for the
unsprung and sprung weights. Fortunately, the
necessary c.g. coordinates for the unsprung weights
may be estimated with reasonable accuracy as
follows:

UNSPRUNG WEIGHT C.G. COORDINATES

SPRUNG WEIGHT C.G. COORDINATES
Once the unsprung weight c.g. coordinates
have been reasonably estimated, then the
following formulae may be used to determine
the sprung weight c.g. coordinates:
Xs = (Wt Xt - Wusf Xusf - Wusr Xusr) / Ws
Ys = Yt
Zs = (Wt Zt - Wusf Zusf - Wusr Zusr) / Ws

TOTAL VEHICLE MOMENTS OF INERTIA
Various formulae are presented in the published
literature for the estimation of the total vehicle Ix, Iy,
and Iz. This existing methodology is fragmentary
and inaccurate, although modification by this
instructor has improved the accuracy of some of
these formulations to reasonable levels. Even so,
most of these formulations should be utilized
only if nothing better is available (however, this
instructor has worked diligently to provide that
“something better”; the resulting formulae from that
effort will be dealt with later).

Allen, Rosenthal, and Szostak “validated”, supposedly with
accuracy of “±20%” (!), the following estimation formula for the
vehicle roll inertia (“Ix”):
Ix = (Wt / 12) x (Woa
2 + Havg
2)
Where:
Ix is the weight moment of inertia of the vehicle about a
longitudinal axis that passes through the vehicle c.g.
Wt is the vehicle weight.
Woa is the overall vehicle width.
Havg is an “average height dimension” of the
vehicle as explained on the next vu-graph.

The “Havg
2” term requires a little explaining. This is a
“weighted” average height-squared characteristic of the
vehicle which is determined as follows:
Havg
2 = {x1 h1
2 + x2 h2
2 + x3 h3
2 + …..xnhn
2}
The “hi” terms are themselves average dimensions, while the
“xi” terms are fractions of the sprung weight overall length;
for clarification:

Allen, Rosenthal, and Szostak also put forward a technique for
the estimation of the sprung vehicle yaw inertia (“Iy”). Their
yaw inertia method was based on the obvious relation:
Izs =Wt kz
2
Where:
Izs is the sprung weight moment of inertia
about the vertical axis.
Wt is the total vehicle weight.
kz is the vehicle sprung weight radius of
gyration about the vertical axis.

The sprung weight radius of gyration (“kz”) is estimated per the
following formulation:
kz = √abWhere:
a is the longitudinal distance from the vehicle
front axle line to the c.g.
b is the longitudinal distance from the vehicle
c.g. to the rear axle line.
So combining these two equations becomes the Allen,
Rosenthal, and Szostak sprung weight yaw inertia estimation
relation:
Izs = Wt a b

It’s hard to see how the previous method could have a ±10%
accuracy as it fails to adequately take the mass distribution
into account; two vehicles could have the same “Wt”, “a”, and
“b” values but vastly different mass distribution. The noted
engineer and author Colin Campbell has suggested a means to
rectify this somewhat. From a paper presented to the 1977 SAE
World Congress in Detroit by Jaguar Motor Company
engineers Bob Knight and Jim Randle, Colin extracted a table
of relative yaw inertias for eight of the most common drive train
configurations (thereby accounting in a general way for the
mass distribution):

If we call this relative yaw inertia factor “fconfig”, then the Allen,
Rosenthal, and Szostak yaw inertia estimation relation can be
refined to include this accounting for mass distribution:
Iz = fconfig Wt a b
All of this is very much like a scheme proposed in the August
1985 issue of Road & Track magazine for estimation of the yaw
inertia. The essence of this method was proposed by a Dr. John
Ellis, ex-Director of the School of Automotive Studies at the
Cranfield Institute of Technology (UK).

This R&T method, expressed in as a single formula, is:
Iz = L (F2 Wf + R2 Wr)
Where:
Iz is the vehicle yaw inertia.
L is a drive train configuration factor.
F is a dimension from the vehicle c.g.
to the front tire contact patch.
R is a dimension from the vehicle c.g.
to the rear tire contact patch.
Wf is the vehicle weight on the front axle.
Wr is the vehicle weight on the rear axle.

For a clearer understanding of the dimensions “F” and “R” the
following figure is provided:

The original Road & Track values for the factor “L” seem to have
been based on either Campbell’s book or the paper by Knight and
Randle. However, the quantity “F2 Wf + R2 Wr” is much larger than
“Wt a b”, such that the original values required considerable
modification to produce reasonable results. Also, the original
R&T set of “L” factors recognized the existence of only 5
configurations; that set was extended to include the full set of 8
configurations as shown in the previous figure. The resulting set
of modified and extended “L” factors to be used in conjunction
with the R&T equation, along with the original factors plus
Campbell’s factors for comparison, is shown in the following
table…

Although none of the sources cited so far put forth a specific
method for vehicle pitch inertia estimation, en total they make it
clear that pitch inertia would be estimated in a manner very similar
to yaw inertia estimation, only the result would be somewhat
smaller. This would seem to be by virtue of the fact that for most
vehicles the profile (X-Z) area is a good deal smaller than the plan
view (X-Y), and the mass moment inertias tend to roughly mimic
the area moment inertias. A study by this instructor of pitch inertia
to yaw inertia ratios for a wide sampling (20) of modern vehicles
(1976-1995, RWD & FWD, Front Engine) resulted in:

A large number of other sources give further competing formulations
for the estimation of the total vehicle mass moments of inertia, but
probably the best set of equations for inertia estimation was
developed by this instructor via the now familiar (hopefully)
methodology of regression analysis. This regression analysis was
based only on the sort of parameters (which statisticians call
“regressors”) readily available via common published literature, with
the observed data values used drawn from SAE 1999-01-1336.
The formulae for the inertias of a
homogeneous rectangular prism
provide clues as to what sort of
parameters to consider for
automotive inertia determination:

Frt. Eng./FWD:
Ix = 7.120451981 Wt
0.570329247 Hoa
-0.187177974 Hcg
1.853542789 Tf
0.141330237 Tr
0.508012356
Woa
1.894887688 (R2 = 0.9810, r = 0.9905, DOF = 3, “Satisfactory”)
Iy = 0.011195547 Wt
1.778282559 Wb
0.014013849 Loa
-0.734193509 Lcg
0.752034094 Hoa
-1.314225187
Hcg
-1.499655458 (R2 = 0.9953, r = 0.9977, DOF = 3, “Satisfactory”)
Iz = 0.474091893 Wt
1.039784091 Wb
1.112079573 Loa
-0.521240418 Lcg
0.117581336 Tf
1.006287567
Tr
-0.011915341 Woa
Frt. Eng./RWD:
Ix = 355.9594941 Wt
0.334010047 Hoa
-1.835110223 Hcg
4.163285414 Tf
-4.266393542 Tr
7.780953344
Woa
Iy = 0.672185766 Wt
0.663883755 Wb
8.527931156 Loa
0.988282414 Lcg
-2.609737538 Hoa
-14.46773139
Hcg
Iz = 1305.504824 Wt
-1.158870843 Wb
11.19966833 Loa
-0.040260416 Lcg
-4.895023757 Tf
-3.718063977
Tr
9.31885401 Woa
The best set of total vehicle inertia estimation equations obtained
by this instructor (or anyone else); for front engine, FWD and RWD:

UNSPRUNG AND SPRUNG MOMENTS OF
INERTIA
Once the total vehicle weight, c.g. coordinates,
and inertias have been determined, and the division of that total
vehicle weight into unsprung and sprung weights with associate
c.g.s has been accomplished, it is a relatively simple matter to
determine the sprung weight moments of inertia (“Ixs, Iys, Izs”).
This is accomplished in accord with the mathematical logic of
mass properties accounting:

INERTIA
Note that the front and rear axle unsprung
weight inertias (“Ixusf, Iyusf, Izusf” and “Ixusf, Iyusf, Izusf”, respectively)
are used in this determination of the sprung weight inertias. No
means of determining those unsprung quantities has been
introduced so far; the unsprung weight per wheel inertias can be
estimated using the formulae for the inertias of a solid
homogeneous disk:

INERTIA
An alternative method for determining the “iYusf”
and “iYusr” values has been determined by researchers Metz,
Akouris, Agney, and Clark in a 1990 paper SAE 900760:
Where the symbolism is:
IYus = Tire and wheel rotational inertia (lb-in-sec2)
W = Tire and wheel weight (lb)
d = Tire inflated (no load) outer diameter (in)
This method was derived via regression analysis from a database
(road tires mounted on conventional pressed steel wheels), so it
should be more accurate than the previous solid disk method.

INERTIA
Once the unsprung inertias per wheel have been
estimated, the unsprung inertias per axle must be determined.
First, the front axle unsprung inertias:
Ixusf = 2 ixusf + 2 wusf (Tf / 2)2
Iyusf = 2 iyusf
Izusf = 2 izusf + 2 wusf (Tf / 2)2
Then the rear axle unsprung inertias:
Ixusr = 2 ixusr + 2 wusr (Tr / 2)2
Iyusr = 2 iyusr
Izusr = 2 izusr + 2 wusr (Tr / 2)2
The unsprung weight products of inertia will be taken as zero as
there is no easy way to estimate those product values, and they
tend to be small.

VEHICLE PRODUCTS OF INERTIA
However, we still need to estimate at least one of the three total
vehicle products of inertia: “Pxz”; the “Pxy” and “Pyz” products of
inertia may be taken as zero by virtue of symmetry:

VEHICLE PRODUCTS OF INERTIA
So, how do we estimate the total vehicle “Pxz” product of
inertia? G.L. Basso in his 1974 paper “Functional Derivation of
Vehicle Parameters for Dynamic Studies” used:
Pxz = ½ (Iz – Ix) tan 2λ
Where:
Pxz is the X-Z product of inertia (slug-ft2, lb-ft2, etc.)
Ix is the total vehicle inertia about the longitudinal axis (slug-ft2,
lb-ft2, etc.).
Iz is the total vehicle inertia about the vertical axis (slug-ft2,
lb-ft2, etc.).
λ is the angle between the vehicle longitudinal reference
axis and the vehicle longitudinal principal axis (degrees,
radians).

& CONTROL: ESTIMATION OF TOTAL VEHICLE
PRODUCTS OF INERTIA
A possible
way of
estimating
the total
vehicle
product of
inertia “Pxz”
would be to
estimate a
likely value
for “λ” from
this chart
and then use
the previous
equation.

VEHICLE MASS PROPERTIES
So, now the complete vehicle mass properties are determined, at
least to the extent most commonly needed for conceptual
design:

SPRUNG ROLL MOMENT OF INERTIA
Iroll = Ixs Cos2ф – 2 Pxzs Sinф Cosф + Izs Sin2ф + Ws hr
2

& CONTROL: CONCLUSIONS
• A NUMBER OF COMPUTER PROGRAMS FOR
MASS PROPERTIES ANALYSIS & CONTROL HAS
BEEN WRITTEN BY THE INSTRUCTOR. THOSE
PROGRAMS PLUS COPIES OF “THE BASIC
ALGORITHMS OF MASS PROPERTIES ANALYSIS &
CONTROL” AND “AUTOMOTIVE MASS
PROPERTIES ESTIMATION” WILL BE PROVIDED
EVERY STUDENT STILL CONSCIOUS.
• MASS PROPERTIES ANALYSIS & CONTROL
PROGRAMS (WRITTEN BY THE INSTRUCTOR):
MASSACCT.BAS, UNCERT.BAS,
STDDEV.BAS,
CORRELTN.BAS, XFORM.BAS

5- MASS PROPERTIES ANALYSIS and CONTROL

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