Preliminary Vehicle Mass Estimation
Using Empirical Subsystem
Influence Coefficients
www.a-sp.org
Report on
Preliminary Vehicle Mass Estimation
Using Empirical Subsystem Influence Coefficients
by
Donald E Malen, Adjunct Assistant Professor
Kundan Reddy, Masters Candidate
InterProfessional Programs
University of Michigan
Ann Arbor, MI
Prepared for
The FGPC-Mass Compounding Project Team
Auto/Steel Partnership
2000 Town Center, Suite 320
Southfield, Michigan
May 9, 2007
(Revised June 26, 2007)
Acknowledgments
The authors deeply appreciate the guidance and support provided by the FGPC-Mass Compounding
project team:
J. R. Shaw, US Steel Corp. (Chairman); H. Ezzat , T. M. Lee, C. C. Leung, J. Polewarczyk, General Motors
Corp., V. C. Shah, DaimlerChrysler Corp.; R. Sohmshetty, S. Das, Ford Motor Co.; C. Potter, A.I.S.I.; and
R. A. Heimbuch, P. J Villano, Auto/Steel Partnership.
Special appreciation is due to the diligent efforts of C. Leung and V. Shah in providing vehicle mass data
and their patience in the several iterations of consolidating the data. The authors also appreciate the data
provided by L. Pecar of the GM Benchmarking Activity.
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Preliminary Vehicle Mass Estimation
Malen and Reddy
Executive Summary
Vehicle design engineers intuitively know that an unplanned mass increase in a component during
vehicle design has a ripple effect throughout the vehicle; other components need to be resized increasing
vehicle mass even more. The phrase mass begets mass describes this phenomenon. A more encouraging
view of this behavior is considering a reduction in the mass of a component enabled by a new technology
resulting in a greater mass saving for the overall vehicle. These secondary mass changes can be
considerable—estimated at an additional 0.7 to 1.8 times the initial mass change.
This mass compounding behavior may be modeled using subsystem mass influence coefficients—the
incremental change in subsystem mass for a unit change in gross vehicle mass. Published data on
influence coefficients is sparse, and that published are based on vehicles in the 1975–1981 model years.
The purpose of this study is to update influence coefficients using contemporary vehicles.
In this study, mass data from 35 vehicles have been analyzed to determine mass influence coefficients.
The vehicles in the study covered Sedans, SUVs, Pick Ups, and Vans (See Appendix C for the specific
models used). Linear regression was used to determine the influence coefficients.
The primary results of this study are summarized in the illustration below. The secondary mass change
is that additional change due to resizing components when a vehicle mass change occurs during design
(the primary change). This secondary change depends on the vehicle mass influence coefficient—the sum
of coefficients for the subsystems. Indicated are results of previous studies from 1975 and 1981, along
with results of this study for All vehicles, the SUV group, and the Sedan group. (The Pick Up and Van
categories had an insufficient number of vehicles to be represented alone.)
When all subsystems can be resized, the secondary mass savings is from 0.8 to 1.5 kg/kg (1.25kg/kg is
the estimate for the All vehicles group). When the powertrain has been fixed and is not available for
resizing, the secondary mass savings is from 0.4 to 0.5 kg/kg (0.5kg/kg is the estimate for the All vehicles
group).
2.5
2.0
Secondary
Mass 1.5
Savings as
Fraction of
1.0
Original
Change
0.5
Γ
V
1981
0.0
1975
All Sedan
SUV
0
0.1
0.2
0.3
0.4
0.5
0.6
Vehicle Mass Influence γ V
All subsystems free to resize
All subsystems less powertrain
Section 8 provides an in depth example application of this data.
3
0.7
0.8
Table of Contents
1. Acknowledgements
2
2. Executive Summary
3
3. Preliminary Mass Estimation in Vehicle Design
5
4. Mass Compounding Model
5
5. Previous work on Mass Influence Coefficients
6
6. Description of this study
7
7. Summary of Results
8
8. Example Applications
9
9. Bibliography
14
Appendices
Technical Appendix A Derivation of mass model with influence coefficients
15
Technical Appendix B Estimation of subsystem influence coefficients
18
Data Appendix C
Vehicles used in analysis
22
Functional Subsystem Mass Categories
23
•
•
•
•
•
•
Gross Vehicle Mass vs. Plan View Area
Curb Mass vs. Plan View Area
System Mass as a percentage of Curb Mass
Summary of Influence Coefficients
Subsystem Mass vs. Gross Vehicle Mass-Linear
Subsystem Mass vs. Gross Vehicle Mass-Log
repeated for
1) All Vehicles
2) Sedans
3) SUVs
4) Pick Ups
24
29
34
39
Tabular Summary of Subsystem Influence Coefficients
44
Selection of Influence Coefficients for Inclusion in Model
45
4
Preliminary Vehicle Mass Estimation
Malen and Reddy
3. Preliminary Mass Estimation in Vehicle Design
Vehicle mass is an important parameter during design. Subsystems are sized based on vehicle mass, and
important product characteristics such as acceleration performance and fuel economy depend on vehicle
mass. Given this importance of mass in vehicle design, mass estimation becomes a critical activity. The
purpose of mass estimation varies with the design stage.
During the product planning stage, or pre-configuration stage, the purpose of mass estimation is to
determine general feasibility of a program and to set initial mass goals for vehicle subsystems. At this
stage, only a broad vehicle mission is identified; for example vehicle type and number of passengers.
Because of this sparse data, mass estimation is often based on first order statistical models based on
contemporary vehicles.
In the next stage of design, the configuration stage, the purpose of mass estimation is to aid in decision
making about which alternative technologies to include in the program, and also to add additional
precision to subsystem mass goals. In this stage, mass estimation is based on both semi-empirical models
and statistical data.
During the detail design stage, post-configuration stage, the goal of mass estimation is monitor mass
growth and to control to target. As the vehicle configuration is set, mass estimation is based on actual
subsystem designs and prototypes.
In this paper, we focus on the pre-configuration and configuration stages. These are particularly significant
as lasting decisions are made about which technologies and subsystem will be used. The implications of
these decisions shadow all subsequent work.
4. Mass Compounding Model
Vehicle design engineers intuitively know that an unplanned mass increase in a component during
vehicle design has a ripple effect throughout the vehicle; other components need to be resized increasing
vehicle mass even more. The phrase mass begets mass describes this phenomenon. A more encouraging
view of this behavior is considering a reduction in the mass of a component enabled by a new technology
resulting in a greater mass saving for the overall vehicle. These secondary mass changes can be
considerable—estimated at an additional 0.7 to 1.8 times the initial—primary--mass change.
A means to quantify the secondary mass change is with a mass compounding model. In this model, each
subsystem (denoted by i) is assigned an influence coefficient, γi. The influence coefficient is the change in
the subsystem mass when gross vehicle mass undergoes a unit change. The physical interpretation of the
influence coefficient is each subsystem is sized to some degree by the mass of the vehicle, and as the
vehicle mass changes the subsystem must also be resized.
The typical question we are interested in answering is;
Given a balanced vehicle under design, a primary mass change now occurs
during design. What is the final vehicle mass after resizing subsystems?
The answer depends on the vehicle mass influence coefficient, γVEH. This the sum of coefficients for all
subsystems which may undergo resizing. The resulting vehicle mass is given by Equation 4.1 (see
Appendix A for a derivation).
5
Preliminary Vehicle Mass Estimation
Malen and Reddy
WV∞ = W0 + ∆ + ∆ΓV
Equation 4.1
⎡ γV ⎤
ΓV = ⎢
⎥ Secondary Mass Coefficient for Vehicle
⎣ (1 − γ V ) ⎦
where
Equation 4.2
W0
is the initial vehicle mass for which the subsystems are sized
∆
is the initial mass change ( primary mass change)
is the resized, compounded vehicle mass
WV∞
∆ΓV
is the additional (secondary) mass change
n
γV
is the mass influence coefficient for the vehicle given by γ V = ∑ γ i
γI
is the mass influence coefficient for subsystem i
i =1
The resulting mass for subsystem i due to an initial increase of ∆;
wi ∞ = wi + ∆Γi
Equation 4.3
⎡ γi ⎤
Γi = ⎢
⎥ Secondary Mass Coefficient for Subsystem i
⎣ (1 − γ v ) ⎦
is the initial subsystem mass
wi
wi∞
∆Γi
where
Equation 4.4
is the resized subsystem mass
is the additional secondary change
γV , γi and ∆ are given above
5. Previous work on Mass Influence Coefficients
Published data on influence coefficients is sparse, and those published are based on vehicles in the 1975–
1981 model years [1, 2, 3]. Figure 5.1 summarizes the subsystem mass influence coefficients for two of
these studies. Both studies used linear regression of measured subsystem mass data as described in the
next section.
0 .3
0 .2 5
0 .2
0 .1 5
0 .1
1 9 7 5 -1 3 v e h ic le s , G M H , X , A , B , C p la tfo rm s
1 9 8 1 -1 8 v e h ic le s , ra n g e o f m a n u fa c tu re s , c o m p a c t fro n t w h e e l d riv e
Summary of Previous Mass Influence Coefficients
Figure 5.1
6
rs
Bu
m
pe
ca
tri
ec
El
Ti
g
rin
ee
St
re
s
W &
he
el
s
l
in
tr a
er
w
Po
a
S y k in g
st
em
si
Br
en
Su
sp
r
ea
R
Su t
sp
e
on
Fr
Bo
d
St y
ru
ct
ur
ns
e
io
n
0
on
0 .0 5
γ v e h = .6 4 4
γ v e h = .4 2 0
Preliminary Vehicle Mass Estimation
Malen and Reddy
Based on these influence coefficients, the secondary vehicle mass savings using Equation 4.2 is 1.81 for
the 1975 study, and 0.72 for the 1981 study. Note that resizing the body structure and powertrain
contribute the most to this secondary mass savings.
6. Description of this study
In this study, we identify subsystem influence coefficients using regression. A function is fit relating the
mass of subsystem i to gross vehicle mass, Figure 6.1. The influence coefficient is the slope of the fit curve.
Data from 32 vehicles were used for the fit. The 32 vehicles were further divided into Sedan, SUV, and
Pick up groups. Appendix B describes the model and the method used for estimation. Appendix C
shows the scatter plots for each subsystem for each vehicle group.
Subsystem
Mass
Wi
γ
Gross Vehicle Mass, WV
Influence Coefficient By Linear Regression
Figure 6.1
Figure 6.2 shows a comparison of influence coefficients by subsystem. Shown are the results for each
vehicle group for both a linear function and also a power function, Equation B.3. A more complete table
of results is contained in Appendix C-Tabular Summary of Subsystem Influence Coefficients.
0.6
0.5
0.4
0.3
0.2
0.1
No
n-s
tr u
Bo
ctu
dy
re
St
ruc
Fr
on
tu r
tS
e
us
pe
Re
ns
ar
ion
Su
sp
en
sio
n
Br
ak
i ng
Po
we
rT
Fu
rai
el
n
&
Ex
ha
us
t
St
e
er i
Tir
ng
es
&
W
he
els
Ele
c tr
ica
l
Co
ol i
ng
Bu
mp
er s
Cl
os
ur e
s
0.0
1975 Study
1981 Study
All Vehicles (Linear)
Sedan (Linear)
SUV (Linear)
Pick Up (Linear)
All Vehicles (Power
@ 3000kg)
Sedan (Power
@ 3000kg)
SUV (Power
@ 3000kg)
Pick Up (Power
@ 3000kg)
Subsystem Mass Influence Coefficients Determined in this Study
Figure 6.2
7
An alternative means to estimate the mass influence coefficient is by the Ratio Method [7]. In this method,
only the single reference vehicle data is needed. The subsystem mass-to-gross vehicle mass relationship is
assumed as shown in Figure 6.3. In this case, the influence coefficient is given by Equation 5.1.
γ = wi / WV
Subsystem
Mass
wi
Gross Vehicle Mass WV
Influence Coefficient By Ratio
Figure 6.3
Ratio method for influences coefficient determination,
γi =
where
wi
Wv
Equation 6.1
wi
current subsystem mass
WV
original gross vehicle mass
7. Summary of Results
The results of this study are summarized in Table 7.1 and graphically in Figure 7.1.
All
Sedan
SUV
Ratio
(sedan)
.562
.599
.463
.532
.332
.331
.290
.391
1.28
Times primary
mass change
0.50
times
1.49
times
0.86
times
1.14
times
0.49
times
0.41
times
0.64
times
Vehicle Influence Coefficient
All subsystems free to resize
Vehicle Influence Coefficient
All subsystems free to resize Except Powertrain
Secondary Mass savings
All subsystems free to resize
Secondary Mass Savings
All subsystems free to resize Except Powertrain
Secondary Mass Savings
Table 7.1
Indicated on Figure 7.1 are results of previous studies from 1975 and 1981, along with results of this study
for All vehicles, the SUV group, and the Sedan group. (The Pick Up and Van categories had an insufficient
number of vehicles to be represented alone.) When all subsystems can be resized, the secondary mass
savings is from 0.8 to 1.5 kg/kg (1.25kg/kg is the estimate for the All vehicles group). When the
powertrain is not available for resizing, the secondary mass savings is from 0.4 to 0.5 kg/kg (0.5kg/kg is
the estimate for the All vehicles group).
8
Preliminary Vehicle Mass Estimation
Malen and Reddy
2.5
2.0
Secondary
Mass
1.5
Savings as
Fraction of
Original 1.0
Change
0.5
ΓV
1981
0.0
1975
All Sedan
SUV
0
0.1
0.2
0.3
0.4
0.5
Vehicle Mass Influence
0.6
0.7
Pick Up
0.8
0.9
γV
All subsystems free to resize
All subsystems less powertrain
Secondary Mass Savings
Figure 7.1
8. Example Application
A new vehicle is in the planning stage (pre-configuration). It is targeted at 5 passengers with a 120kg cargo
capacity. The target test weight class for fuel economy evaluation is 2250Lb (1950Lb or 886kg curb mass).
Mass reduction is a primary goal and several technologies are under investigation for application. The
objectives of mass analysis at this stage are to
1) determine an initial mass estimate for the vehicle based on conventional design
2) determine the most effective mass reduction technologies to apply
3) estimate the vehicle mass when these technologies are used
4) set subsystem mass goals for the detail design stage which are consistent with 1) , 2), & 3).
Step 1: Pre Configuration Mass Estimation
From human accommodation and dimensional benchmarking, the vehicle length is estimated at 4.732 m
and width at 1.815m. From this sparse data, the vehicle curb mass can be estimated given the plan view
area (relationship in Appendix C, Sedan section).
curb mass (kg) = 147.75 A(m 2 ) + 229.59
A = (1.815m)(4.732m) = 8.58858m 2
curb mass = 1498.55kg
Adding passenger mass and cargo mass, the gross vehicle mass is found. Finally, based on the Subsystem
Mass fraction of Curb Mass (Appendix C in the Sedan Section) the mass of each subsystem may be
estimated. For example,
(Body Non-structure mass) = 0.204 (Curb Mass)
(Body Non-structure mass) = 305.70 kg
These steps are automated in the spreadsheet provided with this project, Figure 8.1.
9
Preliminary Vehicle Mass Estimation
Malen and Reddy
Pre-Configuration Vehicle Mass Estimation
Figure 8.1
The above mass estimate is for a nominal vehicle as defined by the data set used in this paper given the
footprint. This estimate is greater than the desired 886kg curb mass by 633kg.
Step 2: Identify Mass Reduction Technologies
With the nominal vehicle identified, potential mass reduction technologies may be identified for each
subsystem. For consistent evaluation of technologies, these steps are suggested;
2a) The initial mass for each subsystem is given by the nominal vehicle of Step 1 (those identified in
Figure 8.1 in this example).
Example: Front Suspension subsystem mass=73.43kg
2b) Now for a vehicle of the gross vehicle mass identified in Step 1 (1968.55kg) determine the new
subsystem mass enabled by a particular technology.
Example technology: Shape optimization arms
Front Suspension mass with technology=66.09kg
>>Note that this new front suspension is sized for the gross mass of the vehicle
identified in Step 1. This is required for the later mass compounding analysis.<<
2c) The mass savings is the difference between 2a) and 2b).
Example mass savings=73.43-66.09=7.34kg
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Preliminary Vehicle Mass Estimation
Malen and Reddy
These steps are summarized for several hypothetical technologies in Table 8.1 Columns 1, 2, 3.
subsystem
mass reduction technology
mass savings
cost
kg
cost/unit mass (mc)
$
Tires & Wheels
Minimum capacity wheels and tires
20.00
-20
non structure
reduced sound treatment with
19.45
-5
Braking
optimized pedal bracket
3.00
0
rear suspension
shape optimization
6.59
0
body structure
joint improvements
15.00
0
closures
hardware: optimization for bar
10.00
0.1
front suspension
shape optimization
7.34
0.2
body structure
AHSS optimization
70.00
3
non structure
reduce glass thickness
5.00
0.3
non structure
IP substrate optimization
21.43
2
Braking
tubular pedals
4.00
0.4
non structure
seat frame shape optimization
40.00
5
closures
AHSS optimization
12.46
2
Fuel and Exhaust
lower gage of exhaust
5.99
1
bumper
shape optimization
4.95
1
steering
tubular rack
2.00
0.5
powertrain
reduce wall thickness
27.72
7
non structure
carpet material
15.00
5
non structure
new seat system technology
5.00
5
Fuel and Exhaust
New muffler technology
10.00
50
body structure
carbon fiber underbody
60.00
500
Potential Mass Reduction Technologies (Hypothetical Examples)
Table 8.1
$/kg
-1.000
-0.257
0.000
0.000
0.000
0.010
0.027
0.043
0.060
0.093
0.100
0.125
0.161
0.167
0.202
0.250
0.253
0.333
1.000
5.000
8.333
It is important to note again that the mass reduction in Column 3 must be based on the vehicle mass
identified in Step 1 to meet the assumptions of the mass compounding model.
Step 3: Sort Mass Reduction Technologies by Cost
This marginal cost of mass, mc, (cost per unit mass reduction) may be used to sort which technologies to
include in the new vehicle program.
3a) Identify the cost difference to provide this technology in this specific vehicle. (Note, this may be a cost
reduction and a negative value)
Example technology: Shape optimization arms
Front Suspension cost increase with technology=$0.20
3b) Identify the marginal cost to provide the technology.
Example: marginal cost=mc=($0.20)/(7.34kg reduction)=0.027$/kg
3c) From all identified mass reduction technologies, sort by increasing marginal cost
Example: Table 8.1 is sorted for 21 potential technologies
3d) Filter technologies using marginal cost for inclusion in the vehicle configuration
Example: Include all technologies with mc ≤ 0.4 $/kg
The 18 technologies at the top of Table 8.1 are carried forward.
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Preliminary Vehicle Mass Estimation
Malen and Reddy
3e) For those technologies passing the filter, sum the total mass reduction for each subsystem.
Example;
Subsystem
Mass reduction using technologies passing mc filter
Non Structure
19.45+5+21.43+40+15=100.88kg
Body Structure
15+70=85kg
Front Suspension
7.34kg
Rear Suspension
6.59kg
Braking
3+4=7.00kg
Powertrain
27.72kg
Fuel and exhaust
5.99kg
Steering
2.00kg
Tire & Wheels
20kg
Bumper
4.95kg
Closures
10.0+12.46=22.46kg
These technologies provide a primary mass reduction total of 289.93kg.
Step 4: Estimate Vehicle Mass Using Mass Compounding
The mass savings opportunities found in Step 3, are primary mass reductions. They are sized for the
vehicle mass found in Step 1; 1500kg curb mass. Now as each technology is applied to the vehicle, the
curb mass is reduced. Due to this reduction, the subsystems may be resized. This resizing results in a
secondary mass savings given by Equation 4.2,
Primary mass savings ∆=
289.93kg
Secondary mass savings =
⎡ γV ⎤
∆⎢
⎥ , where the vehicle influence coefficient, γV, may be based
(
1
γ
)
−
V ⎦
⎣
on the results of this study shown in Figure 7.1. Taking a nominal value for the vehicle influence
γV ⎤
⎥ =1.1368 or a secondary mass savings=329.61kg.
⎣ (1 − γ V ) ⎦
⎡
coefficient; γV = 0.532 gives ⎢
Thus the total mass reduction when the mass reduction technologies are applied is the compounded
vehicle mass which is the sum of the primary reduction (289.93kg) and the secondary reduction (329.61kg)
due to the subsystems being resized for the new reduced vehicle mass; 289.93kg + 329.61kg=619.54kg. To
see how this compounded reduction is distributed across the subsystems, Equation 4.3 and 4.4 may be
used. These steps are automated in the spreadsheet provided with this project, Figure 8.2.
The spreadsheet of Figure 8.2 applies the mass influence coefficients found in this work to arrive at the
compounded masses. The user may select from the All Vehicles, Sedan, SUV groups, use the Ratio method,
Equation 6.1, or input a User Defined set of coefficients. Selection of which to use is based on the subject
vehicle type and the predicted gross vehicle mass. The predicted mass should be within the range of data
used to estimate the influence coefficients. Interpolation limits for the influence coefficients are;
All Vehicles group
Sedan group
SUV group
1600<GVM<3150kg
1600<GVM<2225kg
1920<GVM<3150kg
If a prediction beyond these ranges is necessary, the Ratio method is recommended.
12
Preliminary Vehicle Mass Estimation
Malen and Reddy
Compounded Mass Estimation (including powertrain resizing)
Figure 8.2
Step 5: Adjust Subsystem Mass to arrive at Design Targets
In Steps 1-4, we have used a rational procedure to estimate a vehicle's mass based on current practice,
and then to estimate the influence of mass reduction technologies using a mass compounding model. It is
important during this stage to investigate alternative configurations. For example, below is the
compounding model as in Figure 8.2 except the powertrain has not been allowed to resize, Figure 8.2.
Compounded Mass Estimation (powertrain not resized)
Figure 8.2
These mass estimation tools are intended to be highly iterative allowing rapid sensitivity studies to arrive
at robust design targets for vehicle and subsystem mass.
13
Preliminary Vehicle Mass Estimation
Malen and Reddy
Bibliography
Past Influence Coefficient Studies
1. Internal GM communication, Report of the Cost of Weight Task Force, January, 1975
2. Internal GM communication, Weight Influence Coefficients, November, 1981.
3. Anderson, M., Engine and Transmission Mass as a Fraction of Vehicle Mass, Integrated Vehicle
System Design Course Notes, University of Michigan, January, 2007.
Independent Mass Data Sources
4. www.autobytel.com, Autobytel, 18872 MacArthur Boulevard, Irvine, California 92612-1400.
5. www.cars.com, Classified Ventures, LLC, 175 W. Jackson Blvd., Suite 800, Chicago, IL 60604.
6. www.internetautoguide.com, Internet Auto Guide.
Methodology for Mass Analysis
7. Society of Allied Weight Engineers, Aircraft Weight Engineering, 1996
8. Tornbeek, E., Synthesis of Subsonic Airplane Design, Delft University Press, Boston, 1988
9. de Weck, O. L., A Systems Approach To Mass Budget Management, Paper AIAA 2006-7055, 11th
AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, September 2006.
10. Spacecraft Preliminary Mass Estimation & Allocation, Technical University of Delft,
http://www.lr.tudelft.nl/live/pagina.jsp?id=10d9b8b2-e85b-48e7-94f2462356f1d730&lang=en
11. Piano Software For Preliminary Design Of Commercial Aircraft Including Mass
Estimation, http://www.lissys.demon.co.uk/
Statistical Methods
12. Box, G., and Draper, N., Empirical Model-Building and Response Surfaces, Wiley & Sons,
NY, 1987
13. Mendenhall, W., et al, Mathematical Statistics with Applications, Duxbury Press, Boston,
1986.
14. Neter, J., et al., Applied Linear Statistical Models, IRWIN Publishing, Boston, 1990
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Preliminary Vehicle Mass Estimation
Malen and Reddy
Technical Appendix A
Derivation of mass model and influence coefficients
Vehicle mass is the sum of the masses for n subsystems,
WV = w1 + w2 + ... + wi + ... + wn
Equation A.1
Where WV is the vehicle mass
wi is the mass of subsystem i
For convenience, we include as subsystems the passengers, and cargo, so WV represents the maximum
vehicle mass (gross vehicle mass).
The mass of each subsystem depends upon the mass of the vehicle and other functional parameters.
wi = f (WV , functional parameters)
Equation A.2
Note that some subsystems will have a strong vehicle mass dependence—for example, the powertrain
subsystem—while for others the dependence on vehicle mass will be weak—for example, interior
ventilation.
Mass Influences Coefficients
Imagine subsystem i is designed to function within a vehicle mass of WV with the resulting subsystem
mass of wi. Now imagine a second vehicle which is slightly heavier at mass WV+dWV. Subsystem i is now
sized for this slightly larger mass with the resulting subsystem mass of wi+dwi. We define the subsystem
mass influence, γi, as the change in subsystem mass required when the vehicle mass is increased by one
unit;
γi =
∂wi
∂WV
Equation A.3
Consider a vehicle design with initial mass W0 and all subsystems sized for this mass. Now during
preliminary design, an unexpected small mass increase of δ occurs in a subsystem. All subsystems must
now be resized for this mass increase. The resulting change in vehicle mass is dW which we would like to
determine. This can be quantified using subsystem mass influence coefficients;
dW =
∂wi
∂w
∂w
∂w1
δ + 2 δ + ... +
δ + ... + n δ
∂WV
∂WV
∂WV
∂WV
dW = δ (γ 1 + γ 2 + ... + γ i + ... + γ n )
dW
δ
Equations A.4
= (γ 1 + γ 2 + ... + γ i + ... + γ n )
The vehicle mass influence coefficient, γV, is defined as the change in mass of the vehicle, dW, due to a
resizing of all the subsystems for a unit increase in mass.
γV =
dW
δ
15
Equation A.5
Preliminary Vehicle Mass Estimation
Malen and Reddy
Comparing the last of Equations A.4 with A.5, we see that the vehicle mass influence coefficient, γV, is the
sum of the influence coefficients for all subsystems.
n
γ V = ∑γ i
Equation A.6
i =1
Mass Change during the Configuration Design Stage
Now consider a balanced vehicle design with initial mass W0. It is now learned that the mass of a
subsystem has changed by an amount ∆ for a new vehicle mass of W0+∆.
Since the subsystems were sized for vehicle mass W0, they must be redesigned to function with the
additional mass ∆:
Resizing 1
mass change : ∆
resize subsystems due to mass change causes additional mass : ∆γ V
new vehicle mass W1 = (W0 + ∆) + ∆γ V
But now the subsystems are sized for vehicle mass (W0+∆), and they must be resized for the additional
mass ∆γV:
Resizing 2
mass change : ∆γ V
resize subsystems due to mass change causes additional mass : ∆γ V
W2 = (W0 + ∆ + ∆γ V ) + ∆γ V
2
2
In a similar way, the subsystems are sized for vehicle mass (W0+∆+∆γV), and they must be resized for the
additional mass ∆γV2:
Resizing 3
mass change : ∆γ V
2
resize subsystems due to mass change causes additional mass : ∆γ V
W3 = (W0 + ∆ + ∆γ V + ∆γ V ) + ∆γ V
2
3
3
This resizing continues until we have:
Resizing m
Wm = W0 + ∆ + ∆γ V + ∆γ V + ∆γ V + ... + ∆γ V + ... + ∆γ V , m → ∞
2
3
j
m
Equations A.7
Wm = W0 + ∆ + ∆ (γ V + γ V + γ V + ... + γ V + ... + γ V ),
2
3
j
m
Recognizing the sum in the parentheses is a geometric series we have;
16
m→∞
Preliminary Vehicle Mass Estimation
Malen and Reddy
S = (γ V + γ V + γ V + ... + γ V + ... + γ V ),
2
3
j
m→∞
m
S = γ V (1 + γ V + γ V + γ V + ... + γ V + ... + γ V ), m → ∞
2
3
j
m
S = γ V (1 + S )
Equations A.8
S (1 − γ V ) = γ V
S=
γV
, when 0 < γ V < 1
(1 − γ V )
Substituting the last of Equations A.8 into A.7 gives the mass of the vehicle after all resizing;
⎡ γV ⎤
Wm→∞ = W0 + ∆ + ∆ ⎢
⎥
⎣ (1 − γ V ) ⎦
Equations A.9
Where ∆ is the original mass change
⎡ γV ⎤
∆⎢
⎥ is the secondary mass change due to resizing subsystems.
⎣ (1 − γ V ) ⎦
We define the Secondary mass coefficient for the vehicle, ΓV ;
⎡ γV ⎤
ΓV = ⎢
⎥
⎣ (1 − γ V ) ⎦
Equation A.10
⎡ γi ⎤
Γi = ⎢
⎥
⎣ (1 − γ v ) ⎦
Equation A.11
and for each subsystem, Γi, as;
So the final vehicle mass due to an unexpected mass increase of ∆ is
and the final mass for subsystem i is;
WV∞ = W0 + ∆ + ∆ΓV
Equation A.12
wi ∞ = wi + ∆Γi
Equation A.13
To prove that the sum of subsystem influence coefficients is the vehicle influence coefficient, begin with
Equation A.12;
Wm → ∞ = W0 + ∆ + ∆ΓV
WV = w1 + w2 + ... + wi + ... + wn
Wm → ∞ = ∆ + ( w1 + ∆Γ1 ) + ( w2 + ∆Γ2 ) + ... + ( wi + ∆Γi ) + ...( wn + ∆Γn )
Wm → ∞ = ∆ + ( w1 + w2 + ... + wi + ... + wn ) + ∆(Γ1 + Γ2 + ... + Γi + ... + Γn )
Wm → ∞ = ∆ + W0 + ∆(Γ1 + Γ2 + ... + Γi + ... + Γn )
∆ΓV = ∆(Γ1 + Γ2 + ... + Γi + ... + Γn )
⎡ γV ⎤
⎡ γ1
γi
γn ⎤
γ2
∆⎢
+
+ ... +
+ ... +
⎥ = ∆⎢
⎥
(1 − γ V )
(1 − γ V ) ⎦
⎣ (1 − γ V ) ⎦
⎣ (1 − γ V ) (1 − γ V )
γ V = γ 1 + γ 2 + ... + γ i + ... + γ n
17
Equation A.14
Preliminary Vehicle Mass Estimation
Malen and Reddy
Technical Appendix B
Modeling Subsystem Mass
We assume that the mass of each subsystem depends on the gross vehicle mass and on other functional
parameters [8];
wi = f (WV , functional parameters)
α
β
β
Equation B.1
β
wi = C iWV ( Pa a Pb b ...Pr r )
Where wi is a subsystem mass
WV is the gross vehicle mass
Pa,b,..r are functional parameters some of which are performance measures
For example, consider the powertrain subsystem mass, wi. Its mass depends on the overall vehicle mass,
α
β
CiWV . The mass also depends on vehicle acceleration performance; Pa a , where Pa is the required 060mph acceleration time. It also depends on the powertrain layout; TFWD, LRWD, AWD; Pb. Here Pb is
an indicator variable: Pb =+1 for TFWD, Pb =0 for LWD, Pb =-1 for AWD.
This relationship may be visualized graphically;
Log
Subsystem
Weight
Log
Functional
Parameter
Surface defined
by Equation B.1
Log
Gross Vehicle
Weight
Subsystem Mass Relationship
Figure B.1
With the model defined by Equation B.1, the subsystem influence coefficient is given by;
γi =
∂wi
∂WV
Equation B.2
WV
Estimation of subsystem influence coefficients
To estimate the relationship between subsystem mass and vehicle mass, we fit observed data with both a
linear and power model. In both cases, the fit was made using linear regression.
α
wˆ i = C iWV
wˆ i = β 0 + β 1WV
18
power
linear
Equations B.3
Preliminary Vehicle Mass Estimation
Malen and Reddy
In this study, we have not accounted for the subsystem mass dependence on functional parameters and
have only considered the dependence on gross vehicle mass. Thus we are looking at a projection of the
true relationship onto an x-y plane, Figure B.2.
Log
Subsystem mass
wi
Log
Functional Performance
Surface defined by
α
wi = (CW V ) (functional performance) β
1
Projection of data when functional
performance not accounted for
Log
Gross Vehicle Mass
WV
Projection of data on to x-y plane for curve fitting
Figure B.2
This necessity, has the practical effect of increasing the apparent lack of fit (measured by smaller R2
values). Figure B.3 illustrates this by showing a typical residual error—the difference between a data
point and the fit function. In our case, this residual is comprised of both pure error—randomness, along
with the ignored dependence on functional parameters. Despite this assumption, the data fit is deemed to
be satisfactory even though the R2 values are somewhat low due to this effect.
Pure Error, ε, due to
Subsystem
mass
wi
actual data
Pure Error ε
residual
•Misclassification of parts
•Incorrect vehicle
model used
r
Variation
in subsystem
Performance
prediction
•Clerical: recording
•Weighing parts
GVM WV
Components of Residual Error
Figure B.3
For all fit equations, the R2 values are reported. Below is a brief summary of the meaning of this
coefficient [12, 14].
The variation of the un-fit data is measured by the sum of the squared deviations from the average value,
Figure B.4. This sum is the total sum of squares, SSTO.
19
Preliminary Vehicle Mass Estimation
Subsystem
mass
wi
Malen and Reddy
Sum of squares; total
ρi
SSTO = ∑ ρ i
ρi is residual from mean
2
GVM
WV
Total Sum of Squares
Figure B.4
Now a function is fit to the data as illustrated in Figure B.5. The remaining variation in the data is
measured by the sum of the squared deviations from the function. This sum is the error sum of squares,
SSE.
Subsystem
mass
wi
ri
Sum of squares; error
SSE = ∑ ri
2
ri is residual from model
GVM
WV
Error Sum of Squares
Figure B.5
The R2 coefficient of determination is the fraction of variation explained beyond that using the average value only,
and is given by;
R2 =
( SSTO − SSE )
SSTO
Equation B.4
Thus any non-zero R2 indicates a reduction in variation provided by the fit function. Again in our case, the residual
error, ri, contained in the SSE contains both pure error as well as the variation due to functional performance
differences between the various vehicles as shown in Figure B.3.
While R2 measures the overall goodness of fit of the model, we are interested in the influence coefficient. For a
linear model, the influence coefficient is the slope of the fit line, β1. We can place a confidence interval around this
parameter using Equation B.5 [13];
data : x1 , x2 , x3 ,...xn ; y1 , y2 , y3 ,... yn ;
yˆ = β 0 + β1 x
β1 ~ N [β1 ,Var ( β1 )]
Equations B.5
⎛ 1 ⎞ ∑ ( yˆ − y )
Var ( β1 ) = ⎜
⎟
2
⎝ n − 2 ⎠ ∑ (x − x)
2
i =1, n
For example, looking at the All Vehicles group and the Body Structure subsystem, we have an influence
coefficient β1=0.1758 and R2=0.4542. Applying Equations B.5,
20
Preliminary Vehicle Mass Estimation
Malen and Reddy
yˆ = β 0 + β1 x
yˆ = −1.5294 + 0.1758 x
Equation B.6
n = 33
Var ( β1 ) = .001198
The 90% confidence interval is shown in Figure B.6.
0.3
90% confidence
interval for influence
coefficient
Body Structure 0.2
Mass
0.1
Influence
0.0
Confidence Interval for an Influence Coefficient
Figure B.6
21
Preliminary Vehicle Mass Estimation
Malen and Reddy
Data Appendix C
Vehicles used in analysis
Mass data were collected for 35 vehicles representing Sedans, SUVs, Pickups and Vans. For each vehicle,
the mass of each functional subsystem (see next page for subsystem definitions) was calculated.
Considerable effort was placed on consistency of the component masses contained in each subsystem
across all vehicle sources. In addition to the subsystem mass, Gross Vehicle Mass, Curb Mass, and Number
of Passengers were identified for each vehicle using independent sources [4, 5, 6].
The vehicles in this study are shown below. Due to the propriety nature of subsystem mass data, specific
numerical values are not provided.
Sedans:
SUV:
Pick Up:
Van:
15 (of which there is 1 duplicate)
12
5
Note: Due to the small sample size, Pick Up results
are less reliable than for Sedans and SUVs.
Note: Due to the insufficient sample size, vans were
not included as a vehicle group.
Note: For the 'All Vehicles' group, all nonduplicate vehicles were included
3 (of which there is 1 duplicate)
35 (33 vehicles used in study)
2004 VW Touareg
2004 Mazda 3
2004 Nissan Murado
2004 Toyota Sienna
2004 Hundai XG350
2004 Toyota Prius
2003 Lexus ES300
2003 Toyota Camry (US)
2003 BMW 330i
2003 Infiniti G35
2003 Honda Accord
2003 Toyota Corolla Sedan
2002 Audi A4
(SUV)
(Sedan)
(SUV)
(Van) (Dup.)
(Sedan)
(Sedan)
(Sedan)
(Sedan)
(Sedan)
(Sedan)
(Sedan)(Dup.)
(Sedan)
(Sedan)
2007 Model Year
Cadillac SRX
Chevrolet HHR
Saturn Outlook
GMC Denali Sierra Crew Cab
Chevrolet Colorado
Chevrolet Impala
Pontiac G6 SE1
Cadillac STS
GMC Yukon
Saturn Vue
(SUV)
(SUV)
(SUV)
(Pick Up)
(Pick Up)
(Sedan)
(Sedan)
(Sedan)
(SUV)
(SUV)
2002 Honda Civic LX
2003 Honda Accord EX
2003 PT Cruiser
2003 Toyota Matrix XRS
2003 Toyota Tacoma 4x2
2004 Dodge Ram 4x4 Light Duty
2004 Nissan Titan LE
2004 Toyota Highlander Premium
2004 Toyota Sienna
2005 Honda Odyssey Touring
2005 Jeep Liberty
2005 Jeep Wrangler
22
(Sedan)
(Sedan)
(SUV)
(SUV)
(Pick Up)
(Pick Up)
(Pick Up)
(SUV)
(Van)
(Van)
(SUV)
(SUV)
Functional Subsystem Mass Categories
Each category below contains the set of components which provide a specific vehicle function. The assumption is
that the mass required to provide that function varies in part with the gross vehicle mass.
1. Body Non-structural
Sheet Metal
Glass
Seats
Insulation
Trim
Heating and Ventilation
Exterior Lighting
Wiper
2. Body Structure
Body Shell (Body-in-White less closures)
Frame (if present)
Engine Cradle (if present)
3. Front Suspension
Spring
Control Arms
Knuckle
Stabilizer Bar
4. Rear Suspension
5. Braking
Disc/Drum
Caliper
Hydraulic cylinder
6a. Engine
Engine
Engine Cooling
Starting System
6b. Transmission
Transmission
Drive Shafts
Note:
Engine and Transmission subsystems were combined to
form the Powertrain Subsystem used in the final analysis.
7. Fuel System and Exhaust
8. Steering
Rack
Column
Tie rods
Power assist
9. Tires & Wheels
10. Electrical
Entertainment, navigation
Lighting
Wiring
11. Cooling
Air Conditioning components
12. Bumpers
13. Body Closures
Note:
Closures include hardware and door trim, with the
exception of the GM Benchmark data where it is the door
shell only. These cases are noted on the graphs.
23
Preliminary Vehicle Mass Estimation
Malen and Reddy
ALL VEHICLES
GVM / PLAN VIEW AREA
y = 326.15x 0.8647
y = 202.34x + 371.22
CURB / PLAN VIEW AREA
2
R = 0.6338
2
R = 0.6309
4000
Curb Mass (Kg)
y = 169.07x + 143.61
2
R = 0.5544
Sedan
SUV
Pick Up
Linear
Power
2000
1000
2000
Sedan
SUV
Pick Up
Linear
Power
1000
0
0
5
10
Plan View Area (m2)
15
5
10
Plan View Area (m2)
Subsystem Mass as a Fraction of Curb Mass
GVM / Cargo Mass
Cargo
Mass (kg)
Body Non-structural 0.185
500
250
125
0
4000
3000
2000
1000
0
1000
2000
3000
Sum of Subsystem + Passenger Mass
4000
Body Structure
0.231
Front Suspension
0.047
Rear Suspension
0.039
Braking
0.029
Engine
0.126
Transmission
0.079
Power Train
0.205
Fuel and Exhaust
0.039
Steering
0.013
Tires & Wheels
0.067
Electrical
0.037
Cooling
0.024
Bumpers
0.020
Closures
0.064
Body Non structural
600
400
200
1000
2000
0
y = 0.1988x0.9507
2
R = 0.3637
800
0
All Vehicles
0.3
0.2
0.1
Body Non structural
Body Non structural Mass
Body Non structural Mass
y = 0.1445x - 8.4657
15
No
B ns
Fr ody truc
on S tu
r
t
Re t Sus ruct al
ar
p ur
Su ens e
sp ion
en
Br sion
ak
in
Tr
an Eng g
sm ine
Po is
Fu we sio
n
el
r
&E Tra
xh in
Tir
a
es Steeust
&
W ring
Ele heel
ct s
Co rica
o l
Bu ling
mp
Cl ers
os
ure
s
GVM (Kg)
2
R = 0.5488
3000
3000
Reported GVM
y = 213.66x0.9299
3000
4000
2
R = 0.2974
1000
100
1000
10000
GVM
GVM
24
Preliminary Vehicle Mass Estimation
Malen and Reddy
ALL VEHICLES
Body structural
y = 0.1758x - 1.5294
Body structural
y = 0.325x 0.9176
2
R = 0.4542
1000
Body Structural Mass
Body Structural Mass
800
600
400
200
0
1000
2000
3000
100
1000
4000
10000
GVM
GVM
Front Suspension
Front Suspension
y = 0.0137x1.1176
2
R = 0.4096
160
Front Suspension Mass
Front Suspension Mass
y = 0.0454x - 21.537
120
80
40
0
1000
2000
3000
100
10
1000
4000
10000
GVM
Rear Suspension
y = 0.0354x - 12.061
R
2
Rear Suspension
y = 0.0072x1.172
= 0.1912
200
2
R = 0.2062
1000
Rear Suspension Mass
Rear Suspension Mass
2
R = 0.3067
1000
GVM
150
100
50
0
2
R = 0.4672
1000
2000
3000
4000
100
10
1000
10000
GVM
GVM
25
Preliminary Vehicle Mass Estimation
Malen and Reddy
ALL VEHICLES
Braking System
y = 0.0077x + 30.355
Braking System
y = 1.421x 0.4422
2
R = 0.0311
100
Braking System Mass
Braking System Mass
100
80
60
40
20
0
10
1000
2000
3000
4000
1000
GVM
Powertrain
Powertrain
2
R = 0.6687
y = 0.004x1.4741
800
1000
600
400
200
1000
2000
3000
100
1000
4000
10000
GVM
GVM
Fuel & Exhaust System
Fuel & Exhaust System
2
R = 0.3357
y = 0.0103x1.1318
Fuel & Exhaust System Mass
Fuel & Exhaust System Mass
y = 0.0332x - 7.612
160
120
80
40
0
2
R = 0.7265
Powertrain Mass
Powertrain Mass
10000
GVM
y = 0.2302x - 162.07
0
2
R = 0.0269
1000
2000
3000
4000
2
R = 0.3723
1000
100
10
1000
10000
GVM
GVM
26
Preliminary Vehicle Mass Estimation
Malen and Reddy
ALL VEHICLES
Steering
y = 0.0056x + 8.8642
Steering
y = 0.1265x 0.6583
2
R = 0.1418
100
30
Steering Mass
Steering Mass
40
20
10
0
1000
2000
3000
10
1000
4000
10000
GVM
GVM
Tires & Wheels
y = 0.0488x + 4.2581
Tires & Wheels
y = 0.058x0.9818
2
R = 0.7106
150
100
50
1000
2000
3000
100
10
1000
4000
10000
GVM
GVM
Electrical
y = 0.0143x + 30.235
Electrical
y = 0.7054x0.5751
2
R = 0.0688
2
R = 0.1142
1000
120
Electrical Mass
Electrical Mass
160
80
40
0
2
R = 0.7354
1000
Tires & Wheels Mass
Tires & Wheels Mass
200
0
2
R = 0.0961
1000
2000
3000
100
10
1000
4000
GVM
10000
GVM
27
Preliminary Vehicle Mass Estimation
Malen and Reddy
ALL VEHICLES
Cooling
y = 0.0206x - 5.7438
Cooling
2
R = 0.2651
y = 0.0225x 0.9627
100
75
Cooling Mass
Cooling Mass
100
50
25
0
1000
2000
3000
10
1000
4000
GVM
Bumpers
Bumpers
y = 0.2118x0.6462
2
R = 0.2139
80
100
60
40
20
1000
2000
3000
10
1000
4000
10000
GVM
GVM
Closures
y = 0.0867x - 82.945
1000
Doors with
components
200
150
100
Doors in
white
50
1000
2000
2
R = 0.2438
y = 8x10-6x 2.1099
Closures Mass
Closures Mass
Closures
2
R = 0.3037
250
0
2
R = 0.0918
Bumpers Mass
Bumpers Mass
10000
GVM
y = 0.0129x + 3.9693
0
2
R = 0.2013
3000
100
10
1000
4000
10000
GVM
GVM
28
Preliminary Vehicle Mass Estimation
Malen and Reddy
SEDAN
CURB MASS / PLAN VIEW AREA
GVM / PLAN VIEW AREA
y = 163.14x + 561.69
y = 376.41x 0.7676
2
R = 0.5502
2
R = 0.5545
2000
2000
1600
GVM (kg)
Curb Mass (kg)
2500
1500
y = 147.75x + 229.59
2
R = 0.5128
y = 206.11x 0.9212
2
R = 0.5034
1200
1000
500
800
400
0
2
4
6
8
Plan View Area (m2)
10
0
12
2
Subsystem Mass as a Fraction of Curb Mass
0.044
Braking
0.032
Engine
0.118
Transmission
0.067
Power Train
0.185
Fuel and Exhaust
0.040
Steering
0.014
Tires & Wheels
0.065
Electrical
0.046
Cooling
0.027
Bumpers
0.022
Closures
0.046
0.5
0.4
Sedan
0.3
0.3
0.2
0.1
0.2
0.1
0
0.0
No
n-s
tru
Bo
ctu
dy
re
St
ruc
Fr
on
t
u
re
tS
us
pe
Re
ns
ar
ion
Su
sp
en
sio
n
Br
ak
i
Po
ng
we
rT
Fu
r
a
el
in
&
Ex
ha
us
t
St
ee
Tir
rin
es
g
&
W
he
els
Ele
ctr
ica
l
Co
oli
ng
Bu
mp
ers
Cl
os
ure
s
0.049
Rear Suspension
1975 Study
1981 Study
Sedan (Linear)
Sedan (Power
@ 3000kg)
Body Non Structural
R
2
Body Non Structural
y = 0.1465x1.0054
= 0.2058
Body Non structural Mass
y = 0.1419x + 26.297
600
Body Non structural
Mass
12
0.6
No
B ns
Fr ody truc
on S tu
r
t
Re t Sus ruct al
ar
p ur
Su ens e
sp ion
en
Br sion
ak
in
Tr
an Eng g
sm ine
P
i
Fu owe ssio
n
el
r
&E Tra
xh in
Tir
a
es Steeust
&
W ring
h
Ele eel
ct s
Co rica
o l
Bu ling
m
Cl pers
os
ure
s
0.227
Front Suspension
10
Influence Coefficient Summary
Sedans
Body Non-structural 0.204
Body Structure
4
6
8
Plan View Area (m2)
400
200
0
1000
2000
3000
GVM
2
R = 0.2263
1000
100
1000
10000
GVM
29
Preliminary Vehicle Mass Estimation
Malen and Reddy
SEDAN
Body Structural
y = 0.1317x + 78.114
Body Structural
y = 0.8247x 0.7928
2
R = 0.6626
2
R = 0.6934
1000
Body Structural Mass
Body Structural Mass
500
400
300
200
100
100
1000
0
1000
2000
3000
10000
GVM
GVM
Front Suspension
y = 0.631x 0.622
2
R = 0.1057
120
Front Suspension Mass
Front Suspension Mass
y = 0.0333x + 8.0864
Front Suspension
90
60
30
1000
0
1000
2000
100
10
1000
3000
10000
GVM
GVM
Rear Suspension
Rear Suspension
y = 2x10-7 x2.5895
2
R = 0.337
160
Rear Suspension Mass
Rear Suspension Mass
y = 0.0894x - 108.43
120
80
40
0
1000
2
R = 0.0488
2000
2
R = 0.3659
1000
100
10
1000
3000
GVM
10000
GVM
30
Preliminary Vehicle Mass Estimation
Malen and Reddy
SEDAN
Braking System
y = 0.0177x + 12.206
Braking System
y = 0.5579x0.5719
2
R = 0.0378
100
Braking System Mass
Braking System Mass
80
60
40
20
0
1000
2000
10
1000
3000
GVM
Powertrain
Powertrain
y = 0.0002x1.8598
2
R = 0.8126
1000
400
300
200
100
1000
2000
100
1000
3000
10000
GVM
GVM
Fuel and Exhaust System
Fuel and Exhaust System
y = 0.0197x1.0525
2
R = 0.0835
Fuel and Exhaust System Mass
y = 0.0272x + 6.3007
Fuel and Exhaust System Mass
2
R = 0.8586
Powertrain Mass
Powertrain Mass
500
0
10000
GVM
y = 0.2703x - 249.45
120
90
60
30
0
2
R = 0.0154
1000
2000
3000
GVM
2
R = 0.1445
1000
100
10
1000
10000
GVM
31
Preliminary Vehicle Mass Estimation
Malen and Reddy
SEDAN
Steering
y = -0.0023x + 25.304
Steering
y = 373.16x -0.3868
2
R = 0.0084
100
Steering Mass
Steering Mass
30
20
10
0
1000
2000
10
1000
3000
10000
GVM
GVM
Tires & Wheels
y = 0.0395x + 18.528
Tires & Wheels
y = 0.1496x0.8524
2
R = 0.4535
120
90
60
30
1000
2000
100
10
1000
3000
10000
GVM
GVM
Electrical
y = 0.0139x + 40.455
Electrical
2
R = 0.0115
y = 0.0276x1.0228
Electrical Mass
Electrical Mass
2
R = 0.1023
1000
160
120
80
40
0
2
R = 0.5074
1000
Tires & Wheels Mass
Tires & Wheels Mass
150
0
2
R = 0.0179
1000
2000
100
10
1000
3000
10000
GVM
GVM
32
Preliminary Vehicle Mass Estimation
Malen and Reddy
SEDAN
Cooling
y = 0.0446x - 46.748
Cooling
y = 3x10-6 x2.1611
2
R = 0.5784
100
60
Cooling Mass
Cooling Mass
80
40
20
0
1000
2000
10
1000
3000
10000
GVM
GVM
Bumpers
y = 0.0174x - 0.9741
Bumpers
2
R = 0.2799
y = 0.015x1.0134
40
Bumpers Mass
Bumpers Mass
2
R = 0.2845
100
50
30
20
10
0
1000
2000
10
1000
3000
10000
GVM
GVM
Closures
y = 0.1028x - 131.1
Closures
2
R = 0.1394
y = 5x10-9 x3.0387
2
R = 0.1636
1000
200
Doors with
components
150
Closures Mass
Closures Mass
2
R = 0.5661
100
50
100
Doors in
white
10
1000
0
1000
2000
3000
10000
GVM
GVM
33
Preliminary Vehicle Mass Estimation
Malen and Reddy
SUV
GVM / PLAN VIEW AREA
CURB MASS / PLAN VIEW AREA
2
y = 205.28x + 489.31
y = 402.26x0.7981
y = 150.16x + 417.36
y = 324.28x 0.767
R = 0.5458
2
R = 0.544
4000
Curb Mass (kg)
GVM (kg)
2000
1000
0
3
6
9
Plan View Area (m2)
12
2000
1000
0
15
3
Subsystem Mass as a Fraction of Curb Mass
0.048
Rear Suspension
0.040
Braking
0.027
Engine
0.138
Transmission
0.077
Power Train
0.215
Fuel and Exhaust
0.035
Steering
0.011
Tires & Wheels
0.067
Electrical
0.033
Cooling
0.024
Bumpers
0.017
Closures
0.077
15
0.5
0.4
0.3
SUV
0.3
0.2
0.1
0.2
0
0.0
0.1
No
n-s
tru
Bo
ctu
dy
re
St
ruc
Fr
on
t
u
re
tS
us
pe
Re
ns
ar
ion
Su
sp
en
sio
n
Br
ak
i
Po
ng
we
rT
Fu
r
a
el
in
&
Ex
ha
us
t
St
ee
Tir
rin
es
g
&
W
he
els
Ele
ctr
ica
l
Co
oli
ng
Bu
mp
ers
Cl
os
ure
s
0.229
Front Suspension
12
0.6
No
B ns
Fr ody truc
on S tu
r
t
Re t Sus ruct al
ar
p ur
Su ens e
sp ion
en
Br sion
ak
in
Tr
an Eng g
sm ine
P
i
Fu owe ssio
n
el
r
&E Tra
xh in
Tir
a
es Steeust
&
W ring
h
Ele eel
ct s
Co rica
o l
Bu ling
m
Cl pers
os
ure
s
Body Structure
6
9
Plan View Area (m2)
Influence Coefficient Summary
SUV
Body Non-structural 0.177
1975 Study
1981 Study
SUV (Linear)
SUV (Power
@ 3000kg)
Body Non Structural
Body Non Structural
y = 0.0008x1.6506
2
R = 0.6766
y = 0.2373x - 236.91
800
Body Non Structural Mass
Body Non Structural Mass
2
R = 0.3978
3000
3000
600
400
200
0
2
R = 0.3965
1000
2000
3000
4000
GVM
2
R = 0.7013
1000
100
1000
10000
GVM
34
Preliminary Vehicle Mass Estimation
Malen and Reddy
SUV
Body Structural
Body Structural
y = 12.215x 0.4485
2
R = 0.33
y = 0.0765x + 218.34
1000
Body Structural Mass
Body Structural Mass
600
450
300
150
0
1000
2000
3000
100
1000
4000
10000
GVM
GVM
Front Suspension
Front Suspension
y = 7x10-6x 2.1045
2
R = 0.8176
200
Front Suspension Mass
Front Suspension Mass
y = 0.0787x - 96.804
150
100
50
0
1000
2000
3000
4000
100
10
1000
10000
GVM
Rear Suspension
Rear Suspension
1.9817
y = 1x10-5x
2
R = 0.4068
200
Rear Suspension Mass
Rear Suspension Mass
y = 0.0645x - 78.811
150
100
50
1000
2000
2
R = 0.7236
1000
GVM
0
2
R = 0.3319
3000
4000
2
R = 0.3817
1000
100
10
1000
10000
GVM
GVM
35
Preliminary Vehicle Mass Estimation
Malen and Reddy
SUV
Braking System
y = 0.0036x + 38.73
Braking System
y = 2.2981x0.3756
2
R = 0.0075
100
Braking System Mass
Braking System Mass
80
60
40
20
0
1000
2000
3000
10
1000
4000
GVM
Powertrain
Powertrain
y = 0.0549x1.1371
2
R = 0.4237
800
1000
600
400
200
1000
2000
3000
100
1000
4000
10000
GVM
GVM
Fuel and Exhaust System
Fuel and Exhaust System
2
R = 0.3483
y = 0.0175x
Fuel and Exhaust System Mass
Fuel and Exhaust System Mass
y = 0.0254x + 2.0453
100
80
60
40
20
0
2
R = 0.4822
Powertrain Mass
Powertrain Mass
10000
GVM
y = 0.173x - 22.151
0
2
R = 0.0188
1000
2000
3000
4000
1.0487
2
R = 0.432
100
10
1000
10000
GVM
GVM
36
Preliminary Vehicle Mass Estimation
Malen and Reddy
SUV
Steering
y = 0.0085x + 0.0729
Steering
y = 0.001x1.2618
2
R = 0.2486
100
30
Steering Mass
Steering Mass
40
20
10
0
1000
2000
3000
10
1000
4000
10000
GVM
GVM
Tires & Wheels
y = 0.0374x + 30.485
Tires & Wheels
y = 0.3292x0.7581
2
R = 0.7306
Tires & Wheels Mass
Tires & Wheels Mass
120
80
40
1000
2000
3000
100
10
1000
4000
10000
GVM
GVM
Electrical
y = 0.0455x - 47.102
Electrical
y = 0.0004x1.5455
2
R = 0.6354
2
R = 0.7031
1000
Electrical Mass
160
Electrical Mass
2
R = 0.7207
1000
160
0
2
R = 0.2185
120
80
40
10
1000
0
1000
2000
3000
100
4000
GVM
10000
GVM
37
Preliminary Vehicle Mass Estimation
Malen and Reddy
SUV
Cooling
y = 0.0401x - 50.975
Cooling
y = 5x10-6 x 2.0598
2
R = 0.7354
100
75
Cooling Mass
Cooling Mass
100
50
25
0
1000
2000
3000
10
1000
4000
10000
GVM
GVM
Bumpers
y = 0.0193x - 15.349
Bumpers
y = 0.0018x1.2394
2
R = 0.3833
Bumpers Mass
60
40
20
1000
2000
3000
Closures
Closures
y = 453.49x -0.1772
2
R = 0.0246
200
150
100
Doors in
white
50
2000
3000
2
R = 0.0031
1000
Doors with
components
1000
10000
GVM
250
Closures Mass
10
1000
4000
GVM
y = 0.0204x + 84.476
0
2
R = 0.2462
100
Closures Mass
Bumpers Mass
80
0
2
R = 0.6324
100
10
1000
4000
10000
GVM
GVM
38
Preliminary Vehicle Mass Estimation
Malen and Reddy
Pick Up
GVM / PLAN VIEW AREA
CURB MASS / PLAN VIEW AREA
2
y = 155.34x + 869.82
R = 0.6882
y = 478.35x0.702
2
R = 0.7078
2
R = 0.585
y = 98.88x1.2301
2
R = 0.6458
3000
Curb Mass (kg)
4000
3000
GVM (kg)
y = 194.79x - 218.79
2000
1000
0
4
8
Plan View Area (m2)
12
2000
1000
0
16
4
Subsystem Mass as a Fraction of Curb Mass
0.021
Engine
0.124
Transmission
0.122
Power Train
0.247
Fuel and Exhaust
0.046
Steering
0.012
Tires & Wheels
0.075
Electrical
0.027
Cooling
0.019
Bumpers
0.023
Closures
0.072
0.5
0.4
Pick up
0.3
0.3
0.2
0.1
0.2
0.1
0
0.0
No
n-s
tru
Bo
ctu
dy
re
St
ruc
Fr
on
t
u
re
tS
us
pe
Re
ns
ar
ion
Su
sp
en
sio
n
Br
ak
i
Po
ng
we
rT
Fu
r
a
el
in
&
Ex
ha
us
t
St
ee
Tir
rin
es
g
&
W
he
els
Ele
ctr
ica
l
Co
oli
ng
Bu
mp
ers
Cl
os
ure
s
0.031
Braking
1975 Study
1981 Study
Pick Up (Linear)
Pick Up (Power
@ 3000kg)
Body Non Structural
Body Non Structural
y = 4x10-5 x 2.0202
2
R = 0.3014
Body Non structural Mass
y = 0.2124x - 248.57
Body Non structural Mass
16
0.6
No
B ns
Fr ody truc
on S tu
r
t
Re t Sus ruct al
ar
p ur
Su ens e
sp ion
en
Br sion
ak
in
Tr
an Eng g
sm ine
P
i
Fu owe ssio
n
el
r
&E Tra
xh in
Tir
a
es Steeust
&
W ring
h
Ele eel
ct s
Co rica
o l
Bu ling
m
Cl pers
os
ure
s
0.047
Rear Suspension
12
Influence Coefficient Summary
Pick Ups
Body Non-structural 0.156
Body Structure
0.223
Front Suspension
8
Plan View Area (m2)
600
450
300
150
0
1000
2000
3000
4000
2
R = 0.3321
1000
100
1000
10000
GVM
GVM
39
Preliminary Vehicle Mass Estimation
Malen and Reddy
Pick Up
Body Structural
y = 0.4453x - 719
Body Structural
y = 3x10-7x 2.6754
2
R = 0.5678
1000
Body Structural Mass
Body Structural Mass
800
600
400
200
0
1000
2000
3000
4000
100
1000
10000
GVM
GVM
Front Suspension
Front Suspension
y = 0.0315x1.0098
2
R = 0.2669
150
Front Suspension Mass
Front Suspension Mass
y = 0.0354x - 1.6075
120
90
60
30
0
1000
2000
3000
4000
100
10
1000
10000
GVM
Rear Suspension
y = 0.0287x - 15.192
Rear Suspension
2
R = 0.9861
y = 0.003x 1.2584
Rear Suspension Mass
80
60
40
20
0
1000
2000
2
R = 0.2636
1000
GVM
Rear Suspension Mass
2
R = 0.6374
3000
4000
GVM
2
R = 0.9853
100
10
1000
10000
GVM
40
Preliminary Vehicle Mass Estimation
Malen and Reddy
Pick Up
Braking System
y = 0.0339x - 46.856
Braking System
y = 6x10-7x2.2871
2
R = 0.3216
100
Braking System Mass
Braking System Mass
80
60
40
20
10
1000
0
1000
2000
3000
4000
GVM
Powertrain
Powertrain
y = 0.0022x1.5596
2
R = 0.7712
Powertrain Mass
600
400
200
1000
2000
3000
100
1000
4000
10000
GVM
GVM
Fuel and Exhaust System
Fuel and Exhaust System
2
R = 0.5274
y = 2x10-5x1.9676
Fuel and Exhaust System Mass
Fuel and Exhaust System Mass
y = 0.0621x - 71.059
160
120
80
40
0
2
R = 0.8142
1000
800
Powertrain Mass
10000
GVM
y = 0.2953x - 289.5
0
2
R = 0.3138
1000
2000
3000
4000
2
R = 0.536
1000
100
10
1000
10000
GVM
GVM
41
Preliminary Vehicle Mass Estimation
Malen and Reddy
Pick Up
Steering
y = 0.0056x + 8.8616
Steering
y = 0.0309x0.8408
2
R = 0.1363
100
30
Steering Mass
Steering Mass
40
20
10
0
1000
2000
3000
10
1000
4000
10000
GVM
GVM
Tires and Wheels
y = 0.0356x + 52.678
Tires and Wheels
2
R = 0.2622
y = 0.3396x0.7689
Tires & Wheels Mass
Tires & Wheels Mass
2
R = 0.3161
1000
200
150
100
50
1000
2000
3000
100
10
1000
0
4000
10000
GVM
GVM
Electrical
y = 0.0189x + 3.8375
Electrical
2
R = 0.4488
y = 0.0376x0.9209
80
2
R = 0.4722
100
Electrical Mass
Electrical Mass
2
R = 0.207
60
40
20
10
1000
0
1000
2000
3000
4000
10000
GVM
GVM
42
Preliminary Vehicle Mass Estimation
Malen and Reddy
Pick Up
Cooling
y = 0.0181x - 10.62
Cooling
y = 0.0312x0.8853
2
R = 0.0829
100
75
Cooling Mass
Cooling Mass
100
50
25
10
1000
0
1000
2000
3000
4000
10000
GVM
GVM
Bumpers
y = 0.0198x - 7.7917
Bumpers
y = 0.0067x1.1153
2
R = 0.4043
100
60
40
20
1000
2000
3000
10
1000
4000
10000
GVM
GVM
Closures
Closures
y = 5x10-7x2.4574
2
R = 0.701
y = 0.1276x - 190.53
200
Closures Mass
Closures Mass
2
R = 0.6981
1000
250
150
100
50
0
2
R = 0.3722
Bumpers Mass
Bumpers Mass
80
0
2
R = 0.0662
1000
2000
3000
100
10
1000
4000
10000
GVM
GVM
43
0.176
0.045
0.035
Front
0.021 0.029
Suspension
Rear
0.013 0.012
Suspension
0.006
0.011 0.031
0.036 0.022
0.006
Steering
Tires &
Wheels
Electrical
0.069
0.087
Closures
0.562
0.010
0.013
Bumpers 0.048 0.006
0.016
0.021
0.017
0.050
0.006
0.031
0.206
0.010
0.031
0.037
0.161
Cooling
0.014
0.049
0.033
0.230
Powertrain 0.183 0.131
Fuel and
Exhaust
0.008
0.038 0.016
Braking
System
All Vehicles
0.122
0.008
0.016
0.014
0.049
0.005
0.033
0.262
0.007
0.033
0.039
0.154
α=2.1099
c=8E-06
α=0.6462
c=0.2118
c=0.0225
α=0.9627
α=0.5751
c=0.7054
c=0.058
α=0.9818
α=0.6583
c=0.1265
c=0.0103
α=1.1318
α=1.4741
c= 0.004
c=1.421
α=0.4422
c=0.0072
α=1.172
c=0.0137
α=1.1176
c=0.325
α=0.9176
0.599
0.103
0.017
0.045
0.014
0.045
0.017
0.032
0.033
0.043
-0.006
(0)
-0.002
(0)
0.040
0.030
0.200
0.014
0.058
0.025
0.144
0.153
0.027
0.270
0.018
0.089
0.033
0.132
0.142
Power
Linear @ 1800
kg
Sedan
Influence Coefficients
Power Power Power
1981 Linear @ 1800 @ 3000 Equation
kg
kg
Y=C Xα
c=0.1988
0.145
0.131
0.127
α=0.9507
Body
0.294 0.166
Structure
Body Nonstructural
1975
Historical
Sum of
recommended 0.644 0.413
coefficients
44
α=3.0387
c=5E-09
α=1.0134
c=0.015
c=3E-06
α=2.1611
α=1.0228
c=0.0276
c=0.1496
α=0.8524
c=373.16
α=-0.3959
c=0.0197
α=1.0525
c= 0.0002
α=1.8598
c=0.5579
α=0.5719
c=2E-07
α=2.5895
c=0.631
α=0.622
c=0.8247
α=0.7928
Power
Equation
Y=C Xα
c=0.1465
α=1.0054
0.463
0.020
0.019
0.040
0.046
0.037
0.009
0.025
0.173
0.004
0.065
0.079
0.077
0.237
Linear
-0.006
0.015
0.050
0.049
0.036
0.010
0.027
0.187
0.006
0.051
0.102
0.066
0.242
Power
@ 3000
kg
SUV
c=453.49
α=-0.1772
α=1.2394
0.900
0.128
0.020
0.018
c=5E-06
α=2.0598
c=0.0018
0.019
0.036
0.006
c=0.0004
α=1.5455
α=0.7581
c=0.3292
c=0.001
0.062
c=0.0175
α=1.0487
α=1.2618
0.295
0.034
0.029
0.035
0.445
0.144
0.019
0.011
0.018
0.041
0.007
0.091
0.303
0.041
0.030
0.034
0.537
c=5E-07
α=2.4574
c=0.0067
α=1.1153
c=0.0312
α=0.8853
c=0.0376
α=0.9209
c=0.3396
α=0.7689
c=0.0309
α=0.8408
c=2E-05
α=1.9676
c= 0.0022
α=1.5596
α=2.2871
c=6E-07
c=0.003
α=1.2584
c=0.0315
α=1.0098
c=3E-07
α=2.6754
Pick Up
(Not signifcant as a group)
Power
Power
@ 3000 Equation
Linear
kg
Y=C Xα
c=4E-05
0.212 0.285
α=2.0202
c= 0.0549
α=1.1371
α=0.0188
c=2.2981
c=1E-05
α=1.9817
c=7E-06
α=2.1045
c=12.215
α=0.4485
Power
Equation
Y=C Xα
c=0.0008
α=1.6506
Recommended subsystem coefficients
shown in bold with border
Preliminary Vehicle Mass Estimation
Malen and Reddy
Selection of Influence Coefficients for Inclusion in Model
Inclusion
Criteria
Physical
link with
Included
vehicle
in prior
γ
mass Magnitude work
Regression Coefficient R2
All
Sedan SUVs
Pick
Average
Non structure 0.36
Body structure 0.47
Front
0.41
Rear
0.21
Braking
0.03
Powertrain
0.73
0.23
0.69
0.70
0.33
0.33
0.64
0.41*
0.53
Little
Strong
0.20
0.15
0.11
0.37
0.04
0.86
0.82
0.41
0.02
0.48
0.27
0.99
0.32
0.81
0.40
0.49
0.10
0.72
Strong
Strong
Strong
Strong
0.03
0.03
0.02
0.20
Fuel & exhaust 0.37
Steering
0.14
Tire & wheels 0.74
Electrical
0.11
Cooling
0.27
0.14
0.02
0.43
0.25
0.54
0.21
0.37
0.15
0.51
0.10
0.57
0.73
0.70
0.74
0.32
0.47
0.08
0.57
0.35
0.41
Bumpers
Closures
0.28
0.16
0.38
0.02
0.40
0.70
0.32
0.30
0.21
0.30
Key
0.25
0.00-0.25
0.25
0.50 -0.50
1.00
0.50-1.00
Include
Little
0.03
Moderate <0.01
Strong
Little
Moderate
Strong
Little
0.04
0.02
0.02
0.01
0.09
No
Yes
Yes
Yes
Yes
Yes
No
Yes
Yes
Yes/No
No
Yes
No
No
Yes
Yes
Yes
Yes
Yes
No
(Yes)
Yes
No
No
Yes
No
*note: High R2 coefficient for Non-structure subsystem
assumed to be due to co linearity of plan view area with GVM:
wnon structure → (interior space) → (plan view area) →(GVM)
45
2000 Town Center, Suite 320
Southfield, Michigan 48075
Tel: 248.945.4777
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