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. 

2
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). 

Section 8 provides an in depth example application of this data. 

1981 1975 0.0

0.5

1.0

1.5

2.0

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Vehicle Mass Influence 

Secondary 

Mass 

Savings as 

Fraction of 

Original 

Change 

SUV Sedan All

V γ

ΓV

All subsystems free to resize 

All subsystems less powertrain 

Preliminary Vehicle Mass Estimation Malen and Reddy

3
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 24 

 2) Sedans 29 

 3) SUVs 34 

 4) Pick Ups 39

 

 Tabular Summary of Subsystem Influence Coefficients 44 

 Selection of Influence Coefficients for Inclusion in Model 45 

4
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). 

Preliminary Vehicle Mass Estimation Malen and Reddy

5
WV∞ = W0 + ∆ + ∆ΓV Equation 4.1

⎥

⎦

⎤ ⎢

⎣

⎡

− Γ = (1 ) V

V

V

γ

γ Secondary Mass Coefficient for Vehicle Equation 4.2 

 where W0 is the initial vehicle mass for which the subsystems are sized 

 ∆ is the initial mass change ( primary mass change) 

 WV∞ is the resized, compounded vehicle mass 

 ∆ΓV is the additional (secondary) mass change 

 γV is the mass influence coefficient for the vehicle given by ∑=

=

n

i

V i

1

γ γ 

 γI is the mass influence coefficient for subsystem i 

The resulting mass for subsystem i due to an initial increase of ∆; 

wi ∞ = wi + ∆Γi Equation 4.3 

⎥

⎦

⎤ ⎢

⎣

⎡

− Γ = (1 ) v

i

i

γ

γ Secondary Mass Coefficient for Subsystem i Equation 4.4 

 where wi is the initial subsystem mass 

 wi∞ is the resized subsystem mass

 ∆Γi 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

0.05

0.1

0.15

0.2

0.25

0.3

Body

Structure

Front

Suspension

Rear

Suspension

Braking

System

Powertrain

Steering

Tires &

Wheels

Electrical

Bumpers

1975-13 vehicles, G M H, X, A, B, C platform s

1981-18 vehicles, range of m anufactures, com pact front wheel drive γ veh=.420

γ veh=.644

0

0.05

0.1

0.15

0.2

0.25

0.3

Body

Structure

Front

Suspension

Rear

Suspension

Braking

System

Powertrain

Steering

Tires &

Wheels

Electrical

Bumpers

1975-13 vehicles, G M H, X, A, B, C platform s

1981-18 vehicles, range of m anufactures, com pact front wheel drive γ veh=.420

γ veh=.644

Summary of Previous Mass Influence Coefficients 

Figure 5.1 

Preliminary Vehicle Mass Estimation Malen and Reddy

6
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. 

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. 

All Vehicles (Linear)

All Vehicles (Power 

@ 3000kg)

Sedan (Linear)

Sedan (Power 

@ 3000kg)

SUV (Linear)

SUV (Power 

@ 3000kg)

Non-structure

Body Structure

Front Suspension

Rear Suspension

Braking

Power Train

Fuel & Exhaust

Steering

Tires & Wheels

Electrical

Cooling

Bumpers

Closures

1975 Study

1981 Study

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Pick Up (Linear)

Pick Up (Power 

@ 3000kg)

All Vehicles (Linear)

All Vehicles (Power 

@ 3000kg)

Sedan (Linear)

Sedan (Power 

@ 3000kg)

SUV (Linear)

SUV (Power 

@ 3000kg)

Non-structure

Body Structure

Front Suspension

Rear Suspension

Braking

Power Train

Fuel & Exhaust

Steering

Tires & Wheels

Electrical

Cooling

Bumpers

Closures

Non-structure

Body Structure

Front Suspension

Rear Suspension

Braking

Power Train

Fuel & Exhaust

Steering

Tires & Wheels

Electrical

Cooling

Bumpers

Closures

1975 Study

1981 Study

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Pick Up (Linear)

Pick Up (Power 

@ 3000kg)

Subsystem Mass Influence Coefficients Determined in this Study 

Figure 6.2

Gross Vehicle Mass, WV 

Subsystem 

Mass 

Wi

γ

Preliminary Vehicle Mass Estimation Malen and Reddy

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. 

Influence Coefficient By Ratio 

Figure 6.3 

Ratio method for influences coefficient determination,

v

i

i W

w

γ = Equation 6.1 

 where 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) 

Vehicle Influence Coefficient 

All subsystems free to resize .562 .599 .463 .532 

Vehicle Influence Coefficient 

All subsystems free to resize Except Powertrain .332 .331 .290 .391 

 

Secondary Mass savings 

All subsystems free to resize

1.28 

Times primary 

mass change 

1.49 

times 

0.86 

times 

1.14 

times 

Secondary Mass Savings 

All subsystems free to resize Except Powertrain

0.50 

 times 

0.49 

times 

0.41 

times 

0.64 

times 

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). 

Gross Vehicle Mass WV 

Subsystem γ = wi / WV

Mass

wi

8
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 1498.55kg

(1.815 )(4.732 ) 8.58858m

curb mass(kg) 147.75 ( ) 229.59

2

2

=

= =

= +

A m m

A m

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. 

1981 1975

0.0

0.5

1.0

1.5

2.0

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Vehicle Mass Influence 

Secondary 

Mass 

Savings as 

Fraction of 

Original 

Change 

SUV Sedan All

0.9

Pick Up 

V γ

ΓV

All subsystems free to resize 

All subsystems less powertrain 

Preliminary Vehicle Mass Estimation Malen and Reddy

9
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 

Preliminary Vehicle Mass Estimation Malen and Reddy

10
These steps are summarized for several hypothetical technologies in Table 8.1 Columns 1, 2, 3. 

subsystem mass reduction technology mass savings cost cost/unit mass (mc)

 kg $ $/kg 

Tires & Wheels Minimum capacity wheels and tires 20.00 -20 -1.000 

non structure reduced sound treatment with 19.45 -5 -0.257 

Braking optimized pedal bracket 3.00 0 0.000 

rear suspension shape optimization 6.59 0 0.000 

body structure joint improvements 15.00 0 0.000 

closures hardware: optimization for bar 10.00 0.1 0.010 

front suspension shape optimization 7.34 0.2 0.027 

body structure AHSS optimization 70.00 3 0.043 

non structure reduce glass thickness 5.00 0.3 0.060 

non structure IP substrate optimization 21.43 2 0.093 

Braking tubular pedals 4.00 0.4 0.100 

non structure seat frame shape optimization 40.00 5 0.125 

closures AHSS optimization 12.46 2 0.161 

Fuel and Exhaust lower gage of exhaust 5.99 1 0.167 

bumper shape optimization 4.95 1 0.202 

steering tubular rack 2.00 0.5 0.250 

powertrain reduce wall thickness 27.72 7 0.253 

non structure carpet material 15.00 5 0.333 

non structure new seat system technology 5.00 5 1.000 

Fuel and Exhaust New muffler technology 10.00 50 5.000 

body structure carbon fiber underbody 60.00 500 8.333 

Potential Mass Reduction Technologies (Hypothetical Examples)

Table 8.1 

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. 

Preliminary Vehicle Mass Estimation Malen and Reddy

11
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 = ⎥

⎦

⎤ ⎢

⎣

⎡

− ∆

(1 ) V

V

γ

γ , where the vehicle influence coefficient, γV, may be based 

on the results of this study shown in Figure 7.1. Taking a nominal value for the vehicle influence 

coefficient; γV = 0.532 gives ⎥

⎦

⎤ ⎢

⎣

⎡

(1− ) V

V

γ

γ =1.1368 or a secondary mass savings=329.61kg. 

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 1600<GVM<3150kg 

Sedan group 1600<GVM<2225kg 

SUV group 1920<GVM<3150kg 

If a prediction beyond these ranges is necessary, the Ratio method is recommended. 

Preliminary Vehicle Mass Estimation Malen and Reddy

12
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. 

Preliminary Vehicle Mass Estimation Malen and Reddy

13
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-48e7-94f2-

462356f1d730&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 

Preliminary Vehicle Mass Estimation Malen and Reddy

14
Technical Appendix A 

Derivation of mass model and influence coefficients 

Vehicle mass is the sum of the masses for n subsystems, 

WV = w + w + + wi + + wn ... ... 1 2 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. 

w f (W , functional parameters) i = V 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; 

V

i

i W

w

∂

∂

γ = 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; 

( ... ... )

( ... ... )

... ...

1 2

1 2

1 2

i n

i n

V

n

V

i

V V

dW

dW

W

w

W

w

W

w

W

w dW

γ γ γ γ δ

δ γ γ γ γ

δ δ δ δ

= + + + + +

= + + + + +

∂

∂

+ +

∂

∂

+ +

∂

∂

+

∂

∂ =

 Equations A.4 

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. 

δ γ

dW

V = Equation A.5 

Preliminary Vehicle Mass Estimation Malen and Reddy

15
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

i

V i

1

γ γ Equation A.6 

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

V

V

W W γ

γ

= + ∆ + ∆

∆

∆

new vehiclemass ( )

resize subsystems due to mass change causes additional mass:

mass change :

1 0

But now the subsystems are sized for vehicle mass (W0+∆), and they must be resized for the additional 

mass ∆γV: 

 Resizing 2

2

2 0

2

( )

resize subsystems due to mass change causes additional mass:

mass change :

V V

V

V

W W γ γ

γ

γ

= + ∆ + ∆ + ∆

∆

∆

In a similar way, the subsystems are sized for vehicle mass (W0+∆+∆γV), and they must be resized for the 

additional mass ∆γV

2

: 

 Resizing 3

2 3

3 0

3

2

( )

resize subsystems due to mass change causes additional mass:

mass change :

V V V

V

V

W W γ γ γ

γ

γ

= + ∆ + ∆ + ∆ + ∆

∆

∆

This resizing continues until we have: 

 Resizing m

= + ∆ + ∆ + + + + + + → ∞

= + ∆ + ∆ + ∆ + ∆ + + ∆ + + ∆ → ∞

W W m

W W m

m

V

j

m V V V V

m

V

j

m V V V V

( ... ... ), 

... ... , 

2 3

0

2 3

0

γ γ γ γ γ

γ γ γ γ γ

 Equations A.7 

Recognizing the sum in the parentheses is a geometric series we have; 

 

Preliminary Vehicle Mass Estimation Malen and Reddy

16
, when 0 1

(1 )

(1 )

(1 )

(1 ... ... ), 

( ... ... ), 

V

2 3

2 3

< < − =

− =

= +

= + + + + + + + → ∞

= + + + + + + → ∞

γ

γ

γ

γ γ

γ

γ γ γ γ γ γ

γ γ γ γ γ

V

V

V V

V

m

V

j

V V V V V

m

V

j

V V V V

S

S

S S

S m

S m

 Equations A.8 

Substituting the last of Equations A.8 into A.7 gives the mass of the vehicle after all resizing; 

⎥

⎦

⎤ ⎢

⎣

⎡

− →∞ = + ∆ + ∆

(1 ) 0

V

V Wm W

γ

γ Equations A.9 

 Where ∆ is the original mass change 

 ⎥

⎦

⎤ ⎢

⎣

⎡

− ∆(1 ) V

V

γ

γ is the secondary mass change due to resizing subsystems. 

We define the Secondary mass coefficient for the vehicle, ΓV ;

⎥

⎦

⎤ ⎢

⎣

⎡

− Γ = (1 ) V

V

V

γ

γ Equation A.10 

 and for each subsystem, Γi, as;

⎥

⎦

⎤ ⎢

⎣

⎡

− Γ = (1 ) v

i

i

γ

γ Equation A.11

So the final vehicle mass due to an unexpected mass increase of ∆ is 

WV∞ = W0 + ∆ + ∆ΓV Equation A.12 

and the final mass for subsystem i is; 

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; 

V i n

V

n

V

i

V V V

V

V i n

m i n

m i n i n

m i i n n

V i n

m V

W W

W w w w w

W w w w w

W w w w w

W W

γ γ γ γ γ

γ

γ

γ

γ

γ

γ

γ

γ

γ

γ

= + + + + +

⎥

⎦

⎤ ⎢

⎣

⎡

− + + − + + − + − ⎥ = ∆

⎦

⎤ ⎢

⎣

⎡

− ∆

∆Γ = ∆ Γ + Γ + + Γ + + Γ

= ∆ + + ∆ Γ + Γ + + Γ + + Γ

= ∆ + + + + + + + ∆ Γ + Γ + + Γ + + Γ

= ∆ + + ∆Γ + + ∆Γ + + + ∆Γ + + ∆Γ

= + + + + +

= + ∆ + ∆Γ

→∞

→∞

→∞

→∞

... ...

(1 ) ... (1 ) ... (1 ) (1 ) (1 )

( ... ... )

( ... ... )

( ... ... ) ( ... ... )

( ) ( ) ... ( ) ...( )

... ...

1 2

1 2

1 2

0 1 2

1 2 1 2

1 1 2 2

1 2

0

Equation A.14

Preliminary Vehicle Mass Estimation Malen and Reddy

17
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]; 

( ... )

( , )

a b r

i i V a b r

i V

w C W P P P

w f W functional parameters

α β β β =

= Equation B.1 

 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 Ci

WV . The mass also depends on vehicle acceleration performance; a Pa

β , where Pa is the required 0-

60mph 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; 

Subsystem Mass Relationship 

Figure B.1 

With the model defined by Equation B.1, the subsystem influence coefficient is given by; 

WV V

i

i W

w

∂

∂

γ = Equation B.2 

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. 

ˆ linear

ˆ power

i 0 1 V

i i V

w W

w C W

β β

α

= +

= Equations B.3 

Log 

Subsystem 

Weight 

Log 

Gross Vehicle 

Weight

Log 

Functional 

Parameter

Surface defined 

by Equation B.1 

Preliminary Vehicle Mass Estimation Malen and Reddy

18
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. 

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. 

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. 

Subsystem 

mass 

wi 

GVM WV

residual 

r Variation 

in subsystem 

Performance

Pure Error ε

actual data 

prediction 

Pure Error, ε, due to 

•Misclassification of parts 

•Incorrect vehicle 

model used 

•Clerical: recording 

•Weighing parts 

Log 

Subsystem mass 

wi 

Log 

Gross Vehicle Mass 

WV 

Log 

Functional Performance

Surface defined by 

Projection of data when functional 

performance not accounted for 

1 ( ) ) α β wi = CWV (functional performance

Preliminary Vehicle Mass Estimation Malen and Reddy

19
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. 

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; 

SSTO

SSTO SSE R ( ) 2 − = 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]; 

[ ]

∑

∑

=

−

− ⎟

⎠

⎞ ⎜

⎝

⎛

− =

= +

i n

n n

x x

y y

n

Var

N Var

y x

data x x x x y y y y

1,

2

2

1

1 1 1

0 1

1 2 3 1 2 3

( )

( ˆ )

2

1 ( )

~ , ( )

ˆ

: , , ,... ; , , ,... ;

β

β β β

β β

 Equations B.5 

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, 

Subsystem 

mass 

wi

GVM 

WV

ρi 

2

= ∑ i SSTO ρ

Sum of squares; total

ρi is residual from mean

Subsystem 

mass 

wi

GVM 

WV

2

= ∑ i SSE r

Sum of squares; error 

ri is residual from model 

ri 

Preliminary Vehicle Mass Estimation Malen and Reddy

20
( ) .001198

33

ˆ 1.5294 0.1758

ˆ

1

0 1

=

=

= − +

= +

β

β β

Var

n

y x

y x

 Equation B.6 

The 90% confidence interval is shown in Figure B.6. 

Confidence Interval for an Influence Coefficient 

Figure B.6 

90% confidence 

interval for influence 

coefficient 

0.0

0.1

0.2

0.3

Body Structure 

Mass 

Influence 

Preliminary Vehicle Mass Estimation Malen and Reddy

21
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: 15 (of which there is 1 duplicate) 

SUV: 12 

Pick Up: 5 Note: Due to the small sample size, Pick Up results 

 are less reliable than for Sedans and SUVs. 

Van: 3 (of which there is 1 duplicate) Note: Due to the insufficient sample size, vans were 

 not included as a vehicle group. 

 35 (33 vehicles used in study) Note: For the 'All Vehicles' group, all non- 

 duplicate vehicles were included 

2004 VW Touareg (SUV) 

2004 Mazda 3 (Sedan) 

2004 Nissan Murado (SUV) 

2004 Toyota Sienna (Van) (Dup.) 

2004 Hundai XG350 (Sedan) 

2004 Toyota Prius (Sedan) 

2003 Lexus ES300 (Sedan) 

2003 Toyota Camry (US) (Sedan) 

2003 BMW 330i (Sedan) 

2003 Infiniti G35 (Sedan) 

2003 Honda Accord (Sedan)(Dup.) 

2003 Toyota Corolla Sedan (Sedan) 

2002 Audi A4 (Sedan) 

2007 Model Year

Cadillac SRX (SUV) 

Chevrolet HHR (SUV) 

Saturn Outlook (SUV) 

GMC Denali Sierra Crew Cab (Pick Up) 

Chevrolet Colorado (Pick Up) 

Chevrolet Impala (Sedan) 

Pontiac G6 SE1 (Sedan) 

Cadillac STS (Sedan) 

GMC Yukon (SUV) 

Saturn Vue (SUV) 

2002 Honda Civic LX (Sedan) 

2003 Honda Accord EX (Sedan) 

2003 PT Cruiser (SUV) 

2003 Toyota Matrix XRS (SUV) 

2003 Toyota Tacoma 4x2 (Pick Up) 

2004 Dodge Ram 4x4 Light Duty (Pick Up) 

2004 Nissan Titan LE (Pick Up) 

2004 Toyota Highlander Premium (SUV) 

2004 Toyota Sienna (Van) 

2005 Honda Odyssey Touring (Van) 

2005 Jeep Liberty (SUV) 

2005 Jeep Wrangler (SUV) 

Preliminary Vehicle Mass Estimation Malen and Reddy

22
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 Note: Engine and Transmission subsystems were combined to 

Drive Shafts 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 Note: Closures include hardware and door trim, with the

13. Body Closures exception of the GM Benchmark data where it is the door 

 shell only. These cases are noted on the graphs. 

23
ALL VEHICLES

GVM / PLAN VIEW AREA

y = 202.34x + 371.22 R 2 = 0.6309

y = 326.15x0.8647 R2

= 0.6338

0

1000

2000

3000

4000

5 10 15

Plan View Area (m2)

GVM (Kg)

Sedan

SUV

Pick Up

Linear

Power

CURB / PLAN VIEW AREA

y = 169.07x + 143.61 R2

= 0.5544

y = 213.66x0.9299 R2

= 0.5488

0

1000

2000

3000

5 10 15

Plan View Area (m2)

Curb Mass (Kg)

Sedan

SUV

Pick Up

Linear

Power

Cargo 

Mass (kg)

GVM / Cargo Mass

0

1000

2000

3000

4000

1000 2000 3000 4000

Sum of Subsystem + Passenger Mass

Reported GVM

500

250

125

0

Closures

Bumpers

Cooling

Electrical

Tires & Wheels

Steering

Fuel and Exhaust 

Power Train

Transmission

Engine

Braking

Rear Suspension

Front Suspension

Body Structure 

Body Non-structural

Subsystem Mass as a Fraction of Curb Mass

All Vehicles

0.064

0.020

0.024

0.037

0.067

0.013

0.039

0.205

0.079

0.126

0.029

0.039

0.047

0.231

0.185

0

0.1

0.2

0.3

Tires & Wheels

Nonstructural

Body Structure

Front Suspension

Rear Suspension Braking

Engine

Transmission

Power Train

Fuel &Exhaust Steering

Electrical Cooling

Bumpers

Closures

Body Non structural

y = 0.1445x - 8.4657 R 2 = 0.3637

0

200

400

600

800

1000 2000 3000 4000

GVM

Body Non structural Mass

Body Non structural

y = 0.1988x0.9507 R 2

= 0.2974

100

1000

1000 10000

GVM

Body Non structural Mass

Preliminary Vehicle Mass Estimation Malen and Reddy

24
ALL VEHICLES

Body structural

y = 0.1758x - 1.5294 R 2

= 0.4542

0

200

400

600

800

1000 2000 3000 4000

GVM

Body Structural Mass

Body structural

y = 0.325x 0.9176 R2

= 0.4672

100

1000

1000 10000

GVM

Body Structural Mass

Front Suspension

y = 0.0454x - 21.537 R2

= 0.4096

0

40

80

120

160

1000 2000 3000 4000

GVM

Front Suspension Mass

Front Suspension

y = 0.0137x1.1176 R 2

= 0.3067

10

100

1000

1000 10000

GVM

Front Suspension Mass

Rear Suspension

y = 0.0354x - 12.061 R 2 = 0.1912

0

50

100

150

200

1000 2000 3000 4000

GVM

Rear Suspension Mass

Rear Suspension

10

100

1000

1000 10000

GVM

Rear Suspension Mass

y = 0.0072x1.172 R 2= 0.2062

Preliminary Vehicle Mass Estimation Malen and Reddy

25
ALL VEHICLES

Braking System

y = 0.0077x + 30.355 R 2

= 0.0311

0

20

40

60

80

100

1000 2000 3000 4000

GVM

Braking System Mass

Braking System

y = 1.421x 0.4422 R 2

= 0.0269

10

100

1000 10000

GVM

Braking System Mass

Powertrain

y = 0.2302x - 162.07 R 2

= 0.6687

0

200

400

600

800

1000 2000 3000 4000

GVM

Powertrain Mass

Powertrain

y = 0.004x1.4741 R 2= 0.7265

100

1000

1000 10000

GVM

Powertrain Mass

Fuel & Exhaust System

y = 0.0332x - 7.612 R2

= 0.3357

0

40

80

120

160

1000 2000 3000 4000

GVM

Fuel & Exhaust System Mass

Fuel & Exhaust System

y = 0.0103x1.1318 R 2

= 0.3723

10

100

1000

1000 10000

GVM

Fuel & Exhaust System Mass

Preliminary Vehicle Mass Estimation Malen and Reddy

26
ALL VEHICLES

Steering

y = 0.0056x + 8.8642 R2

= 0.1418

0

10

20

30

40

1000 2000 3000 4000

GVM

Steering Mass

Steering

y = 0.1265x0.6583 R2

= 0.0961

10

100

1000 10000

GVM

Steering Mass

Tires & Wheels

y = 0.0488x + 4.2581 R2

= 0.7106

0

50

100

150

200

1000 2000 3000 4000

GVM

Tires & Wheels Mass

Tires & Wheels

y = 0.058x0.9818 R 2

= 0.7354

10

100

1000

1000 10000

GVM

Tires & Wheels Mass

Electrical

y = 0.0143x + 30.235 R2 = 0.0688

0

40

80

120

160

1000 2000 3000 4000

GVM

Electrical Mass

Electrical

y = 0.7054x0.5751 R2= 0.1142

10

100

1000

1000 10000

GVM

Electrical Mass

Preliminary Vehicle Mass Estimation Malen and Reddy

27
ALL VEHICLES

Cooling

y = 0.0206x - 5.7438 R 2 = 0.2651

0

25

50

75

100

1000 2000 3000 4000

GVM

Cooling Mass

Cooling

y = 0.0225x0.9627 R 2 = 0.2013

10

100

1000 10000

GVM

Cooling Mass

Bumpers

y = 0.0129x + 3.9693 R2 = 0.2139

0

20

40

60

80

1000 2000 3000 4000

GVM

Bumpers Mass

Bumpers

y = 0.2118x0.6462 R2 = 0.0918

10

100

1000 10000

GVM

Bumpers Mass

Closures

y = 0.0867x - 82.945 R 2

= 0.3037

Doors in 

white

Doors with 

components

0

50

100

150

200

250

1000 2000 3000 4000

GVM

Closures Mass

Closures

y = 8x10-6x2.1099 R2

= 0.2438

10

100

1000

1000 10000

GVM

Closures Mass

Preliminary Vehicle Mass Estimation Malen and Reddy

28
SEDAN

GVM / PLAN VIEW AREA

y = 163.14x + 561.69 R 2

= 0.5502

y = 376.41x0.7676 R 2

= 0.5545

0

500

1000

1500

2000

2500

2 4 6 8 10 12

Plan View Area (m2)

GVM (kg)

CURB MASS / PLAN VIEW AREA

y = 147.75x + 229.59 R2 = 0.5128

y = 206.11x0.9212 R2= 0.5034

0

400

800

1200

1600

2000

2 4 6 8 10 12

Plan View Area (m2)

Curb Mass (kg)

0.046

0.022

0.027

0.046

0.065

0.014

0.040

0.185

0.067

0.118

0.032

0.044

0.049

0.227

0.204

Closures

Bumpers

Cooling

Electrical

Tires & Wheels

Steering

Fuel and Exhaust 

Power Train

Transmission

Engine

Braking

Rear Suspension

Front Suspension

Body Structure 

Body Non-structural

Sedan

0

0.1

0.2

0.3

Subsystem Mass as a Fraction of Curb Mass

Tires & Wheels

Nonstructural

Body Structure

Front Suspension

Rear Suspension Braking

Engine

Transmission

Power Train

Fuel &Exhaust Steering

Electrical Cooling

Bumpers

Closures

Influence Coefficient Summary

Sedan (Linear)

Sedan (Power 

@ 3000kg)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

1975 Study

1981 Study

Sedans

Non-structure

Body Structure

Front Suspension

Rear Suspension

Braking

Power Train

Fuel & Exhaust

Steering

Tires & Wheels

Electrical

Cooling

Bumpers

Closures

Body Non Structural

y = 0.1419x + 26.297 R 2 = 0.2058

0

200

400

600

1000 2000 3000

GVM

Body Non structural 

Mass

Body Non Structural

y = 0.1465x1.0054 R2

= 0.2263

100

1000

1000 10000

GVM

Body Non structural Mass

Preliminary Vehicle Mass Estimation Malen and Reddy

29
SEDAN

Body Structural

y = 0.1317x + 78.114 R2

= 0.6626

0

100

200

300

400

500

1000 2000 3000

GVM

Body Structural Mass

Body Structural

y = 0.8247x0.7928 R 2

= 0.6934

100

1000

1000 10000

GVM

Body Structural Mass

Front Suspension

y = 0.0333x + 8.0864 R 2

= 0.1057

0

30

60

90

120

1000 2000 3000

GVM

Front Suspension Mass

Front Suspension

y = 0.631x0.622 R 2

= 0.0488

10

100

1000

1000 10000

GVM

Front Suspension Mass

Rear Suspension

y = 0.0894x - 108.43 R 2

= 0.337

0

40

80

120

160

1000 2000 3000

GVM

Rear Suspension Mass

Rear Suspension

y = 2x10-7 x2.5895 R 2

= 0.3659

10

100

1000

1000 10000

GVM

Rear Suspension Mass

Preliminary Vehicle Mass Estimation Malen and Reddy

30
SEDAN

Braking System

y = 0.0177x + 12.206 R 2

= 0.0378

0

20

40

60

80

1000 2000 3000

GVM

Braking System Mass

Braking System

y = 0.5579x0.5719 R 2

= 0.0154

10

100

1000 10000

GVM

Braking System Mass

Powertrain

y = 0.2703x - 249.45 R 2

= 0.8126

0

100

200

300

400

500

1000 2000 3000

GVM

Powertrain Mass

Powertrain

y = 0.0002x1.8598 R 2

= 0.8586

100

1000

1000 10000

GVM

Powertrain Mass

Fuel and Exhaust System

y = 0.0272x + 6.3007 R 2

= 0.0835

0

30

60

90

120

1000 2000 3000

GVM

Fuel and Exhaust System Mass

Fuel and Exhaust System

y = 0.0197x1.0525 R2

= 0.1445

10

100

1000

1000 10000

GVM

Fuel and Exhaust System Mass

Preliminary Vehicle Mass Estimation Malen and Reddy

31
SEDAN

Steering

y = -0.0023x + 25.304 R 2

= 0.0084

0

10

20

30

1000 2000 3000

GVM

Steering Mass

Steering

y = 373.16x-0.3868 R 2

= 0.0179

10

100

1000 10000

GVM

Steering Mass

Tires & Wheels

y = 0.0395x + 18.528 R 2

= 0.4535

0

30

60

90

120

150

1000 2000 3000

GVM

Tires & Wheels Mass

Tires & Wheels

y = 0.1496x0.8524 R 2

= 0.5074

10

100

1000

1000 10000

GVM

Tires & Wheels Mass

Electrical

y = 0.0139x + 40.455 R 2

= 0.0115

0

40

80

120

160

1000 2000 3000

GVM

Electrical Mass

Electrical

y = 0.0276x1.0228 R 2

= 0.1023

10

100

1000

1000 10000

GVM

Electrical Mass

Preliminary Vehicle Mass Estimation Malen and Reddy

32
SEDAN

Cooling

y = 0.0446x - 46.748 R 2

= 0.5784

0

20

40

60

80

1000 2000 3000

GVM

Cooling Mass

Cooling

y = 3x10-6 x

2.1611 R 2

= 0.5661

10

100

1000 10000

GVM

Cooling Mass

Bumpers

y = 0.0174x - 0.9741 R 2

= 0.2799

0

10

20

30

40

50

1000 2000 3000

GVM

Bumpers Mass

Bumpers

y = 0.015x1.0134 R 2

= 0.2845

10

100

1000 10000

GVM

Bumpers Mass

Closures

y = 0.1028x - 131.1 R 2

= 0.1394

0

50

100

150

200

1000 2000 3000

GVM

Closures Mass

Doors in 

white

Doors with 

components

Closures

y = 5x10-9 x3.0387 R 2

= 0.1636

10

100

1000

1000 10000

GVM

Closures Mass

Preliminary Vehicle Mass Estimation Malen and Reddy

33
SUV

GVM / PLAN VIEW AREA

y = 205.28x + 489.31 R2

= 0.5458

y = 402.26x0.7981 R2

= 0.544

0

1000

2000

3000

4000

3 6 9 12 15

Plan View Area (m2)

GVM (kg)

CURB MASS / PLAN VIEW AREA

y = 150.16x + 417.36 R 2

= 0.3965

y = 324.28x0.767 R 2

= 0.3978

0

1000

2000

3000

3 6 9 12 15

Plan View Area (m2)

Curb Mass (kg)

SUV

0

0.1

0.2

0.3

Closures

Bumpers

Cooling

Electrical

Tires & Wheels

Steering

Fuel and Exhaust 

Power Train

Transmission

Engine

Braking

Rear Suspension

Front Suspension

Body Structure 

Body Non-structural

0.077

0.017

0.024

0.033

0.067

0.011

0.035

0.215

0.077

0.138

0.027

0.040

0.048

0.229

0.177

Subsystem Mass as a Fraction of Curb Mass

Tires & Wheels

Nonstructural

Body Structure

Front Suspension

Rear Suspension Braking

Engine

Transmission

Power Train

Fuel &Exhaust Steering

Electrical Cooling

Bumpers

Closures

Influence Coefficient Summary

SUV (Linear)

SUV (Power 

@ 3000kg)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

1975 Study

1981 Study

SUV

Non-structure

Body Structure

Front Suspension

Rear Suspension

Braking

Power Train

Fuel & Exhaust

Steering

Tires & Wheels

Electrical

Cooling

Bumpers

Closures

Body Non Structural

y = 0.2373x - 236.91 R 2

= 0.6766

0

200

400

600

800

1000 2000 3000 4000

GVM

Body Non Structural Mass

Body Non Structural

y = 0.0008x1.6506 R 2

= 0.7013

100

1000

1000 10000

GVM

Body Non Structural Mass

Preliminary Vehicle Mass Estimation Malen and Reddy

34
SUV

Body Structural

y = 0.0765x + 218.34 R 2

= 0.33

0

150

300

450

600

1000 2000 3000 4000

GVM

Body Structural Mass

Body Structural

y = 12.215x0.4485 R 2

= 0.3319

100

1000

1000 10000

GVM

Body Structural Mass

Front Suspension

y = 0.0787x - 96.804 R 2

= 0.8176

0

50

100

150

200

1000 2000 3000 4000

GVM

Front Suspension Mass

Front Suspension

y = 7x10-6x2.1045 R 2

= 0.7236

10

100

1000

1000 10000

GVM

Front Suspension Mass

Rear Suspension

y = 0.0645x - 78.811 R 2

= 0.4068

0

50

100

150

200

1000 2000 3000 4000

GVM

Rear Suspension Mass

Rear Suspension

y = 1x10-5x

1.9817 R 2

= 0.3817

10

100

1000

1000 10000

GVM

Rear Suspension Mass

Preliminary Vehicle Mass Estimation Malen and Reddy

35
SUV

Braking System

y = 0.0036x + 38.73 R 2

= 0.0075

0

20

40

60

80

1000 2000 3000 4000

GVM

Braking System Mass

y = 2.2981x0.3756 R 2

= 0.0188

10

100

1000 10000

GVM

Braking System Mass

Braking System

y = 0.173x - 22.151 R 2

= 0.4237

0

200

400

600

800

1000 2000 3000 4000

GVM

Powertrain Mass

Powertrain

y = 0.0549x1.1371 R 2

= 0.4822

100

1000

1000 10000

GVM

Powertrain Mass

Powertrain

Fuel and Exhaust System

y = 0.0254x + 2.0453 R 2

= 0.3483

0

20

40

60

80

100

1000 2000 3000 4000

GVM

Fuel and Exhaust System Mass

y = 0.0175x1.0487 R 2

= 0.432

10

100

1000 10000

GVM

Fuel and Exhaust System Mass

Fuel and Exhaust System

Preliminary Vehicle Mass Estimation Malen and Reddy

36
SUV

y = 0.0085x + 0.0729 R 2= 0.2486

0

10

20

30

40

1000 2000 3000 4000

GVM

Steering Mass

Steering

y = 0.001x1.2618 R 2

= 0.2185

10

100

1000 10000

GVM

Steering Mass

Steering

y = 0.0374x + 30.485 R 2

= 0.7306

0

40

80

120

160

1000 2000 3000 4000

GVM

Tires & Wheels Mass

Tires & Wheels

y = 0.3292x0.7581 R 2

= 0.7207

10

100

1000

1000 10000

GVM

Tires & Wheels Mass

Tires & Wheels

y = 0.0455x - 47.102 R 2

= 0.6354

0

40

80

120

160

1000 2000 3000 4000

GVM

Electrical Mass

Electrical

y = 0.0004x1.5455 R 2

= 0.7031

10

100

1000

1000 10000

GVM

Electrical Mass

Electrical

Preliminary Vehicle Mass Estimation Malen and Reddy

37
SUV

y = 0.0401x - 50.975 R 2= 0.7354

0

25

50

75

100

1000 2000 3000 4000

GVM

Cooling Mass

Cooling

y = 5x10-6 x2.0598 R2= 0.6324

10

100

1000 10000

GVM

Cooling Mass

Cooling

y = 0.0193x - 15.349 R 2

= 0.3833

0

20

40

60

80

1000 2000 3000 4000

GVM

Bumpers Mass

Bumpers

y = 0.0018x1.2394 R 2

= 0.2462

10

100

1000 10000

GVM

Bumpers Mass

Bumpers

y = 0.0204x + 84.476 R 2

= 0.0246

0

50

100

150

200

250

1000 2000 3000 4000

GVM

Closures Mass

Closures

Doors in 

white

Doors with 

components

y = 453.49x-0.1772 R 2

= 0.0031

10

100

1000

1000 10000

GVM

Closures Mass

Closures

Preliminary Vehicle Mass Estimation Malen and Reddy

38
Pick Up

y = 155.34x + 869.82 R2

= 0.6882

y = 478.35x0.702 R 2

= 0.7078

0

1000

2000

3000

4000

4 8 12 16

Plan View Area (m2)

GVM (kg)

GVM / PLAN VIEW AREA

y = 194.79x - 218.79

y = 98.88x1.2301

R 2

= 0.585

R2= 0.6458

0

1000

2000

3000

4 8 12 16

Plan View Area (m2)

Curb Mass (kg)

CURB MASS / PLAN VIEW AREA

0.072

0.023

0.019

0.027

0.075

0.012

0.046

0.247

0.122

0.124

0.021

0.031

0.047

0.223

0.156

Closures

Bumpers

Cooling

Electrical

Tires & Wheels

Steering

Fuel and Exhaust 

Power Train

Transmission

Engine

Braking

Rear Suspension

Front Suspension

Body Structure 

Body Non-structural

Pick up

Tires & Wheels

Nonstructural

Body Structure

Front Suspension

Rear Suspension Braking

Engine

Transmission

Power Train

Fuel &Exhaust Steering

Electrical Cooling

Bumpers

Closures

0

0.1

0.2

0.3

Subsystem Mass as a Fraction of Curb Mass Influence Coefficient Summary

Pick Up (Linear)

Pick Up (Power 

@ 3000kg)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

1975 Study

1981 Study

Pick Ups

Non-structure

Body Structure

Front Suspension

Rear Suspension

Braking

Power Train

Fuel & Exhaust

Steering

Tires & Wheels

Electrical

Cooling

Bumpers

Closures

y = 0.2124x - 248.57 R 2

= 0.3014

0

150

300

450

600

1000 2000 3000 4000

GVM

Body Non structural Mass

Body Non Structural

y = 4x10-5 x 2.0202 R 2= 0.3321

100

1000

1000 10000

GVM

Body Non structural Mass

Body Non Structural

Preliminary Vehicle Mass Estimation Malen and Reddy

39
Pick Up

y = 0.4453x - 719 R2

= 0.5678

0

200

400

600

800

1000 2000 3000 4000

GVM

Body Structural Mass

Body Structural

y = 3x10-7x 2.6754 R2

= 0.6374

100

1000

1000 10000

GVM

Body Structural Mass

Body Structural

Front Suspension

y = 0.0354x - 1.6075 R 2

= 0.2669

0

30

60

90

120

150

1000 2000 3000 4000

GVM

Front Suspension Mass

y = 0.0315x1.0098 R2

= 0.2636

10

100

1000

1000 10000

GVM

Front Suspension Mass

Front Suspension

y = 0.0287x - 15.192 R2

= 0.9861

0

20

40

60

80

1000 2000 3000 4000

GVM

Rear Suspension Mass

Rear Suspension

y = 0.003x1.2584 R2

= 0.9853

10

100

1000 10000

GVM

Rear Suspension Mass

Rear Suspension

Preliminary Vehicle Mass Estimation Malen and Reddy

40
Pick Up

y = 0.0339x - 46.856 R 2

= 0.3216

0

20

40

60

80

1000 2000 3000 4000

GVM

Braking System Mass

Braking System

y = 6x10-7x2.2871 R2= 0.3138

10

100

1000 10000

GVM

Braking System Mass

Braking System

y = 0.2953x - 289.5 R2

= 0.7712

0

200

400

600

800

1000 2000 3000 4000

GVM

Powertrain Mass

Powertrain

y = 0.0022x1.5596 R 2

= 0.8142

100

1000

1000 10000

GVM

Powertrain Mass

Powertrain

y = 0.0621x - 71.059 R 2

= 0.5274

0

40

80

120

160

1000 2000 3000 4000

GVM

Fuel and Exhaust System Mass

Fuel and Exhaust System

y = 2x10-5x1.9676 R2= 0.536

10

100

1000

1000 10000

GVM

Fuel and Exhaust System Mass

Fuel and Exhaust System

Preliminary Vehicle Mass Estimation Malen and Reddy

41
Pick Up

y = 0.0056x + 8.8616 R2

= 0.1363

0

10

20

30

40

1000 2000 3000 4000

GVM

Steering Mass

Steering

y = 0.0309x0.8408 R 2

= 0.207

10

100

1000 10000

GVM

Steering Mass

Steering

y = 0.0356x + 52.678 R 2

= 0.2622

0

50

100

150

200

1000 2000 3000 4000

GVM

Tires & Wheels Mass

Tires and Wheels

y = 0.3396x0.7689 R2

= 0.3161

10

100

1000

1000 10000

GVM

Tires & Wheels Mass

Tires and Wheels

y = 0.0189x + 3.8375 R2

= 0.4488

0

20

40

60

80

1000 2000 3000 4000

GVM

Electrical Mass

Electrical

y = 0.0376x0.9209 R2

= 0.4722

10

100

1000 10000

GVM

Electrical Mass

Electrical

Preliminary Vehicle Mass Estimation Malen and Reddy

42
Pick Up

y = 0.0181x - 10.62 R2

= 0.0829

0

25

50

75

100

1000 2000 3000 4000

GVM

Cooling Mass

Cooling

y = 0.0312x0.8853 R2

= 0.0662

10

100

1000 10000

GVM

Cooling Mass

Cooling

y = 0.0198x - 7.7917 R2

= 0.4043

0

20

40

60

80

1000 2000 3000 4000

GVM

Bumpers Mass

Bumpers

y = 0.0067x1.1153 R 2

= 0.3722

10

100

1000 10000

GVM

Bumpers Mass

Bumpers

y = 0.1276x - 190.53 R2

= 0.701

0

50

100

150

200

250

1000 2000 3000 4000

GVM

Closures Mass

Closures

y = 5x10-7x2.4574 R2

= 0.6981

10

100

1000

1000 10000

GVM

Closures Mass

Closures

Preliminary Vehicle Mass Estimation Malen and Reddy

43
Fuel and 

Exhaust 

Steering

Tires & 

Wheels

Electrical

Cooling

Bumpers

Closures

Powertrain

Braking 

System

Rear 

Suspension

Front 

Suspension

Body 

Body Nonstructural

Structure 

c=5E-09 

α=3.0387 0.045 0.103 c=8E-06 α=2.1099 0.122 0.069 0.087

c=0.015 

α=1.0134 0.017 0.017 c=0.2118 α=0.6462 0.008 0.010 0.013

c=3E-06 

α=2.1611 0.032 0.045 c=0.0225 α=0.9627 0.016 0.016 0.021

c=0.0276 

α=1.0228 0.033 0.014 c=0.7054 α=0.5751 0.014 0.017 0.014

c=0.1496 

α=0.8524 0.043 0.040 c=0.058 α=0.9818 0.049 0.050 0.049

c=373.16 

α=-0.3959

-0.006

(0)

-0.002

(0)

c=0.1265 

α=0.6583 0.005 0.006 0.006

c=0.0197 

α=1.0525 0.030 0.027 c=0.0103 α=1.1318 0.033 0.031 0.033

c= 0.0002 

α=1.8598 0.200 0.270 c= 0.004 α=1.4741 0.262 0.206 0.230

c=0.5579 

α=0.5719 0.014 0.018 c=1.421 α=0.4422 0.007 0.010 0.008

c=2E-07 

α=2.5895 0.058 0.089 c=0.0072 α=1.172 0.033 0.031 0.035

c=0.631 

α=0.622 0.025 0.033 c=0.0137 α=1.1176 0.039 0.037 0.045

c=0.8247 

α=0.7928 0.144 0.132 c=0.325 α=0.9176 0.154 0.161 0.176

c=0.1465 

α=1.0054 0.153 0.142 c=0.1988 α=0.9507 0.127 0.131 0.145

Power 

Equation 

Y=C Xα

Power 

@ 1800 

kg

Linear

Power 

Equation 

Y=C Xα

Power 

@ 3000 

kg

Power 

@ 1800 

kg

Linear

Sedan All Vehicles

0.006 0.0480.0220.006

0.036

0.031 0.0110.131 0.1830.016 0.0380.012 0.0130.029 0.0210.166 0.294

1981 1975

c=5E-07 

α=2.4574 0.144 0.128 c=453.49 α=-0.1772 -0.006 0.020

c=0.0067 

α=1.1153 0.019 0.020 c=0.0018 α=1.2394 0.015 0.019

c=0.0312 

α=0.8853 0.011 0.018 c=5E-06 α=2.0598 0.050 0.040

c=0.0376 

α=0.9209 0.018 0.019 c=0.0004 α=1.5455 0.049 0.046

c=0.3396 

α=0.7689 0.041 0.036 c=0.3292 α=0.7581 0.036 0.037

c=0.0309 

α=0.8408 0.007 0.006 c=0.001 α=1.2618 0.010 0.009

c=2E-05 

α=1.9676 0.091 0.062 c=0.0175 α=1.0487 0.027 0.025

c= 0.0022 

α=1.5596 0.303 0.295 c= 0.0549 α=1.1371 0.187 0.173

c=6E-07 

α=2.2871 0.041 0.034 c=2.2981 α=0.0188 0.006 0.004

c=0.003 

α=1.2584 0.030 0.029 c=1E-05 α=1.9817 0.051 0.065

c=0.0315 

α=1.0098 0.034 0.035 c=7E-06 α=2.1045 0.102 0.079

c=3E-07 

α=2.6754 0.537 0.445 c=12.215 α=0.4485 0.066 0.077

c=4E-05 

α=2.0202 0.285 0.212 c=0.0008 α=1.6506 0.242 0.237

Linear

Power 

@ 3000 

kg Linear

Pick Up 

(Not signifcant as a group) SUV Historical

Power 

Equation 

Y=C Xα

Power 

Equation 

Y=C Xα

Power 

@ 3000 

kg

Influence Coefficients

Sum of 

recommended 

coefficients

0.644 0.413 0.562 0.599 0.900 0.463

Recommended subsystem coefficients shown in bold with border

44
Selection of Influence Coefficients for Inclusion in Model 

1.00 0.50-1.00 

0.25 -0.50 0.50 

0.2500-0.25 

Closures 0.30 0.16 0.02 0.70 0.30 Little 0.09 No 

Bumpers 0.21 0.28 0.38 0.40 0.32 Strong 0.01 Yes 

Cooling 0.27 0.57 0.74 0.08 0.41 Moderate 0.02 No 

Electrical 0.11 0.10 0.70 0.47 0.35 Little 0.02 No 

Tire & wheels 0.74 0.51 0.73 0.32 0.57 Strong 0.04 Yes 

Steering 0.14 0.02 0.25 0.21 0.15 Moderate <0.01 (Yes)

Fuel & exhaust 0.37 0.14 0.43 0.54 0.37 Little 0.03 No 

Powertrain 0.73 0.86 0.48 0.81 0.72 Strong 0.20 Yes 

Braking 0.03 0.04 0.02 0.32 0.10 Strong 0.02 Yes 

Rear 0.21 0.37 0.41 0.99 0.49 Strong 0.03 Yes 

Front 0.41 0.11 0.82 0.27 0.40 Strong 0.03 Yes 

Body structure 0.47 0.69 0.33 0.64 0.53 Strong 0.15 Yes 

Non structure 0.36 0.23 0.70 0.33 0.41* Little 0.20 No 

γ 

Magnitude 

All Sedan SUVs Pick Average 

Regression Coefficient R2 

Key *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) 

Inclusion

Criteria 

Physical 

link with 

vehicle 

mass 

Included 

in prior 

work 

No 

Yes 

No 

Yes/No 

Yes 

Yes 

No 

Yes 

Yes 

Yes 

Yes 

Yes 

No 

Include 

Preliminary Vehicle Mass Estimation Malen and Reddy

45
2000 Town Center, Suite 320

Southfield, Michigan 48075

Tel: 248.945.4777

www.a-sp.org