# Development of a Modal Emissions Model Using Data from the Cooperative Industry/Government Exhaust Emission Test Program {#development-of-a-modal-emissions-model-using-data-from-the-cooperative-industrygovernment-exhaust-emission-test-program .Awma-Title}

#  {#section .Awma-Head-2}

# 00-Abstract-64 {#abstract-64 .Awma-Head-2}

Roger L. Wayson

Brian Y. Kim

Civil and Environmental Engineering

University of Central Florida

P.O. Box 162450

Orlando, FL 32816-2450

\(407\) 823-2480

wayson@pegasus.cc.ucf.edu

**Keywords: emission, modal, mode, multiplier, mobile, model**

ABSTRACT

The Environmental Protection Agency's (EPA's) recommended model,
MOBILE5a, has been used extensively to predict emission factors based on
average speeds for each fleet type. Because average speeds are not
appropriate in modeling intersections or other scenarios involving
intermittent travel, emission factors that are specific for vehicle
operating modes (acceleration, deceleration, free-flow, and idle) have
been studied in the past. Several models have been developed that use
the concept of acceleration-speed products to serve as input variables
to determine multipliers that can be used to modify constant speed
emission factors. Although relatively simplistic, this process provides
results that are considered more accurate than constant speed emission
factors. The Comprehensive Modal Emissions Model (CMEM) developed under
sponsorship by the National Cooperative Highway Research Program (NCHRP)
is based on a parameterized physical approach. While anticipated to be
more accurate, the input requirements to the model are necessarily more
complicated.

This paper presents a new modal emissions model based on existing
emissions data from the early 1990s revision efforts of the Federal Test
Procedure (FTP). The model updates the older acceleration-speed product
models that were based on data from the 1970s. Using second-by-second
emissions data, several different forms of the modeling equations were
developed and statistically analyzed for predicting multiplying factors
for CO. A goal in developing this model is to serve as a comparison
basis for the NCHRP model. The second and more important goal is to use
the model as part of a microscale traffic simulation model that predicts
air quality near roadways.

INTRODUCTION

A new vehicle emissions model has been developed at the University of
Central Florida (UCF) to predict vehicle emissions that are specific to
operating modes (e.g. idle, cruise, acceleration, and deceleration).
This modal emissions model uses the product of speed and acceleration as
the input variable and a modal multiplier as the independent variable.
The modal multiplier is used to convert a constant speed emission factor
for Carbon Monoxide (CO), such as that obtained from the MOBILE series
models, into a modal emission factor.

The new model updates and expands upon previous models by using
emissions data from the 1993-1994 tests conducted at the General Motors
Proving Ground (1). The other models use older databases, typically
gathered in the 1970s. A goal in developing the model was to serve as a
comparison for existing modal emission models including the
Comprehensive Modal Emissions Model (CMEM) developed at the University
of California at Riverside (UCR) under sponsorship by the National
Cooperative Highway Research Program (NCHRP) (2). However, the main
reason was to use the model as part of an overall air quality model that
predicts CO concentrations near roadways (3). This air quality model,
also developed at UCF, uses a microscale traffic movement algorithm and
Gaussian puff equations to model each vehicle as a moving point source.
This simulation approach requires a model that can provide
second-by-second modal emission factors. Although CMEM can produce
second-by-second emission factors, its use in the air quality model
would be difficult partly due to data requirements, but also because the
CMEM software was not designed to be used in a simulation environment.
However, it may be possible to incorporate the main CMEM code(s) into
the air quality model such that communication between the two is
internal rather than through input and output files. Due to time
constraints, CMEM incorporation into the air quality model is beyond the
scope of this project and is a consideration for future work.

BACKGROUND

Emission models have typically been based on regressions of emissions
data collected for driving cycles such as the Federal Test Procedure
(FTP). Since the condition of a vehicle in the hot and cold stabilized
phases are considered to be similar, bag 4 measurements are usually not
taken. The basic emission rate (BER) determined by the EPA recommended
MOBILE series models is weighted by using the default start mode
fractions of 0.43 for cold start and 0.57 for hot start conditions. The
emission rates determined by these models are based on average speeds
for each fleet type. Because average speeds are not appropriate in
modeling intersections or other scenarios involving intermittent travel,
emission factors that are specific for vehicle operating modes
(acceleration, deceleration, free-flow, and idle) have been studied.

In the past, several models have been developed that use the concept of
speed-acceleration products to serve as input variables to determine
multipliers that can be applied to constant speed emission factors. The
reasoning behind this approach is that vehicles are assumed to
accelerate at a constant rate of power input. This translates into the
product of speed and acceleration being equivalent to power divided by
mass which can easily be proven by unit analysis. Therefore, the
underlying thought is that as the speed-acceleration product, and hence,
power demand increases, so will the emission rate for CO (4).

One of the studies was performed by the Colorado Department of Highways
(CDOH) (4). They developed a relatively simple model to predict the
multipliers that could be used to convert constant speed emission
factors to modal emission factors. A multiplier is actually a normalized
value derived by dividing a vehicle's emission rate for a particular
modal activity (i.e. combination of acceleration and speed of vehicle)
by its average 75 FTP emission rate. Using a total of 45 vehicles (1975
model year) and 39 modal activities in the Surveillance Driving Sequence
(SDS) for each vehicle, best fit curves for the data were developed that
correlate multipliers for CO, HC, and NO~x~ with the product of speed
and acceleration. The corresponding quadratic equation for CO is
presented below:

> M = 0.182 - 7.9776 x 10^-3^ (AS) + 3.6227 x 10^-4^ (AS)^2^ (1)
>
> M = multiplier for CO
>
> AS = speed-acceleration product (ft^2^/sec^3^)

Since the emission rate and the speed-acceleration product are believed
to be linearly related, the second order term in equation 1 has been
attributed to factors such as the air/fuel mixture enrichment under load
and reaching the limits of the catalytic converter. Due to the fact that
the modal emission rates were normalized with respect to the 75 FTP rate
obtained from bag data, the multipliers are only applicable to emission
factors based on a constant speed of 19.6 mph which is the average speed
for the 75 FTP cycle.

The CALINE4 model uses a similar approach to determining multipliers.
However, the data sets used to derive regression equations were
different than the ones use in the CDOH study. CALINE4 uses data
specific to California (5). The two equations employed in CALINE4 are
presented as equations 2 and 3:

EFA = (BAG2)(0.75)e^(0.0454)(AS)^ (2)

EFA = (BAG2)(0.027)e^(0.098)(AS)^ (3)

> EFA = modal emission factor

BAG2 = constant speed emission factor

Equation 2 is applicable for vehicles accelerating from rest to 45 mph,
and equation 3 corresponds to vehicles accelerating from speeds greater
than 15 mph to 60 mph. These equations are similar to the CDOH equation
except that the constant speed emission factor (BAG2) has been included
so that the equations represent modal emission factors rather than a
multiplying factor. Unlike the CDOH model, CALINE4 uses only bag 2
(stabilized mode) emissions data to derive the multipliers. Therefore,
CALINE4 requires that constant speed emission factors be based at 16.2
mph which is the average speed for the bag 2 stage of the FTP cycle.

The new NCHRP model (CMEM) is based on a parameterized physical approach
(2). While anticipated to be more accurate than the speed-acceleration
product approach, the input requirements are also more involved. The
inputs for this model can be grouped into two broad categories: input
operating variables and model parameters. Examples of input operating
variables include second-by-second speed, grade, and accessory use
information (e.g. air conditioning). Model parameters include public
domain or generic types (e.g. vehicle mass, engine displacement, tire
rolling resistance, transmission efficiencies, etc.) and measured types
(e.g. engine friction factor, thermal efficiency, catalyst pass
fraction, etc.). The model not only determines composite emission
factors but also provides second-by-second tailpipe emissions. The
current condition of the software does not realistically allow it to be
used in a simulation environment where function calls to CMEM would have
to be made during each simulated time step. Therefore, the core
algorithm in CMEM would need to be reproduced and implemented internally
for it to be used in an air quality model that requires modal emission
factors during a simulation.

METHODOLOGY

The development of modal multipliers is based on the comprehensive
emissions data from the 1993-1994 tests conducted at the General Motors
Proving Ground. The tests arose out of efforts to review and revise the
FTP and was jointly executed by the EPA, the California Air Resources
Board (CARB) and the automotive industry. For simplicity, this
cooperative industry/government exhaust emissions data will henceforth
be referred to as the CIGEE data which includes both cumulative bag data
and second-by-second emissions data for several different test cycles.

The method of analysis essentially mirrors those used in the CDOH and
CALINE4 studies. Second-by-second emissions data are correlated with the
products of average speeds and accelerations. The steps involved in
determining multipliers are outlined in Figure 1. Only the CIGEE data
corresponding to the FTP test cycle (also known as the LA4) was used. A
total of 83 possible acceleration/deceleration ranges were identified
from this cycle. A difficulty in developing these ranges was that the
ranges for each tested vehicle did not perfectly correlate time-wise
with those for the others. This is understandable since tolerances along
the speed versus time curve are allowed during the FTP test. Since the
differences between these ranges were small, the derived acceleration
values from each vehicle were considered comparable to those from other
vehicles. The list of speed ranges shown in Table 1 were derived based
on the requirements that the ranges are common to all vehicles tested
and that the list contains a sufficient mixture of speed ranges
producing varying acceleration values. Such a list is similar to the
lists of SDS modes used in the CDOH and CALINE4 studies. Speed ranges
corresponding to deceleration were not used since it was assumed that
emissions during deceleration are similar to idle emissions. Also, any
emissions data corresponding to vehicles that were calibrated for
stoichiometric combustion were eliminated since they would not allow
those vehicles to produce emissions under commanded enrichment.
Stoichiometric calibration was done for some vehicles in order to
compare them with production vehicles experiencing commanded enrichment.

The most significant difference between the new emissions model and the
previously developed models is that the hot and cold mode percentages
were incorporated into the new model. Using composite bag data, separate
equations were developed for vehicles in the hot transient, cold
transient, and stabilized modes. In order to use these equations,
constant speed emission rates specific to each of the modes must be
determined. If using MOBILE5a, these hot, cold, and stable emission
rates can be determined by using the percentages shown in Table 2. These
rates can be considered "pure" since the percentages in Table 2 force
MOBILE5a to produce unweighted, bag-specific values. In addition to
modal activity, equations were also categorized by vehicle type. The
descriptions of each vehicle type and vehicle information are presented
in Table 3. Since the CIGEE database did not include heavy-duty
vehicles, no equations for this vehicle-type could be developed.

RESULTS

The modal multiplier regression coefficients and goodness of fit
criteria are presented in Table 4. The model types are presented as
equations 4-6:

Polynomial: y = ax^2^ + bx + c (4)

Power Series: y = ax^b^ (5)

Exponential: y = ae^bx^ (6)

The coefficients in Table 4 indicate that for the stabilized modes, the
regression equations increase with the product of speed and
acceleration. This is the opposite case for the transient (hot and cold)
modes. The reasons for the decreasing functions are not clear since
several factors could account for this. One reason may be that a few
non-representative data points (i.e. outliers) may have skewed the
results. Another reason may be that for the vehicles tested, increasing
the speed of the vehicles under transient conditions could have improved
engine combustion (e.g. by increasing engine temperature) resulting in
lower CO emisisons. This effect could have overcome the effect of
increasing CO emissions from an increase in fuel consumption (i.e.
enrichment of the air-fuel mixture).

Except for cars and heavy-light trucks under the stabilized condition
and light trucks under the hot transient condition, the coefficient of
determination (R^2^) appears reasonable for most of the regression
results. The root MSEs appear to be relatively small for most of the
models except for those in the cold transient mode. However, all of
their p-values are well below the significance level of 0.05 (95%
confidence). Taking into account all of the goodness of fit criteria,
the recommended equations for combinations of vehicle type and mode are
indicated in bold in Table 4 and plotted in Figures 2-4.

In an effort to compare the current model with existing models, the
model developed for "cars" under stabilized conditions was plotted
against the models used in CALINE4 and CMEM (NCHRP model). In order to
simplify the comparisons, only the regression equations for the
stabilized (bag 2) phase were used. Since this phase of the FTP cycle
starts from the rest condition (0 mph), the appropriate CALINE4 equation
was used:

> EFA = (BAG2)(0.75)e^(0.0454)(AS)^ (7)

This equation corresponds to vehicles starting from rest and
accelerating up to 45 mph. Since modal multipliers were modeled, the
actual equation used to represent CALINE4 was:

M = EFA/BAG2 = (0.75)e^(0.0454)(AS)^ (8)

Since CMEM does not contain regression equations to directly provide
modal multipliers, the velocity activity data for the FTP bag 2 phase
was used to run CMEM and obtain second-by-second emissions data. The
default vehicle parameter data supplied with the CMEM software was used
during the run. The second-by-second emissions data was used to obtain
average emission rates (g/s) for each of the selected speed ranges in
the stabilized mode shown in Table 1. Dividing these rates by the
average emission rate (converted to g/s by multiplying g/mile by the
average speed, 16.2 mph) for the stabilized phase resulted in modal
multipliers. Regression analysis of the multipliers with the product of
speed and acceleration resulted in equation 9.

M = (0.919)e^(0.0461)(AS)^ (9)

The plots of each of the three models are shown in Figure 5. The model
developed using the CIGEE data provided the highest values and
therefore, appears to be the most conservative. This model corresponds
to the cars ("C") category and is presented as equation 10.

M = (1.300)e^(0.0430)(AS)^ (10)

The ratio between the CIGEE multipliers and the CALINE4 multipliers
decreases from 1.73 at 1 mph to 1.58 at 45 mph. Similarly, the CIGEE
multipliers to CMEM multipliers range from 1.40 at 1 mph to 1.24 at 45
mph. While these comparisons are valid for CALINE4, they may not be
entirely accurate for CMEM. The aforementioned regression analysis using
the second-by-second emissions output from CMEM only involved one
vehicle's parameter data (i.e. default data). This was done due to a
lack of data. Therefore, a more comprehensive study would involve the
use of several different vehicle parameter data to obtain a more
statistically valid regression equation. An alternative but equal method
of comparison between the models would have involved the plotting of
actually emission values instead of multipliers. But since the
multipliers and emission rates are directly related, the results would
have led to similar conclusions.

CONCLUSION

The new modal emissions model presented in this paper is an update to
the older emissions models developed by the CDOH and CALINE4 studies.
The use of modal multipliers is significantly different than the
physical modeling approach employed in the NCHRP model (CMEM). A simple
comparison analysis appears to indicate that the new model provides
similar results to that of CMEM. The simplicity of the new model allows
easy integration into a traffic simulation model where functions calls
to the modal emissions algorithm can be made directly within the native
code.

REFERENCES

1.  Haskew, Harold M., Kevin Cullen, Thomas F. Liberty, and William K.
    Langhorst. "The Execution of a Cooperative Industry/Government
    Exhaust Emission Test Program." Society of Automotive Engineers. SAE
    Technical Paper 94C016, Presented at Convergence 1994.

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2.  Barth, Mathew, Feng An, Theodore Younglove, George Scora, Carrie
    Levine, Marc Ross, and Thomas Wenzel. "Comprehensive Modal Emissions
    Modal (CMEM), Version 2.0, User's Guide." National Cooperative
    Highway Research Program, Project 25-11. January 3, 2000.

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3.  Wayson, Roger L., Brian Y. Kim, C. David Cooper, and John M.
    MacDonald. "Predicting Air Quality Near Roadways Through the
    Application of a Gaussian Puff Model to Moving Sources." A Paper
    Presented to the Air & Waste Management Association, 93^rd^ Annual
    Meeting, Salt Lake City, Utah, June 18-22, 2000.

4.  Griffin, Richard. "Air Quality Impact of Signaling Decisions."
    Colorado Department of Highways. CDOH-DTP-R-80-11. October 1980.

5.  Benson, Paul. "CALINE4 -- A Dispersion Model For Predicting Air
    Pollutant Concentrations Near Roadways." State of California
    Department of Transportation. Final Report November 1984, Revised
    June 1989.

Table 1

FTP Speed Ranges and Corresponding Speed-Acceleration Products

+---------+------+-------+-------+--------+----------+----------------+
|         | *    | **    | *     | **Ave  | **A      | **Speed-       |
|         | *Sta | End** | *Time | rage** | verage** | Acceleration** |
|         | rt** |       | in**  |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| **Cycle | *    | **Sp  | **M   | **S    | **Accele | # Product      |
| Mode**  | *Spe | eed** | ode** | peed** | ration** |                |
|         | ed** |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| *       | *    | **(m  | **(s  | **(    | **(mp    | **             |
| *Name** | *(mp | ph)** | ec)** | mph)** | h/sec)** | (mph^2^/sec)** |
|         | h)** |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
|         |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| FTP3^a^ | 5.9  | 22.5  | 11    | 14.2   | 1.51     | 21.43          |
+---------+------+-------+-------+--------+----------+----------------+
| FTP7^a^ | 6.6  | 47.5  | 42    | 27.05  | 0.97     | 26.34          |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 4.3  | 34.9  | 20    | 19.6   | 1.53     | 29.99          |
| TP11^a^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 5.9  | 30    | 13    | 17.95  | 1.85     | 33.28          |
| TP15^a^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 6.6  | 36    | 17    | 21.3   | 1.73     | 36.84          |
| TP18^a^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
|         |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 3.5  | 25    | 19    | 14.25  | 1.13     | 16.13          |
| TP22^b^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 6.6  | 17    | 8     | 11.8   | 1.30     | 15.34          |
| TP26^b^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 4.5  | 26    | 15    | 15.25  | 1.43     | 21.86          |
| TP30^b^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 3.2  | 28.6  | 18    | 15.9   | 1.41     | 22.44          |
| TP35^b^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 6.3  | 28.9  | 17    | 17.6   | 1.33     | 23.40          |
| TP37^b^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 0    | 28.5  | 18    | 14.25  | 1.58     | 22.56          |
| TP40^b^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 4    | 28    | 18    | 16     | 1.33     | 21.33          |
| TP44^b^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 0.6  | 24.9  | 22    | 12.75  | 1.10     | 14.08          |
| TP46^b^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 5.4  | 23    | 9     | 14.2   | 1.96     | 27.77          |
| TP50^b^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 1.5  | 21.8  | 20    | 11.65  | 1.0      | 11.82          |
| TP53^b^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 4.8  | 22    | 14    | 13.4   | 1.23     | 16.46          |
| TP61^b^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
|         |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 3    | 22.4  | 11    | 12.7   | 1.76     | 22.40          |
| TP66^c^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 3.3  | 47.5  | 42    | 25.4   | 1.05     | 26.73          |
| TP70^c^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 1    | 34.6  | 20    | 17.8   | 1.68     | 29.90          |
| TP74^c^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 2.6  | 30.1  | 13    | 16.35  | 2.12     | 34.59          |
| TP78^c^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+
| F       | 3.3  | 36.1  | 17    | 19.7   | 1.93     | 38.01          |
| TP81^c^ |      |       |       |        |          |                |
+---------+------+-------+-------+--------+----------+----------------+

^a^Cold Transient Mode.

^b^Stabilized Mode.

^c^Hot Transient Mode.

**Table 2**

**Vehicle Mode Percentages Required to Obtain Bag-Specific Emission
Rates**

  ------------------------------ --------------- -------------- -------------
                                 **Vehicle Mode                 
                                 Percentages**                  

                                 **Bag 1**       **Bag 2**      **Bag 3**

  PCCN                           100             0              0

  PCHC                           0               0              100

  PCCC                           100             0              0

                                                                

  Vehicle Speed (mph)            25.6            16.2           25.05
  ------------------------------ --------------- -------------- -------------

Note: PCCN = Cold Transient Mode Percentage without a catalyst;

PCHC = Hot Transient Mode Percentage with a catalyst;

PCCC = Cold Transient Mode Percentage with a catalyst.

**Table 3**

**Vehicle Categories**

+-----------------------+----------------------------+----------------+
| ## Vehicle            | **Make, Model,**           | **Age of       |
|                       |                            | Catalytic**    |
| ## Type               | **and Year**               |                |
|                       |                            | **Converter    |
|                       |                            | (miles)**      |
+-----------------------+----------------------------+----------------+
| Cars (C)              | Ford Escort '93            | 50,000         |
+-----------------------+----------------------------+----------------+
|                       | Ford Taurus LX '93         | 50,000         |
+-----------------------+----------------------------+----------------+
|                       | Ford Mustang LX '93        | 50,000         |
+-----------------------+----------------------------+----------------+
|                       | Oldsmobile Cutlass Supreme | 100,000        |
|                       | '94                        |                |
+-----------------------+----------------------------+----------------+
|                       | Pontiac Grand Prix '94     | 50,000         |
+-----------------------+----------------------------+----------------+
|                       | Oldsmobile 98 '94          | 50,000         |
+-----------------------+----------------------------+----------------+
|                       | Cadillac Seville '94       | 100,000        |
+-----------------------+----------------------------+----------------+
|                       | Oldsmobile Custom Cruiser  | 50,000         |
|                       | '92                        |                |
+-----------------------+----------------------------+----------------+
|                       | Saturn Saturn '94          | 50,000         |
+-----------------------+----------------------------+----------------+
|                       | Geo Metro '93              | 36,000         |
+-----------------------+----------------------------+----------------+
|                       | Pontiac Grand Am '93       | 50,000         |
+-----------------------+----------------------------+----------------+
|                       | Honda Civic '92            | 50,000         |
+-----------------------+----------------------------+----------------+
|                       | Mitsubishi Mirage '93      | 4,000          |
+-----------------------+----------------------------+----------------+
|                       | Toyota Camry '93           | 50,000         |
+-----------------------+----------------------------+----------------+
|                       | Toyota Corolla '93         | 50,000         |
+-----------------------+----------------------------+----------------+
|                       | Mercedes 420 SEL '92       | 50,000         |
+-----------------------+----------------------------+----------------+
|                       |                            |                |
+-----------------------+----------------------------+----------------+
| Light Trucks (LT)     | Jeep Cherokee '94          | 50,000         |
+-----------------------+----------------------------+----------------+
|                       | Ford Ranger 4x2 XLT '93    | 50,000         |
+-----------------------+----------------------------+----------------+
|                       | Ford F150 4x2 '93          | 50,000         |
+-----------------------+----------------------------+----------------+
|                       | GMC Sonoma P/U '93         | 100,000        |
+-----------------------+----------------------------+----------------+
|                       | Chevrolet C10 P/U '94      | 126,000        |
+-----------------------+----------------------------+----------------+
|                       | Mazda MPV '92              | 90,000         |
+-----------------------+----------------------------+----------------+
|                       | Nissan Pathfinder '92      | 35,000         |
+-----------------------+----------------------------+----------------+
|                       |                            |                |
+-----------------------+----------------------------+----------------+
| Heavy Light Trucks    | Ford F250 4x2 '93          | 50,000         |
| (HLT)                 |                            |                |
+-----------------------+----------------------------+----------------+
|                       | Chevrolet Suburban '92     | 30,000         |
+-----------------------+----------------------------+----------------+
|                       | Chevrolet G30 Van '93      | 30,000         |
+-----------------------+----------------------------+----------------+
|                       | Chevrolet C30/K30 Duallie  | 150,000        |
|                       | '94                        |                |
+-----------------------+----------------------------+----------------+

**Table 4**

### **Modal Multiplier Regression Results**

<table style="width:100%;">
<colgroup>
<col style="width: 9%" />
<col style="width: 9%" />
<col style="width: 8%" />
<col style="width: 5%" />
<col style="width: 11%" />
<col style="width: 9%" />
<col style="width: 11%" />
<col style="width: 8%" />
<col style="width: 8%" />
<col style="width: 9%" />
<col style="width: 11%" />
</colgroup>
<tbody>
<tr class="odd">
<td><p><strong>Mode</strong></p>
<p><strong>Type</strong></p></td>
<td><p><strong>Vehicle</strong></p>
<p><strong>Type<sup>a</sup></strong></p></td>
<td><p><strong>Model</strong></p>
<p><strong>Type<sup>b</sup></strong></p></td>
<td><strong>n<sup>c</sup></strong></td>
<td><strong>A</strong></td>
<td><strong>b</strong></td>
<td><strong>c</strong></td>
<td><strong>R<sup>2</sup></strong></td>
<td><p><strong>Root</strong></p>
<p><strong>MSE</strong></p></td>
<td><strong>MSE</strong></td>
<td><strong>p-value</strong></td>
</tr>
<tr class="even">
<td>Cold</td>
<td>HLT</td>
<td>Pow</td>
<td>5</td>
<td>1.55X10<sup>13</sup></td>
<td>-8.571</td>
<td>n/a</td>
<td>0.996</td>
<td>1.867</td>
<td>3.487</td>
<td>0.0001</td>
</tr>
<tr class="odd">
<td>Transient</td>
<td></td>
<td><strong>Exp</strong></td>
<td><strong>5</strong></td>
<td><strong>271388</strong></td>
<td><strong>-0.392</strong></td>
<td><strong>n/a</strong></td>
<td><strong>0.997</strong></td>
<td><strong>1.630</strong></td>
<td><strong>2.656</strong></td>
<td><strong>7.00E-05</strong></td>
</tr>
<tr class="even">
<td></td>
<td></td>
<td>Poly</td>
<td>5</td>
<td>0.50315</td>
<td>-32.775</td>
<td>529.929</td>
<td>0.962</td>
<td>7.177</td>
<td>51.506</td>
<td>0.03838</td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td></td>
<td>LT</td>
<td>Pow</td>
<td>5</td>
<td>1.11X10<sup>13</sup></td>
<td>-8.256</td>
<td>n/a</td>
<td>0.997</td>
<td>3.123</td>
<td>9.753</td>
<td>7.00E-05</td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td><strong>Exp</strong></td>
<td><strong>5</strong></td>
<td><strong>161332</strong></td>
<td><strong>-0.339</strong></td>
<td><strong>n/a</strong></td>
<td><strong>0.998</strong></td>
<td><strong>2.681</strong></td>
<td><strong>7.186</strong></td>
<td><strong>5.00E-05</strong></td>
</tr>
<tr class="even">
<td></td>
<td></td>
<td>Poly</td>
<td>5</td>
<td>0.88438</td>
<td>-58.235</td>
<td>952.377</td>
<td>0.979</td>
<td>10.019</td>
<td>100.4</td>
<td>0.0215</td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td></td>
<td>C</td>
<td>Pow</td>
<td>5</td>
<td>4.52X10<sup>14</sup></td>
<td>-9.377</td>
<td>n/a</td>
<td>0.998</td>
<td>2.951</td>
<td>8.707</td>
<td>3.00E-05</td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td><strong>Exp</strong></td>
<td><strong>5</strong></td>
<td><strong>615340</strong></td>
<td><strong>-0.388</strong></td>
<td><strong>n/a</strong></td>
<td><strong>0.999</strong></td>
<td><strong>2.846</strong></td>
<td><strong>8.102</strong></td>
<td><strong>2.00E-05</strong></td>
</tr>
<tr class="even">
<td></td>
<td></td>
<td>Poly</td>
<td>5</td>
<td>1.22165</td>
<td>-79.779</td>
<td>1292.240</td>
<td>0.967</td>
<td>16.273</td>
<td>264.8</td>
<td>0.03261</td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td>Stabilized</td>
<td>HLT</td>
<td>Pow</td>
<td>11</td>
<td>0.1784</td>
<td>0.875</td>
<td>n/a</td>
<td>0.421</td>
<td>0.628</td>
<td>0.395</td>
<td>0.03077</td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td><strong>Exp</strong></td>
<td><strong>11</strong></td>
<td><strong>0.9331</strong></td>
<td><strong>0.047</strong></td>
<td><strong>n/a</strong></td>
<td><strong>0.454</strong></td>
<td><strong>0.610</strong></td>
<td><strong>0.372</strong></td>
<td><strong>0.02298</strong></td>
</tr>
<tr class="even">
<td></td>
<td></td>
<td>Poly</td>
<td>11</td>
<td>0.007789</td>
<td>-0.198</td>
<td>3.124</td>
<td>0.473</td>
<td>0.636</td>
<td>0.404</td>
<td>0.077</td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td></td>
<td>LT</td>
<td>Pow</td>
<td>11</td>
<td>0.4085</td>
<td>0.832</td>
<td>n/a</td>
<td>0.688</td>
<td>0.687</td>
<td>0.472</td>
<td>0.00159</td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td><strong>Exp</strong></td>
<td><strong>11</strong></td>
<td><strong>1.949</strong></td>
<td><strong>0.045</strong></td>
<td><strong>n/a</strong></td>
<td><strong>0.759</strong></td>
<td><strong>0.605</strong></td>
<td><strong>0.366</strong></td>
<td><strong>0.00048</strong></td>
</tr>
<tr class="even">
<td></td>
<td></td>
<td>Poly</td>
<td>11</td>
<td>0.01876</td>
<td>-0.529</td>
<td>7.601</td>
<td>0.825</td>
<td>0.545</td>
<td>0.297</td>
<td>0.00093</td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td></td>
<td>C</td>
<td>Pow</td>
<td>11</td>
<td>0.3038</td>
<td>0.782</td>
<td>n/a</td>
<td>0.516</td>
<td>0.594</td>
<td>0.353</td>
<td>0.01273</td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td><strong>Exp</strong></td>
<td><strong>11</strong></td>
<td><strong>1.3001</strong></td>
<td><strong>0.043</strong></td>
<td><strong>n/a</strong></td>
<td><strong>0.582</strong></td>
<td><strong>0.552</strong></td>
<td><strong>0.305</strong></td>
<td><strong>0.00634</strong></td>
</tr>
<tr class="even">
<td></td>
<td></td>
<td>Poly</td>
<td>11</td>
<td>0.0135</td>
<td>-0.404</td>
<td>5.542</td>
<td>0.665</td>
<td>0.524</td>
<td>0.275</td>
<td>0.01264</td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td>Hot</td>
<td>HLT</td>
<td><strong>Pow</strong></td>
<td><strong>5</strong></td>
<td><strong>23336412</strong></td>
<td><strong>-4.886</strong></td>
<td><strong>n/a</strong></td>
<td><strong>0.885</strong></td>
<td><strong>0.835</strong></td>
<td><strong>0.698</strong></td>
<td><strong>0.01714</strong></td>
</tr>
<tr class="odd">
<td>Transient</td>
<td></td>
<td>Exp</td>
<td>5</td>
<td>456.2</td>
<td>-0.194</td>
<td>n/a</td>
<td>0.855</td>
<td>0.937</td>
<td>0.879</td>
<td>0.02447</td>
</tr>
<tr class="even">
<td></td>
<td></td>
<td>Poly</td>
<td>5</td>
<td>0.04636</td>
<td>-3.065</td>
<td>51.237</td>
<td>0.938</td>
<td>0.751</td>
<td>0.565</td>
<td>0.06201</td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td></td>
<td>LT</td>
<td>Pow</td>
<td>5</td>
<td>488.98</td>
<td>-1.635</td>
<td>n/a</td>
<td>0.323</td>
<td>1.237</td>
<td>1.530</td>
<td>0.31772</td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td><strong>Exp</strong></td>
<td><strong>5</strong></td>
<td><strong>11.73</strong></td>
<td><strong>-0.060</strong></td>
<td><strong>n/a</strong></td>
<td><strong>0.347</strong></td>
<td><strong>1.215</strong></td>
<td><strong>1.475</strong></td>
<td><strong>0.29602</strong></td>
</tr>
<tr class="even">
<td></td>
<td></td>
<td>Poly</td>
<td>5</td>
<td>-0.001006</td>
<td>-0.066</td>
<td>4.938</td>
<td>0.365</td>
<td>1.467</td>
<td>2.151</td>
<td>0.63473</td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td></td>
<td>C</td>
<td><strong>Pow</strong></td>
<td><strong>5</strong></td>
<td><strong>6758</strong></td>
<td><strong>-2.332</strong></td>
<td><strong>n/a</strong></td>
<td><strong>0.843</strong></td>
<td><strong>0.634</strong></td>
<td><strong>0.402</strong></td>
<td><strong>0.02777</strong></td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td>Exp</td>
<td>5</td>
<td>29.54</td>
<td>-0.082</td>
<td>n/a</td>
<td>0.795</td>
<td>0.725</td>
<td>0.526</td>
<td>0.04221</td>
</tr>
<tr class="even">
<td></td>
<td></td>
<td>Poly</td>
<td>5</td>
<td>0.02697</td>
<td>-1.814</td>
<td>32.112</td>
<td>0.956</td>
<td>0.411</td>
<td>0.169</td>
<td>0.0439</td>
</tr>
</tbody>
</table>

^a^"HLT"=Heavy Light Truck; "LT"=Light Truck; "C"=Car.

^b^"Pow"=power series; "Exp"=Exponential; "Poly"=Polynomial.

^c^Number of data points used in regression.

Note: Selections in bold are considered best choices for modeling
according to goodness of fit criteria.

### **Figure 1**

**Methodology for Derivation of Modal Multipliers**

![](media/image1.wmf)

**Figure 2**

**Regression Analysis for the Cold Transient Mode**

![](media/image2.wmf)

**Figure 3**

### **Regression Analysis for the Stabilized Mode**

![](media/image3.wmf)

**Figure 4**

### **Regression Analysis for the Hot Transient Mode**

![](media/image4.wmf)

**Figure 5**

**Comparisons of Modal Multipliers Obtained from Different Models**