LDTP 78-03
Technical Support Report
A Brief Treatise on the Problems
Associated with Using One Vehicle to Determine the Dynamometer
Power Absorption for a Second Similar Vehicle
Glenn D. Thompson
March 1978
Notice
Technical support reports do not necessarily represent the final EPA
decision. They are intended to present a technical analysis of an issue
and recommendations resulting from the assumptions and constraints of
that analysis. Agency policy constraints or data received subsequent to
the date of release of this report may alter the conclusions reached.
Readers are cautioned to seek the latest analysis from EPA before using
the information contained herein.
Standards Development and Support Branch
Emission Control Technology Division
Office of Mobile Source Air Pollution Control
Office of Air and Waste Management
U.S. Environmental Protection Agency
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Abstract
The current EPA regulations for exhaust emission certification and
fuel economy measurements allow a manufacturer the option of road
testing a vehicle to determine a dynamometer adjustment which will
subsequently be used for all vehicles of a similar class. In the pro-
cess of using the road test results from one vehicle to determine the
dynamometer adjustment for a second vehicle there is, of course, the
potential for error. This report briefly discusses the magnitude of
these possible errors.
Two specific cases are considered. In the first case the error is
assumed to be random, unintentional and it is assumed that good en-
gineering practices are used. In this instance it is estimated that the
error associated with using a dynamometer adjustment obtained from test
results of one vehicle to represent a second vehicle should be less than
about one-half horsepower. This error is about 5 percent of the typical
dynamometer adjustment, and is considered acceptable.
In the second case it is assumed that the potential for error is
used in a systematic manner to result in a reduced loading for the
second vehicle. In this instance when the possible errors are maximized
to result in the optimum beneficial dynamometer loading of the second
test vehicle, the possible errors are much greater. In this case it is
concluded that a total error of 3.4 horsepower at 50 mi/hr is possible.
The largest single contribution, 1.3 horsepower, is introduced by pos-
sible variations in the road versus dynamometer rolls behavior of the
tire. The second largest possible contribution, 1.0 horsepower, may be
introduced by brake drag effects.
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Purpose
The purpose of this report is to discuss the technical problems
associated with using one vehicle to determine the dynamometer adjust-
ment to simulate the road experience of a second similar vehicle.
Specifically, the magnitude of the errors which can occur in this in-
stance are investigated. Two general cases are considered; the case
where the errors are random and unintentional, and then secondly, the
case where the potential for error is used in a systematic manner to
result in a reduced loading for the second vehicle.
Background
The current EPA regulations for exhaust emission certification and
fuel economy measurements allow a manufacturer the option.of road test-
ing a vehicle to determine a dynamometer adjustment which will subse-
quently be used for all vehicles of a similar class. In the process of
using the road test results from one vehicle to determine the dynamo-
meter adjustment for a second vehicle there is, of course, the potential
for error. This report briefly discusses the magnitude of these pos-
sible errors.
Discussion
This section is divided into three subsections. The first section
derives an expression for the possible error associated with using the
dynamometer adjustment appropriate for one vehicle when testing a simi-
lar,.vehicle. The subsequent sections discuss the magnitudes of this
error. In the first subsection the error is assumed to be random,
unintentional and it is assumed that good engineering practices are
used. In the second case it is assumed that the error is intentionally
maximized to result in an optimum dynamometer load on the second test
vehicle.
A. Development of an Expression for the Potential Error
When the vehicle is operated on road, the force on the vehicle is
the sum of the aerodynamic drag, the tire rolling resistance and the
drive train-chassis dissipative forces. This may be expressed as:
FR = Aero + 4(Tire) + Dtrain + NonDAxle (1)
where:
FR = the force experienced by the vehicle on the road
Aero = the aerodynamic drag of the vehicle
Tire = the tire rolling resistance force
Dtrain = the drive train dissipative forces
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NonDAxle = the dissipative forces of the non-driving axle
The tire rolling resistance force is quite nearly proportional to
the vertical vehicle load on the tire, at least in free rolling or
nearly free rolling situations. For this reason it is frequently ex-
pressed as the product of a dimensionless rolling resistance coefficient
times the vertical tire load. In this notation equation 1 becomes:
FR = Aero + 2[(Rr)(LDAxle)/2] + 2[(Rr)(LNDAxle)/2] 4- Dtrain + (2)
NonDAxle
= Aero + (Rr)(LDAxle) + (Rr)(LNDAxle) + Dtrain + NonDAxle
where
Rr = the tire rolling resistance coefficient on a road surface
LDAxle = the vertical load on the driving axle
LNDAxle = the vertical load on the non-driving axle
The sum of LDAxle and LNDAxle is, of course, the total vehicle weight.
Therefore, equation 2 may be written as:
FR = Aero + (Rr)(W) + Dtrain + NonDAxle (3)
where
W = the vehicle weight
On the dynamometer, the force acting on the vehicle is the sum of
the dynamometer force and the drive train and drive tire losses. This
may be expressed in the manner of equation 1 as:
FD = Dyno + 2[(Rd)(LDAxle)/2] + Dtrain (4)
= Dyno + (Rd)(LDAxle) + Dtrain
FD = the total force acting on the vehicle when on the dynamometer
Rd = the tire rolling resistance coefficient for the tire
on the dynamometer
Dyno = the total dynamometer force acting on the vehicle
If the dynamometer experience of the vehicle is to be equivalent to the
road experience, FR of equation 2 must equal FD of equation 4 or:
Dyno + (Rd)(LDAxle) +Dtrain =
Aero + (Rr)(LDAxle) + (Rr)(LNDAxle) + Dtrain + NonDAxle (5)
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Therefore the dynamometer force for exact dynamometer simulation of the
vehicle road experience is:
Dyno = Aero + (Rr)(LDAxle) - (Rd)(LDAxle) + (Rr)(LNDAxle) +
NonDAxle (6)
The rolling resistance on the dynamometer (Rd) is frequently expressed
in terms of the ratio of tire-dynamometer rolling resistance to the
tire-road rolling resistance. That is, in the form of the usual "one on
the rolls equals two on the road". This may be written, without specifying
the proportionality constant (x) , as:
Rd = xRr (7)
Substituting equation 7 into equation 6:
Dyno = Aero + (Rr)(LDAxle) - (Rr)(x)(LDAxle)
+ (Rr)(LNDAxle) + NonDAxle (8)
= Aero + NonDAxle + (Rr)[(1-x)LDAxle + LNDAxle]
Equations 2 and 3 describe the road experience of the vehicle while
equation 4 describes the dynamometer experience. In addition, equations
8 describe the dynamometer force when the dynamometer is correctly
adjusted to simulate the road experience. These equations can now be
used to discuss the errors which can occur when the dynamometer adjustment
appropriate for one vehicle is used to represent a different vehicle.
This is, of course, the usual case when an alternate dynamometer adjust-
ment is requested for a certification vehicle, based on road measurements
of a prototype vehicle.
Equation 8 may be considered as the appropriate dynamometer adjust-
ment, determined from the prototype vehicle. The appropriate dynamometer
adjustment for the second, certification vehicle, may be expressed as
Dyno1 = Aero' + NonDAxle1 + (Rr1)[(1-x')LDAxle' + LNDAxle'] (9)
The error, if the dynamometer adjustment for the first vehicle is used when
testing the second vehicle is the difference between equations 8 and 9.
That is
Error = Dyno - Dyno1
= Aero-Aero1 + NonDAxle - NonDAxle' + (10)
(Rr)[(1-x)LDAxle + LNDAxle] - (Rr')[(1-x')LDAxle' +
LNDAxle']
The first two terms (Aero, NonDAxle) are simple vehicle-to-vehicle
variations. The remaining terms represent a vehicle dynamometer inter-
action error, which is of course, influenced by the variations between
the two vehicles. Labeling the last two terms as a vehicle-dynamometer
interaction error:
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Error = Aero - Aero' + NonDAxle - NonDAxle' + VDIerror (11)
where
VDIerror = the vehicle dynamometer interaction error
= (Rr)[(l-x)LDAxle + LNDAxle] - (Rr1)[(l-x')LDAxle' (12)
+ LNDAxle1]
It is desirable, from the EPA standpoint, to have the vehicle dyna-
mometer interaction error, equation 12, be zero. The error can vanish
in several ways. First, the two major terms can be equal in magnitude
but opposite in sign. In general, the manner in which this would most
likely occur would be if the tires of the two vehicles are functionally
equivalent. That is:
Rr = Rr'
and
x = x' (13)
In this case
VDIerror = (Rr)[(1-x)(LDAxle - LDAxle') + (LNDAxle - LNDAxle')] (14)
The error subsequently vanishes if the axle loads are equivalent.
,4. '
The other possibility for zero error occurs when both major terms
vanish independently. That is:
(1-x)LDAxle + LNDAxle =0
and
(l-x')LDAxle' + LNDAxle' =0 (15)
This requires
LDAxle + LNDAxle = (x)LDAxle
and
LDAxle' + LNDAxle' = (x')LDAxle' (16)
But the sum of the two axle loads is the vehicle weight, therefore the
requirement is:
x = W/LDAxle (17)
x1 = W/LDAxle'
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This is the exact mathematical expression necessary for the dynamometer
tire force dissipation assumption "one on the rolls equals two on the
road" to be valid.
In the subsequent investigation of the dynamometer simulation
error, it is convenient to express the prime quantities in terms of the
change between the two vehicles. In this case:
Rr1 = Rr + ARr
x' = x + Ax
LDAxle1 = LDAxle + ALDAxle
LNDAxle' = LNDAxle + ALNDAxle (18)
In equation 18, ARr, Ax, ALDAxle and ALNDAxle represent the changes
in these quantities between the two vehicles. These changes can, of
course, be positive or negative. Inserting equation 18 into equation 12
gives the dynamometer interaction error as:
VDIerror = - Rr[(l-x) ALDAxle + ALNDAxle
-Ax(LDAxle + ALDAxle)]
-ARr [(1-x)LDAxle + LNDAxle
+(l-x)ALDAxle + ALNDAxle
-Ax(LDAxle + ALDAxle)] (19)
The purpose of this report is to investigate the magnitude of the
typical error, therefore neglecting terms which are products of the
changes in the quantities is an acceptable approximation. This ap-
p^roximation is valid as long as the changes in the quantities are small
compared to the original values. Using this approximation, the error
can be expressed as:
VDIerror * -Rr[(l-x) ALDAxle + ALNDAxle - AxLDAxle]
-AR[(l-x) LDAxle + LNDAxle] (20)
Regrouping:
VDIerror = -(Rr)(1-x)(ALDAxle)
-(Rr)(ALNDAxle)
+(Rr)(LDAxle)(Ax) (21)
-[(1-x)(LDAxle) + LNDAxle](ARr)
The terms of equation 21 may be considered independently as:
LDAxle Error = -(Rr)(1-x)(ALDAxle)
LNDAxle Error = -(Rr)(ALNDAxle) (22)
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Road-Rolls Error = (Rr)(LDAxle)(Ax)
Rolling Resistance Error = [(l-x)LDAxle + LNDAxle]ARr
The error, as expressed by equations 10 and 20 will be investigated
in two approaches. In the first case, the magnitude of the probable
errors when reasonable, "good engineering" practices are followed will
be considered. In this case the errors result from minor vehicle or
component variations. In the second case, the intent to propagate
errors is assumed and the maximum possible errors will be considered.
B. The Magnitude of Probable Errors Expected with Reasonable
"Good Engineering Practice"
If good engineering practice is used, the vehicle will be similar
and, in general, the errors will be small.
Under the constraints of the applicability of coast down results to
different vehicles, the first vehicle should represent the appropriate
aerodynamics of the production vehicle. Therefore:
Aero' = Aero (23)
The wheel bearing, and brake drag of the vehicles should be similar,
therefore:
. NonDAxle,= NonDAxle' . (24)
Assuming normal production tolerances a variation of + 10 percent would
seem reasonable for the brake drag. Recent EPA measurements have indi-
cated the total non-driving wheel bearing and brake drag of a vehicle
may be as much as one horsepower at 50 mph. A 10 percent variation
would therefore be + 0.1 horsepower at 50 mph.
In the case of the dynamometer simulation error, VDIerror, the
previously discussed necessary condition for this error to vanish will
be approximately met and the error is expected to be small. It would
seem reasonable to expect a variation of 10 percent in nominally iden-
tical tires. Assuming this variation:
Ax = + 0.1 x
ARr = + 0.1 Rr (25)
The load on the non-driving axle might change by 100 pounds because of
engine variation. Likewise the drive axle load might vary by 50 pounds
because of minor body variations or the amount of fuel in the vehicle.
Therefore, it is assumed:
ALDAxle = + 100 pounds
ALNDAxle = ±'.50: pounds (26)
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Recent EPA investigations indicate appropriate values for the
necessary tire parameters are:
x * 1.7 (27)
Rr - 10 pounds force/1000 pounds load
- 10 Ib/klb
In the case of a typical 4000 pound vehicle, it is reasonable to
expect axle loads of approximately 2100 pounds and 1900 pounds for the
non-drive axle and drive axles respectively. Therefore:
LDAxle = 1900 pounds
LNDAxle = 2100 pounds (28)
Substituting the values given in equations 25 through 28 into
equation 22 gives_tbe_dynaiQOjmeter..simulation errors .as:
LDAxle Error = 0.35 pounds
LNDAxle Error = -1.0 pounds
Road-Rolls Error = 3.23 pounds
Rolling Resistance Error = 0.77 pounds (29)
. The worst case error occurs when all of the above error components
have the same sign. In this case, the maximum error is 5.35 pounds
force. The probable error is the RMS value of 3.48 pounds. At 50 mph
these errors would be 0.71 and 0.46 horsepower respectively.
v
Including the possible brake drag error of 0.1 horsepower, the
experimental error associated with minor vehicle variations should in
all cases be less than 0.9 horsepower and would be expected to be less
than about 0.5 horsepower.
In the calculation of the dynamometer simulation error, it should
be noted that the third term dominated. This term originates from
variations in the road versus rolls behavior of the tire. Therefore,
the tire must be considered as an important aspect of the vehicle.
C. The Magnitude of Potential Intentional Errors
The previous section assumes reasonable good engineering judgment
is used with the intent to keep errors small. This section examines the
possibilities of errors deliberately propagated to achieve a low dyna-
mometer adjustment. In this case it will be assumed that all errors are
uniformly in the direction to optimize the dynamometer load on the
second vehicle. In addition, maximum variations of the parameter are
assumed. The first obvious possibility occurs if the first vehicle is
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more aerodynamic than the second. Constraints on vehicle type, frontal
area and protuberance group reduce the possibility. However, by optimum
selection of trim packages to minimize aerodynamic drag, it may still be
possible to induce errors of 5 percent. In the case of a typical ve-
hicle with an aerodynamic power requirement of about 10 horsepower at 50
mph this would be about 0.5 horsepower.
The second obvious place to introduce error is to eliminate the
non-driving axle drag on the first vehicle. EPA measurements have indi-
cated non-drive axle brake drag may dissipate as much as one horsepower.
If this is typical, elimination of the drag would induce a one horse-
power error.
Optimization of the vehicle dynamometer interaction error is a
logical method to introduce additional errors. Errors can be introduced
by optimizing the tire selection, the vehicle weight distribution, and
the tire inflation pressure. The possible magnitude of each of these
errors will be discussed independently.
1. Tire Selection
The obvious approach to achieve a low dynamometer adjustment is to
optimize the vehicle tires on the first vehicle. By careful selection
of tires within a generic tire type, it is probably possible to locate
tires which have a rolling resistance 30 percent below the rolling re-
sistance of the intended production tire. Equation 22 gives the error
associated with the change in tire rolling resistance as:
Tire Error = [(1-x) (LDAxle) + LNDAxle](ARr) (30)
Using AR = 0.3R, and the previously used parameters:
Tire Error =2.31 Ibs
= 0.31 horsepower at 50 mi/hr (31)
It is likewise assumed that careful tire selection could result in
tires which differed in the road to rolls characteristics by 30 percent.
This effect could also be introduced or greatly increased by variations
in tire operating pressures. The error introduced by this effect is:
Road Rolls Error = (Rr) (LDAxle)Ax
= 9.69 pounds
= 1.29 horsepower at 50 mi/hr (32)
2. Weight Distribution
Changes in weight distribution induce additional possible errors.
If the vehicle weight distribution changed by 200 pounds, for example,
one vehicle had an axle load which differed by 100 pounds from the
design load and the second which also differed by 100 pounds in the
opposite direction, the error would be:
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LDAxle Error = Rr(l-x)(ALDAxle)
=1.4 pounds
= .18 horsepower at 50 mi/hr (33)
Likewise:
LNDAxle Error = (Rr)(ALNDAxle)
= 2.0 pounds
=0.26 horsepower at 50 mi/hr (34)
The maximum dynamometer simulation error is the sum of the absolute
values of the individual errors. This sum is 15.4 pounds force or 2.0
horsepower at 50 mph. Including the maximum anticipated errors from
aerodynamic effects, 0.5 horsepower and brake drag elimination, 1.0
horsepower, the total possible error is 3.4 horsepower at 50 mi/hr.
Conclusions
The error associated with using a dynamometer adjustment obtained
from test results of one vehicle to represent a second vehicle is ex-
pected to be less than about one-half horsepower if "good engineering"
practice and intent is used. This error is about 5 percent of the
typical dynamometer adjustment, and is considered acceptable.
If the possible errors are consistently maximized to result in
optimum beneficial .dynamometer loading of the second test vehicle, the
possible errors are much greater. In this case a total error of 3.4
horsepower at 50 mi/hr is possible. Of this total error, the largest
single contribution, 1.3 horsepower, is from variations in the tire-
rolls behavior of the tires. The second largest possible contribution,
1.0 horsepower, may be introduced by brake drag effects.
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