Citation
Gerneralized deformation and total velocity change analysis system of equations (G-DaTaDeltaVâ„¢)

Material Information

Title:
Gerneralized deformation and total velocity change analysis system of equations (G-DaTaDeltaVâ„¢)
Creator:
Ogden, Jerry Scott ( author )
Language:
English
Physical Description:
1 electronic file (276 pages) : ;

Subjects

Subjects / Keywords:
Deformations (Mechanics) -- Mathematical models ( lcsh )
Motor vehicles -- Collision damage ( lcsh )
Metals -- Impact testing ( lcsh )
Collisions (Physics) ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Review:
Current methods for analyzing motor vehicle deformation utilize a force-deflection analysis for determining deformation work energy, which relies on vehicle-specific structural stiffness coefficients determined from full-scale impact testing. While the current database is quite extensive for frontal stiffness values for passenger cars and many light trucks, vans and SUVs from the 1970’s up to modern day, the database is devoid of specific crash tests needed for deformation analysis of rear and/or side structures of many vehicles. Additionally, there exists very few structural stiffness coefficients for heavy commercial vehicles, buses, recreational vehicles, heavy equipment or motorcycles necessary for application with the current force-deflection analysis methods. ( ,, )
Review:
The primary goal of this research is to develop an accurate, reliable and broadly applicable deformation analysis method that requires the structural stiffness coefficients for only one collision involved vehicle. The developed methodology expands the application of deformation analysis to include unconventional vehicles and other objects and surfaces not supported by the current structural stiffness coefficient database. The G-DaTADeltaVâ„¢ System of Equations incorporates linear and rotational effects, as well as impact restitution resulting from conservative forces acting during a given collision impulse. Additionally, the G-DaTADeltaVâ„¢ System of Equations accounts for tire-ground forces and inter-vehicular friction, non-conservative force contributions acting on the collision system that are commonly present during offset and oblique non-central collision configurations.
Review:
Correlation and descriptive statistics, as well as the raw analysis results, indicate a highly reliable and significantly improved degree of precision and accuracy achieved through the application of the G-DaTADeltaVâ„¢ System of Equations when determining vehicular total velocity changes for oblique and offset non-central impacts.
Thesis:
Thesis (Ph.D) - University of Colorado Denver.
Bibliography:
Includes bibliographic references
System Details:
System requirements: Adobe Reader.
General Note:
Department of Civil Engineering
Statement of Responsibility:
by Jerry Scott Ogden.

Record Information

Source Institution:
|University of Colorado Denver
Holding Location:
|Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
944967825 ( OCLC )
ocn944967825
Classification:
LD1193.E53 2015d O33 ( lcc )

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Full Text
GENERALIZED DEFORMATION AND TOTAL VELOCITY CHANGE ANALYSIS
SYSTEM OF EQUATIONS (G-DaTAzlF)
by
JERRY SCOTT OGDEN
B.S., Eastern Oregon University, 1988
M.S., University of Colorado Denver, 1995
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Engineering and Applied Sciences
Civil Engineering
2015


2015
JERRY SCOTT OGDEN
ALL RIGHTS RESERVED
11


This thesis for the Doctor of Philosophy degree by
Jerry Scott Ogden
has been approved for the
Engineering and Applied Sciences Program
by,
Wesley Marshall, Chair
Bruce Janson, Advisor
Peter Jenkins
Ronald Rorrer
Apostol Panayotov
m
Date: October 15, 2015


Ogden, Jerry Scott (PhD, Engineering and Applied Sciences)
Generalized Deformation and Total Velocity Change Analysis System of Equations (G-
DaTAzlE)
Thesis directed by Professor Bruce Janson
ABSTRACT
Current methods for analyzing motor vehicle deformation utilize a force-deflection
analysis for determining deformation work energy, which relies on vehicle-specific
structural stiffness coefficients determined from full-scale impact testing. While the current
database is quite extensive for frontal stiffness values for passenger cars and many light
trucks, vans and SUVs from the 1970s up to modern day, the database is devoid of specific
crash tests needed for deformation analysis of rear and/or side structures of many vehicles.
Additionally, there exists very few structural stiffness coefficients for heavy commercial
vehicles, buses, recreational vehicles, heavy equipment or motorcycles necessary for
application with the current force-deflection analysis methods.
The primary goal of this research is to develop an accurate, reliable and broadly
applicable deformation analysis method that requires the structural stiffness coefficients
for only one collision involved vehicle. The developed methodology expands the
application of deformation analysis to include unconventional vehicles and other objects
and surfaces not supported by the current structural stiffness coefficient database. The G-
DaTAzlT System of Equations incorporates linear and rotational effects, as well as
impact restitution resulting from conservative forces acting during a given collision
impulse. Additionally, the G-DaTAzU System of Equations accounts for tire-ground
forces and inter-vehicular friction, non-conservative force contributions acting on the
collision system that are commonly present during offset and oblique non-central collision
configurations.
Correlation and descriptive statistics, as well as the raw analysis results, indicate a
highly reliable and significantly improved degree of precision and accuracy achieved
IV


through the application of the G-DaTAzU System of Equations when determining
vehicular total velocity changes for oblique and offset non-central impacts.
The form and content of this abstract are approved. I recommend its publication.
Approved: Bruce Janson
v


ACKNOWLEDGMENTS
I would like to thank the Colorado State Patrol (www.colorado.gov/csp) and the
Colorado Department of Transportation (www.coloradodot.info) for providing collision
history data for the State of Colorado used in Appendix A of this study. I would like to
express my appreciation to James Neptune of Neptune Engineering, Inc.
(www.neptuneeng.com). for granting permission to utilize the extensive database of
vehicle stiffness coefficients that he tirelessly updates with new testing results to keep
everyone current. Additionally, the Faro HD and Faro Reality CAD collision
reconstruction programs were utilized to produce the high impact diagrams appearing
within this study.
I would especially like to thank the engineers of OEC Forensics
(www.OEC4N6.com). Mathew Martonovich and Courtney Engle. The insight, criticism
and critical thinking where needed by other professionals and trusted colleagues is crucial
to the success of such an endeavor. No accomplishment of this scope and magnitude is
possible without the support of an understanding, patient and loving family. I want to
express my deepest appreciation to my wife Cherie, and my children Carson, Natasha and
Adriana. Their encouragement to make this happen, undying support and understanding
regarding the extreme time commitment necessary for this project, along with their loving
confidence made this dream a reality.
vi


TABLE OF CONTENTS
Chapter
1: STUDY MOTIVATION AM) OBJECTIVES.........................................1
1.1 Study Motivation.....................................................1
1.2 Statement of Study Objectives.......................................6
2: LITERATURE RESEARCH.....................................................9
2.1 Background of Collision Analysis.....................................9
2.1.1 Basic Collision Analysis Methods.................................9
2.1.2 Passenger Vehicle Event Data Recording Systems..................9
2.1.3 Heavy Vehicle Event Data Recording Systems......................13
2.1.4 Vehicle Tracking Systems........................................17
2.2 Vehicle Deformation Analysis.......................................18
2.2.1 Pioneering Studies and Computer Programs........................18
2.2.2 Modem Damage Deformation Studies................................23
3: COLLINEAR CENTRAL IMPACTS; DEVELOPING BASIC DEFORMATION
ANALYSIS PRINCIPLES.......................................................26
3.2 Equation of Motion for Collinear Central Impacts...................27
3.2.1 Conservation of Energy..........................................28
3.3.1 Lagranges Equations............................................30
3.3.2 Deriving the System Momentum....................................31
3.3.3 Equation of Motion..............................................32
3.4 Impulse-Momentum of the System.....................................34
3.5 Collision Force from Vehicle Deformation...........................36
3.6 Central Impact Work/Energy Model...................................43
3.7 Determination of A and B Stiffness Coefficients....................46
3.7.1 Frontal Stiffness Coefficients..................................46
vii


3.7.2 Off-Set Frontal, Side, and Rear Surface Stiffness Coefficients.......48
3.7.3 Commercial Vehicle Stiffness Coefficients............................48
3.8 Central Impact Vehicle-to-Vehicle Velocity Change........................49
3.8.1 Developing a Force-Deflection/Velocity Change Relationship...........49
3.8.2 Force-Deflection/Velocity Change Considering Restitution Effects.....53
3.8.3 Force-Deflection Velocity Change Considering Restitution and Non-
Conservative Tire-Ground Forces............................................56
3.9 Missing Vehicle Parameters...............................................58
3.9.1 Background...........................................................59
3.9.2 Impact Force Balancing Crosscheck....................................60
3.9.3 Determination Missing Deformation Depths.............................61
3.9.4 Application of Work/Energy Principles for Unknown Stiffness Coefficients 62
4: OFF-SET AND OBLIQUE NON-CENTRAL IMPACTS: GENERALIZED
DEFORMATION AND TOTAL VELOCITY CHANGE ANALYSIS (G-DaTAAV)
SYSTEM OF EQUATIONS.............................................................67
4.1 Objectives...............................................................67
4.2 Oblique and Offset Impact Momentum Principles............................69
4.2.1 Linear Momentum......................................................69
4.2.2 Rotational Momentum..................................................77
4.2.3 Mass Moment of Inertia...............................................82
4.3 Principle Direction of Force from Damage Profiles........................84
4.3.1 Determining Damage Centroid..........................................85
4.3.2 Maximum Engagement...................................................89
4.4 Oblique Impact Generalized Force-Deflection Model........................90
4.4.1 Oblique Deformation Principles.......................................91
4.4.2 Developing the Generalized Force-Deflection Model....................92
4.4.3 Generalized Force-Deflection Principles..............................94
viii


4.5 Oblique Impact Work/Energy Model........................................95
4.5.1 Contributions of Oblique Angle to Vehicle Deformation................95
4.5.2 Contributions of Non-Conservative Inter-vehicular Frictional Forces.96
4.5.3 Generalized Impact Work/Energy Model................................99
4.6 Generalized Impact Force-Deflection/Velocity Change Model...............100
4.6.1 Rotational Contributions...........................................100
4.6.2 Generalized Impulse Time..........................................103
4.7 Generalized Newtonian Prediction of Missing Vehicle Parameters..........103
4.7.1 Generalized Impact Force Balancing Crosscheck........................104
4.7.2 Determining Missing Deformation Depths............................105
4.7.3 Generalized Work/Energy Principles and Unknown Stiffness Coefficients.. 107
4.8 Summary of Findings; G-DaTAAE System of Equations......................109
5: GENERALIZED DEFORMATION AND TOTAL VELOCITY CHANGE ANALYSIS
(G-DaTAzlF) SYSTEM OF EQUATIONS APPLICATION AND EVALUATION. 114
5.1 Overview of G-DaTAAV System of Equations Development....................114
5.1.1 Chapter 2 Contributions............................................114
5.1.2 Chapter 3 Contributions...........................................115
5.1.3 Chapter 4 Contributions...........................................115
5.2 Anatomy of G-DaTAAV System of Equations................................116
5.2.1 Non-central Impact Work/Energy Sink Contributions..................117
5.2.3 Non-central Impact Non-Conservative Force Contributions..............118
5.3 G-DaTAAF System of Equations Evaluation..............................120
5.3.1 RICSAC Testing.....................................................120
5.3.2 RICSAC Original Study Findings....................................123
5.3.3 RICSAC G-DaTAAV System of Equations Application Approach....127
5.3.4 RICSAC G-DaTAAV System of Equations Analysis Results........129
IX


5.4 National Automotive Sampling System Real-World Collisions
133
5.4.1 NASS G-DaTAAV System of Equations Application Approach......134
5.4.2 NASS G-DaTAAV System of Equations Analysis Results..........136
5.4.3 Overall G-DaTAAV System of Equations Evaluation.............140
5.5 Findings and Conclusions.........................................144
5.5.1 G-DaTAAV System of Equations Applications...................144
5.5.2 G-DaTAAV System of Equations Limitations....................145
5.5.3 Future Research..............................................146
5.5.4 Conclusions..................................................147
REFERENCES.............................................................152
APPENDICES
A: COLLISIONS ON COLORADO HIGHWAYS AND WORK ZONES FROM 2007 TO
2011........................................................................160
A.l Objectives.............................................................160
A.2 Collision Data for all State Highways from 2007 to 2011................160
A.3 Collision Data for Construction Zone Collisions 2007 to 2011...........165
A.3.1 Overview of Crash Reported Data...................................165
A.3.2 Passenger Vehicle Construction Zone Collisions....................169
AAA Sport Utility Vehicle, Pickup and Van Construction Zone Collisions..171
A.3.4 Heavy Vehicle Construction Zone Collisions.........................173
A.3.5 Motorcycle Construction Zone Collisions............................175
A. 4 Summary..............................................................177
B: G-DaI A 1/ ANALYSIS OF INDIVIDUAL RICSAC TESTS.........................179
B. l RICSAC 1 Broadside Oblique Impact....................................179
B.2 RICSAC 2 Broadside Oblique Impact......................................182
x


B.3 RICSAC 3 Front to Rear Oblique Offset Impact..........................185
B.4 RICSAC 4 Front to Rear Oblique Offset Impact..........................189
B.5 RICSAC 5 Front to Rear Oblique Offset Impact..........................192
B.6 RICSAC 6 Front to Side Oblique Offset Impact..........................196
B.7 RICSAC 7 Front to Side Oblique Offset Impact..........................200
B.8 RICSAC 8 Perpendicular Broadside Offset Impact.......................203
B.9 RICSAC 9 Perpendicular Broadside Offset Impact.......................207
B. 10 RICSAC 10 Perpendicular Broadside Offset Impact....................210
B. 11 RICSAC 11 Front-to-Front Offset Impact.............................214
B.12 RICSAC 12 Front-to-Front Offset Impact..............................217
C: G-DaTAzlF ANALYSIS OF INDIVIDUAL NASS TESTS............................222
Cl.l 2010-08-037..........................................................223
C1.2 2010-12-154..........................................................224
C1.3 2011-04-127..........................................................226
C1.4 2011-08-107..........................................................228
C1.5 2011-08-112..........................................................229
C1.6 2011-09-075..........................................................231
C1.7 2011-09-091..........................................................232
C1.8 2011-11-085..........................................................234
C1.9 2011-12-049..........................................................235
C1.10 2011-12-189.........................................................237
C1.5 2012-08-064..........................................................238
Cl. 12 2012-08-080........................................................240
C F 13 2012-12-016........................................................241
Cl. 14 2012-41-024........................................................243
xi


Cl.15 2012-43-014.......................................................244
Cl.16 2012-43-026.......................................................246
Cl.17 2012-43-106.......................................................247
Cl.18 2012-48-106.......................................................249
Cl.19 2013-12-059.......................................................250
Cl.20 2013-12-106.......................................................252
Cl.21 2013-12-112.......................................................253
Cl.22 2013-43-152.......................................................255
Cl.23 2013-76-094.......................................................256
Cl.24 2013-76-165.......................................................258
Cl.25 2013-79-139.......................................................259
xii


LIST OF FIGURES
Figure
1.1 Total crashes by vehicle type for 2007 to 2011 (429,013 vehicles)......................2
1.2 Colorado construction zone collisions by vehicle type for 2007 to 2011 (4316 vehicles).2
2.1 Linear approximation of collision force (adapted from Figure 2.3 [6])................19
2.2 Damage profde approximation (adapted from Figure 2.5 [6])............................20
3.1(a) SAE Conventional vehicle coordinate system...........................................27
3.1(b) Earth-fixed oriented vehicle coordinate system.......................................27
3.2 Central collinear impact.............................................................28
3.3 Collision deformation as a linear spring system......................................34
3.4 Collision pulse......................................................................36
3.5 Two-dimensional, single DOF vehicle-barrier collision model..........................37
3.6 Measured damage dimensions...........................................................40
3.7 Relationship between force per unit width and residual deformation...................43
3.8 Relationship between barrier impact velocity and residual deformation................47
3.9 Collision as a simple harmonic oscillator............................................51
3.10 Impact with restitution effects (adapted from Figures 3.3 and 3.9)....................54
3.11 Braking force contribution to collision impulse.......................................57
3.12 Paired force regions from impact deformation..........................................64
4.1 Non-central oblique and off-set impacts................................................68
4.2(a) Planar collision trajectory angle determination....................................71
4.2(b) Oblique collision rotation angle determination.......................................71
4.3 Linear momentum vector parallelogram.................................................74
4.4 Linear momentum velocity change and PDOF...............................................74
4.5 Surface slope and friction relationships...............................................77
4.6 Moment arm applied to produce rotation about mass center...............................79
4.7 Rotational post-impact motions ......................................................81
4.8 Damage profile measurements..........................................................86
4.9 Damage centroid match at maximum engagement for PDOF...................................90
4.10 Vehicle spring continuum............................................................91
4.11 Oblique impact PDOF acting at damage centroid..........................................93
4.12 Friction of extended contact, scraping impacts.........................................97
5.1 Regression graph for original CRASH deformation based analysis.........................124
xiii


5.2 Regression graph for original SMAC momentum based analysis....................127
5.3 G-DaTAAV piecewise damage match values versus RICSAC tests...................132
5.4 G-DaTAAV weighted average damage values versus RICSAC test...................133
5.5 G-DaTAAV piecewise damage match versus NASS Bosch CDR data...................138
5.6 G-DaTAAV weighted average damage versus NASS Bosch CDR data...................139
A. 1 Total crashes by vehicle type for 2007 to 2011 (429,013 vehicles).............161
A.2 Colorado State Highway crashes by collision type 2007 to 2011 (235,884 impacts involving
429,013 vehicles)..................................................................162
A.3 Colorado State Highway crashes by roadway type for 2007 to 2011................163
A.4 Colorado State Highway crashes by severity for 2007 to 2011....................163
A.5 Colorado construction zone collisions by vehicle type for 2007 to 2011 (4316 total vehicles)
...................................................................................166
A.6 Colorado construction zone collision by collision type for 2007 to 2011........167
A.7 Colorado construction zone collisions by roadway type for 2007 to 2011.........168
A.8 Colorado construction zone collisions by severity for 2007 to 2011.............169
A.9 Passenger vehicle construction zone collisions 2007 to 2011....................170
A. 10 Passenger vehicle construction zone collision severity 2007 to 2011..........170
A.ll Sport utility vehicle construction zone collisions 2007 to 2011...............171
A. 12 Sport utility vehicle construction zone collision severity 2007 to 2011......172
A. 13 Pickup and full sized van construction zone collisions 2007 to 2011..........172
A. 14 Pickup and van construction zone collision severity 2007 to 2011.............173
A. 15 Heavy vehicle construction zone collisions 2007 to 2011......................174
A. 16 Heavy vehicle construction zone collision severity 2007 to 2011..............174
A. 17 Motorcycle construction zone collisions 2007 to 2011.........................176
A. 18 Motorcycle construction zone collision severity 2007 to 2011................176
B. l Maximum engagement PDOF diagram for RICSAC 1.................................179
B.2 Maximum engagement PDOF diagram for RICSAC 2...................................183
B.3 Maximum engagement PDOF diagram for RICSAC 3...................................186
B.4 Maximum engagement PDOF diagram for RICSAC 4...................................190
B.5 Maximum engagement PDOF diagram for RICSAC 5...................................193
B.6 Maximum engagement PDOF diagram for RICSAC 6...................................197
B.7 Maximum engagement PDOF diagram for RICSAC 7...................................201
B.8 Maximum engagement PDOF diagram for RICSAC 8...................................204
B.9 Maximum engagement PDOF diagram for RICSAC 9...................................208
xiv


B. 10 Maximum engagement PDOF diagram for RIC SAC 10.............................211
B. 11 Maximum engagement PDOF diagram for RIC SAC 11.............................215
B.12 Maximum engagement PDOF diagram for RICSAC 12...............................218
Cl NASS 2010-08-037..............................................................223
C2 NASS 2010-12-154..............................................................225
C3 NASS 2011-04-127..............................................................226
C4 NASS 2011-08-107..............................................................228
C5 NASS 2011-08-112..............................................................229
C6 NASS 2011-09-075..............................................................231
C7 NASS 2011-09-091..............................................................232
C8 NASS 2011-11-085..............................................................234
C9 NASS 2011-12-049..............................................................235
C10NASS 2011-12-189..............................................................237
Cl 1 NASS 2012-08-064............................................................238
C12 NASS 2012-08-080.............................................................240
C13 NASS 2012-12-016.............................................................241
C14 NASS 2012-41-024.............................................................243
C15 NASS 2012-43-014.............................................................244
C16 NASS 2012-43-026.............................................................246
C17 NASS 2012-43-106.............................................................247
C18 NASS 2012-48-106.............................................................249
C19 NASS 2013-12-059.............................................................250
C20 NASS 2013-12-106.............................................................252
C21 NASS 2013-12-112.............................................................253
C22 NASS 2013-43-152.............................................................255
C23 NASS 2013-76-094.............................................................256
C24 NASS 2013-76-165.............................................................258
C25 NASS 2013-79-139.............................................................259
xv


LIST OF TABLES
Table
2.1 Reproduced Table I from 49CFR Part 561................................. 10
2.2 Reproduced Table II from 49CFR Part 561..................................11
2.3 Reproduced Table 1 from SAE J2728, June 2010.............................14
4.1 Yaw moment of inertia empirical constants................................84
5.1 Summary of RICSAC tests with coordinate transformation..................122
5.2 Original CRASH based deformation analysis results.......................123
5.3 Original SMAC based momentum analysis results...........................126
5.4 RICSAC test values versus G-DaTAAV analysis............................131
5.5 Summary of statistics...................................................132
5.6 NASS reported Bosch CDR Tool data versus G-DaTAAV analysis.............137
5.7 Summary of Statistics...................................................138
A. 1 Highway construction zone collisions by vehicle type and collision type 2007 to 2011
as reported by the Colorado State Patrol....................................165
xvi


CHAPTER 1: STUDY MOTIVATION AND OBJECTIVES
1.1 Study Motivation
Advancements in highway geometric design, traffic control systems, intelligent
vehicles and highways and vehicle crashworthiness, as well as automotive handling,
stability and control have progressed significantly from the introduction of the automobile
as a viable means of transportation in the early 20th century. However, to continue the trend
of increasing transportation safety and decreasing severe and fatal injury crash events, it is
imperative that Civil Engineers and Mechanical Engineers work together as a multi-
disciplinary team. Cooperative engineering between disciplines provides the most logical
approach for ensuring the necessary analytical tools. Robust methodologies should be
developed and uniformly utilized to properly analyze collision events as an engineering
team, and for each discipline's unique applications and research interests.
Traditionally, passenger car classification vehicles over the past 50 years have
comprised the vast majority of full-scale vehicle impact testing for crashworthiness and
research regarding collision dynamics. The consideration of light trucks, vans and SUVs
into vehicle safety research has only taken place within the past few decades. Accordingly,
the remaining approximate half of collision-involved vehicles not part of the passenger car
classification have comparatively limited scope and breadth of impact testing conducted to
date.
1


Motorcycle
Bicycle
Semi, Bus 6992 (1.6%)
RV
19,562 (4.6%)
Other
Unknown
10,188 (2.4%)
Sport Utility
Vehicle
92,326 (21.5%)
Passenger Vehicle
221,022 (51.5%)
Pickup and
Full Size Van
78,923 (18.4%)
Data Source COOT Crashes and Rates on State Highways 2007 to 2011
Figure 1.1 Total crashes by vehicle type for 2007 to 2011 (429,013 vehicles)
Motorcycle
Bicycle
68 (1.6%)
Semi, Bus, RV
366 (8.5%)
Other or Unknown
50(1.2%)
Sport Utility
Vehicle
862 (20.0%)
Passenger Vehicle
2153 (49.9%)
Pickup and
Full Size Van
817 (18.9%)
Data Source Colorado State Patrol
Figure 1.2 Colorado construction zone collisions by vehicle type for 2007 to 2011
(4316 vehicles)
Appendix A of this study investigates collision data collected from the Colorado
Department of Transportation and the Colorado State Patrol for State highways. Appendix
2


A focuses on the overall crash statistics for State of Colorado highway system for the study
years of2007 to 2011, as well as the collisions related to highway construction zones during
that same period. The following charts show the distribution of collision involved vehicles
by vehicle type for all State highways and State highway construction zones for the study
years 2007 to 2011.
The collision data contained in Figure 1.1 and Figure 1.2, demonstrate that about
half of all vehicles involved in collisions within the State of Colorado fit the passenger
vehicle classification, which is comprised of coupes, sedans, sports cars, and wagons. The
remaining approximate half of all involved vehicles include sport utility (SUV), light trucks
and vans, heavy vehicles, recreational vehicles (RV), heavy machinery, motorcycles,
bicycles, pedestrians and other types of transportation modes. The study data does not
indicate that half of all collisions are passenger-car-to-passenger-car collisions. Instead,
the data simply indicates that half of all vehicles involved in public roadway collision
events are passenger cars, which are two very different conditions to consider. The study
data does not exclude the obvious consequence of mismatched vehicle size and vehicle
classifications interacting in highway collisions. Mismatched vehicles at impact impose the
consideration of a whole new set of variables by a multi-disciplinary engineering research
or investigative team.
For the Civil Engineer, the ability to study vehicle collisions has important
implications from a general statistical standpoint regarding traffic safety studies. However,
collision analysis provides equal contributions towards collision mitigation measures and
traffic management practices. Highway construction zones add a complication to the
roadway system in that entering, navigating through and exiting from construction
activities are unique to the activity of the construction project rather than roadway
classification. When entering a construction zone, motorists typically must comply with
incremental reductions in speeds resulting in a pace well below the normal regulatory speed
limit for the roadway. Depending upon the complexity of the construction activities speed
reductions may be fluid in order to manage traffic negotiating the buffer zone and work
arrea. Throughout the work zone, drivers may encounter reduced roadway width merge
tapers, lane shifts, and detours or diversions, as well as temporary stoppages controlled by
3


temporary signalization or flagging operations. To complicate the driving task even further,
work zone activities can last in duration from a few hours to years. The dynamic or
changing nature of work zones require the Civil Engineer to understand the stability and
handling characteristics of the traffic stream, as well as the ability to assess the safety and
effectiveness of work zone temporary traffic control treatments.
Often the statistical analysis of collisions for highway safety purposes relies upon
reported vehicle speeds or collision severity parameters from police agency incident
reports. However, the accuracy of those reported speed values can easily be called into
question. Several uncertainties regarding the accuracy of reported speeds and collision
parameters loom as follows:
Were the speeds and severity levels determined by proper analysis
procedures with proper data?
Were speed recording devices such as speed traps, laser or radar
measurement utilized?
Were estimates determined by a trained investigating officer or authority?
Were estimates provided by lay witnesses or collision-involved drivers?
Each layer of reported data where these questions are unknown can add a new
source of unacceptable error. As such, the necessity for a reliable, accurate and broadly
applicable generalized analysis methodology that does not require extensive field
investigation becomes critical for proper traffic safety evaluations.
The understanding of vehicle velocities, velocity changes, accelerations and the
effects of collision forces from a real-world perspective are critical for the Mechanical
Engineer studying vehicle dynamics, crashworthiness and mechanical designs and failures.
As an example, the use of controlled impact testing provides a means for experimentally
evaluating algorithms for supplemental restraint system deployments and the performance
of primary occupant restraint systems. Controlled vehicle testing also provides many of the
initial evaluations of advanced warning and collision avoidance countermeasure systems.
Computer analysis can model vehicle stability and handling, but ultimately controlled track
testing determines whether design features perform as intended under normal operation. A
4


Mechanical Engineer should have at least a working knowledge of highway design and
traffic control operations to understand fully, predict and design for vehicle stability and
handling characteristics. Knowledge of traffic flow fundamentals provides the Mechanical
Engineer with a knowledge set to assess vehicle performance under realistic conditions the
motorist will encounter on public roadways. Furthermore, a Mechanical Engineer that
studies crashworthiness and safety system performance of real-world collision events must
have reliable, accurate and relatively precise analysis methods for determining collision
kinematics and dynamics. The need for a reliable, accurate and broadly applicable
generalized analysis methodology for determining impact velocities and severity levels of
real-world collision events is necessary for assessing the performance of safety design
features, improving vehicle handling, and optimizing crashworthiness and occupant safety.
For the Forensic Engineer and the collision researcher, knowledge regarding
vehicle velocities assists with determining factors related to collision causation,
synchronicity, and contributions arising from the human-vehicle-roadway-environment
interface. However, unless at-scene field investigators responding to a collision event are
aware of the types of collision scene data necessary, planar dynamics trajectory-based
analysis may not be feasible. Therefore, the need for a reliable, accurate and broadly
applicable generalized model based upon vehicle deformation is necessary for a thorough
analysis of many real-world vehicle collision events.
The advent of vehicle Event Data Recorders (EDR) in light trucks, vans and cars,
along with Electronic Control Modules (ECM) in heavy vehicles has come a considerable
way towards providing instrumented data for real-world collision events. Telematics and
GPS tracking systems provide some data regarding position and vehicle speeds, but do not
provide high enough resolution for those interested in studying collision forces and
severity. However, at the time of this study, not every vehicle is equipped with an EDR,
ECM, or GPS tracking system. Additionally, the uniformity of the data recorded by even
the best systems available remain insufficiently reliable enough to provide the data
necessary for all types of collision events and combinations of colliding vehicles.
Therefore, for at least the foreseeable future the need for a reliable, accurate and broadly
applicable generalized model for vehicle deformation analysis is necessary for the proper
study of many real-world vehicle collision events.
5


Currently accepted practices for motor vehicle collision-related deformation
analysis have developed into reasonably accurate, reliable and commonly used
methodologies for determining collision severity levels and collision velocities. However,
applicability of the current models require proper structural stiffness values for each
colliding vehicle, characteristic not only of the vehicles analyzed but also specifically to
the impacted surface. Fairly extensive test data from vehicle manufacturers and test
laboratories provide frontal stiffness coefficients for many passenger cars and light trucks.
A much smaller number of tests support deformation analysis regarding side and rear
surface stiffness coefficients. Additionally, only a limited number of heavy vehicle frontal
barrier impacts, to include semi-tractors and buses, support frontal stiffness coefficient
determination. Accordingly, either conducting additional expensive and time consuming
full-scale barrier impact tests must fill the data gaps, or more realistically, another reliable,
accurate and broadly applicable generalized method provides the means for analyzing
collisions between passenger vehicle and non-passenger vehicle impacts.
1.2 Statement of Study Objectives
In response to the current shortfalls in modem vehicle deformation analysis
techniques, the primary objectives of this study are to provide the following advancements
to the current body of knowledge regarding vehicular impact deformation analysis:
Develop reliable, accurate and broadly applicable generalized vehicle
deformation methodologies eliminating the dependence on multiple structural
stiffness coefficients, regardless of the impacted surface and vehicle type
involved.
Develop and incorporate inter-vehicular non-conservative frictional forces due
to the colliding surfaces of vehicles sliding during the approach velocity change
of an impact into a reliable, accurate and broadly applicable generalized vehicle
deformation model.
Develop numerical algorithms allowing for input of deformation depths and
widths at intervals which more accurately explain and follow the unique
deformation profile of a collision involved vehicle, thereby eliminating the
6


trapezoidal rule reliance upon evenly spaced deformation measurements of 2, 4
or 6 intervals.
Establish important relationships regarding impact forces as they relate to motor
vehicle collisions and vehicle deformation properties.
Provide future researchers with enhanced analytical tools necessary for the
analysis of traffic collision events for the purpose of enhancing traffic safety,
collision dynamics and vehicle design, as well as crashworthiness and occupant
protection.
Further the body of knowledge regarding the behavior of motor vehicles during
real-world collision events.
During the process of developing the generalized analysis methods within this
study, the following advancements from the past 30 years are incorporated into forming
the final, comprehensive generalized methods for determining impact velocity change:
Consideration of external impulses to the impact produced by tire-ground non-
conservative forces during the approach velocity change of an impact.
Consideration of rotational effects produced by oblique or offset collisions that
result in principle directions of force that do not pass through the mass centers
of vehicles.
When developing any engineering model, a balance between a comprehensive
consideration of every feasible condition or potential variable versus a simplistic model
that depends upon only a few discrete and easily determined variables must be achieved.
The risk of an overly comprehensive model lies in the dependence upon too many input
parameters. Too comprehensive of a model becomes narrowly applicable to only a few
conditions where the model variables are determinable. An approach that is too
comprehensive can create undue analysis complexity, not to mention the potential for
multiple sources of random error due to variability within or between input parameters.
However, too simplistic of a model likewise becomes narrowly applicable and may not
take into consideration all of the main factors that contribute to the event or process
modeled, thus introducing systematic error to the analysis.
History reveals that with the development of any analysis methodology or scientific
discovery, there will be individuals whom will criticize and refuse to accept results due to
7


personal biases or lack of understanding of the methodological principles. For these
reasons, the generalized models developed and presented in this study were tested using
industry standard collision analysis validation tests (RICSAC)[100], Additionally, the
developed models were tested against other independently reported real-world vehicle
impact data (NASS) [57], In this way, the engineering community should have the
necessary information to evaluate the accuracy, precision and efficacy of this study.
Chapter 5 presents the evaluation of the generalized and comprehensive vehicle
deformation analysis methods developed during the course and scope of this study. The
study culminates into reliable, accurate and broadly applicable generalized vehicle
deformation algorithms that do not require the reliance upon multiple variables or analysis
parameters that are often unknown or unknowable for real-world collision conditions.
The development of reliable, accurate and broadly applicable generalized models
sufficiently comprehensive in nature so as not to over-simplify collision dynamics, but
straightforward and practical enough to consider the important known or knowable
variables regarding both controlled environment and real-world collision events, form the
overwhelming impetus of this study.
8


CHAPTER 2: LITERATURE RESEARCH
2.1 Background of Collision Analysis
2.1.1 Basic Collision Analysis Methods
The knowledge of vehicle collision velocities may be helpful in determining factors
related to collision causation and timing for both criminal and civil liability issues, and
research interests in collision avoidance and mitigation. The determination of velocity
change, peak acceleration, and impulse applications during a collision event are crucial to
understanding collision severity. Determining these values assists in the evaluation of
occupant kinematics, injury potential and injury severity studies, as well as provide
valuable data for research regarding occupant protection systems. However, unless field
investigators responding to a collision event are aware of the necessary data for a trajectory-
based collision analysis and record the data promptly, a trajectory-based analysis may not
be feasible.
The lack of properly documented scene data remains the common conditional
limitation for engineers and researchers called upon to analyze collision events days,
months or even years after the event occurred. Therefore, the need for a reliable, accurate
and broadly applicable generalized analysis model outside of the traditional trajectory-
based momentum analysis is necessary for many real-world vehicle collision events.
2.1.2 Passenger Vehicle Event Data Recording Systems
On-board event data recording (EDR) systems as part of airbag system controls,
record data that may provide speeds for finite periods of time at a low resolution of usually
1 to 2 Hz leading up to the impact (depending upon model year of the vehicle). Some
vehicle EDR systems may capture imited acceleration and velocity change vector data from
an impact event. The National Highway Traffic Safety Administration (NHTSA) final rule
on 49CFR Part 563 specifies the uniform minimum requirements for accuracy, collection,
storage, survivability and ability to image data from onboard motor vehicle collision EDR
systems.
9


It is important to note Part 561 does not require vehicles to have EDRs or EDR-like
devices. Instead, Part 561 only specifies requirements for what must be recorded and in
what manner should the vehicle have an EDR or EDR-like device. The intent of the final
rule initially mandated compliance of vehicles manufactured for sale in the United States
on or after September 1, 2010, later extended to September 1, 2012.
The following are Tables I and II from the rule that provide the data elements
required for all light vehicles equipped with an EDR (tables extracted from 49 CFR Part
561). [1]
Table 2.1 Reproduced Table I from 49CFR Part 561
TABLE I DATA ELEMENTS REQUIRED FOR ALL VEHICLES EQUIPPED
WITH AN EDR
Data Element Recording Interval / Time1 (Relative to time zero) Data Sample Rate Samples per Second
Delta-V, longitudinal 0 to 250 ms 100
Maximum delta-V, longitudinal 0-300 ms n_a.
Time, maximum delta-V 0-300 ms oa.
Speed, vehicle indicated -5.0 to 0 sec 2
Engine throttle, % full (or accelerator pedal, % full) -5.0 to 0 sec 2
Service brake, on/off -5.0 to 0 sec 2
Ignition cycle, crash -1.0 sec n.a.
Ignition cycle, download At time of download n.a.
Safety belt status, driver -1.0 sec n.a.
Frontal air bag warning lamp, on/off -1.0 sec a a.
Frontal air bag deployment, time to deploy, in the case of a single stage air bag, or time to first stage deployment, in the case of a multi-stage airbag, driver Event n.a.
Frontal air bag deployment, time to deploy, in the case of a single stage air bag, or time to first stage deployment, in the case of a multi-stage air bag, right front passenger Event n.a.
10


Table 2.1 continued
Multi-event, number of events (1,2) Event na
Time from event 1 to 2 As needed n.a.
Complete file recorded (yes, no) Following other data na.
1 Pre-crash data and crash data are asynchronous. The sample time accuracy
requirement for pre-crash time is -0.1 to 1.0 sec (e.g., T = -1 would need to occur
between1.1 and0 seconds.)
Table 2.2 Reproduced Table II from 49CFR Part 561
TABLE H-DATA ELEMENTS REQUIRED FOR VEHICLES UNDER
SPECIFIED CONDITIONS
Data Element Name Condition for Requir ement Recording Interval/ Time1 (Relative to time zero) Data Sample Rate (Per Second)
Lateral acceleration If recorded* 0-250 ms 500
Longitudinal acceleration If recorded 0-250 ms 500
Normal acceleration If recorded 0-250 ms 500
Delta-V, lateral If recorded 0-250 ms 100
Maximum delta-V, lateral If recorded 0-300 ms n.a.
Time maximum delta-V, lateral If recorded 0-300 ms n_a.
Time for maximum delta-V, resultant If recorded 0-300 ms n_a.
Engine rpm If recorded -5.0 to 0 sec 2
Vehicle roll angle If recorded -1.0 up to 5.0 sec3 10
ABS activity" (engaged, non-engaged) If recorded -5.0 to 0 sec 2
Stability control (on, off, engaged) If recorded -5.0 to 0 sec 2
Steering input If recorded -5.0 to 0 sec 2
Safety belt status, right front passenger (buckled, not buckled) If recorded -1.0 sec n.a.
Frontal air bag suppression switch status, right front If recorded -1.0 sec n_a.
11


Table 2.2 continued
passenger (on, off, or auto)
Frontal air bag deployment, time to n& stage, driver4 If equipped with a drivers frontal air bag with a multi- stage inflator. Event n.a.
Frontal air bag deployment, time to n& stage, right front 4 passenger If equipped with a right front passengers frontal air bag with a multi-stage inflator. Event n.a.
Frontal air bag deployment, n1*1 stage disposal, driver, Y/N (whether the nth stage deployment was for occupant restraint or propellant disposal purposes) If recorded Event n.a.
Frontal air bag deployment, n* stage disposal, right front passenger, Y/N (whether the nth stage deployment was for occupant restraint or propellant disposal purposes) If recorded Event n.a.
Side air bag deployment, time to deploy, driver If recorded Event n.a.
Side air bag deployment, time to deploy, right front passenger If recorded Event n.a.
Side curtain/tube air bag deployment, time to deploy, driver side If recorded Event n.a.
Side curtain/tube air bag deployment, time to deploy, right side If recorded Event n.a.
Pretensioner deployment, time to fire, driver If recorded Event n.a.
Pretensioner deployment, time to fire, right front passenger If recorded Event n.a.
Seat track position switch, foremost, status, driver If recorded -1.0 sec n_a.
Seat track position If recorded -1.0 sec n.a.
12


Table 2.2 continued
switch, foremost, status, right front passenger
Occupant size classification, driver If recorded -1.0 sec n.a.
Occupant size classification, right front passenger If recorded -1.0 sec ii a
switch, foremost, status, right front passenger
Occupant size classification, driver If recorded -1.0 sec n.a.
Occupant size classification, right front passenger If recorded -1.0 sec ii a
Occupant position classification, driver If recorded -1.0 sec ii a
Occupant position classification, right front passenger If recorded -1.0 sec ii a
1 Pre-crash data and crash data are asynchronous. The sample time accuracy
requirement for pre-crash time is -0.1 to 1.0 sec (e.g. T = -1 would need to occur between
1.1 and 0 seconds.)
If recorded means if the data is recorded in non-volatile memory for the
purpose of subsequent downloading.
3 vehicle roll angle may be recorded in any time duration, -1.0 sec to 5.0 sec is
suggested.
4 List this element n-1 times, once for each stage of a multi-stage air bag system.
2.1.3 Heavy Vehicle Event Data Recording Systems
Engine manufacturers have equipped many heavy commercial vehicles with event
data recording systems through the engine electronic control systems that may provide
some collision related data, usually at low resolutions of around 1Hz. SAE J2728 is a
recommended practice document published by the Society of Automotive Engineers that
applies to Heavy Vehicle Event Data Recorders (HVEDR). SAE J2728 applies to heavy-
duty (HD) ground wheeled vehicles over 4545 kg (10,000 ETS pounds), commonly referred
to as Class 3-8, which are intended to be compliant with current Federal Motor Vehicle
Safety Standards (FMVSS) and/or Federal Motor Carrier Safety Regulations (FMCSR).
J2728 defines the term heavy vehicle as a motor vehicle equipped with vehicle
communication networks SAE J1708/J1587 and or SAE J1939. J2728 focuses primarily
13


on wheeled ground vehicles with standard on-board power supplies (batteries). The intent
of J2728 is to address the needs of OEM (original equipment manufacturer) original, OEM
modified/adaptive, and non-OEM aftermarket systems and does not specifically exclude
trailers and similar non-engine powered vehicles. The following set of tables from J2728
outline the performance requirements of this recommended practice [2] [3],
Table 2.3 Reproduced Table 1 from SAE J2728, June 2010
TABLE 1 LIST OF DATA ELEMENTS
Data Etm*nt Description Continent; Example; Alternate element names Conflg item Recorded
Alternate vehicle ID vehicle-unique, alpna-numene dentifier substitute tor the- vin . =cr mase situations where a standard VIN is unavalable, not access, tile, not req ured. or has been Swinged (e.g.,as on happen with a solvate tile), then Hie HVEDR System shall utilize a veftlcle-umque. ilbhanumenc idenbTier substitute for the VIN. User Header
Event Data Recording Complete This data indicates whether or not a romplele set of data hat tie event data recording device Is designed to capture was successtoiy recorded By snd stored in die device. MD Header
Event Date Tre dale when the event occutred. Date MM'DCmYY MO Header
Event Time Tlie time wtien the event occurred Time HH I.1M.SS GMT: 2WLf dock me HVEDR mus1 provide Its own real-time clock tababiilv. indudiiv) battery backup Mo Header
HVEDR Make vianulacturer name for hvedr. Mir Header
HVEDR Model Model number lor HVEDR Mir Header
HVEDR Serial Number Serial number tor HVEDR MtT Header
Pie-event Buffet Size iSamples} Defines how many dala sarnies ae stored in the pre-event buffer. Tier 1 minimum sample rate is 10 Hz, for total cd 15 s, therefore 150 samples MIT Header
^i-Eveni Buffer size iSamcfest Delines how many dala samples are stored Irtlhe cost-event buffer. ner 1 minimum sample rate Is 10 Hz. for total of 15 s. .herefore 150 samples MIT Header
Rear Axte Ratio ^lu d Transmasm output shaft soeed lo Tire revolution rote User Header
Tire see Tire size in Resotunons dm hm. user Hewer
Total Event Records HVEDR Supports Total number of event records the -TVEDR supports in non-vofalle Ttemorv. Mir Header
Trigger Thresholds Jststhe currently conllgured Ingger nresnoldfs). 2 or mare sub-stnngs corUaning the tolowng data items Trigger Data Item Trigger comparison v, eic.) Trigger Threshold Value] Tngger thresholds tomwitea as sent colon (:) delimited 1st At mmmum, wll contain one threshold for acceleration rigger and cos threshold tor last step iriqqer MIT/User Header
Trigger Threshold Activated radicates which Trigger Threshold .vas adrvaied to cause the recording he event. SubElring from Trigga Thresholds fer adnrated lugger No Header
Trigger Threshold Count radicates now many ingger firesholds the device has beer xntvni.ired with Calculated value based on Trigger Threshold^. Cainl>- Z No Header
VIM indicates the Vehicle Identification '(umber (VIN) assigned by the vehicle manufacturer. 3iD. 237, mid. Varies by rr% Transmission Rate: on request PCM 65260 (pre-ZOtD) Engine orfy for pcst-20 10 Transmission Rate: on request /IN Mil nol be reported lo otrer ECUs by HVEDR, but tvl be provided by the HVEDR to the Extraction Tool Mfr/ User Header
Veh cle Configuration A Iree-rorm text held for vehicle rfnflni iratinn Mir; User Fooler
ABS Retarder Status ndicates the status d the ASS Retarder. ABS Retarder Statue MO Pre-event Post- ewsil
ABS Brake Control Status Trader ndicates u>e status oT the ABS Brake xntrol system on the vehicle/trador, active or not active. ABS Brake Control ABS Retarder Control (SAEJ1587 only) Mo Rre-everit Post- event
ABS Warning Lamp Status Tractor rwicatw me status d the abs warning light on he vetiicleitcmtcc nn or off ABS Warning Lamp No pre-event Post- event
14


Table 2.3 continued
SAE________________________J2728 Issued JUN2010____________________Page 9 of 48
Data Clement Description Comment 1 Example f Alternate element names Confia item Recorded
4BS Brafce Control status Trailer Indicates the statu s of the A6S Brake control system on Trailers), active or not active. Acbve if ABB Brake control is active foranv trailer. ABS Brake Control it active noun iwktipie trailers, warrants, additional irwestigallon. No Pne-eirem Post- event
4BS Warning Lamp Status Trailer Indicates the status orihe ABS naming tigf* on Tralerfs). on or off. On ir ABS warning ligru is on lor any trailer ABS Warning Lamp If cn whh multple tralers. warrants additional invesbgalian. No Pre-evenl PUS1- evenl
fl.ccelerok* Pedal Position Percentage ratio o* Che Chrome pedol opening (drivers operation) Accelerator Pedal Position 1 No Pro-event Pcst- evenl
Brane Status- Earning Indicates the slatus or Ihe switch that installed to detect whether or noi the caiklna brake has been acclied Parking Brake swan NO Pre-evenl Pcst- evenl
Brake Staius- Service Indicates the slaSus or Ihe switch that is rhstaaed in bra*e system to detect whether the service brake has been applied. This switch is usually used to turn on the brake lamps service Brake Peak NO Pre-event Pcst- evenl
Clinch Switch Indicates the stalus of the switch that ts usually installed m or connected to ihe clutch pedal to detect whether or not the clutch pedal is depressed. NO Pre-evenl PCSt- evenl
Cruise Control Active Indicates whether Cruise Control is active NO Prc-cvcnl PCSt- etrem
Cruise Control Set- Speed The speed to which the Cruise Control is set No Pre-event Pcst- eveni
Cruise Ccmfrcl states The cunent sta&e, or mode, or operation by the cruise control device No Pre-eveni Pcst- event
Engine Hours Number of hours that the engine has been operated irom tiitre of control uni fir si use la 1he lime or the event lrlqqer Engine Total Hours of Operation NO Pre-event Pcst- eirenl
Engine Retortte* Percent Torque Braking torque orihe relarder as a percent or retarder coftiiguratioa reference torque Engine Retarder Percent (SAE J1667) Actual Retarder Percent Torque (SAE J1939) NO Pre-eveM Post- eirem
Engine Retarder Stalus indicates the status or Ihe Engine Retarder For SAE J1S39 networks it may provide retarder data Tor odier coni do hems Engine Retarder stalus (SAE J1587) Retarder EnaNe Brake Assist Swncti Engine Speed Rotaiionai speed or me engine output snan. NO Pre-evenl PCSt- evem
Event Buffer Number In systems with multiple buffers, this will Hanliry which buffer the even! dalo cam? horn NO Pre-event Pcst- evenl
Transmisscn Geer Currant Tranamsslcn Gear NO Pre-eveni Pcst- evenl
Two Speed Axle switch Indication of current axle slate. Axfe Range Indication! NO Pre-evem Pcst- evem
Total Vehicle Distance Accumulated distance traveled try vehicle at (he Ume of the event lrlqqer Total Veh.de Distance No Pre-event Pcst- evem
Vehicle Speed [Preferred souce for ^eTiicle speed} The long itudinal speed of the vehicle that is calculated or estimated from Ihe vehicle speed sensor (VSS). Also referred to as Road Speed in scene standards. Note that the V33 is not always an accurate representation or the velocity of the chassis relative to ihe around No Pre-mmt Pcst- evem
Wheel Based Vehicle Speed ABB Wheel Based Vehicle Speed. Vehicle Speed No ?re-event Post- event
The difficulty with heavy vehicle EDR (HVEDR) systems, aside from the absence
of mandatory requirements for recordable HVEDR systems, sets squarely within the lack
of uniformity in data recorded, if any, and how or whether the data can be commercially or
15


privately imaged for analysis. Additionally, HVEDR systems are secondary to the control
and operation of the various control systems for the engine and EPA required emissions
systems. HVEDR system may also monitor operational controls of ancillary components
of the drivetrain and braking systems, as outlined in the previous tables from J2728.
HVEDR data availability is also dependent on the year and engine manufacturer rather than
the chassis model of heavy vehicle. As such, each engine manufacturer will have a different
electronic control module (ECM) or a combination of modules (CPC, MCU, ACU), which
may or may not have HVEDR capability. Imaging of some HVEDR modules takes place
through the main communication data link port or through a direct connection with the
module if configured as a single ECM and HVEDR unit. However, the design of HVEDR
modules as a combination of different control units functioning together makes imaging
from the main communication data link port imperative in order to eliminate false fault
codes generated if the system does not remain connected. Removing each control unit of
the engine control system from the vehicle and connecting the units with specialized
harnesses while imaging the HVEDR and control systems can eliminate false fault codes.
Connecting the HVEDR component of a collision involved heavy vehicle to a surrogate
vehicle for download is the last option to image data, but may bypass fault codes that would
have otherwise been present on the collision-involved vehicle. Regardless of the HVEDR
data imaging process, expensive and specialized CAN bus communication systems, cables
and software are necessary to access and image any data stored.
Even if a commercial vehicle is equipped with a HVEDR systems that provides
event triggers such as sudden acceleration events, last stop records or safety restraint
system triggers, there is no guarantee that data imaged is related to a particular collision of
interest without proper analysis or study of the record. Even then, some collision events
that may be catastrophic to a passenger vehicle colliding with a heavy vehicle may be
insufficient to produce an accelerating event trigger for the much more massive heavy
vehicle involved in the event. As such, the development of engineering analysis methods
applicable to large vehicles has become necessary for the determination of collision
velocities and severity levels produced by impact events.
16


2.1.4 Vehicle Tracking Systems
Commercial vehicle dispatchers have traditionally used telematics systems to track
the progress of shipments or passengers. Some Telematic systems provide not only GPS
coordinate locations but GPS speeds of vehicles along a travel path, with some advanced
systems even providing flash updates of vehicle maintenance schedules and monitored
systems within the vehicle drivetrain. A telematics system could randomly collect vehicle
data at frequencies of several seconds to even hours depending upon the system
configuration and service plan. Some systems gather data at specified checkpoints along a
route or transportation corridor, or at higher frequency intervals for local operations. The
highest typical resolution for telematics systems used by local delivery, transit, refuse or
other similar commercial operations record position, direction and speed at up to 1 Hz,
again depending upon the system used.
Many modern passenger vehicles have GPS navigation systems, or even onboard
GPS alert systems capable of tracking locations, provide speeds and warn a central
monitoring location of an emergency event such as an airbag deployment. Many onboard
systems actively monitor reported stolen vehicles, so as to provide law enforcement with
the location or even a description of the occupants of a stolen vehicle if equipped with
video monitoring. Such system also allows a monitoring station to control the drivers
ability to control the vehicle, as well as lock the doors to prevent escape once the vehicle
is stopped, and power terminated.
However, the vast majority of passenger vehicle telematics and GPS systems, to
include portable GPS systems, sample the position, distance and speed of a vehicle at no
greater than 1 Hz. Many systems would not store data unless an active route guidance
operation was functioning at the time of a collision event. With the relatively low
resolution, lack of uniformity in telematics and GPS information, and relative lack of wide
use of these systems, the need for broadly applicable generalized engineering analysis
methods applicable to large and small vehicles is again necessary for the determination of
collision velocities and severity levels.
17


2.2 Vehicle Deformation Analysis
2.2.1 Pioneering Studies and Computer Programs
Permanent vehicle collision related deformation analysis has been part of technical
collision investigations since the 1950's. In 1952, the Automobile Crash Injury Research
Program (ACIR) was initiated. The purpose of ACIR was in documenting injury causation
factors for vehicle occupants involved in traffic collisions, with the primary objective of
the program being that of injury mitigation and prevention. By the mid-1960s, program
participants consisted of 31 states providing over 50,000 cases for further research and
analysis. The collision damage analysis methods of this early study consisted of comparing
the permanent deformations of colliding vehicles resulting from a subject traffic collision
event to similar damages from tests conducted at known impact speeds. These methods
produced results that were rarely entirely representative of a particular collision event and
were justly considered only first approximation methods. In time, the anecdotal approach
to collision severity analysis was finally discarded altogether by properly trained analysts.
[4]
In September of 1966, President Lyndon Johnson signed the National Traffic and
Motor Vehicle Safety Act and the National Highway Safety Act. This action established
the authority to develop both the Federal Motor Vehicle Safety Standards (FMVSS) and
the National Traffic Safety Agency, now known as the National Highway Traffic Safety
Administration (NHTSA). During the signing of these acts President Johnson stated, Auto
accidents are the biggest cause of death and injury among Americans under 35. In 1965,
50,000 people were killed on the nations highways in automobile collisions. [4]
Work done by Campbell in the early 1970's developed the earliest analytical
approaches focusing on vehicle damage. [5] Campbell observed a linear relationship
between fixed barrier impact speeds and residual deformation of a vehicle structure during
full-scale impact testing using General Motors vehicles. Campbell recognized a linear
collision force relationship between the resistance of a vehicle structure to deformation per
unit width and the residual measured deformation. The linear relationship equates to the
following equation relating average deformation depth and width to a very simplistic
idealized spring model:
18


Figure 2.1 Linear approximation of collision force (adapted from Figure 2.3[6])
w ( B C2 ^
E = H A- CH------vG dw CampbellsEquation 2.1
Where, A, B and G are stiffness factors for a vehicle group or category
C = Average crush or damage depth across contact damage region
w = overall width of contact damage region
McHenry and others at Cornell Aeronautical Lab (currently known as CALSPAN)
conducted further research in the development of SMAC (Simulation Model for
Automobile Collisions) which improved upon Campbell's earlier observations.
Specifically, McHenry also noted that like Campbells initial discovery, vehicles behave
like linear energy dissipating springs. McHenry developed equations to consider the energy
principles developed by Campbell relating them to the kinetic energy change produced by
a plastic impact between colliding vehicles as they relate to the resultant vehicle velocity
changes. Later adaptations of the work by Campbell and McHenry resulted in a Fortran-
based mainframe computer analysis program known as the CALSPAN Reconstruction of
Accident Speeds on Highways, or CRASH. CRASH provided a first approximation of
vehicle velocity change for input into the more detailed momentum based SMAC
(Simulation Model for Automobile Collisions) computer analysis program, which was
19


intended to predict impact speeds. CRASH III was the last CALSPAN produced edition of
the CRASH program. [6] [7] [8]
CRASH HI and its predecessor versions restricted the measurement of damage
profiles to 2, 4 or 6 evenly spaced deformation depth measurements over the contact
damage width on a vehicle surface. The trapezoidal rule was utilized to estimate the
damaged area over these arbitrary widths and evenly spaced measurement intervals. The
following formulation determined the deformation work for a damage profile of two, four
and six evenly spaced deformation depths:
Figure 2.2 Damage profile approximation (adapted from Figure 2.5 [6])
Two measurement points:
E = L
A
2
Cy CyC'y
2.2
Four measurement points:
3
(q + 2c2 + 2c3 +c4) +
B
2.3
H---+ 2C2 + 2C^ + C4 C^C2 + ^2^3 ^3^4 )
20


Six measurement points:
5 B Cj + 2c2 + 2c3 + 2c4 + 2cs + +
6 v.+cic2 + c2c3 + c3c4 + c4c5 + c5c6 y
2.4
The trapezoidal methodology was troublesome in the fact that vehicles involved in
collision events outside of full overlap barrier tests rarely exhibited damage profiles
effectively measured using evenly spaced intervals. The resulting error in deformation
measurements resulted in over or under-approximations of damage energy determinations.
Several commercial versions became available over the approximate 40 years since the
original study by Campbell. However, with only marginal improvements in many of the
restrictions of the CRASH III program, and a continued reliance on the trapezoidal rule for
deformation energy determination.
The full version of the CRASH III and SMAC programs calculate impact speed and
velocity change for the crash vehicle, or vehicles, using both damage-based and
momentum-based priciples. With only the knowledge of vehicle damage profiles, CRASH
III can only determine velocity change of two impacting vehicles or the barrier equivalent
velocity (speed related to impacting a fixed barrier) of a single vehicle. An important
assumption of the CRASH III program is that the energy dissipated through work during
the approach phase of an impact can be approximated from the residual deformation
measurements using the trapezoidal methodology. The following are the basic equations
for determining the velocity change of colliding vehicles:
Current CRASH-based programs cannot determine vehicle impact speeds. Post-
impact trajectory histories are needed to determine impact speeds using a planar
momentum-based analysis once the velocity changes for the vehicles were determined
using the CRASH-based algorithms. The CRASH III limits analysis to two vehicles
involved in a single impact, or one vehicle colliding with a fixed and non-yielding object.
2.5
2.6
21


In the mid-1980s McHenry et al., proposed upgrades to the CRASH III algorithms to
include restitution effects to allow for a broader application of the CRASH algorithms to
low velocity impact events. A CRASH IV version, which was never adopted or completed,
was intended to consider restitution effects. [9] The inclusion of restitution effects allowed
for the consideration of the velocity change elements that resulted during the separation
phase of an impact. McHenry proposed the following equations that originally appeared in
the 1981 edition of the CRASH III Users Manual:
EtOTAL
'Crush
yy I ^EtotaiJ 2
(ml (nil + m2 ]
yy I ^^TOTAL^h
Vm2(m1+m2)
2.7
2.8
Due to the limitations of a Fortran-based punch card process for 1970s and 1980s
vintage main-frame computations, damage profiles remained limited to even numbers of
uniformly spaced deformation depths of either 2, 4 or 6 measurements in order to save
computing time. The establishment of damage measurement protocols standardized the
measurements and interpretation of damages for CRASH related program use. The
differentiation between contact and induced damages was established. Protocols addressed
how to account for effects of bumper system underride/override. Additionally
considerations were made accounting for bowing of a vehicle structure that typically occurs
for high-speed lateral impacts, or shifting of front and rear structures due to high-speed
oblique collisions. [10] Many of the original measurement protocols remain applicable.
However, one objective of the current study is to eliminate the arbitrary evenly spaced
measurement protocol in lieu of measurement that accurately describe damage shape with
as many or as few measurements and at any width interval necessary to adequately describe
the deformation to a vehicle structure.
A significant limitation of the CRASH III program rested on the assumption that
all vehicles within discrete wheelbase dimensions possessed the same dimensional, inertial
and structural stiffness characteristics. Tabulated dimensional, inertial and stiffness data
into separate vehicle categories within internal default data tables were used during the
analysis process. However, there is no apparent justification for this arbitrary grouping by
22


wheelbase, and testing has shown that wide variations of such properties exist between and
within vehicles of similar wheelbase ranges. In particular, the assigned rear-end and side
vehicle stiffness coefficients were incorrectly determined and do not resemble actual rear-
end or side stiffness characteristics of production vehicles.
In an attempt to evaluate the velocity change from rotation, the CRASH III
algorithms incorporated an effective mass concept to account for rotational effects resulting
from the moment about the vehicle mass center created by an offset application of the
principle direction of force. An offset application of the principle direction of force, or
PDOF, occurs during oblique and/or offset collisions. The effective mass concept has been
demonstrated to produce more reliable estimates of velocity change since its introduction.
[11] [12][13]
In general, the limiting assumptions and analytical constraints of CRASH III and
similar CRASH-based programs are as follows: [14] [15] [16]
Deformation energy is equal to the impact kinetic energy loss.
Collisions are inelastic, and the centroids of damage reach a common velocity.
Sliding between vehicles occurs during the separation phase of the impact and not
during the approach velocity change phase, and, therefore, is not accounted for in
the velocity change analysis.
Tire-ground forces are negligible (non-conservative forces external to the impact)
or minuscule as compared to the collision force.
Damage profile measurements are limited by evenly spaced measurements of 2, 4
or 6 deformation depths over uniform spaced measurement widths across the
contact damage width, excluding induced damage regions.
Vehicle structural stiffness defined by categories of vehicles by type (i.e., car, truck,
van), and wheelbase lengths, all assumed to have similar inertial and structural
stiffness characteristics.
2.2.2 Modern Damage Deformation Studies
In the 1990s topics regarding vehicle deformation analysis again became a new
focus of research. Largely due to the power of personal computing and an increased market
for vehicle collision analysis software, researchers again focused on improvements
23


augmented by the already accepted engineering fundamentals that comprise the
foundations of vehicle deformation analysis. One of the original and primary research
focuses surrounded the elimination of arbitrary groupings of A and B stiffness coefficients
by wheelbase and/or vehicle classification. CRASH-based analysis advanced towards
relying on a robust library of vehicle and surface specific structural stiffness coefficients
for many cars. However, as extensive the library to date may be, many vehicles have either
a dearth of structural stiffness information or none whatsoever. In response to the database
deficiencies, studies also developed methods for determining crash specific stiffness
coefficients. Many of these attempts rely upon simplifying assumptions to complete a
deformation-based analysis when stiffness data was not otherwise available for one of the
involved vehicles. [4][15] [17] [18] [19]
Additional research spanning from the 1990s to present day have provided further
adaptations of the original CRASH-based algorithms. The formulation of reliable
analytical tools for determining the velocity change and peak acceleration levels of minor
damage, low-velocity impacts (AV <16 kph (10 mph)) became a major focus. Velocity
change equations derived from linear momentum, energy, and restitution principles, also
known as MER methods, considered the contributions of collision restitution for low-
velocity impacts, often ignored for higher velocity impacts. The developed models were
primarily applicable to collinear and central impacts having a PDOF passing through the
mass centers of the involved vehicles. [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
Research models to date have incorporated not only restitution effects but also
tire/ground non-conservative force contributions. This engineer authored multiple research
projects regarding not only the development of collinear low-velocity impact collision
deformation methodologies but also tested the accuracy of the algorithm accuracies against
full-scale vehicle-to-vehicle collision tests. The models developed by this engineer and
[31] [32]
2.9
Afj =^AF2
2.10
m
24


other researchers provide reliable methods for predicting the velocity change and peak
acceleration levels of primarily collinear central impacts, or minor impact events where the
struck vehicle is stationary at impact.
Currently accepted practices for motor vehicle deformation analysis have
developed into reasonably accurate, reliable and commonly used techniques for predicting
collision severity levels and collision velocities, dependent upon the availability of
structural stiffness values for each colliding vehicle. Although extensive test data is
presently available for frontal stiffness coefficients for many cars and light trucks, few tests
are available for side and rear surface vehicle specific stiffness coefficients for any vehicle
type. Additionally, only a limited number of heavy vehicle frontal barrier impacts, to
include semi-tractors and buses, are available for frontal stiffness coefficient
determination, and none from the side or rear structures of these unique vehicles.
Accordingly, either additional expensive and time-consuming barrier tests are needed to
fill these gaps, or another method of analysis must be developed.
25


CHAPTER 3: COLLINEAR CENTRAL IMPACTS; DEVELOPING BASIC
DEFORMATION ANALYSIS PRINCIPLES
To achieve the objectives of this study, it becomes necessary to derive the
foundational principles behind vehicle deformation analysis. The single-degree-of-
freedom collinear collision represents the most simplified collision event, as well as the
most logical starting point. Derivation and presentation of the fundamental principles allow
for the expansion of the physics behind deformation analysis to account for more complex
and generalized formulations, the overwhelming impetus of this study.
3.1 Coordinate System
A vehicle-fixed coordinate system is commonly used to describe motion in vehicle
dynamics. SAE J670 establishes a commonly used vehicle-fixed coordinate system as
shown in Figure 3.1(a) [33], The vehicle-fixed coordinate system follows the right-hand-
rule; positive x-axis oriented from the vehicle mass center forward (longitudinal axis), the
positive y-axis from the vehicle mass center and towards the right side (passenger side) of
the vehicle (lateral axis), and the positive z-axis from the vehicle mass center and
downward (vertical axis). Rotation about the x-axis towards the right side of the vehicle
produces positive roll (y xz=p). Rotation about the y-axis that results in raising the front
of the vehicle produces positive pitch (z x x = q). Clockwise rotation about the z-axis
produces positive yaw (xxj = r).
The SAE vehicle-fixed coordinate system, however, creates confusion when
analyzing collision dynamics. Collision analysis typically utilizes an inertial earth-fixed
Cartesian coordinate systems with a conceptually more logical vertical (+) z-axis oriented
upwards from the earths surface. An earth-fixed vehicle local coordinate system has the
(+) vertical axis pointing upwards from the vehicles center of mass. By applying the right-
hand rule, the (+) longitudinal x-axis orients towards the vehicle front and the lateral (+)>-
axis towards the drivers side or left side of the vehicle. The earth-fixed coordinate system
has its roots in aviation, where pilots prefer to measure altitude with positive rather than
negative z-axis values. For this same reasoning, an earth-fixed oriented local coordinate
system for a vehicle has wide acceptance for motor vehicle collision analysis. This study
26


will utilize the earth-fixed oriented vehicle coordinate system as shown in Figure 3.1(b)
unless otherwise specified.
Figure 3.1(a) SAE Conventional vehicle coordinate system
Figure 3.1(b) Earth-fixed oriented vehicle coordinate system
3.2 Equation of Motion for Collinear Central Impacts
The basic equation of motion for a simple single-degree-of-freedom, collinear and
central impact between two vehicles assists in developing the basic equations needed for
vehicle deformation analysis. Straight-forward adaptation of the equation of motion from
27


a single degree of freedom, collinear system into a multiple degree-of-freedom, multi-
dimensional system results from recognizing forces, accelerations and velocities are
vectors. However, the vast majority of motor vehicle collisions are easily modeled using
planar dynamic principles without introducing significant error or questionable accuracy
into the analysis results. The basic principles equations developed in this chapter translate
to collision forces acting oblique to the and mass centers of the colliding vehicles, thus
resulting in post-impact rotational effects.
3.2.1 Conservation of Energy
Figure 3.2 depicts a planar, collinear, head-on, single degree of freedom (DOF)
impact event, in which motion only occurs within a single horizontal coordinate of the
system. Figure 3.2 depicts the collision impulse acting along a line between the mass
centers of the colliding vehicles, such that all energy is expended through vehicle
deformation and horizontal motion. Such an impact results in rectilinear motion along only
one generalized coordinate, qi. For rectilinear motion, all other potential coordinates, or
qi+i .v, for the system result in arbitrary constraint equations equal to zero in their
respective directions; i.e., qi= real equation of motion, and qt+i=0 (for i=l,...,2n-l
coordinates). Figure 3.2 could easily depict a collinear rear-end collision event, or a moving
vehicle striking any stationary vehicle in any orientation, as long as the collision impulse
results in a principal direction of force (PDOF) that acts along the mass centers of the
colliding vehicles.
Figure 3.2 Central collinear impact
28


The collision depicted in Figure 3.2 results in a change in the overall kinetic energy,
T, for each vehicle. Additionally, work done by the system in the form of vehicle damage,
or potential energy, V. The work done by the system in creating permanent vehicle
deformation can be effectively modeled as two linear springs in series compressing against
each other while exerting an equal but opposite force in accordance with Newtons third
law. The Conservation of Energy for the collinear, central, single DOF vehicle-to-vehicle
collision system as shown in Figure 3.2 can be represented in a generalized coordinate
system, q: [34][35][49]
Where, mi and m2 are the masses of vehicles 1 and 2
initial ar|d initial are the initial generalized linear velocities of vehicles 1
and 2
q\finalm& q2final are the final generalized linear velocities of vehicles 1
and 2
kl and Jc2 are the spring constants representing each the structural stiffness
for each vehicle
q\ and q2 are the inward deformations to each vehicle resulting from the
impact
Equation 3.1 simply states the kinetic energy brought into the system by two
colliding vehicles is equal to the kinetic energy of the vehicles following the impact, as
well as the potential energy stored in each vehicle structure spring during deformation.
However, potential energy is not actually stored during deformation. Otherwise each
vehicle would rebound back to their original shapes following an impact event. Instead, the
stored energy is converted to work according to the structural resistance characteristics
of each vehicle, producing permanent deformation while the collision impulse exerts
maximum compression upon each vehicle structure.
Applying Hookes Law as stated in Equation 3.2, allows for the relationship
between the maximum deformation of a vehicle structure and impact energy. [35][49]
29


3.2
-Jr rf
spring 2 *'/ spring ^spring
F =-k
is svnnv o 'V
Where, k^ng = spring constant; springs ability to resist changing length
d = linear compression of the spring
3.3.1 Lagranges Equations
Analytical dynamics treats two-vehicle collision systems as a whole, allowing for
the analysis of the system using scalar quantities such as kinetic energies, potential energies
and work when determining the equations of motion for a dynamical system. Powerful
methods of formulating equations of motion for various mechanical systems were
developed by Lagrange (1736-1813) thorugh the application of DAlemberts Principle
expressed in terms of generalized coordinates. The application of Lagranges equations
bypasses and/or eliminates many of the tedious aspects of vector dynamics. Vector
dynamics based directly upon the applications of Newtons second law of motion
concentrates on forces and motions that may not be easily determined form a free-body-
diagram of many dynamical systems.
Lagranges equations are differential equations which consider the total energy of
the system and virtual work instantaneous in time when developing the equations of motion
as they relate to the conservative and non-conservative forces acting on a system. The step-
wise application of Lagranges equations results in the intermediate determination of the
system linear momentum, but also allows for the indirect determination of reaction forces
acting within a dynamic system through the use of Lagrange multipliers. The general form
of Lagranges equations requires the partial differentiation of each term of the
Conservation of Energy statement for each generalized displacement and generalized
velocity of the system, as expressed in Equation 3.1. [34] [35]
f \ f \
d 5L 5L
dt l5(iu
Qc + Qnc
3.3
Where, L = T-V is the Lagrangian, (for i=0,...,n generalized coordinates)
T= kinetic energy
V= potential energy
30


Qc= generalized conservative forces (due to potentials)
Qnc= generalized non-conservative forces
Respective to a simple holonomic impact, the generalized conservative forces result
from the potentials that are expressed by the Lagrangian, and therefore, Qc = 0. Initial
considerations in this chapter assume generalized the non- conservative forces due to inter-
vehicular friction and tire-roadway forces are also zero; Q nc 0. The application of
Lagranges equations to this simple holonomic system starts with the grouping of like terms
for the kinetic energy, T, contributions to the system as follows:
The potential energy, V, contribution to the system is due to deformation of the
involved vehicles, which is the final statement in Equation 3.1 as follows:
The kinetic energy term, T, of Equation 3.4 does not contain any generalized
displacements, or contains no generalized velocities, or qt terms. Solving the Lagrangian with respect to
the first element of Equation 3.3 results in only the partial differentials of the kinetic energy
statement, T, for the system returning non-zero values when differentiated with respect to
generalized velocities, qt, of the system. With respect to the second element of Equation
3.3, only the potential energy statement, V, for the system will return non-zero values when
differentiated with respect to generalized displacements, qu of the system.
3.3.2 Deriving the System Momentum
Solving the partial differential equations of the Lagrangian with respect to the
generalized velocities results in the following:
T = -
3.4
3.5
3.6
31


When evaluating the Lagrangian with respect to the generalized velocities of the
system produces the following useful statement for the generalized linear momentum of
the system:
For a conservative system, Equation 3.7 equals zero and provides the statement of
the Conservation of Linear Momentum for the two-vehicle, single DOF collision system
shown in Figure 3.2. The generalized velocities ai and ai represent the initial
1 initial 1 final
and final velocities of the ith vehicle, respectively. The quantity (qiinitial qifinai) represents
the velocity change due to the exchange of linear momentum, or Ay of the ith vehicle for
the two-vehicle, single DOF collision system. Equation 3.7 can be rearranged to show the
relationship between the linear momentum related velocity changes of a two-vehicle, single
DOF collision system, establishing important impulse-momentum considerations for the
system.
m' J = ~q2fJ 3 8
"vAvi = -"vAv2
3.3.3 Equation of Motion
Solving the time derivative of the generalized momentums for the system produces
the forces with respect to the time-rate-change of linear momentum for the collision event
as follows:
d
5L
(/??l [fi^initial fincd) + m2 initial final)]
=nc-'fif J+m '(<72 ,,tar 3.9
When solving the position partial derivatives, qt, of the Lagrangian for the potential
energy terms of Equation 3.1 yields the impact force with respect to the vehicle
deformations as follows:
32


f \
kbql -k2 q2 3.10
Summing the results of the partial differential equations in accordance with
Equation 3.3 results in the final form of the Lagrange determination of the generalized
equation of motion for a two-vehicle, collinear single DOF holonomic central impact
collision system:
Equation 3.11 is the same equation of motion resulting from a vector dynamics
derivation approach using Newtons second law of motion for this single DOF system (one
coordinate along the In the vast majority of vehicle-to-vehicle collisions it is reasonable to assume that
the initial accelerations for both vehicles are equal to zero. Since impulse has yet to initiate
any action upon either vehicle at initial contact when t = 0, the generalized equation of
motion for a two-vehicle, collinear single DOF collision system as expressed in Equation
3.11 can also be expressed in terms of the time-rate-change of the velocity changes of both
colliding vehicles, and the deformation forces as follows:
where, kl and k2 are the spring constants for each vehicle during deformation
cl and c2 are the inward deformation, or crush, extents for each vehicle
The linear spring terms of Equation 3.11, 3.12 and 3.13 represent a two vehicle
collinear impact that behaves as two linear springs in series under compression.
3.12
3.13
33


3.4 Impulse-Momentum of the System
The following is the general expression of the relationship between force applied
and the change in spring length from Hookes Law for a linear spring: [34][35][49]
F =i J- spring * \ '(3
V spring cy spring
F =t -L impact / ^vehicle Cdamage
3.14
3.14
The spring deformation, Sspring, in Equation 3.14 is analogous to deformation to a
motor vehicle, Cdamage, in Equation 3.15. The spring constant, kspimg, in Equation 3.14
represents a linear springs resistance characteristics to change in overall length during
force application. The spring constant for a motor vehicle, kvehicie, in Equation 3.15 similarly
represents a vehicles structural characteristics for resisting damage deformation when
subjected to the impulse of a collision event. An ideal un-damped linear spring behaves as
a simple harmonic oscillating system. However, a vehicle compresses to its maximum
deformation at t= ^, or 1/4 of a full period of oscillation, and remains damaged under
sufficient force without continuing through an entire period or repetitive periods of
oscillation as shown in Figure 3.3.
Ax = Damage Depth
^final
Vfinal

*
/ 2it
8 = A sin (cot) + B cos (cot)
Figure 3.3 Collision deformation as a linear spring system
Equation 3.13 provides the Newtons second law statement for a conservative
system, in that the sum of the external forces acting upon the system must be equal to the
sum of the conservative forces due to potentials. From the Conservation of Linear
Momentum statements of Equation 3.8, the sum of the momentum changes of the vehicles
must also be equal to zero since momentum is conserved. Therefore, the sum of the forces
34


due to the Hookes Law representation of the deformation forces (potentials or
conservative forces) in Equation 3.13 must also equal zero since momentum is conserved.
From this understanding, the impulse-momentum relationship for the system in terms of
the impact deformation forces are represented in Equation 3.16 with respect to the ith
vehicle of the collison system of Figure 3.3.
mr
Avi
dt
Avi Av2
dt
ffl:
(ki'crk2'C2)-0
r
TF external = 0 = mi"
V
{F Internal-F 2 external) {kvci-ki'ci)
dt
m2"
Av;
dt
t final
J fdt = fi-A/ =mrAv=krc1-At =Jimpuhe 3-15
t initial
Where, tinitial = 0 at the initiation of the collision event
tfind = time at peak impulse of the collision event
final tinitial) = Al = time interval to reach peak impulse
and, J= peak collision impulse
The external forces of a collision increase to their peak values from the initiation of
the impulse at t=initial or 0, until the maximum force is reached at t=final or peak, as
shown in Figure 3.4. The area under the collision pulse curve represents the momentum
change as a result of the collision impulse. The time for impulse to reach its peak, At =
(tfnai- tinitial), provides the necessary information for determining the peak collision
acceleration at the mass centers of the vehicles. Determiing the peak acceleration of a
vehicle resulting from an impact is essential for evaluationg the inertial response of vehicle
occupants during any given collision event.
35


Figure 3.4 Collision pulse
Approximating the general shape of a collision pulse as triangular demonstrates
how the acceleration increases towards the peak where maximum force acts equal-and-
opposite between each vehicle at t=final. From the impulse-momentum relationship of the
impact event, Equation 3.17 determines the time to reach the maximum force due to the
collision between the two vehicles.
At
"VAV,-
F
impact
At, is the same for both vehicles 3.16
3.5 Collision Force from Vehicle Deformation
Figure 3.5 demonstrates the linear spring model relationship of a vehicle colliding
with a barrier. The kinetic energy of a vehicle as it approaches the barrier transfers to work
to compress the vehicle structural spring, synonymous with producing permanent
deformation to the vehicle structure. First, the simplest of models will be developed where
the vehicle strikes a non-deformable, infinite mass barrier squarely, producing a uniform
damage profile across the vehicle width.
36


8x = Damage Depth
Figure 3.5 Two-dimensional, single DOF vehicle-barrier collision model
The relationship between permanent residual deformation and the barrier impact
velocity is the barrier impact velocity, BIV, or barrier equivalent velocity, BEV. A
vehicles struck surface can be considered as a continuum of j=n linear springs in the form
of Equation 3.15 oriented perpendicular to the contact surface with the barrier. Each of the
j=n springs has a resistance to compression, kj, and its own characteristic change in spring
length as a function of the collision time, Ac/t), when subjected to a collision force with
respect to time at each of the springs in the continuum, Fj(t). Defining the compression
of the jth spring across the vehicle width as the average between the two damage depths
that bound the confines of the damaged region provides the following expression:
Where, q and q+i are successive measurements taken parallel to the damaged
surface of the vehicle from its undamaged position, for j=n measurements
across the full damage width
Ac,(l) = the average deformation depth between measured points q and q+i
The total of all forces acting upon the vehicle during the barrier collision phase,
while neglecting non-conservative forces, is represented by Equation 3.18.
2
3.17
37


3.18
n
n
EF/0 = E/t/Ac/0
Maximum engagement and, therefore, maximum deformation to the vehicle
colliding with the barrier occurs at the peak impulse time, tfmai, as expressed in Equation
3.17. Maximum engagement with the barrier coincides with the moment where the kinetic
energy of the vehicle is completely absorbed by the compression of the colliding vehicle,
resulting in maximum deformation of the vehicle at maximum impulse. The ideal linear
spring system completely converts the kinetic energy of the collision into work to compress
the array of springs across the contact surface during the barrier impact, resulting in
permanent plastic deformation with no restitution. Integrating both sides of Equation 3.19
with respect to compression depth, dAcj, results in the expression for the kinetic energy
dissipated as work to compress the continuum of y ideal springs of the system.
Where, 7} = system kinetic energy compressing the jth simple spring element
Uspring = total work to produce the total deformation to the vehicle as a
simple spring
The derivation of Equation 3.19 is important in establishing the relationship
between the peak collision force, F\ and the work, Utotal, done to produce permanent
deformation to the vehicle/spring continuum at maximum impulse.
Except under the most severe of collisions and deformation patterns, the structure
of a vehicle involved in a barrier or vehicle-to-vehicle impact will experience some degree
of restitution as shown in Figure 3.4. The residual deformation extent measured
perpendicular to the impact surface after a collision, AcR, is typically documented after
restitution has occurred. Each Acj of Equations 3.19 and 3.20 represents the maximum
deflection of the vehicle/spring system, or Ac7, at maximum engagement. When
3.19
38


considering restitution, the maximum deformation of the vehicle determines the work to
produce permanent deformation of Equation 3.20. Equation 3.21 expresses the relationship
between the compression of the vehicle structure at maximum impulse and the residual
damage measured at some time after the impact:
ACy = ACy +ACy 3.20
Where, Ac, = maximum deformation of spring} during the impact at time = tfmai
ACy = residual/measured deformation of spring] at some time following the
impact measured perpendicular to the impact surface
Ac = material restitution effect not measured for spring]
At this juncture, it is appropriate to describe the measurement of Ac* values for a
given deformation pattern. Figure 3.6 shows an exemplary damage profile for a frontal
impact. Measurements are taken from a deformation point on the vehicle surface and
perpendicular to the theoretical undamaged position of the vehicle. Measurements initiate
from either the left front (drivers side front) or right front (passenger side) of the vehicle
along the damage width.
Ideally, damage measurements should identify locations of linearity within the
deformed region, or regular intervals that best describe the deformation profile. Each Aw
is the distance between points measured parallel to the impacted surface of the vehicle. A
table, such as that shown in Figure 3.6 can facilitate the recording of deformation points
measured along the total width of contact and induced bending to the vehicle structure and
surfaces resulting from the impact event.
39


Description of point measured Distance from Left Front (0) Distance to undamaged profile
Left front comer wO: +0 cm (0 inches) cO: +81 cm (32 inches)
Left front frame rail wl: +46 cm (18 inches) cl: +81 cm (32 inches)
Center bumper reinforcement bar w2: +92 cm (36 inches) c2: +71 cm (28 inches)
C3...C(n-l) w3...w(n-l) c3...c(n-l)
Right front comer w(n): +183 cm (72 inches) c(n): +51 cm (20 inches)
Figure 3.6 Measured damage dimensions
So that, 2^X = ^cm+^cm} =81 cm (32 inches)
^2 = (81cm + 71cm) = q^cm (30 inches), and so forth.
And, AWi = W\ ~ Oo= (46cm Ocm) = 46cm (18 inches)
= w -w = (92cm -46cm) = 46cm (18 inches), and so forth.
Now consider the continuum of springs across the total width of damage, from wo
to wn, as shown in Figure 3.6. The maximum deformation profile becomes a function of
the damage width, c(Aw). Equation 3.22 expresses the total force applied per unit width of
deformation by dividing both sides of Equation 3.18 by the width differential.
1 z(e-a4 1
Y.F
7=0
Aw,
Aw,
3.21
40


Rearranging Equation 3.21 provides the total force applied per unit of total damage
width to the differential system spring constant per unit width, Equation 3.22 is
expressed only in relation to the total compression of the vehicle structure, Ac across the
total width of the vehicle as a uniform spring constant k value for the ith vehicle, h, over
the entire width of deformation for the ith vehicle, Wi.
F
A Wi

\WrJ
Ac
3.22
The relationship for Ac', as expressed in Equation 3.20 is then substituted into
Equation 3.22 to produce the relationship between force per unit width and total
deformation of the vehicle structure, shown in Equation 3.23.
F1
Awi
rir\ rir\
W'A c+ik

\WU
Ac
3.23
Equation 3.23 considers the deformation, or crush, measured following an impact
and the restitution of the materials following the impact that cannot be directly measured.
Equation 3.24 accounts for the maximum deformation of the impact at the moment of peak
force application.
At this point, it is appropriate to define components of Equation 3.24 in terms of
vehicle-spring stiffness coefficients A and B for the unique impacted surface of a
vehicle as follows:
1 (force)/(length), which is the force per unit depth to initiate
i
damage to the vehicle and applied throughout the application of
external forces resulting from the collision
(force)/(area), the generalized spring constant associated with
resistance to continued deformation/spring compression of the
vehicle structure as a result of the external forces of the collision
41


The presented definitions of A and B stiffness coefficients unique to the impacted
surface, provide an expression for the maximum deformation perpendicular to the impacted
surface at the fh location for the vehicle at the moment of peak force application to
a vehicle, F1.
Ac
i
j
A r+A'
Ac _l
7 Bo
3.24
Equation 3.25 establishes the following important characteristic regarding
deformation produced by an impact event as it relates to a vehicles unique stiffness
parameters:
The maximum deformation at application of peak impulse is a function of
the measurable residual damage following the impact event and the ratio
between the force per unit width necessary to initiate deformation and the
force per unit area to continue permanent deformation to the structure.
The Central Impact Force-Deflection Model of Equation 3.25 is defined by
substituting the value for Ac) from Equation 3.24 into Equation 3.21 and solving for the
peak collision force. Figure 3.7 provides a graphical representation of the linear properties
of Equation 3.25.
IF' = I
f
}=0
y=o
B,
Ac A
A
B
\\
iJ)
Awj = 'L(Ai+Bi-Acj)-Awj 3-25
y=0
42


Figure 3.7 Relationship between force per unit width and residual deformation
3.6 Central Impact Work/Energy Model
The Central Impact Force-Deflection Model of Equation 3.25 provides the peak
external force acting on a vehicle while colliding with a barrier or another vehicle. The
peak external force of Equation 3.25 acts equal in magnitude but opposite in direction of
application to the peak external force applied to the barrier or opposing vehicle in
accordance with Newtons third law. Using the expression in Equation 3.19, the work
required to produce permanent deformation to a motor vehicle structure, or any other obj ect
for that matter that behaves in a similar manner, can be determined based on the A and B
stiffness characteristics of the ith vehicle involved in an impact event by substituting
Equation 3.24 into Equation 3.19. The resultant expression of Equation 3.26 provides the
Central Impact Work/Energy Model.
43


f
U, = Z
j=o
At-Ac* +
B^cf)
a;
2 B,
'Aw,
3.26
Where, Ai and Bi = unique structural stiffness values for the ith vehicles impacted
surface
Acf = the residual deformation, or crush, of the jth deformation measured
on the vehicle perpendicular to the damaged surface from its undamaged
dimensions
Aw,= width to the jth deformation, measured parallel to the damaged surface
The same expression for the Central Impact Energy/Work Model can be derived
from Work/Energy principles using Equation 3.25 as follows:
ZF' = ZU+5,-Ac;)-Ah/
J= 0 J= 0
. * . -B
n &CJ n &CJ
UrE J F dAc Z J ((A,+B, AcR,) Aw]) dAcJ
i=o o
f
i=o o
U=Z
J=0
ArAc,+
Br(AcA
+G,
Aw,
3.27
Where, Ai and Bi = unique structural stiffness values for the ith vehicles impacted
surface
A cf =the residual deformation, or crush, of the jth location measured on
the vehicle perpendicular to the damaged surface from the original
undamaged dimensions
Wj= width to the jth crush location measured parallel to the damaged surface
G i= the constant of integration for the ith vehicle, A
2-4
Equations 3.26 and 3.27 recognize that every impact event that behaves in this manner,
regardless of the presence or absence of permanent deformation, has the following
properties:
44


An elastic range exists (q.wJJ-\w) that in the presence of any applied force, a finite

quantity of energy will be returned to the system regardless of permanent
deformation conditions. This value describes the restitution constant for a given
structure, regardless of impact severity, but not the coefficient of restitution of an
impact event between two objects.
A constant energy for each unit of deformation depth and width (4 ^ Aw,) must be
applied to the system during the application of external forces from the collision in
order to continue producing deformation damage.
After achieving the first two conditiooons, the vehicle structure will continue to
absorb energy and damage at a rate one-half the product of the structural spring
constant, deformation width, and the square of the deformation depth (vAf AT' A
) across the width of contact.
For a fixed barrier or a collinear vehicle-to-vehicle collision producing uniform damage
profiles of average deformation depth, c = ^____1_, the Central Impact Work/Energy Model
n
takes on the familiar form derived by Campbell as initially presented in Chapter 2 as
Equation 2.1.
w i
E= J| A-C + ^ + G
\
dw
3.28
Where, C=average of the deformation depths across the contact damage width
measure perpendicular to the damaged surface from the original undamaged
dimensions
H=width of contact damage only measured parallel to the damaged surface,
does not include induced damage associated with bending of the structure
outside of the contact region
While simplistic to todays standards, Equation 3.28 was an initial breakthrough in
collision related vehicle deformation analysis. However, the numerical expression derived
here as Equation 3.26 allows for deformation depth measurements taken at locations along
the width appropriate for describing the deformation profile, no matter how many
measurements or the interval spacing. Outside of flat fixed barrier impacts, deformation
45


profiles are rarely adequately described by a single average deformation value without
profile shape modifying factors, over-simplifications, or unless utilizing a weighted
average deformation depth as developed later in this study [14],
3.7 Determination of A and B Stiffness Coefficients
The methods for determining the A and B stiffness coefficients for a given vehicle
are well established in the literature [36] [37] [38] [39] [40] [20], It is important to
understand that A and B stiffness coefficients are unique to a particular vehicle make,
model and production years, as well as the impact surface; i.e., front, side or rear. Many
vehicles have clone or sister (family) vehicles that share the same structures and/or
components, and, therefore, comparable structural stiffness properties, as addressed in the
cited literature. Additionally, many vehicles have production cycles that can span a decade
or more in which the structure of the vehicle remains unchanged. Establishing stiffness
properties of a given vehicle through full-scale barrier impacts at known barrier impact
velocities, and where the resultant deformation profiles (both cR and w) are measured,
provides the appropriate A and B stiffness characteristics for a vehicle structure. The use
of commercially available vehicle specific A and B values or direct analysis of tests
spanning the production year of a vehicle family provide accurately determined stiffness
parameters. Stiffness values are determined using the total width of damage and represent
the average stiffness for the contact surface, indicating that A and B values are functions of
the damage width, w. For most cases, this assumption does not produce significant
divergence from the actual vehicle structural response.
3.7.1 Frontal Stiffness Coefficients
Vehicle deformation generated by an impact event is modeled as compression of a
linear spring as shown in Figure 3.5. The determination of A and B stiffness values is
straight forward for uniform deformation depth profiles having full barrier overlap, as
demonstrated by the following derivation.
46


Figure 3.8 Relationship between barrier impact velocity and residual deformation
v =bo+bi-c
3.29
T =- 2
-L barrier -tm vehicle V barrier
2
l_
2
1
m vehicle
(bo+b\-cR>)
f
Wl vehicle
bo+2-bobiC +\b\-c
V
2\
R C
A-C + ^- + G
'W 2 Wlvehicle'
bo+2-bo'brc +\b\-c
2\
Solving like terms:
(mvehicle 'bo 'b\)
A =
3.30
W
B
, in
vehicle
w
3.31
Equations 3.30 and 3.31 provide the means to determine A and B stiffness
coefficients for the fully overlapping frontal impacts into a non-deformable barrier.
Common values for the bo value for frontal impacts is 2.0 m/s (4.5 mph). Equation 3.30 for
the known barrier impact velocity (BIV) and measured average residual deformation
allows for the determination of the bi value.
47


3.7.2 Off-Set Frontal, Side, and Rear Surface Stiffness Coefficients
Frontal offset barrier impacts, side and rear impacts with deformable barriers or
movable barriers are not as straight forward for determining vehicle stiffness data.
References [35] [36] [37] [41] [42], as well as earlier work done by the author of this study,
provide established and generally accepted methods for determining stiffness coefficients
for vehicle surfaces under these conditions.
Another concern regarding stiffness coefficient determination lies in the fact that
many federally mandated full scale tests did not account for an air gap present between
a flexible bumper cover that rebounds out to near its original position as opposed to the
deformable bumper reinforcement bar. As such, an air gap adjustment must be made on
a test specific basis to account for the actual deformation of the vehicle, not some arbitrary
position of a detached plastic bumper cover. Accordingly, the use of commercially
available vehicle stiffness coefficients has become more prevalent since 2000, to where the
use of commercially available resources is the norm rather than the exception. One of the
most commonly utilized and trusted resource will be referenced and utilized when
appropriate for vehicle stiffness coefficients for this study [43],
3.7.3 Commercial Vehicle Stiffness Coefficients
Recent research has developed limited frontal crash stiffness coefficients for select
models of motor coaches, school buses, and commercial semi-tractors, for a total of 9 heavy
vehicles from full frontal barrier impact tests [44], These tests are quite insightful as to the
structural resistance of the tested heavy vehicles, in that the A and B stiffness values do not
diverge appreciably from the higher values seen for most light trucks and vans. However,
the current information regarding heavy vehicle stiffness values only provides a small
sample of data and should be considered as such when used. Accordingly, the need exists
for a comprehensive analysis method that does not strictly rely on stiffness factors for all
impacting vehicles, and especially when considering heavy vehicles. That again is a
primary focus of this study and will be addressed in the following chapter.
48


3.8 Central Impact Vehicle-to-Vehicle Velocity Change
An often crucial objective of collision analysis is the determination of impact
severity as it relates to vehicle occupants. The forces acting on a vehicle during any given
collision event produce an impulse that can change the direction and magnitude of the
approach velocity of vehicles, thus producing a momentum exchange as previously
discussed. It is this very momentum exchange that is of interest, typically for the
determination of collision severity as it relates to the vehicle and its occupants. The
understanding of collision severity requires knowledge regarding velocity change, impulse
time, peak acceleration and the principal direction of force (PDOF) applied during the
impulse. The collinear single DOF central impacts of this chaper apply to collisions where
the PDOF acts collinear to the impact velocities, or approach velocities of the vehicles, as
well as through the mass centers of the colliding vehicles. The Chapter 4 will address
conditions where the PDOF is oblique to the mass centers and non-collinear impacts.
3.8.1 Developing a Force-Deflection/Velocity Change Relationship
The derivation of simple velocity change equations from force-deflection principles
for completely plastic impacts without external impulses produced by tire forces have been
well document in literature, and will be presented for background knowledge
[6] [7] [9] [ 14] [21 ].
Collision configurations considered to this point have included single vehicle fixed
barrier, and collinear central impacts where the principal direction of force (PDOF) acts
through the mass centers of the colliding vehicles. While this is indeed the case for most
rear-end and frontal full-overlap collision events and many broadside perpendicular
impacts with aligned mass centers at impact, such conditions are often not the case for
many conceivable collision configurations. However, the development of the simple model
allows for the development of more complex collision models, since the physical laws of
motion do not change.
Figure 3.9 represents a simple harmonic oscillator according to the equation
expressed in the illustration. Recall the basic equation of motion for the system as
expressed by Equation 3.11. Since the initial consideration assumes the impact is a
conservative system, the sum of all external forces acting on the system during the impulse,
49


At, is equal to 0 in compliance with Newtons second law. Therefore, the relationship
between the two impacting vehicles can be rewritten in the form of Equations 3.32 and
3.33 as follows:
m,
+ m^
Av9
Avj
dt dt
\
= -(k\-c\ + k2-c2)
mx -m2
\m\ +m2 j
Avi | Av2
dt dt
k\ k2
Itx k2
I (Ac/ +Ac^)
mx -m2
^ f d2Sx | d252}
ymx+m2j
dt
dt
y
f kx-k2 ^
\kx +k2j
A,
So that,
0 = tl +
dt2
r mx +m2's f kx-k2 ^
l mx-m2 J Kkx+k2^
A
3.32
d2S mx +m2 ^ f kl'k2 1
dt2 { mx-m2 J
A,
3.33
From the equation for a simple harmonic oscillating system that applies to this
collision model, the second derivative of the position function is as follows:
7 C 7
2^ = (A sin (mAt) + B cos (mAt)) = Aco cos {mAt) Bm sin {mAt)
dt dt
^/2Anlax _ -Am2 sin
dt2
'mx +m2 ^ f k\'k2 1
{ mx-m2 J Kk\ +k2;
fmAt) Bm2 cos {mAt)
Amax = Am2 sin (oAt) + Bm2 cos (oAt)
3.34
50


Figure 3.9 Collision as a simple harmonic oscillator
Consider maximum engagement between the vehicles, and therefore cw. The
period of oscillation occurs at t=7i/2, where cos(coAt) = 0 and sin(a>At) = 1, establishing a
boundary value condition for the system. Equation 3.34 reduces to the following when
equating like terms:
Aco2 =
^ ml + m2 ^ f h'k2 1
{ mx-m2 J
3.35
Where, A=&mx
1
2
3.36
In the development of Equation 3.34, the rate of deformation between the colliding
vehicles was the first element of the derivation. The rate at which the colliding vehicles
start deflecting/deforming at 1=0 is the closing velocity between the contact surfaces of the
vehicles where sin(a>t) = 0, establishing another boundary value condition.
iv =
^ml+m2')f kx-k2
mx-m2 J
k | + k
2)
51


d5
dt
= {V\nWal ~ v2tnWal) = A(0 COs(firf) = A(D
A 8 (yljnjtial v2initial)
ml -m2
\ml+m2j
^kx +k2^
kx k,
2 y
(vl
initial v ^initial )
-v2.
(wi +OT2)(^r^2)
(m]-m2)(k] +k2)

3.37
Energy of a linear spring vehicle structure is dissipated as work producing
maximum deformation at peak force as previously determined in Equation 3.19. Therefore,
Equation 3.37 expresses the closing velocity between the impacting vehicles.
( ^ iniilial ^2initial)
(^+w2)-2-([/1+(/2)
(ml -m2)
3.38
At maximum engagement, the centroid of damage for each vehicle reaches a
common velocity, Vc, allowing for the following relationship:
vc >?, + m2) = m] v\mitial + m2 v2initial
w =
h-^initial + m2 'V2initial
ml +m2
3.39
The change in kinetic energy of the system during the period of maximum
instantaneous engagement during the approach period can be stated as follows:
T -T -T
Approach initial common
= Tl1h VlLfcrf +Tm2- v2l,t,ai +mi)- Vc
T =1.
approach 2
ml m2
\m\ +m2 j
ip^initial ^2initial)
The plastic deformation impact of a simple two-vehicle collision system during the
approach phase of the impact results in velocity change magnitudes of colliding vehicles
as follows:
52


approach ^'c ^ approach
f mi-V\nmal+m2-v2mHM^
ml + m2
-vl
initial
m,
mx+m2
{^initial V^initial)
approach T? approach
f mCV\mtial+m2-v2tmtial^
mx +m2
- v2
initial
m,
mx+m2 j
(V1initial V^initial)
Avl = .. V 2 m2 (^1 damage + damage ) mx {ml +mf)
Av2 = \ 2-mx- (Uldamage + U2damage) m2 (m\ + m2 )
Equations 3.40 and 3.41 apply only to collinear or central impacts where restitution
effects and tire forces are negligible between the vehicles during the approach phase of an
impact event.
3.8.2 Force-Deflection/Velocity Change Considering Restitution Effects
The collision represented in Figure 3.9 considers only the linear momentum
velocity change leading up to the maximum engagement, or the approach phase of the
impact. However, as depicted in Figure 3.10, at least some form of restitution may occur
for the vast majority of vehicle impact events. The portion of the velocity change following
the application of maximum impulse occurs during the collision separation phase, hereto
referred to as the separation velocity change. Figure 3.10 illustrates the relationship
between the approach and separation velocity changes upon the total velocity change of a
vehicle during an impact event.
53


Ax = Damage Depth
e = restitution range
Figure 3.10 Impact with restitution effects (adapted from Figures 3.3 and 3.9)
At higher closing speed impacts in excess of 56 kph (35 mph), ignoring collision
restitution has been assumed to produce negligible error using Equations 3.40 and 3.41.
Potential exceptions to this common and otherwise reasonable assumption occurs under
the folowing conditions:
Impact involving an axle and/or wheel/tire
Collision resulting in compression of engine components into the firewall
Fixed object impacts or collisions between heavy and light vehicles.
Under these condistions, restitution may significantly affect the separation phase of
the collision so as warranting consideration. Figure 3.9 depicts a collinear head-on
collision, but easily adapted to represent a collinear same direction collision event.
The common problem for collinear central motor vehicle impacts involves finding
the pre-impact velocities of two colliding bodies. Post-collision velocities are usually
determinable for many real-world collision events. Therefore, the impact system has two
unknown values, requiring two equations for determination of the unknown velocities.
Initially, consider a perfectly elastic collinear impact between two objects through the
statement of the Conservation of Linear Momentum and the Conservation of Energy for a
collinear impact as follows:
'initial
3.42
3.43
54


Equations 3.42 and 3.44 are rearranged and grouped by related terms for the
purpose of establishing the relationship between two impacting bodies as follows:
mx (V\nrUal ~ Vlfinal ) = ' (V2final ~ V2lmtial ) 3 44
initial ~ 14 final ) ^2 i^2 final ~ V2initial )
m\ {y\nitial 14 final ) {f^initial ~ 14 final ) ^2 i^2final V2initial ) i^2final ~ V2initial ) 3 '43
Dividing Equation 3.45 by Equation 3.44 and rearranging terms with respect to
initial and final velocities of the system results in the traditional form for expressing the
momentum-energy-restitution (MER) relationship as follows:
{v^initial final ) (^1initial ^ final) ^2 (p2 final ^2initial ) (p2 final ^2initial)
mi (^initial ~ Vlfinal) m2 (v2final ~ v2tnWal)
(y^-initial ^final ) (p2 final ^2initial)
1 =
v2
final
-vl
final
e =
^initial ^2initial
v2 final ~ V1 final
V^initial ~ v2initial
Perfectly elastic collision velocity ratio; e = \
Partially elastic/plastic collision velocity ratios; 0 < e < 1
Avl
TOTAL
TOTAL
= (l + e)kvlapproach
approach
Av2 TnTAT =(l + e)Av2
The derivation defines the coefficient of restitution as the ratio between the
separation and approach velocities of colliding objects. Restitution values range as 0 < e <
1; indicating restitution ranges from completely plastic deformation (e=0), and
approaching a completely elastic deformation (e vehicle impacts result in impact restitutions that range from 0 < e < 0.6. Lower restitution
values correlate to high velocity impacts, while larger restitution values correlate to low-
velocity impacts. Equations 3.46 and 3.47 express the velocity change magnitudes for
partially-elastic collisions when considering impact restitution [6][16]:
55


Avl = (l + e)^ 2' In2 {Uldamage + U2damage j 3.46
ml (mj +m2 )

Av2 = (l + e)^ 2'm]' (Uldamage + U2damage j 3.47
m2 (m\ + m2 )

If restitution effects are negligible then Equations 3.46 and 3.47 revert to their
parent and most fundamental form of Equations 3.40 and 3.41.
3.8.3 Force-Deflection Velocity Change Considering Restitution and Non-
Conservative Tire-Ground Forces
External tire forces may act upon two colliding vehicles during the impulse leading
up to maximum engagement, producing effects on the velocity change of each vehicle
during the impact. Tire-ground interaction produces a non-conservative constraining force
external to the collision impulse during the approach phase of the impact, as represented
as Q nc in Lagranges Equation 3.3. Tire force contribution can be a significant at lower
velocity change levels, but become increasingly insignificant as the velocity change levels
increase for vehicles of similar mass.[21][23][28] [45] [46] Examples of when non-
conservative tire forces may produce motion constraints are as follows:
Collinear rear-end collision event where struck vehicle (target vehicle) has
applied braking at contact by the striking vehicle (bullet vehicle).
Broadside impact where the target vehicle slides broadside against the roadway
surface during the approach phase.
Broadside impact where the bullet vehicle is considerably less massive than the
target vehicle (i.e., passenger vehicle striking heavy commercial vehicle).
Any collision configuration having a motion constraint at the target vehicles
tire/ground interface, such as wheel blocking, curbs, wall or barrier.
A stopped vehicle with brakes applied produces a motion constraint in the (-)cc
direction during the aproach phase of a collision event. The imposed motion contraint
56


reduces the target vehicles positive-direction velocity change, and increases to the bullet
vehicles negative-direction velocity change as shown in Figure 3.11.
mivl=Bullet
FI brake
m2v2=Target
F2brake
Flbrake= -m2(g|J.n)
F2brake=-m2(g|J.n)
Figure 3.11 Braking force contribution to collision impulse
Tire-ground contribution vehicle 1:______Tire-ground contribution vehicle 2:
(gMn)
F1brake = mi ^brake = m2 feH F2brake m2 a2brake m2
% ^V^rake = -m2 (gpn) m2' Av^rake = -m2-(gpn)
a, _-m2-(gpn)-At ZWibrake a..o _-m2-(gpn)-At_ AVZbrake
mi m2
= -(gpn)-At
Tire/ground constraint forces resist motion of the struck vehicle during the time of
the approach phase of an impact, At. Equation 3.16 determines the impulse time period
resulting from the change in linear momentum, which is equivalent to the At for the braking
constraint impulse. The velocity change for each vehicle resulting from the linear
momentum exchange from the conservative impact force, and non-conservative tire-
ground constraint force, therefore, both occur during the approach phase of the collision.
Equations 3.49 and 3.50, hereto known as the Central Impact Force-Deflection
Velocity Change Equations, incorporate the consideration of tire-ground non-
conservative constraint force contributions to the velocity change magnitude for each
vehicle of a collinear central impact. Equations 3.48 and 3.49 mathematically state that if
tire/ground constraint force contributions are considered, the net result is an increase in the
velocity change magnitude of the bullet vehicle, and a net decrease in the total velocity
change magnitude for the target vehicle. When tire/ground constraint force contributions
57


are negligible or not present, and if restitution is also negligible or not present, then
Equations 3.48 and 3.49 revert to their parent fundamental form of Equations 3.40 and
3.41.
Avl = (l + e)^ 2' In2 {Uldamage + U2damage j ^ f m2-(g-p-n)-At^ 3.48
ml (m1 + m2) l m\ 2
Av2 = (l + 2 m1- {Uldamage + U2damage j _(£ M-n)-At] 3.49
I m2-(ml+m2)
Where, Avl = total velocity change of striking (bullet) vehicle; m/sec (ft/sec)
Av2 = total velocity change of struck (target) vehicle; m/sec (ft/sec)
e= coefficient of restitution; collision level dependent, unitless
g= gravity constant 9.81 m/sec2 (32.2 ft/sec2)
fj= roadway coefficient of friction; unitless
n= braking efficiency and/or brake force distribution as a decimal
percentage (0 < n < 1.0)
At= impulse time period during approach velocity change phase; sec
3.9 Missing Vehicle Parameters
This chapter has developed the necessary principals and equations for determining
velocity change resulting from collinear central impacts. Development of the Central
Impact Force-Deflection Model and the Central Impact Work/Energy Model allow for the
determination of velocity changes resulting from collinear central impacts from measured
deformation profiles consisting of piecewise width and depth measurements for both
vehicles, as well as known structural stiffness coefficients for each unique vehicle structure
involved. The deformaton and stiffness parameters are usually known or knowable for most
collision safety research and testing applications. Instrumentation of the test vehicles
negates the need for calculating collision severity using the methods presented. However,
58


for a real-world collision event it is far more likely than not that many test-measurable
variables may be missing that cannot otherwise be recreated or properly accounted for
without additional analysis methods. Therefore, when reconstructing real-world collision
events, the methods presented provide at least a foundation for velocity change and
acceleration determination.
The need for accounting for potentially missing data leads to one of the main focuses
of this study; eliminating the reliance upon structural stiffness values for both impacting
vehicles. Therefore, this section of the study will focus on the development of analytical
models to satisfy two probable and frequent scenarios of missing data from real-world
collisions:
Known deformation profile of one of the involved vehicles, but unknown
deformation profile for the associated vehicle in any given impact event.
Known structural stiffness coefficients for one of the involved vehicles, unknown
structural stiffness coefficients for the associated vehicle in any given impact event.
3.9.1 Background
Previous studies have presented methodologies for estimating collision specific
structural stiffness coefficients when only one vehicles properties are known or knowable.
Neptune and Flynn developed a method for determining collision specific stiffness
coefficients for missing values related to one of the collision vehicles.[17] While this
method has utility and can produce reasonably accurate results, the analyst must insert
certain assumptions regarding the vehicles resistance to collision forces that are typically
only determined through full-scale testing. These assumptions may be reasonable in some
circumstances, but not applicable for collisions where estimation of some of the critical
guess values cannot be determined.
The vast majority of full-scale tests have been barrier impacts to the front of
vehicles; some with full overlap and some with partial overlap. Most full-scale testing is
designed to meet compliance with FMVSS 208 Occupant Crash Protection. [47] The
purpose of FMVSS 208 is reducing the number of traffic fatalities and the severity of
collision-related injuries by specifying vehicle crashworthiness in terms of forces and
59


accelerations measured on anthropomorphic test devices in test crashes for passenger
vehicles, and later to add trucks, buses and multipurpose passenger vehicles with a gross
vehicle weight rating of 3856 kg (8,500 US pounds) or less. Tested vehicles are
appropriately instrumented and subjected to 48 kph (30 mph) barrier impacts, although
industry standards are now at 56 kph (35 mph) barrier impacts. Since many vehicles known
as sisters or clones of the same design exist between models of the same corporate
manufacturer, many tests are conducted with only one of the sister/clone family.
3.9.2 Impact Force Balancing Crosscheck
The following is a representation of the Newtons third law expression for the
collision of two vehicles with respect to the Central Impact Force-Deflection Model of
Equation 3.25.
The above expressions simply state that the force of impact acting upon each
vehicle is equal in magnitude, but opposite in direction of application. During an impact
the principal direction of force vectors, or PDOF, acting upon each vehicle during the
impulse, or approach phase of the collision, are equal in magnitude and anti-parallel in
direction. Therefore, the absolute values of the PDOF magnitudes of the two colliding
bodies are equal. Equation 3.50 provides a crosscheck method to ensure that the damages
considered for each vehicle are appropriate for the collision event, and hereto defined as
the Newtonian Central Impact Force-Balance Relationship. Measurements are only as
exact as the methodology used. Limitations are also present in the significant digits of
inertial, structural and dimensional data for each vehicle. Therefore, a vehicle-to-vehicle
collision should be considered to be within force balance if the conservative forces
calculated for each vehicle using Equation 3.50 are within 10%.
n
n
3.50
60


The force balance crosscheck should be used to ensure that Newtons third law is
not violated during the analysis process. However, Newtons third law only applies to the
generalized conservative forces, Qc, of the system at impact, and must exclude the non-
conservative generalized and/or constraint forces, Qnc, acting upon the system external to
the collision impulse.
Forcing a Newtons third law compliance using Equation 3.50 produces one of the
most useful principles for verification of other parameters, or determination of unknown
parameters such as deformation properties or structural stiffness properties.
3.9.3 Determination Missing Deformation Depths
Due to the passage of time a vehicle may become unavailable for inspection.
Photographic evidence of a vehicles deformation profile may be insufficient or for other
reasons outside the control of an analyst, instances arise where one vehicle associated with
a collision event may be unavailable for direct inspection or determination of deformation
profile. In some cases, photographs may depict only the overall width and shape of
deformation without clear evidence for accurate measurements of deformation depth.
The Central Impact Force-Deflection Model (Equation 3.25) and the Central
Impact Energy/Work Model (Equation 3.26), require knowledge of structural A and B
stiffness coefficients and deformation profiles (AcR and Aw) for both vehicles. However,
the Newtons third law expression of Equation 3.50 allows for not only the Newtonian
Central Impact Force-Balance Relationship crosscheck, but also for a Newtons third law
prediction of the damage profile. Solving Equation 3.50 for AF for the vehicle with
unknown deformation depths but a known or knowable width, results in the Newtonian
Central Impact Deformation Prediction Model of Equation 3.51. [18] [48]
tp1 I = I tp1
Jf knowm L* unknown
Tf1 dC known A
A/ = unknown jcLunknown Z-^ I unknown 3.51
Bunknown

61


Where, Ac^nknown = Newtonian predicted residual inward deformation for vehicle
of unknown deformation depth
Fknown = Impact peak force calculated for the vehicle of known deformation
width and depth profile
Aunknown= A stiffness coefficient for vehicle of unknown damage depth
Bunknown= B stiffness coefficient for vehicle of unknown damage depth
Aw^nknown = total deformation width for the vehicle of unknown
deformation (known or knowable)
The Newtonian Central Impact Deformation Prediction Model of Equation 3.51
also allows for the piece-wise determination of each force element on the vehicle of known
deformation measurements, to predicted the piece-wise deformation for each associated
width for the vehicle of unknown deformation depth. Likewise, using the weighted average
deformation on the vehicle of known deformation depth, the Newtonian Central Impact
Deformation Prediction Model determines the weighted average deformation depth for the
vehicle of unknown values.
Utilizing the Newtonian Central Impact Deformation Prediction Model of Equation
3.51 for either a weighted average or piece-wise formulation allows for the prediction of a
damage profile for a vehicle that has not been or cannot be measured. The determination
of the deformation profile is facilitated by relying upon Newtons third law. Forcing a
Newtons third law compliance between the impact forces of the two colliding vehicles or
objects relies upon the reasonable addumption that momentum is conserved.
3.9.4 Application of Work/Energy Principles for Unknown Stiffness Coefficients
All models presented to this point require knowledge of the structural
characteristics of each vehicle. Most collisions will have at least one set of vehicle
structural stiffness coefficients; usually the frontal factors for the bullet vehicle with respect
to collinear central impacts. However, vehicle structural stiffness coefficients may be
limited for every vehicle or vehicle family, front/rear/side structures, heavy vehicles and
62


motorcycles. Therefore, the need exists for a different strategy by which knowledge of the
collision deformation and only one set of structural stiffness values result in an accurate
determination of velocity change levels.
Work/Energy principles provides a direct correlation between the work to produce
permanent vehicle deformation and the applied conservative force producing those
damages during the approach phase of the impact. The force necessary to move some object
over a distance is equal to the work done upon the object, regardless of its path. In other
words, when the sum of all conservative forces does work upon a rigid body, a change in
kinetic energy occurs. The Central Impact Force-Deflection Model provides the means by
which the sum of the conservative forces are determine, and the measurable deformation
to a vehicle provides the distance of application of the total conservative forces.
|Fdx = Uwork where / is the force applied to peak
Applying Work/Energy principles allows for the determination of the work, Umissing,
when the A and B structural stiffness coefficients for one of the vehicle are unknown or
not knowable. Newtons second law alios for determination of the impact force from the
deformation properties of the vehicle of known A and B structural stiffness coefficients.
Deformation profiles for both vehicles must be known for this application. The application
of Work/Energy principals leads to a major contribution of this study expressed by
Equation 3.52, hereafter known as the Central Impact Piecewise Work/Energy Missing
Stiffness Equation.
Where, (Ut) , = Work on vehicle with unknown A/B values at damage
V K /unknown ^
interval k, where k /...//values of deformation intervals
dx is the distance over which the force is applied to the system
(Equation 3.25)
3.52
63


[Fj) = force applied to known A/B values vehicle at interval j, where
V J /known
j=l...n values for deformation intervals
(Ac^ ) = deformation depth on vehicle with unknown A/B values at
damage interval k, where k /...//values of deformation intervals
Ideally for Equation 3.53 to produce the most accurate results, the damage intervals,
k, for the unknown vehicle should be paired with damage intervals, j, for the known vehicle
in the equation. Therefore, damage profiles for each vehicle must have the same number
of corresponding deformation widths; or j=k intervals of damage measurements. However,
the width of each interval are not necessarily equivalent, but can be paired to discrete
deformation zones between the colliding vehicles. Figure 3.12 illustrates examples of
damage profiles zone matching for the direct application in Equation 3.52.
Figure 3.12 Paired force regions from impact deformation
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It is not uncommon for the damage profiles between two colliding vehicles to exist
making it difficult to partition damage zones into neat paired packages as shown in Figure
3.12. Therefore, when pardoning of damage is difficult, the elimination of the associated
damage width partitioning reduces the potential for error. Collisions where deformation
zone partitioning may be difficult are as follows:
Narrow object impact producing wide contact and/or induced damage profiles on
the bullet vehicle and narrowly focused damage on the target object
motorcycle-to-vehicle collisions
Small vehicles striking large commercial vehicles or trailers
Oblique impacts prodding a wide contact area on only one fo the colliding vehicles
As an alternative, the weighted average damage depth on the vehicle of unknown
structural stiffness coefficients along with the total impact force determined from the
vehicle of known structural stiffness coefficients, S t'Lown must be considered when
determining the total work of the impact conservative forces upon the vehicle of unknown
structural stiffness coefficients.
The total damage width is the sum of its individual parts, and the sum of the ratio
of each width segment with respect to the total damage width = 1:
n
W total = ZAWJ
7=1
Awl Avr2
^total ^total

= 1
w
total
The weighted average of inward deformation for the unknown vehicle is therefore
determined by the following:
SAlVAcy
Cunknown = ------ Weighted average of deformation width 3.53
iAwj
7=1
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The substitution of Equation 3.53 into Equation 3.52 for determining the work to
produce damages to the vehicle of unknown structural stiffness produces the Central
Impact Weighted Average Work/Energy Missing Stiffness Equation of Equation 3.54.
unknown
F
I
known
nR
v' unknown
3.54
Where, Fwork)unknown = total work to produce permanent inward deformation
Fnown = imPact force calculated from vehicle of known structural stiffness
C unknown =weighted average deformation of vehicle of unknown structural
stiffness (from Equation 3.53)
By now it should be intuitive that the PDOF acting upon any vehicle during any
given impulsive impact event acts through the centroid of the damage profile. The utility
of Equation 3.54 lies in the removal of interpretation of deformation intervals that are
associated with the colliding vehicles. Removal of interpretation allows the weighted
average of deformation of the vehicle of unknown structural A and B stiffness values (i.e.,
centroid of damage) to act as the distance over which work is done on the vehicle structure
as a whole, thus satisfying the requirements of Newtons third law. Equation 3.54 also
establishes an important relationship useful for determining the work done to damage any
composite structure having a complex and/or piecewise linear deformation profile.
For a given applied peak impact force, the dissipated energy doing work to
produce damage to a motor vehicle or other composite structure is
determinable using the weighted average deformation depth for any
complex damage profile.
This final statement and the development of Equations 3.52 and 3.54 mark the
capstone of the study objectives with respect to collinear, central impacts. The following
chapter will address these same principles as they relate to oblique, non-central collisions
where rotational effects and inter-vehicular friction contributions may be significant and
which are not present during the simple impact configurations presented thus far.
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CHAPTER 4:OFF-SET AND OBLIQUE NON-CENTRAL IMPACTS:
GENERALIZED DEFORMATION AND TOTAL VELOCITY CHANGE
ANALYSIS (G-DATAAV) SYSTEM OF EQUATIONS
4.1 Objectives
Up to this point, the equations developed have been used to solve the
relatively simple conditions of collinear central impacts, which result in no rotational
effects upon either colliding vehicle. It is important to remember that motor vehicles are a
collection of relative fixed points, or rigid bodies, and behave as rigid bodies when
subjected to collision forces. The vast majority of vehicular collisions, outside of rollover
events and falls, flips or vaults, fit into planar two-dimensional problems with rotation
restricted to yaw about the z-axis. The Conservation of Linear Momentum provides a
simplified solution to collision events producing limited or negligible rotation. However,
as will be described and developed in this chapter, non-central impacts can produce
sufficient rotational effects which cannot be ignored. Oblique impacts produce PDOFs that
are not perpendicular to the impacted surface, which complicates the interpretation of
residual deformation values, c and Ac used in the analysis procedures presented in
Chapter 3. Vehicular collision configurations producing rotation are illustrated in Figure
4.1 and described as follows:
Oblique non-central, or angled and non-collinear collisions where the principal
direction of force (PDOF) does not act through the mass center of one or more
involved vehicle in a collision event
Off-set collinear where the PDOF acts parallel to, but not through the mass centers
of the vehicles
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Figure 4.1 Non-central oblique and off-set impacts
The objectives of this chapter are to accomplish the following with respect to
vehicle deformation analysis:
Develop methods for determining the principal direction of force (PDOF) applied
between colliding vehicles from vehicle deformation profdes.
Develop generalized models for analyzing vehicle deformation that build upon the
Central Impact Force-Deflection Model Equation 3.25 of Chapter 3.
Incorporate rotational contributions to the development of generalized models for
analyzing total velocity change magnitudes using the Central Impact Work/Energy
Model and the Central Impact Force-Deflection/Velocity Change equations of
Chapter 3.
Develop a generalized model for determining the total velocity change levels for
colliding vehicles that include central and oblique collisions, and where only one
vehicle has known structural stiffness coefficients.
To eliminate confusion, the earth-fixed vehicle coordinate system of Figure 3.1(b) will
be used for this chapter unless otherwise specifically stated.
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4.2 Oblique and Offset Impact Momentum Principles
4.2.1 Linear Momentum
Chapter 3 developed the linear momentum, P, and equation of motion for a system
of two colliding collinear vehicles. Recall the linear momentum, P, for the system was
derived from the partial differentiation of the Lagrangian with respect to the generalized
velocities, of the system of generalized coordinates. Even though the Lagrangian starts
with the consideraton of scalar quantities, namely kinetic energy, T, the analytical approach
of utilizing Lagranges Equations results in equations of motion for the system that are
vector quantities.
For a conservative system, the sum of the generalized momentum before and after
impact is equal to zero, (Pj +P2)-(pj +P2) = 0, and therefore, Equation 3.7 becomes the
following:
[M. 1 + [M2 ] = [M, ] [vl],,, + [M2 ]. [ 4.1
Where, [M.l an d[M;] are the mass matrix for vehicles 1 and 2 respectively
fvll...,,[v2l...,,[ vll and f v2l are the vector arrays for the initial
L Jinitial7 L Jinitial 7 L Jfinal L J final J
and final velocities of vehicle 1 and 2 respectively
The vast majority of vehicle collisions are planar except rollover collisions, falls,
flips or vaults. Only rotational effects in yaw about the vertically (z-axis) may be present
for planar impacts. The following describes the total planar mass matrix and the velocity
vector arrays for a planar impact, where (ij,k) are unit vectors in the (x,y,z) coordinate
system:
mlxx mlxy IK m2xx m2xy I2xz
mlyx mlw I\z [M;] = m2yx m2yy I2yz
71 71 IK 72 7r 12 1277

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xlj r y xi; x2t r x2i
II S l > l y}j ii | > | y^i IV2L tfa/H y2j II s: i > | y2j
h\ ft*. Aj K,
The local coordinate systems for each vehicle pass through their respective mass
centers, leaving only the diagonal terms of the total planar mass matrix for each vehicle as
a non-zero value. As such, the total planar mass matrix for each vehicle reduces to the
following:
K]
ml 0 0 m2 0 0
0 ml 0 [M2] = 0 m2 0
0 0 71 0 0 I277

Placing the inertial coordinate system at the mass center of vehicle 1, mass 1 of the
two colliding vehicles, results in the following expressions for the planar equations
regarding the linear momentum and moment of momentum of the system:
x-direction linear momentum:
ml vlmitiai' cos(6>l) + ml v2imtial cos(6>2) = ml vlfinal cos(6l ) + m2-v2final cos(6>2) 4.2
v-direction linear momentum:
m\-vlimtial sin(#l) + m2 v2initial sin(#2) = ml v\final sin(6d ) + m2-v2final sin(#2 ) 4.3
Rotation in yaw about z-direction:
I\zz 0 + I2ZZ (j)2 = I\zz $ + I2ZZ (f)2 4.4
Figure 4.2(a) illusrates planar linear motion of a two-vehicle oblique impact system
described by Equations 4.2 and 4.3, and Figure 4.2(b) illustrates the rotation of the vehicles
as expressed by Equation 4.4.
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y
Total trajectories:
y
y
Departure trajectory, mr.
y
Departure trajectory, mr.
Figure 4.2(a) Planar collision trajectory angle determination
Figure 4.2(b) Oblique collision rotation angle determination
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By placing the origin of the global coordinate system at the mass center of vehicle
1 and the x-axis parallel to the approach direction of vehicle 1, the first term in Equation
4.3 is zero. This simplification allows for the determination of v2inwai if the approach and
departure angles, as well as vehicle masses, are known or knowable. The determination of
vlmitiai is accomplished by substituting the results for v2inmai into Equation 4.2 and solving
for Vlinitial.
The velocity change equation for each vehicle is determined mathematically from
the linear momentum by separating the velocity change into its (x,y) components as
follows:
AEc = Vfinal COS (O) ~ VlmtM COs(#)
Av>> = Vfinal Sin {?) ~ ^initial sin (#)
Av = VAvx +Avl
Equations 4.5 and 4.6 are expression for the velocity change magnitudes resulting
from the linear momentum of the system.
A graphical determination of the linear momentum determined mathematically by
Equations 4.2 and 4.3 is accomplished by plotting a set of scaled parallelograms using
vector diagramming properties as shown in Figure 4.3. Graphically, the momentum of the
system is plotted using the momentum magnitudes as the vector length and measured
angles as the vector direction while utilizing the tip-to-tail vector diagram methodology.
The velocity change magnitude is equivalent to the momentum unit length of the line
between the tip of the approach momentum vector directed towards the departure
momentum vector. The sense of the line results in a vector pointing in the direction of the
72


velocity change for each particular vehicle, which will be opposite in sense for the
momentum change vector of the opposing vehicle.
The graphical solution is accomplished by first plotting the scaled departure
momentum vectors to create an inside departure parallelogram that allows for the
determination of the total momentum of the system. Secondly and while knowing the
approach momentum vector directions, the outside approach parallelogram common to
the total momentum vector previously determined is then plotted. The common diagonal
between each parallelogram is the total linear momentum brought into the impact,
(p, + P2 ) The total linear momentum brought into the system must be equal and opposite
to the total linear momentum leaving the impact, for collisions producing
negligible rotation. Completion of the parallelogram provides for the determination of the
approach velocity vectors and their magnitude determinations.
It should be noted that for a conservative system that obeys Newtons third law
involving conservative forces only, the velocity change vectors for each vehicle will be of
equivalent linear momentum units. The vector magnitude lengths must be oriented parallel
and opposite in direction from each other. However, if non-conservative forces act upon
the vehicles during the collision, and/or if rotation results, a linear momentum analysis
cannot determine the total velocity change for the vehicles. Total velocity change is the
cumulative velocity change of colliding vehicles due to conservative and non-conservative
forces; i.e., linear momentum change, rotational momentum change, tire/ground constraint
force and inter-vehicular dissipative friction. The determination of these additional effects
upon the total velocity change will be addressed later in this chapter.
For the earth-fixed Cartesian coordinate system chosen, the PDOF is measured
counterclockwise from the x-axis or a line parallel to the x-axis. Figures 4.3 and 4.4 show
the graphic representation method for solving the momentum solutions for the linear
momentum velocity change and PDOF direction for the collision event.
73


74


The PDOF acting upon the vehicles are determined by first recognizing the angle
of the PDOF acting upon vehicle 1, IJpdofi and the angle of the PDOF acting upon vehicle
2, IJpdof2, are measured counterclockwise to the approach path of vehicle 1, or the x-axis,
in an earth-fixed Cartesian coordinate system.
tan (A#) = ^ = v^rMV')-''uuuarMV)
Avx 17final COS(6') - COS(6)
The following determines the PDOF acting upon each vehicle with respect to the
x-axis:
A 01 = tan-1 ' vlfinal sin (£!') vhnmal sin {0\) ^ 4.7
v Vl final COS (01') vlinitial COS (0l) ,
A02 = tan-1 ' v2final sin (02') v2imtial sin (02) ^ 4.8
v v2fmal cos (02') v2imtial cos (02) y
When post-impact trajectories and resultant surface drag factors are known or
knowable, reasonable estimates regarding the post-impact velocities of vehicles involved
in a collision event can be made from the energy principles of Chapter 3. As a vehicle
slows, work is done between the tires, body components or side of a vehicle that is sliding
on other a surface. The total kinetic energy of the vehicle at the start of the negative
acceleration due to braking, spinning, sliding or even coasting is the sum of the final kinetic
energy and work done while slowing.
T =T +77
A initial final w work
1 2 1 2
VMtial =-'m'Vfinal + Ffind,on 4 9
The force due to friction can be expressed using Newtons second law and
substituted into Equation 4.9, allowing for a relationship between the initial and final
velocities as follows:
75


1
Vinitial ~ 2 'm Vfinal +
,2 ,
initial
+ m-afnction-d
2 m ''
Vfinal + 2 a friction ^
^initial
yjV final + 2
4.10
The acceleration due to friction, apicuon, is expressed as the ratio between the force
necessary to cause the vehicle to slide against the contact surface with the normal force due
friction coefficient, p. However, the surface slope may also affect the rate at which the
acceleration due to friction occurs and should be accounted for when known. The
gravitational influence of surface slope is based upon the tangent of the angle of the surface
the vehicle is in contact with to the earth-fixed horizontal axis, as shown in Figure 4.4.
Another term for the slope of the surface is the grade of the surface; whether an uphill
grade for a positive slope or a downhill grade for a negative slope. If the final velocity of
a vehicle is zero, or the vehicle comes to a complete stop following the impact, then
Equation 3.10 while accounting for surface slope is simplified as follows:
to gravity that holds the vehicle to that surface, which is often referred to as the surface
4.11
Where, p = surface friction (level surface, no adjustments)
\|/ = surface slope angle; radian (degree)
d = acceleration/braking/slowing distance; m (ft)
/= effective surface drag factor adjusted for roadway slope; unitless
76


Run
Figure 4.5 Surface slope and friction relationships
Equation 4.12 can be used for a vehicle that travels over multiple surfaces with
multiple different drag factors and surface slopes as follows:
^'initial
2-g-T.
i=1
// tan (i//l)
A/l+tan(///)2
d
= J2-g-'Zfrd,
i=i
4.12
4.2.2 Rotational Momentum
Off-set and oblique collisions result in the PDOF acting away from the mass center
of one or both vehicles, thus producing a moment about the respective mass centers as a
result of the collision. Equations 4.2, 4.3 and 4.4 state that the linear and rotational
77


momentum (or moment of momentum) of the colliding system is conserved. The Newtons
second law statement for rotation with respect to Figure 4.2(b) is as follows:
x = rxF7
4.13
^magnitude ~ ^ ^magnitude ~ ^cm$
Where, t = torque
r = moment arm vector from rotation at mass center to applied force
Fi = applied force vector from collision
/ = polar mass moment of inertia (about axis through mass center)
= rotational acceleration
hi = perpendicular moment arm for impact induced moment
The torque about the mass center of the vehicle is equal to the angular inertia of the
vehicle. For the vehicle in Figure 4.6, the applied force at the right front of the vehicle
produces a counterclockwise rotation. Unless the vehicle is setting on a frictionless surface,
such as wet ice, frictional forces will act counter to the rotation of the vehicle during the
collision impulse, just as occurred when braking forces were considered for a collinear
impact in Chapter 3.
Ff =m-g
f \
// tan (\j/)
-Jl + tan (///)2
= m- g-f -n
n
4.14
Where,Ff= Force due to friction between vehicle tires and roadway surface
p = surface friction (level surface, no adjustments)
\|i = surface slope
n = braking/sliding efficiency (ratio of mass on sliding tire(s))
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/= effective surface drag factor
Figure 4.6 Moment arm applied to produce rotation about mass center
The net torque applied to the vehicle in Figure 4.6 is the difference between the
moment produced by the applied force and the moment about the rotation point for the
opposing frictional forces acting at the vehicle/roadway interface, which for small
displacements at a single axle the rotation point for the moment arm of the friction may be
the center of the axle furthest from the impact location, and for large displacements the
rotation point for the opposing friction will be about the vehicle mass center.
= / = hi -pi ~hf -pf =hi -pi ~hf (&)(/) 415
Expressing the force acting upon the vehicle due to the impact impulse by Newtons
second law and the accelerations expressed in terms of velocity changes over the impulse
79


time period, the system can be solved for the linear velocity change of the vehicle as shown
in the steps to derive Equation 4.16.
m-a = -

hr
m
At h,
For most vehicle collision reconstructions, the initial rotation of a vehicle leading
into an impact event is negligible or zero, leaving only the post-collision rotation of a
vehicle as the necessary value when determining A. Figure 4.7 shows a vehicle rotating
to final rest post-impact. The rotation of the vehicle is determined by the rotation of the
local coordinate system of the vehicle oriented in an earth-fixed orientation as the vehicle
is placed incrementally upon its post-collision tire marks. In Figure 4.8 the individual post-
collision tire trajectories of the vehicle are plotted using different colored lines to facilitate
ease of vehicle incremental placements on the scene evidence. The center of mass distance
travel and change in vehicle heading is determined for increments along the rotating post-
collision path, so that the effective drag factor for the vehicle, assuming no braking post-
impact, can be determined. If wheels are locked due to damage, the post-collision friction
can also account for the locked wheels braking contribution based upon the weight
distribution of the vehicle upon the locked and unlocked wheels during the rotation towards
final rest.
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Figure 4.7 Rotational post-impact motions
The vehicle in Figure 4.7 has an initial heading of its local coordinate ^-axis in line
with the earth-fixed inertial coordinate system x-axis. However, the initial heading could
be any angle where the vehicles 2,-axis is rotated from the earth fixed inertial coordinate
system x-axis. The vehicle rotates in yaw without braking about its local vertical (zkaxis
either clockwise or counterclockwise, so that the effective deceleration rate, again without
braking over the entire distance, regardless of rotation direction is given by the following:
rotate _ ju tan (y/) sin(^ + 1 2 ^l + tan(y/) y 2
\|/i = surface slope at the ith interval
81


$ = rotation of the vehicle axis from the inertial coordinate system at ith position
f irotate= effective surface drag factor for ith interval
When the surface slope is small, as is usually the case for roadways and roadside
geometric features (\j/ < 20, n/9 radians), Equation 4.17 can be simplified as follows:
fre=(u tanW)-
sin(
<4 + A
7+1 '
The determination of the vehicles rotational velocity change between each spin
interval is determined from basic kinematic equations for determining the time period for
each rotational slowing interval.
Av; . Av
ai = ~ and, Atf =----------
At, a,
g ( \ // tan (///.) sin(^+^>) /* rotate o J i
vV' +tan ('//,) J 2
4.19
A&i
Aj Q;+i
At,
,)§ f,r
v, V
;+1
4.20
4.2.3 Mass Moment of Inertia
The mass moment of inertia of an object is its measure of resisting rotational
acceleration, just as mass is the measure of a body to resist linear acceleration. The moment
of inertia of an object is a function of shape and mass. If the moment of inertia is determined
about an axis that passes through the mass center of an object, it is called a polar moment
of inertia.
82


Using polar moments of inertia tends to make analysis much easier, and is routinely
used for motor vehicle collision analysis with few exceptions. The moment of inertia about
any axis, to include those that do not pass through the mass center of a body can be
determined using the parallel axis theorem if the polar moment of inertia is known.
Likewise, the polar moment of inertia of a composite or oddly shaped object can be
determined using the parallel axis theorem with respect to each of the individual moments
of inertia of the geometric shapes that make up the body. In general, the moment of inertia
is determined as the sum of the product of all the differential mass elements of the body,
dm, and the square of its distance from the axis of rotation, r.
dl = r2 dm
n 4.21
I = Yjr?-ml
i=1
Another method of describing the polar moment of inertia of an object with great
utility is considering the mass, m, is concentrated within an equivalent radius about a
primary axis that passes through the mass center, known as the radius of gyration, kg. [49]
/ = m-k^ 4.22
Since passenger vehicles, light trucks and vans are non-homogeneous complex
geometric shapes, the mass moment of inertia is determined experimentally using tilt table
measurements, or more commonly from best-fit equations derived from experimental data.
Garrott presented data from the NHTSA Light Vehicle Inertial Parameter Data Base
containing measured vehicle inertial parameters of 356 tested vehicles, plus tilt table data
for 168 vehicles [50] as a follow up to an initial paper presenting an algorithm for
determining moments of inertia for the curb weight of unloaded vehicles by distinct vehicle
classifications [51], Neptune presented a method for determining the yaw moment of
inertia (hz) based upon the method presented by Garrott for the curb weight of an unloaded
vehicle, but allowing for the addition of occupant and cargo weights [52], providing greater
utility for collision analysis. Equation 4.23 from the Neptune paper reduces to the best fit
algorithm developed by Garrott when the vehicle is unloaded.
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f f XX
mcurb (L2+b2)- km- ^loaded ^curb
Kg v ^loaded ))
Where,/zz = yaw moment of inertia (about z-axis)
mCurb= curb mass of vehicle (unloaded)
mioaded= loaded mass of vehicle (curb plus occupants and cargo)
L = total length of vehicle
b = maximum width of vehicle
KgKm = geometric empirically determined constants (Table 4.1)
Table 4.1 Yaw moment of inertia empirical constants
Vehicle Type Kg Km R2
All combined 13.1 0.696 0.85
Passenger Car 13.8 0.769 0.86
Light Truck 13.4 0.750 0.92
SUV 12.2 0.656 0.76
Light Van 12.3 0.642 0.90
4.3 Principle Direction of Force from Damage Profiles
In Chapter 3, the focus was on collinear central impacts. As such, the principle
direction of force applied by the striking vehicle upon the struck vehicle was along either
the longitudinal or lateral primary axis of a vehicle. However, oblique and off-set collisions
result in a principle direction of force that does not act along a primary axis of the vehicle
nor through the vehicle center of mass. Instead, oblique and offset collisions result in a
PDOF that is angular to a primary axis, or parallel to a primary axis but offset from the
mass center, thus resulting in rotation. In order to account for the effects of an oblique
impact damage profile and rotation, the principle direction of force (PDOF) must be
determined.
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Full Text

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GENERALIZED DEFORMATION AND TOTA L VELOCITY CHANGE ANALYSIS SYSTEM OF EQUATIONS (G-DaTAV ) by JERRY SCOTT OGDEN B.S., Eastern Oregon University, 1988 M.S., University of Colorado Denver, 1995 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Engineering and Applied Sciences Civil Engineering 2015

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ii JERRY SCOTT OGDEN ALL RIGHTS RESERVED

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iii This thesis for the Doctor of Philosophy degree by Jerry Scott Ogden has been approved for the Engineering and Applied Sciences Program by, Wesley Marshall, Chair Bruce Janson, Advisor Peter Jenkins Ronald Rorrer Apostol Panayotov Date: October 15, 2015

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iv Ogden, Jerry Scott (PhD, Engin eering and Applied Sciences) Generalized Deformation and Total Velocity Change Analysis System of Equations (GDaTAV ) Thesis directed by Professor Bruce Janson ABSTRACT Current methods for analyzing motor vehicl e deformation utilize a force-deflection analysis for determining deformation work energy, which relies on vehicle-specific structural stiffness coefficients determined from full-scale im pact testing. Wh ile the current database is quite extensive for frontal sti ffness values for passenger cars and many light trucks, vans and SUVs from the 1970s up to mo dern day, the database is devoid of specific crash tests needed for deformation analysis of rear and/or side struct ures of many vehicles. Additionally, there exists very few structural stiffness coeffi cients for heavy commercial vehicles, buses, recreational vehicles, hea vy equipment or motorcycles necessary for application with the current force-deflection analysis methods. The primary goal of this research is to develop an accurate, reliable and broadly applicable deformation analysis method that re quires the structural stiffness coefficients for only one collision involved vehicle. The developed methodology expands the application of deformation analysis to incl ude unconventional vehicles and other objects and surfaces not supported by the current structural stiffness coefficient database. The GDaTAV System of Equations incorporates linea r and rotational effects, as well as impact restitution resulting from conservati ve forces acting during a given collision impulse. Additionally, the G-DaTAV System of Equations accounts for tire-ground forces and inter-vehicular friction, non-cons ervative force contributions acting on the collision system that are commonly present during offset and oblique non-central collision configurations. Correlation and descriptive stat istics, as well as the raw an alysis results, indicate a highly reliable and significantly improved de gree of precision and accuracy achieved

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v through the application of the G-DaTAV System of Equations when determining vehicular total velocity changes for oblique and offset non-central impacts. The form and content of this abstract are approved. I reco mmend its publication. Approved: Bruce Janson

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vi ACKNOWLEDGMENTS I would like to thank th e Colorado State Patrol ( www.colorado.gov/csp) and the Colorado Department of Transportation ( www.coloradodot.info) for providing collision history data for the State of Colorado used in Appendix A of this study. I would like to express my appreciation to James Neptune of Neptune Engineering, Inc. (www.neptuneeng.com), for granting permission to utilize the extensive database of vehicle stiffness coefficients that he tireles sly updates with new testing results to keep everyone current. Additionally, the Faro HD and Faro Reality CAD collision reconstruction programs were utilized to pr oduce the high impact diagrams appearing within this study. I would especially like to thank the engineers of OEC Forensics (www.OEC4N6.com), Mathew Martonovich and Courtney Engle. The insight, criticism and critical thinking where needed by other pr ofessionals and trusted colleagues is crucial to the success of such an endeavor. No accomp lishment of this scope and magnitude is possible without the support of an understanding, patient and loving family. I want to express my deepest appreciati on to my wife Cherie, and my children Carson, Natasha and Adriana. Their encouragement to make th is happen, undying support and understanding regarding the extreme time commitment necessary for this project, along with th eir loving confidence made this dream a reality.

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vii TABLE OF CONTENTS Chapter 1: STUDY MOTIVATION AND OBJECTIVES ...............................................................11.1 Study Motivation ....................................................................................................... 11.2 Statement of Study Objectives .................................................................................. 62: LITERATURE RESEARCH ...........................................................................................92.1 Background of Collision Analysis ............................................................................ 92.1.1 Basic Collision Analysis Methods ...................................................................... 92.1.2 Passenger Vehicle Event Da ta Recording Systems ............................................ 92.1.3 Heavy Vehicle Event Data Recording Systems ............................................... 132.1.4 Vehicle Tracking Systems ................................................................................ 172.2 Vehicle Deformation Analysis ................................................................................ 182.2.1 Pioneering Studies and Computer Programs .................................................... 182.2.2 Modern Damage Deformation Studies ............................................................. 233: COLLINEAR CENTRAL IMPACTS; DEVELOPING BASIC DEFORMATION ANALYSIS PRINCIPLES ................................................................................................263.2 Equation of Motion for Collinear Central Impacts ................................................. 273.2.1 Conservation of Energy .................................................................................... 283.3.1 Lagranges Equations ....................................................................................... 303.3.2 Deriving the System Momentum ...................................................................... 313.3.3 Equation of Motion ........................................................................................... 323.4 Impulse-Momentum of the System ......................................................................... 343.5 Collision Force from Vehicle Deformation ............................................................ 363.6 Central Impact Work/Energy Model ....................................................................... 433.7 Determination of A and B Stiffness Coefficients .................................................... 463.7.1 Frontal Stiffness Coefficients ........................................................................... 46

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viii 3.7.2 Off-Set Frontal, Side, and Rear Surface Stiffness Coefficients ....................... 483.7.3 Commercial Vehicle Sti ffness Coefficients ..................................................... 483.8 Central Impact Vehicle-to-V ehicle Velocity Change ............................................. 493.8.1 Developing a Force-Deflection/Velocity Change Relationship ....................... 493.8.2 Force-Deflection/Velocity Change Considering Restitution Effects ............... 533.8.3 Force-Deflection Velocity Change Considering Restitution and NonConservative Tire-Ground Forces ............................................................................. 563.9 Missing Vehicle Parameters .................................................................................... 583.9.1 Background ....................................................................................................... 593.9.2 Impact Force Balancing Crosscheck ................................................................ 603.9.3 Determination Missing Deformation Depths ................................................... 613.9.4 Application of Work/Energy Principles for Unknown Stiffness Coefficients 624:OFF-SET AND OBLIQUE NON-CENTRA L IMPACTS: GENERALIZED DEFORMATION AND TOTAL VELOCI TY CHANGE ANALYSIS (G-DaTA V) SYSTEM OF EQUATIONS..674.1 Objectives ................................................................................................................ 674.2 Oblique and Offset Impact Momentum Principles ................................................. 694.2.1 Linear Momentum ............................................................................................ 694.2.2 Rotational Momentum ...................................................................................... 774.2.3 Mass Moment of Inertia ................................................................................... 824.3 Principle Direction of For ce from Damage Profiles ............................................... 844.3.1 Determining Damage Centroid ......................................................................... 854.3.2 Maximum Engagement..................................................................................... 894.4 Oblique Impact Generalized Force-Deflection Model ............................................ 904.4.1 Oblique Deformation Principles ....................................................................... 914.4.2 Developing the Generalized Force-Deflection Model ...................................... 924.4.3 Generalized Force-De flection Principles.......................................................... 94

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ix 4.5 Oblique Impact Work/Energy Model ...................................................................... 954.5.1 Contributions of Oblique Angl e to Vehicle Deformation ................................ 954.5.2 Contributions of Non-C onservative Inter-vehicular Frictional Forces ............. 964.5.3 Generalized Impact Work/Energy Model ......................................................... 994.6 Generalized Impact Force-Deflec tion/Velocity Change Model ............................ 1004.6.1 Rotational Contributions................................................................................. 1004.6.2 Generalized Impulse Time.............................................................................. 1034.7 Generalized Newtonian Prediction of Missing Vehicle Parameters ..................... 1034.7.1 Generalized Impact Force Balancing Crosscheck .......................................... 1044.7.2 Determining Missing Deformation Depths .................................................... 1054.7.3 Generalized Work/Energy Principles and Unknown Stiffness Coefficients .. 1074.8 Summary of Findings; G-DaTAV System of Equations ................................. 1095: GENERALIZED DEFORMATION AND TO TAL VELOCITY CHANGE ANALYSIS (G-DaTAV ) SYSTEM OF EQUATIONS A PPLICATION AND EVALUATION .1145.1 Overview of G-DaTA V System of Equations Development .......................... 1145.1.1 Chapter 2 Contributions.................................................................................. 1145.1.2 Chapter 3 Contributions.................................................................................. 1155.1.3 Chapter 4 Contributions.................................................................................. 1155.2 Anatomy of G-DaTA V System of Equations ................................................. 1165.2.1 Non-central Impact Work/Energy Sink Contributions ................................... 1175.2.3 Non-central Impact Non-Conser vative Force Contributions.......................... 1185.3 G-DaTAV System of Equations Evaluation ................................................... 1205.3.1 RICSAC Testing ............................................................................................. 1205.3.2 RICSAC Original Study Findings .................................................................. 1235.3.3 RICSAC G-DaTA V System of Equations Application Approach .......... 1275.3.4 RICSAC G-DaTA V System of Equations Analysis Results ................... 129

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x 5.4 National Automotive Sampling System Real-World Collisions ........................... 1335.4.1 NASS G-DaTA V System of Equations Application Approach ............... 1345.4.2 NASS G-DaTA V System of Equations Analysis Results ........................ 1365.4.3 Overall G-DaTA V System of Equations Evaluation ............................... 1405.5 Findings and Conclusions ..................................................................................... 1445.5.1 G-DaTA V System of Equations Applications ......................................... 1445.5.2 G-DaTA V System of Equations Limitations ........................................... 1455.5.3 Future Research .............................................................................................. 1465.5.4 Conclusions .................................................................................................... 147REFERENCES ................................................................................................................152 APPENDICES A: COLLISIONS ON COLORADO HIGHWAYS AND WORK ZONE S FROM 2007 TO 2011..................................................................................................................................160A.1 Objectives ............................................................................................................. 160A.2 Collision Data for all State Highways from 2007 to 2011 ................................... 160A.3 Collision Data for Constructi on Zone Collisions 2007 to 2011 ........................... 165A.3.1 Overview of Crash Reported Data ................................................................. 165A.3.2 Passenger Vehicle Construction Zone Collisions .......................................... 169A.3.3 Sport Utility Vehicle, Pickup and Van Construction Zone Collisions .......... 171A.3.4 Heavy Vehicle Construction Zone Collisions ............................................... 173A.3.5 Motorcycle Construction Zone Collisions ..................................................... 175A.4 Summary .............................................................................................................. 177B: G-DaTAV ANALYSIS OF INDIVIDUAL RICSAC TESTS ..............................179B.1 RICSAC 1 Broadside Oblique Impact.................................................................. 179B.2 RICSAC 2 Broadside Oblique Impact.................................................................. 182

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xi B.3 RICSAC 3 Front to Rear Oblique Offset Impact ................................................. 185B.4 RICSAC 4 Front to Rear Oblique Offset Impact ................................................. 189B.5 RICSAC 5 Front to Rear Oblique Offset Impact ................................................. 192B.6 RICSAC 6 Front to Side Oblique Offset Impact .................................................. 196B.7 RICSAC 7 Front to Side Oblique Offset Impact .................................................. 200B.8 RICSAC 8 Perpendicular Broadside Offset Impact ............................................. 203B.9 RICSAC 9 Perpendicular Broadside Offset Impact ............................................. 207B.10 RICSAC 10 Perpendicular Broadside Offset Impact ......................................... 210B.11 RICSAC 11 Front-to-Front Offset Impact .......................................................... 214B.12 RICSAC 12 Front-to-Front Offset Impact .......................................................... 217C: G-DaTAV ANALYSIS OF INDIVIDUAL NASS TESTS ..................................222C1.1 2010-08-037........................................................................................................ 223C1.2 2010-12-154........................................................................................................ 224C1.3 2011-04-127........................................................................................................ 226C1.4 2011-08-107........................................................................................................ 228C1.5 2011-08-112........................................................................................................ 229C1.6 2011-09-075........................................................................................................ 231C1.7 2011-09-091........................................................................................................ 232C1.8 2011-11-085........................................................................................................ 234C1.9 2011-12-049........................................................................................................ 235C1.10 2011-12-189...................................................................................................... 237C1.5 2012-08-064........................................................................................................ 238C1.12 2012-08-080...................................................................................................... 240C1.13 2012-12-016...................................................................................................... 241C1.14 2012-41-024...................................................................................................... 243

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xii C1.15 2012-43-014...................................................................................................... 244C1.16 2012-43-026...................................................................................................... 246C1.17 2012-43-106...................................................................................................... 247C1.18 2012-48-106...................................................................................................... 249C1.19 2013-12-059...................................................................................................... 250C1.20 2013-12-106...................................................................................................... 252C1.21 2013-12-112...................................................................................................... 253C1.22 2013-43-152...................................................................................................... 255C1.23 2013-76-094...................................................................................................... 256C1.24 2013-76-165...................................................................................................... 258C1.25 2013-79-139...................................................................................................... 259

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xiii LIST OF FIGURES Figure 1.1 Total crashes by vehicle type for 2007 to 2011 (429,013 vehicles) ........................................... 2 1.2 Colorado construction zone collisions by vehicle type for 2 007 to 2011 (4316 vehicles) ........ 2 2.1 Linear approximation of collision force (adapted from Figure 2.3[6]) .................................... 19 2.2 Damage profile approximation (a dapted from Figure 2.5 [6]) ................................................ 20 3.1(a) SAE Conventional vehi cle coordinate system ..................................................................... 27 3.1(b) Earth-fixed oriented ve hicle coordinate system .................................................................. 27 3.2 Central collinear impact .................................................................................................. ......... 28 3.3 Collision deformation as a linear spring system ...................................................................... 34 3.4 Collision pulse ......................................................................................................................... 36 3.5 Two-dimensional, single DOF ve hicle-barrier collision model ............................................... 37 3.6 Measured damage dimensions ................................................................................................ 40 3.7 Relationship between force per un it width and residual deformation ...................................... 43 3.8 Relationship between barrier impact velocity and residual deformation ................................. 47 3.9 Collision as a simple harmonic oscillator ................................................................................ 51 3.10 Impact with restitution effects (adapted from Figures 3.3 and 3.9) ....................................... 54 3.11 Braking force contribution to collision impulse .................................................................... 57 3.12 Paired force regions from impact deformation ...................................................................... 64 4.1 Non-central oblique and off-set impacts .................................................................................. 68 4.2(a) Planar collision trajectory angle determination ................................................................... 71 4.2(b) Oblique collision rota tion angle determination ................................................................... 71 4.3 Linear momentum vector parallelogram .................................................................................. 74 4.4 Linear momentum velocity change and PDOF ........................................................................ 74 4.5 Surface slope and fri ction relati onships .................................................................................. 77 4.6 Moment arm applied to produ ce rotation about mass center ................................................... 79 4.7 Rotational post-impact motions ........................................................................................... 81 4.8 Damage profile measurements ............................................................................................... .. 86 4.9 Damage centroid match at ma ximum engagement for PDOF ................................................. 90 4.10 Vehicle spring continuum ...................................................................................................... 91 4.11 Oblique impact PDOF acting at damage centroid ................................................................. 93 4.12 Friction of extended contact, scraping impacts ...................................................................... 97 5.1 Regression graph for original CRASH deformation based analysis ...................................... 124

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xiv 5.2 Regression graph for original SMAC momentum based analysis ......................................... 127 5.3 G-DaTA V piecewise damage match values versus RICSAC tests ................................. 132 5.4 G-DaTA V weighted average damage values versus RICSAC test ................................. 133 5.5 G-DaTA V piecewise damage match ve rsus NASS Bosch CDR data ............................. 138 5.6 G-DaTA V weighted average damage ve rsus NASS Bosch CDR data ........................... 139 A.1 Total crashes by vehicle type for 2007 to 2011 (429,013 vehicles) ...................................... 161 A.2 Colorado State Highway crashes by collision type 2007 to 2011 (235,884 impacts involving 429,013 vehicles) ............................................................................................................. ............ 162 A.3 Colorado State Highway crashes by roadway type for 2007 to 2011 ................................... 163 A.4 Colorado State Highway crashes by severity for 2007 to 2011 ............................................ 163 A.5 Colorado construction zone collisions by vehicle type for 2007 to 2011 (4316 total vehicles) .............................................................................................................................. ....................... 166 A.6 Colorado construction zone collision by collision type for 2007 to 2011............................. 167 A.7 Colorado construction z one collisions by roadway type for 2007 to 2011 ........................... 168 A.8 Colorado construction zone collisio ns by severity for 2007 to 2011 .................................... 169 A.9 Passenger vehicle construction zone collisions 2007 to 2011 ............................................... 170 A.10 Passenger vehicle construction zone collision severity 2007 to 2011 ................................. 170 A.11 Sport utility vehicle constructi on zone collisions 2007 to 2011 ......................................... 171 A.12 Sport utility vehicle construction zone collision severity 2007 to 2011 ............................. 172 A.13 Pickup and full sized van constructi on zone collisions 2007 to 2011 ................................. 172 A.14 Pickup and van cons truction zone collision sev erity 2007 to 2011 .................................... 173 A.15 Heavy vehicle construction z one collisions 200 7 to 2011 .................................................. 174 A.16 Heavy vehicle construction zone collision severity 2007 to 2011 ...................................... 174 A.17 Motorcycle construction zone collisions 2007 to 2011 ....................................................... 176 A.18 Motorcycle construction zone collision severity 2007 to 2011 ........................................... 176 B.1 Maximum engagement PDOF diagram for RICSAC 1 ......................................................... 179 B.2 Maximum engagement PDOF diagram for RICSAC 2 ......................................................... 183 B.3 Maximum engagement PDOF diagram for RICSAC 3 ......................................................... 186 B.4 Maximum engagement PDOF diagram for RICSAC 4 ......................................................... 190 B.5 Maximum engagement PDOF diagram for RICSAC 5 ......................................................... 193 B.6 Maximum engagement PDOF diagram for RICSAC 6 ......................................................... 197 B.7 Maximum engagement PDOF diagram for RICSAC 7 ......................................................... 201 B.8 Maximum engagement PDOF diagram for RICSAC 8 ......................................................... 204 B.9 Maximum engagement PDOF diagram for RICSAC 9 ......................................................... 208

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xv B.10 Maximum engagement PDOF diagram for RICSAC 10 ..................................................... 211 B.11 Maximum engagement PDOF diagram for RICSAC 11 ..................................................... 215 B.12 Maximum engagement PDOF diagram for RICSAC 12 ..................................................... 218 C1 NASS 2010-08-037 ........................................................................................................... ..... 223 C2 NASS 2010-12-154 ........................................................................................................... ..... 225 C3 NASS 2011-04-127 ........................................................................................................... ..... 226 C4 NASS 2011-08-107 ........................................................................................................... ..... 228 C5 NASS 2011-08-112 ........................................................................................................... ..... 229 C6 NASS 2011-09-075 ........................................................................................................... ..... 231 C7 NASS 2011-09-091 ........................................................................................................... ..... 232 C8 NASS 2011-11-085 ........................................................................................................... ..... 234 C9 NASS 2011-12-049 ........................................................................................................... ..... 235 C10 NASS 2011-12-189 .......................................................................................................... .... 237 C11 NASS 2012-08-064 .......................................................................................................... .... 238 C12 NASS 2012-08-080 .......................................................................................................... .... 240 C13 NASS 2012-12-016 .......................................................................................................... .... 241 C14 NASS 2012-41-024 .......................................................................................................... .... 243 C15 NASS 2012-43-014 .......................................................................................................... .... 244 C16 NASS 2012-43-026 .......................................................................................................... .... 246 C17 NASS 2012-43-106 .......................................................................................................... .... 247 C18 NASS 2012-48-106 .......................................................................................................... .... 249 C19 NASS 2013-12-059 .......................................................................................................... .... 250 C20 NASS 2013-12-106 .......................................................................................................... .... 252 C21 NASS 2013-12-112 .......................................................................................................... .... 253 C22 NASS 2013-43-152 .......................................................................................................... .... 255 C23 NASS 2013-76-094 .......................................................................................................... .... 256 C24 NASS 2013-76-165 .......................................................................................................... .... 258 C25 NASS 2013-79-139 .......................................................................................................... .... 259

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xvi LIST OF TABLES Table 2.1 Reproduced Table I from 49CFR Pa rt 561 10 2.2 Reproduced Table II from 49CFR Part 561 11 2.3 Reproduced Table 1 from SAE J2728, June 2010 14 4.1 Yaw moment of inertia empirical constants 84 5.1 Summary of RICSAC tests with coordinate transformation 122 5.2 Original CRASH based deformation analysis results 123 5.3 Original SMAC based momentum an alysis results 126 5.4 RICSAC test valu es versus G-DaTA V analysis 131 5.5 Summary of statistics 132 5.6 NASS reported Bosch CDR Tool data versus G-DaTA V analysis 137 5.7 Summary of Statistics 138 A.1 Highway construction zone collisions by vehicle type and collision type 2007 to 2011 as reported by the Colorado State Patrol 165

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1 CHAPTER 1: STUDY MOTI VATION AND OBJECTIVES 1.1 Study Motivation Advancements in highway geometric design, traffic control systems, intelligent vehicles and highways and vehicle crashwor thiness, as well as automotive handling, stability and control have progr essed significantly from the introduction of the automobile as a viable means of transportation in the early 20th century. However, to continue the trend of increasing transportation safety and decreasing severe and fatal injury crash events, it is imperative that Civil Engineers and Mechanic al Engineers work together as a multidisciplinary team. Cooperative engineering between disciplines provi des the most logical approach for ensuring the necessary analytic al tools. Robust methodologies should be developed and uniformly utilized to properly analyze collision events as an engineering team, and for each discipline's unique app lications and research interests. Traditionally, passenger car classification vehicles over the past 50 years have comprised the vast majority of full-scale vehicle impact testing for crashworthiness and research regarding collision dynamics. The cons ideration of light tr ucks, vans and SUVs into vehicle safety research has only taken pl ace within the past few decades. Accordingly, the remaining approximate half of collision-invo lved vehicles not part of the passenger car classification have comparativel y limited scope and breadth of impact testing conducted to date.

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2 Figure 1.1 Total crashes by vehicle type for 2007 to 2011 (429,013 vehicles) Figure 1.2 Colorado construction zone co llisions by vehicle type for 2007 to 2011 (4316 vehicles) Appendix A of this study investigates collision data collected from the Colorado Department of Transportation and the Colora do State Patrol for St ate highways. Appendix

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3 A focuses on the overall crash statistics for St ate of Colorado highway system for the study years of 2007 to 2011, as well as the collisions related to highway c onstruction zones during that same period. The following charts show the distribution of collision involved vehicles by vehicle type for all State highways and St ate highway construction zones for the study years 2007 to 2011. The collision data containe d in Figure 1.1 and Figure 1.2, demonstrate that about half of all vehicles involved in collisions within the State of Colorado fit the passenger vehicle classification, which is comprised of coupes, sedans, sports cars, and wagons. The remaining approximate half of all involved vehi cles include sport utility (SUV), light trucks and vans, heavy vehicles, re creational vehicles (RV), he avy machinery, motorcycles, bicycles, pedestrians and othe r types of transportation mode s. The study data does not indicate that half of all collisions are passenger-car-to-passenger-car collisions. Instead, the data simply indicates that half of all vehicles involved in public roadway collision events are passenger cars, which are two very different conditions to consider. The study data does not exclude the obvious consequenc e of mismatched vehi cle size and vehicle classifications interacting in highway collisi ons. Mismatched vehicles at impact impose the consideration of a whole new set of variables by a multi-disciplinary engineering research or investigative team. For the Civil Engineer, the ability to study vehicle collisi ons has important implications from a general sta tistical standpoint regarding tra ffic safety studies. However, collision analysis provides equal contributions towards collision mitigation measures and traffic management practices. Highway cons truction zones add a complication to the roadway system in that entering, naviga ting through and exiting from construction activities are unique to the activity of the construction project rather than roadway classification. When entering a construction zone, motorists typically must comply with incremental reductions in speeds resulting in a pace well below the normal regulatory speed limit for the roadway. Depending upon the comple xity of the construction activities speed reductions may be fluid in order to manage traffic negotiating the buffer zone and work arrea. Throughout the work zone, drivers may encounter reduced roadway width merge tapers, lane shifts, and detours or diversions, as well as tem porary stoppages controlled by

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4 temporary signalization or flaggi ng operations. To complicate the driving task even further, work zone activities can last in duration from a few hours to years. The dynamic or changing nature of work zones require the Ci vil Engineer to understa nd the stability and handling characteristics of the traffic stream, as well as the ability to assess the safety and effectiveness of work zone temporary traffic control treatments. Often the statistical analys is of collisions for high way safety purposes relies upon reported vehicle speeds or col lision severity parameters from police agency incident reports. However, the accuracy of those repor ted speed values can easily be called into question. Several uncertainties regarding the accuracy of reported speeds and collision parameters loom as follows: Were the speeds and severity levels determined by proper analysis procedures with proper data? Were speed recording devices such as speed traps, laser or radar measurement utilized? Were estimates determined by a trained investigating officer or authority? Were estimates provided by lay witne sses or collision-i nvolved drivers? Each layer of reported data where th ese questions are unknown can add a new source of unacceptable error. As such, the n ecessity for a reliable, accurate and broadly applicable generalized analysis methodol ogy that does not require extensive field investigation becomes critical for proper traffic safety evaluations. The understanding of vehicle velocities, velocity cha nges, accelerations and the effects of collision forces from a real-world perspective are critical for the Mechanical Engineer studying vehicle dynamics, crashworthiness and mechanical designs and failures. As an example, the use of controlled impact testing provides a mean s for experimentally evaluating algorithms for supplemental restrain t system deployments and the performance of primary occupant restraint systems. Contro lled vehicle testing also provides many of the initial evaluations of advanced warning and collision avoidance countermeasure systems. Computer analysis can model vehicle stability and handling, but ultimately controlled track testing determines whether design features perform as intended under normal operation. A

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5 Mechanical Engineer should have at least a working know ledge of highway design and traffic control operations to understand fully, predict and design for vehicle stability and handling characteristics. Knowle dge of traffic flow fundament als provides the Mechanical Engineer with a knowledge set to assess vehicl e performance under real istic conditions the motorist will encounter on public roadways. Furthermore, a Mechanical Engineer that studies crashworthiness and safety system perf ormance of real-world collision events must have reliable, accurate and relatively precis e analysis methods for determining collision kinematics and dynamics. The need for a reliable, accurate and broadly applicable generalized analysis methodology fo r determining impact velocities and severity levels of real-world collision events is necessary for assessing the performance of safety design features, improving vehicle handling, and optim izing crashworthiness and occupant safety. For the Forensic Engineer and the collision researcher, knowledge regarding vehicle velocities assists with determin ing factors related to collision causation, synchronicity, and contributions arising fr om the human-vehicle-roadway-environment interface. However, unless at-scene field inve stigators responding to a collision event are aware of the types of collision scene data necessary, planar dynamics trajectory-based analysis may not be feasible. Therefore, the need for a reliable, accurate and broadly applicable generalized model based upon vehicl e deformation is necessary for a thorough analysis of many real-world vehicle collision events. The advent of vehicle Event Data Recorders (EDR) in light trucks, vans and cars, along with Electronic Control M odules (ECM) in heavy vehicles has come a considerable way towards providing instrumented data fo r real-world collision events. Telematics and GPS tracking systems provide some data rega rding position and vehi cle speeds, but do not provide high enough resolution for those inte rested in studying collision forces and severity. However, at the time of this study, not every vehicl e is equipped with an EDR, ECM, or GPS tracking system. Additionally, th e uniformity of the data recorded by even the best systems available remain insuffici ently reliable enough to provide the data necessary for all types of collision events and combinations of colliding vehicles. Therefore, for at least the foreseeable future the need for a reliable, accurate and broadly applicable generalized model fo r vehicle deformation analysis is necessary for the proper study of many real-world ve hicle collision events.

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6 Currently accepted practices for motor vehicle collision-related deformation analysis have developed into reasonably accurate, reliable and commonly used methodologies for determining collision severity levels and collision velocities. However, applicability of the current models require pr oper structural stiffness values for each colliding vehicle, characteristic not only of the vehi cles analyzed but also specifically to the impacted surface. Fairly extensive test data from vehicle manufacturers and test laboratories provide frontal sti ffness coefficients for many pa ssenger cars and light trucks. A much smaller number of tests support deform ation analysis regarding side and rear surface stiffness coefficients. Additionally, only a limited number of heavy vehicle frontal barrier impacts, to include semi-tractors a nd buses, support frontal stiffness coefficient determination. Accordingly, either conductin g additional expensive and time consuming full-scale barrier impact tests must fill the data gaps, or more realistically, another reliable, accurate and broadly applicable generalized method provides the means for analyzing collisions between passenger vehicle and non-passenger vehicle impacts. 1.2 Statement of Study Objectives In response to the current shortfalls in modern vehicle deformation analysis techniques, the primary objectives of this st udy are to provide the fo llowing advancements to the current body of knowledge regarding vehicular impact deformation analysis: Develop reliable, accurate and broadly applicable generalized vehicle deformation methodologies eliminating th e dependence on multiple structural stiffness coefficients, regardless of the impacted surface and vehicle type involved. Develop and incorporate in ter-vehicular non-conservati ve frictional forces due to the colliding surfaces of vehicles slid ing during the approach velocity change of an impact into a reliable, accurate a nd broadly applicable generalized vehicle deformation model. Develop numerical algorithms allowing for input of deformation depths and widths at intervals which more accu rately explain and follow the unique deformation profile of a collision invol ved vehicle, thereby eliminating the

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7 trapezoidal rule reliance upon evenly spaced deformation measurements of 2, 4 or 6 intervals. Establish important relationships regarding impact forces as they relate to motor vehicle collisions and vehi cle deformation properties. Provide future researchers with enhan ced analytical tools necessary for the analysis of traffic collision events for the purpose of enhancing traffic safety, collision dynamics and vehicle design, as well as crashworthiness and occupant protection. Further the body of knowledge regarding the be havior of motor vehicles during real-world collision events. During the process of devel oping the generalized analys is methods within this study, the following advancements from the past 30 years are incorporated into forming the final, comprehensive generalized methods for determining impact velocity change: Consideration of external impulses to the impact produced by tire-ground nonconservative forces during the approach velocity change of an impact. Consideration of rotational effects produced by oblique or offset collisions that result in principle directions of for ce that do not pass through the mass centers of vehicles. When developing any engineering model, a balance between a comprehensive consideration of every feasib le condition or potential variable versus a simplistic model that depends upon only a few discrete and easily determined variables must be achieved. The risk of an overly comprehensive mode l lies in the dependence upon too many input parameters. Too comprehensive of a model becomes narrowly applicable to only a few conditions where the model variables are de terminable. An approach that is too comprehensive can create undue analysis complexity, not to mention the potential for multiple sources of random error due to vari ability within or between input parameters. However, too simplistic of a model likewis e becomes narrowly applicable and may not take into consideration all of the main fact ors that contribute to the event or process modeled, thus introducing systematic error to the analysis. History reveals that with th e development of any analys is methodology or scientific discovery, there will be individuals whom will criticize and refuse to accept results due to

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8 personal biases or lack of understanding of the methodological principles. For these reasons, the generalized models developed and presented in this study were tested using industry standard collision analysis valida tion tests (RICSAC)[100 ]. Additionally, the developed models were tested against othe r independently reported real-world vehicle impact data (NASS) [57]. In this way, the engineering community should have the necessary information to evaluate the accu racy, precision and efficacy of this study. Chapter 5 presents the evaluation of the generalized and comprehensive vehicle deformation analysis methods developed durin g the course and scope of this study. The study culminates into reliable, accurate a nd broadly applicable generalized vehicle deformation algorithms that do not require th e reliance upon multiple variables or analysis parameters that are often unknown or unknowable for real-world collision conditions. The development of reliable, accurate and broadly applicable generalized models sufficiently comprehensive in nature so as not to over-simplify collision dynamics, but straightforward and practical enough to consider the important known or knowable variables regarding both controlled environment and real-w orld collision events, form the overwhelming impetus of this study.

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9 CHAPTER 2: LITERATURE RESEARCH 2.1 Background of Collision Analysis 2.1.1 Basic Collision Analysis Methods The knowledge of vehicle collision velocitie s may be helpful in determining factors related to collision causation and timing for bo th criminal and civil liability issues, and research interests in collision avoidance and mitigation. The determination of velocity change, peak acceleration, and impulse applicatio ns during a collision event are crucial to understanding collision severit y. Determining these values assists in the evaluation of occupant kinematics, injury potential and in jury severity studies, as well as provide valuable data for research regarding occupa nt protection systems. However, unless field investigators responding to a co llision event are aware of the necessary data for a trajectorybased collision analysis and record the data promptly, a trajectory-based analysis may not be feasible. The lack of properly documented scene data remains the common conditional limitation for engineers and researchers calle d upon to analyze col lision events days, months or even years after the event occurred. Therefore, the need for a reliable, accurate and broadly applicable generali zed analysis model outside of the traditional trajectorybased momentum analysis is necessary for many real-world vehicl e collision events. 2.1.2 Passenger Vehicle Event Data Recording Systems On-board event data recording (EDR) system s as part of airbag system controls, record data that may provide speeds for finite periods of time at a lo w resolution of usually 1 to 2 Hz leading up to the impact (depe nding upon model year of the vehicle). Some vehicle EDR systems may capture imited acceleration and velocity change vector data from an impact event. The National Highway Traffi c Safety Administrati on (NHTSA) final rule on 49CFR Part 563 specifies th e uniform minimum requirements for accuracy, collection, storage, survivability and ability to image data from onboard motor vehicle collision EDR systems.

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10 It is important to note Part 561 does not require vehicles to have EDRs or EDR-like devices. Instead, Part 561 only specifies requirements for what must be recorded and in what manner should the vehicle have an EDR or EDR-like device. The intent of the final rule initially mandated compliance of vehicles manufactured for sale in the United States on or after September 1, 2010, later extended to September 1, 2012. The following are Tables I and II from th e rule that provide the data elements required for all light vehicles equipped with an EDR (tables extracted from 49 CFR Part 561). [1] Table 2.1 Reproduced Table I from 49CFR Part 561

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11 Table 2.1 continued Table 2.2 Reproduced Table II from 49CFR Part 561

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12 Table 2.2 continued

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13 Table 2.2 continued 2.1.3 Heavy Vehicle Event Data Recording Systems Engine manufacturers have equipped many heavy commercial vehicles with event data recording systems through the engine electronic contro l systems that may provide some collision related data, usually at lo w resolutions of around 1Hz. SAE J2728 is a recommended practice document published by th e Society of Automotive Engineers that applies to Heavy Vehicle Event Data Reco rders (HVEDR). SAE J2728 applies to heavyduty (HD) ground wheeled vehicles over 4545 kg (10,000 US pounds), commonly referred to as Class 3-8, which are inte nded to be compliant with current Federal Motor Vehicle Safety Standards (FMVSS) and/or Federal Mo tor Carrier Safety Regulations (FMCSR). J2728 defines the term heavy vehicle as a motor vehicle equipped with vehicle communication networks SAE J1708/J1587 an d or SAE J1939. J2728 focuses primarily

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14 on wheeled ground vehicles with standard on-bo ard power supplies (batteries). The intent of J2728 is to address the needs of OEM (ori ginal equipment manufacturer) original, OEM modified/adaptive, and non-OEM aftermarket systems and does not specifically exclude trailers and similar non-engine powered vehicl es. The following set of tables from J2728 outline the performance requirements of this recommended practice [2] [3]. Table 2.3 Reproduced Table 1 from SAE J2728, June 2010

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15 Table 2.3 continued The difficulty with heavy vehicle EDR (H VEDR) systems, aside from the absence of mandatory requirements for recordable HVEDR systems, sets squarely within the lack of uniformity in data recorded, if any, and how or whether the data can be commercially or

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16 privately imaged for analysis. Additionally, HVEDR systems are secondary to the control and operation of the various control system s for the engine and EPA required emissions systems. HVEDR system may also monitor ope rational controls of ancillary components of the drivetrain and braking systems, as outlined in the previous tables from J2728. HVEDR data availability is also dependent on the year and engine ma nufacturer rather than the chassis model of heavy vehicle. As such, each engine manufacturer will have a different electronic control module (ECM) or a combination of modul es (CPC, MCU, ACU), which may or may not have HVEDR capability. Imag ing of some HVEDR modules takes place through the main communication data link port or through a direct connection with the module if configured as a single ECM and HVEDR unit. However, the design of HVEDR modules as a combination of different control units functioning together makes imaging from the main communication data link port impe rative in order to eliminate false fault codes generated if the system does not remain connected. Removing each control unit of the engine control system from the vehicl e and connecting the un its with specialized harnesses while imaging the HVEDR and control systems can eliminate false fault codes. Connecting the HVEDR component of a collisio n involved heavy vehicle to a surrogate vehicle for download is the last option to im age data, but may bypass fault codes that would have otherwise been present on the collision-involved vehicle. Regardless of the HVEDR data imaging process, expens ive and specialized CAN bus communication systems, cables and software are necessary to access and image any data stored. Even if a commercial vehicle is equippe d with a HVEDR systems that provides event triggers such as sudden acceleration even ts, last stop records or safety restraint system triggers, there is no guarantee that data imaged is related to a particular collision of interest without proper analysis or study of th e record. Even then, so me collision events that may be catastrophic to a passenger vehi cle colliding with a heavy vehicle may be insufficient to produce an accelerating event trigger for the much more massive heavy vehicle involved in the event. As such, the development of engineering analysis methods applicable to large vehicles has become necessary for the determination of collision velocities and severity levels produced by impact events.

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17 2.1.4 Vehicle Tracking Systems Commercial vehicle dispatchers have traditionally used telematics systems to track the progress of shipments or passengers. So me Telematic systems provide not only GPS coordinate locations but GPS speeds of vehicl es along a travel path, with some advanced systems even providing flash updates of ve hicle maintenance schedules and monitored systems within the vehicle drivetrain. A tele matics system could randomly collect vehicle data at frequencies of several seconds to even hours depending upon the system configuration and service plan. Some systems gather data at specified chec kpoints along a route or transportation corridor, or at higher frequency inte rvals for local operations. The highest typical resolution for telematics system s used by local delivery, transit, refuse or other similar commercial operations record position, direction and speed at up to 1 Hz, again depending upon the system used. Many modern passenger vehicles have G PS navigation systems, or even onboard GPS alert systems capable of tracking loca tions, provide speeds and warn a central monitoring location of an emergency event such as an airbag deployment. Many onboard systems actively monitor reported stolen vehicl es, so as to provide law enforcement with the location or even a descrip tion of the occupants of a stol en vehicle if equipped with video monitoring. Such system also allows a monitoring station to control the drivers ability to control the vehicle, as well as lock the doors to prevent escape once the vehicle is stopped, and power terminated. However, the vast majority of passenger vehicle telematics and GPS systems, to include portable GPS systems, sample the posit ion, distance and speed of a vehicle at no greater than 1 Hz. Many systems would not store data unless an active route guidance operation was functioning at the time of a collision event. With the relatively low resolution, lack of uniformity in telematics and GPS information, and relative lack of wide use of these systems, the need for broadly applicable generalized engineering analysis methods applicable to large and small vehicles is again necessary for the determination of collision velocities and severity levels.

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18 2.2 Vehicle Deformation Analysis 2.2.1 Pioneering Studies and Computer Programs Permanent vehicle collision related deforma tion analysis has been part of technical collision investigations since the 1950's. In 1952, the Automobile Crash Injury Research Program (ACIR) was initiated. The purpose of ACIR was in documenting injury causation factors for vehicle occupants i nvolved in traffic collisions, wi th the primary objective of the program being that of in jury mitigation and prevention. By the mid-1960s, program participants consisted of 31 states providing over 50,000 cases for further research and analysis. The collision damage analysis methods of this early study consisted of comparing the permanent deformations of colliding vehicles re sulting from a subject traffic collision event to similar damages from tests conduc ted at known impact speeds. These methods produced results that were rarely entirely representative of a particular collision event and were justly considered only first approxima tion methods. In time, the anecdotal approach to collision severity analysis was finally di scarded altogether by pr operly trained analysts. [4] In September of 1966, President Lyndon Johnson signed the National Traffic and Motor Vehicle Safety Act and the National Highway Safety Act. This action established the authority to develop both the Federal Mo tor Vehicle Safety Standards (FMVSS) and the National Traffic Safety Agency, now know n as the National Highway Traffic Safety Administration (NHTSA). During the signing of these acts President Johnson stated, Auto accidents are the biggest cause of death and injury among Americans under 35. In 1965, 50,000 people were killed on the nations highways in automobile collisions. [4] Work done by Campbell in the early 1970' s developed the earliest analytical approaches focusing on vehicle damage. [5 ] Campbell observed a linear relationship between fixed barrier impact speeds and residual deformation of a ve hicle structure during full-scale impact testing using General Moto rs vehicles. Campbell recognized a linear collision force relationship betw een the resistance of a vehicl e structure to deformation per unit width and the residual measured deforma tion. The linear relationship equates to the following equation relating average deformatio n depth and width to a very simplistic idealized spring model:

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19 Figure 2.1 Linear approximation of collision force (adapted from Figure 2.3[6]) 2 02wBC EACGdw Campbells Equation 2.1 Where, A, B and G are stiffness f actors for a vehicle group or category C = Average crush or damage de pth across contact damage region w = overall width of contact damage region McHenry and others at Cornell Aeronautic al Lab (currently known as CALSPAN) conducted further research in the devel opment of SMAC (Simulation Model for Automobile Collisions) which improved upon Campbell's earlier observations. Specifically, McHenry also noted that like Cam pbells initial discovery, vehicles behave like linear energy dissipating springs. McHenry developed equations to consider the energy principles developed by Campbell relating them to the kinetic energy change produced by a plastic impact between colliding vehicles as th ey relate to the resultant vehicle velocity changes. Later adaptations of the work by Campbell and McHenry resulted in a Fortranbased mainframe computer analysis program known as the CALSPAN Reconstruction of Accident Speeds on Highways, or CRASH. CRASH provided a first approximation of vehicle velocity change for input into the more detailed momentum based SMAC (Simulation Model for Automobile Collisions) computer analysis program, which was

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20 intended to predict impact speeds. CRASH III was the last CALSPAN produced edition of the CRASH program. [6] [7] [8] CRASH III and its predecessor versions restricted the measurement of damage profiles to 2, 4 or 6 evenly spaced deform ation depth measurements over the contact damage width on a vehicle surface. The trapez oidal rule was utilized to estimate the damaged area over these arbitrary widths and evenly spaced measurement intervals. The following formulation determined the deformation work for a damage profile of two, four and six evenly spaced deformation depths: Figure 2.2 Damage profile approximati on (adapted from Figure 2.5 [6]) Two measurement points: 22 12112226 AB ELccccccG 2.2 Four measurement points: 1234 2222 123412233422 2 3 223 6 A cccc L E B ccccccccccG 2.3

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21 Six measurement points: 123456 222222 123456 12233445562222 2 2222 5 5 6 A cccccc L E cccccc B G cccccccccc 2.4 The trapezoidal methodology was troublesome in the fact that vehi cles involved in collision events outside of full overlap barrier tests rare ly exhibited damage profiles effectively measured using evenly spaced inte rvals. The resulting error in deformation measurements resulted in over or under-approxi mations of damage energy determinations. Several commercial versions became availabl e over the approximate 40 years since the original study by Campbell. However, with only marginal improvements in many of the restrictions of the CRASH III program, and a continued reliance on the trapezoidal rule for deformation energy determination. The full version of the CRASH III and SMAC programs calculate impact speed and velocity change for the crash vehicle, or vehicles, using both damage-based and momentum-based priciples. With only the know ledge of vehicle damage profiles, CRASH III can only determine velocity change of two impacting vehicles or the barrier equivalent velocity (speed related to impacting a fixe d barrier) of a single vehicle. An important assumption of the CRASH III program is that the energy dissipated through work during the approach phase of an impact can be approximated from the residual deformation measurements using the trapezoidal met hodology. The following are the basic equations for determining the velocity change of colliding vehicles: 2 1 1122CrushEm V mmm 2.5 1 2 2122CrushEm V mmm 2.6 Current CRASH-based programs cannot determine vehicle impact speeds. Postimpact trajectory histories are needed to determine impact speeds using a planar momentum-based analysis once the velocity changes for the vehicles were determined using the CRASH-based algorithms. The CRASH III limits analysis to two vehicles involved in a single impact, or one vehicle colliding with a fixed and non-yielding object.

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22 In the mid-1980s McHenry et al., proposed upgrades to the CRASH III algorithms to include restitution effects to allow for a br oader application of th e CRASH algorithms to low velocity impact events. A CRASH IV ve rsion, which was never adopted or completed, was intended to consider restitu tion effects. [9] The inclusion of restitution effects allowed for the consideration of the ve locity change elements that resulted during the separation phase of an impact. McHenry proposed the following equations that originally appeared in the 1981 edition of the CR ASH III Users Manual: 2 2 1 1121 2TOTAL Crush TOTALEeE E m V mmm 2.7 1 2 2122TOTALEm V mmm 2.8 Due to the limitations of a Fortranbased punch card process for 1970s and 1980s vintage main-frame computations, damage profiles remained limited to even numbers of uniformly spaced deformation depths of either 2, 4 or 6 measurements in order to save computing time. The establishment of damage measurement protocols standardized the measurements and interpretation of dama ges for CRASH related program use. The differentiation between contact and induced damages was established. Protocols addressed how to account for effects of bumper system underride/override. Additionally considerations were made account ing for bowing of a vehicle stru cture that typically occurs for high-speed lateral impacts, or shifting of front and rear structures due to high-speed oblique collisions. [10] Many of the original measurement protocols remain applicable. However, one objective of the current study is to eliminate the arbitrary evenly spaced measurement protocol in lieu of measurement th at accurately describe damage shape with as many or as few measurements and at any widt h interval necessary to adequately describe the deformation to a vehicle structure. A significant limitation of the CRASH III program rested on the assumption that all vehicles within discrete wheelbase dimensions possessed the same dimensional, inertial and structural stiffness characteristics. Tabulat ed dimensional, inertial and stiffness data into separate vehicle categories within internal default data tables were used during the analysis process. However, there is no appare nt justification for th is arbitrary grouping by

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23 wheelbase, and testing has shown that wide variations of such properties exist between and within vehicles of similar wheelbase ranges. In particular, the assigned rear-end and side vehicle stiffness coefficients were incorrectly determined and do not resemble actual rearend or side stiffness character istics of production vehicles. In an attempt to evaluate the veloc ity change from rotation, the CRASH III algorithms incorporated an effective mass concept to account for rotational effects resulting from the moment about the vehicle mass center created by an offset application of the principle direction of force. An offset applic ation of the principle di rection of force, or PDOF, occurs during oblique and/or offset co llisions. The effective mass concept has been demonstrated to produce more reliable estimate s of velocity change since its introduction. [11] [12] [13] In general, the limiting assumptions and analytical constraints of CRASH III and similar CRASH-based programs are as follows: [14] [15] [16] Deformation energy is equal to the impact kinetic energy loss. Collisions are inelastic, and the centroids of damage reach a common velocity. Sliding between vehicles o ccurs during the separation pha se of the impact and not during the approach velocity change phase, and, therefore, is not accounted for in the velocity change analysis. Tire-ground forces are negligible (non-cons ervative forces external to the impact) or minuscule as compared to the collision force. Damage profile measurements are limited by evenly spaced measurements of 2, 4 or 6 deformation depths over uniform spaced measurement widths across the contact damage width, excludi ng induced damage regions. Vehicle structural stiffn ess defined by categories of vehicl es by type (i.e., car, truck, van), and wheelbase lengths, all assumed to have similar inerti al and structural stiffness characteristics. 2.2.2 Modern Damage Deformation Studies In the 1990s topics regarding vehicle deformation analysis again became a new focus of research. Largely due to the power of personal compu ting and an increased market for vehicle collision analysis software, researchers again focused on improvements

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24 augmented by the already accepted engineer ing fundamentals that comprise the foundations of vehicle deforma tion analysis. One of the orig inal and primary research focuses surrounded the elimination of arbitrary groupings of A and B stiffness coefficients by wheelbase and/or vehicle classification. CRASH-based analysis advanced towards relying on a robust library of vehicle and surf ace specific structural stiffness coefficients for many cars. However, as extensive the library to date may be, many vehicles have either a dearth of structural stiffness information or none whatsoever. In re sponse to the database deficiencies, studies also de veloped methods for determining crash specific stiffness coefficients. Many of these attempts rely upon simplifying assumptions to complete a deformation-based analysis when stiffness data was not otherwise available for one of the involved vehicles. [4][1 5] [17] [18] [19] Additional research spanning from the 1990s to present day have provided further adaptations of the original CRASH-based algorithms. The formulation of reliable analytical tools for determining the velocity change and peak acceleration levels of minor damage, low-velocity impacts ( V 16 kph (10 mph)) became a major focus. Velocity change equations derived from linear momentum, energy, and restitutio n principles, also known as MER methods, considered the cont ributions of collision restitution for lowvelocity impacts, often ignored for higher ve locity impacts. The developed models were primarily applicable to collinear and central impacts having a PDOF passing through the mass centers of the involved vehicles [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] 12 2 2 2 12 12 1 1 1CrushEmm e V m mme m 2.9 2 12 1m VV m 2.10 Research models to date have incorporat ed not only restituti on effects but also tire/ground non-conservative force contributions. This engineer authored multiple research projects regarding not only the development of collinear low-velocity impact collision deformation methodologies but also tested the accuracy of the algorithm accuracies against full-scale vehicle-to-vehicle collision tests. The models developed by this engineer and

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25 other researchers provide reliable methods fo r predicting the veloc ity change and peak acceleration levels of primarily collinear central impacts, or minor impact events where the struck vehicle is stationary at impact. Currently accepted practices for motor vehicle deformation analysis have developed into reasonably accurate, reliable and commonly used techniques for predicting collision severity levels and collision velo cities, dependent upon the availability of structural stiffness values for each colliding vehicle. A lthough extensive test data is presently available for frontal stiffness coeffi cients for many cars and light trucks, few tests are available for side and rear surface vehicle specific stiffness coefficients for any vehicle type. Additionally, only a limite d number of heavy vehicle fr ontal barrier impacts, to include semi-tractors and buses, are ava ilable for frontal stiffness coefficient determination, and none from the side or rear structures of these unique vehicles. Accordingly, either additional expensive and time-consuming barrier tests are needed to fill these gaps, or another method of analysis must be developed.

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26 CHAPTER 3: COLLINEAR CENTRAL IMPACTS; DEVELOPING BASIC DEFORMATION ANALYSIS PRINCIPLES To achieve the objectives of this st udy, it becomes necessary to derive the foundational principles behi nd vehicle deformation anal ysis. The single-degree-offreedom collinear collision represents the most simplified collision event, as well as the most logical starting po int. Derivation and presentation of the fundamental principles allow for the expansion of the physics behind deform ation analysis to account for more complex and generalized formulations, the ov erwhelming impetus of this study. 3.1 Coordinate System A vehicle-fixed coordinate system is commonly used to describe motion in vehicle dynamics. SAE J670 establishes a commonly used vehicle-fixed coor dinate system as shown in Figure 3.1(a) [33]. The vehicle-fixe d coordinate system follows the right-handrule; positive x-axis oriented from the vehicle mass center forward (longit udinal axis), the positive y-axis from the vehicle mass center and towa rds the right side (passenger side) of the vehicle (lateral axis), and the positive z-axis from the vehicle mass center and downward (vertical axis ). Rotation about the x-axis towards the right side of the vehicle produces positive roll (y x z = p). Rotation about the y-axis that results in raising the front of the vehicle produces positive pitch (z x x = q). Clockwise rotation about the z-axis produces positive yaw (x x y = r). The SAE vehicle-fixed coordinate syst em, however, creates confusion when analyzing collision dynamics. Collision analysis typically utilizes an inertial earth-fixed Cartesian coordinate systems with a conceptually more logical vertical (+) z-axis oriented upwards from the earths surface. An earth-fixed vehicle local coordinate system has the (+) vertical axis pointing upwards from the ve hicles center of mass. By applying the righthand rule, the (+) longitudinal x-axis orients towards the vehicle front and the lateral (+) yaxis towards the drivers side or left side of the vehicle. The earth-fixed coordinate system has its roots in aviation, where pilots prefer to measure altitude with positive rather than negative z-axis values. For this same reasoning, an earth-fixed oriented local coordinate system for a vehicle has wide acceptance for motor vehicle collision analysis. This study

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27 will utilize the earth-fixed oriented vehicle coordinate system as shown in Figure 3.1(b) unless otherwise specified. Figure 3.1(a) SAE Conventional vehicle coordinate system Figure 3.1(b) Earth-fixed oriented vehicle coordinate system 3.2 Equation of Motion for Collinear Central Impacts The basic equation of motion for a simple single-degree-of-freedom, collinear and central impact between two vehicles assists in developing the basic equations needed for vehicle deformation analysis. Straight-forward adaptation of the e quation of motion from X Z y +Roll, p +Yaw, r +Pitch, q Vertical cm (x) (z) (y) Vertical +Roll, +Yaw, +Pitch, cm

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28 a single degree of freedom, collinear system into a multiple degree-of-freedom, multidimensional system results from recognizing forces, accelerations and velocities are vectors. However, the vast majority of moto r vehicle collisions are easily modeled using planar dynamic principles without introducing significant error or questionable accuracy into the analysis results. The basic principles equations devel oped in this chapter translate to collision forces acting oblique to the and mass centers of the colliding vehicles, thus resulting in post-impact rotational effects. 3.2.1 Conservation of Energy Figure 3.2 depicts a planar, collinear, h ead-on, single degree of freedom (DOF) impact event, in which motion only occurs w ithin a single horizonta l coordinate of the system. Figure 3.2 depicts the collision im pulse acting along a line between the mass centers of the colliding vehicles, such that all energy is expended through vehicle deformation and horizontal motion. Such an im pact results in rectilinear motion along only one generalized coordinate, q1. For rectilinear motion, all ot her potential coordinates, or qi+1s , for the system result in arbitrary constraint equations equal to zero in their respective directions; i.e., q1= real equation of motion, and qi+1=0 (for i=1,,2n-1 coordinates). Figure 3.2 could easily depict a collinear rear-end collision event, or a moving vehicle striking any stationary vehicle in any orientation, as long as the collision impulse results in a principal direction of force (P DOF) that acts along the mass centers of the colliding vehicles. Figure 3.2 Central collinear impact

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29 The collision depicted in Figure 3.2 results in a change in the overall kinetic energy, T for each vehicle. Additionally, work done by th e system in the form of vehicle damage, or potential energy, V The work done by the system in creating permanent vehicle deformation can be effectively modeled as two linear springs in series compressing against each other while exerting an equal but opposit e force in accordance with Newtons third law. The Conservation of Energy for the collinear, central, single DOF vehicle-to-vehicle collision system as shown in Figure 3.2 can be represented in a generalized coordinate system, q : [34][35][49] 222222 1212111 12 222121212initial initial final finalmmmmkkqqqqqq 3.1 Where, m1 and m2 are the masses of vehicles 1 and 2 1initialq and 2initialq are the initial generalized lin ear velocities of vehicles 1 and 2 1 f inalqand 2 f inalq are the final generalized linear velocities of vehicles 1 and 2 1 k and 2 k are the spring constants representi ng each the structural stiffness for each vehicle 1 q and 2 q are the inward deformations to each vehicle resulting from the impact Equation 3.1 simply states the kinetic energy brought into the system by two colliding vehicles is equal to the kinetic energy of the vehicles following the impact, as well as the potential energy stored in each vehicle structure spring during deformation. However, potential energy is not actually s tored during deforma tion. Otherwise each vehicle would rebound back to thei r original shapes following an impact event. Instead, the stored energy is converted to work according to the structural resi stance characteristics of each vehicle, producing permanent deform ation while the collision impulse exerts maximum compression upon each vehicle structure. Applying Hookes Law as stated in Equa tion 3.2, allows for the relationship between the maximum deformation of a vehicle structure and impact energy. [35][49]

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30 21 2spring s pringspringk E 3.2 Where, s pringk = spring constant; sp rings ability to re sist changing length = linear compression of the spring 3.3.1 Lagranges Equations Analytical dynamics treats two-vehicle co llision systems as a whole, allowing for the analysis of the system using scalar quantities such as kine tic energies, potential energies and work when determining the equations of motion for a dynamical system. Powerful methods of formulating equations of moti on for various mechanical systems were developed by Lagrange (17361813) thorugh the application of DAlemberts Principle expressed in terms of generali zed coordinates. The applica tion of Lagranges equations bypasses and/or eliminates many of the tedi ous aspects of vector dynamics. Vector dynamics based directly upon the applicatio ns of Newtons second law of motion concentrates on forces and motions that may not be easily determined form a free-bodydiagram of many dynamical systems. Lagranges equations are differential equa tions which consider the total energy of the system and virtual work instantaneous in time when developing the equations of motion as they relate to th e conservative and non-c onservative forces acting on a system. The stepwise application of Lagranges equations resu lts in the intermediate determination of the system linear momentum, but also allows for th e indirect determination of reaction forces acting within a dynamic system through the use of Lagrange multipliers. The general form of Lagranges equations requires the partial differentiation of each term of the Conservation of Energy statement for each generalized displacement and generalized velocity of the system, as expres sed in Equation 3.1. [34] [35] cnc iidLL QQ dtqq 3.3 Where, L TV is the Lagrangian, (for i=0,...,n generalized coordinates) T = kinetic energy V = potential energy

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31 Qc= generalized conservative forces (due to potentials) Qnc= generalized non-conservative forces Respective to a simple holonomic impact, th e generalized conserva tive forces result from the potentials that are expresse d by the Lagrangian, and therefore, Qc = 0. Initial considerations in this chapter assume generali zed the nonconservative forces due to intervehicular friction and tire-roa dway forces are also zero; Qnc = 0. The application of Lagranges equations to this simple holonomic system starts with th e grouping of like terms for the kinetic energy, T contributions to the system as follows: 22 22 12 11' 22'11 22 Tmmqqqq 3.4 The potential energy, V contribution to the system is due to deformation of the involved vehicles, which is the final statement in Equation 3.1 as follows: 221 12 212Vkkqq 3.5 The kinetic energy term, T of Equation 3.4 does not contain any generalized displacements, or qi terms. Likewise, the potential energy term, V of Equation 3.5 contains no generalized velocities, or iq terms. Solving the Lagrangian with respect to the first element of Equation 3.3 results in only the partial differentials of the kinetic energy statement, T for the system returning non-zero values when differentiate d with respect to generalized velocities, iq of the system. With respect to the second element of Equation 3.3, only the potential energy statement, V for the system will return non-zero values when differentiated with respect to generalized displacements, qi, of the system. 3.3.2 Deriving the System Momentum Solving the partial differential equations of the Lagrangian with respect to the generalized velocities re sults in the following: 22 11 22 2211 1212 11 222211 22initial final initial final i initial final initial finalL mm qq mm qqqq q qq 3.6

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32 When evaluating the Lagrangian with resp ect to the generalized velocities of the system produces the following useful statemen t for the generalized linear momentum of the system: 121122initialfinal initialfinal iL mmqqqq q 3.7 For a conservative system, Equation 3.7 equa ls zero and provides the statement of the Conservation of Linear Momentum for the two-vehicle, single DOF collision system shown in Figure 3.2. The generalized velocities initialqi and f inalqi represent the initial and final velocities of the ith vehicle, respectively. The quantity initialfinalqiqi represents the velocity change due to the exchange of linear momentum, or iv, of the ith vehicle for the two-vehicle, single DOF collision system. Equation 3.7 can be rearranged to show the relationship between the linear momentum relate d velocity changes of a two-vehicle, single DOF collision system, establishing important impulse-momentum considerations for the system. 12 12 121122initialfinial initialfinalmm mmqqqq vv 3.8 3.3.3 Equation of Motion Solving the time derivative of the genera lized momentums for the system produces the forces with respect to the time-rate-change of linear momentum for the collision event as follows: 12 121122 1122initialfinal initialfinal i initialfinal initialfinaldLd mm dtdt mmqqqq q qqqq 3.9 When solving the position partial derivatives, qi, of the Lagrangian for the potential energy terms of Equation 3.1 yields the impact force with respect to the vehicle deformations as follows:

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33 22 1211 1212 221212iL kkkkqqqq qqq 3.10 Summing the results of the partial di fferential equations in accordance with Equation 3.3 results in the final form of th e Lagrange determination of the generalized equation of motion for a two-vehicle, coll inear single DOF holonom ic central impact collision system: 1201 2112212initialfinal initialfinalmmk kqqqqqq 3.11 Equation 3.11 is the same equation of motion resulting from a vector dynamics derivation approach using Newt ons second law of motion for th is single DOF system (one coordinate along the q1 direction). In the vast majority of vehicle-to-vehicle collisions it is reasonable to assume that the initial accelerations for both vehicles are e qual to zero. Since impulse has yet to initiate any action upon either vehicl e at initial contact when t = 0, the generalized equation of motion for a two-vehicle, collinear single DOF collision system as expressed in Equation 3.11 can also be expressed in terms of the tim e-rate-change of the velocity changes of both colliding vehicles, and the deformation forces as follows: 12 1201122mm dtdtvv kckc 3.12 12 121122mm dtdtvv kckc 3.13 where, k1 and k2 are the spring constants for each vehicle during deformation c1 and c2 are the inward deformation, or crush, extents for each vehicle The linear spring terms of Equation 3.11, 3.12 and 3.13 represent a two vehicle collinear impact that behaves as two lin ear springs in series under compression.

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34 3.4 Impulse-Momentum of the System The following is the general expression of the relationship between force applied and the change in spring length from Hooke s Law for a linear spring: [34][35][49] spring springspringk F 3.14 impact vehicledamagekc F 3.14 The spring deformation, spring, in Equation 3.14 is analogous to deformation to a motor vehicle, cdamage, in Equation 3.15. The spring constant, kspirng, in Equation 3.14 represents a linear springs resistance characteristics to change in overall length during force application. The spring c onstant for a motor vehicle, kvehicle, in Equation 3.15 similarly represents a vehicles structural characteris tics for resisting damage deformation when subjected to the impulse of a collision event. An ideal un-damped linear spring behaves as a simple harmonic oscillating system. Howe ver, a vehicle compresses to its maximum deformation at t= 2, or 1/4 of a full period of osci llation, and remains damaged under sufficient force without continuing through an entire period or repetitive periods of oscillation as shown in Figure 3.3. Figure 3.3 Collision deformation as a linear spring system Equation 3.13 provides the Newtons second law statement for a conservative system, in that the sum of the external for ces acting upon the system must be equal to the sum of the conservative forces due to potentials. From the Conservation of Linear Momentum statements of Equation 3.8, the su m of the momentum changes of the vehicles must also be equal to zero since momentum is conserved. Therefore, the sum of the forces Simple harmonic oscillation of the form: sincos A tBt

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35 due to the Hookes Law representation of the deformation forces (potentials or conservative forces) in Equation 3.13 must al so equal zero since momentum is conserved. From this understanding, the impulse-momentum relationship for the system in terms of the impact deformation forces are represen ted in Equation 3.16 with respect to the ith vehicle of the collison system of Figure 3.3. 12 112212 12 12 1211220 0external externalexternaldtdt dtdtvv kckcmm vv mm F kckc FF final initialii iiiiimpulset dt ttmvkctJ FF 3.15 Where, initialt = 0 at the initiation of the collision event f inalt= time at peak impulse of the collision event finalinitialttt = time interval to reach peak impulse and, J = peak collision impulse The external forces of a collision increase to their peak values from the initiation of the impulse at t=initial or 0, until the maximum force is reached at t=final or peak as shown in Figure 3.4. The area under the collis ion pulse curve represents the momentum change as a result of the collision impulse The time for impulse to reach its peak, t = ( tfinaltinitial), provides the necessary information for determining the peak collision acceleration at the mass centers of the vehicles. Determiing the peak acceleration of a vehicle resulting from an impact is essentia l for evaluationg the iner tial response of vehicle occupants during any given collision event.

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36 Figure 3.4 Collision pulse Approximating the general shape of a collision pulse as triangular demonstrates how the acceleration increases towards the peak where maximum force acts equal-andopposite between each vehicle at t=final From the impulse-momentum relationship of the impact event, Equation 3.17 determines the ti me to reach the maximum force due to the collision between the two vehicles. i i impactmv t F t is the same for both vehicles 3.16 3.5 Collision Force from Vehicle Deformation Figure 3.5 demonstrates the linear spring m odel relationship of a vehicle colliding with a barrier. The kinetic energy of a vehicle as it approaches the barr ier transfers to work to compress the vehicle stru ctural spring, synonymous with producing permanent deformation to the vehicle struct ure. First, the simplest of m odels will be developed where the vehicle strikes a non-deformable, infinite mass barrier squarely, producing a uniform damage profile across the vehicle width.

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37 Figure 3.5 Two-dimensional, single DO F vehicle-barrier collision model The relationship between permanent residual deformation and the barrier impact velocity is the barrier impact velocity, BIV, or barrier equivalent velocity, BEV. A vehicles struck surface can be considered as a continuum of j=n linear springs in the form of Equation 3.15 oriented perpendicular to the c ontact surface with the barrier. Each of the j=n springs has a resistance to compression, kj, and its own characteristic change in spring length as a function of the collision time, cj(t) when subjected to a collision force with respect to time at each of the springs in the continuum, Fj(t) Defining the compression of the jth spring across the vehicle width as the average between the two damage depths that bound the confines of the damaged region provides the following expression: 12()jj jcc t c 3.17 Where, cj and cj+1 are successive measurements taken parallel to the damaged surface of the vehicle from its undamaged position, for j=n measurements across the full damage width cj(t) = the average deformation depth between measured points cj and cj+1 The total of all forces acting upon the vehicle during the barr ier collision phase, while neglecting non-conservative forces, is represented by Equation 3.18.

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38 00() ()nn j jj jjttkc F 3.18 Maximum engagement and, therefore, maximum deformation to the vehicle colliding with the barrier occurs at the peak impulse time, tfinal, as expressed in Equation 3.17. Maximum engagement with the barrier co incides with the moment where the kinetic energy of the vehicle is completely absorbed by the compression of th e colliding vehicle, resulting in maximum deformation of the ve hicle at maximum impulse. The ideal linear spring system completely converts the kinetic en ergy of the collision into work to compress the array of springs across the contact surf ace during the barrier impact, resulting in permanent plastic deformation with no restitution. Integrating both sides of Equation 3.19 with respect to compression depth, dcj, results in the expression for the kinetic energy dissipated as work to compress the continuum of j ideal springs of the system. 00 00 2 001 2 jjnn j j jjj jj nn j jjj jjccdckcdc F ckc F 2 111 2nn j s pring jj jjUkc T 3.19 Where, Tj = system kinetic energy compressing the jth simple spring element Uspring = total work to produce the total deformation to the vehicle as a simple spring The derivation of Equation 3.19 is important in establishing the relationship between the peak collision force, FI, and the work, Utotal, done to produce permanent deformation to the vehicle/spring continuum at maximum impulse. Except under the most severe of collisions and deformation patterns, the structure of a vehicle involved in a barri er or vehicle-to-vehicle imp act will experience some degree of restitution as shown in Figure 3.4. The residual deformation extent measured perpendicular to the impact surface after a collision, cR, is typically documented after restitution has occurred. Each cj of Equations 3.19 and 3.20 represents the maximum deflection of the vehi cle/spring system, or cI, at maximum engagement. When

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39 considering restitution, the maximum deformati on of the vehicle determines the work to produce permanent deformation of Equation 3. 20. Equation 3.21 expresses the relationship between the compression of the vehicle stru cture at maximum impulse and the residual damage measured at some time after the impact: I Re jjjccc 3.20 Where, I jc maximum deformation of springj during the impact at time = tfinal R jc residual/measured deformation of springj at some time following the impact measured perpendicu lar to the impact surface e j cmaterial restitution effect not measured for springj At this juncture, it is appropriate to describe the measurement of R j cvalues for a given deformation pattern. Figure 3.6 shows an exemplary damage profile for a frontal impact. Measurements are taken from a defo rmation point on the vehicle surface and perpendicular to the theoretical undamaged pos ition of the vehicle. Measurements initiate from either the left front (drivers side front ) or right front (passenger side) of the vehicle along the damage width. Ideally, damage measurements should iden tify locations of lin earity within the deformed region, or regular intervals that best describe the defo rmation profile. Each w is the distance between points measured parallel to the impacted surface of the vehicle. A table, such as that shown in Figure 3.6 can facilitate the recording of deformation points measured along the total width of contact and induced bending to the vehicle structure and surfaces resulting from the impact event.

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40 Description of point measured Distance from Left Front (0) Distance to undamaged profile Left front corner w0: +0 cm (0 inches) c0: +81 cm (32 inches) Left front frame rail w1: +46 cm (18 inches) c1: +81 cm (32 inches) Center bumper reinforcement bar w2: +92 cm (36 inches) c2: +71 cm (28 inches) C3C(n-1) w3w(n-1) c3c(n-1) Right front corner w(n): +183 cm (72 inches) c(n): +51 cm (20 inches) Figure 3.6 Measured damage dimensions So that, 01 18181 81 22 RR Rcmcm cmcc c(32 inches) 12 28171 76 22 RR Rcmcm cmcc c (30 inches), and so forth. And, 110(460)460cmcmcmww (18 inches) 221(9246)46 cmcmcmwww (18 inches), and so forth. Now consider the continuum of springs across the total width of damage, from w0 to wn, as shown in Figure 3.6. The maximum deformation profile becomes a function of the damage width, cw. Equation 3.22 expresses the total force applied per unit width of deformation by dividing both sides of Equation 3.18 by the width differential. 0011nn I I j jj jj jjkc F ww 3.21 0 R c 1 R c 2 Rc R nc 0w 1w 2w nw 2 R c R nc 1 w 2 w nw 1Rc

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41 Rearranging Equation 3.21 provides the total force applied per unit of total damage width to the differential system spring constant per unit width, j j k w. Equation 3.22 is expressed only in relation to the total compression of the vehicle structure, I c across the total width of the vehicle as a uniform spring constant k value for the ith vehicle, ki, over the entire width of deformation for the ith vehicle, wi. I I i i i ik c wF w 3.22 The relationship for I j c as expressed in Equation 3. 20 is then substituted into Equation 3.22 to produce the relationship between force per unit width and total deformation of the vehicle stru cture, shown in Equation 3.23. I R IRI iii iiii iii iwwwkkk F cccc w 3.23 Equation 3.23 considers the deformation, or crush, measured following an impact and the restitution of the mate rials following the impact that cannot be directly measured. Equation 3.24 accounts for the maximum deformati on of the impact at the moment of peak force application. At this point, it is appropriate to define components of Equation 3.24 in terms of vehicle-spring stiffness coefficients A a nd B for the unique im pacted surface of a vehicle as follows: I i i i ik wc A (force)/(length), which is the force per unit depth to initiate damage to the vehicle and applied throughout the application of external forces resulting from the collision i i ik wB (force)/(area), the generalized spring constant associated with resistance to continued deform ation/spring compression of the vehicle structure as a result of the external forces of the collision

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42 The presented definitions of A and B stiffness coefficients unique to the impacted surface, provide an expression for the maximum deformation perpendicular to the impacted surface at the jth location for the ith vehicle at the moment of peak force application to a vehicle, FI. IR i jj i A cc B 3.24 Equation 3.25 establishes the following important characteristic regarding deformation produced by an impact event as it relates to a vehicles unique stiffness parameters: The maximum deformation at application of peak impulse is a function of the measurable residual damage following the impact event and the ratio between the force per unit width necessa ry to initiate deformation and the force per unit area to co ntinue permanent deforma tion to the structure. The Central Impact Force-Deflection Model of Equation 3.25 is defined by substituting the value for I j c from Equation 3.24 into E quation 3.21 and solving for the peak collision force. Figure 3.7 provides a gra phical representation of the linear properties of Equation 3.25. 00 0 nn n RR I i ji ii jjjj jj j iA cwcw FBAB B 3.25

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43 Figure 3.7 Relationship between force per unit width and residual deformation 3.6 Central Impact Work/Energy Model The Central Impact Force-Deflection Model of Equation 3.25 provides the peak external force acting on a vehicle while collid ing with a barrier or another vehicle. The peak external force of Equation 3.25 acts equal in magnitude but opposite in direction of application to the peak external force applied to the barrier or opposing vehicle in accordance with Newtons third law. Using the expression in Equation 3.19, the work required to produce permanent deformation to a motor vehicle structure, or any other object for that matter that behaves in a similar manner, can be determined based on the A and B stiffness characteristics of the ith vehicle involved in an impact event by substituting Equation 3.24 into Equation 3.19. The resultant expression of Equation 3.26 provides the Central Impact Work/Energy Model. 2 01 2 n ij jI jUk c and, IR i jj iA cc B and, i ik B 2 01 2 n i ii ji R j iA Ucw B B IF Bc A w I c Rc Fpeak

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44 2 2 022n i R i i ijj j iR jc B A Ucw A B 3.26 Where, Ai and Bi = unique structural stiffness values for the ith vehicles impacted surface R jc the residual deformation, or crush, of the jth deformation measured on the vehicle perpendicular to the damaged surface from its undamaged dimensions wj = width to the jth deformation, measured para llel to the damaged surface The same expression for the Central Impact Energy/Work Model can be derived from Work/Energy principles us ing Equation 3.25 as follows: 00 00 00RR jjnn R I jii jj jj nn RRR I jii ijjjj jjcccw FAB Udccwdc FAB 2 02n i R i ijij jR jc B UcGw A 3.27 Where, Ai and Bi = unique structural stiffness values for the ith vehicles impacted surface R jcthe residual deformation, or crush, of the jth location measured on the vehicle perpendicular to the damaged surface from the original undamaged dimensions wj= width to the jth crush location measured para llel to the damaged surface G i= the constant of integration for the ith vehicle, 22i i A B Equations 3.26 and 3.27 recognize that every im pact event that behaves in this manner, regardless of the presence or absence of permanent deformation, has the following properties:

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45 An elastic range exists ( 22 i iii iA Gww B) that in the presence of any applied force, a finite quantity of energy will be returned to the system regardless of permanent deformation conditions. This value descri bes the restitution constant for a given structure, regardless of imp act severity, but not the coe fficient of restitution of an impact event between two objects. A constant energy for each unit of deformation depth and width (R ijj A cw) must be applied to the system during the application of external forces fr om the collision in order to continue producing deformation damage. After achieving the first two conditiooons, th e vehicle structure will continue to absorb energy and damage at a rate one-h alf the product of the structural spring constant, deformation width, and the square of the deformation depth ( 2 1 2 R ijj B cw ) across the width of contact. For a fixed barrier or a collinear vehicle-to -vehicle collision producing uniform damage profiles of average deformation depth, 0 n R j jnc C, the Central Impact Wo rk/Energy Model takes on the familiar form derived by Campbell as initially presented in Chapter 2 as Equation 2.1. 2 02WBC E ACGdw 3.28 Where, C =average of the deformation depths across the contact damage width measure perpendicular to the damaged surface from the original undamaged dimensions w =width of contact damage only measured parallel to the damaged surface, does not include induced damage associ ated with bending of the structure outside of the contact region While simplistic to todays standards, E quation 3.28 was an initial breakthrough in collision related vehicle deformation analysis However, the numerical expression derived here as Equation 3.26 allows for deformation depth measurements taken at locations along the width appropriate for describing th e deformation profile, no matter how many measurements or the interval spacing. Outsid e of flat fixed barrier impacts, deformation

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46 profiles are rarely adequately described by a single average deformation value without profile shape modifying factor s, over-simplifications, or unless utilizing a weighted average deformation depth as deve loped later in this study [14]. 3.7 Determination of A and B Stiffness Coefficients The methods for determining the A and B s tiffness coefficients for a given vehicle are well established in the literature [36] [3 7] [38] [39] [40] [20] It is important to understand that A and B stiffne ss coefficients are unique to a particular vehicle make, model and production years, as we ll as the impact surface; i. e., front, side or rear. Many vehicles have clone or sister (family) vehicl es that share the same structures and/or components, and, therefore, comparable structur al stiffness properties, as addressed in the cited literature. Additionally, many vehicles have production cy cles that can span a decade or more in which the structure of the vehicle remains unchanged. Establishing stiffness properties of a given vehicle through full-scale ba rrier impacts at known barrier impact velocities, and where the result ant deformation profiles (both cR and w ) are measured, provides the appropriate A and B stiffness characteristics for a vehicle structure. The use of commercially available vehicle specific A and B values or direct analysis of tests spanning the production year of a vehicle fam ily provide accurately determined stiffness parameters. Stiffness values are determined us ing the total width of damage and represent the average stiffness for the co ntact surface, indicating that A and B values are functions of the damage width, w For most cases, this assumpti on does not produce significant divergence from the actual vehicle structural response. 3.7.1 Frontal Stiffness Coefficients Vehicle deformation generated by an impact event is modeled as compression of a linear spring as shown in Figure 3.5. The determination of A and B stiffness values is straight forward for uniform deformation depth profiles having full barrier overlap, as demonstrated by the following derivation.

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47 Figure 3.8 Relationship between barrier impact velocity and residual deformation 01Rvbbc 3.29 2 21 2 1 01 2barrier vehiclebarrier vehicleRmv T mbbc 2 2 0011 1 22R R vehiclec mbbbc b 2 2 2 0011 1 222R R vehicleBC ACGwc mbbbc b Solving like terms: 01 vehiclembb A w 3.30 2 1 vehiclemb B w 3.31 Equations 3.30 and 3.31 provide the means to determine A and B stiffness coefficients for the fully overlapping frontal impacts into a non-deformable barrier. Common values for the b0 value for frontal impacts is 2.0 m/s (4.5 mph). Equation 3.30 for the known barrier impact velocity (BIV) and measured average residual deformation allows for the determination of the b1 value. 01Rvbbc

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48 3.7.2 Off-Set Frontal, Side, and Rear Surface Stiffness Coefficients Frontal offset barrier impacts, side and rear impacts with deformable barriers or movable barriers are not as straight forwar d for determining vehicle stiffness data. References [35] [36] [37] [41] [42], as well as earlier work don e by the author of this study, provide established and genera lly accepted methods for determ ining stiffness coefficients for vehicle surfaces unde r these conditions. Another concern regarding sti ffness coefficient determinatio n lies in the fact that many federally mandated full scale tests did no t account for an air gap present between a flexible bumper cover that rebounds out to near its origin al position as opposed to the deformable bumper reinforcement bar. As such, an air gap adjustme nt must be made on a test specific basis to account for the actual deformation of the vehicle, not some arbitrary position of a detached plastic bumper cover. Accordingly, the use of commercially available vehicle stiffness coefficients has b ecome more prevalent since 2000, to where the use of commercially available resources is th e norm rather than the exception. One of the most commonly utilized and trusted resource will be referenced and utilized when appropriate for vehicle stiffness coefficients for this study [43]. 3.7.3 Commercial Vehicle St iffness Coefficients Recent research has developed limited frontal crash stiffness coefficients for select models of motor coaches, school buses, and comme rcial semi-tractors, for a total of 9 heavy vehicles from full frontal barrier impact tests [44]. These tests are quite insightful as to the structural resistance of the test ed heavy vehicles, in that the A and B sti ffness values do not diverge appreciably from the higher values seen for most light trucks and vans. However, the current information regarding heavy vehi cle stiffness values only provides a small sample of data and should be considered as such when used. Accordingly, the need exists for a comprehensive analysis method that does not strictly rely on stiffness factors for all impacting vehicles, and especially when c onsidering heavy vehicl es. That again is a primary focus of this study and will be addressed in the following chapter.

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49 3.8 Central Impact Vehicle-to-V ehicle Velocity Change An often crucial objective of collision analysis is the determination of impact severity as it relates to vehi cle occupants. The forces acti ng on a vehicle during any given collision event produce an impulse that can change the direction and magnitude of the approach velocity of vehicles, thus producing a momentum exchange as previously discussed. It is this very momentum exchange that is of interest, typically for the determination of collision severity as it rela tes to the vehicle and its occupants. The understanding of collision severity requires knowledge regarding velocity change, impulse time, peak acceleration and the principal dir ection of force (PDOF) applied during the impulse. The collinear single DOF central impacts of this chaper apply to collisions where the PDOF acts collinear to the impact velocities, or approach velocities of the vehicles, as well as through the mass centers of the colliding vehicles. The Chapter 4 will address conditions where the PDOF is oblique to the mass centers and non-collinear impacts. 3.8.1 Developing a Force-Deflection/Velocity Change Relationship The derivation of simple ve locity change equations from force-deflection principles for completely plastic impacts without extern al impulses produced by tire forces have been well document in literature, and will be presented for background knowledge [6][7][9][14][21]. Collision configurations considered to this point have included single vehicle fixed barrier, and collinear central impacts where the principa l direction of force (PDOF) acts through the mass centers of the colliding vehicl es. While this is indeed the case for most rear-end and frontal full-overlap collision events and many broadside perpendicular impacts with aligned mass centers at impact such conditions are often not the case for many conceivable collision configurations. Ho wever, the development of the simple model allows for the development of more complex collision models, since the physical laws of motion do not change. Figure 3.9 represents a simple harmonic oscillator according to the equation expressed in the illustration. Recall the ba sic equation of motion for the system as expressed by Equation 3.11. Since the initia l consideration assumes the impact is a conservative system, the sum of all external fo rces acting on the syst em during the impulse,

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50 t is equal to 0 in compliance with Newtons second law. Therefore, the relationship between the two impacting vehicles can be re written in the form of Equations 3.32 and 3.33 as follows: 12 12 121212 12 1212 22 121212 max 22 12121122IIvv mmkckc dtdt mmvvkk cc mmdtdtkk mmddkk mmkk dtdt So that, 2 1212 max 2 12120 mmkk d mmkk dt 3.32 2 1212 max 2 1212mmkk d mmkk dt 3.33 From the equation for a simple harmonic oscillating system that applies to this collision model, the seco nd derivative of the positi on function is as follows: max 2 22 max 2sincos cos sin sin cos dd A tBtAtBt dtdt d AtBt dt 22 1212 max 1212sin cos mmkk AtBt mmkk 3.34

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51 Figure 3.9 Collision as a simple harmonic oscillator Consider maximum engagement betw een the vehicles, and therefore max. The period of oscillation occurs at t=/2 where cos(t) = 0 and sin(t) = 1, establishing a boundary value condition for the system. Equa tion 3.34 reduces to the following when equating like terms: 2 1212 max 1212mmkk A mmkk 3.35 Where, A=max 1 21212 1212mmkk mmkk 3.36 In the development of Equation 3.34, the rate of deformation between the colliding vehicles was the first element of the derivation. The rate at which the colliding vehicles start deflecting/deforming at t =0 is the closing velocity betw een the contact surfaces of the vehicles where sin(t) = 0, establishing another boundary value condition. maxsincos A tBt

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52 1 2max 1212 121212cos 12initialinitial initialinitiald vvAtA dt mmkk Avv mmkk 2 1212 max 121212initialinitialmmkk vv mmkk 3.37 Energy of a linear spring vehicle stru cture is dissipated as work producing maximum deformation at peak force as previous ly determined in Equation 3.19. Therefore, Equation 3.37 expresses the closing velo city between the impacting vehicles. 1212 122 12initialinitialmmUU vv mm 3.38 At maximum engagement, the centroid of damage for each vehicle reaches a common velocity, Vc, allowing for the following relationship: 1212 12 1212 12c initial initial initial initial cvmmmvmv mvmv v mm 3.39 The change in kinetic energy of th e system during the period of maximum instantaneous engagement during the appro ach period can be stated as follows: 222 111 1212 22212 Approachinitialcommon initialinitialcTTT mvmvmmV 2 12 1 2 1212approach initialinitialmm Tvv mm The plastic deformation imp act of a simple two-vehicle collision system during the approach phase of the impact results in velo city change magnitudes of colliding vehicles as follows:

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53 12 12 2 1212 111 12 initial initial approachcapproach initial initialinitialmvmv vvvv mm m vv mm 12 12 1 1212 222 12 initial initial approachcapproach initial initialinitialmvmv vvvv mm m vv mm 2 112212 1damagedamagemUU v mmm 3.40 1 212212 2damagedamagemUU v mmm 3.41 Equations 3.40 and 3.41 apply only to collin ear or central impacts where restitution effects and tire forces are negligible between the vehicles during the approach phase of an impact event. 3.8.2 Force-Deflection/Velocity Change Considering Restitution Effects The collision represented in Figure 3.9 considers only the linear momentum velocity change leading up to the maximum engagement, or the approach phase of the impact. However, as depicted in Figure 3.10, at least some form of restitution may occur for the vast majority of vehicle impact events The portion of the velo city change following the application of maximum impulse occurs during the collision separation phase, hereto referred to as the separation velocity change. Figure 3.10 illustrates the relationship between the approach and separation velocity changes upon the total velocity change of a vehicle during an impact event.

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54 Figure 3.10 Impact with restitution effects (adapted from Figures 3.3 and 3.9) At higher closing speed impacts in exce ss of 56 kph (35 mph), ignoring collision restitution has been assumed to produce neglig ible error using Equa tions 3.40 and 3.41. Potential exceptions to this common and ot herwise reasonable assumption occurs under the folowing conditions: Impact involving an axle and/or wheel/tire Collision resulting in compression of engine components into the firewall Fixed object impacts or collisions between heavy and light vehicles. Under these condistions, rest itution may significantly affect the separation phase of the collision so as warranting considerati on. Figure 3.9 depicts a collinear head-on collision, but easily adapted to represent a collinear same direction collision event. The common problem for coll inear central motor vehicle impacts involves finding the pre-impact velocities of two colliding bodies. Post-collision velocities are usually determinable for many real-world collision even ts. Therefore, the impact system has two unknown values, requiring two equations for determination of the unknown velocities. Initially, consider a perfectly elastic collinear impact between two objects through the statement of the Conservation of Linear Mome ntum and the Conservation of Energy for a collinear impact as follows: 12121212initial initial final finalmvmvmvmv 3.42 2222 121211 1212 22initial initial final finalmvmvmvmv 3.43 Vapproach Vseparation VTOTAL Vapproach+ Vseparation

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55 Equations 3.42 and 3.44 are rearrange d and grouped by related terms for the purpose of establishing the relationship be tween two impacting bodies as follows: 121122initialfinal finalinitialmvvmvv 3.44 2222 121122initialfinal finalinitialmvvmvv 1211112222initialfinalinitialfinal finalinitialfinalinitialmvvvvmvvvv 3.45 Dividing Equation 3.45 by Equation 3.44 and rearranging terms with respect to initial and final velocities of the system results in the traditional form for expressing the momentum-energy-restitution (MER) relationship as follows: 12 1211112222 1122 1122 21 1 12 21initialfinalinitialfinal finalinitialfinalinitial initialfinal finalinitial initialfinalfinalinitial finalfinal initialinitial finalfimvvvvmvvvv mvvmvv vvvv vv vv vv e 12 111 212nal initialinitial TOTALapproach TOTALapproachvv vev vev The derivation defines the coefficient of restitution as the ratio between the separation and approach velocities of colliding objects. Restitution values range as 0 e < 1; indicating restitution ra nges from completely plastic deformation (e=0), and approaching a completely elastic deformation (e<1). Full-scale testing for vehicle-tovehicle impacts result in impact re stitutions that range from 0 < e 0.6. Lower restitution values correlate to high velocity impacts, wh ile larger restitution values correlate to lowvelocity impacts. Equations 3.46 and 3.47 ex press the velocity change magnitudes for partially-elastic collisions when c onsidering impact restitution [6][16]: Perfectly elastic collis ion velocity ratio; e = 1 Partially elastic/plastic collision velocity ratios; 0 e < 1

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56 2 112212 11damagedamagemUU ve mmm 3.46 1 212212 21damagedamagemUU ve mmm 3.47 If restitution effects are negligible then Equations 3.46 and 3.47 revert to their parent and most fundamental fo rm of Equations 3.40 and 3.41. 3.8.3 Force-Deflection Velocity Change Considering Restitution and NonConservative Tire-Ground Forces External tire forces may act upon two colliding vehicles durin g the impulse leading up to maximum engagement, producing effects on the velocity change of each vehicle during the impact. Tire-ground interaction pr oduces a non-conservativ e constraining force external to the collision impulse during the a pproach phase of the impact, as represented as Qnc in Lagranges Equation 3.3. Tire force contribution can be a significant at lower velocity change levels, but become increasingl y insignificant as the velocity change levels increase for vehicles of similar mass.[21 ][23][28] [45] [46] Examples of when nonconservative tire forces may produce motion constraints are as follows: Collinear rear-end collision event where struck vehicle (target vehicle) has applied braking at contact by the striking vehicle (bullet vehicle). Broadside impact where the target vehicl e slides broadside against the roadway surface during the approach phase. Broadside impact where the bullet vehicle is considerably less massive than the target vehicle (i.e., passenger vehicl e striking heavy commercial vehicle). Any collision configuration having a motion constraint at the target vehicles tire/ground interface, such as wheel blocking, curbs, wall or barrier. A stopped vehicle with brakes applied pr oduces a motion constraint in the (-)x direction during the aproach phase of a collision event. The imposed motion contraint

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57 reduces the target vehicles positive-direction velocity change, and increases to the bullet vehicles negative-direction velocity change as shown in Figure 3.11. Figure 3.11 Braking force contri bution to collision impulse Tire-ground contribution vehicle 1: Tire-ground contribution vehicle 2: brake1brake2 brake 12 2 brake 1F1ma1mgn v1 mmg n t mgnt v1 m brake2brake2 brake 22 2 brake 2F2ma2mgn v2 mmg n t mgnt v2 gnt m Tire/ground constraint forces resist motion of the struck vehicle during the time of the approach phase of an impact, t Equation 3.16 determines the impulse time period resulting from the change in linear momentum, which is equivalent to the t for the braking constraint impulse. The velocity change for each vehicle resulting from the linear momentum exchange from the conservative impact force, and non-conservative tireground constraint force, therefore, both occur during the approach phase of the collision. Equations 3.49 and 3.50, hereto known as the Central Impact Force-Deflection Velocity Change Equations, incorporate the consid eration of tire-ground nonconservative constraint force contributions to the velocity change magnitude for each vehicle of a collinear central impact. Equations 3.48 and 3.49 mathematically state that if tire/ground constraint force contribu tions are considered, the net re sult is an increase in the velocity change magnitude of the bullet vehicle, and a net decrease in the total velocity change magnitude for the target vehicle. When tire/ground constraint force contributions + + m2v2=Target m1v1=Bullet F2brake F1brake F1Impact= F2Impact F2brake=-m2(g n) F1brake= -m2(g n)

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58 are negligible or not present, and if restitution is also negligible or not present, then Equations 3.48 and 3.49 revert to their parent fundamental form of Equations 3.40 and 3.41. 2 2 1121212 11damagedamagemUU mgnt ve mmmm 3.48 1 212212 21damagedamagemUU vegnt mmm 3.49 Where, v1 = total velocity change of striking (bullet) vehicle; m/sec (ft/sec) v2 = total velocity change of struck (target) vehicle; m/sec (ft/sec) e= coefficient of restitution; co llision level dependent, unitless g = gravity constant 9.81 m/sec2 (32.2 ft/sec2) = roadway coefficient of friction; unitless n = braking efficiency and/or brake force distribution as a decimal percentage (0 n 1.0) t= impulse time period during appro ach velocity change phase; sec 3.9 Missing Vehicle Parameters This chapter has developed the necessary principals and equations for determining velocity change resulting from collinear central impacts. Development of the Central Impact Force-Deflection Model and the Central Impact Work/Energy Model allow for the determination of velocity changes resulting fr om collinear central im pacts from measured deformation profiles consisting of piecewis e width and depth measurements for both vehicles, as well as known structural stiffness coefficients for each unique vehicle structure involved. The deformaton and stiffness paramete rs are usually known or knowable for most collision safety research and testing applicat ions. Instrumentation of the test vehicles negates the need for calculati ng collision severity using the methods presented. However,

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59 for a real-world collision event it is far more likely than not that many test-measurable variables may be missing that cannot otherw ise be recreated or properly accounted for without additional analysis methods. Therefor e, when reconstructi ng real-world collision events, the methods presented provide at least a foundation for velocity change and acceleration determination. The need for accounting for potentially missing data leads to one of the main focuses of this study; eliminating the reliance upon st ructural stiffness values for both impacting vehicles. Therefore, this sect ion of the study will focus on the development of analytical models to satisfy two probable and frequent scenarios of missing data from real-world collisions: Known deformation profile of one of the involved vehicles, but unknown deformation profile for the associated vehicle in any given impact event. Known structural stiffness coefficients for one of the involved vehicles, unknown structural stiffness coefficients for the asso ciated vehicle in any given impact event. 3.9.1 Background Previous studies have presented met hodologies for estimating collision specific structural stiffness coefficients when only one vehicles properties are known or knowable. Neptune and Flynn developed a method for determining collision specific stiffness coefficients for missing values related to one of the collision vehicles.[17] While this method has utility and can produce reasonably accurate results, the analyst must insert certain assumptions regarding the vehicles resist ance to collision for ces that are typically only determined through full-scale testing. Th ese assumptions may be reasonable in some circumstances, but not applicable for collisi ons where estimation of some of the critical guess values cannot be determined. The vast majority of full-scale tests have been barrier impacts to the front of vehicles; some with full overlap and some with partial overlap. Most full-scale testing is designed to meet compliance with FMVSS 208 Occupant Crash Pr otection. [47] The purpose of FMVSS 208 is reducing the number of traffic fatalities and the severity of collision-related injuries by specifying vehicle crashworthiness in terms of forces and

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60 accelerations measured on anthropomorphic test devices in test crashes for passenger vehicles, and later to add tr ucks, buses and multipurpose passenger vehicles with a gross vehicle weight rating of 3856 kg (8,500 US pounds) or less. Tested vehicles are appropriately instrumented and subjected to 48 kph (30 mph) barrier impacts, although industry standards are now at 56 kph (35 mph) barrier impacts. Since many vehicles known as sisters or clones of the same design exist between models of the same corporate manufacturer, many tests are conducted with only one of the sister/clone family. 3.9.2 Impact Force Balancing Crosscheck The following is a representation of the Newtons third law expression for the collision of two vehicles with respect to the Central Impact Force-Deflection Model of Equation 3.25. 0012nn I I j k jkFF 11 22 001122nn RR j jk k jkcwcw ABAB 3.50 The above expressions simply state th at the force of impact acting upon each vehicle is equal in magnitude, but opposite in direction of ap plication. During an impact the principal direction of force vectors, or PDOF, acting upon each vehicle during the impulse, or approach phase of the collision, are equal in magnitude and anti-parallel in direction. Therefore, the absolute values of the PDOF magnitudes of the two colliding bodies are equal. Equation 3.50 provides a cros scheck method to ensure that the damages considered for each vehicle are appropriate fo r the collision event, and hereto defined as the Newtonian Central Impact Force-Balance Relationship. Measurements are only as exact as the methodology used. Limitations are al so present in the significant digits of inertial, structural and dimensional data for each vehicle. Therefore, a vehicle-to-vehicle collision should be considered to be within force balance if the conservative forces calculated for each vehicle using Equation 3.50 are within 10%.

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61 The force balance crosscheck should be us ed to ensure that Newtons third law is not violated during the analysis process. However, Newtons third law only applies to the generalized conservative forces, Qc, of the system at impact, and must exclude the nonconservative generalized and/or constraint forces, Qnc, acting upon the system external to the collision impulse. Forcing a Newtons third law compliance using Equation 3.50 produces one of the most useful principles for verification of other parameters, or determination of unknown parameters such as deformation propertie s or structural stiffness properties. 3.9.3 Determination Missing Deformation Depths Due to the passage of time a vehicle may become unavailable for inspection. Photographic evidence of a vehicles deformati on profile may be insufficient or for other reasons outside the control of an analyst, inst ances arise where one ve hicle associated with a collision event may be unavailable for direct inspection or determination of deformation profile. In some cases, photographs may depi ct only the overall width and shape of deformation without clear evidence for accurate measurements of deformation depth. The Central Impact Force-Deflection Model (Equation 3.25) and the Central Impact Energy/Work Model (Equation 3.26), require knowledge of structural A and B stiffness coefficients and deformation profiles (cR and w ) for both vehicles. However, the Newtons third law expression of Equation 3.50 allows for not only the Newtonian Central Impact Force-Balance Relationship crosscheck, but also for a Newtons third law prediction of the damage prof ile. Solving Equation 3.50 for cR for the vehicle with unknown deformation depths but a known or knowable width, results in the Newtonian Central Impact Deformation Prediction Model of Equation 3.51. [18] [48] II knownunknownFF I known I unknownunknown R unknown unknownF wA c B 3.51

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62 Where, R unknownc Newtonian predicted residual in ward deformation for vehicle of unknown deformation depth I knownF Impact peak force calculated fo r the vehicle of known deformation width and depth profile Aunknown= A stiffness coefficient for vehicle of unknown damage depth Bunknown= B stiffness coefficient for vehicle of unknown damage depth I unknownw = total deformation width for the vehicle of unknown deformation (known or knowable) The Newtonian Central Impact Defo rmation Prediction Model of Equation 3.51 also allows for the piece-wise determination of each force el ement on the vehicle of known deformation measurements, to predicted the piece-wise deformation for each associated width for the vehicle of unknown deformation de pth. Likewise, using the weighted average deformation on the vehicle of known deformation depth, the Newtonian Central Impact Deformation Prediction Model determines the weighted average deformation depth for the vehicle of unknown values. Utilizing the Newtonian Central Impact Deformation Prediction Model of Equation 3.51 for either a weighted average or piece-wis e formulation allows for the prediction of a damage profile for a vehicle that has not been or cannot be measured. The determination of the deformation profile is facilitated by relying upon Newtons third law. Forcing a Newtons third law compliance between the impact forces of the two colliding vehicles or objects relies upon the reasona ble addumption that momentum is conserved. 3.9.4 Application of Work/Energy Principles for Unknown Stiffness Coefficients All models presented to this point require knowledge of the structural characteristics of each vehicle. Most collis ions will have at least one set of vehicle structural stiffness coefficients; usually the frontal factors for the bullet vehicle with respect to collinear central impacts. However, vehicl e structural stiffness coefficients may be limited for every vehicle or vehicle family, fr ont/rear/side structures heavy vehicles and

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63 motorcycles. Therefore, the need exists for a different strategy by which knowledge of the collision deformation and only one set of structural stiffness values result in an accurate determination of velocity change levels. Work/Energy principles pr ovides a direct correlation between the work to produce permanent vehicle deformation and the ap plied conservative fo rce producing those damages during the approach phase of the impact The force necessary to move some object over a distance is equal to the work done upon the object, regardless of its path. In other words, when the sum of all conservative forces does work upon a rigid body, a change in kinetic energy occurs. The Central Impact Force-Deflection Model provides the means by which the sum of the conservative forces are determine, and the measurable deformation to a vehicle provides the dist ance of application of the total conservative forces. workFdxU where F is the force applied to peak dx is the distance over which the fo rce is applied to the system Applying Work/Energy principles allows for the determination of the work, Umissing, when the A and B structural stiffness coeffi cients for one of the vehicle are unknown or not knowable. Newtons second law allos for determination of the impact force from the deformation properties of the vehicle of known A and B struct ural stiffness coefficients. Deformation profiles for both vehi cles must be known for this application. The application of Work/Energy principals leads to a majo r contribution of this study expressed by Equation 3.52, hereafter known as the Central Impact Piecewise Work/Energy Missing Stiffness Equation. 00 nn R I jii jj jjcw FAB (Equation 3.25) 00 0 nn n R I j kk unknown known unknown kk jUc F 3.52 Where, k unknownU = Work on vehicle with unknown A/B values at damage interval k, where k=1n values of deformation intervals

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64 I j knownF = force applied to known A/B values vehicle at interval j where j=1n values for deformation intervals R k unknownc = deformation depth on vehicle with unknown A/B values at damage interval k where k=1n values of deformation intervals Ideally for Equation 3.53 to produce the most accurate results, the damage intervals, k for the unknown vehicle should be paired with damage intervals, j for the known vehicle in the equation. Therefore, damage profiles fo r each vehicle must have the same number of corresponding deformation widths; or j=k intervals of damage measurements. However, the width of each interval are not necessarily equivalent, but can be paired to discrete deformation zones between the colliding vehicles. Figure 3.12 illustrates examples of damage profiles zone matching for the direct application in Equation 3.52. Figure 3.12 Paired force regions from impact deformation j2 = k2 j3 = k3 j4 = k4 j1 = k1 j2 = k2 j3 = k3

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65 It is not uncommon for the damage profiles between two colliding vehicles to exist making it difficult to partition damage zones into neat paired packages as shown in Figure 3.12. Therefore, when partioning of damage is difficult, the elimina tion of the associated damage width partitioning reduces the potential for error. Collisions where deformation zone partitioning may be difficult are as follows: Narrow object impact producing wide cont act and/or induced damage profiles on the bullet vehicle and narrowly focu sed damage on the target object motorcycle-to-vehicle collisions Small vehicles striking large comm ercial vehicles or trailers Oblique impacts prociding a wide contact ar ea on only one fo the colliding vehicles As an alternative, the weighted averag e damage depth on the vehicle of unknown structural stiffness coefficien ts along with the total impact force determined from the vehicle of known structural stiffness coefficients, I knownF, must be considered when determining the total work of the impact c onservative forces upon the vehicle of unknown structural stiffness coefficients. The total damage width is the sum of its individual parts, and the sum of the ratio of each width segment with respect to the total damage width = 1: 1n totalj jww 121 n totaltotaltotalwww www The weighted average of inward deform ation for the unknown vehicle is therefore determined by the following: 1 1 n jj R j unknown n j jwc C w Weighted average of deformation width 3.53

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66 The substitution of Equation 3.53 into Equation 3.52 for determining the work to produce damages to the vehicle of unknow n structural stiffness produces the Central Impact Weighted Average Work/Energy Missing Stiffness Equation of Equation 3.54. R I known Work unknown unknownU F C 3.54 Where, Work unknownU = total work to produce permanent inward deformation I knownF = impact force calculated from ve hicle of known stru ctural stiffness R unknownCweighted average deformation of vehicle of unknown structural stiffness (from Equation 3.53) By now it should be intuitive that th e PDOF acting upon any vehicle during any given impulsive impact event acts through the centroid of the damage profile. The utility of Equation 3.54 lies in the removal of interpretation of deformati on intervals that are associated with the colliding vehicles. Rem oval of interpretation allows the weighted average of deformation of th e vehicle of unknown structural A and B stiffness values (i.e., centroid of damage) to act as the distance ove r which work is done on the vehicle structure as a whole, thus satisfying the requirement s of Newtons third law. Equation 3.54 also establishes an important relati onship useful for determining the work done to damage any composite structure having a complex and/or piecewise linear deformation profile. For a given applied peak im pact force, the dissipated energy doing work to produce damage to a motor vehicle or other composite structure is determinable using the weighted average deformation depth for any complex damage profile. This final statement and the developmen t of Equations 3.52 and 3.54 mark the capstone of the study objectives with respect to collinear, central impacts. The following chapter will address these same principles as they relate to oblique, non-central collisions where rotational effects and in ter-vehicular friction contribut ions may be significant and which are not present during the simple imp act configurations pr esented thus far.

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67 CHAPTER 4: OFF-SET AND OBLI QUE NON-CENTRAL IMPACTS: GENERALIZED DEFORMATION AND TOTAL VELOCITY CHANGE ANALYSIS (G-DATAV) SYSTEM OF EQUATIONS 4.1 Objectives Up to this point, the equations developed have been used to solve the relatively simple conditions of collinear centr al impacts, which result in no rotational effects upon either colliding vehi cle. It is important to remember that motor vehicles are a collection of relative fixed poi nts, or rigid bodies, and be have as rigid bodies when subjected to collision forces. The vast major ity of vehicular collisions, outside of rollover events and falls, flips or vaults, fit into planar two-dimensional problems with rotation restricted to yaw about the z-axis The Conservation of Linear Momentum provides a simplified solution to collision events producin g limited or negligible rotation. However, as will be described and developed in this chapter, non-central impacts can produce sufficient rotational effects which cannot be ignored. Oblique impacts produce PDOFs that are not perpendicular to the impacted surf ace, which complicates the interpretation of residual deformation values, RcandRc, used in the analysis procedures presented in Chapter 3. Vehicular collision configurations producing rotation are illustrated in Figure 4.1 and described as follows: Oblique non-central, or angled and non-co llinear collisions where the principal direction of force (PDOF) does not act th rough the mass center of one or more involved vehicle in a collision event Off-set collinear where the PDOF acts para llel to, but not through the mass centers of the vehicles

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68 Figure 4.1 Non-central oblique and off-set impacts The objectives of this chapter are to accomplish the following with respect to vehicle deformation analysis: Develop methods for determining the prin cipal direction of force (PDOF) applied between colliding vehicles from vehicle deformation profiles. Develop generalized models for analyzi ng vehicle deformation that build upon the Central Impact Force-Deflection M odel Equation 3.25 of Chapter 3. Incorporate rotational contri butions to the development of generalized models for analyzing total velocity change magnitude s using the Central Impact Work/Energy Model and the Central Impact Force-Defl ection/Velocity Change equations of Chapter 3. Develop a generalized model for determining the total velocity change levels for colliding vehicles that include central and oblique collisions, and where only one vehicle has known structural stiffness coefficients. To eliminate confusion, the earth-fixed vehicl e coordinate system of Figure 3.1(b) will be used for this chapter unless otherwise specifically stated.

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69 4.2 Oblique and Offset Impa ct Momentum Principles 4.2.1 Linear Momentum Chapter 3 developed the linear momentum, P and equation of motion for a system of two colliding collinear vehicl es. Recall the linear momentum, P, for the system was derived from the partial differentiation of the Lagrangian with respect to the generalized velocities,iq, of the system of generalized coordina tes. Even though the Lagrangian starts with the consideraton of scalar qua ntities, namely kinetic energy, T the analytical approach of utilizing Lagranges Equations results in equations of motion for the system that are vector quantities. For a conservative system, the sum of the generalized momentum before and after impact is equal to zero, 1220' 1PPPP, and therefore, Equation 3.7 becomes the following: initial initial fiinal final1212Mv1Mv2Mv1Mv2 4.1 Where, 1M and 2M are the mass matrix for vehicles 1 and 2 respectively initialv1, initialv2, f inalv1and f inalv2 are the vector arrays for the initial and final velocities of vehi cle 1 and 2 respectively The vast majority of vehicle collisions are planar except rollover collisions, falls, flips or vaults. Only rotational eff ects in yaw about the vertically ( z-axis ) may be present for planar impacts. The following describes th e total planar mass matrix and the velocity vector arrays for a planar impact, where (i,j,k) are unit vectors in the ( x,y,z) coordinate system: 111 111 111 x xxyxz yxyyyz zxzyzzmmI mmI I II 1M 222 222 222 x xxyxz yxyyyz zxzyzzmmI mmI III 2M

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70 1 1 1i j initial k x y v1 ' '1 1 1i j final k x y v1 2 2 2i j initial kx y v2 ' '2 2 2i j final kx y v2 The local coordinate systems for each ve hicle pass through their respective mass centers, leaving only the diagonal terms of the total planar mass matrix for each vehicle as a non-zero value. As such, the total planar mass matrix for each vehicle reduces to the following: 100 010 001zzm m I 1M 200 020 002zzm m I 2M Placing the inertial coordinate system at the mass center of vehicle 1, mass 1 of the two colliding vehicles, results in the follo wing expressions for the planar equations regarding the linear momentum and mo ment of momentum of the system: x -direction linear momentum: ''11cos(1)22cos(2)11cos(1)22cos(2)initial initial final finalmv mv mv mv 4.2 y -direction linear momentum: ''11sin(1)22sin(2)11sin(1)22sin(2)initial initial final finalmv mv mv mv 4.3 Rotation in yaw a bout z-direction: ''11221122zz zz zz zzIIII 4.4 Figure 4.2(a) illusrates planar linear moti on of a two-vehicle oblique impact system described by Equations 4.2 and 4. 3, and Figure 4.2(b) illustrate s the rotation of the vehicles as expressed by Equation 4.4.

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71 Figure 4.2(a) Planar collision trajectory angle determination Figure 4.2(b) Oblique collisio n rotation angle determination = x-axis Total trajectories: Approach trajectories: Departure trajectory, m1: Departure trajectory, m2:

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72 By placing the origin of the global coordinate system at the mass center of vehicle 1 and the x-axis parallel to the approach direction of vehicle 1, the first term in Equation 4.3 is zero. This simplification allows for the determination of v2initial if the approach and departure angles, as well as vehicle masses, are known or knowable. The determination of v1initial is accomplished by substituting the results for v2initial into Equation 4.2 and solving for v1initial. The velocity change equation for each ve hicle is determined mathematically from the linear momentum by separating the veloci ty change into its (x,y) components as follows: 22cos cos sin1sin xfinal initial yfinal initial xyvvv vvv vvv Equations 4.5 and 4.6 are expression for the velocity change magnitudes resulting from the linear momentum of the system. 2211cos11cos11sin11sin1final initial final initialvvv vv 4.5 2222cos22cos22sin22sin2final initial final initialvvv vv 4.6 A graphical determination of the linear momentum determined mathematically by Equations 4.2 and 4.3 is accomplished by plotting a set of scaled parallelograms using vector diagramming properties as shown in Figure 4.3. Grap hically, the momentum of the system is plotted using the momentum magn itudes as the vector length and measured angles as the vector direction while utilizi ng the tip-to-tail vector diagram methodology. The velocity change magnitude is equivale nt to the momentum unit length of the line between the tip of the approach momentum vector directed towards the departure momentum vector. The sense of the line results in a vector pointing in the direction of the

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73 velocity change for each particular vehicl e, which will be opposite in sense for the momentum change vector of the opposing vehicle. The graphical solution is accomplished by first plotting the scaled departure momentum vectors to create an inside depa rture parallelogram that allows for the determination of the total momentum of the system. Secondly and while knowing the approach momentum vector directions, the outside approach parallelogram common to the total momentum vector previously determ ined is then plotted. The common diagonal between each parallelogram is the total linear momentum brought into the impact, 12PP. The total linear momentum brought into the system must be equal and opposite to the total linear momentum leaving the impact,'' 12PP for collisions producing negligible rotation. Completion of the parall elogram provides for the determination of the approach velocity vectors and th eir magnitude determinations. It should be noted that for a conservativ e system that obeys Newtons third law involving conservative forces only, the velocity change vectors for each vehicle will be of equivalent linear momentum units. The vector magnitude lengths must be oriented parallel and opposite in direction from each other. Ho wever, if non-conservative forces act upon the vehicles during the collision, and/or if rotation results, a linear momentum analysis cannot determine the total velocity change for the vehicles. Total velocity change is the cumulative velocity change of colliding vehicles due to conservative and non-conservative forces; i.e., linear momentum change, rotational momentum change, tire/ground constraint force and inter-vehicular dissipative friction. The determination of these additional effects upon the total velocity change will be addressed later in this chapter. For the earth-fixed Cartesian coordinate system chosen, the PDOF is measured counterclockwise from the x-axis or a line parallel to the x-axis. Figures 4.3 and 4.4 show the graphic representation method for solvi ng the momentum solutions for the linear momentum velocity change and PDOF direction for the collision event.

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74 Figure 4.3 Linear momentum vector parallelogram Figure 4.4 Linear momentum velocity change and PDOF x y m1v1initial PDOF1 PDOF2 m v1y m1 v1x m2 v2y m2 v2x 1 2 PDOF1 PDOF2 y x

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75 The PDOF acting upon the vehicles are de termined by first recognizing the angle of the PDOF acting upon vehicle 1, PDOF1, and the angle of the PDOF acting upon vehicle 2, PDOF2, are measured counterclockwise to the approach path of vehicle 1, or the x-axis in an earth-fixed Cartesian coordinate system. sin sin tan cos cos yfinal initial xfinal initialvvv vvv The following determines the PDOF acting upon each vehicle with respect to the x-axis : 11sin11sin1 1tan 1cos11cos1final initial final initialvv vv 4.7 12sin22sin2 2tan 2cos22cos2final initial final initialvv vv 4.8 When post-impact trajectories and resu ltant surface drag factors are known or knowable, reasonable estimates re garding the post-impact veloci ties of vehicles involved in a collision event can be made from the en ergy principles of Chapter 3. As a vehicle slows, work is done between the tires, body comp onents or side of a vehicle that is sliding on other a surface. The total kinetic energy of the vehicle at the start of the negative acceleration due to braking, spinning, sliding or even coasting is the sum of the final kinetic energy and work done while slowing. initialfinalworkTTU 2211 22initial finalfrictionmvmvFd 4.9 The force due to friction can be e xpressed using Newtons second law and substituted into Equation 4.9, allowing for a relationship between the initial and final velocities as follows:

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76 22 2211 22 2 initial finalfriction initialfinalfrictionmvmvmad vvad 22initialfinalfrictionvvad 4.10 The acceleration due to friction, afriction, is expressed as the ratio between the force necessary to cause the vehicle to slide agains t the contact surface with the normal force due to gravity that holds the vehicle to that surf ace, which is often referred to as the surface friction coefficient, However, the surface slope may also affect the rate at which the acceleration due to friction occurs and should be accounted for when known. The gravitational influence of surface slope is base d upon the tangent of the angle of the surface the vehicle is in contact with to the earth-fixed horizontal axis, as shown in Figure 4.4. Another term for the slope of the surface is the grade of the surface; whether an uphill grade for a positive slope or a downhill grade for a negative slope. If the final velocity of a vehicle is zero, or the vehicle comes to a complete stop following the impact, then Equation 3.10 while accounting for surface slope is simplified as follows: 2tan 22 1taninitialvgdgfd 4.11 Where, = surface friction (level surface, no adjustments) = surface slope angle; radian (degree) d = acceleration/braking/sl owing distance; m (ft) f = effective surface drag factor ad justed for roadway slope; unitless

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77 Figure 4.5 Surface slope and friction relationships Equation 4.12 can be used for a vehicle that travels over multiple surfaces with multiple different drag factors and surface slopes as follows: 2 1 1tan 2 1tan 2n i initiali i i n ii ivgd gfd 4.12 4.2.2 Rotational Momentum Off-set and oblique collisions result in the PDOF acting away from the mass center of one or both vehicles, thus producing a mo ment about the respective mass centers as a result of the collision. Equa tions 4.2, 4.3 and 4.4 state that the linear and rotational Ffriction Ffriction mg mg Run Run Rise Rise wz wx wz wx 2tan 1tan frictionag 2tan 1tan frictionag tan R ise R un tan R ise R un

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78 momentum (or moment of momentum) of the co lliding system is conserved. The Newtons second law statement for rotation with resp ect to Figure 4.2(b) is as follows: I magnitudemagnitudecmhFI rF 4.13 Where, = torque r = moment arm vector from rotation at mass center to applied force FI = applied force vector from collision I = polar mass moment of inertia (about axis through mass center) = rotational acceleration hI = perpendicular moment arm for impact induced moment The torque about the mass center of the vehi cle is equal to the a ngular inertia of the vehicle. For the vehicle in Figure 4.6, the a pplied force at the right front of the vehicle produces a counterclockwise rotation. Unless th e vehicle is setting on a frictionless surface, such as wet ice, frictional forces will act c ounter to the rotation of the vehicle during the collision impulse, just as occurred when brak ing forces were considered for a collinear impact in Chapter 3. 2tan 1tanf F mg n mgfn 4.14 Where, Ff = Force due to friction between vehicle tires and roadway surface = surface friction (level surface, no adjustments) = surface slope n = braking/sliding efficiency (r atio of mass on sliding tire(s))

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79 f = effective surface drag factor Figure 4.6 Moment arm applied to produce rotation about mass center The net torque applied to the vehicle in Figure 4.6 is the difference between the moment produced by the applied force and th e moment about the rotation point for the opposing frictional forces acting at the vehicle/roadway interface, which for small displacements at a single axle the rotation poin t for the moment arm of the friction may be the center of the axle furthest from the impact location, and for large displacements the rotation point for the opposing friction will be about the vehicle mass center. zzIIffIIfIhFhFhFhmgfn 4.15 Expressing the force acting upon the vehicl e due to the impact impulse by Newtons second law and the accelerations expressed in terms of velocity changes over the impulse FI r hI Ff I ..

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80 time period, the system can be solved for the lin ear velocity change of the vehicle as shown in the steps to derive Equation 4.16. zzff I zzf IIhF ma h Ihmgfn v t m th zz f I zzff I I hmgfnt v mh IhFt mh 4.16 For most vehicle collision reconstructions, the initial rotation of a vehicle leading into an impact event is negligible or zero leaving only the post-co llision rotation of a vehicle as the necessary value when determining Figure 4.7 shows a vehicle rotating to final rest post-impact. The rotation of th e vehicle is determined by the rotation of the local coordinate system of the vehicle oriented in an earth-fixed orientation as the vehicle is placed incrementally upon its post-collision tire marks. In Figure 4.8 the individual postcollision tire trajectories of th e vehicle are plotted using differe nt colored lines to facilitate ease of vehicle incremental placements on the scene evidence. The center of mass distance travel and change in vehicle heading is dete rmined for increments along the rotating postcollision path, so that the effective drag fact or for the vehicle, assuming no braking postimpact, can be determined. If wheels are lock ed due to damage, the post-collision friction can also account for the locked wheels braking contribution based upon the weight distribution of the vehicle upon the locked a nd unlocked wheels during the rotation towards final rest.

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81 Figure 4.7 Rotational post-impact motions The vehicle in Figure 4. 7 has an initial heading of its local coordinate -axis in line with the earth-fixed inertial coordinate syst em x-axis. However, the initial heading could be any angle where the vehicles -axis is rotated from the earth fixed inertial coordinate system x-axis. The vehicle rotates in yaw without braking about its local vertical -axis either clockwise or counterclockwise, so that the effective deceleration rate, again without braking over the entire distance, regardless of rotation direction is given by the following: 1 2tan sin() 2 1tan rotate ii if 1 2 0 0tan 2sin() 2 1tan 2n i ii rotationi i i n rotate ii ivg d gfd 4.17 Where, i = surface friction (level surface, no adjustments) for the ith interval i = surface slope at the ith interval

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82 i = rotation of the vehicle axis from the inertial coordinate system at ith position f i rotate= effective surface drag factor for ith interval When the surface slope is small, as is usually the case for roadways and roadside geometric features ( 20o, /9 radians), Equation 4.17 can be simplified as follows: 1tansin() 2 rotate ii if 1 02tansin() 2n ii rotationii ivg d 4.18 The determination of the vehicles rotati onal velocity change between each spin interval is determined from basic kinematic equations for determining the time period for each rotational slowing interval. i i iv a t and, i i iv t a 11 1 2tan sin() 2 1tanii ii i rotate i i ii ivv vv t gf g 4.19 1 1 rotate iii i i iiigf tvv 4.20 4.2.3 Mass Moment of Inertia The mass moment of inertia of an objec t is its measure of resisting rotational acceleration, just as mass is the measure of a body to resist linear acceleration. The moment of inertia of an object is a func tion of shape and mass. If the moment of inertia is determined about an axis that passes through the mass center of an object, it is called a polar moment of inertia

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83 Using polar moments of inertia tends to make analysis mu ch easier, and is routinely used for motor vehicle collision analysis with few exceptions. The moment of inertia about any axis, to include those that do not pass through the mass center of a body can be determined using the parallel axis theorem if the polar moment of inertia is known. Likewise, the polar moment of inertia of a composite or oddly shaped object can be determined using the parallel axis theorem w ith respect to each of the individual moments of inertia of the geometric shap es that make up the body. In ge neral, the moment of inertia is determined as the sum of the product of a ll the differential mass el ements of the body, dm and the square of its distance from the axis of rotation, r. 2 2 1 n ii idIrdm I rm 4.21 Another method of describing the polar mome nt of inertia of an object with great utility is considering the mass, m is concentrated within an equivalent radius about a primary axis that passes through the mass center, known as the radius of gyration, kg. [49] 2 g I mk 4.22 Since passenger vehicles, light trucks and vans are non-homogeneous complex geometric shapes, the mass moment of inertia is determined experimentally using tilt table measurements, or more commonly from best-fit equations derived from experimental data. Garrott presented data from the NHTSA Light Vehicle Inertial Pa rameter Data Base containing measured vehicle inertial parameters of 356 tested vehicles, plus tilt table data for 168 vehicles [50] as a follow up to an initial paper presenting an algorithm for determining moments of inertia for the curb wei ght of unloaded vehicles by distinct vehicle classifications [51]. Neptune presented a method for dete rmining the yaw moment of inertia ( Izz) based upon the method presented by Garrott for the curb weight of an unloaded vehicle, but allowing for the addition of occ upant and cargo weights [52], providing greater utility for collision analysis. Equation 4.23 from the Neptune paper reduces to the best fit algorithm developed by Garrott when the vehicle is unloaded.

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84 22 curbloadedcurb zzM Gloadedmm m ILbK Km 4.23 Where, Izz = yaw moment of inertia (about z-axis ) mcurb= curb mass of vehicle (unloaded) mloaded= loaded mass of vehicle (cur b plus occupants and cargo) L = total length of vehicle b = maximum width of vehicle KG KM = geometric empirically determ ined constants (Table 4.1) Table 4.1 Yaw moment of in ertia empirical constants Vehicle Type KG KM R2 All combined 13.1 0.696 0.85 Passenger Car 13.8 0.769 0.86 Light Truck 13.4 0.750 0.92 SUV 12.2 0.656 0.76 Light Van 12.3 0.642 0.90 4.3 Principle Direction of Fo rce from Damage Profiles In Chapter 3, the focus was on collinear central impacts. As such, the principle direction of force applied by the striking vehicle upon the stru ck vehicle was along either the longitudinal or lateral primary axis of a ve hicle. However, oblique and off-set collisions result in a principle direction of force that does not act along a primary axis of the vehicle nor through the vehicle center of mass. Instead, oblique and offset collisions result in a PDOF that is angular to a primary axis, or parallel to a primary axis but offset from the mass center, thus resulting in rotation. In or der to account for the effects of an oblique impact damage profile and rotation, the pr inciple direction of force (PDOF) must be determined.

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85 4.3.1 Determining Damage Centroid In general, the planar centroid of a shape is described by the following equations for finding the (x,y) coordinates of the centroid as x y: 0 0 n ii i n i i x A x A 4.24 0 0 n ii i n i iyA y A 4.25 Figure 4.9 is a simple damage profile depiction with residual deformation measurements, R nc, and width measurements from the zer o point at the origin and measured for each location to its resp ective residual deformation m easurement point. The area bounded on two sides by residual deformation measurements R ic and 1 R ic along the cR-axis, 1 iiww at the base along the w -axis, and bordering the top described by the damage profile can be approximated as a trapezo id of area described by Equation 4.26. 11 122RR RR ii ii ii iicc cc A ww w 4.26 In order to determine the centroid of each differential trapezoidal deformation area, each differential trapezoid is broken into two composite parts consisting of a rectangle and a right triangle as shown in Figure 4. 8, with areas described as follows: 1 RR rectiiiii A cwwcw 4.27 11122RR RR iiiiiii triangleccwwccw A 4.28

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86 Figure 4.8 Damage profile measurements Each differential trapezoid within the deformation profile has its own unique centroid coordinates with respect to crush depth, ic, and width, iw, generally described by the following equations: rectrecttriangletriangle i recttrianglecAcA c AA 4.29 rectrecttriangeltriangle i recttrianglewAwA w AA 4.30 Solving for iw, or the deformation width position of each differential trapezoid starts with substitution of geometric values into Equations 4.30 and simplifying to find the final equation. c w

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87 1 232 1 1 2 ii irectrecttriangletriangle i recttriangle ww w R RR iiii RRR iiiiiwAwA w AA cwcc cwccw 22 3 1 1 66 1 1 2 1 12 3RRR iiiii RR iii RR iii RR iicwccw wcc wcc cc 11 12 3RR iiii i RR iiwwcc w cc differential trapezoid cent roid width coordinate 4.31 Repeating the same procedure for each differential ic, or crush depth centroid of the deformation profile, results in Equation 4.32 below. 11 1 11 23 2 1 1 2 22 1 11 3 1 22 111 11 333 1 rectrecttriangletriangle i recttriangle RRRR R iiiiiiiii iii RRRRRR iiiiii RR ii RRRR iiii RR iicAcA c AA ccwcccwcc wcc cccccc cc cccc cc 22 11 13RRRR iiii i RR iicccc c cc differential trapezoid cent roid crush coordinate 4.32 By inspection of Equations 4.31 and 4.32, the determination of the differential centroid coordinates remains tr ue whether the deformation profile of any differential

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88 segment is a trapezoid, rectangle, square, tria ngle or zero value, as well as whether the deformation values increase or decrease between differential deformation area bounded sides. The final determination of the deform ation profile centroid coordinates is completed by substituting the differential i x and iy values calculated in the previous steps for the xi and yi values of Equations 4.24 and 4.25, and the Ai values are the areas of the differential trapezoids determined using Equa tion 4.26. The total area is determined by summing all of the differential areas of the deformation profile. Equations 4.33 and 4.34 are the final coordinates for the total deformation centroid with respect to the (width, crush), or ( w,c ) coordinate system established for the measurements, which are essentially weighted averages between the di fferential centroids across the deformation profile. 1122 0 12 0 n ii iiiiiinn i n iiin i iwA wAwAwAwA w AAAA A 4.33 1122 0 12 0 n ii iiiiiinn i n iiin i icA cAcAcAcA c AAAA A 4.34 For deformation profiles with multiple deformation depth measurements, R ic, and multiple widths,iw, it may be useful to set up a char t of values for the incremental calculations when determining the final deform ation profile centroid coordinates. For complex deformation profiles, the use of a C AD program to determine the centroid of the deformation streamlines the process and elimin ates the need for the manual calculations since they are performed inte rnally by the CAD program.

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89 4.3.2 Maximum Engagement Previously in this chapter, methods fo r calculating and measuring the angle which the PDOF acts upon a vehicle, PDOF, were developed from linear momentum principles as expressed in Equations 4.7 and 4.8, as well as the vector diagramming methodology. Many collisions are either not thoroughly documented to include the data necessary for solving the momentum solutions of Equations 4. 7 and 4.8, or insufficient physical evidence is present on the roadway for such a dete rmination with reasonable accuracy due to minimal roadway evidence or the vehicles bein g moved from their rest locations before measurements are taken. However, the prof ile of a vehicle and determination of the centroid of the damage profile allows for th e determination of the PDOF through damage matching. The PDOF acts through the centroid of each vehicle damage profile at maximum engagement and peak impulse according to Equa tion 3.25. Since force is a vector, the line connecting the damage centroid profile of each vehicle at maximum engagement is the line of action with which the force is applied to do work in deforming each vehicle, or PDOF. The determination of the PDOF from matchi ng the deformation prof iles after determining the deformation centroids for each vehicle is represented by Figure 4.9. For impacts resulting in less severe damage profiles, it is often easier to plot th e damage centroids on each vehicle and approximate the PDOF by conn ecting the two centroids with a line while at initial contact, which again would be the line representing the PDOF resulting for the impact. The steps for determining the PDOF from damage matching are as follows: Create scaled deformation diagrams with centroid coordinates plotted on each vehicle. Position vehicles together at maximum e ngagement with the respective centroids remaining positioned with respect to each vehicle damage profile. When completed, the centroids will pass through each other and lie within the boundaries of the opposing vehicle. A line connecting the two centroids at ma ximum engagement represents the PDOF acting upon the vehicles duri ng the peak impulse.[6]

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90 Measure the PDOF for each respective vehi cle with respect to a perpendicular datum line from the side of contact which is also parallel to the primary axis of the vehicle emanating from the contact surface (i.e., front, side, rear), and passing through the damage centroid at maximum engagement.[6] Figure 4.9 Damage centroid match at maximum engagement for PDOF Using a CAD program greatly enhances the ab ility to physically po sition the vehicles at maximum engagement when the PDOF acts upon each vehicle, so that the moment arm for rotational affects can also be measured on the scaled diagram. 4.4 Oblique Impact Generalized Force-Deflection Model Chapter 3 developed the Central Impact Force-Deflection Model For central impacts, the PDOF acts perpendicular to the im pacted surface and para llel to either the primary x -axis or y -axis, while passing through the mass centers of the involved vehicles. The force necessary to deform any vehicle unde r these conditions is directly determined by Equation 3.26. However, an oblique impact results in a PDOF that does not act parallel to a primary axis of one or both impacting vehicle, nor through the mass center. An offset impact results in forces which act parallel to a primary axis, but do not pass through a vehicles mass center. Each of these impact configurations form a force couple and moment about the mass centers of each vehicle, thus resulting in the potential for torque to the impacting vehicles that was not considered for collinear impacts addressed in Chapter 3. PDOF2 PDOF1 Centroid Centroid h h

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91 4.4.1 Oblique Deformation Principles In Chapter 3, the force applied to a vehicle structure during an impact was perpendicular to the vehicle surface. However, many collision configurations result in a PDOF that is oblique, or angled to the co llision surface, which affects how the applied impact force is determined and/or analyzed. As with the deformation analysis of ma ny composite structures, a motor vehicle can be considered as a continuum of springs along each surface. Figure 4.11 shows the spring continuum representation of a motor vehi cle, diagrammed such that the front, rear and sides are represented by springs of diffe rent colors, different line thicknesses and therefore, different spring stiffness factors, kfront, krear and kside. Frontal and rear structures tend to be homogeneous in structural stiffne ss due to intentional design consequences in order to meet federal impact requirements. However, side structures, while relatively uniform along body panels, have hardened zones at the axles and B-pillar (door lock posts), thus depicted as stiffer springs through thic ker lines relative to the other side spring components within Figure 4.10. Figure 4.10 Vehicle spring continuum kfront = front stiffness krear = rear stiffness kside = side stiffness cos cosfrontal frontalFi kik F sin sinside sideFj kjk F Where = deformation distance along

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92 Figure 4.10 depicts a force, F, applied oblique to the vehi cle structure at an angle, to the longitudinal x-axis of the vehicle. This force will be considered as applied at a corner or across the frontal surface, such that the resultant deflection of the structure occurs in both the i and j directions. The total force applied to the vehicle structure in Figure 4.11 is mathematically described as follows: FiFj F 2 2cos sinmagnitudefrontal sideFkk 4.35 Full scale barrier impact testing of vehi cle structures reveals a general trend between the frontal/rear structures with respect to the side st ructure of any given vehicle, in that front and rear A/B stru ctural stiffness coefficients ar e generally 3 to 4 times greater than side structure A/B stiffness coeffici ents. Inspection of Equation 4.35 clearly establishes the stiffness and compression of the frontal structure is th e overriding factor in determining the force magnitude applied to th e structure. Equal contribution between the frontal and side struct ural stiffness contributions does not occur until the PDOF angle from the longitudinal axis of the vehicle reaches approximately 1.3 radians ( 75o). For corner impacts behind the front bumper or in front of the rear bumper, the vehicle deformation should be primarily lateral into the vehicle st ructure, and primarily frontal/rear for corner impacts into either bumper system at angles less than 1.3 radi ans from the longitudinal axis of the vehicle for the respective impacted surface. The following section will address how an oblique impact deformation is considered using only a singl e structural stiffness value for the vehicle thus eliminating the need fo r two dimensions of stiffness for oblique impacts. 4.4.2 Developing the Generalized Force-Deflection Model Consider the vehicle in Figure 4.11. The PDOF acting upon the centroid of the damaged frontal surface is angled from the residual deformation measurements, R nc, which are recorded parallel to the primary longitu dinal axis of the ve hicle for a frontal deformation profile. Therefore, without accounting for the PDOF acting upon the vehicle,

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93 only the longitudinal component of the force is accounted for, which may lead to an underapproximation of the force acting upon the ve hicle structure for an oblique impact depending upon the PDOF angle. Therefor e, the total force of the impact, FPDOF, must be determined. Figure 4.11 Oblique impact PDOF acting at damage centroid The compression of the vehicle structure during deformation occurs along the line of action described by the PDOF into the imp acted surface. Therefore, the compression of the impact surface spring occurs over a grea ter distance than what is measured from the perpendicular of the impacted surface, cR. The following derivation for FPDOF when considering the foundation established through the Central Impact Force-Deflection Model of Equation 3.26 results in the Generalized Impact Force-Deflection Model. cos() 1 cos() xPDOFPDOF PDOFx PDOFFF FF cos() 1 cos() RPDOF xPDOF PDOFR x PDOFCC CC 0cos() n R Gen j PDOF ii j PDOF jFw c FAB 4.36 The Generalized Impact Force-Deflection Model of Equation 4.36 provides the magnitude of the external force acting upon a ve hicle during an oblique or offset impact, which is equal in magnitude but opposite in direction of application to the external force PDOF PDOFX PDOFY PDOF R nc

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94 applied to the opposing vehicle in accordance wi th Newtons third law. Equation 4.36 is a more complete and generalized statement of the force-deflection model developed in Chapter 3, in that as PDOF approaches 0 for Equation 4.36, then FGen 0 n I j j F of Equation 3.26. Equation 4.36 has broader application to a multitude of varying impact configurations that include central, collinear, o ffset and oblique impacts. 4.4.3 Generalized Force-Deflection Principles The Generalized Impact Force-Deflection Model of Equation 4.36 establishes three important relationships between collision force, deformation dimensions and the structural design characteristics of a vehicle, which can be stated as follows: Force-Deflection Principle 1: Im pulse-Deformation Relationship The peak resultant force applied to a vehicle structure during an impact event is equivalent to the sum of the fo rces necessary to initiate permanent damage and the forces which produce permanent residual damage to the structure during the approach phase of the impact. Force-Deflection Principle 2: Deform ation Depth and Width Relationship Deformation intrusion into an impacted st ructure or surface of a vehicle is a function of the ability of a stru cture to resist permanent inward deformation over the width of applied forces. The final force-deflection relationship consid eration is an intuitive consideration of the previous two rather simplistic but important relationships. When considering ForceDeflection Principles 1 and 2 together, a very crucial and key generalized fundamental design consideration for the management of collision forces as they relate to vehicle structural design can be stated as follows:

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95 Impact Force-Deflection Principle 3: Ge neralized Impact Force Management Vehicle structural designs which manage impact forces in order to maximize the width of force distribution will minimize inward deformation into the occupant compartment, thus providing additional protection for vehicle occupants from secondary im pacts from intruding components associated with the vehi cles exterior shell and/ or interior components within the deformation region. 4.5 Oblique Impact Work/Energy Model 4.5.1 Contributions of Oblique An gle to Vehicle Deformation The Generalized Force-Deflection Model provides the means by which the total force acting upon a vehicle during an impact, regardless of whet her the impact is a central or oblique collision event, is determined. Si milar methods to those in formulating the expression for the work necessary to deform the vehicle spring system resulting from a central impact that developed in Chapter 3 are employed when determining the work necessary to produce permanent deformation to the ith vehicle involved in an oblique or central impact event when the deformation de pth and width measurements for each vehicle are known or knowable, which results in the Oblique Impact Work/Energy Model represented by Equation 4.37. 00 00cos() PDOF PDOF jjCC nn j GenPDOFRPDOF jjj ii PDOF jjFdC c dCw AB Complete the integration and substitute the value for 2 2 01 22cos()cos() R n j j i ObliqueR i ij i iPDOFPDOF jc w A Uc BB A

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96 2 2 2 01tan 22 n i ObliqueR i PDOFi i ijj j iR jc B A Ucw A B 4.37 Where, Ai and Bi = unique structural stiffness values for the ith vehicles impacted surface R jc the jth deformation depth measured on the vehicle perpendicular to the damaged surface from its undamaged dimensions wj = width to the jth deformation, measured para llel to the damaged surface PDOFi = angle of the PDOF acting upon the ith vehicle 4.5.2 Contributions of Non-Conservative Inter-vehicular Frictional Forces For collisions that result in sliding rela tive to the contact surfaces, additional work is done upon the system result ing from non-conservative for ces due to inter-vehicular friction. The non-conservative inter-vehicular frictional force occurs during th e application of the impact impulse and during the approach phase of the impact. The nonconservative force due to inter-vehicular friction results in an increase in the overall impulse time period due to the extended contact between the vehicles as they impact and slide against each other. Work due to inter-vehicular friction ofte n results from oblique impacts where the corner or a narrow contact region on a st riking vehicle slides along the extended surface of a struck vehicle, so that the ove rall comparative contact and damage regions of the vehicles are dissimilar. The resultant force due to impact determined in Equation 4.36 only changes as a result of the conser vative force from the additional deformation width and depth to either or both vehicles, but does not ch ange as a resu lt of the nonconservative sliding between th e two surfaces. However, the ex tended contact area results in an extended impulse contact time period between the vehicles as they engage, slide against each other and separa te during the collision event. The inter-vehicular friction

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97 creates another energy sink to the vehicle structure that must be considered for the total velocity change determination for each vehicle. Consider the contact between the two vehi cles shown in Figure 4.12. The width of contact on the striking vehicle, m1, is concentrated on the left front corner, while the width of contact on the struck vehicle, m2, extends across a much wider width of contact, which is due to scraping between the vehicles during impact. The width of scraping may be difficult to directly measure on each vehicle, but is easily and accurately approximated by the difference in contact widths. 21 scrapewww 4.38 Figure 4.12 Friction of extended contact, scraping impacts The work done within the region of scra ping is dissipated energy due to kinetic friction as the two surfaces slide against each other, as well as continued deformation resulting from an extended impact impulse un til final separation occurs. The basic equation for kinetic friction between any two surfaces is as follows:

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98 f riction Normal kFF 4.39 Where, k = coefficient of kinetic friction FNormal = normal force acting betw een two sliding surfaces Since the force due to friction is acti ng between the two slid ing vehicle surfaces while in contact, the fricti on force acting upon vehicle 1, m1, must be equal in magnitude but opposite in application to the friction force acting upon vehicle 2, m2. 1122 1122 21 12 1212 kk kk kkmama mgnmgn mm mgmg mmmm 2 11 12 friction km Fmg mm 4.40 1 22 12 friction km Fmg mm 4.41 The work done due to friction is the integr al of the force over the work distance, wscrape which results in the additional work due to friction that must be accounted for during the approach phase of the collision. 112 111 12 00 1 222 12 00scrape scrapeww Friction kscrape ww Friction kscrapem Fdwmgdw mm m Fdwmgdw mm 2 1 1 12 friction k scrapem mgw mmU 4.42 1 2 2 12 friction k scrapem mgw mmU 4.43 Studies by [53] and [54] ha ve reported the coefficient of kinetic friction for vehicleto-vehicle contact ranges between 0.3 k 1.1 (default k = 0.5), depending upon the

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99 angle of impact. The highest friction levels were associated with impacts nearing parallel approach angles at contact (sideswipe) or when snagging occurred while the lower friction levels were associated with oblique impacts. By inspection of Equations 4.42 and 4.43, the work due to inter-vehicular frictional effects is the same for both vehicles. Additionally, inter-vehicular contributions may be a significant considerat ion for impacts that produce large discrepancies between the sliding contact widths of the colliding vehicles. Intuitively, the longer the vehicles remain in contact dur ing the approach velocity change phase, the longer the external tire/surface impulse affects the overall veloc ity change levels for both vehicles. Therefore, the consid eration of work due to nonconservative forces producing inter-vehicular friction if evidence of sliding wi thin the contact surfaces of the vehicles is present should provide a more accurate analysis of the total velocity changes for the colliding vehicles. 4.5.3 Generalized Impact Work/Energy Model The development of the oblique impact and friction work equations allow for a more generalized analysis of work produc ing permanent deformation. A complete generalized model should account for the effects of an oblique collision upon the residual damage approximations across the damage width, as well as the contributions of friction between sliding surfaces in cont act during the approach phase of the impact. The total work done on the system is the sum of the work due to conservative and non -conservative forces acting during the approach phase, which is the Generalized Impact Work/Energy Model of Equations 4.44 and 4.45. 111 2 2 1 2 1 11 01 12 121 11 1 t a n 22GenObliquefriction R n j R jj P D O F j ks c r a p eBc A Acw B mm gw mmUUU 4.44

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100 222 2 2 2 2 2 2 2 0 2 12 122 221tan 22GenObliquefriction R n j R jjPDOF j kscrapeBc A Acw B mm gw mmUUU 4.45 4.6 Generalized Impact Force-Deflection/Velocity Change Model In Chapter 3, the Central Impact Wo rk/Energy Model considered the work to produce damages to vehicles from central imp acts, including restitution and tire forces generated during the collision impulse. This chapter has considered the effects of an oblique impact in developing the Generalized Impact Force-Deflection Model and the Oblique Impact Work/Energy Model A generalized model for determining the velocity change of each vehicle should consider all of the significant conservative and nonconservative effects of the impact in order to develop a universal model applicable to a wider variety of damage analys is conditions. Therefor e, a universal velo city change model should consider the effects of the impact work, work due to friction and the effects of tire forces as they relate to each vehicle in the collision event, as well as rotational contributions. 4.6.1 Rotational Contributions Previous studies have derived equations and evaluated velocity change effects due to the moment created by offset and oblique impacts.[6][8][12] [55] Figure 4.2 shows an oblique collision between two ve hicles and the moment arms, h1 and h2, for the torque applied to vehicles 1 and 2 respectively. The torque applied to each vehicle is given by Equation 4.13, and Newtons second law states the sum of the torques for a conservative system must be equal to zero. Equation 4. 22 also provided an important relationship between the yaw moment of inertia of a ve hicle about the primar y vertical axis, Izz, and the radius of gyration of the effective mass as it rotates around the primary vertical axis. magnitudemagnitudecmhFI rF (restatement of 4.13)

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101 2zzg I mk (restatement of 4.22) Additionally, the acceleration at the point of contact between the vehicles are equivalent and have the following relati onships with respect to each vehicle: 111commoncmaah and, 222commoncmaah 1 11commoncmaa h and, 2 22commoncmaa h The rotational acceleration of each vehicl e produced by the force couple can be related to Equations 3.12 and 3.21 in the following manner. 11 2 1 11 1 1Gen zz commoncm ghFI aa mk h and, 22 2 2 22 2 2Gen zz commoncm ghFI aa mk h Substituting the Newtons second law statement for the generalized force allows for the solution for the acceleration at the mass center of the vehicle. 2 111 11 commoncm cmgaa ak h and, 2 222 22 commoncm cmgaa ak h 2 22 11 1 1g cmcommon gk aa kh and, 2 22 22 2 2g cmcommon gk aa kh 11cmcommonaa and, 22cmcommonaa Where, 1 2 111 1zz zzI I mh and, 2 2 222 2zz zzI I mh 111 1common cmv Fm t and, 222 2common cmv Fm t

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102 The values of 1 and 2 are defined here as the effective rotational (dynamic) mass ratio for vehicle 1 and vehicle 2, respectively. The accelerations are the time rate change of velocity for the respective vehicles at their respective ma ss centers and for the respective common velocity changes as calc ulated in Chapter 3 using the Central Impact ForceDeflection Velocity Change Models However, the models deve loped in Chapter 3 do not consider the effects of non-conservative forces external to the collision impulse, or rotational effects. As defined previously, the total velocity change is the cumulative velocity change of colliding vehicles due to conservative and non-conservative forces; i.e., linear momentum change, rotational mome ntum change, tire/ground forces and intervehicular friction. The culmination of the cons ideration of restituti on, tire/ground forces, inter-vehicular friction and rotational c ontributions to the collision leads to the Generalized Total Velocity Change Model for each of the impacting vehicles as follows: 2212 2 1 11112212 1GenGen GenmUU mgnt e mmmmv 4.46 1112 2 2211222 1 GenGen GenmUU e g nt mmmv 4.47 Where, e= coefficient of restitution for collision level 1 and 2 = effective rotational (dynamic) mass ratios = roadway coefficient of friction n= brake force distribution as a decimal (0 n 1.0) t= impulse time period during approach velocity change 1 GenU and 2 GenU = the total work determined by Equations 4.44 and 4.45

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103 4.6.2 Generalized Impulse Time The time period of the braking impulse co ntribution to the total vehicle velocity change of each vehicle occurs during the a pproach velocity change, which includes the scraping contact between the vehicles while da mage is still being produced but before restitution, if any, occurs. Friction is occurri ng essentially parallel to the vehicle surfaces by definition, and therefore does not contribute to the generalized force of the impulse, which is applied perpendicular to the contact surface. Therefore, the introduction of friction to the impact results in an extended impulse time period that is a function of the work due to impact and inter-vehicular friction, as we ll as rotational effects as shown in Equation 4.48. 12 1122 2 11222 GenGen Gen GenUU mm mm Ft 4.48 Inspection of Equation 4.46 through Equati on 4.48 reveals that in the absence of inter-vehicular friction and mass adjusting rotation al effects, each of these equations reduce to their parent forms developed for central impacts in Chapter 3. 4.7 Generalized Newtonian Prediction of Missing Vehicle Parameters As addressed in Section 3.8 of Chapter 3, one of the main focuses of this study is the elimination of the reliance upon structural stiffness values for both impacting vehicles, or knowledge of the deformation profiles for both impacting vehicles when determining the work to produce permanent deformation to each unique vehicle, as well as unique impacted surface of a vehicle during a collision event. Wh ile these principles were developed for collinear central impacts in Chap ter 3, this section of the study will focus on the development of analytical models to satisfy the same two following probable and frequent scenarios of missing data from real-world collisi on events regarding oblique collisions: Known deformation profile of one of the involved vehicles, but unknown deformation profile for the associ ated vehicle in the collision.

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104 Known structural stiffness coefficients for one of the involved vehicles, unknown structural stiffness coefficients for th e associated vehicle in the collision. 4.7.1 Generalized Impact Force Balancing Crosscheck The Newtons third law crosscheck of collis ion forces from vehicle deformation of Section 3.8.2 of Chapter 3 for central impact s. The following is a representation of the Newtons third law expression for the collision of two vehicles wi th respect to the Generalized Impact Force-Deflection Model of Equation 4.36. The expressions in Equation 4.49 are a statement of the Generalized Newtonian Force-Balance Crosscheck when the vehicle structural stiffness values and damage profiles for both vehicles are known. 0012nn g en Gen jk jkFF 11 22 00 12cos() cos()nn RR jj jj jj PDOFPDOFww cc ABAB 4.49 Equation 4.49 simply states the impact force acting upon each vehicle is equal in magnitude, but opposite in direction for each vehicle during the impulse, which occurs during the approach phase of the collision. Th e relationship of Equation 4.49 allows for the same crosscheck method as developed for collin ear central impacts of Chapter 3, in order to ensure that the damages considered and us ed in the equations leading to the severity determination of Equations 4.44 and 4.45 are appropriate for the collision event. Since measurements are only as exact as the methodology used, and since limitations in the significant digits of inertial, st ructural and dimensional data fo r vehicles are also present, a vehicle-to-vehicle col lision should be considered to be within balance if the forces calculated for each vehicle using Equation 4.47 are within 10%. Whenever possible, the force balance cro sscheck should be used when deformation analysis with known A and B stru ctural stiffness values for both vehicles are utilized in order to ensure that Newtons third law is not inadvertently being violated by incorrect variable usage during the analysis process. Howe ver, just as demonstrated in Chapter 3 for

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105 collinear central impacts, a Newtons third la w compliance can be forced between vehicles of an oblique or offset impact so that other parameters can be verifi ed and/or determined. 4.7.2 Determining Missing Deformation Depths Due to the passage of time a vehicle may become unavailable for inspection, photographic evidence of a vehicles deformation profile may be insufficient or for other reasons outside the control of an analyst, instances arise where one ve hicle associated with a collision event may be unavailable for direct inspection and measurement, or determination of deformation profile. In some cases, only a few photographs generally depicting deformation patterns may be availa ble, so that only the overall width of deformation is determinable on one of the colliding vehicles. The Generalized Impact Force-Deflection Model (Equation 4.36) and the Generalized Impact Work/Energy Model (Equations 4.44 and 4.45) in their present form as derived require know ledge of structural A and B stiffness coefficients and deformation profiles, cR and w for both vehicles. However, the Newtons third law expression of Equation 4.49 allows for not only the Generalized Newtonian Force-Balance Crosscheck but also for a Newtons third law predicti on of the damage profile by solving for cGen for the vehicle with unknown deformation dept hs, which is determined from the Generalized Force-Deflection Model which considers the PDOF ac ting upon the vehicle during an oblique collision. Equation 4.50 w ill hereto be referred to as the Generalized Newtonian Deformation Prediction Model: Gen known unknownGenGen knownunknown unknown Gen unknown unknownF wFF A c B 4.50 Where, Gen knownF generalized peak force calcu lated for the vehicle of known deformation Gen unknownF generalized peak force for vehicle of unknown deformation

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106 Gen unknownc Newtonian predicted generalized deformation for vehicle of unknown deformation unknownw = total deformation width for the vehicle of unknown deformation depth (quantity must be known or knowable) unknownA and unknownB = structural stiffness va lues for vehicle of unknown deformation depth (these values must be known or knowable) The Generalized Newtonian Deformation Prediction Model of Equation 4.50 also allows for the determination of each force element determined on the vehicle of known multiple deformation measurements, Gen known j Ffor j=1n and predicted onto each associated width for the ve hicle of unknown deformation depth to produce a piecewise damage profile, Gen unknown kc for k=1..n However, if the width of damage between the two vehicles are significantly different due to scraping or induced damage, then the total force, Gen knownF, determined from the Generalized Impact Force-Deflection Model (Equation 4.36) can be used to determine what is essentially a weighted average generalized deformation depth, Gen unknown averagec across the deformation width, unknown totalw, for the vehicle of unknown deformation depth. While a comparative deformation profile would not be produced, the final results from either calculation method w ith respect to the Generalized Impact Energy/Work Model and the Generalized Force-Deflection Velocity Change equations will be the same. Utilizing the Generalized Newtonian Deformation Prediction Model of Equation 4.50 to predict the damage profile or weight ed average deformation depth for a vehicle allows for the completion of the values necessary to determine the generalized velocity changes of the vehicles wit hout the need for direct defo rmation depth measurements on one of the vehicles involved in an impact event. By forcing a Ne wtons third law force balance between the two colliding vehicles or objects, momentum and energy are conserved.

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107 The Generalized Newtonian Defo rmation Prediction Model of Equation 4.50 establishes the following important characteristic regarding deformation patterns: For short duration impacts of known applied force between composite structures where the structural proper ties and deformatio n profile of at least one of the interacti ng structures is known or knowable, the extent of inward deformation of the opposing inte racting composite structure can be determined when only the width of c ontact between the two surfaces is known or knowable. 4.7.3 Generalized Work/Energy Principles and Unknown Stiffness Coefficients To this point, all of the models presente d in this chapter and Chapter 3 require for a given known force, the structural stiffness characteristics of each vehicle must be known to determine the work and collision severity produced by an impact. However, as indicated in Chapter 3, vehicle structural stiffness co efficients may be limited by vehicle or vehicle family, front/rear/side structures and for heavy vehicles. This leads to the application of Work/Energy principles as they relate vehi cle deformation analys is, which should be a familiar and intuitive approach at this point. The Generalized Force-Deflection Model provides the means by which the net generalized force of a central or oblique impact can be de termine, and the measureable deformation to a vehicle with PDOF consider ations provides the distance over which the net generalized force is applied. GenGen workFdxU Where, FGen is the generalized force magnitude applied at peak impulse, and dx is the distance over which the work on the system occurs Applying Work/Energy principles to a generalized vehicle-to-vehicle collision as was demonstrated for a central impact in Sect ion 3.8.3, now allows for the development of the Generalized Work/Energy Missing Stiffness Equation as it applies to the vehicle of unknown structural stiffness characteristics.

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108 00cos()nn R Gen j jknownknown j jj knownw c FAB 21tan()Gen Gen R friction unknownunknown unknown knownCUU F 4.51 Where, Gen unknownU = Generalized work on vehicle with unknown A/B Gen knownF = Generalized force applied to the vehicle of known A/B 1 1 n j j R j unknown n j jwc C w = weighted average deformation depth on vehicle with unknown A/B values The Generalized Work/Energy Missing Stiffness Equation of Equation 4.51 allows the weighted average of residual deformation with respect to the PDOF applied to the vehicle of unknown structural A and B stiffness values to act as the distance over which work is done on the vehicle structure at the damage centroid. The Generalized Work/Energy Missing Stiffness Equation establishes no vehicletype-specific conditions or rest rictions that are characteri stic of the use of A and B stiffness factors. By using the Generalized Work/Energy Missing Stiffness Equation for the determination of the generalized work done on the system, the analysis of impacts with vehicles of unknown stiffness due to the lack of testing, lack of model year overlap, or lack of adequate information regarding structural characteristics is no longer a limiting factor. As long as one of the vehicl e surfaces involved in an im pact has known or knowable structural stiffness charact eristics, any vehicle or object of unknown structural characteristics but known or knowable deform ation profile can be analyzed using the deformation analysis methods contained in Chapters 3 and 4 of this study.

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109 The greatest significance for Generalized Work/Energy Mi ssing Stiffness Equation application involves the analysis of the following impacts where vehicle and surface specific structural stiffness characteri stics are either sc arce or non-existent: Broadside or oblique side impacts Rear end impacts Impacts involving light trucks, vans and s port utility vehicles where vehicle and surface specific structural s tiffness values are scarce. Impacts involving heavy vehicles, buses, RVs and other similar vehicles with few vehicle and surface specific data. Impacts with non-vehicular objects, or unique vehicles such as trailers or heavy equipment that deform when struck, but have no known structural stiffness data. Later work in Chapter 5 will de monstrated that utilizing the Generalized Work/Energy Missing Stiffness of Equation 4.51 under all circumstan ces, regardless of whether or not the A and B structural stiffn ess values for both vehicles are known, will produce greater accuracy and precision when determining total velocity change levels for the involved vehicles. 4.8 Summary of Findings; G-DaTAV System of Equations The culmination of the gene ralized models deve loped in this study comprise what is hereto defined as the Generalized Deformation and Tota l Velocity Change Analysis (G-DaTAV) System of Equations. By formulating generalized models for deformation analysis, a broader application of the fundamental principles developed in Chapters 3 has been developed that apply to not only central im pacts but non-central oblique impacts that produce rotation. When non-conservative forces act upon the vehicles during the approach and/or depart ure phase of the collision, with or without rotation due to a non-central collision configuration, lin ear momentum is not conserved. The GDaTAV System of Equations is necessary for the determination of the total velocity change of an impact, which is defined as follows:

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110 Total Velocity Change: The Total Velocity Change is the cu mulative velocity change of colliding vehicles due to the conservative and non-conservative forces acting upon the system during the approach and departure phases of an impact event. The G-DaTAV System of Equations includes a series of generalized equations that determine the total velocity change produced by a central or non-central collision configuration. The Generalized Impact Force-Deflection Model of Equation 4.36 establishes three important rela tionships between collision fo rce, deformation dimensions and the structural design charac teristics of a vehicle, which can be stated as follows: Force-Deflection Principle 1: Im pulse-Deformation Relationship The peak impulse applied to a vehicle structure during an impact event is equivalent to the sum of the forces necessary to initiate permanent damage and the forces which produce permanent residual damage to the structure during the impact approach phase. Force-Deflection Principle 2: Deform ation Depth and Width Relationship Deformation intrusion into an impacted st ructure or surface of a vehicle is a function of the structures ability to resist permanent inward deformation over the contact and induced damage width of applied forces. The final force-deflection relationship consid eration is an intuitive consideration of the previous two rather simplistic but signif icant relationships. Wh en considering ForceDeflection Principles 1 and 2 together, a dditional very crucial and key generalized fundamental design considerations for the management of collision forces as they relate to vehicle structural design can be stated as follows:

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111 Impact Force-Deflection Principle 3: Ge neralized Impact Force Management Vehicle structural desi gns that manage impact forces by maximizing the distribution width will minimize inward deform ation into the occupant compartment. Such designs provid e additional protection for vehicle occupants from secondary impacts from intruding components associated with the vehicles exterior shell and/ or interior components within the deformation region. The Generalized Newtonian Defo rmation Prediction Model of Equation 4.50 establishes the following important characteristic regarding deformation patterns: Impact Force-Deflection Principle 4: Gene ralized Impact Deformation Prediction The extent of inward structural defo rmation of a composite structure of known stiffness characte ristics is determinable from the known applied forces determined from known or knowable structural properties and deformation profile of an interacting composite structure. Equation 4.51, the Generalized Work/Energy Mi ssing Stiffness Equation, represents one of the most significant cap stone developments of this st udy. The use of Equation 4.51 permits for the analysis of the following impacts where vehicle and surface specific structural stiffness characteristics are either scarce or non-existent: Broadside or oblique side impacts Rear end impacts Impacts involving light trucks, vans and s port utility vehicles where vehicle and surface specific structural s tiffness values are scarce. Impacts involving heavy vehicles, buses, RVs and other similar vehicles with few vehicle and surface specific data. Impacts with non-vehicular objects, or unique vehicles such as trailers or heavy equipment that deform when struck, but have no known structural stiffness data.

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112 The Generalized Work/Energy Missing Stiffness Equation provides the equivalency between the work done to damage a vehicle s composite structure resulting in a complex and/or piecewise linear deformation profile, and the approximation of the same work when considering the force acting through the we ighted average deformation depth for the conplex deformation profile. For a given applied peak im pact force, the dissipated energy doing work to produce damage to a moto r vehicle or another co mposite structure is determinable using the weighted average deformation depth for any complex damage profile. The above final statement marks the capstone of the study objectives with respect to vehicle impact deformation analysis. Th e final declaration allows for a broader application of the developed generalized models to offset and oblique, multiple degree-offreedom impact systems. The equations and principles developed within this study have primary applicability to automobile collisions. Ve hicular impacts typically invol ve the interactions between composite non-homogeneous structures. Therefore, the principles and concepts developed in Chapters 3 and 4 have the potential for broad-based utility for the design and experimental performance prediction of comp osite materials and structures outside of automotive design. The principles developed in this study should also apply to composite materials intended to deform when subjected to brief impulse impacts such as athletic protective equipment, safety shielding and accessories, or other similar composite material products designed for impact resistance or protection. Chapter 5 outlines the results of applying the G-DaTAV System of Equations developed in Chapter 4 to actual vehicle-to-v ehicle instrumented testing. Utilizing the twelve staged RICSAC (Research Input for Computer Simulation of Automobile Collisions) impact tests comprises the initial verification of the generalized methods developed in this study. The RICSAC testing is the earliest st udy still used to date for the validation of computer simulation models of automobile coll isions. The final verification involves application of the generalized analys is methodologies to vehi cle-to-vehicle real-

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113 world collisions having airbag system event da ta recorder (EDR) data that recorded the total velocity changes of the vehicles as reported by the National Automotive Sampling System database maintained by the National Highway Traffic Safety Administration. The aforementioned validation procedures determine the accuracy, precision, applicability, efficacy and limitations of the generalized an alysis methodologies wh ile also considering under-represented vehicle stiffness categorie s such as light trucks, SUVs and vans.

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114 CHAPTER 5: GENERALIZED DEFO RMATION AND TOTA L VELOCITY CHANGE ANALYSIS (G-DATAV) SYSTEM OF EQUATIONS APPLICATION AND EVALUATION The following chapter summarizes the results of applying the G-DaTAV System of Equations developed in Chapter 4 to industry standard staged co llisions as a first level of accuracy, precision and efficacy assessmen t. As expected, the highest degree of correlation and linear test-of-fit was achieved when applying the gene ralized algorithms developed in this study to carefully stag ed collisions. Additional validation of the GDaTAV System of Equations tests the accuracy, precision and efficacy when applied to real-world collision events. Finally, co mparing the results of the original CRASH formulations introduced in Chapter 2 with resp ect to the industry sta ndard staged collision tests provides a full appreciation of the contributions made by the G-DaTAV System of Equations. The study results provide insight into the accuracy, precisio n, an efficacy of the G-DaTAV System of Equations; a head-to-head comparison between the old methods and the new generalized algo rithms developed during this study. 5.1 Overview of G-DaTAV System of Equations Development 5.1.1 Chapter 2 Contributions Chapter 2 provides a historical summary of research regarding the fundamental analysis principles of vehi cle deformation from Campbells initial insights to the CALSPAN development of th e CRASH and SMAC programs. The derivation of the original equations in Chapter 2, specifically Equations 2.7 through 2.9, rest firmly on physics and engineering dynamic principles as they apply to the mechanical system of a vehicular impact. However, the applicability of the original equations remain largely limited to central, inelastic collisions producing negligible rotation. Accordingly, the equations introduced in Chapter 2 have somewh at simplistic and restrictive applications related to real-world collision events.

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115 5.1.2 Chapter 3 Contributions Chapter 3 introduces the reader to the foundational equa tions of motion and system momentum for a central impact utilizing La grangian analytical dynamics. Analytical dynamics aids in identifying conservative and non-conservative generalized forces acting on the collision system that may not be determinable using the intuitive classical Newtonian dynamics approach. Ut ilizing analytical dynamics a llows for the inclusion of tire/ground constraint forces ac ting external to the collision impulse for central impacts. Chapter 3 develops the followi ng basic central impact vehicle deformation analysis equations that form the foundation for the generalized methodologies developed as the targeted scope of the study: The Central Impact Force-Deflection Model (Equation 3.25) The Central Impact Energy/Work Model (Equation 3.26) The Central Impact Force-Deflectio n Velocity Change Models (Equations 3.48 and 3.49) The Newtonian Central Impact Fo rce-Balance Relationship (Equation 3.50) The Newtonian Central Impact Defo rmation Prediction Model (Equation 3.51) The Central Impact Piecewise Wo rk/Energy Missing Stiffness (Equation 3.52) The Central Impact Weighted Average Work/Energy Missing Stiffness (Equation 3.54) The Central Impact Piecewise and Weighted Average Work /Energy Missing Stiffness methods of Equation 3.52 and 3.54 represen t a major contributi on of this study by eliminating the reliance upon structural sti ffness factors for both colliding vehicles. 5.1.3 Chapter 4 Contributions Chapter 4 derives the capstone objectives by formulating the G-DaTAV System of Equations utilized when determining a new de scriptive concept regarding vehicle collision analysis defined as the total velo city change Non-central impacts result from a collision impulse offset from the vehicle mass centers. The offset PDOF creates a moment about the vehicle mass center, resulting in rotatio n. The system mo mentum following a non-central impact is comprised of linear and ro tational components as a result of the offset

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116 applied impact forces. Attempts to analyze and/or determine the total velocity change resulting from an oblique non-central impact utilizing only the principles of the Conservation of Linear Momentum will lead to an under-approximation of the total velocity change values for each vehicle. The under-prediction error of the total velocity change increases as the moment arm of the applied force to the vehicle mass center increases. Additionally, linear momentum is fundamentally derived from impulse-momentum principles resulting from the conservative forces acting during a given impact event. Conservative impact forces result from potentials (thus the modeling of the system as a continuum of linear springs) which act equal in magnitude but opposite in direction along the PDOF for each vehicle in accordance with Newtons third law. However, for other than simple central impacts, non-conservative forces due to ti re/ground friction constraint forces and inter-vehicular friction dissipative for ces can act upon the syst em during an impact event. Without the understanding of the system equations of motion, the contributions of non-conservative forces cannot be accounted for with strictly a linear or rotational momentum based determination of velocity change. Accordingly, Chapter 4 develops and introduces methods which determine the total velocity change for vehicles involved in non-central impact configurations. In the absence of non-conservative force contributions these same equations simplify to the central impact algorithms developed in Chapter 3. The comp rehensive analytical methods developed in Chapter 4 incorporate restitution, rotation, tir e/ground constraint forces, and inter-vehicular friction dissipative forces pres ent during the collis ion event within a suite of non-conditionspecific, analytically de rived equations; thus generalized equations defined as the Generalized Deformation and Total Velocity Change Analysis (G-DaTAV) System of Equations. 5.2 Anatomy of G-DaTAV System of Equations The main objectives of Chapter 4 developed the G-DaTAV System of Equations as a comprehensive set of generalized algorithms applicable to non-central and central impacts. Separation of additional velocity change contributions to the total velocity

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117 change produced during non-central impacts fall in to two distinct ca tegories; additional non-conservative force contributions resu lting in dissipative and constraint energy sinks, and the addition of applied conservative force contributions due to restitution and rotation. 5.2.1 Non-central Impact Work/Energy Sink Contributions Full scale testing conducted in accordance with federal motor vehicle safety standards for crashworthiness and safety as sessments provide the data necessary to determine the unique A and B structural stiffne ss coefficients for a colliding vehicle (sister vehicles from related model year runs and clone vehicles from partner manufacturers included). [56] Test vehicles collide with a fixed barrier at known velocities, or a moving barrier collides with a stati onary test vehicle. Vehicles are instrumented, acceleration histories and velocity histories recorded, a nd deformation profiles measured. The kinetic energy of the moving test vehi cle or moving barrier dissipate s as deformation work during the collision impulse with/by the barrier. The Generalized Impact Work/Energy Model expressed by Equations 4.44 and 4.45 provide for the determination of the work due to impact forces that produce deformation to both vehicle structures along their respec tive PDOFs, as well as the addition of the dissipated energy sink contri bution due to inter-vehicular friction. The following presentation of the generalized form of thes e equations provides insight into the anatomy of the work/energy considerations resulting from applied forces during a collision event. Each of the elements within the Generalized Impact Work/Energy Model, UOblique and Ufriction, are in units for work/energy (ft.lb for US units, J or kg.m2.s-2 for SI units). 2 2 2 12 0 1221tan 22GenObliquefriction R n j R j j PDOF k scrape jBc Amm Acwgw BmmUUU Inspection of the Generalized Impact Work/Energy Model clearly demonstrate that as the width of scraping between two colliding vehicles approaches 0 (zero), the contribution of dissipated work due to inter-v ehicular friction also approaches 0 (zero). UOblique, collision impulse work Ufriction, inter-vehicular dissipated work

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118 Additionally, as an oblique impact PDOF approa ches collinear, the PDOF is parallel to a primary axis of the vehicle so that the m easured deformation approaches the measured compression of the vehicle structure. As the PDOF approaches where it acts through the mass center of a vehicle, the effec tive rotational (dynamic) mass ratio, component of the Generalized Impact Work/Energy Model approaches 1 (one). Therefore, as inter-vehicular friction, rotation, damage angul ar offset and tire/ground cons traint forces decrease, the Generalized Impact Work/Energy Model equations revert toward s the conditions of the Collinear Impact Work/Energy Model equations. The suite of generalized equations was intended to achieve app licability to all forms of planar central and non-central impacts. Therefore, the Generalized Impact Work/Energy Model Equations 4.44 and 4.45 achieves the study objective. 5.2.3 Non-central Impact Non-Conservative Force Contributions Lagranges Equation (Equation 3.3 from Chap ter 3) simply states that the total energy of a system results from the conserva tive and non-conservative generalized forces acting upon the system. Chapter 3 develops the relationship betw een the generalized conservative forces acting during an imp act and the potentials resulting from the deformation to each colliding vehicle. Also in Chapter 3, the deformation potentials were modeled according to Hookes la w and Newtons second law for the approach phase of the velocity change. Chapter 3 ac counts for the restitution effe cts derived from the system energy and linear momentum during the separa tion phase of the collision event. The effective rotational (dynamic) mass ratio, for each vehicle within Equations 4.46 and 4.47 of Chapter 4 account for the impact momentum change due to the moment of momentum (rotational momentum ) following the application of the collision impulse offset from the vehicle mass centers for non-central impacts. These contributions to the total velocity change are directly due to the conservative forces acting during an impact event. Broadside and oblique impacts can experien ce impulsive contribu tions external to the impact resulting from tire/ ground constraint for ces acting upon the target vehicle as the tires resist motion while the ve hicles are in contact. This same external impulse occurs during many collinear central im pact events as well. The im pulse produced by tire/ground friction, a vector, oppose s the impact impulse for the oblique ly or broadside target vehicle

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119 (opposite in direction of impact impulse), and acts in the sa me direction as the impact impulse acting upon the bullet vehicle (same direction of the impact im pulse). Accordingly, the velocity change contribution due to non-conservative tire/ground constraint forces produces an additive component to the total velocity change for the bullet vehicle, and a reduction in total impulse for determining the total velocity change for the target vehicle. The Generalized Total Ve locity Change Model expressed as Equations 4.46 and 4.47 provide for the determination of the c onservative change in linear and rotational momentum (velocity change), as well as the non-conservative velocity change contributions resulting from tire/ ground constraint forces extern al to the impact impulse. 2212 2 1 11112212 1GenGen GenmUU mgnt e mmmmv Equation 4.46 1112 2 2211222 1GenGen GenmUU egnt mmmv Equation 4.47 Inspection of Equations 4.46 and 4.47 again clearly demonstrates the contributions of the conservative and non-conservative ge neralized forces acting during the impact impulse. Each of the elements within the Generalized Total Ve locity Change Model, represented as vConservative and vNon-conservative, are in units for velocity (ft/sec or mph for US units, m/sec or kph for SI units) and clea rly reflect impulse/momentum contributions. The Generalized Total Velocity Change Model demonstrates that as the tire/roadway constraint approaches 0 ( zero), the contributi on of non-conservative tire/ground constraint forces to the total velocity change also approaches 0 (zero); reverting back to the conditions for the Collinear Velocity Change Model of Chapter 3. As a noncentral impact approaches the condition wher e the PDOF acts through the mass centers of the colliding vehicles no mome nt is produced by the impact impulse. As the moment arm vConservative vNon-conservative vConservative vNon-conservative

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120 approaches 0 (zero), the effectiv e rotational (dynamic) mass ratio, approaches the value of 1 (one). Additionally as the collision approaches on ly plastic deformation with no restitution effects, then the Generalized Total Velocity Change Model of Equations 4.46 and 4.47 approach their parent forms as developed in the earliest of studies by McHenry for the CRASH-based analysis, which was established in Chapter 2 as applicable only to central impacts.[6][7][8] The intent for the de veloped generalized e quations was for broadbased applicability to all forms of planar cen tral and non-central impacts. Therefore, the Generalized Total Velocity Change Model of Equations 4.46 and 4.47 achieves the intended derivation design. 5.3 G-DaTAV System of Equations Evaluation The G-DaTAV System of Equations as developed in Chapter 4 consist of the following suite of equations: The Generalized Impact Force-Deflection Model (Equation 4.36) The Generalized Impact Work/Energy Model (Equations 4.44 and 4.45) The Generalized Work/Energy Missing Stiffness Equation (Equation 4.51) The Generalized Total Velocity Change Model (Equations 4.46 and 4.47) The detailed mathematical application of the G-DaTAV System of Equations to industry testing and real-world collisions was accomplished using PTC MathCAD Prime 3.0. MathCAD completes all dime nsional conversions and significant figure considerations, eliminating mathematical unit conversion or truncation errors. Appendix B provides complete individual G-DaTAV System of Equations calculations for each of the twelve industry staged collision tests. Appendix C contains the detailed mathematical application of these principles to each real-world collision as a secondary validation of the G-DaTAV System of Equations. 5.3.1 RICSAC Testing The Research Input for Computer Simula tion of Automobile Collisions (RICSAC) [100] was a federally funded research pr oject completed by CA LSPAN Corporations Advanced Technology Center in Buffalo, New York. Twelve fully instrumented staged

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121 vehicle-to-vehicle collisions provided critical data for validating existing and future computer analysis and simula tion programs. The collision configurations consisted of oblique and offset non-central impacts spec ifically pairing large and small passenger vehicles for each test. The following list presents the five collision c onfigurations selected by the researchers: Configuration A: Frontal Oblique Offset Configuration B: Rear Oblique Offset Configuration C: Side Perpendicular Offset Configuration D: Front Side Oblique Offset Configuration E: Rear Side Oblique Offset The researchers affixed accelerometers to th e four corners of each vehicle with an additional accelerometer cente red on each vehicles firewa ll between the engine and occupant compartments. Vehicle impact sp eeds were obtained by pulling the vehicles together into impact using a trolley cable syst em. The trolley attachment released prior to impact so as not to introduce additional external contributions to the impact tests. Each test was documented using high-speed film, direct measurements of vehicle deformation patterns, and still photography. Integration of the acceleration curves from the firewall accelerometer data provided the initial RICSAC determination of the vehicle total velocity change testing levels. McHenry, who was one of the primary CALSPAN research engineers on the project, later determined the reported velocity changes were incorrect. Each firewall accelerometer position was different that the respective vehicle mass centers. As such, McHenry completed coordinate transformatio ns of the data to provide the total velocity change test values appropriate at each test vehicles mass center [57]. The CALSPAN test facility had an effectiv e dry surface skid resistance value at 87, which is equivalent to a drag factor of = 0.87. Each RICSAC test provides the vehicle masses, damage profiles, impact configurations and data necessary for the determination of the total velocity changes for each vehicle-to-vehicle impact test using the G-

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122 DaTAV System of Equations. The following table lists the properly transformed total velocity changes for the vehicles resulting from all twelve RICSAC tests. Table 5.1 Summary of RICSAC tests with coordinate transformation RICSAC Test and Vehicle Vx Transform kph Vy Transform kph Vtotal Transform kph Vtotal Transform mph 1-1974 Chevrolet Malibu -18.2 7.7 19.8 12.3 1-1974 Ford Pinto 25.7 -8.5 27.2 16.3 2-1974 Chevrolet Malibu -27.8 14.1 31.2 19.9 2-1974 Ford Pinto -30.1 -28.3 41.3 26.0 3-1974 Ford Torino -15.3 -1.1 15.3 9.3 3-1974 Ford Pinto 25.1 3.5 25.4 15.3 4-1974 Ford Torino -30.1 -0.5 30.1 17.4 4-1974 Ford Pinto 35.7 1.0 35.7 22.1 5-1974 Ford Torino -26.1 -0.2 26.1 16.4 5-1975 Honda Civic 41.0 2.4 41.0 25.4 6-1974 Chevrolet Malibu -14.2 3.9 14.6 9.5 6-1975 Volkswagen Rabbit 22.0 -6.8 23.0 14.2 7-1974 Chevrolet Malibu -18.8 4.0 19.3 12.2 7-1975 Volkswagen Rabbit 31.4 -5.6 31.9 18.3 8-1974 Chevrolet Chevelle -21.6 10.6 24.0 15.1 8-1974 Chevrolet Chevelle 13.2 -11.9 17.7 11.4 9-1974 Honda Civic -29.0 13.7 32.0 20.5 9-1974 Ford Torino 11.1 -7.1 13.2 8.3 10-1974 Honda Civic -46.3 29.1 54.7 34.8 10-1974 Ford Torino 15.9 -12.4 20.1 12.4 11-1974 Chevrolet Vega -39.3 2.6 39.4 25.0 11-1974 Ford Torino 25.3 0.8 25.3 16.4 12-1974 Chevrolet Vega -65.7 -1.3 65.7 44.7 12-1974 Ford Torino 43.0 -2.4 43.0 29.7

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123 5.3.2 RICSAC Original Study Findings Part of the RICSAC study evaluated the pr ecision and accuracy of the originally formulated CRASH III and SMAC based com puter programs. Chapter 2 of this study briefly introduced each analysis program. Table 5.2 outlines the comparison between the original CRASH III formulation and the Table 5.1 coordinate transformed RICSAC tests. Figure 5.1 graphically demonstrates the linear comparison plot of the data in Table 5.2. Table 5.2 Original CRASH based deformation analysis results

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124 Figure 5.1 Regression graph for origin al CRASH deformation based analysis The RICSAC test data versus the orig inal CRASH III formulation show poor correlation, having an R2 value of 0.33. The 2 test-of-fit value of 70.2 ( p =0.0001, n =23), indicates the differences between the total velocity change of the individual RICSAC tests versus the calculated values using the orig inal CRASH III formulation are statistically significant. The overall precision of the analysis results was .45% to within one standard deviation of the mean. The percen t error ranged from an under-approximation of -69.0% (-10.9 mph difference) for vehicle 2 fo r RICSAC Test 3, to an over-approximation of +79.2% (15.7 mph difference) for vehicl e 2 of RICSAC Test 7. Additionally, the descriptive statistics indicate the applicati on of the original CRASH III formulation to oblique or offset impacts should be rej ected as producing statistically significant differences between the calculated total velocity change and test reported total velocity change values. RICSAC Volume IV, Table 3-1 summari zed the original SMAC formulation results for total velocity change determination utilizing vehicle pre-collision velocities to +10% -10%

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125 predict post-collision motion a nd departure velocity directi ons. SMAC is an iterative solving system based upon planar dynamics. SMAC requires knowledge of vehicle impact velocities or total velocity change as input into the equations and does not rely upon vehicle deformation principles. Real-w orld collision anal ysis rarely begins with known impact velocities for the involved vehicl es. Accordingly, SMAC is prim arily used as a verification tool only after determining collision veloci ties from a reconstruction analysis. Table 5.3 summarizes the results of the total velocity change determination using the original SMAC formulation. Figure 5.2 graphically illustrates the linearity of the SMAC analysis versus the total velocity change data of Table 5.1. The original SMAC formulation showed an acceptable correlation w ith the test data having an R2 value of 0.941. The 2 test of fit value of 8.17 ( p =0.9981, n =23) indicates the difference between the test and calculated valu es for the original SMAC formulation are not statistically significant. While correlati on and descriptive sta tistics indicate that reasonable overall accuracy is achieved using th e original SMAC formulations, the overall precision of the results varies by 11.77% for errors within one sta ndard deviation of the mean. The percentage differences between the ca lculated and test results varied at -12.7% (-2.5 mph absolute difference) for vehicle 2 of RICSAC Test 7 up to +35.8% (4.0 mph absolute difference) for vehi cle 2 of RICSAC Test 8.

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126 Table 5.3 Original SMAC based momentum analysis results

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127 Figure 5.2 Regression graph for origin al SMAC momentum based analysis The relative accuracy and precision of th e original CRASH III vehicle deformation analysis results of Table 5.2 form the baseline for comparison with the improvements and contributions to the G-DaTAV System of Equations developed in this study. 5.3.3 RICSAC G-DaTAV System of Equations Application Approach Appendix B contains the individual calcu lations for each of the twelve RICSAC tests using the G-DaTAV System of Equations developed as the objective of this study. The benefit of using the staged RICS AC testing as a means of evaluating the accuracy and precision of a collision analysis model rests on the instrumentation and thorough documentation for each of the collis ions. Each instrumented RICSAC test measured the total velocity change acting upon each vehicle resu lting from the collision impulses and external non-conservative fo rces acting during the test impacts. +10% -10%

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128 Each RICSAC test contains the following information necessary for the evaluation of the total velocity change using the G-DaTAV System of Equations, and no interpretation or estimation of thes e crucial variables are necessary: Vehicle weights, front and rear axle weight distributions, vehi cle mass centers Vehicle collision deformati on width and depth profiles Diagrams at impact, post-collision trajecto ries and tire marks, and the final rest locations Contact with wheel/tire hard zones for rest itution considerations when appropriate Drag factor for the test site surface Scaled diagrams of the vehicles were pos itioned together at maximum engagement with the PDOF passing through the damage centroids as discussed in Chapter 4. The (x,y) velocity change vector data in Table 5.1 provided the information necessary to determine the PDOF acting upon each vehicle.[56] The mo ments of inertia for the vehicles were determined using Equation 4.23 from Chapter 4. Since the RICSAC tests reported all measurements in US units, the moment arms for the offset and oblique impacts were measured using the Faro Reality CAD program to within 0.1 feet. The RICSAC testing provided all deformation widt h and depth dimensions to the nearest inch. With the exception of the transformed total velocity change data reported in Mc Henry [56], all test related data recorded within the individual RICSAC reports as summarized in RICSAC Volumes 2 and 3 were used. PTC MathCAD Pr ime 3.0 was used for the complete set of calculations contained in Appendix B, which completes all unit conversions and significant digit truncations inte rnally to eliminate the potenti al for unit conversion errors. The initial evaluation using the industry standard RICSAC testing determines the accuracy and precision of the G-DaTAV System of Equations developed in this study as they relate to idealized variable inputs. The capstone contribution of this study involves the incorporation of all of the following contributions to the approximation of the total velocity change produced by a non-central impact: Collision restitution effects (Equations 4.46 and 4.47) Tire/ground friction contributions (Equations 4.46 and 4.47)

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129 Oblique impact rotational mass ef fects (Equations 4.46 and 4.47) Inter-vehicular frictional eff ects (Equations 4.44 and 4.45) Reliance upon frontal structural stiffness coefficients [ 19] of striki ng vehicle only and using Generalized Work/Energy Missing Stiffness Equation 4.51 for determining the work to damage the struck vehicle for RICSAC collision configurations C, D and E. Reliance upon frontal structur al stiffness coefficients [19] of only one colliding vehicle (switched between impacts to test for sensitivity) and using Generalized Work/Energy Missing Stiffness Equation 4.51 for determining the work to damage the other collision associated vehicle for RICSAC collision configurations A and B. 5.3.4 RICSAC G-DaTAV System of Equations Analysis Results Table 5.4 summarizes the raw results from determining the vehicle total velocity change from the G-DaTAV System of Equations. The results are plotted for linearity in Figure 5.3 regarding the piecewise damage profile analysis, and Figure 5.4 for the weighted average damage profile methodological approaches. The G-DaTAV System of Equations showed excellent correlation to the RICSAC test data with an R2 = 0.989 for piecewise damage profile analysis, and an R2 = 0.991 for the weighted average damage profile analysis. The 2 = 1.06 for the piecewise and 2 = 1.08 for the weighted average methodologies (two-tailed =0.99, n =23) indicate no statistica lly significant difference between the total velocity changes for the RICSAC tests and cal culated values using either G-DaTAV System of Equations. The overall precision when applying the G-DaTAV System of Equations varies by .0% (.1 mph) for the piecew ise method and .8% (.9 mph) for the weighted average method for errors within one standard deviation of the mean. The greatest percentage differences between the calculate d and test results for the piecewise method varied between -8.5% (-1.5 mph absolute diffe rence) for vehicle 2 of RICSAC Test 7 to +9.9% (2.9 mph absolute difference) for vehi cle 2 of RICSAC Test 12. The precision of the weighted average method improved the ma ximum overall deviations to -2.8% (-0.5

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130 mph absolute difference) for vehicle 2 of RI CSAC Test 1, and +8.9% (0.9 mph absolute difference) for vehicle 2 of RICSAC Test 3. A clear evaluation of the improvements made by the G-DaTAV System of Equations versus the original CRASH III models lies in the comparison between the absolute average percentage differences (AAPD) between the two datasets. The AAPD improved by a factor of more than 10, with 34.9% for the original CRASH III evaluation, to an AAPD of 3.1% for the piecewise and 3.4% for the weighted average approaches of the G-DaTAV System of Equations. The marked improvements in AAPD provide demonstrative evidence to the efficacy of the methodologies deve loped in this study. Correlation, descriptive statistics, and the raw analysis results indicate a significant degree of precision and accu racy when applying the G-DaTAV System of Equations for determining the total velocity change levels of a non-central im pact configuration when compared to the RICSAC dataset. Careful st ating, instrumentation and documentation of each of the twelve RICSAC tests precisely dete rmined many of the variables required as input into the G-DaTAVSystem of Equations. Therefore, a high degree of correlation should be expected when comparing the staged RICSAC test results to any set of equations that properly consider and/ or describe the physics of a non-central impact event.

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131 Table 5.4 RICSAC test values versus G-DaTA V analysis

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132 Table 5.5 Summary of statistics Figure 5.3 G-DaTA V piecewise damage match values versus RICSAC tests +10% -10%

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133 Figure 5.4 G-DaTAV weighted average damage values versus RICSAC test 5.4 National Automotive Samplin g System Real-World Collisions The National Automotive Sampling System (NASS) provides the NHTSA with a comprehensive compilation of real-world co llision events represen ting a broad-based collection of collision configur ations from across the country. This data represents a reusable source of information that was collected utilizing standardi zed field techniques implemented by NASS trai ned field technicians. Through using a core set of crash data components , NASS has demonstrated its utility and applicability to a vast array of statistical and analytical studi es regarding traffic safety and vehicle collision dynamics. [58] Real-world oblique or offset collisions involving light trucks and SUVs from the NASS database further evaluate the accuracy, precision and efficacy of the G-DaTAV System of Equations developed in this study. The onl y records selected for evaluation contained EDR data imaged from the airbag system of at least one vehicle for oblique or offset collisions involving at least one light truck or SUV. The EDR systems, just like the +10% -10%

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134 accelerometers in the staged RICSAC collision te sting, record the tota l acceleration pulses, which integrates to the total velocity changes of the impacted vehicles. Even with the relative prevalence of mode rn airbag system equipped vehicles, many of the hundreds of collisions involving light trucks or SUVs within the current NASS database either did not have EDR data, or contained inco mplete data records detailing the deformation profiles of each vehicle. The NASS records contained 44 light trucks or SUVs as part of 25 real-world ob lique collisions from 2010 to 2013 NASS data having complete EDR and deformation profile measurements and photographs. The fact that the database contained so few collisions that met the compre hensive criteria further reinforces the need for the G-DaTAV System of Equations in order to facilitate accurate analysis of the multitudes of other real-world collision ev ents occurring daily involving non-EDR equipped vehicles. 5.4.1 NASS G-DaTAV System of Equations Application Approach Twenty-five (25) collisions from the NASS Crashworthiness Data System (CDS) Case Viewer from the 2004 to 2013 approved data se t met the following specific criteria for consideration: Two vehicle collisions involving at least one light truck/van or one SUV category vehicle, with a preference to collision s involving only these category vehicles. At least one vehicle must have a complete EDR imaged report using the Bosch Crash Data Retrieval Tool (Bosch CDR Tool) without ev idence of sign ificant data clipping or incomplete data records, w ith preference upon collisions involving both vehicles having a CDR report. Both colliding vehicles have complete measured damage profiles consistent with photographs documenting the post-collision condition of each vehicle. One vehicle must have Ne ptune Engineering NEI da tabase reported A and B structural stiffness coeffici ent values specific to the ve hicle and impacted surface, or applicable to sister model year runs or corporate manufacturer clones. The NASS database provides the year, ma ke and model of each colliding vehicle and the standard curb weight from various sour ces, as well as the occupant and cargo load

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135 at the time of impact when known. However, NASS field investigat ors do not weigh the vehicles, determine the mass center or the weight distribution for any vehicles. In order to replicate real-world analysis procedures likely followed by individual collision analysts or researchers, the standard curb weight and distribution were determined using an industry resource [59], and while adding occupant and cargo loads. Additionally, the NASS database does not provide a measured drag fa ctor for the individual roadway surfaces of the reported collisions. Accordingly, a uniform approximation of a dry roadway drag factor of = 0.80 was used as the baseline roadway fr iction for each analysis. Structural stiffness data for one of the colliding vehicles wa s obtained through the NEI database.[19] The following additional variables necessary for the G-DaTAV System of Equations analysis were available directly from the NA SS database for each collision as follows: Vehicle collision deforma tion width and depth profile s (measured in SI units) Diagrams at impact, post-collision trajectorie s and tire marks, a nd vehicle final rest locations Contact with wheel/tire hard zones for re stitution considerati ons provided through vehicle photographic eviden ce, when appropriate EDR output images using the Bosch CDR To ol for at least one of the vehicles, having both longitudinal and lateral total velocity change recordings The vehicles were positioned together at maximum engagement with the PDOF passing through the damage centroids as discussed in Chapter 4. The total velocity change (x,y) vectors from the total velocity change records from the vehicle EDRs provided the means for determining the PDOF acting upon ea ch vehicle. Equation 4.23 determined the moments of inertia for the vehicles. The NAS S collision data is reported in SI units. Therefore, the Faro Reality CAD program was utilized to measure the moment arm for the offset, and oblique impacts to within 0.1 meters. The verifi cation analysis only used the NASS reported damage width and depth dimensi ons measured to the nearest centimeter. All data recorded in the NA SS reports of each real-world collision event was used as reported with no interpretation or modificat ions. PTC MathCAD Prime 3.0 was used for the calculations contained in Appendix C, which completes all unit conversions and significant digit truncations inte rnally, eliminating the potenti al for unit conversion errors.

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136 The final evaluation using the NASS reported real-world collision data allows for the determination of the accuracy and precision of the G-DaTAV System of Equations. The benefit of a secondary an alysis utilizing the NASS da ta lies in the fact that NASS studies represent data t ypically available or obtained during a real-wor ld collision investigation. The NASS validation incorpor ates the following contributions to the total velocity change produced by an oblique, offset and noncentral impact applied to collisions involving SUVs and light: Collision restitution effects (Equations 4.46 and 4.47) Tire/ground friction contributions (Equations 4.46 and 4.47) Oblique impact rotational mass ef fects (Equations 4.46 and 4.47) Inter-vehicular frictional eff ects (Equations 4.44 and 4.45) Reliance upon frontal structur al stiffness coefficients [19] of only one colliding vehicle while using the Generalized Work/Energy Missing Stiffness Equation 4.51 for determining the work to damage the opposing colliding vehicle for each NASS two-vehicle collision event. 5.4.2 NASS G-DaTAV System of Equations Analysis Results Table 5.6 summarizes the raw results from determining the vehicle total velocity change of the NASS data using the G-DaTAV System of Equations. The results are plotted for linearity in Figure 5.5 regarding the piecewise damage profile analysis, and Figure 5.6 for the weighted average da mage profile analysis approaches.

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137 Table 5.6 NASS reported Bosch CDR Tool data versus G-DaTA V analysis

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138 Table 5.7 Summary of Statistics Figure 5.5 G-DaTA V piecewise damage match versus NASS Bosch CDR data

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139 Figure 5.6 G-DaTAV weighted average damage versus NASS Bosch CDR data The G-DaTAV System of Equations again showed excellent correlation to the NASS total velocity change Bosch CDR data with an R2 = 0.979 for piecewise analysis and an R2 = 0.975 for the weighted average analysis. The 2 = 2.92 for the piecewise and 2 = 2.98 for the weighted average damage profile analysis methodologies (two-tailed =0.99, n =43) indicate no statistically significant difference between the total velocity changes for the NASS Bosch CDR reported real-world collisions and calculated values using either G-DaTAV System of Equations methodological approach. The overall precision of the results vari es by .3% (.1 mph) for the piecewise method and .7% (.1 mph) for the weighted average method for errors within one standard deviation of the mean. The greatest percentage differences between the calculated and NASS real-world collision results using the piecewise damage profile method varied between -12.9% (-2.5 mph absolute differe nce) for vehicle 1 of NASS 2013-76-094 to +14.4% (1.3 mph absolute difference) fo r vehicle 1 of NASS 201312-059. The precision utilizing the weighted average damage pr ofile method improved to -12.8% (-2.5 mph absolute difference) for vehicle 1 of NASS 2013-76-094 to +10.0% (0.9 mph absolute

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140 difference) for vehicle 1 of NASS 2013-12-059. The fact that the extremes of both methods involved the same vehicles for the same reported collisions could be random, but is probably due to a systematic error in the data reported within the particular NASS files. The relatively high degree of co rrelation between the NASS reported total velocity changes from the vehicle EDR data and the G-DaTAV System of Equations indicates the suite of equations produces reasonable precision and accuracy for determining the total velocity change resulting from these real-world collision events. Additionally, the evaluation results from utilizing the G-DaTAV System of Equations indicates NASS training regarding vehicle de formation documentation app ears adequate for reducing systematic errors between investigators. 5.4.3 Overall G-DaTAV System of Equations Evaluation Correlation and descriptive statistics, as well as the raw analysis of the RICSAC and NASS real-world collisi on data when applying the G-DaTAV System of Equations indicates a highly reliable and signific antly improved degree of precision and accuracy over previous methods for determining vehicular total velocity change levels for oblique and offset non-central impacts. Unlik e the twelve RICSAC tests, the NASS realworld collisions were not carefully staged a nd instrumented, nor were the many variables input into the G-DaTAV System of Equations documented to the precision of the RICSAC tests. The RICSAC testing documentation was completed by the same team of researchers at one test facility, while each NASS real-world collisions occurred at varying locations across the United States over a th ree year span, and documented by different NASS trained investigators. The NASS real-world collision data set represents a realistic comparison to field collected data that a co llision investigator would encounter when tasked with reconstructing a real-world collision event. Stark and pronounced differences between the validation findings when applying the G-DaTAV System of Equations to the controlled RICSAC staged testing ve rsus the random NASS documented real-world collisions should have been present if the algorithms developed within this study had systematic errors or violations in the physics of oblique and offset non-central impacts. Instead, the results of the RICS AC and NASS evaluations of the G-DaTAV System

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141 of Equations demonstrate a reasonable degree of da ta correlation, accuracy, precision and efficacy when applied to real-world collision events. 5.4.3.1 G-DaTAV System of Equations Sensitivity Analysis Every model developed and intended to eval uate the behavior of a mechanical or physical condition is an approximation no matte r how precise, detailed or descriptive. Therefore, it is important to evaluate su ch models for accuracy and precision through application comparisons with controlled test ing. The RICSAC and NASS evaluations of the G-DaTAV System of Equations provide the comparative assessment of the accuracy, precision and efficacy of the approximations of total velocity change for noncentral impacts when analyzing vehicle de formation profiles utilizing the derived algorithms. Regardless of the relative high de gree of accuracy, it is equally important to determine where variable sensitivities to the accuracy of the approximations may exist. As a result of analyzing the RICSAC and NASS data general observations regarding variables with little sensitivity while applying the G-DaTAV System of Equations are as follows: The G-DaTAV System of Equations is not sensitive to reasonable random differences between collision deforma tion measurements obt ained by different, properly trained investigator s. Differences in deformation depth measurements of % generally resulted in no more than a % difference in the total velocity change results for all RICSAC tests comb ined. The greatest deviation for a systematic increase or decrease in deformation depth measurements for both involved vehicles of % was a difference in total velocity change of .6% (greatest deviation in RICSAC 2). The G-DaTAV System of Equations is not sensitive to the inertial properties approximated by using commercially availabl e data in the absence of directly measured vehicle weights and weight di stributions. Varying vehicle masses by % resulted in approximately a .1% difference in total velocity change results across the board for all RICSAC tests. The G-DaTAV System of Equations is not sensitive to the choice of A and B stiffness coefficients obtain ed through the NEI database [19], as long as they are

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142 for the appropriate impacted surface (i.e., front, rear or side), and the test is for sister vehicles that are within the manufacture year range for the same vehicle or its corporate clones. Varying A/B stiffness va lues by % resulted in approximately a .1% difference in total velocity change results across the board for all RICSAC tests. The G-DaTAV System of Equations is not sensitive to the effects of intervehicular friction, since the majority of the work/energy contributions result from the impact impulse. Varying inter-vehicular fiction values by % from a default k = 0.5 g produced no more than a .1% difference in total velocity change results (greatest deviation in RICS AC 2). However, ignoring in ter-vehicular friction for collisions with scraping of 0.75 meters (30 inches) or more resulted in an underapproximation of total velocity change by as much as -9.4% (greatest deviation in RICSAC 6). The G-DaTAV System of Equations is not sensitive to the choice of drag factor for the roadway, as long as the chos en drag factor is within reason for the particular roadway surface; i.e., asphalt, concrete, dry, wet, etc. Varying roadway fiction or braking efficiency values by % generally resulted in no more than a .5% difference in total velocity change calculations (gr eatest deviation in RICSAC 1). Ignoring braking effects for broadside offset and oblique impacts resulted in errors in total velocity change up to approximately .1% (greatest deviation in RICSAC 1). The most critical elements of the G-DaTAV System of Equations having the greatest potential for affecting the accuracy of the total velocity change approximations lay in the determination of the restitution coefficient, the PDOF acting upon each vehicle during the impact, and the resu ltant moment arm a bout the vehicle mass centers. The PDOF angle contribution affects the total deformation depth and, therefore, the total work due to impact forces. Additionally, the direction a nd location of the application of the PDOF determines the moment arm created by an appl ied force offset from the vehicle mass center, and thus the rotational contributions to the total velocity change resulting from a noncentral impact condition.

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143 Neglecting restitution can had as much as a -19.8% under-approximation of the total velocity change for the vehicle of the least ma ss with respect to collisions involving impacts with wheels, tires and axles where the coefficient of restitution ranges from e = 0.2 to 0.4 (great est deviation in RICSAC 3). Ignoring the principle directi on of force correction to the deformation depth had as much as an approximate -33.0% effect upon the determination of total velocity change (greatest deviation in RICSAC 2). Ignoring the dynamic mass ratio rotational effect s can result in as much as a -24.4% effect (greatest deviation in RICSAC 8) upon the total velocity change determination, with the most significant in fluence associated w ith oblique angles with moment arms approaching 1 meter. As demonstrated in the original CRASH an alysis of the RICSAC data as presented in Section 5.2.1.1, errors as high 79.2% (g reatest deviation in RICSAC 7) resulted when the PDOF adjustment, dynamic mass ra tio for rotation, restitution, intervehicular friction and tire/ground force contributions within the G-DaTAV System of Equations are not considered. 5.4.3.2 Mitigation Measures for Sensitivities If a collision event results in a non-central configuration, the following steps should significantly reduce systematic errors introduced into the G-DaTAV System of Equations: Produce scaled diagrams of the vehicles and damage profiles resulting from the impact, including contact and induced damages. Position colliding vehicles together at either initial contact or at maximum engagement for determining the location and direction of the PDOF application upon each vehicle as demons trated in Chapter 4. Unless accurately and precisely determined range the measured values for the PDOF angle and the moment arm for determ ining the effective rotational (dynamic) mass ratio, for both vehicles.

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144 Unless directly measured, range the effective roadway net drag factor when tire/ground impulse contributions should be considered, and varying inter-vehicular friction can enhance accuracy. Following these simple procedures when determining the total velocity changes time to impulse peak while determining peak accelerations, and ultimately when determining vehicle impact ve locities, random and/or syst ematic errors should be significantly reduced, providing the engineer analyst or re searcher with reasonable confidence in the accuracy and precision of the G-DaTAV System of Equations. 5.5 Findings and Conclusions 5.5.1 G-DaTAV System of Equations Applications The RICSAC testing considers small, medium and large passenger vehicles involved in oblique non-central impacts. The intended imp act configurations paired dissimilar sized vehicles duri ng the testing so that rotati onal external non-conservative force contributions could be evaluated. The St ate of Colorado collision statistics analyzed in Appendix A and summarized in Chapter 1 indica te that approximately half of all vehicles involved in highway collisions, to include highway construction zones, fit into the passenger vehicle classification. Light trucks, vans, and SUVs make up the majority of the remaining half of vehicles involved in co llisions in Colorado between 2007 and 2012. Furthering the validation of the G-DaTAV System of Equations to include impacts involving light trucks, vans a nd SUVs further expands the validation of the analysis methodology developed in this study to addi tional non-passenger vehicle classifications. All that is needed to apply the G-DaTAV System of Equations to a collision event are the variables identified in Chapter 4, and A and B structural stiffness coefficients of one of the involved vehicles Therefore, the determinatio n of deformation energy for other vehicle types such as heavy vehicles and motorcycles that have limited or no structural stiffness values, is plausible as long as the other involved vehicle has A and B stiffness data.

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145 Application of the G-DaTAV System of Equations with respect to the RICSAC and NASS data also revealed the following obser vations regarding collision restitution considerations: For collisions producing deformation depths averaging 0.3 m (12 inches) or more over the deformation width, ranging restitution between 0 e 0.1 will provide accurate consideration of restitution effects. For impacts into wheels/axles of at least one vehicle, even when deformation is greater than 0.3 m, a restitution range between 0.2 e 0.40 will provide accurate consideration of restitution effects. For low velocity impacts where vTotal 4.5 m/sec (approximately 10 mph), the restitution varies between e = 0.6 at very low velocities (vTotal 0.9 m/sec, approximately 2 mph) to e = 0.3 at the upper end of the range. Selecting an appropriate restitution value is often an iterative process, but ranging the restitution is expected to provide greater assurance of an accurate consideration of restitution effects. All RICSAC and the NASS collisions analyzed relied upon an inter-vehicular friction of scrape=0.3 to 0.5 for collisions resulting in scraping between the colliding vehicles surfaces. However, if evidence of snagging between the sliding surfaces is present, such as body panels pulled in the directi on of sliding between vehicles, then consideration of higher inter-vehicular friction values may be appropriate. Again, ranging inter-vehicular friction for snagging conditions is likely to produce greater accuracy, but less precision in the analysis results. However, the contribution of inter-vehicular friction is the least significant of all other energy sinks, or impulse and rotational contributions to total velocity change 5.5.2 G-DaTAV System of Equations Limitations The damage analysis methods in existence pr evious to those deve loped in this study required knowledge of structural A and B stiffness data for both vehicles involved in a given collision event, limiting their application. However, fo rmulation of the analysis so that structural stiffness data for only one of the involved vehicle in a collision event allows for a much broader application to include those vehicle clas sifications with limited or no

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146 structural stiffness data. The only major limitation of the G-DaTAV System of Equations is that it remains reliant upon full-scal e impact testing for determining the A and B structural stiffness values for one involved vehicle. The continued reliance on structural stiffness values re quires continued full-scale imp act testing, and may require testing of non-conventional vehicles or impact conditions when structural stiffness data for at least one of the vehicl es is not available. Additional limitations result when ve hicle deformation profiles cannot be reasonably measured directly or indirectly by photographic evidence, or when the analyst has limited training or understanding regardi ng proper deformation profile measurements. Even though the G-DaTAV System of Equations is not particularly sensitive to minor deformation profile measurement fluctuations, unrealistic approximations of deformation width and/or depth will have an effect on the accuracy of the model. Damage profile width and depth determination is quite intuitive and is also the subject of collision investigation training courses. The NASS data analysis dem onstrates that deformation profiles collected by different properly trained inve stigators do not produce random or systematic errors that significantly affect the analysis of total velocity change when using the G-DaTAV System of Equations. If critical variables are unknown or unknowable, the use of the G-DaTAV System of Equations may be limited or unreliable. Exercising proper engineering judgment when applying these algorithms or a ny other form of analysis is critical for reducing error or uncertainty when va riables for the analysis are missing. 5.5.3 Future Research Based on the results of this study, futu re research should include testing the GDaTAV System of Equations with data regarding the following: Motorcycle-to-vehicle impacts. Heavy vehicles, equipment a nd unconventional vehicles. Impacts with objects, both fixed obj ects and those that are movable.

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147 Additional testing of the G-DaTAV System of Equations to real-world impacts with EDR data to verify the total veloc ity changes of the impacted vehicles. Additional staged collision testing with modern vehicles to augment and/or replace the RICSAC test data. It is unlikely that eliminating the reliance upon structural stiffness values or other forms of structural behavior will ev er result in accurate vehicle deformation analysis. The key to understanding the impact resi stance performance of any ma terial or composite when subjected to an impulsive for ce requires at least some know ledge of the ability of the material or composite to resist permanent de formation. Since motor vehicles are fairly complex composite structures, full-scale test ing will likely remain one of the major methods of obtaining structural data for the foreseeable future With this said, additional future studies could also focus on the following: Finite element methods of determining deformation work. Development of vehicle deformation anal ysis methods that are not reliant upon structural stiffness values obtaine d from full-scale impact testing. 3-dimensional methods that determine work using deformation volumes, and PDOF considerations for volumetric analysis of total collision work. 5.5.4 Conclusions The development intent for one comprehens ive suite of generalized equations for determining total velocity change from vehicle deformation profiles applicable to central and non-central impacts successfully culminates into the G-DaTAV System of Equations. The suite of equations developed in th is study require knowledge of structural A and B stiffness coefficients for only one colliding vehicle surface. Current models require knowledge of both vehicle structural stiffness values specific to the vehicle surface involved (i.e., front, rear or side structures). The need fo r only one set of A and B stiffness values represents the primary advancement to the analysis of vehicle deformation profiles arising from the G-DaTAV System of Equations. The developed generalized approach allows for a broader, more universal application of vehicle deformation principles

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148 as they relate to the determination of total velocity change peak impulse time and, therefore, the peak accelerati ons applied during a central or non-central impact event. The G-DaTAV System of Equations is inclusive of vehicle types having little or no A and B structural stiffness data, or when impacted surfaces such as the side or rear are involved where again limited stiffness data exists. In this way, the G-DaTAV System of Equations can be applied to light trucks, SUVs, commercial vehicles, motorcycles or even collisions with objects. The stated objectives of this study in Chapter 1 identified the following main objectives as they relate to the G-DaTAV System of Equations developed in Chapter 4: Develop reliable, accurate and broadly a pplicable generalized vehicle deformation methodologies that eliminate the depende nce on multiple structural stiffness coefficients, regardless of the impacted surface and vehicle type involved. Develop and incorporate inter-vehicular non -conservative frictional forces due to the colliding surfaces of vehicles sliding during the approach velocity change of an impact into a reliable, accurate and broadly applicable generalized vehicle deformation model. Develop numerical algorithms allowing for input of deformation depths and widths at intervals which more accurately e xplain and follow the unique deformation profile of a collision involved vehicle, thereby eliminating the trapezoidal rule reliance upon evenly spaced deformation m easurements of 2, 4 or 6 intervals. Establish important relationships regarding impact forces as they relate to motor vehicle collisions and vehicle deformation properties. Provide future researchers with enhanced an alytical tools necessary for the analysis of traffic collision events for the purpose of enhancing traffic safety, collision dynamics and vehicle design, as well as cr ashworthiness and occupant protection. Further the body of knowledge regarding the behavior of motor vehicles during real-world collision events. Development of reliable, accurate and broadly applicable generalized models sufficiently comprehensive in nature so as not to over-simplify collision dynamics,

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149 but straightforward and practical enough to consider the important known or knowable variables regarding both controll ed environment and real-world collision events, forms the overwhelming impetus of this study. Chapters 3 and 4 developed the physic s and foundational jus tification for the GDaTAV System of Equations, through complete and t horough derivations of the underlying equations. Earlier in this chapter, the developed methodol ogies were tested against industry standard staged collisions (RICSAC) as well as real-world collisions (NASS). Chapter 4 developed the G-DaTAV System of Equations which provides a broad-based application to impact configuratio ns and vehicle combinations that were not subject to analysis with previous deformati on profile analysis met hods. The application of the developed algorithms to both staged, and real-world collision data produced accurate and relatively precis e determinations of total velocity change produced by central and noncentral collision events. A greater understand ing of conservative and non-conservative forces acting on a vehicle-to -vehicle collision system establishes the foundation for considering the contributions of restitution, rotation, inter-vehicu lar dissipative friction forces and tire/ground co nstraint forces. The ability to determine the total velocity change resulting from a collision event provides the researcher with the tools to determine cause and effect relationships: Peak collision accelerations necessary to determine occupant kinematics related to occupant protection measures and vehicle structural de sign considerations for the occupant compartment to reduce and/or e liminate severe and/or fatal occupant injury potentials. Collision response with fixed objects and road side design features such as impact attenuators, guard rail systems, and tr affic control devices to improve highway safety and reduce vehicle occupant injuries from collisions with roadway and roadside design features. Analysis of vehicle velocities for forensic determination of collision causation and contributions.

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150 Determination of loading upon vehicle com ponents and structures for design and failure considerations. Additional principles regarding vehicl e deformation and composite material deformation properties were developed and stated in Chapters 3 and 4 of this study, which provide researchers with a dditional relationships and c onsiderations as follows: Total Velocity Change: The Total Velocity Change is the cu mulative velocity change of colliding vehicles due to the conservative and non-conservative forces acting upon the system during the approach and departure phases of an impact event. Applied Force and Deformation Area Principle: For a given applied peak im pact force, the dissipated energy doing work to produce damage to a moto r vehicle or another co mposite structure is determinable using the weighted average deformation depth for any complex damage profile. Force-Deflection Principle 1: Im pulse-Deformation Relationship The peak impulse applied to a vehicle structure during an impact event is equivalent to the sum of the forces necessary to initiate permanent damage and the forces which produce permanent residual damage to the structure during the impact approach phase. Force-Deflection Principle 2: Deform ation Depth and Width Relationship Deformation intrusion into an impacted st ructure or surface of a vehicle is a function of the structures ability to resist permanent inward deformation over the contact and induced damage width of applied forces.

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151 Impact Force-Deflection Principle 3: Ge neralized Impact Force Management Vehicle structural desi gns that manage impact forces by maximizing the distribution width will minimize inward deform ation into the occupant compartment. Such designs provid e additional protection for vehicle occupants from secondary impacts from intruding components associated with the vehicles exterior shell and/ or interior components within the deformation region. Impact Force-Deflection Principle 4: Gene ralized Impact Deformation Prediction The extent of inward structural defo rmation of a composite structure of known stiffness characte ristics is determinable from the known applied forces determined from known or knowable structural properties and deformation profile of an interacting composite structure. The development of the G-DaTAV System of Equations satisfies the objectives outlined in Chapter 1 of this st udy and restated in this section. The GDaTAV System of Equations provides a comprehensive yet straightforward consideration of additional energy sinks and ex ternal impulses that act during a non-central impact. The results of this study present a unifying system of equations that consider the impact contributions of central an d non-central impacts. Moreover, the G-DaTAV System of Equations provides significant improvements over the current state-of-the-art analysis of vehicle deformation, as well as a greater understanding of impact mechanics.

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152 REFERENCES[1] USDOT, NHTSA, 49 CFR Part 563 EVENT DATA RECORDERS, August 28, 2006. [2] SAE International, "J2728 Heavy Vehicl e Event Data Recorder (HVEDR) Standard Tier 1," June 2010. [3] J. Steiner, T. Cheek and D. Plant, "Module 1: HVEDR Da ta Networks," in Accessing & Interpreting Heavy Vehi cle Event Data Recorders (C1022) 2014. [4] B. G. McHenry, "The Algorithms of CRASH," McHenry Software, Inc., 2001 SECCC, August 2001. [5] K. Campbell, "Energy Ba sis for Collision Severity," 740565 Society of Automotive Engineers, 1974. [6] "CRASH III User's Guide and Tec hnical Manual," United States DOT, National Highway Traffic Safety Admi nistration, National Center for Statistics and Analysis, Accident Investigation Division, Washington DC. [7] R. McHenry, "A Comparison of Results Obtained With Different Analytical Techniques for Reconstruction of Highway Accidents," 750893 Society o f Automotive Engineers, 1970. [8] R. R. McHenry, "Computer Program for Reconstruction of Highway Accidents," 730980 Society of Automotive Engineers, 1973. [9] R. McHenry and B. McHenry, "A Re vised Damage Analysis Procedure for the CRASH Computer Program," 861894 Society of Automotive Engineers, 1986. [10] N. Tumbas and R. Smith, "Measuri ng Protocol for Quantif ying Vehicle Damage from an Energy Basis Point of View," 880072 Society of Automotive Engineers, 1988. [11] J. A. Neptune, G. Y. Blair and J. E. Flynn, "A Method for Quantifying Vehicle Crush Stiffness Coefficients," 920607 Society of Automotive Engineers, 1992. [12] J. Lawrence, R. Fix, A. Ho, D. King and P. D'Addario, "Fro nt and Rear Car Crash Coefficients for Energy Calculations," 2010-01-0069, Society of Automotive Engineers Intern ational 2010. [13] N. Rose, S. Fenton a nd R. Ziernicki, "An Examina tion of the CRASH3 Effecrtive Mass Concept," 2004-01-1181, Society of Automotive Engineers International 2004.

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153 [14] T. Day and R. Hargens, "An Overvi ew of the Way EDCRASH Computes Delta-V," 870055 Society of Automotive Engineers, 1987. [15] G. Smith, M. James, T. Peri and D. Struble, "Frontal Cr ush Energy and Impulse Analysis of Narrow Objects," 87-WA/SAF-5 American So ciety of Mechanical Engineers, December 1987. [16] J. Neptune, J. Flynn, H. Underwood and P. Chavez, "Impact Analysis Based Upon the CRASH3 Damage Algorithm," 950358 Society of Automotive Engineers, 1995. [17] J. Kerkhoff, "An Inve stigation into Vehicle Frontal Impact Stiffness, BEV and Repeated Testing for Reconstructing Accidents," 930899 Society of Automotive Engineers, 1993. [18] J. Neptune and J. Flynn, "A Meth od for Determining Accident Specific Crush Stiffness Coefficients," 940913 Society of Automotive Engineers, 1994. [19] Neptune Engineering, Inc., NEI Data Store, "https://www.neptuneeng.com/datastore/index.cfm," [Online]. [20] M. Bailey, B. Wong and J. Lawren ce, "Data and Methods for Estimating the Severity of Minor Impacts," 950352 Society of Automotive Engineers, 1995. [21] J. S. Ogden, Methods of Investig ating and Reconstructin g Minor Damage Low Velocity Motor Vehicle Accidents, Master of Science in Civil Engineering Thesis, University of Colorado Denver, October 1995. [22] J. S. Ogden and D. M. Sprague, "Mot or Vehicle Damage Fo rce Balancing Methods for Accident Reconstruction," 1998 Compendium of Techni cal Papers, Institute o f Transporation Engineers District VI 68th Annual Meeting, 1998. [23] J. S. Ogden, "Forensic Engineering Analysis of Damage and Restitution in Low Velocity Impacts," The Journal of the National Acade my of Forensic Engineers, Vol. XVI, No. 2, December 1999. [24] P. M. Burkhard, "Delta-V, BEV and Coefficient of Restitution Relationships as Applied to the Interpretation of Vehicle Crash Test Data," 2001-01-0499 Society o f Automotive Engineers International, 2001. [25] N. J. Carpenter and J. B. Welcher, "Stiffness and Crush Ener gy Analysis for Vehicle Collision and its Relationship to Barrier Equivalent Velocity (BEV)," 2001-010500, Society of Automotive Engineers International 2001. [26] B. E. Heinrichs, J. M. Lawrence, B. D. Allin, J. J. Bowler, C. C. Wilkinson, K. W. Ising, D. J. King and S. J. Ptucha, "Low-Speed Impact Testing of Pickup Truck Bumpers," 2001-01-0893 Society of Automoti ve Engineers International, 2001.

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154 [27] A. L. Cipriani, A. D. Woodhouse, M. L. Woodhouse, A. D. Cornetto, A. P. Dalton, C. B. Tanner and T. A. Timbario, "Low Speed Collinear Impact Severity: A Comparison Between Full Scale Testing a nd Analytical Pred iction Tools with Restitution Analysis," 2002-01-0540 Society of Automotive Engineers International, 2002. [28] J. S. Ogden, "Forensic Engineer ing Analysis of Multiple Vehicle Impact Sequences," Journal of the National Academy of Forensic Engineers, Vol. XXV, No. 2, December 2008. [29] A. J. Happer, M. C. Hughes, M. D. Peck and S. M. Boehme, "Practical Analysis Methodology for Low Speed Vehicle Collisi ons Involving Vehicles with Modern Bumper Systems," 2003-01-0492 Society to Automoti ve Engineers International, 2003. [30] N. A. Rose, G. Beauchamp and W. Bo rtles, "Quantifying the Uncertainty in the Coefficient of Restitution Obtained with Accelerometer Data from a Crash Test," 2007-01-0730 Society of Automoti ve Engineers International, 2007. [31] W. R. Scott, C. Bain, S. J. Manoogian J. M. Cornier and J. R. Funk, "Simulation Model for Low-Speed Bump er-to-Bumper Crashes," 2010-01-0051 Society o f Automotive Engineers International, 2010. [32] M. A. Ivory, C. J. Furbish, M. R. Ho ffman, E. R. Miller, R. L. Anderson and R. D. Anderson, "Brake Pedal Response and Occupant Kinematics During Low Speed Rear-End Collisions," 2010-01-0067 Society of Automotive Engineers International, 2010. [33] 2008-01-24, "SAE J670 Vehicle Dynami cs Terminology," Society of Automotive Engineers Internationa l, Warrendale, PA, 2008. [34] D. T. Greenwood, Advanced Dynamics New York: Cambridge University Press, 2006. [35] M. D. Ardema, Analytical Dynamics: Theory and Applications, New York, Boston, Dordrecht, London, Moscow: Klwer Acan emic / Plenum Publishers, 2005. [36] A. D. Conetto, J. Su way, R. Wahba and F. Bayan, "Calculating Three Dimensional Stiffness Coefficients for Use in Th ree Dimensional Simulation Modeling for Accident Reconstruction," 2014-01-0473, 01 April Society of Automotive Engineers Intern ational 2014. [37] B. Gilbert, R. Jadischke and J. McCarthy, "Calculating Non-Li near Frontal Stiffness Coefficients," 2014-01-0474, 01 April Society of Automotive Engineers International 2014.

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155 [38] J. A. Neptune, "A Comparison of Cr ush Stiffness Characteristics from PartialOverlap and Full-Overlap Frontal Crash Tests," 1999-01-0105, Society o f Automotive Engineers In ternational 1999. [39] J. A. Neptune and J. E. Flynn, "A Method for Determining Crush Stiffness Coefficients from Offset Fr ontal and Side Crash Tests," 980024, Society o f Automotive Engineers In ternational 1998. [40] J. Kerkhoff, "An Inve stigation into Vehicle Frontal Impact Stiffness, BEV and Repeated Testing for Rec onstructing Accidents," 930899, Society of Automotive Engineers 1993. [41] J. Lawrence, R. Fix, A. Ho, D. J. King and P. D'Addario, "Front and Rear Car Crush Coefficients for Energy Calculations," 2010-01-0069, Society of Automotive Engineers Intern ational 2010. [42] M. Wood, V. Shekhawat, T. Kubose and R. Kelkar, "Prediction of Stiffness Coefficients for Frontal Imp acts in Passenger Vehicles," 2014-01-0466, Society o f Automotive Engineers In ternational 2014. [43] Neptune Engineering, Inc., "NEI Data Store," [Online]. Available: https://www.neptuneeng.com/DataStore/index.cfm. [44] N. Poirette, F. Bayan, J. Suway, A. Cornetto, A. Cipriani and R. Wahba, "Stiffness Coefficients of Heavy Commercial Vehicles," 2013-01-0796, 08 April Society o f Automotive Engineers In ternational 2013. [45] J. S. Ogden, "Forensic Engineering Analysis of Damage and Restitution in Low Velocity Impacts," The Journal of the National Acade my of Forensic Engineers, Vol 16, No. 2, December 1999. [46] J. P. Singh and N. J. Carpenter, "W ork-Enery Relationships for the Collinear Single Degree of Freedom Impact Model under th e Case of Net Unbalanced Externally Applied Forces," 2013-01-0794, Society of Automotive Engineers International 2013. [47] F. US Dot, "Part 571.208 'Occupant Crash Protection'," 49 CFR Part 571 National Highway Traffic Safety Administration, Federal Motor Vehicle Safety Standards. [48] J. S. Ogden and D. M. Sprague, "Mot or Vehicle Damage Fo rce Balancing Methods for Accident Reconstruction," 1998 Compendium of Technica l Papers, ITE District VI 68th Annual Meeting, 1998. [49] R. C. Hibbeler, Engineering Mechanics: Statics and Dynamics, 3rd ed., New York, New York: Macmillan Publishing Company, 1983.

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156 [50] W. R. Garrott, "Measured Vehicle In ertial Parameters NHTSA's Data Through September 1992," 930897, Society of Automotive Engineers 1993. [51] W. R. Garrott, M. W. Monk and J. P. Chrstos, "Vehicle Inertial Parameters Measured Values and Approximations," 881767, Society of Automotive Engineers 1988. [52] J. A. Neptune, "Overvie w of an HVE Vehicle Database," 960896, Society o f Automotive Engineers 1998. [53] C. Y. Warner, G. C. Smith, M. B. Ja mes and G. J. Germane, "Friction Applications in Accident Reconstruction," 830612, Society of Automotive Engineers 1983. [54] M. C. Marine, "On the Concept of In ter-Vehicle Friction and its Application in Automobile Accident Reconstruction," 2007-01-0744, Society of Automotive Engineers Intern ational 2007. [55] R. Fix, D. King and T. Fricker, "Estimating the Speed Change and Relative Approach Speed of Aligned Offset Impacts using CRA SH3 Techniques," 2014-010472, Society of Automotive Engineers International 2014. [56] N. P. 5. US DOT, "FMVSS 208 O ccupant Crash Protect ion, FMVSS 214 Side Impact Protection, FMVSS 301 Fuel Sy stem Integrity," Code of Federal Regulations, Title 49. [57] B. G. McHenry and R. R. McHenry, "RICSAC-97 A Reevaluation of the Reference Set of Full Scale Crash Tests," 970961, Society of Automotive Engineers 1997. [58] Department of Transporation, Nati onal Highway Traffic Sa fety Administration, "www.nhtsa.gov/NASS, Na tional Automotive Sampling System," 2010-2013. [Online]. [59] 4N6XPRT Systems, "Expert Autostats, ver. 5.5," 2015. [60] Colorado Department of Transportation, Safety and Engineer ing Branch, Accident Data Management Unit, "Crashes an d Rates on State Highways," CDOT, 2011. [61] Colorado Department of Transportation, Safety and Engineer ing Branch, Accident Data Management Unit, "Crashes an d Rates on State Highways," CDOT, 2010. [62] Colorado Department of Transportation, Safety and Engineer ing Branch, Accident Data Management Unit, "Crashes an d Rates on State Highways," CDOT, 2009. [63] Colorado Department of Transportation, Safety and Engineer ing Branch, Accident Data Management Unit, "Crashes an d Rates on State Highways," CDOT, 2008.

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157 [64] Colorado Department of Transportation, Safety and Engineer ing Branch, Accident Data Management Unit, "Crashes an d Rates on State Highways," CDOT, 2007. [65] J. S. Ogden, M. Martonovich, Z. Weimer and K. M. Kloberdanz, "Information Analysis for the Collision Analyst," Collision: The International Compendium for Crash Research, vol. 7, no. 1, 2012. [66] Northwestern University Center for Public Safety, "Chapter 2 Information From and About People," in Traffic Collision Iinvestigation, Tenth Edition 2006. [67] U.S. Department of Transportation, Federal Highway Administration, "Part VI Temporary Traffic Control," in M anual on Uniform Traffi c Control Devices for Streets and Highways, 2009 Edition, Revisions 1 and 2, May 2012. [68] Colorado State Patrol, R aw collision statistics provided for the study years 2007 to mid-2014 detailing all construction zone related collisions reported within the State, 2007 to mid-2014. [69] National Automotive Sampling System, "2010-08-037, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [70] National Automotive Sampling System, "2010-12-154, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [71] National Automotive Sampling System, "2011-04-127, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [72] National Automotive Sampling System, "2011-08-107, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [73] National Automotive Sampling Syst em, "2011-08-112, el v 1.0.112008," National Highway Traffic Safe ty Administration. [74] National Automotive Sampling System, "2011-09-075, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [75] National Automotive Sampling System, "2011-09-091, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [76] National Automotive Sampling System, "2011-11-085, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [77] National Automotive Sampling System, "2011-12-049, rel v 1.0.112008," National Highway Traffic Safe ty Administration.

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158 [78] National Automotive Sampling System, "2011-12-189, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [79] National Automotive Sampling System, "2012-08-064 rel v 1.0.112008," National Highway Traffic Safe ty Administration. [80] National Automotive Sampling System, "2012-08-080, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [81] National Automotive Sampling System, "2012-12-016, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [82] National Automotive Sampling System, "2012-041-024, rel v 1.0.112008," National Highway Traffic Safety Administration. [83] National Automotive Sampling System, "2012-43-014, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [84] National Automotive Sampling System, "2012-43-026, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [85] National Automotive Sampling System, "2012-43-106, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [86] National Automotive Sampling System, "2012-48-106. rel v 1.0.112008," National Highway Traffic Safe ty Administration. [87] National Automotive Sampling System, "2013-12-059, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [88] National Automotive Sampling System, "2013-12-106, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [89] National Automotive Sampling System, "2013-12-112, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [90] National Automotive Sampling Syst em, "2013-43-152, rel v. 1.0.112008," National Highway Traffic Safe ty Administration. [91] National Automotive Sampling System, "2013-76-094, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [92] National Automotive Sampling Syst em, "2013-76-165, rel v. 1.0.112008," National Highway Traffic Safe ty Administration.

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159 [93] National Automotive Sampling System, "2013-79-139, rel v 1.0.112008," National Highway Traffic Safe ty Administration. [94] J. S. Ogden, Methods of Investig ating and Reconstructin g Minor Damage Low Velocity Motor Vehicle Accidents, Master of Science Thesis, University o f Colorado Denver, October 1995. [95] J. A. Neptune and J. E. Flynn, "A Me thod for Determining Accident Specific Crush Stiffness Coefficients," Society of Automotive Engineers 940913, 1993. [96] D. T. Greenwood, Classical Dy namics, Vols. Unabridged and corrected republication of the work first published by Prentice-Hall, Inc., Englewood Cliffs, NJ, 1977, Mineola, NY: Dove r Publications, 1997. [97] R. M. Brach and R. M. Brach, Vehi cle Accident Analysis and Reconstruction Methods, 2nd ed., Warrendale, PA : SAE International, 2011. [98] D. T. Greenwood, Advanced Dynamics New York: Cambridge University Press, 2006. [99] N. Poirette, F. P. Ba yan, J. Suway, A. Cornetto, A. Cipriani and R. Wahba, "Stiffness Coefficients of H eavy Commercial Vehicles," SAE Technical Paper 2013-01-0796, 08 April 2013. [100] US DOT HS-7-01511, McHenry, Lynch, Segal, "Research Input for Computer Simulation of Automobile Co llisions, Volumes 1-4," 1978. [101] J. Kerkhoff, "An Invest igation into Vehicle Frontal Stiffness, BEV and Repeated Testing for Reconstr ucting Accidents," 930899 Society of Automotive Engineers, 1993. [102] R. V. Dukkipati, J. Pang, M. S. Qa tu, G. Sheng and Z. Shuguang, "Chapter 10 Accident Reconstruction," in Road Vehicle Dynamics SAE International, 2008. [103] J. Y. Wong, "Chapter 1 M echanics of Pneumatic Tires," in Theory of Ground Vehicles, Fourth Edition John Wiley & Sons, Inc., 2008.

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160 APPENDIX A: COLLISIONS ON CO LORADO HIGHWAYS AND WORK ZONES FROM 2007 TO 2011 A.1 Objectives Each year, the Accident Data Manageme nt Unit of the Colorado Department of Transportation, Safety and Tra ffic Engineering Branch compile s the crash reports for State highways into the Crashes and Rates on St ate Highways compendium. This data is currently available up to and including the year 2011. Information contained within the body of a crash report, along the outside margin s and within specializ ed sections provide information by which the location, type, date/time and factors relate d to collision events occurring allows for the statistical analysis of trends and reoccurring circumstances subject to remedial countermeasures. Often, the inve stigating authorities will enter information regarding vehicle speeds, driv er actions, vehicle movements, mechanical/human factors, toxicology and causative factors. This appendix explores the overa ll collision data for State highways and trends in collisions by vehicle and roadway characteristics, as well as collision severity, and compare those results to collisions occurring specifically within highway construction zones during the years 20 07 to 2011 [60] [61] [6 2] [63] [64]. From this data, the need for a generalized vehicle deformation model will be established, thus providing the justification and necessity for the remaining scope of this study. A.2 Collision Data for all State Highways from 2007 to 2011 Statistics gathered by th e State of Colorado elucidat e the involvement of some 429,013 motor vehicles in 235,884 separate collision events on State highways across Colorado for the years 2007 to 2011. The statistics result in a ra tio of 1.82 vehicles to every collision during the study peri od, indicating that most coll isions on State highways are multiple vehicle collision events. Figure A.1 shows the distribution of vehicles involved in collision events, pointing out that more th an half (51.5%) of all collisions involved passenger vehicles such as sedans, coupes, hatchbacks, and wagons. The remaining 48.5% of all collisions incl ude vehicles other than standard passenger vehicles, which also includes those that have more or less than four road wheels.

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161 Figure A.1 Total crashes by vehicle type for 2007 to 2011 (429,013 vehicles) Of the total vehicles involved in collisi ons on State highways, the vast majority iinvolved vehicle-to-vehicle re ar-end impacts (38.5%), while side impacts (13.9%) and sideswiping impacts (13.9%) made up a distan t second and third place, respectively. The remaining nearly 34% of collisions on State highways involved vehicle-to-vehicle or vehicle-to-object, and single ve hicle collision events. Figure A.2 shows the distribution of collisions by type for the study years 2007 to 2011.

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162 Figure A.2 Colorado State Highway cr ashes by collision type 2007 to 2011 (235,884 impacts involving 429,013 vehicles) Collisions occurred predominantly on principle arterial roadways, many of which are multi-lane divided highways with relatively high posted regulatory speed limits. Traffic volumes on principle arterial roadways are generally ab ove 80 kph (50 mph), with generally higher peak hour traffic volumes due to proximities in and around major population centers within the State. Figure A.3 shows the distribution of collisions by roadway type within the State of Colorado fo r the years 2007 to 2011. The vast majority of all reported collisions on State highways are property damage only (88.8%), with the remaining 11.2% of collisions distributed am ongst injury crashes and fatalities. It is difficult to separate injury-only and fatality-onl y collisions since injuries and fatalities can and do occur during the same collision event. Figure A.4 shows the distribution of injury related criteria for collisions on State highways for 2007 to 2011.

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163 Figure A.3 Colorado State Highway cras hes by roadway type for 2007 to 2011 Figure A.4 Colorado State Highway cr ashes by severity for 2007 to 2011

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164 The collision data presented indicates a preponderance of co llisions with in the State of Colorado primarily limited to property-damage-only events, at least at the time of the reporting. It is also this authors experien ce that many crash reports do not consider the potential for injuries that may manifest themse lves later after involve d parties have cleared the scene, or those injuries reported or alleged at a later date after finalization of the report and field investigation by authorities. The potential for a clai m or manifestation of minor injuries following the completion of a report becomes especially true when considering minor collision events associated with rear-e nd and sideswipe impacts. Due to safety or time constraints, field investig ators choose not to record the necessary information for more accurate and/or precise determination of vehicl e speeds and rely largely on their own visual estimations or witness and driver statements regarding speeds and timing of collision events. These estimations can be problematic when attempting to determine collision speeds and severity levels for more detail ed statistical or engineering studies. Highway safety has improved since the a dvent of the intersta te highway system, largely due to detailed and timely engineeri ng studies. In spite of these imporvements, public roadways designed a nd intended to function under normal operational conditions, absent special events, emergencies and highw ay construction, stil l produce significant collision events within the State. Collision causation determination oftentimes require engineering analysis related to the driver, roadway geometric and de sign features, traffic control and vehicle mechanical contributions. All of these areas of study rely upon accurate data recorded through timely and proper inves tigations. Investigations are often cost prohibitive to recreate under test conditions, a nd may be challenging or even impossible to reconstruct. Reliance upon witness and driver accounts is riddled with potential pitfalls that range from bias, experience level and understa nding of collision dynamics of the witness, to false memories and intentional misleading statements [65] [66]. The collision physical evidence that remains the longest is often the information that is available to the engineer tasked with the analysis. T ypically, the physical evidence that rema ins for the longest duration most often consists of the damage d vehicles and/or phot ographic documentation of the damaged vehicles and the scene. Under such circumstances, accurate methods of analyzing vehicle deformation characteristics b ecome crucial to a successful and reliable engineering analysis.

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165 A.3 Collision Data for Construction Zone Collisions 2007 to 2011 A.3.1 Overview of Crash Reported Data Highway construction zones are a natura l consequence of transportation on public roadways. Whether due to new construction or the rehabilitation and/or improvement of existing infrastructure, construc tion zones result in the inevitab le interruption of the traffic stream. Part VI of the Manual on Uniform Tr affic Control Devices (MUTCD) covers the principles of temporary traffic control for de aling with adequate a nd standardized methods of handling traffic leading into and within highway constructions zones. [67] Modern advances in temporary traffic control procedures are not the subject of this study. However, highway construction zone co llisions for the study years of 2007 to 2011 accounted for only 1.0% (4316 total involved vehicles) of th e vehicles involved in reported collisions according to the crash report data for State highways from the Colorado State Patrol [68]. Table A.1 details the reported co llision data by vehicle type i nvolved, as well as collision type. Table A.1 Highway construction zone collisions by vehicle type and collision type 2007 to 2011 as reported by the Colorado State Patrol

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166 The distribution of vehicle types involved in highway construction zone collisions in the State varies little from the overall di stribution for all State hi ghway collisions from 2007 to 2011. The construction zone dataset dem onstrates passenger vehicles still make up approximately half (49.9%) of all vehicles involved in collision-related events. The percentages of SUVs involved in construction zone co llisions versus th e overall collision percentage (20.0% versus 21.5% ) are likewise similar, as is the case with pickups and fullsized vans (18.9% versus 18.4%). Motorcycle collision experience also remained the same (1.6% versus 1.6%). However, the biggest change in collision experience involved heavy vehicles such as semi-tractor/trailers, buses and RVs (8.5% versus 4.6%). The characteristics of reduced lanes and lane widt hs, decrease in curve radius, lane drops and shifts and restrictions upon other roadway geometric feat ures produce navigational challenges for large profile vehicles, as well as other roadway user s as they negotiate construction zones adjacent to or while passing larger profile vehicles. Figures A.5 and A.6 outline the vehicle and collision type charac teristics of Colorado highway construction zones from 2007 to 2011. Figure A.5 Colorado construction zone collisions by vehicle type for 2007 to 2011 (4316 total vehicles)

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167 Figure A.6 Colorado construction zone collision by collision type for 2007 to 2011 The majority of collisions by roadway type shifted from prin ciple arterial roadways for the overall State highway system collision trends to interstate highways (48.2%) for the majority of vehicles involved in construction zone collisions. While principle arterial and other US and State highways made up only 35.5% of construction zone collisions, the mix for overall State highway collision even ts between 2007 and 2011 for these roadways accounted for approximately 61.4% of collision involved vehicl es. Although not a specific focus of this study, this shift towards a great er occurrence of constr uction zone collisions for interstate highways c ould be attributable to the following factors: Higher reduction in constr uction zone speed limits from posted speed limits between roadway types for interstate hi ghways versus US and State highways. More detailed and long-term infrastructu re projects on inters tate highways as compared to US and State highways during the period studied, placing a higher demand upon attention for roadway users.

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168 Figure A.7 Colorado construction zone co llisions by roadway type for 2007 to 2011 Construction zones during 2007 to 2011 expe rienced a reduction in the percentage of injury and fatality occurrences (9.5%) as compared to the overall State highway occurrences during the same period (11.2%). For construction zone collisions, 90.5% of collision involved vehicles were reported as property-damage-only events. However, the same caveat regarding the reporting of injuries for minor events may not be accurately reflected by the crash reporting system. Figure A.8 illustrates the trends for collision severity for construction zone crash events for 2007 to 2011.

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169 Figure A.8 Colorado construction zone collisions by severity for 2007 to 2011 A.3.2 Passenger Vehicle Construction Zone Collisions Passenger vehicles comprised 49.9% of ve hicles involved in construction zone related collision events (3336 vehicles). Consistent with the overall collision data for the State, most passenger-vehicle-related collisions occurred on interstate facilities (52.0%), resulting in 696 injuries and 158 total fatali ties between 2007 and 2011. Rear-end impacts comprised the vast majority of collisions involving passenger vehicles (2222 vehicles, 66.7%), followed by sideswiping (321 vehicl es, 9.6%) and side impacts (218 vehicles, 6.5%). The analysis of rear-end, sideswipe a nd side impact collisions may become complicated by the lack of rear and side sti ffness data for many vehicles. The application of current vehicle deformation analysis models require the use of stru ctural stiffness data for both involved vehicles; each structural stiffne ss characteristic to the front, side or rear of each involved vehicle.

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170 Figure A.9 Passenger vehicle constr uction zone collisions 2007 to 2011 Figure A.10 Passenger vehicle construction zone collision severity 2007 to 2011

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171 A.3.3 Sport Utility Vehicle, Pickup an d Van Construction Zone Collisions Sport utility vehicles (SUV), light pickups, and full-sized vans comprise the next highest collision vehicle t ype frequency for the 2007 to 2011 highway construction zone data. Rear-end impacts again accounted for the vast majority of collisions involving SUVs (1020 vehicles, 66.4%). SUV impacts were pr imarily property-dam age-only collisions (1393 vehicles, 90.6%), with only 118 reported in juries and 19 fatalities for the remaining 144 involved vehicles Pickups and full sized vans also had a high percentage of rear-end collisions (818 vehicles, 61.9%). As with SUVs, property damage only collisions predominated (1145 vehicles, 86.7%), with the remaining 176 collisi on involved vehicles within this class having 221 reported injuries and 13 fatalities. Regardless of the type of collision configuration for SUVs, light pickups and full sized vans, stru ctural stiffness data available for these vehicles is significantly limited as compared to passenger vehicles. Figure A.11 Sport utility vehicle cons truction zone collisions 2007 to 2011

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172 Figure A.12 Sport utility ve hicle construction zone co llision severity 2007 to 2011 Figure A.13 Pickup and full sized van construction zone collisions 2007 to 2011

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173 Figure A.14 Pickup and van construction zone collision severity 2007 to 2011 A.3.4 Heavy Vehicle Construction Zone Collisions Heavy vehicles, buses and recreational ve hicles (RV) had fewer occurrences of rear-end impacts (227 vehicles, 40.5%) with an overall increa se in sideswiping and side impact events (206 vehicles, 36.7% combined). Increased incidents of sideswipe and side collisions most likely results from the increase d width of this vehicle category as compared to the roadway and lane width restrictions common present in many highway construction zones. Heavy vehicles had predominantly more property-damage-only collisions (494 vehicles, 88.2%), than injury and fatal collisions (66 vehicl es), but had the second highest gross total of fatalities (41) of all vehicl e categories. Regardless of the collision configuration for heavy vehicles, essentially no structural stiffn ess data is available for any impacted surface.

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174 Figure A.15 Heavy vehicle construction zone collisions 2007 to 2011 Figure A.16 Heavy vehicle construction zone collision severity 2007 to 2011

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175 A.3.5 Motorcycle Construction Zone Collisions The remaining class of vehicles in this portion of the study involves two and threewheeled motorized vehicles, such as motorcyc les and scooters. Twowheeled vehicles are especially susceptible to long itudinal disturbances, abrupt ch anges in surface elevation and roadway surface irregularities such as those created by roto-milling operations, resurfacing and overlays on roadways open to public travel. Part VI of the MUTCD, Section 6F.54 describes the use of the Motorcycle Plaque (W8-15P) as it relates to providing additional advanced warning for motorcycle traffic to augment other temporary traffic control signs such as the Shoulder Drop-Off (W8-17P) Uneven Lanes (W8-11), Rough Road (W8-8), Loose Gravel (W8-7) and the Grooved Pavement (W8-15) temporary advanced warning signs. The logic behind utilizing the supplemental Motorcycle Plaque (W8-15P) is due to the added risk for motorcycles to common surf ace irregularities resulting from construction zone activities. While the overall frequency of motorcycle collisions within construction zones were low (101 motorcycles), the vast majority of collisions were overturn non-contact events, consistent with the surface irregula rity conditions described in the previous paragraph. Also in stark contrast to all ot her vehicle types, 73.3% of all motorcycle collisions involved injuries or fatalities (78 injuries and 12 fatalities), while only 26.7% of motorcycle involved collision events within highway construction zones were propertydamage-only events. The relative vulnerabil ity of motorcycle riders and motorcycle passengers when experiencing an instability or collision event due to the lack of external vehicular protection and restra int systems most likely expl ains the increased relative frequency of serious injury or death.

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176 Figure A.17 Motorcycle construc tion zone collisions 2007 to 2011 Figure A.18 Motorcycle construction zone collision severity 2007 to 2011

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177 A.4 Summary The engineering analysis of the miriad of identified vehicles within the traffic stream throughout this chapter require sufficient and proper data in order to determine contributions to collision causation due to th e driver, roadway, vehicle and environment. Without proper measurements and documentation of the collision scene physical evidence, a trajectory-based analysis may not be possible, or may be called into question. Engineering analysis of collision events largely functions upon the quality, not necessarily the quantity, of physical evidence, as well as the skill of the analyst. Many engineering analyses of collision events occur after sufficient time has passed that much of the scene related physical evidence may no longer be available for consideration. The reliance on detailed photographs, if available, may be the only sc ene-related evidence available. However, vehicle deformation, in particular for the most serious of collision events, often remains available for the longest period of time, and may be the only physical evid ence the engineer has available regarding a collision event. Ther efore, the need for an accurate and precise vehicle deformation based analysis methodol ogy is essential for many collision analysis projects and research related activities. Additionally, since the passage of time degrades roadway evidence, it is all too often the case that the passage of time may result in unavailability of collision involved vehicles for detailed measurements of deformation profiles and documentation of mechanical components. Therefore, the need exists for many collision analysis projects and research related activities for an accurate and precise vehicle deformation based analysis methodology that allows for the pred iction of missing vehicl e deformation profiles with all other factors known. Lastly, the lack of comprehe nsive front, rear and side st ructural data coverage for all roadway vehicles creates a problem w ith current vehicle deformation models. Therefore, the need for an accurate and pr ecise vehicle deformation based analysis methodology that is not reliant upon having structural stiffness values for all involved vehicles is essential for many collision analysis projects and research related activities. The current vehicle deformation models may fall short in applicability to many vehicle collisions outlined by the State of Co lorado vehicle collisi on data. Accordingly, this study focused on the development of ge neralized models that apply to multiple

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178 collision conditions, not only broadening the sc ope of application of vehicle deformation based analysis, but also providing more accurate and precise determinations of collision severity, which is a crucial part of comple ting the missing informa tion for a trajectorybased speed analysis, as well as the studie s of vehicle crashwor thiness and occupant protection.

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179 APPENDIX B: G-DATAV ANALYSIS OF INDIVIDUAL RICSAC TESTS B.1 RICSAC 1 Broadside Oblique Impact RICSAC 1 staged collision involved the front of a 1974 Chevrolet Malibu (Vehicle 1, m1) colliding with the right front side of a 1974 Ford Pinto (Vehicle 2, m2) in accordance with Configuration D. The reported impact ve locity for both vehicles was 19.8 mph. The Chevrolet test weight was 4460 pounds, and th e Ford test weight was 3110 pounds. Each vehicle contained two 49CFR Part 572 50th percentile anthropomorphic test devices (ATD). The ATDs in the Ford were instrumented while the ATDs in the Chevrolet were un-instrumented during the collision. The following is the test specific data and analysis results for this impact test using the G-DaTAV System of Equations. Figure B.1 Maximum engagement PDOF diagram for RICSAC 1 Variables for the analysis were obtained from reported mass and deformation profiles, and data extrapolated from the collis ion diagram. The A and B structural stiffness data was obtained with permission from Nept une Engineering NEI Data Store for a 1970 Chevrolet Malibu four-door sedan.

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180 The following are the calculation results using matched segment piecewise analysis between vehicles, single stiffness A/B values and forced Newtons third law compliance outlined in Chapter 4.

PAGE 197

181 The following are calculation results usi ng weighted average deformation depth analysis between vehicles, single stiffne ss A/B values and forced Newtons third law compliance outlined in Chapter 4.

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182 B.2 RICSAC 2 Broadsi de Oblique Impact RICSAC 2 staged collision involved the front of a 1974 Chevrolet Malibu (Vehicle 1, m1) colliding with th e right front side of a 1974 Ford Pinto (Vehicle 2, m2) in accordance with Configuration D. The re ported impact velocity of both vehicles was 31.5 mph. The Chevrolet test weight was 4710 pounds, and th e Ford test weight was 3260 pounds. Each vehicle contained two 49CFR Part 572 50th percentile anthropomorphic test devices (ATD). The ATDs in the Ford were instrume nted while the ATDs in the Chevrolet were un-instrumented during the collision. The followi ng is the test specific data and analysis results for this impact test using the ge neralized methods deve loped in this study.

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183 Figure B.2 Maximum engagement PDOF diagram for RICSAC 2 Variables for the analysis were obtained from reported mass and deformation profiles, and data extrapolated from the collis ion diagram. The A and B structural stiffness data was obtained with permission from Neptune Engineering NEI Da ta Store for a 1970 Chevrolet Malibu four-door sedan.

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184 The following are the calculation results using matched segment piecewise analysis between vehicles, single stiffness A/B values and forced Newtons third law compliance outlined in Chapter 4.

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185 The following are calculation results usi ng weighted average deformation depth analysis between vehicles, single stiffne ss A/B values and forced Newtons third law compliance outlined in Chapter 4. B.3 RICSAC 3 Front to Rear Oblique Offset Impact RICSAC 3 staged collision involved the front of a 1974 Ford Torino (Vehicle 1, m1) colliding with the rear of a 1974 Ford Pinto (Vehicle 2, m2) in accordance with Configuration B. The reported impact veloc ity of the Ford Torino was 21.2 mph and the Ford Pinto was stopped. The Ford Torino test weight was 4980 pounds, and the Ford Pinto

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186 test weight was 3140 pounds. Each vehicle contained two 49CFR Part 572 50th percentile anthropomorphic test devices (ATD). The ATDs in the Ford Pinto were instrumented while the ATDs in the Ford Torino were un-instrumented during the collision. The following is the test specific data and analysis results for this impact test using the generalized methods developed in this study. Figure B.3 Maximum engagement PDOF diagram for RICSAC 3 Variables for the analysis were obtained from reported mass and deformation profiles, and data extrapolated from the collis ion diagram. The A and B structural stiffness data was obtained with permission from Nept une Engineering NEI Data Store for a 1974 Ford Torino four-door sedan.

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187

PAGE 204

188 The following are the calculation results using matched segment piecewise analysis between vehicles, single stiffness A/B values and forced Newtons third law compliance outlined in Chapter 4.

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189 The following are calculation results usi ng weighted average deformation depth analysis between vehicles, single stiffne ss A/B values and forced Newtons third law compliance outlined in Chapter 4. B.4 RICSAC 4 Front to Rear Oblique Offset Impact RICSAC 4 staged collision involved the front of a 1974 Ford Torino (Vehicle 1, m1) colliding with the rear of a 1974 Ford Pinto (Vehicle 2, m2) in accordance with Configuration B. The reported impact veloc ity of the Ford Torino was 38.7 mph and the Ford Pinto was stopped. The Ford Torino test weight was 4980 pounds, and the Ford Pinto test weight was 3190 pounds. Each vehicl e contained two 49CFR Part 572 50th percentile anthropomorphic test devices (ATD). The ATDs in the Ford Pinto were instrumented while the ATDs in the Ford Torino were un-instrumented during the collision. The following is the test specific data and analysis results for this impact test using the generalized methods developed in this study.

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190 Figure B.4 Maximum engagement PDOF diagram for RICSAC 4 Variables for the analysis were obtained from reported mass and deformation profiles, and data extrapolated from the collis ion diagram. The A and B structural stiffness data was obtained with permission from Nept une Engineering NEI Data Store for a 1974 Ford Torino four-door sedan.

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191 The following are the calculation results using matched segment piecewise analysis between vehicles, single stiffness A/B values and forced Newtons third law compliance outlined in Chapter 4.

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192 The following are calculation results usi ng weighted average deformation depth analysis between vehicles, single stiffne ss A/B values and forced Newtons third law compliance outlined in Chapter 4. B.5 RICSAC 5 Front to Rear Oblique Offset Impact RICSAC 5 staged collision involved the front of a 1974 Ford Torino (Vehicle 1, m1) colliding with the rear of a 1975 Honda Civic (Vehicle 2, m2) in accordance with Configuration B. The reported impact veloc ity of the Ford was 39.7 mph and the Honda was stopped. The Ford test weight was 4600 pounds, and the Honda te st weight was 2530

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193 pounds. Each vehicle contained two 49CFR Part 572 50th percentile anth ropomorphic test devices (ATD). The ATDs in the Honda were instrumented while the ATDs in the Ford were un-instrumented during the collision. The following is the test specific data and analysis results for this impact test using the generalized methods developed in this study. Figure B.5 Maximum engagement PDOF diagram for RICSAC 5 Variables for the analysis were obtained from reported mass and deformation profiles, and data extrapolated from the collis ion diagram. The A and B structural stiffness data was obtained with permission from Neptune Engineering NEI Da ta Store for a 1974 Ford Torino four-door sedan.

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194 The following are the calculation results using matched segment piecewise analysis between vehicles, single stiffness A/B values and forced Newtons third law compliance outlined in Chapter 4.

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195 The following are calculation results usi ng weighted average deformation depth analysis between vehicles, single stiffne ss A/B values and forced Newtons third law compliance outlined in Chapter 4.

PAGE 212

196 B.6 RICSAC 6 Front to Side Oblique Offset Impact RICSAC 6 staged collision involved the front of a 1974 Chevrolet Malibu (Vehicle 1, m1) colliding with the right front side of a 1975 Volkswagen Rabbit (Vehicle 2, m2) in accordance with Configuration D. The reported impact velocity for both vehicles was 21.5 mph. The Chevrolet test weight was 4310 poun ds, and the Volkswagen test weight was 2640 pounds. Each vehicle contai ned two 49CFR Part 572 50th percentile anthropomorphic test devices (ATD). The ATDs in the Volksw agen were instrumented while the ATDs in the Chevrolet were un-instrumented during the collision. The following is the test specific data and analysis results for this impact test us ing the generalized methods developed in this study.

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197 Figure B.6 Maximum engagement PDOF diagram for RICSAC 6 Variables for the analysis were obtained from reported mass and deformation profiles, and data extrapolated from the collis ion diagram. The A and B structural stiffness data was obtained with permission from Nept une Engineering NEI Data Store for a 1970 Chevrolet Malibu four-door sedan.

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198 The following are the calculation results using matched segment piecewise analysis between vehicles, single stiffness A/B values and forced Newtons third law compliance outlined in Chapter 4.

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199 The following are calculation results usi ng weighted average deformation depth analysis between vehicles, single stiffne ss A/B values and forced Newtons third law compliance outlined in Chapter 4.

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200 B.7 RICSAC 7 Front to Side Oblique Offset Impact RICSAC 7 staged collision involved the front of a 1974 Chevrolet Malibu (Vehicle 1, m1) colliding with the right front side of a 1975 Volkswagen Rabbit (Vehicle 2, m2) in accordance with Configuration E. The reported impact velocity for both vehicles was 29.1 mph. The Chevrolet test weight was 3310 poun ds, and the Volkswagen test weight was 2610 pounds. Each vehicle contai ned two 49CFR Part 572 50th percentile anthropomorphic test devices (ATD). The ATDs in the Volksw agen were instrumented while the ATDs in the Chevrolet were un-instrumented during the collision. The following is the test specific data and analysis results for this impact test us ing the generalized methods developed in this study.

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201 Figure B.7 Maximum engagement PDOF diagram for RICSAC 7 Variables for the analysis were obtained from reported mass and deformation profiles, and data extrapolated from the collis ion diagram. The A and B structural stiffness data was obtained with permission from Nept une Engineering NEI Data Store for a 1970 Chevrolet Malibu four-door sedan.

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202 The following are the calculation results using matched segment piecewise analysis between vehicles, single stiffness A/B values and forced Newtons third law compliance outlined in Chapter 4.

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203 The following are calculation results usi ng weighted average deformation depth analysis between vehicles, single stiffne ss A/B values and forced Newtons third law compliance outlined in Chapter 4. B.8 RICSAC 8 Perpendicular Broadside Offset Impact RICSAC 8 staged collision involved the front of a 1974 Chevrolet Malibu (Vehicle 1, m1) colliding with the right side of a 1974 Chevrolet Malibu (Vehicle 2, m2) in accordance with Configuration C. The reported impact velocity for both vehicles was 20.7 mph. The Vehicle 1 test we ight was 4480 pounds, and the Vehi cle 2 test weight was 4710 pounds. Each vehicle contai ned two 49CFR Part 572 50th percentile anth ropomorphic test

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204 devices (ATD). The ATDs in Vehicle 2 were instrumented while the ATDs in Vehicle 1 were un-instrumented during the collision. The following is the test specific data and analysis results for this impact test using the generalized methods developed in this study. Figure B.8 Maximum engagement PDOF diagram for RICSAC 8 Variables for the analysis were obtained from reported mass and deformation profiles, and data extrapolated from the collis ion diagram. The A and B structural stiffness data was obtained with permission from Nept une Engineering NEI Data Store for a 1970 Chevrolet Malibu four-door sedan.

PAGE 221

205 The following are the calculation results using matched segment piecewise analysis between vehicles, single stiffness A/B values and forced Newtons third law compliance outlined in Chapter 4.

PAGE 222

206 The following are calculation results usi ng weighted average deformation depth analysis between vehicles, single stiffne ss A/B values and forced Newtons third law compliance outlined in Chapter 4.

PAGE 223

207 B.9 RICSAC 9 Perpendicular Broadside Offset Impact RICSAC 9 staged collision involved the front of a 1975 Honda Civic (Vehicle 1, m1) colliding with the right front side of a 1974 Ford Torino (Vehicle 2, m2) in accordance with Configuration C. The repor ted impact velocity for both vehicles was 21.2 mph. The Honda test weight was 2270 pounds, and the Ford test weight was 4930 pounds. Each vehicle contained two 49CFR Part 572 50th percentile anthropomorphic test devices (ATD). The ATDs in the Ford were instrume nted while the ATDs in the Honda were uninstrumented during the collision. The following is the test specific data and analysis results for this impact test usi ng the generalized methods developed in this study.

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208 Figure B.9 Maximum engagement PDOF diagram for RICSAC 9 Variables for the analysis were obtained from reported mass and deformation profiles, and data extrapolated from the collis ion diagram. The A and B structural stiffness data was obtained with permission from Nept une Engineering NEI Data Store for a 1974 Honda Civic two-door coupe.

PAGE 225

209 The following are the calculation results using matched segment piecewise analysis between vehicles, single stiffness A/B values and forced Newtons third law compliance outlined in Chapter 4.

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210 The following are calculation results usi ng weighted average deformation depth analysis between vehicles, single stiffne ss A/B values and forced Newtons third law compliance outlined in Chapter 4. B.10 RICSAC 10 Perpendicular Broadside Offset Impact RICSAC 10 staged collision involved th e front of a 1975 Honda Civic (Vehicle 1, m1) colliding with the right front side of a 1974 Ford Torino (Vehicle 2, m2) in accordance with Configuration C. The repor ted impact velocity for both vehicles was 33.3 mph. The Honda test weight was 2320 pounds, and the Ford test weight was 4750 pounds. Each

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211 vehicle contained two 49CFR Part 572 50th percentile anthropomorphic test devices (ATD). The ATDs in the Ford were instrumented while the ATDs in the Honda were uninstrumented during the collision. The following is the test specific data and analysis results for this impact test using the generalized methods developed in this study. Figure B.10 Maximum engagement PD OF diagram for RICSAC 10 Variables for the analysis were obtained from reported mass and deformation profiles, and data extrapolated from the collis ion diagram. The A and B structural stiffness data was obtained with permission from Nept une Engineering NEI Data Store for a 1974 Honda Civic two-door coupe.

PAGE 228

212 The following are the calculation results using matched segment piecewise analysis between vehicles, single stiffness A/B values and forced Newtons third law compliance outlined in Chapter 4.

PAGE 229

213 The following are calculation results usi ng weighted average deformation depth analysis between vehicles, single stiffne ss A/B values and forced Newtons third law compliance outlined in Chapter 4.

PAGE 230

214 B.11 RICSAC 11 Front-to-Front Offset Impact RICSAC 11 staged collision involved the front of a 1974 Chevrolet Vega (Vehicle 1, m1) colliding with the front of a 1974 Ford Torino (Vehicle 2, m2) in accordance with Configuration A. The reported impact veloc ity for both vehicles was 20.4 mph. The Chevrolet test weight was 3060 pounds, and th e Ford test weight was 4480 pounds. Each vehicle contained two 49CFR Part 572 50th percentile anthropomorphic test devices (ATD). The ATDs in the Chevrolet were instrumented while the ATDs in the Ford were un-instrumented during the collision. The following is the test specific data and analysis results for this impact test using the generalized methods developed in this study.

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215 Figure B.11 Maximum engagement PD OF diagram for RICSAC 11 Variables for the analysis were obtained from reported mass and deformation profiles, and data extrapolated from the collis ion diagram. The A and B structural stiffness data was obtained with permission from Nept une Engineering NEI Data Store for a 1974 Ford Torino four-door sedan.

PAGE 232

216 The following are the calculation results using matched segment piecewise analysis between vehicles, single stiffness A/B values and forced Newtons third law compliance outlined in Chapter 4.

PAGE 233

217 The following are calculation results usi ng weighted average deformation depth analysis between vehicles, single stiffne ss A/B values and forced Newtons third law compliance outlined in Chapter 4. B.12 RICSAC 12 Front-to-Front Offset Impact RICSAC 12 staged collision involved the front of a 1974 Chevrolet Vega (Vehicle 1, m1) colliding with the front of a 1974 Ford Torino (Vehicle 2, m2) in accordance with Configuration A. The reported impact veloc ity for both vehicles was 20.4 mph. The Chevrolet test weight was 3060 pounds, and th e Ford test weight was 4480 pounds. Each vehicle contained two 49CFR Part 572 50th percentile anthropomorphic test devices (ATD). The ATDs in the Chevrolet were instrumented while the ATDs in the Ford were un-instrumented during the collision. The following is the test specific data and analysis results for this impact test using the generalized methods developed in this study.

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218 Figure B.12 Maximum engagement PD OF diagram for RICSAC 12 Variables for the analysis were obtained from reported mass and deformation profiles, and data extrapolated from the collis ion diagram. The A and B structural stiffness data was obtained with permission from Neptune Engineering NEI Da ta Store for a 1974 Chevrolet Vega two-door hatchback in or der to show the methodology across vehicle structures different from what was used for the same vehicle combination in RICSAC 11.

PAGE 235

219 The following are the calculation results using matched segment piecewise analysis between vehicles, single stiffness A/B values and forced Newtons third law compliance outlined in Chapter 4.

PAGE 236

220 The following are calculation results usi ng weighted average deformation depth analysis between vehicles, single stiffne ss A/B values and forced Newtons third law compliance outlined in Chapter 4.

PAGE 237

221

PAGE 238

222 APPENDIX C: G-DATAV ANALYSIS OF INDIVIDUAL NASS TESTS The National Automotive Sampling System (NASS) provides the NHTSA with a comprehensive compilation of real-world co llision events represen ting a broad-based collection of collision configur ations from across the country. This data represents a reusable source of information that was collected utilizing standardi zed field techniques implemented by trained field technicians. By using a core set of crash data components, NASS has demonstrated its utility and applicab ility to a vast array of statistical and analytical studies regarding traffic safety and vehicle collision dynamics. Twenty-five collisions were selected from the NASS CD S Case Viewer for the 2004 to 2013 data set under the following criteria: Two vehicle collisions involving at least one light truck/van or one SUV category vehicle, with a preference to collision s involving only these category vehicles. At least one vehicle must have a complete Crash Data Retrieval (Bosch CDR Tool) report without evidence of significant data c lipping or incomplete recording of data, with preference upon collisions involving bot h vehicles having a Bosch CDR Tool report. Both colliding vehicles have complete measured damage profiles consistent with photographs of vehicles. One vehicle must have a Neptune Engineering NEI da tabase applicable crush stiffness coefficient applicable for sister/clone or model year run. NASS reported collision data that met the established criteria for this study came from report years 2010 to 2013 due to NASS data collec tion practices or the lack of vehicles involved with EDRs capable of recordi ng acceleration collision pulses. Some NASS reported collisions involved one vehicle with complete longitudinal and lateral collision data with the other involved vehicle limited to an earlier genera tion EDR providing only one direction of collision pulse. This problem was easily resolved by determining the principal direction of force acting on the ve hicle with a complete Bosch CDR Tool report by resolving the longitudinal and lateral data in to the collision vector, placing the vehicles together using a collision diagram as discusse d in Chapter 4 and determining then the total velocity change for the collision from its single reported value and the angle of the PDOF to the vehicle.

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223 The following are the raw application of the G-DaTAV System of Equations developed in this study for the NASS data set meeting the criteria as outlined. C1.1 2010-08-037 Collision 2010-08-037 involved a 2009 Toyota Tacoma pickup striking the left rear of a 2009 Pontiac G6. Bosch CDR Tool data wa s available only for the Pontiac. [69] Figure C1 NASS 2010-08-037

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224 Piecewise analysis results: Weighted average results: C1.2 2010-12-154 Collision 2010-12-154 involved a 2009 Pontiac G6 colliding with the right side of a 2010 Ford F150 pickup. Bosch CDR Tool data was available for both vehicles. [70]

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225 Figure C2 NASS 2010-12-154

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226 Piecewise analysis results: Weighted average results: C1.3 2011-04-127 Collision 2010-04-127 involved a 2005 Chevrolet Equinox SUV striking the left side of a 2002 Ford Explorer SUV. Only Bosch CDR Tool data for the Chevrolet was available. [71] Figure C3 NASS 2011-04-127

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227 Piecewise analysis results: Weighted average results:

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228 C1.4 2011-08-107 Collision 2011-08-107 involved a 2001 Buick LeSabre striking the left side of a 2008 Chrysler Aspen SUV. Bosch CDR Tool data was available for both vehicles. [72] Figure C4 NASS 2011-08-107

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229 Piecewise analysis results: Weighted average analysis results: C1.5 2011-08-112 Collision 2011-08-112 involved a 2005 Chevrolet Equinox SUV striking the front left side of a 2006 Chevrolet Impala. Bosch CD R Tool data was available for both vehicles. [73] Figure C5 NASS 2011-08-112

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230 Piecewise analysis results: Weighted average analysis results:

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231 C1.6 2011-09-075 Collision 2011-09-075 involved a 2010 Buick L acrosse SUV striking the rear left side of a 2008 Honda Ridgeline pickup. Bosch CDR Tool data was available only for the Buick. [74] Figure C6 NASS 2011-09-075

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232 Piecewise analysis results: Weighted average analysis results: C1.7 2011-09-091 Collision 2011-09-091 involved a 2003 GMC 2500 pickup striking the front left side of a 2007 Acura MDX SUV. Bosch CDR T ool data was available only for the GMC. [75] Figure C7 NASS 2011-09-091

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233 Piecewise analysis results: Weighted average analysis results:

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234 C1.8 2011-11-085 Collision 2011-11-085 involved a 2004 Hyundai Sonata striking the front left side of a 2011 Ford Escape SUV. Bosch CDR Tool data was available only for the Ford. [76] Figure C8 NASS 2011-11-085

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235 Piecewise analysis results: Weighted average analysis results: C1.9 2011-12-049 Collision 2011-12-049 involved a 2007 Chevrolet Equinox SUV striking the front right side of a 2007 GMC 1500 pickup. Bosch CDR Tool data was available for both vehicles. [77] Figure C9 NASS 2011-12-049

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236 Piecewise analysis results: Weighted average analysis results:

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237 C1.10 2011-12-189 Collision 2011-12-189 involved a 2011 Chevrolet Equinox SUV left front striking the right side of a 2006 Chevrolet Impala. Bosch CDR Tool data was available for both vehicles. [78] Figure C10 NASS 2011-12-189

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238 Piecewise analysis results: Weighted average analysis results: C1.5 2012-08-064 Collision 2012-08-064 involved a 2009 Cadillac Escalade SUV left front striking the front left corner of a 1998 Honda Accord. Bosch CDR Tool data was available only for the Cadillac. [79] Figure C11 NASS 2012-08-064

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239 Piecewise analysis results: Weighted average analysis results:

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240 C1.12 2012-08-080 Collision 2012-08-080 involved a 2010 GMC Yukon SUV left front corner sideswiping the left side of a 2012 Toyota 4Runne r SUV. Bosch CDR Tool data was available for both vehicles. [80] Figure C12 NASS 2012-08-080

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241 Piecewise analysis results: Weighted average analysis results: C1.13 2012-12-016 Collision 2012-12-016 involved a 2008 Cadillac CTS striking the right side of a 2008 Chevrolet Trailblazer SUV. Bosch CDR Tool data was available for both vehicles. [81] Figure C13 NASS 2012-12-016

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242 Piecewise analysis results: Weighted average analysis results:

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243 C1.14 2012-41-024 Collision 2012-41-024 involved a 2005 Toyota Camry left front striking the right front of a 2010 Toyota Tundra pickup. Bosch CDR Tool data was available for both vehicles. [82] Figure C14 NASS 2012-41-024

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244 Piecewise analysis results: Weighted average analysis results: C1.15 2012-43-014 Collision 2012-43-014 involved a 2011 Jeep Liberty SUV striking the right side of a 2011 Ford F250 pickup. Bosch CDR Tool data was available for both vehicles. [83] Figure C15 NASS 2012-43-014

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245 Piecewise analysis results: Weighted average analysis results:

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246 C1.16 2012-43-026 Collision 2012-43-026 involved a 2009 Lexus RX350 SUV striking the left front of a 2005 Toyota Camry. Bosch CDR Tool data was available for both vehicles. [84] Figure C16 NASS 2012-43-026

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247 Piecewise analysis results: Weighted average analysis results: C1.17 2012-43-106 Collision 2012-43-106 involved a 2001 Lincoln Navigator SUV striking the left side of a 2011 Dodge Durango S UV. Bosch CDR Tool data was available for both vehicles. [85] Figure C17 NASS 2012-43-106

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248 Piecewise analysis results: Weighted average analysis results:

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249 C1.18 2012-48-106 Collision 2012-48-106 involved a 2007 Toyota FJ Cruiser SUV striking the left side of a 2007 Toyota RAV4 SUV. Bosch CDR T ool data was available for both vehicles. [86] Figure C18 NASS 2012-48-106

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250 Piecewise analysis results: Weighted average analysis results: C1.19 2013-12-059 Collision 2013-12-059 involved a 2006 Chevrolet K1500 striking the left side of a 2005 Chevrolet Malibu. Bosch CDR Tool data was available for both vehicles. [87] Figure C19 NASS 2013-12-059

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251 Piecewise analysis results: Weighted average analysis results:

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252 C1.20 2013-12-106 Collision 2013-12-106 involved a 2012 Chevrolet Equinox SUV striking the right side of a 2008 GMC C2500. Bosch CDR Tool data was available for both vehicles. [88] Figure C20 NASS 2013-12-106

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253 Piecewise analysis results: Weighted average analysis results: C1.21 2013-12-112 Collision 2013-12-112 involved a 2004 Chevrolet Venture SUV striking the right side of a 2012 Chevrolet Equinox SUV. Bosch CDR Tool data was available for both vehicles. [89] Figure C21 NASS 2013-12-112

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254 Piecewise analysis results: Weighted average analysis results:

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255 C1.22 2013-43-152 Collision 2013-43-152 involved a 1997 Chevrolet C1500 pickup striking the left side of a 2011 Ford Ranger pickup. Bosch CDR Tool data was available for both vehicles. [90] Figure C22 NASS 2013-43-152

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256 Piecewise analysis results: Weighted average analysis results: C1.23 2013-76-094 Collision 2013-76-094 involved a 2010 Dodge J ourney SUV striking the right side of a 2007 Pontiac Torrent SUV. Bosch CDR Tool data was available for both vehicles. [91] Figure C23 NASS 2013-76-094

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257 Piecewise analysis results: Weighted average analysis results:

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258 C1.24 2013-76-165 Collision 2013-76-165 involved a 2013 Ford F150 crew cab pickup striking the left side of a 2013 Ford F150 regular cab pickup. Bosch CDR Tool data was available for both vehicles. [92] Figure C24 NASS 2013-76-165

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259 Piecewise analysis results: Weighted average analysis results: C1.25 2013-79-139 Collision 2013-79-139 involved a 2004 Toyota Pr ius striking head-on offset with a 2007 Toyota Highlander SUV. Bosch CDR Tool data was available for both vehicles. [93] Figure C25 NASS 2013-79-139

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260 Piecewise analysis results: Weighted average analysis results: