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An examination of the effective mass concept for eccentric motor vehicle impacts in the crash3 algorithm

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Title:
An examination of the effective mass concept for eccentric motor vehicle impacts in the crash3 algorithm
Creator:
Rose, Nathan Andrew
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English
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121 leaves : ; 28 cm

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Subjects / Keywords:
Automobiles -- Collision damage -- Mathematical models ( lcsh )
Automobiles -- Crashworthiness -- Mathematical models ( lcsh )
Traffic accidents -- Mathematical models ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references (leaves 117-121).
General Note:
Department of Mechanical Engineering
Statement of Responsibility:
by Nathan Andrew Rose.

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|University of Colorado Denver
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|Auraria Library
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53905930 ( OCLC )
ocm53905930
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LD1190.E55 2003m R67 ( lcc )

Full Text
AN EXAMINATION OF THE EFFECTIVE MASS CONCEPT
FOR ECCENTRIC MOTOR VEHICLE IMPACTS IN
THE CRASH3 ALGORITHM
by
Nathan Andrew Rose
B.S., Colorado School of Mines, 1998
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Mechanical Engineering
2003
nt
! All


This thesis for the Master of Science
degree by
Nathan Andrew Rose
has been approved
by
Sean Wright


Rose, Nathan Andrew (M.S., Mechanical Engineering)
An Examination of the Effective Mass Concept for Eccentric
Motor Vehicle Impacts in the CRASH3 Algorithm
Thesis Directed by Professor John Trapp
ABSTRACT
The CRASH 3 damage algorithm, developed in the 1970s, allows documented vehicle
damage to be correlated to the change in velocity that a vehicle experienced during an
impact. The model was originally derived with a two-degree of freedom mass spring
system appropriate for analysis of central collisions, where the line of action of the
resultant collision forces is directed through the centers of gravity of the vehicles. This
central collision model produced simple analytic expressions relating the total vehicle
damage energy to the vehicle changes in velocity.
The model was extended to the general case of eccentric collisions by the introduction of
the effective mass concept. The effective mass concept was based on the idea that when
the collision force does not pass through the center of gravity of a vehicle, the full weight
of the vehicle does not participate in the collision. The effective mass concept represents
a simplifying assumption that allowed an analytic solution for a six degree-of-ffeedom
system with non-linear equations of motion.
Questions have been raised within the automotive engineering community regarding the
accuracy of the effective mass concept. The research described in this report tested the
accuracy of the effective mass concept by comparing the results of calculations with the
CRASH 3 algorithm to the results of numerical simulations with a three degree-of-
ffeedom impact model for offset barrier impacts and a six degree-of-ffeedom impact
model for offset vehicle-to-vehicle impacts. A key accomplishment of this research was
to separate a single assumption of the CRASH 3 algorithm and to test it in isolation from
other assumptions invoked by the algorithm. The results of this research have shown that
the effective mass concept accurately models the effects of collision eccentricity, as long
as the collision force moment arm at maximum deformation is used in the analysis.
This abstract accurately represents the content of the candidates thesis. I recommend its
publication.
Signed
John Trapp
in


In Loving Memory of
Larry Cook
Wonderful Father-in-Law and Friend
With Special Thanks to
Curtis and Jeane Rose
Sources of Constant Encouragement


ACKNOWLEDGEMENT
I offer a heartfelt thanks to many who have been an integral part of this project. I have
a wonderful wife and two year old twins that fill my life with wonder and joy. Soon
my wife and I will add a third child. The patience of my family during the many
hours that this project has consumed was essential. I especially commend my wife
who showed tremendous patience in the midst of my neglect all the more noticeable
because of the pregnancy that she is in the midst of as I write. She is worth far more
than rubies (Psalm 31:10).
I am blessed to work at a job that is intellectually stimulating, with people that I enjoy
- many of whom I am privileged to call friends. Among those, several deserve special
mention. William Neale is a great friend and has influenced and challenged me in
ways that few others have. William was also a great help with the figures for this
report. Dr. Richard Ziemicki and Steve Fenton have built a corporate environment
that values creativity, entrepreneurship, and innovation an environment that was
necessary for the completion of this project. Richard and Steve deserve additional
thanks for funding my graduate study and research.
Finally, I have had a great experience at the University of Colorado at Denver. My
advisor, Dr. John Trapp has given me tremendous freedom to pursue my research and
has been patient with my intellectual meanderings. He has had many helpful
suggestions and saved me from at least one fundamental error during this project.
Both Dr. Trapp and Dr. Sam Welch have shown tremendous flexibility with me in
allowing me to tailor my academic program to my interests. I am thankful to both of
them.
A few final acknowledgements are in order. A conversation with Bruce Barnes of
Knott Laboratory early in this project led my thinking in the right direction. The
writings of Raymond McHenry and Ronald Woolley proved enjoyable and thought
provoking throughout this project, continually reminding me of how much I have left
to learn. Though I have met neither of them, they have both challenged me and
stimulated my thinking in new directions. And finally, I offer my thanks to Starbucks
- my office away from home and, at times, the source of my perseverance. Caffeine is
a wonderful thing!
Any errors or omissions in the text that follows are, of course, my responsibility.
Many people have provided thoughts and suggestions, but none of them are
responsible for the shortcomings of this work.


CONTENTS
Figures...................................................................viii
Tables....................................................................x
Chapter
1. CRASH 3 A Model for Crash Severity Analysis.......................1
1.1 Introduction........................................................1
1.1.1 The CRASH 3 Damage Algorithm........................................2
1.2 Notation............................................................8
1.3 Emoris Model for Central Collisions...............................10
1.4 Campbells Determination of Deformation Energy.....................14
1.5 McHenrys Extension of Emori and Campbell CRASH 3................20
1.5.1 Determination of Deformation Energy................................20
1.5.2 Determination of Crash Severity....................................24
1.5.3 Eccentric Collisions...............................................33
1.6 Previous Validation of the CRASH 3 Algorithm.......................41
1.7 Chapter 1 References...............................................47
2. Impact Models for Eccentric Collisions.............................52
2.1 Introduction.......................................................52
2.2 Eccentric Barrier Impacts..........................................52
vi


2.3 Vehicle-to-Vehicle Eccentric Impacts..............................61
2.4 Chapter 2 References..............................................68
3. Examining the Fidelity of the Effective Mass Concept..............71
3.1 Numerical Solution Technique Euler Method.......................71
3.2 Establishing the Accuracy of the Numerical Solution...............74
3.3 Simulation Results for Eccentric Barrier Impacts..................77
3.4 Simulation Results for Vehicle-to-Vehicle Impacts.................87
3.5 Discussion of Results.............................................91
3.5.1 Y-Direction Force Reversal........................................92
3.5.2 Constantly Increasing Moment Arm..................................94
3.5.3 Assumptions of the Effective Mass Concept.........................96
3.6 Potential Criticism..............................................101
3.7 Chapter 3 References.............................................102
Appendix
A. MATLAB Code for Barrier Impact Case..............................103
B. MATLAB Code for Car-to-Car Impact Case...........................109
References..............................................................117
Vll


FIGURES
Figure
1.1 The CRASH 3 Impact Model.............................................3
1.2 Eccentric Barrier Impact Model.......................................6
1.3 Eccentric Vehicle-to-Vehicle Model...................................7
1.4 Emoris Barrier Impact Model........................................11
1.5 Emoris Vehicle-to-Vehicle Model....................................11
1.6 Linearly Varying Crush Profile......................................19
1.7 McHenrys Linear F orce-Deflection Relationship.....................21
1.8 McHenrys Model for Central Impacts.................................25
1.9 Single Mass with Springs in Series..................................26
1.10 McHenrys Eccentric Collision Configuration.........................34
2.1 Eccentric Barrier Impact Model......................................53
2.2 Barrier Impact Free-Body Diagram....................................55
2.3 Eccentric Vehicle-to-Vehicle Model..................................62
2.4 Vehicle-to-Vehicle Free-Body Diagram (Body 1).....................64
2.5 Vehicle-to-Vehicle Free-Body Diagram (Body 2).....................64
3.1 Deformation (Spring Compression) v. Time Initial Velocity
of 30ft/s, Various Initial Collision Force Offsets.................78
3.2 Deformation (Spring Compression) v. Time Initial Velocity
of 40ft/s, Various Initial Collision Force Offsets.................79
vm


3.3 Deformation Energy Variation with Time Initial Velocity
of 30ft/s, Various Initial Collision Force Offsets.................80
3.4 Deformation Energy Variation with Time Initial Velocity
of 40ft/s, Various Initial Collision Force Offsets.................81
3.5 Modified Barrier Impact Model......................................93
IX


TABLES
Table
3.1 Inertial and Stiffness Parameters for Barrier Impact Simulations....77
3.2 Maximum Deformations for Barrier Impact Simulations.................80
3.3 Maximum Deformation Energies for Barrier Impact Simulations.........82
3.4 Actual Velocity Changes for Barrier Impact Simulations..............83
3.5 Moment Arms and Effective Mass Factors for 40ft/s Barrier
Impact Simulations.................................................85
3.6 AV Results for Different Average Moment Arm Definitions.............86
3.7 Moment Arm and AV Results for 30ft/s Barrier Impact Simulations.....87
3.8 Inertial and Stiffness Parameters for Car-to-Car Simulations........88
3.9 Inertial and Stiffness Parameters for Vehicle 2, Final Car-to-Car
Simulations........................................................88
3.10 Maximum Deformation Energies for Car-to-Car Simulations.............89
3.11 Resultant Velocity Changes for Car-to-Car Simulations...............89
3.12 Moment Arm and Effective Mass Factors for Car-to-Car Simulations....90
3.13 Calculated AVs and Percent Difference from Actual...................90
3.14 Max Total Versus Max Centripetal Acceleration, 40ft/s Simulations...99
3.15 Centripetal and Center of Gravity y-Accelerations, 40ft/s
Simulations.......................................................100
x


1.
CRASH 3 A Model for Crash Severity Analysis
1.1 Introduction
The development of mathematical models for understanding the mechanics of motor
vehicle crashes began with models that allowed documented vehicle damage to be
correlated to the sudden change in velocity that a vehicle experienced during an
impact. Documented damage thus became a tool for obtaining a quantitative
measure of the severity of a crash, that in turn, could be correlated to the potential
for occupant injury during that crash [6,14,20,41].1 The development of these
models was, in large part, driven by the need of the automotive industry for real-
world crash data that could be used in the design process. In order for injuries to be
prevented through improved vehicle design, the crash environment in which those
injuries occurred needed to be understood.
The early mathematical models that related vehicle damage to the severity of the
crash were simple, in the sense that they required only a gross, overall
characterization of the vehicle structural properties. With time, models used in
automotive design became more sophisticated, incorporating finite element analysis,
multibody models, and requiring detailed definition of the vehicle structure.
However, the original models that utilize a simple structural characterization have
not lost their usefulness. These models continue to be the most effective models for
generating real-world crash data, since they provide the greatest potential for
uniformity [6,22,23,25,26,41]. A large number of researchers and crash
1 McHenry [20] notes: In particular, the speed-change, AV, and its direction are believed to be the
best indicators of exposure severity for the vehicle occupants.
1


investigators, with varying skill levels and backgrounds can be trained to collect
vehicle damage data and from that data generate an estimate of crash severity and
occupant injury potential. There remains an extensive need for quality real-world
crash data since vehicle designers and safety researchers need to understand the
actual effects of vehicle design changes.
In addition to their usefulness to crash researchers generating empirical data, models
that utilize a simple structural characterization are also useful to engineers analyzing
motor vehicle crashes in a forensic setting. Engineers working outside the context of
design cannot obtain data sufficient for the detailed definition of the vehicle
structure that the more sophisticated models require. On the other hand, empirical
data for the application of simpler models is widely available.
1.1.1 The CRASH 3 Damage Algorithm
Since the mid to late 1970s, the CRASH 3 damage analysis algorithm has been one
of the dominant models within the contexts of the real-world crash research and
forensic engineering. The National Highway Traffic Safety Administration
(NHTSA) began using and distributing the CRASH 3 algorithm in 1981. The
algorithm, as a whole, contains a number of models, including a post-impact
trajectory model, a conservation of linear momentum impact model, and the vehicle
damage analysis model examined in this study.
The CRASH 3 damage algorithm is a closed-form solution for a two degree-of-
freedom system based on the principle of work and energy3 that quantifies crash
2 CRASH 3 is an acronym denoting the third revision of Calspan Reconstruction of Accident Speeds
on the Highway.
3 The principle of work and energy provides a relationship between the depth of deformation to the
vehicle, the force that caused that deformation, and the energy expended in creating that deformation
[2],
2


severity by relating it to the vehicle damage. The input parameters necessary for the
algorithm are limited to those that are available to researchers and forensic
engineers working on crash analysis outside the context of vehicle design. Namely,
only a gross, overall characterization of the vehicle structure is necessary. The
damage algorithm is based on the two-body mass-spring system shown in Figure
1.1. While the system shown in the figure is applicable to collisions where the line
of action of the collision forces travels through the vehicle centers of gravity, the
damage algorithm has been generalized for use with eccentric impact
configurations, where the collision forces do not act through the vehicle centers of
gravity.
X,
Figure 1.1 The CRASH 3 Impact Model
In order to generate a closed-form solution for eccentric collisions, the CRASH 3
damage algorithm relies on an approximation for the effect of vehicle rotation
during the impact. The validity of this approximation for the effects of collision
eccentricity on crash severity is the primary topic of the present research.
The remainder of this chapter gives a detailed discussion of the development of the
CRASH 3 damage algorithm, including a description of how the algorithm was
extended to handle the case of eccentric collisions. This extension involved
3


multiplying the vehicle masses by correction factors that are supposed to yield the
effective mass of each vehicle at the point of application of the resultant collision
force. The effective mass concept is based on the idea that when the collision forces
do not act through the centers of gravity of the vehicles, the entire masses of the
vehicles do not participate in the collision.
Ronald Woolley questioned the validity of the effective mass concept in a 1985
paper titled Inaccuracies in the CRASH3 Program [44]. He stated that:
... the masses in the CRASH equations have been modified in an
attempt to compensate for the rotation effects, the missing terms of
the energy equation. It should be observed that an energy equation
that holds only for the special case of no rotational kinetic energy
and no separation velocity can not be corrected into the proper
form for a general problem having both rotation and separation
velocities, by simply changing the masses into pseudo-masses.
Woolley raised a conceptual problem with the effective mass concept in the CRASH
3 damage algorithm. Unfortunately, he misconstrued the problem as a
misapplication of the principle of conservation of energy. As Day pointed out in a
1987 paper [7], the derivation of the damage analysis algorithm of CRASH 3 never
invoked the principle of conservation of energy. Woolleys mischaracterization of
the derivation of the CRASH 3 equations is understandable considering that the
relationship obtained from the CRASH 3 derivation that relates damage energy to
closing speed (relative speed) at impact can be derived for the central case using the
principle of conservation of energy.
A more accurate description of the potential problem with the effective mass
concept in the CRASH 3 damage algorithm would be to note that the CRASH 3
model, shown in Figure 1.1, does not directly incorporate a collision force that is
4


offset from the center of gravity of the vehicle. Therefore, the equations of motion
for the system do not include rotational terms. Instead, the derivation accounts for
the effects of collision eccentricity after the fact, by introducing the mass correction
factors at the end. There is no precedent in Newtonian mechanics for converting the
solution for a two-degree of freedom system like the one shown in Figure 1.1 -
into the solution for a six degree of freedom system, with two translational and one
rotational degree of freedom per body, by simply adjusting the masses of the bodies.
The effective mass concept in the CRASH 3 damage algorithm has received little
attention since Woolleys criticism partly due to Woolleys misconstrual of the
problem and partly due to the nonlinear nature of a system that includes an offset
collision force. Despite Woolleys mischaracterization of the problem, the effective
mass concept remains an after-the-fact addition to the CRASH 3 model and an
approximate method for incorporating collision forces that are offset from the
vehicle center of gravity. The fidelity of this approach remains an interesting and
unanswered question for the CRASH 3 model. If it is shown that the CRASH 3
damage algorithm is faithful to the physics of eccentric impacts, it remains to clearly
articulate the assumptions underlying the effective mass concept. To this authors
knowledge, these assumptions have not been articulated in the literature.
A discussion of the fidelity of the effective mass concept weaves its way through the
chapters that follow. Chapter 2 discusses an impact model for eccentric barrier
collisions and an impact model for car-to-car collisions that incorporate an offset
collision force. These models, shown in Figure 1.2 for the barrier impact case and in
Figure 1.3 for the car-to-car case, allow the collision force direction to vary with
time.
5


6


7


The equations of motion for these systems are nonlinear and they cannot be reduced
to produce a closed-form solution like that contained in the CRASH 3 damage
algorithm. Further, neither of these systems leads, in a straightforward manner, to
the effective mass concept contained in the CRASH 3 algorithm.
Chapter 3 discusses numerical solutions to the equations of motion for the systems
shown in Figures 1.2 and 1.3. Solutions are presented for both the barrier impact
and vehicle-to-vehicle cases. Parameter studies are conducted to examine the
fidelity of the mass correction concept in the CRASH 3 damage analysis algorithm.
1.2 Notation
For the most part, the present treatment adopts the notation of the authors that are
cited. That notation will be described as it is introduced. In chapter 2, when original
work is presented, the notation used is, in general, consistent with the notation used
in the original derivation of the CRASH 3 damage analysis algorithm. That notation
will also be described when it is introduced. Nonetheless, for ease of reference, a
listing of the notation utilized in this chapter is included below. For those symbols
that are specific to one or two models, those models are listed in parenthesis after
the definition of the symbol.
A, B Force-Crush Curve Stiffness Coefficients (McHenry)
Ai, Bi Coefficients Used in CRASH 3 Derivation
bo, bi Speed-Crush Curve Stiffness Coefficients (Campbell)
C Residual Crush (Campbell)
f Force Per Unit Width (Campbell, McHenry)
Fx Collision Force (McHenry)
Ei, E2 Deformation Energy for Vehicles 1 and 2
8


EBS Equivalent Barrier Speed (Campbell)
g Gravitational Constant
hi,h2 Offset of Collision Forces (McHenry)
ki,k2 Vehicle Structural Stiffness Coefficients (Emori)
ki,k2 Radii of Gyration (McHenry)
Ki, K2 Vehicle Structural Stiffness Coefficients (McHenry)
Ineffective Effective Stiffness of Springs in Series
mi,m2 Masses of Vehicles 1 and 2 (Emori, Campbell)
Mi, M2 Masses of Vehicles 1 and 2 (McHenry)
P Force
t Time
V Impact Speed into Barrier (Campbell)
Vc Common Velocity (McHenry)
VlO, V20 Initial Vehicle Velocities (Emori)
w Width (Campbell)
Wo Total Damage Width (Campbell)
Xl,X2 Vehicle Displacements (Emori)
Xnorm Normal Crush Depth (Woolley)
XOBL Oblique Crush Depth (Woolley)
Xi,X2 Vehicle Displacements (McHenry)
x{,x2 Vehicle Velocities (McHenry)
x x2 Vehicle Accelerations (McHenry)
Xp Acceleration of Point of Common Velocity
(McHenry)
ai, a2 Vehicle Accelerations (Emori)
8 Relative Vehicle Displacement (McHenry)
8X, 8 2 Individual Spring Compressions (McHenry)
9


AVi, AV2
AWnorm
AOBL
lull
¥i> ¥2
V\> ¥2
Changes in Velocity During Impact (McHenry)
Normal Crush Width (Woolley)
Oblique Crush Width (Woolley)
Effective Mass Multipliers (McHenry)
Vehicle Rotation Angles (McHenry)
Angular Accelerations (McHenry)
1.3 Emoris Model for Central Collisions
In 1968, Richard Emori of UCLA suggested a simple model for relating vehicle
deformation and impact force in frontal impacts [10]. He separated the vehicle into
two regions: a rigid mass, which contained the total vehicle mass and which
represented the undamaged portion of the vehicle, and a spring, which represented
the portion of the vehicle that deforms and absorbs energy during the impact. This
separation of the vehicle structure into a deforming region and a rigid region was
made possible by Emoris observation from crash test data that the time histories of
deceleration as measured at various locations within the vehicle does (sic) not
depend, as a first approximation, on the location of transducers if the measurement
is made beyond a certain distance away from the colliding end. This means that the
intact portion of the vehicle may be considered a rigid body [10] and the kinetic
energy of involved vehicles, just before the collision, is mainly absorbed by the
crushed portion of the vehicle [10].4
Figure 1.4 depicts Emoris model for vehicle-to-barrier impacts and Figure 1.5
depicts his model for vehicle-to-vehicle impacts.
4 Also inherent in Emoris model is the assumption that external forces, such as tire forces, can be
neglected during the impact [11]. This assumption is almost always invoked in the impact models
employed for crash reconstruction.
10


Figure 1.4 Emoris Barrier Impact Model
Figure 1.5 Emoris Vehicle-to-Vehicle Model
Emori argued that, when impacted, the front structure of the vehicle resists
deformation in the same way that a spring resists compression. That is, the force
required to crush the vehicle is proportional to the magnitude of the crush. In
general, this proportionality could be nonlinear. However, Emori concluded based
on crash test data, generated largely by Severy of UCLA [18, 30, 36, 37, 38, 39],5
that for frontal impacts the vehicle force-deflection characteristic was linear
5 Cited in Reference [9].
11


(constant stiffness). As a consequence of this linear relationship between force and
deformation, Emori showed that there was also a linear relationship between crush
depth and impact speed for frontal barrier impacts.
For Emoris car-to-car impact model, the equations of motion for the individual
masses are given by6
and
mxax + kxSx 0
(1.1)
m2a2 + k2d2 = 0.
(1.2)
In equations (1.1) and (1.2), mi and m2 are the masses of the bodies, ai and (X2 are
the accelerations of the masses, ki and k2 are the stiffnesses of the springs, and
8X and S2 are the compression of the individual springs.
In Emoris analysis, the time history of acceleration for each vehicle was known
from the staged collision accelerometer data that he was examining. The
displacement of each vehicle could be obtained from a double integration of the
acceleration data. However, Emori did not know the vehicle stiffness characteristics,
ki and k2. So, the time history of deformation for each individual vehicle could not
be determined using equations (1.1) and (1.2). The relative stiffness of the vehicle
structures would have to first be known, since the relative stiffness of the vehicle
structures involved in the crushing determine how the relative displacement of the
bodies is split between deformation to each vehicle [25].
6 Equations (1.1) and (1.2) are not included in Emoris original work. They are included in the
present study for clarity.
12


In cases where the displacement of the undamaged portion of the vehicle is equal to
the deformation namely barrier impacts and in cases where the relative stiffness
of the colliding structures are known namely head-on collisions of identical
vehicles delineation of the individual vehicle deformation and, therefore, the
individual stiffness characteristics of the vehicle structures becomes possible. In
these cases, the individual deformation time history is derivable from the
accelerometer data and the stiffness could, therefore, be determined from this data
by using equations (1.1) and (1.2), or by plotting force (mass times acceleration)
versus deformation and determining the slope. Emori, thus, provided a method for
determining vehicle structural stiffness using barrier impact data.
For the barrier impact case, Emori integrated the individual equations of motion,
(1.1) and (1.2), to yield the equation describing the individual deformation time
histories:
In equation (1.3), i equals 1 or 2 and corresponds to the vehicle being considered, x*
is the displacement of the vehicle and also the deformation of the spring
representing the deforming portion of the vehicle structure v;o is the initial velocity
of the mass representing the intact portion of the vehicle, and k,- is the stiffness of
the crushing structure.
If the collision is assumed to be perfectly plastic (zero restitution), then the
maximum deformation and the permanent deformation are equivalent and equal to
(1.3)
13


(1.4)
Since Emori showed that the vehicle structural stiffness was approximately
constant, and the mass is assumed constant, Equation (1.4) displays the linear
relationship between permanent vehicle damage and impact speed into a barrier.
1.4 Campbells Determination of Deformation Energy
In his doctoral thesis of 1972 [4] and in a technical paper in 1974 [5], Kenneth
Campbell of General Motors proposed a method for quantifying the energy
absorbed in crushing of a vehicle structure and for relating that energy to the
severity of the collision that caused the crushing. Campbell argued that for
convenience, it is desirable to rate collision severity in a manner which is
comparable to an existing test condition like the barrier impact [5]. Campbell
proposed the equivalent barrier speed (EBS) as that basis for rating collision
severity and for predicting occupant injury potential. The equivalent barrier speed is
a measure of the energy absorbed in crushing the vehicle structure and is,
conceptually, the speed at which an accident-involved vehicle would have to be
impacted into a rigid barrier to realize the same deformation energy that was
expended during crashing of the vehicle structure in the actual impact.
Campbell related the energy absorbed in crashing the vehicle structure to the
equivalent barrier speed through equation (1.5).
E-1Z-
2 g
14


In equation (1.5), E is the energy absorbed in crushing of the vehicle structure and
W is the weight of that vehicle. Campbell argued, regarding the implications of
equation (1.5), that the use of energy absorbed per unit weight as a basis for
equivalence with the barrier test, provides a foundation for the development of
objective estimation techniques [5].
Campbell noted that to obtain the equivalent barrier speed it is only necessary to
quantify the energy expended in crushing the single vehicle of interest, while to
determine the sudden change in velocity (AV) experienced by a vehicle during an
impact required quantifying the damage energy for both accident-involved vehicles.
This gave the equivalent barrier speed an advantage over the AV, since determining
the AV required more extensive data, and therefore, would be more costly to
determine.7
Expanding on the model proposed by Emori, Campbell treated the vehicle surface
and structure as a series of dissipative springs of differential width, dw, each with a
linear force-deflection characteristic. Employing a series of springs across the front
of the vehicle allowed Campbell to analyze impacts where the damage was not
uniform across the width of the vehicle namely, the type of crush profiles that
resulted from partial overlap barrier impacts and partial overlap car-to-car frontal
collisions. This was a significant step forward from Emoris model, since actual
frontal impacts rarely involve the entire front end of the vehicle and, therefore,
typically produce non-uniform damage across the front of the vehicle. Campbells
model did, however, assume that the collision damage was uniform vertically and
7 In the end, the AV was adopted by the industry as the best quantifier of collision severity. This was
largely due to the work of McHenry, described in the next section.
15


that the stiffness characteristics of the vehicle were constant across the width and
height of the vehicle.8
Campbell used barrier impact data for 1971 through 1974 GM vehicles to
substantiate the linear force-deflection characteristic employed in his model. In this
barrier impact data, Campbell observed a linear relationship between residual crush
and impact speed. Campbell described the general form of this linear relationship
with equation (1.6).
V = b0 +bxC (1.6)
In equation (1.6), V is the impact speed into the barrier, C is the residual crush in a
barrier impact, the residual crush across the front of the vehicle will be
approximately uniform bo is the impact speed which produces no residual crush,
and bi is the slope of the linear relationship between crush and impact speed.
From this linear relationship between barrier impact speed and residual crush,
Campbell inferred a linear force-deflection characteristic for the dissipative springs
employed in his model. Campbell observed that the energy absorbed (work done)
in plastic deformation of a structure can be obtained by integrating the local force
per unit area over the volume of deformation. If the damage is uniform over the
vertical dimension, then this integration can be eliminated, leaving only the
integrations with respect to crush and width..[5]. The energy absorbed in
crushing of the vehicle structure was, therefore, given by
8 This final assumption was not inherent in the form of the model, however, and current derivatives
of Campbells work allow the stiffness characteristics to be varied by zone.
16


(1.7)
E = JJ(/ dC dw)+Cj
where f is the local force per unit width, dC is a differential element of crush, dw is
a differential element of vehicle width, and ci is a constant that accounts for the
initial energy absorbed before the onset of residual crushing, as implied by the
coefficient bo in equation (1.6).
For the barrier impact cases, for which data was available, Campbell proceeded by
letting f = ao + aiC and then carried out the integration in equation (1.7). Then he
equated the result with the kinetic energy of the vehicle dissipated during the barrier
impact, written in terms of equation (1.6).9 Based on this equality, Campbell was
able to relate the linear force deflection characteristic to the linear relationship
between impact speed and residual crush and showed that the force per unit width
could be written as
/ =
(1.8)
where W is the vehicle weight and wo is the vehicle width. Further, he showed that
the constant, ci, in equation (1.7) could be written as

(1.9)
Therefore, for any vehicle that had corresponding barrier impact data, the force-
deflection characteristic for that vehicle could be defined by plotting the residual
9 Neglecting structural restitution, the energy dissipated during a barrier impact is equal to 'A mV2. In
terms of equation (1.6), this is written as 'A m (b0 + b]C)2.
17


crush that occurred during the barrier impacts against the impact speeds
corresponding to the crush. One only needed to pick the impact velocity intercept
(bo) and the slope of the crush-speed relationship (bi) off of this plot and then plug
them into equations (1.8) and (1.9). Equations (1.8) and (1.9) could then be used to
analyze the energy absorbed in the crushing of a similar vehicle in a real-world
crash with a generalized crush profile.
This absorbed energy was given by equation (1.10), and when integrated with
respect to the crush, C, simplified to equation (1.11).
E =
W
---ff(V>i +b?c)-dC-dw + bl
gw0 33 2 g
(1.10)
E =
W
gwQ
b0bxC +
tic2 A
. W U2
dw +b0
2 g
(1.11)
Campbell then equated equation (1.11) to equation (1.5) through the absorbed
energy to generate the following relationship for determining the equivalent barrier
speed:10
[ebs]2 =
1 w/ \
\\2b0b,C + b2C2)iw+bl
(1.12)
Equation (1.12) can be integrated if the vehicle damage profile can be expressed as
a function of the width, C(w). Campbell carried out this integration for a number of
10 Equation (1.12) assumes that the weight of the test vehicle used to establish the coefficients b0 and
bi is equal to the weight of the accident-involved vehicle, so that the weight will cancel out on both
sides of the equation. If these weights are not equal, then the equations are modified slightly. See
Campbells paper, reference 4, for a discussion of the modification.
18


crash profiles. For instance, for a crash profile that varies linearly over the width of
the vehicle, as shown in Figure 1.6 and described by equation (1.13), Campbell
arrived at equation (1.14).
C
= c,-(c,-cj
f \
W
v^oy
(1.13)
EBS =
bl + b0b{ (Ci +C2)+
(1.14)
Figure 1.6 Linearly Varying Crush Profile
19


1.5 McHenrys Extension of Emori and Campbell -
CRASH 3
Since Campbells landmark paper, research in the area of crash deformation analysis
- specifically, analysis employing a gross overall structural characterization has
focused on connecting, extending and refining the work of Emori and Campbell.
The most widely accepted embodiment and extension of their work is found in the
CRASH 3 damage algorithm, which is based on research conducted by Raymond
McHenry of Calspan Corporation11, under contract with the National Highway
Traffic Safety Administration (NHTSA). Raymond McHenry first presented the
CRASH damage algorithm in 1975 [19]. Since then, the algorithm has undergone a
number of revisions and improvements [21,22,23,24,25,26, 27], the most recent
being CRASH 3. The CRASH 3 damage algorithm is implemented in a number of
commercially available software packages with minor modifications and can be
implemented on a scientific calculator [13].
1.5.1 Determination of Deformation Energy
To quantify the energy expended in deforming a vehicle structure, McHenrys
CRASH damage algorithm employed a model similar to Campbells, although
McHenrys notation departed from Campbell [20]. Just as Campbell had done,
McHenry assumed that the crushing vehicle structure could be represented as a
series of dissipative springs [35], each with a linear force-deflection characteristic.
McHenry represented the stiffness properties for these linear dissipative springs by
two parameters, A and B, rather than Campbells bo and bi. Whereas Campbells
11 Previously named Cornell Aeronautical Lab.
20


coefficients were derived from the crush-impact speed plot, McHenrys A and B
coefficients specified the intercept and the slope of the force-deflection plot.
Figure 1.7 below depicts graphically how the A and B values define the linear force-
deflection relationship. The horizontal axis in the figure represents the crush depth
and the vertical axis represents the force per unit width of damage. The B value, the
slope of the linear force-deflection curve, governs the proportionality between
residual crush and impact force. The A value, the force intercept on the force-
deflection plot, has traditionally been interpreted as the force that can be applied to
the vehicle structure before the onset of permanent (plastic) deformation [44].
=A+BC
w
B
Figure 1.7 McHenrys Linear Force-Deflection Relationship
An alternative interpretation of the A value is offered by Woolley [45]. Woolley
observed that the force-deflection curve intersects the crush axis at a value of-
(A/B). Woolley argued that this negative intercept on the crush axis could be
F
w
21


interpreted as the elastic recovery of the vehicle structure or the difference between
the maximum dynamic crush and the permanent crush. Woolley noted that the
physics of a Campbell-type crush energy model requires that the crush considered
be the dynamic maximum rather than the residual maximum crush. Under
Woolleys interpretation, shifting the force-residual crush curve over a distance A/B
would yield the dynamic force-crush curve. Under this interpretation, the A value
would arise because of the use of residual crush in the model.
The force-deflection characteristic in McHenrys CRASH damage algorithm was,
therefore, given by equation (1.15) below.
/ = A + BC (1.15)
Again, f is the impact force per unit width applied to the vehicle structure.
As in Campbells model, McHenry established general A and B stiffness parameters
from barrier impact and staged collision data. McHenry divided the vehicle
population into six classes delineated by wheelbase and within each class
quantified A and B stiffness coefficients for the front, rear, and side structures. The
CRASH 3 damage algorithm contains the most recent update of McHenrys
stiffness parameter values.12 Analysis for an individual accident-involved vehicle
using the CRASH 3 damage algorithm will, therefore, not consider the force-
deflection characteristic of that individual vehicle. Instead, the algorithm will assign
stiffness parameters to the vehicle structure that are representative of the class into
which the vehicle fits [40,41,44].
12 The CRASH 3 algorithm also assumes generalized vehicle dimensions and inertial properties for
each vehicle category [22, 39]. If known, actual vehicle weight can be entered. The categories
include minicar, subcompact, compact, intermediate, fiillsize, and large.
22


A lack of widely available crash test data for many vehicle models motivated
McHenry to establish these generalized, category stiffnesses for the CRASH 3
damage algorithm. As Smith and Noga note, it would require an extraordinary
number of tests to establish and maintain such coefficients for so many vehicles.
Still, crash test data for a number of vehicle models was available and the
availability of data has increased significantly with time. With this in mind,
commercially available derivatives of the CRASH 3 algorithm allow the user to
define stiffness parameters for specific individual vehicles. Methods for determining
stiffness parameters for an individual vehicle, based on barrier impact data, are
outlined in publications by Neptune, Strother, and Woolley [16, 31, 32, 33,42,
45].13
After defining the force per unit width, McHenry set up an integral over the crush
depth and the vehicle width to determine the deformation energy, just as Campbell
had done in equation (1.7) above. However, McHenry made explicit the origin of
the constant in Campbells equation by setting his limits of integration to include the
energy absorbed before the onset of plastic deformation. Instead of integrating over
the crush depth only, McHenry integrated over the crush depth plus and additional
depth of A/B. Geometrically, this took the integration back to the place on the force-
deflection curve where the force was zero (refer to Figure 1.7). McHenrys equation
for the deformation energy is shown below in equation (1.16).
Wo c
+ BC)dC-dw (1-16)
0 A
B
13 The effective mass concept is employed in the derivation of stiffness parameters from offset barrier
impact data. Therefore, the conclusions of the present study have implications for these methods.
23


Integrating equation (1.16) over the crush depth, plus the preload A/B, yields
equation (1.17).
Application of equation (1.17) to quantify the deformation energy requires a
quantitative description of the vehicle damage surface. This is accomplished in
practice by physical examination of the accident involved-vehicle. Generally, crush
depth measurements are taken perpendicular to the original shape of the damaged
surface of the vehicle. This procedure will not be discussed here. The reader is
referred to reference 12 for a complete discussion of crush profile measurement
techniques.
1.5.2 Determination of Crash Severity
McHenrys fundamental extension of Emori and Campbells work was his
derivation of a model that related the energy expended in crushing the vehicle
structure to the sudden change in velocity experienced by the vehicle during the
impact (AV). Campbell had recognized that if the deformation energy for both
accident-involved vehicles could be determined, then the AVs for the vehicles could
be determined. However, Campbell did not present any methodology for carrying
out this calculation.
To relate deformation energy to AV in central collisions, where the resultant
collision forces act through the centers of gravity of the bodies, McHenry
considered the same car-to-car model that Emori had considered. However, he
(1.17)
24


supplemented Emoris derivation by employing Newtons third law and the
principle of conservation of momentum. McHenrys model for central car-to-car
impacts was shown in Figure 1.1. It is repeated here, in Figure 1.8, for convenience.
X X,
Figure 1.8 McHenrys Model for Central Impacts
McHenry began his consideration of this model by writing the equations of motion
for the individual bodies in the following form:
M,X,=-
'_KiKx '
Kt+K2)
(*, -x2)
M2X2 =
KtK2 '
Kt+K2

(1.18)
(1.19)
In equations (1.18) and (1.19), Mi and M2 are the vehicle masses, Xi and X2 are the
X-displacements of the bodies from their original positions, Xx and X2 are the
accelerations of the bodies, and Ki and K2 are the stiffnesses of the springs. The
coefficients on the right hand sides of the equations,
25


(1.20)
f k}k2 '
\K,+K2J
represent the effective stiffiiess of the two springs in series and arise by considering
Newtons third law equal and opposite reaction forces for the springs. To see
this, consider the system shown in Figure 1.9.
8
Figure 1.9 Single Mass With Springs in Series
Assume that a force P is applied to the mass in Figure 1.9, producing a displacement
in the mass toward the wall and, therefore, compressing the springs that lie between
the mass and the wall. Assume, further, that the springs were initially in their
equilibrium positions. Let 8 be the total displacement of the mass to the right 8i the
compression of the first spring and 82 the compression of the second spring. The
force required to produce the displacement of the mass is given by
P ~ ^effective^
(1.21)
By Newtons third law, the same force P is applied to both springs, so that the force
P can also be written as
(1.22)
26


P = Kl8l=K282
Noting that 8 = Sl + S2, we can write the following:
8 =
K, K,
f
V*>+*2.
(1.23)
Combining this with equation (1.21) and solving for Effective, we arrive at equation
(1.24) for the effective stiffness of the springs in series.
K
effective
*,*2
kx+k2
(1.24)
To achieve a solution of equations (1.18) and (1.19), McHenry combined the
equations by subtracting equation (1.19) from equation (1.18). This yields equation
(1.25) below.

M, +M
V
V 2 j
*,*2
*1+*2J
(jr,-x!)=o
(1.25)
Letting 8 = Xx X2, equation (1.25) can be written as
8 +
M, +M2
V MxM2 j
V
*1*2
*1 +*2 J
8 = 0
(1.26)
Equation (1.26) displays simple harmonic motion and has the solution
27


8 = A, sin
'Mx +M2 Y KxK2
K MXM2
\t + Bx cos
KtK, 'I
l M\M2 J
t (1.27)
At t = 0, the springs are uncompressed (8 = 0). This implies that Bi is equal to zero.
Similarly, at t = 0, the relative closing velocity of the bodies, 80, is equal to
Xxo X20. Taking the derivative of equation (1.27), with Bi = 0, the relative closing
velocity at any time t is given by
S = A
M,+M2
v mxm2

KtK2
Kx+K2J
cos.
rMx+M2^
MXM
2 y
KxK2
Kx+K:
2 y
(1.28)
Plugging in the initial condition for equation (1.28) yields equation (1.29) for Ai.
A =(*, -xA
MxM2
Mx +M2
( Kx +K2a
KxK2
(1.29)
Therefore, the relative displacement of the bodies, 8, is given by equation (1.30)
below.
s=(xl0-x20\
MxM2
V
kx+k2
mx+m2\ kxk2 j
sin
Mx+M7

mxm2 Xki+k2
kxk2
t (1.30)
The maximum relative displacement of the bodies and, therefore, the maximum
compression of the springs occurs when
28


M, + M,
A/
V M\M2 jy
\
k1+k2)
K
t = .
2
(1.31)
Therefore, the time at maximum mutual spring compression is given by
.14
_ K
^max 2 '
m,m2
V
M, +M2 ^,/T.
+ /T,
2 y
(1.32)
and the maximum mutual spring compression itself is given by

MlM2
M, +M2
V
K{ +K2
K\K2
(1.33)
McHenry proceeded by letting 8X = Xx X and 82 = X X2. For force equilibrium
it is necessary that
KX8X = K282
(1.34)
And since, by definition 8 = 8l+82, we can write
*i =
(
-£-L
yKl+K2)
(1.35)
Equation (1.33) can be restated as
14 Interestingly, the time to maximum mutual spring compression or rather, maximum mutual
vehicle crush is independent of the initial relative velocity. The time to maximum mutual vehicle
crash is determined only by the relationship of the vehicle masses and stiffnesses [25].
29


(1.36)
*,0 *20 ~
V
m,+m2
V M2 j\
*,*2
Ki+K2)
max
and using equation (1.34), (1.35) and S = <5, + S2, equation (1.36) can be written as
*10 *20
rM,+M2^
\ MiM2 J
0-37)
The energy absorbed in compressing the springs can be expressed as
e,
(1.38)
and

(1.39)
Therefore, equation (1.37) can be written as
*10 *20
Mx +M2
V M\M2 j
2 (E, +E2)
(1.40)
At maximum mutual spring compression, the bodies will reach a common velocity,
Vc. From conservation of momentum, this common velocity can be written as
30


(1.41)
y MxXw +M2X20
c Mi +M2
The changes in velocity experienced by the bodies, AVi and AV2, from the initial
time, t = 0, to the time of maximum mutual spring compression are given by
equations (1.42) and (1.43).
AV1=Xl0-Vc=Xl0-
fMlXl0+M2Xia\
M, +M
2 2
AV2=Vc-X20 =
fMxXw+M2X2^
Mx+M
-X.
20
2 y
(1.42)
(1.43)
Incorporating equation (1.40) and simplifying algebraically, equations (1.42) and
(1.43) reduce to
and
AVX =
1
2 (£,+£,)
M,

V
M
2 y
(1.44)
AV2 =
1
2{E,+E2)
( \
( M2^
m2 1+ 2

(1.45)
31


Thus, McHenry arrived at equations (1.44) and (1.45) that related the energy
expended in crushing the vehicle structure to the change in velocity experienced by
the vehicle during impact.
It should be noted that McHenrys derivation of equations (1.17), (1.44), and (1.45)
neglected the restitution phase of the collision. More recent works by McHenry [29]
and Neptune [34] have presented modifications to the CRASH 3 equations to
incorporate the energy and velocity changes that occur during the restitution phase.
The present research does not utilize the modified form of the CRASH 3 algorithm
with restitution effects. This is for two reasons.
First, including the effects of restitution in this research represents an unnecessary
complication to the question of the fidelity of McHenrys effective mass concept.
This research compares one analytical model to another, and both models can be
used without reference to the physical phenomenon of restitution. Were the present
research to employ staged collision data, then it might have been important to use a
model that incorporated restitution.
Second, McHenrys extension of the CRASH 3 algorithm to include restitution is, in
this authors experience, not widely used within the automotive engineering
community. The extension is still relatively new and there remain questions
regarding the practicality of implementing the techniques in light of limited
empirical data. It is not the aim of the present research to enter into the discussion
regarding the feasibility of considering restitution within the CRASH 3 framework.
32


1.5.3 Eccentric Collisions15
Equations (1.44) and (1.45) assume a central collision, where the line of action of
the resultant collision forces is through the centers of gravity of both bodies and,
therefore, where the centers of gravity of the bodies reach a common velocity during
the collision. During eccentric collisions the centers of gravity of the bodies do not
reach a common velocity during the impact. Instead, a common velocity is achieved
on the contact surface between the vehicles and the assumptions of equations (1.44)
and (1.45) are not satisfied.
To extend his model to the eccentric collision case, McHenry did not go on to
examine a collision model that incorporated an eccentric collision force. Instead,
McHenry extended the CRASH model by developing the effective mass concept.
In describing the effective mass concept in the CRASH 2 users manual, McHenry
and Lynch [26] stated, when the collision force does not pass through the center of
gravity of a vehicle, an effective mass must be determined. The present research
focused on an examination of the validity of the effective mass concept derived by
McHenry.
McHenry began the development of the effective mass concept by considering the
collision model shown in Figure 1.10. In the figure, a number of new symbols are
introduced. L and I2 are the principal (yaw) moments of inertia of the bodies, vj/i and
vj/2 are the angular orientations of the bodies, hi and I12 are the distances that the
collision forces are offset from the centers of gravity of the bodies measured
perpendicular to the line of action of the collision forces and AFj' and AF2' are the
changes in velocity experienced by each body at point P, the point of application of
the resultant collision forces. The xi-yi and x2-y2 frames are body-fixed reference
15 This section relies on McHenrys discussion in References 20, 22, and 26.
33


frames. During the depicted collision, a common velocity would be reached at point
P.
y2
Figure 1.10 McHenrys Eccentric Collision Configuration
McHenry argued that the acceleration of point P on body 1 could be written as
Xp =Xx + hly/l.
(1.46)
By Newtons second law the collision force could then be expressed as
Fx=-M,X,=-M,(xf-W,).
(1.47)
McHenry wrote the corresponding rotational motion equation for body 1 as follows:
Fxh, =-Ixy>x
(1.48)
34


In McHenrys model, beginning here in equation (1.48), the lower case k refers to
the radius of gyration of a body. This should not be confused with the spring
stiffnesses in Emoris model. From equation (1.48), the angular acceleration of body
1 is given by equation (1.49).
Yx=~
3A
Mxk\
Substitution of equation (1.49) into equation (1.47) yields
M,
( 7-2 7.2 \
kx + A]

Using equation (1.50), we can write
*,2
k;+h;
Letting Y\ =

k{ + h[
then equation (1.51) can be written as
(1.49)
(1.50)
(1.51)
Xx=yxXp. (1.52)
Several observations should be made here. First, McHenrys equation for the
acceleration of point P, equation (1.46) above, is incomplete. A complete expression
35


for the acceleration of a point on a rigid body relative to the center of gravity of that
body takes the following form [1]:
In equation (1.53), ap is the acceleration of the point P, aG is the acceleration of the
center of gravity, rP/ is a position vector that locates the point P in a reference
frame attached to the body at the center of gravity, and y/x is the angular velocity of
the body about the center of gravity. For a derivation of equation (1.53), the reader
is referred to reference 1. The first term on the right side of equation (1.53) accounts
for the acceleration of the center of gravity relative to the inertial frame. The second
term arises due to the angular acceleration of the reference frame attached to the
body. And finally, the third term accounts for the centripetal acceleration of point P.
A comparison of equations (1.46) and (1.53) reveals that McHenry has neglected the
centripetal acceleration of the point P. Equation (1.46) necessarily assumes that the
centripetal acceleration of the point P is negligible relative to the other terms in
equation (1.53). This assumption was tested in the numerical simulations presented
in chapter 3 and will be discussed there.
Second, in the derivation of equation (1.52), McHenry treated hi as a constant.
However, if we examine the dynamics of the system shown in Figure 1.10 through
time, the moment arm of the collision force will change. Apparently, McHenry
intended hi and I12 to be average, or representative, moment arms of the collision
forces. Various definitions of this moment arm are plausible. These various
definitions will be explored in chapter 3 and the results obtained using each one will
be compared.
(1.53)
36


Third, eccentric impacts involve acceleration components and velocity changes in
more than one coordinate direction. For the system shown in Figure 1.10, each
vehicle will experience a velocity change in both the x and y coordinate directions.
This is not reflected in McHenrys derivation. Equation (1.46) neglects not only the
centripetal acceleration, but also the y-component of the acceleration. It could be
argued that, in practice, a two-dimensional velocity change is considered, since a
user of the CRASH 3 algorithm must estimate the direction of the resultant collision
force, thus specifying the two-dimensional direction along which the velocity
change occurs. The user is not confined to defining the collision force direction in
the x-direction, as McHenry has done for his derivation. It is unclear to this author
whether this resolves the problem or not. This issue is explored through the
numerical simulations in Chapter 3.
Finally, McHenry interpreted the physical significance of y, by examining its role
in equation (1.50). McHenry argued that in equation (1.50), y, acted as a multiplier
that adjusts the mass of the body. He called the quantity y,M, the effective mass
of body 1 acting at point P during the collision. Similarly, McHenry defined the
effective mass of body 2 acting at point P during the collision as y2M2
It is interesting that McHenry interpreted y, and y2 as adjustments to the masses of
the bodies. An alternative interpretation would be to see y, and y2 as factors
modifying the collision forces to reflect the effective force at point P. This is
perhaps more physically correct since masses do not act, but rather are acted upon
by forces. It is not that there is less mass acting at point P, but rather that the
collision force is more effective at point P than it is at the centers of gravity of the
bodies.
37


McHenry continued his derivation by integrating equation (1.52) from time t = 0 to
the time that the common velocity is reached at point P, producing equation (1.54).
This integration also assumed that hi was a constant.
AFJ =/}AVl', (1.54)
A VS is the velocity change at point P. Thus, McHenry argued that if the change in
velocity at point P on body one could be found, then the change in velocity at the
center of gravity could be obtained through the mass correction factor yx.
In order to find the change in velocity at the point P for each body, McHenry
substituted his effective masses into equations (1.44) and (1.45) to yield equations
(1.55) and (1.56).

av; =
2(£, +E2)
yxMx fi+

2 {Ex+E2)
y2M2
1 +
Y2M 2
yxMi
(1.55)
(1.56)
In accordance with equation (1.54), McHenry multiplied equations (1.55) and (1.56)
by y{ and y2, respectively, and obtained equations (1.57) and (1.58).
38


(1.57)
A V,=
2yl(E1+E1)
AV2 =
Mt \ +r*<) v. r2M2)
2/2 C^i +-^2)
m2 l YrMr )
(1.58)
McHenry, thus, arrived at equations relating damage energy to the vehicle changes
in velocity for the general case of eccentric collisions. The validity of these
equations, and the effective mass concept that forms their base, was the fundamental
question of the present research. The effective mass multipliers represent a
modification to the CRASH 3 equations that are intended to transform them from a
solution to a two degree-of-freedom system into the solution for a nonlinear six
degree-of-freedom system, a system that does not have an analytic solution.
As will become clear in Chapter 2, the equations of motion of a six degree-of-
freedom system, like the one in Figure 1.10, do not lead, in any obvious way, to the
effective mass concept or to equations (1.57) and (1.58). That does not mean that
the effective mass concept is not an adequate approximation to account for the
effects of collision eccentricity. But if the effective mass concept is adequate, it is
because there are certain relationships between the parameters in motor vehicle
crashes that would not necessarily exist for the general case of eccentric impacts.
A final introductory comment, regarding the use of the effective mass multipliers in
the CRASH 3 damage algorithm, is in order. During an eccentric collision, a higher
relative approach velocity is necessary to produce the same damage depth, and
39


therefore the same damage energy, as would be produced in a central collision
(assuming equal direct damage widths).16 For an eccentric collision, as the collision
force builds up by compression of the vehicle structure, the vehicle will begin to
rotate away from the colliding surface, making it more difficult to build up the
collision force. The vehicle structure becomes effectively stiffer in an eccentric
impact, in the sense that it takes greater relative approach velocity to build up the
same collision and therefore to produce the same damage depth and damage energy.
What is the significance of this for the CRASH 3 damage algorithm? The effective
mass concept is intended to account for the effects of collision eccentricity by
adjusting the vehicle masses. However, this is not the only adjustment that takes
place in the equations of the damage algorithm as a result of the effective mass
concept. Equations (1.57) and (1.58) can be written in following form:
AV,=
2y;(E,+E2)
Y\Mx
1+
yxMx
y2M
2j
av2 =
2rl(E,+E2)
i
r2mi
1+
y2M2
Y\M\
1 7
(1.59)
(1.60)
Written in this form, it is clear that in addition to the multiplication of the masses by
the mass correction factors, y, and y2, the total damage energy is also adjusted,
being multiplied by either y2 or y\.
16 This will be demonstrated in Chapter 3 when we examine principles of eccentric collisions using
the models shown in Figures 1.2 and 1.3.
40


It should be noted that the total damage energy would be naturally adjusted for an
eccentric collision through a physical reduction in the level of vehicle deformation.
In other words, an investigator using the CRASH 3 algorithm for analysis of an
eccentric collision will enter physical evidence crush depth measurements that
already reflects a reduction in damage energy over the impacts central collision
counterpart. Thus, the CRASH 3 damage algorithm adjusts the damage energy
above and beyond the natural adjustment that occurs. Further, since the mass
correction factors will be less than 1.0 for eccentric collisions, this reduction of the
overall damage energy will be by a greater percentage than the corrections to the
masses. This adjustment to the total damage energy and its validity have not been
discussed in the literature. The question of the appropriateness of this adjustment
will be taken up in chapter 3.
1.6 Previous Validation of the CRASH 3 Algorithm
To this authors knowledge, the effective mass concept has never been examined
and tested in isolation from other assumptions in the CRASH 3 algorithm. However,
a number of studies have been conducted that have examined the accuracy of the
algorithm with all of its assumptions acting simultaneously. These studies of
CRASH 3 accuracy are discussed in this section to introduce the reader to previous
validation studies of the algorithm.
In a 1975 paper, Raymond McHenry [20] described the use of the original CRASH
program and indicated 12% accuracy of speed estimates could be obtained in
reasonably well-documented cases. However, McHenry acknowledged that, at the
time, results of staged collisions that are suitable for use in evaluation of
41


reconstruction techniques are relatively scarce.17 McHenry made due with the data
that was available, but warned that the available data does not permit levels of
confidence to be established in a rigorous manner. Rather, it merely indicates
approximate error ranges in general applications.
In users manuals for the CRASH algorithm [22,26], McHenry and Lynch
referenced this testing of the original version of the CRASH algorithm, noting that it
showed an overall accuracy of approximately 12%. McHenry and Lynch
admitted a small degradation in solution accuracy due to the use of default vehicle
dimensions, inertial properties, and stiffnesses [26]. In another place, McHenry
observed that obviously, the accuracy of the damage-based CRASH
approximations of speed changes can be substantially improved, within the existing
framework, by the availability and proper utilization of more extensive test data
[22]. McHenry argued, though, that with improvements embodied in the CRASH 2
algorithm and with planned updating of the vehicle parameter and stiffness data,
users could expect accuracy approaching the range of 5% [26].
In 1982, Smith and Noga of the National Highway Traffic Safety Administration
(NHTS A) examined the accuracy and sensitivity of the CRASH 3 damage algorithm
[40,41]. Since McHenrys testing of the original CRASH algorithm, more extensive
staged collision data had accumulated. Smith and Noga were able to compare data
from 27 staged collisions involving 53 vehicles to the AVs calculated by the
CRASH 3 damage algorithm based on vehicle damage data from these staged
collisions. Regarding accuracy, Smith and Noga found that, at low values of AV in
17 In reference 23, McHenry wrote: It is concluded that an urgent need exists for realistic staged
collision experiments to permit quantitative determinations of the accuracies of reconstruction
techniques. Reference 23 consists largely of a sustained argument for the need for more extensive
staged collision data for validation of reconstruction techniques.
42


the range of 0 to 30mph18 the CRASH 3 damage algorithm tended to
underestimate the AV experienced by a vehicle during the collision. At high values
of AV, above 30mph, they found that the algorithm gave accurate predictions of the
AV.
Smith and Noga identified the generalized stiffness coefficients used by the CRASH
3 damage algorithm as the primary source of systematic error in AV estimates
generated by the algorithm. Since the algorithm simply assigned generalized,
category stiffness coefficients to the vehicle structure, there would, in general, be a
difference between the stiffness parameters employed by the algorithm and the
actual stiffness of the individual accident-involved vehicle. Smith and Noga note
that this problem can become more pronounced when the vehicle side structure is
involved in the collision since the stiffness along the vehicle side structure varies in
a way that the CRASH 3 stiffness parameters do not consider. Namely, the side
structure of the vehicle is stiffer in the area of the suspension and wheel
components, but CRASH 3 considers the structure homogeneous.
Smith and Noga found that errors of 10 percent in the A and B stiffness parameters
propagated to errors of 2 to 6 percent in AV estimates generated by the CRASH 3
damage algorithm. They note that the nature of the function itself, specifically that
delta-V is related to the square root of the absorbed energy... makes the unit error
in delta-V somewhat smaller than a unit error in the coefficients. This in part
justifies the desirable simplification associated with the use of classes of stiffness.
Smith and Noga identified measurement of the vehicle deformation profile and
estimation of the direction of the collision force, which must be estimated by the
18 A majority of motor vehicle accidents produce AVs in this range of 0 to 30 mph.
43


user, as the primary sources of random error in AV determination using the CRASH
3 damage algorithm. They note that damage profiles are highly irregular and
subject to variable interpretation both in depth and length. Smith and Noga found
that the AV calculation performed by the CRASH 3 algorithm was most sensitive to
crush profile measurement errors at low AVs. Further, they found that for two-
vehicle accidents, the AV calculation is more sensitive to the estimate of the
direction of the collision force than to measurement errors of the damage profile.19
In single vehicle accidents, they found that the calculation was more sensitive to
measurement errors while quantifying the damage than to estimates of the direction
of the collision force. In general, Smith and Noga found that when potential
variation of the crush measurements and estimation of the collision force direction
were considered, the 95 percent confidence limits on the calculated AV were
between plus or minus 9 and 25 percent of the AV. Smith and Nogas results
demonstrated that random errors alone could cause inaccuracies that surpassed those
originally reported by McHenry.
Woolley, Warner and Tagg proved the most critical evaluators of the CRASH 3
algorithm in their paper Inaccuracies in the CRASH3 Program. Referring to claims
made in the CRASH 3 users manual, Woolley, Warner and Tagg argued that
claims for accuracy of about 10 percent cannot be validated and they went on to
present cases involving error well in excess of 20 percent [44]. They also cited
their previous work with the CRASH 2 algorithm where examples of errors
exceeding 75% were found in simple test cases.
19 Sensitivity to the estimated collision force direction is significant for the present study since the
direction of the collision force affects the magnitude of the mass correction accomplished by the non-
central impact parameters described above.
44


Woolley identified the following key factors in producing these errors: (1)
omission of terms in the formulation of the energy equation, (2) sensitivity of the
solution to the input estimate of principle-direction-of-force, and (3) the use of
generalized, class stiffnesses rather than stiffness parameters for the actual accident-
involved vehicle. The contention of an incomplete formulation of the energy
equation has been criticized, since the derivation of the CRASH 3 damage algorithm
never invoked the principle of conservation of energy [7]. Additionally, Smith and
Noga extensively discussed the problem of the investigator having to estimate the
principle direction of the collision force and the sensitivity of the CRASH 3 model
to this estimation several years before the publication of Woolleys paper. Smith
and Noga acknowledged potential errors up to 25 percent. Smith and Noga also
discussed the problem of category stiffnesses and the potential errors that they could
produce.
The unique contribution of Woolley, Warner, and Tagg seems to lie in their critique
of the CRASH 3 effective mass concept. This author has found no other written
critiques of the effective mass concept in the CRASH 3 damage algorithm, and
while Woolley and his colleagues misconstrue the problem of the effective mass
concept, it was their critique of the concept that led to the present study.
In 1989, Day and Hargens published a study of the accuracy of a commercially
available derivative of the CRASH 3 algorithm based on comparison with staged
collision data produced by Calspan [9]. Their study did not examine the accuracy of
damage-based estimate of the vehicle changes in velocity the focus of the present
- research but only the ability of the algorithm to estimate impact speed. Day and
Hargens explicitly rejected the validity of previous validation studies such as the
study by Woolley, Warner, and Tagg that examined CRASH 3 estimates of the
45


vehicle changes in velocity. In supporting this rejection, they cited errors in the
change in velocity data from available staged collisions:
Several studies published since 1978 have attempted to use the RICSAC
delta-V data as a basis for validation of the CRASH and other programs ...
In all cases, researchers found significant differences between the delta-Vs
computed by the programs and the delta-Vs measured during the RICSAC
study. This has led many researchers to question the accuracy and usefulness
of these programs. However, the literature clearly stated there were problems
with the sophisticated data acquisition systems aboard the vehicles. One of
the major problems encountered during the study was the fact that the
accelerometer data was not taken at the center of gravity (CG), but rather, at
the firewall. Thus, any rotation during impact would cause error in the
measured separation velocities and, therefore, the delta-V.
Day and Hargens found that for determining collision impact speed, the typical
CRASH analysis had a confidence interval of about -6 to +7 percent of the
combined impact speed.
Validation of the CRASH 3 damage algorithm is not a simple matter. All of the
studies discussed above have focused on the algorithm as a whole either the
damage algorithm as a whole or the entire algorithm. None of these comparisons
have attempted to isolate key assumptions within the damage algorithm and test
those assumptions. This approach makes ferreting out the sources of error somewhat
speculative, since it would not be possible to assess what portion of the errors
resulted from assumptions employed by the CRASH 3 damage algorithm and what
portion of the errors result from systematic or random sources of data error. It
20 A further complication in validating the CRASH 3 algorithm is that, as Smith and Noga observe,
the presumption must be maintained that the data gathered from accelerometers in the staged
collisions represents a true description of the accident. In reality, the data from these
accelerometers is subject to experimental error and uncertainty.
46


appears that, with the exception of Woolleys critique, the effective mass concept
has generally been presumed accurate.
Interestingly, the CRASH 3 damage algorithm contains two adjustments that may
tend to offset one another in some, perhaps many, collisions. Consider an eccentric
collision with a resultant collision force that is not perpendicular to the original edge
of the damaged surface. In such a case, the mass correction factors would attempt to
reduce the calculated change in velocity proportional to the degree of offset of the
collision force. At the same time, though, the CRASH 3 damage algorithm contains
an adjustment to the damage energy to reflect the effects of non-perpendicular
collision forces. This adjustment for non-perpendicular collision forces could tend
to offset the reduction for collision eccentricy accomplished through the mass
correction factors. Conceivably, one could obtain good correlation with a number of
staged collision tests and neither of these assumptions would have been tested, since
errors induced by them could simply offset one another. In reality, it is possible that
both assumptions contain conceptual errors. The present treatment attempts to
conceptually isolate the effective mass concept in the CRASH 3 algorithm. A
similar study could be conducted surrounding the CRASH 3 adjustment to the
damage energy for non-perpendicular collision forces.
1.7 Chapter 1 References
The above discussion is intended to be a thorough review of the development of the
CRASH 3 damage analysis algorithm, specifically regarding the effective mass
concept. However, some of the original sources proved hard to come by and,
therefore, the discussion did not directly rely on the references that are marked with
an asterisk (*). These references are cited in the text in appropriate locations and
were included in this section to facilitate further study by the reader who is able to
obtain them.
47


1. Baruh, Haim, Analytical Dynamics, WCB/McGraw-Hill, 1999.
2. Beer and Johnston, Vector Mechanics for Engineers. Fifth Edition, McGraw-
Hill Book Company, New York, 1988.
3. Brach, Raymond M., Energy Loss in Vehicle Collisions, 871993, Society of
Automotive Engineers, Warrendale, PA, 1987.
4. Campbell, K.L., Energy As A Basis For Accident Severity A Preliminary
Study, Doctoral Thesis, University of Wisconsin, 1972.*
5. Campbell, K.L., Energy Basis for Collision Severity, 740565, Society of
Automotive Engineers, Warrendale, PA, 1974.
6. Cheng, Philip H., et al., An Overview of the Evolution of Computer Assisted
Motor Vehicle Accident Reconstruction, 871991, Society of Automotive
Engineers, Warrendale, PA, 1987.
7. Day, Terry D., Hargens, Randall L., An Overview of the Way EDCRASH
Computes Delta-V, 870045, Society of Automotive Engineers, Warrendale,
PA, 1987.
8. Day, Terry D., Hargens, Randall L., An Overview of the Way EDSMAC
Computes Delta-V, 880069, Society of Automotive Engineers, Warrendale,
PA, 1988.
9. Day, Terry D., Hargens, Randall L., Further Validation of EDCRASH Using
the RICSAC Staged Collisions, 890740, Society of Automotive Engineers,
Warrendale, PA, 1989.
10. Emori, Richard I., Analytical Approach to Automobile Collisions, 680016,
Society of Automotive Engineers, Warrendale, PA, 1968.
11. Emori, Richard I., Vehicle Mechanics of Intersection Collision Impact,
700177, Society of Automotive Engineers, Warrendale, PA, 1970.
12. Equi-Distant Crush Measurement Techniques. Society of Automotive Engineers
Recommended Practice, SAE J2433,1998.
13. Fonda, Albert G., CRASH Extended for Desk and Handheld Computers,
870044, Society of Automotive Engineers, Warrendale, PA, 1987.
14. Gimotty, P.A., Campbell, K.L., Chinachavala, T., Carsten, O., ODay, G.,
Statistical Analysis of the National Crash Severity Study Data, DOT-HS-805-
561, August 1981.*
48


15. Hight, Philip V., Lent-Koop, Bruce, Hight, Robert A., Barrier Equivalent
Velocity, Delta V and CRASH3 Stiffness in Automobile Collisions, 850437,
Society of Automotive Engineers, Warrendale, PA, 1985.
16. Hull, Wendell C, Newton, Barry E., Estimating Crush Stiffness when
Reconstructing Vehicle Accidents, 930898, Society of Automotive Engineers,
Warrendale, PA, 1993.
17. Mason and Whitcomb, The Estimation of Accident Impact Speed, Calspan,
1972*
18. Mathewson, J.H., et.al., Head-on Collisions Series HI, Paper 21 ID presented
at SAE West Coast Meeting, August 16-19, I960.*
19. McHenry, The CRASH Program A Simplified Collision Reconstruction
Program, Calspan, 1975.*
20. McHenry, A Comparison of Results Obtained with Different Analytical
Techniques for Reconstruction of Highway Accidents, 750893, Society of
Automotive Engineers, Warrendale, PA, 1975.
21. McHenry, Extension and Refinements of the CRASH Computer Program Part I
- Analytical Reconstruction of Highway Accidents, NTIS PB-252116,1976.*
22. McHenry, Raymond R., Extensions and Refinements of the CRASH Computer
Program Part H, DOT HS-801 838, February, 1976.
23. McHenry, Raymond R., Jones, Ian S., Extensions and Refinements of the
CRASH Computer Program Part HI, Evaluation of the Accuracy of
Reconstruction Techniques for Highway Accidents, DOT HS-801 839,
February, 1976.
24. McHenry and Lynch, Mathematical Reconstruction of Highway Accidents -
Further Extensions and Refinements of the CRASH Computer Program,
Calspan, 1976.
25. McHenry, Raymond R., Computer Aids for Accident Investigation, 760776,
Society of Automotive Engineers, Warrendale, PA, 1976.
26. McHenry, Raymond R., Lynch, James P., CRASH 2 Users Manual, Calspan,
DOT-HS-5-01124,1976.
27. McHenry and Lynch, Revision of CRASH 2 Computer Program, NHTSA,
1979.
28. McHenry and McHenry, A Revised Damage Analysis Procedure for the
CRASH Computer Program, 861894, Society of Automotive Engineers,
Warrendale, PA, 1986.
49


29. McHenry and McHenry, Effects of Restitution in the Application of Crush
Coefficients, 970960, Society of Automotive Engineers, Warrendale, PA, 1997.
30. Moore, J.O., Feasibility Study of New York State Safety Car Program A
Preliminary Report, 660345, Society of Automotive Engineers, Warrendale,
PA, 1966.*
31. Neptune, James A., et al., A Method for Quantifying Vehicle Crush Stiffness
Coefficients, 920607, Society of Automotive Engineers, Warrendale, PA, 1992.
32. Neptune, James A., Flynn, James E., A Method for Determining Accident
Specific Crush Stiffness Coefficients, 940913, Society of Automotive
Engineers, Warrendale, PA, 1994.
33. Neptune, James A., Flynn, James E., A Method for Determining Crush
Stiffness Coefficients from Offset Frontal and Side Crash Tests, 980024,
Society of Automotive Engineers, Warrendale, PA, 1998.
34. Neptune, James A., Crush Stiffness Coefficients, Restitution Constants, and a
Revision of CRASH3 & SMAC, 980029, Society of Automotive Engineers,
Warrendale, PA, 1998.
35. Prasad, Aloke Kumar, CRASH3 Damage Algorithm Reformulation for Front
and Rear Collisions, 900098, Society of Automotive Engineers, Warrendale,
PA, 1990.
36. Severy, D.M., Mathewson, J.H., Technical Findings from Automobile Impact
Studies, SAE Trans. Vol. 65,1957, pp. 70-83.*
37. Severy, D.M., Mathewson, J.H., Siegel, A.W., Automobile Head-on Collisions
- Series II, SAE Trans. Vol. 67,1959, pp. 238-262*
38. Severy, D.M., Mathewson, J.H., Automobile Barrier and Rear-end Collision
Performance, Paper 62C presented at SAE Summer Meeting, June 8-13,1958.*
39. Severy, D.M., et al., Barrier Collisions Series IV, presented at Highway
Research Board Annual Meeting, January 1961.*
40. Smith, R.A., Noga, J.T., Accuracy and Sensitivity of CRASH, Technical
Report DOT-HS-806-152, January 1982.*
41. Smith, Russell A., Noga, J. Thomas, Accuracy and Sensitivity of CRASH,
821169, Society of Automotive Engineers, Warrendale, PA, 1982.
42. Strother, Charles E., Woolley, Ronald L., James, Michael B., Warner, Charles
Y., Crush Energy in Accident Reconstruction, 860371, Society of Automotive
Engineers, Warrendale, PA, 1986.
50


43. Struble, Donald E., Generalizing CRASH3 for Reconstructing Specific
Accidents, 870041, Society of Automotive Engineers, Warrendale, PA, 1987.
44. Woolley, Ronald L., Warner, Charles Y., Tagg, Melaney D., Inaccuracies in
the CRASH3 Program, 850255, Society of Automotive Engineers, Warrendale,
PA, 1985.
45. Woolley, Ronald L., Non-Linear Damage Analysis in Accident
Reconstruction, 2001-01-0504, Society of Automotive Engineers, Warrendale,
PA, 2001.
51


2. Impact Models for Eccentric Collisions
2.1 Introduction
To explore the physical fidelity of the effective mass concept in the CRASH 3
damage algorithm, the present research will examine two impact models that
directly incorporate forces that are offset from the vehicle centers of gravity.
Conceptually, these two models represent a single system concept in two
manifestations. The first contains a single mass and a single spring and is
appropriate for examining the mechanics of offset barrier impacts. The second
contains two masses and two springs, a system appropriate for consideration of the
mechanics of eccentric vehicle-to-vehicle impacts.
Since, the equations of motion of these systems are nonlinear, numerical solutions
and parameter studies are presented in Chapter 3. This chapter focuses on the
analytical preliminaries to that numerical analysis. The assumptions invoked in the
models are discussed and the equations of motion for the systems are written and
prepared for numerical analysis.
2.2 Eccentric Barrier Impacts
The first impact model considered is shown in Figure 2.1 below. The system
consists of a rectangular body of mass Mi, with principle moment of inertia Ii, a
linear spring, with stiffness coefficient ki, and a barrier of infinite mass (or attached
to the ground). The center of gravity of the body is located in the inertial reference
frame with the coordinates xg and yG and the orientation of the body is described by
the angle y/x measured clockwise off of the inertial x-axis. The reference frame x[ -
y[ is attached to the body and the point IP is fixed in this reference frame at the
52


coordinates ai and bi. The spring is connected to the point IP on the body (xip and
yiP in the inertial reference frame) and point B (xB and yB in the inertial reference
frame) on the barrier.
Center of Gravity
Figure 2.1 Eccentric Barrier Impact Model
53


The following assumptions are invoked for the system shown in Figure 1.2:
1. The only force applied to the body is the spring force generated by
relative movement between point IP on the body and point B on the
barrier. Applied to motor vehicle impacts, this assumption amounts to
assuming that external forces, such as tire forces, are negligible
compared to the collision force and that the duration of impact is short
enough that momentum losses from external forces do not become
significant. This is a common assumption for impact models and is
generally considered valid for modeling high-speed motor vehicle
impacts since the impact duration is usually on the order of 100
milliseconds, short enough that momentum losses due to tire forces are
negligible [2, 3,4].
2. The initial translational velocity and the initial orientation of the body
are in the positive x-direction.
3. The body has no initial rotational velocity.
4. At time t = 0, the spring is at its equilibrium length, and therefore,
applies no force to the body.
A free-body diagram for the body is shown in Figure 2.2 below. The body is
depicted at some time t > 0 where the orientation of the body is no longer in line
with the inertial x-axis. In Figure 2.2, the angle formed by the x[ axis and a line
connecting the body center of gravity to the point IP is given by ax. The orientation
of the line of action of the spring force is described by either the angle or fi2.
54


Line of Action
Figure 2.2 Barrier Impact Free-Body Diagram
The magnitude of the spring force, FSPRING, is equal to the spring stiffness ki
multiplied by the compression of the spring beyond its equilibrium position. The
equilibrium length of the spring is equal to the distance separating points IP and B
at time t = 0 and is calculated as shown below in equation (2.1).
L0 = V -^ipo ) + (.Vb ~ -Vipo) ^
The length of the spring at any instant in time, t > 0, is given by equation (2.2)
below.
55


As lixB xip) y\p)
(2.2)
Thus, at any instant in time, the compression of the spring beyond its equilibrium
position is given by equation (2.3).
L = LS-L0 (2.3)
It should be noted that for the conditions and time period of interest, L will be a
negative quantity.
At any instant in time, the line of action of the spring force is defined by a line
connecting points IP and B. This line and the inertial x-direction form an angle /?,
as shown in Figure 2.2. The spring force in the x-direction is, therefore, given by the
following equation:
Fspring,x kxL cos (3x (2.4)
Similarly, the spring force in the y-direction is given by equation (2.5).
FSPRING,y = A^sinA (2.5)
The placement of the inertial reference frame is arbitrary and it is shown in an
arbitrary location in Figure 2.1. In the equations that follow, we will assume that the
inertial reference frame coincides with point B, fixed on the barrier face, so that the
coordinates xb and ye are equal to zero. The sine and cosine terms in equations (2.4)
and (2.5) can be written using the coordinates of the point P as follows:
56


(2.6)
cos/?, =
)2+(yB-ylP)2
(2.7)
In equations (2.6) and (2.7), xjp and yip will be negative quantities, and so the sine
and cosine terms will be positive. Thus, both equations (2.4) and (2.5) will yield
negative values and the resultant collision force will point up and to the left as
shown in Figure 2.2 and as would be expected with an initial velocity along the
positive x-direction.
So far, equations involving the geometry of the system have been written in terms of
the coordinates of the point IP. In terms of the center of gravity coordinates for the
body, xip and yip can be written as shown below.
(2.8)
(2.9)
Invoke the following trigonometric identities:
cosl
i(a, + y/^) = cos a, cos y/x sin a, sin y/x
(2.10)
sin(a, + ) = sin a, cos y/l + cos a, sin \j/x
(2.11)
57


Also, note that
a, = -Jaf+b[ cos a,
(2.12)
and
b, =y/af + b, sin a,. (2.13)
Substituting equations (2.10) through (2.13) into equations (2.8) and (2.9) yields the
following equations for xip and in terms of the center of gravity coordinates and
the rotation angle:
xip = xG+a, cosy/, -b, sin^ (2-14)
yXP = yG+bx cosy/x + a, sin y/x. (2-15)
Now, write the moment about the center of gravity of the body as follows:
Mg = ~FSPR]NGx{jet, +bx sin(a, +y^iij+FSPRINGyi^Jax+bi cos(a, + ^)) (2.16)
Again, invoking equations (2.10) through (2.13), equation (2.16) can be written as
follows:
Mg = -FSPr,ng.x {bx cosy/x + a, sin ^,)+FSPRINGy {a, cos y/x b, sin y/x) (2.17)
58


Now, the Newton-Euler equations of motion for the system shown in Figure 1.1 can
be written as follows:
M 1%G ~ FSPRING,x (2.18)
Mlh = ^SPRING,y (2.19)
t) II '-'T (2.20)
In preparation for numerical solution, write equations (2.18) through (2.20) in state
variable form, as six first-order ordinary differential equations.

= v,
Gx
(2.21)

SPRING, x
Mi
y G =VGy
VGy =
SPRING,y
Mi
W\ = i
Mg
(2.22)
(2.23)
(2.24)
(2.25)
(2.26)
59


For a complete description of the system and for the numerical analysis presented in
the next chapter, equations (2.21) through (2.26) are supplemented with equations
(2.1) through (2.7), (2.14), (2.15), and (2.17).
Now, examine the equations of motion under the special case when bi is equal to
zero. In this case, equations (2.14) becomes
But, the spring force will now have no moment arm about the center of gravity and
so the rotation angle y/x will be zero at all times. Thus, equation (2.27) reduces
further to the following equation:
Combining equations (2.1) through (2.4), (2.7) and (2.28), and recognizing that yip
is equal to zero when bi goes to zero, yields the following equation for the x-
component of the spring force:
Thus, for the case when bi goes to zero, the x-direction equation of motion,
becomes
xlp =xG +ax cos^j.
(2.27)
xxp xG + Cl,
(2.28)
(2.29)
M,
(2.30)
60


which is the equation of motion for a single degree-of-ffeedom mass-spring system
displaying simple harmonic motion. Equation (2.30) has an exact analytical solution
given by equation (2.31),
where we have let 5 = xG -xG0. Equation (2.31) for the case where bi goes to zero
will be used as a benchmark solution to check the accuracy of the numerical
analysis technique employed in Chapter 3.
2.3 Vehicle-to-Vehicle Eccentric Impacts
The second impact model considered is shown in Figure 2.3. This system consists of
two rectangular bodies, with masses Mi and M2 and principle moments of inertia Ii
and I2, and two linear springs, with stiffness coefficients k] and k2. The centers of
gravity of the bodies are located in the inertial reference frame with the coordinates
xio and yio, for the first body, and the coordinates X2G and y2G, for the second body.
The orientations of the bodies are described by the angles y/x and y/2 measured
clockwise off of the inertial x-axis.
(2.31)
61


62


The reference frame -y[ is attached to the first body and the point IP is fixed in
this reference frame at the coordinates ai and bj. The reference frame x'2-y'2 is
attached to the second body and the point 2P is fixed in this reference frame at the
coordinates a.2 and b2. The first spring is connected to the point IP on the first body
(xip and yip in the inertial reference frame) and the second spring is connected to the
point 2P (x2p and y2p in the inertial reference frame) on the second body. The two
springs are connected together at point P. At any instant in time, the spring forces
act along a line connecting points IP and 2P.
The following assumptions are invoked for the system shown in Figure 1.2:
1. The only forces applied to the bodies are the spring forces generated by
relative movement between the points IP and 2P.
2. The initial translational velocities and the initial orientations of the
bodies are parallel to the inertial x-direction.
3. The bodies have no initial rotational velocity.
4. At time t = 0, the springs is at their equilibrium lengths, and therefore,
apply no forces to the bodies.
Free-body diagrams for the bodies are shown in Figure 2.4 and Figure 2.5 below.
The bodies are shown at some time t > 0. In Figure 2.4, the angle formed with the
x[ axis by a line connecting the center of gravity of body 1 to the point IP is given
by a,. Similarly, in Figure 2.5, the angle formed with the x'2 axis by a line
connecting the center of gravity of body 2 to the point 2P is given by a2. The
orientations of the lines of action of the spring forces is described by the angles /?,
and P2. The lines of action of the spring forces are 180 degrees apart, so
trigonometric identities can be used to write the system equations in terms of one of
these angles.
63


Line of Action
of Spring Force
Figure 2.4 Vehicle-to-Vehicle Free-Body Diagram (Body 1)
Figure 2.5 Vehicle-to-Vehicle Free-Body Diagram (Body 2)
64


The magnitudes of the spring forces applied to the bodies are equal to the effective
stiffness of the springs in series,
(2.32)
multiplied by the total compression of the springs beyond their equilibrium lengths.
Equation (2.32) for the effective stiffness of springs in series was derived in Chapter
1.
The total length of the springs at any instant in time is equal to the straight-line
distance between points IP and 2P and is given by
The original total length of the springs in series is, similarly, given by the following
equation:
Thus, the total compression of the springs beyond their equilibrium positions is
given by
(2.33)
(2.34)
L Ls Lq .
(2.35)
65


The spring force acting on body 1 in the x-direction is, therefore, given by the
following equation:
k k
FSPRING,\x =~ fc ZcosA (2.36)
Since /?, = fi2 +180, and cos(/?2 +180) = cos /?2, we can write the spring force
acting on body 1 in the x-direction as
SPRING,lx
Zcos/?,
kx+k 2
(2.37)
Similarly, the spring force acting on body 1 in the y-direction is given by
SPRING,\y
Zsin P2
kx+k 2
(2.38)
The components of the spring force applied to the second body are equal in
magnitude and opposite in direction as their respective components applied to the
first body, i.e.:
FSPRING,2x ~ FSPRING,lx (2.39)
FSPRING,2y ~ ^SPRING,\y (2.40)
The sine and cosine terms in equations (2.37) and (2.38) are given as follows:
66


sin fi2 = y2P y'p Ls (2.41)
cos/32=X2P~X'p Ls (2.42)
Next, write the moments about the centers of gravity of the bodies as follows:
Mxg = ~Fspring,\x(a\ sin^i + bi cosWx)+Fspring,cos^, - - bx sin ) (2.43)
M2G = ~FSPRING,Ix^P 2 sin ty 2 b2 COS ^SPR/NG^y (^2 COS^2 -b2 sin ^2) (2.44)
In terms of the center of gravity coordinates for the bodies, xip, yip, X2p, and y2p can be written as
xip = x\g + a\ cos^, -bx sin^ (2.45)
yxp = ylG +bx cos^, +a, sin^, (2.46)
x2P = x2G +a2 cos y/2 -b2 svn.y/2 (2.47)
y2p = y2G + bi C0W2 + a2 sin^2 (2.48)
At time t = 0, y/{ = 0 and y/2 =180, and therefore
*l/>0 = X1G0 *" ai (2.49)
67


y\po ~ y\Go +
(2.50)
X2P0 ~ X2G0 a2 (2.51)
y 2P0 = y2G0 (2.52)
Finally, the Newton-Euler equations of motion are given by equations (2.53)
through (2.58).
M x =F ivi ,a,g i SPRJNGAx (2.53)
MxyW = FSPRING,\y (2.54)
ii "T (2.55)
M2%2G ~ ^SPRING,\x (2.56)
2~y 2G = ^SPRING,\y (2.57)
I2V2 =M2G (2.58)
In preparation for numerical solution, write equations (2.53) through (2.58) in state
variable form, as twelve first-order ordinary differential equations.
68


(2.59)
XIG ~ V1 Gx (2.59
F 1 SPRING,\x VWx = A4 M, (2.60)
& > li o (2.61)
F 1 SPRING,ly VlGy ~ M, (2.62)
W\ =o)} (2.63)
1 /, (2.64)
X2G ~ V2Gx (2.65)
F 1 SPRING,\x V2 Gx ~ M2 (2.66)
Ag = V2Gy (2.67)
FSPRING,\y VlGy' m2 (2.68)
V2 2 (2.69)
69


(2.70)
d>
2
M2g
h
For the car-to-car case, when bl and b2 both go to zero, the system reduces to a
one-dimensional, two degree-of-freedom mass-spring system with the following
exact analytic solution:
£ = (vu*o-v:
2Gx0
M,M2 kx + k2
M{ + M2 kxk2
sin t
M, + M2 kxk2
MxM2 kx+k2
(2.71)
In equation (2.71), v1Gx0 and v2GxQ are the initial velocities of the bodies. As for the
barrier impact case, this exact solution for the central impact case will be used as a
benchmark solution for checking the accuracy of the numerical technique employed
in Chapter 3.
2.4 Chapter 2 References
1. Baruh, Haim, Analytical Dynamics. WCB/McGraw-Hill, Boston, 1999.
2. Beer and Johnston, Vector Mechanics for Engineers. Fifth Edition, McGraw-
Hill Book Company, New York, 1988.
3. Rose, Nathan A., Fenton, Stephen J., Hughes, Christopher M., Integrating
Monte Carlo Simulation, Momentum-Based Impact Modeling, and Restitution
Data to Analyze Crash Severity, 2001-01-3347, Society of Automotive
Engineers, Warrendale, PA, 2001.
4. Varat, Michael S., Husher, Stein E., Vehicle Crash Severity Assessment in
Lateral Pole Impacts, 1999-01-0100, Society of Automotive Engineers,
Warrendale, PA, 1999.
70


3. Examining the Fidelity of the Effective Mass Concept
3.1 Numerical Solution Technique Euler Method
In chapter 2, the equations of motion for the models depicted in Figures 2.1 and 2.3
were written as systems of first-order ordinary differential equations in preparation
for numerical analysis. For the system shown in Figure 2.1 the equations of motion
were written as six first-order ordinary differential equations, given by equations
(2.21) to (2.26). For the system shown in Figure 2.3 the equations of motion were
written as twelve first-order ordinary differential equations, given by equations
(2.58) to (2.69).
An Eulers method numerical technique was used to solve these systems of first-
order differential equations for relevant parameter sets [1,2]. Eulers method is a
numerical technique of first-order accuracy and is not generally considered accurate
enough for quantitative analysis of dynamic systems. However, for the modeling of
motor vehicle impacts, the time period of interest is on the order of 100 ms. Over
such a short time period, Eulers method gave exceptional accuracy, and so a
higher-order method was found unnecessary. The accuracy of the Euler method
calculations will be demonstrated in the next section.
For the barrier impact model (Figure 2.1) the Eulers method representation of the
equations of motion is written as follows:



(3.2)
y G,t+&1 =VGy,At + yG,t
(3.3)
F,
(3.4)
M,
i,/+a/ = (3.5)
G
(3.6)
To completely define the system, equations (3.1) through (3.6) are supplemented
with equations (2.1) through (2.7), (2.14) and (2.15), and equation (2.17). Solution
of this system of equations was carried out using MATLAB. The MATLAB code
for the barrier impact case is included in Appendix A.
For the car-to-car impact model (Figure 2.3) the Eulers method representation of
the equations of motion is given by equations (3.7) through (3.18).
(3.7)
F<
(3.8)
y\Gj+&t =V, GyAt + yiG.'
(3.9)
72


1 SPR1NG,1 y,t ,
VlGy,,+A, =------& + VlGy,,
(3.10)
Vu+m =<*>uto + Vu
(3-11)
M, Gt
i.f+A/ = 7 to+co ,,
MG
(3.12)
X2G,l+&t ~ V2Gx,^ + X2G,t
(3.13)
MG*,/+Af
= _-rspring,u,, At + v
M,
2Gx,t
(3.14)
^G.r+Af V2Gy,f^ + y2G,/
(3.15)
* SPRING,\y,l A ,
V2G>-,r+Ar =---------^-^to + V2Gy (3.16)
^2,r+A, =(o2Jto + yf2J
(3.17)
M2G,
2,f+A/ = 7 A? + co2l
*2 G
(3.18)
To completely define the system, equations (3.7) through (3.18) are supplemented
with equations (2.33) through (2.35) and (2.37) through (2.48). Again, solution of
73


this complete system of equations was carried out using MATLAB. The MATLAB
code for the car-to-car case is included in Appendix B.
3.2 Establishing the Accuracy of the Numerical Solution
For the barrier impact case, simulations were run with initial x-direction velocities
of 30 and 40 feet per second and with initial collision force offsets (bi) of 0.0, 0.5,
1.0,1.5,2.0, and 3.0 feet. The simulations were carried out over a time period of
150 milliseconds with 1000 time steps, yielding a step duration of approximately
150 microseconds.
The accuracy of the numerical calculations was checked in two ways. First, for the
two cases with 0.0 feet of initial collision force offset, the numerical solution was
compared to the known exact solution for the central impact case. For both barrier
impact simulations with zero collision force offset, the simulations deviated from
the exact solution by a maximum of 0.35%.
Second, for all of the simulations, the solution was checked to ensure that it satisfied
the principle of conservation of energy. Throughout the simulations the total system
energy should have remained constant. Thus, the initial system energy, which
consisted of the bodys initial translational kinetic energy, should have been equal
to, at all subsequent time steps, the sum of the bodys remaining translational kinetic
energy, the bodys rotational kinetic energy, and the energy of spring compression.
This energy balance, for the barrier impact case is given as follows:
^lVlGrO
2
MivL Mivicr W
2 2 2
+ DE
(3.19)
74


In equation (3.19), DE is the energy of spring compression and is given by equation
(3.20) below:
DE = !hL (3.20)
2
For all of the simulation runs, the maximum deviation from the initial system energy
was 1.06%. The largest error was for the case with an initial collision force offset of
3.0 feet and an initial x-direction velocity of 40 feet per second. The Eulers method
solution, thus, yielded exceptional accuracy.
For the car-to-car impact case, simulations were run with both vehicles at initial x-
direction velocities of 30 feet per second one in the positive direction and one in
the negative direction and with initial collision force offsets (bi) of 0.0, 0.25, 0.5,
0.75,1.0, and 2.0 feet. Additional simulations were then run with the same collision
force offsets with one vehicle at positive 30 feet per second and the other at negative
45 feet per second. Again, the simulations were carried out over a time period of
150 milliseconds with 1000 time steps, yielding a step duration of approximately
150 microseconds.
The accuracy of these numerical simulations was checked in the same way as the
barrier impact case was by comparing the numerical solutions for zero initial
collision force offset with the known exact solution for the central impact case,
given by equation (2.71), and by checking to ensure that each simulation satisfied
the principle of conservation of energy.
75


For the case of zero initial collision force offset, the simulation with initial vehicle
speeds of 30 feet per second and the simulation with initial vehicle speeds of 30 and
45 feet per second both deviated from the exact solution by a maximum of 0.35%.
The energy balance equation for the car-to-car case is given as follows:
^IVlC.tO | M2V2GxO MjVjGx M2V1Gx | ^\V\Gy ^2V2Gy , , Ac6*
- +
+ -
- + DE (3.21)
In equation (3.21), DE is the total energy of spring compression and is given by
equation (3.22)
DE = U&-L\
2 kl+k2
(3.22)
or by equation (3.23) as follows:
DE-
2
k2S2
+ _2_L
2
(3.23)
In equation (3.23), Sl and S2 are the individual spring compressions. For all of the
car-to-car simulations, the numerical solutions deviated from the original system
energy by a maximum of 0.96%. This was for the case with an initial collision force
offsets of 2.0 feet for each vehicle and initial x-direction velocities for body 1 and 2
of positive 30 and negative 45 feet per second, respectively. Thus, the Eulers
method solutions for the car-to-car cases yielded exceptional accuracy for the car-
to-car case.
76


3.3 Simulation Results for Eccentric Barrier Impacts
As described above, for the barrier impact case simulations were run with initial x-
direction velocities of 30 and 40 feet per second and with initial collision force
offsets of 0.0, 0.5,1.0,1.5,2.0, and 3.0. For these simulations, the vehicle inertial
and stiffness parameters were set as follows:
Vehicle Weight 3000 lbs.
Principle Moment of Inertia 1750 lb-ft-sec2
Vehicle Stiffness per Unit Width 100 lb/in2
Damage Width 24 inches
Table 3.1 Inertial and Stiffness Parameters For Barrier Impact Simulations
Each of the values in Table 3.1 represents a typical value for motor vehicle
parameters. For all of the barrier impact simulations, ai was set equal to 6 feet, xb
and ye were established as the origin, and the initial spring length was set to 6 feet.
In the accident reconstruction literature, vehicle stiffness data is reported per unit
width. That procedure is followed in Table 3.1. The spring stiffness used in the
simulations, ki, is therefore obtained by multiplying the vehicle stiffness per unit
width by the damage width and then multiplying by 12 to convert the stiffness to
units of lb/ft. To isolate the effects of collision eccentricity, each simulation was run
with the same damage width. In practice, a change in collision force offset is
associated with a corresponding change in damage width. This is one of the factors
that prevents using actual crash test data to isolate the issue of collision eccentricity.
Figure 3.1 below depicts simulation results for deformation (spring compression)
through time for the simulations with an initial vehicle velocity of 30 feet per
second. This figure contains a curve for each of the initial collision force offsets -
0.0, 0.5,1.0,1.5,2.0, and 3.0. The case with an initial collision force offset of 0.0
77


feet exhibits the highest deformation and the case with an initial collision force
offset of 3.0 exhibits the lowest deformation, consistent with the principle that,/or
the same closing speed and same damage width, eccentric impacts result in less
deformation. The case with zero initial collision force offset showed a maximum
deformation of 1.71 feet, while the case with 3.0 feet of initial collision force offset
showed a maximum of 1.35 feet, a 20.6% reduction in deformation due to collision
eccentricity.
Figure 3.1 Deformation (Spring Compression) v. Time
Initial Velocity of 30 ft/s. Various Initial Collision Force Offsets
0.000 0.025 0.050 0.075 0.100 0.125
Time (sec)
Also apparent from Figure 3.1 is the trend that the maximum deformation occurs
sooner as the initial collision force offset increases. At an initial collision force
offset of 0.0 feet, the central impact case, the maximum deformation occurred at
approximately 89 milliseconds. For the case with an initial collision force offset of
3.0 feet, the maximum deformation occurred at approximately 68 milliseconds.
Figure 3.2 shows similar deformation versus time plots for the simulations with an
initial vehicle velocity of 40 feet per second.
78


Figure 3.2 Deformation (Spring Compression) v. Time
Initial Velocity of 40 ft/s. Various Initial Collision Force Offsets
In this case, the maximum deformation for the case with zero initial collision force
offset was 2.28 feet and the maximum deformation for the case with 3.0 feet of
initial collision force offset was 1.77 feet, a 22.4% reduction in deformation due to
collision eccentricity. For zero initial collision force offset, the maximum
deformation again occurred at 89 milliseconds, reminding us that for central impact,
the time to maximum deformation is dependent only on the mass and stiffness
properties of the vehicle, not on the initial closing velocity.1 For an initial collision
force offset of 3.0 feet, the maximum deformation again occurred at 68
milliseconds, indicating that for eccentric collisions the time to maximum
deformation may again be independent of the initial closing velocity between the
vehicle and the barrier. Table 3.2 below summarizes the maximum deformation
obtained for each simulation run.
1 This assumes that the vehicle structural stiffness properties are not rate dependent. Varat and
Husher [4] examined barrier impact data at varying speeds and found that, for the vehicles they
examined, the vehicle structural response was not rate dependent.
79


Initial Velocity (ft/s) b, (ft) Maximum Deformation (ft)
30.0 0.0 1.71
0.5 1.69
1.0 1.64
1.5 1.58
2.0 1.50
3.0 1.35
40.0 0.0 2.28
0.5 2.25
1.0 2.17
1.5 2.08
2.0 1.98
3.0 1.77
Table 3.2 Maximum Deformations for Barrier Impact Simulations
Plots of deformation energy through time show trends that are similar and related to
the trends for the deformation. Deformation energy versus time plots, for the case of
an initial velocity of 30 feet per second, are shown in Figure 3.3.
Figure 33 Deformation Energy Variation with Time
Initial Velocity of 30 ft/s, Various Initial Collision Force Offsets
For an initial collision force offset of 0.0 feet, the maximum deformation energy
was approximately 42,100 foot-pounds, and for an initial collision force offset of
80


3.0 feet the maximum deformation energy was approximately 26,231 foot-pounds.
This represents a reduction in damage energy of 37.7% due to the effects of
collision eccentricity.
Figure 3.4 Deformation Energy Variation with Time
Initial Velocity of 40 ft/s, Various Initial Collision Force Offsets
Figure 3.4 displays plots of deformation energy versus time for the simulations with
an initial vehicle velocity of 40 feet per second. For an initial collision force offset
of 0.0 feet, the maximum deformation energy was approximately 74,844 foot-
pounds, and for an initial collision force offset of 3.0 feet the maximum deformation
energy was approximately 45,347 foot-pounds. This represents a reduction in
damage energy of 39.4% due to the effects of collision eccentricity. It should be
noted that the maximum deformation energy for the case with an initial velocity of
40 feet per second and an initial collision force offset of 3.0 feet is roughly
equivalent to the maximum deformation energy for the case with zero initial
collision force offset with an initial velocity of 30 feet per second. Thus, roughly a
44% increase in initial kinetic energy was necessary to produce the same
81


deformation energy between 0.0 feet of initial collision force offset and 3.0 feet of
initial collision force offset.
The maximum deformation energy for each simulation run is a part of the
calculation of AV using the CRASH 3 algorithm and so the maximum deformation
energy for each simulation is summarized in Table 3.3 below.
Initial bi Maximum Deformation
Velocity (ft/s) (ft) Energy (ft-lb)
30.0 0.0 42099.68
0.5 41182.83
1.0 38828.61
1.5 35748.18
2.0 32458.13
3.0 26231.32
40.0 0.0 74843.89
0.5 72841.97
1.0 68043.39
1.5 62189.48
2.0 56226.05
3.0 45347.49
Table 3.3 Maximum Deformation Energies for Barrier Impact Simulations
In order to address the fidelity of the CRASH 3 effective mass concept, two
additional values were extracted from the simulation runs. First, the resultant change
in velocity experienced by the body from time zero to the time of maximum
deformation was obtained. This is the value that was compared to the CRASH 3
calculated velocity change using the deformation energies given in Table 3.3.
Second, appropriate values for the resultant collision force moment arm were
calculated, since this value is needed to calculate the effective mass factors, y, which
are also used in the calculation of the velocity change with the CRASH 3 algorithm
equations.
82


The first of these, the resultant AV experienced by the vehicle in each simulation
run, is easily obtained from the simulation data. The results are summarized in
Table 3.4 below.
Initial Velocity (ft/s) b, (ft) AV resultant (ft/s)
30.0 0.0 30.00
0.5 29.09
1.0 26.95
1.5 24.39
2.0 21.90
3.0 17.58
40.0 0.0 40.10
0.5 38.35
1.0 34.95
1.5 31.28
2.0 27.91
3.0 22.36
Table 3.4 Actual Velocity Changes for Barrier Impact Simulations
The second, obtaining a representative value for the collision force moment arm is
not as straight forward, since it is not immediately clear what value of the moment
arm is the most representative. A number of possibilities exist for defining this
average moment arm, including the following:
1. The first possibility is to let havg equal the initial offset of the collision force
(bi). We would expect this to be the least accurate, since the initial offset is
the least representative of the moment arm throughout the collision. This
moment arm occurs at the beginning of the impact, when the collision force
is low.
2. The second possibility is to let havg equal the arithmetic mean value of the
instantaneous moment arms (the moment arm at each time step) between
83


time zero and maximum spring compression. This average moment arm is
given by
N
(3.24)
where N is the number of time steps between time zero and the maximum
spring compression. Again, we would not expect this havg to be
representative since it gives equal weight to the h value at every time step.
We suspect that h values associated with higher forces should be more
heavily weighted.
3. Thus, the third possibility is to let havg equal a weighted average of the
instantaneous moment arms between time zero and maximum spring
compression. The weighting could utilize either the instantaneous resultant
collision forces or the instantaneous resultant accelerations. This average
moment arm is given by
where N is again the number of time steps between time zero and the
maximum spring compression and a* is the acceleration at each time step.
N
(3.25)
84


4. The fourth possibility is to let havg equal the collision force moment arm at
the time the maximum force (maximum deformation) is achieved.
Ultimately, the correct choice between these four options will be the moment arm
concept that reduces the error in the AV predicted by the CRASH 3 algorithm. For
the barrier impact simulations, the moment arm at any instant in time is defined as
the perpendicular distance between the collision force line of action and the vehicle
center of gravity. The instantaneous moment arm is given by equation (3.26).
^ = -Jaf+bf sin(^, + ax -/?2) (3.26)
For the simulations with an initial vehicle velocity of 40 feet per second,
calculations were carried out with all four of these definitions for the average
collision force moment arm. For each of these 40 feet per second simulations, the
resulting average moment arms are summarized below, in Table 3.5, along with the
effective mass factors that these moment arms produce.
Definition 1 ^avg bj (ft) y Definition 2 havg hmean (ft) y Definition 3 havg ~ hweighted (ft) y Definition 4 havg h(Fmax) (ft) y
0.0 1.00 0.00 1.00 0.00 1.00 0.00 1.00
0.5 0.99 0.64 0.98 0.71 0.97 1.12 0.94
1.0 0.95 1.24 0.92 1.35 0.91 2.00 0.82
1.5 0.89 1.79 0.85 1.92 0.84 2.69 0.72
2.0 0.82 2.31 0.78 2.46 0.76 3.26 0.64
3.0 0.68 3.32 0.63 3.46 0.61 4.23 0.51
Table 3.5 Moment Arms and Effective Mass Factors for 40ft/s Barrier Impact Simulations
For the barrier impact case, M2 approaches infinity and E2 approaches zero and the
CRASH 3 damage algorithm equations for AV, equations (1.57) and (1.58), reduce
to the following equations:
85


(3.27)
AV>
ty\E\
Mx
AV2=0
(3.28)
Using equations (3.24), (3.25), (3.26) and the data in Tables 3.1, 3.3, and 3.5, the
change in velocity (AV) for each simulation can be calculated for each of the
moment arm definitions. These calculated velocity changes are then compared to the
actual AVs in Table 3.4. Table 3.6 below summarizes the AV results obtained from
these calculations along with the percent difference between the calculated AV and
the actual AV. Again, this is for the simulations with an initial vehicle velocity of 40
feet per second.
Definition 1 AV (fl/s) % Difference Definition 2 AV (fl/s) % Difference Definition 3 AV (fl/s) % Difference Definition 4 AV (fl/s) % Difference
40.08 -0.04 40.08 -0.04 40.08 -0.04 40.08 -0.04
39.28 2.44 39.11 1.99 39.02 1.76 38.29 -0.16
37.24 6.55 36.74 5.13 36.49 4.40 34.70 -0.72
34.53 10.39 33.77 7.96 33.40 6.77 31.04 -0.75
31.54 13.03 30.65 9.81 30.22 8.30 27.76 -0.52
25.65 14.71 24.77 10.76 24.38 9.04 22.33 -0.16
Table 3.6 AV Results for Different Average Moment Arm Definitions
Clearly, the collision force moment arm at maximum force (max deformation) is the
most representative moment arm, yielding accuracy within one percent of the actual
value for AV. The significance of these results will be discussed in section 3.5
below. For analysis of the remaining simulations, barrier impact simulations at 30
feet per second and all of the vehicle-to-vehicle simulations, only the moment arm
of the collision force at maximum force was used for the change in velocity
calculations.
86


Table 3.7 below summarizes the results for the barrier impact simulations at 30 feet
per second, including the moment arm at maximum force, the effective mass factor
that results from this moment arm, the calculated change in velocity, and the percent
difference from the actual change in velocity.
havg h(F max) (ft) Y Calculated AV (ft/s) % Difference
0.00 1.00 30.06 0.21
0.91 0.96 29.10 0.02
1.70 0.87 26.88 -0.26
2.36 0.77 24.33 -0.25
2.94 0.68 21.84 -0.26
3.94 0.55 17.56 -0.12
Table 3.7 Moment Arm and AV Results for 30ft/s Barrier Impact Simulations
Again, the calculated values for AV deviate from the actual values by less than one
percent, and the effective mass concept appears to be performing well.
3.4 Simulation Results for Eccentric
Vehicle-to-Vehicle Impacts
For the car-to-car case, two sets of simulations were run. The first set was with
initial x-direction velocities of 30 feet per second for both vehicles one in the
positive and one in the negative x-direction. The second set was with an initial x-
direction velocity of 30 feet per second for the first vehicle and negative 45 feet per
second for the second vehicle. For each of these sets, simulations were run with
initial collision force offsets of 0.0, 0.25, 0.5, 0.75,1.0, and 2.0. For these
simulations, the vehicle inertial and stiffness parameters were set as follows:
87


Vehicle Weights 3000 lbs.
Principle Moments of Inertia 1750 lb-ft-sec"
Vehicle Stiffnesses per Unit Width 100 lb/in2
Damage Widths 24 inches
Table 3.8 Inertial and Stiffness Parameters For Car-to-Car Simulations
For simplicity, the two vehicles were assumed to have identical inertial and stiffness
properties. To test how this simplifying assumption would affect the results
obtained, one additional simulation was run with the following modified parameters
for the second vehicle:
Vehicle Weight 2000 lbs.
Principle Moments of Inertia 1500 lb-ft-sec2
Vehicle Stiffnesses per Unit Width 75 lb/in2
Damage Widths 24 inches
Table 3.9 Inertial and Stiffness Parameters For Vehicle 2, Final Car-to-Car Simulation
This simulation run was run with initial vehicle speeds of 30 feet per second and
negative 45 feet per second and with an initial collision force offset of 3.0 feet. For
this simulation, the system showed a maximum change in system energy of 1.08
percent. The maximum deformation energies for each car-to-car simulation run are
summarized in Table 3.10 below and the resultant velocity changes for each
simulation are summarized in Table 3.11.
In Table 3.10, it should be noted that regardless of the vehicle impact speeds, if the
vehicle stiffnesses are equal, then the damage energy is equally split between the
vehicles. It is not until the last simulation, where the vehicle stiffnesses were
different, that the damage energies were different.
88


Vehicle 1 Initial Velocity (fl/s) Vehicle 2 Initial Velocity (fl/s) bi (ft) b2 (ft) Vehicle 1 Max Deformation Energy (ft-lb) Vehicle 1 Max Deformation Energy (ft-lb)
30 -30 0.00 0.00 42099.58 42099.58
0.25 0.25 41862.46 41862.46
0.50 0.50 41182.68 41182.68
0.75 0.75 40139.20 40139.20
1.00 1.00 38828.41 38828.41
2.00 2.00 32458.00 32458.00
30 -45 0.00 0.00 65780.59 65780.59
0.25 0.25 65343.59 65343.59
0.50 0.50 64113.03 64113.03
0.75 0.75 62276.35 62276.35
1.00 1.00 60036.86 60036.86
2.00 2.00 49748.70 49748.70
3.00 3.00 29237.69 38983.59
Table 3.10 Maximum Deformation Energies for Car-to-Car Simulations
Vehicle 1 Initial Velocity (fl/s) Vehicle 2 Initial Velocity (fl/s) b]and b2 (ft) AN l .resultant (fl/s) AN 2,resultant (fl/s)
30 -30 0.00 30.03 30.03
0.25 29.76 29.76
0.50 29.13 29.13
0.75 28.15 28.15
1.00 26.98 26.98
2.00 21.94 21.94
30 -45 0.00 37.54 37.54
0.25 37.20 37.20
0.50 36.10 36.10
0.75 34.66 34.66
1.00 33.02 33.02
2.00 26.49 26.49
3.00 18.23 27.36
Table 3.11 Resultant Velocity Changes for Car-to-Car Simulations
Tables 3.12 and 3.13 summarize the results for all of the car-to-car simulations,
including the moment arm at maximum force, the effective mass factor that results
from this moment arm, the calculated change in velocity, and the percent difference
from the actual change in velocity.
89


Vehicle 1 Initial Velocity (ft/s) Vehicle 2 Initial Velocity (ft/s) b] &b2 (ft) havg, 1 hi(F max) (ft) 7i ftavg.1 hi(Fmax) (ft) 72
30 -30 0.00 0.00 1.00 0.00 1.00
0.25 0.46 0.99 0.46 0.99
0.50 0.91 0.96 0.91 0.96
0.75 1.32 0.92 1.32 0.92
1.00 1.70 0.87 1.70 0.87
2.00 2.94 0.68 2.94 0.68
30 -45 0.00 0.00 1.00 0.00 1.00
0.25 0.55 0.98 0.55 0.98
0.50 1.06 0.94 1.06 0.94
0.75 1.52 0.89 1.52 0.89
1.00 1.92 0.84 1.92 0.84
2.00 3.18 0.65 3.18 0.65
3.00 4.05 0.53 4.12 0.59
Table 3.12 Moment Arm and Effective Mass Factors for Car-to-Car Simulations
Vehicle 1 Initial Velocity (ft/s) Vehicle 2 Initial Velocity (ft/s) bi & b2 (ft) Calculated AV, (ft/s) % Difference Calculated av2 (ft/s) % Difference
30 -30 0.00 30.06 0.11 30.06 0.11
0.25 29.81 0.17 29.81 0.17
0.50 29.10 -0.12 29.10 -0.12
0.75 28.08 -0.23 28.08 -0.23
1.00 26.88 -0.37 26.88 -0.37
2.00 21.84 -0.44 21.84 -0.44
30 -45 0.00 37.58 0.10 37.58 0.10
0.25 37.15 -0.13 37.15 -0.13
0.50 36.04 -0.17 36.04 -0.17
0.75 34.50 -0.46 34.50 -0.46
1.00 32.82 -0.59 32.82 -0.59
2.00 26.35 -0.55 26.35 -0.55
3.00 18.19 -0.26 27.28 -0.30
Table 3.13 Calculated AVs and Percent Difference from Actual
Thus, when used with the moment arm corresponding to the maximum collision
force, the effective mass concept performs well for the vehicle-to-vehicle impact
model, with the calculated velocity changes deviating from the actual values by less
than one percent.
90