Publications 2001-01-3347 Monte Carlo Momentum

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SAE TECHNICAL

PAPER SERIES 2001-01-3347

Integrating Monte Carlo Simulation, Momentum-

Based Impact Modeling, and Restitution

Data to Analyze Crash Severity

Nathan A. Rose, Stephen J. Fenton and Christopher M. Hughes

Knott Laboratory, Inc

Reprinted From: ATTCE 2001 ProceedingsVolume 1: Safety

(P-367)

Automotive & Transportation TechnologyCongress & Exhibition

October 1- 3, 2001Barcelona, Spain


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2001-01-3347

Integrating Monte Carlo Simulation, Momentum-Based Impact

Modeling, and Restitution Data to Analyze Crash Severity

Nathan A. Rose, Stephen J. Fenton and Christopher M. Hughe

Knott Laboratory, In

ABSTRACT

Crash severity is quantified by the change in velocityexperienced by a vehicle during an impact along with thetime duration over which that change in velocity occurs.Since the values of the input parameters for calculating

the change in velocity are not known exactly, there isuncertainty associated with the calculated change invelocity. Accurate evaluation of the crash severity will,therefore, include analysis of the effect that uncertaintiesin the values of the input parameters have on thecalculated change in velocity. Monte Carlo simulation, astatistical technique, enables the reconstructionist toevaluate the effect of uncertainty on the analysis of crashseverity.

Use of the Monte Carlo simulation technique is beneficialsince a reconstructionist can enter a range of values foreach input parameter. A probability distribution can be

assigned to the range of values, which indicates thelikelihood that any value in that range corresponds to theactual value of the parameter. The simulation generatesthousands of possible combinations of the inputparameters selected from the specified ranges, monitorsthe results of the combinations and analyzes themstatistically. Application of the Monte Carlo technique isintended to improve the legitimacy of crash severityanalysis by helping the reconstructionist consider a widerange of possible solutions within the bounds of theimperfect data and report statistically meaningful rangesfor the change in velocity.

This paper demonstrates the application of the MonteCarlo technique to impact severity analysis using aderived two-dimensional, rigid body, momentum-basedimpact model. Thorough guidance is given to aid thereconstructionist in integrating the momentum modelwith the Monte Carlo simulation technique and thismethod is illustrated with a case study. Since the impactmodel employs restitution constraints in the normal andtangential directions, the effect of uncertainty informulating appropriate ranges for the values of therestitution coefficients is discussed.

NOTATION

m massI moment of inertiaV velocity

angular velocity

V change in velocity

change in angular velocityP impulsea X-coordinate of C.G.b Y-coordinate of C.G.c tangential coordinate of C.G.d normal coordinate of C.G.e coefficient of restitution

orientation of the slip plane

Subscripts

1, 2 vehicle numbersx X-directiony Y-directiont tangential directionn normal directioni pre-impactf post-impact

INTRODUCTION

Crash severity, quantified by the change in velocityexperienced by each vehicle during the impact

1, is a key

parameter for assessing occupant injuries in motor

vehicle accidents [10, 18, 25, 33]. The interest olitigators in knowing the potential for occupant injuriesmakes the accurate assessment of crash severityessential. However, the data necessary to evaluate thechange in velocity experienced by each vehicle is no

1 Strictly speaking, crash severity is quantified by the change in

velocity experienced by the vehicle during impact along with the timeduration over which that change in velocity occurs [15]. In thediscussion that follows, crash severity is used interchangeably withchange in velocity. This is only to improve readability, and comes withthe recognition that the time duration of the change in velocity is alsoan important parameter in assessing the crash severity.

Copyright 2001 SAE International and Messe Dsseldorf.


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known with complete certainty [5, 19, 20, 21, 24, 28, 29,30, 31, 32]. The location of the point of impact and thepoints of rest, the orientation of the vehicle velocityvectors before and after impact, and the location wherethe resultant collision force is exchanged cannot beestimated or measured perfectly. Estimation ofreasonable values for the coefficients of restitution atvarying levels of crash severity and varying impactconfigurations is also fraught with uncertainty.

Coefficients of friction are typically unmeasured andmust be estimated. Even when they are measured, therewill remain questions about the fidelity of themeasurement in relation to the conditions at the time ofthe crash. This uncertainty associated with the inputparameters results in subsequent uncertainty that isassociated with the calculated change in velocity.

Monte Carlo simulation provides a statistical analysistechnique to analyze the propagation of uncertainty fromthe input parameters to the final result, leading tostatistically relevant conclusions regarding the probable

V experienced by a vehicle during an impact. The

Monte Carlo technique allows a range of values to bespecified for each input parameter of the impact modelreflective of the level of certainty associated with thatparameter. A probability distribution, that indicates thelikelihood that any value in that range corresponds to theactual value, is attached to each of these ranges, andthe simulation then generates possible combinations ofthese parameters based on the ranges and distributions.The results of these combinations are tracked andanalyzed statistically. The final result is a probabilitydistribution that expresses the likelihood that any value ofthe change in velocity corresponds to the actual value.

While application of Monte Carlo simulation to certainaspects of crash reconstruction has been addressed inthe literature [19, 20, 30], previous papers haveaddressed trivial applications. This paper presentsthorough and systematic guidance for the application ofMonte Carlo simulation to the non-trivial case of impactanalysis using a momentum-based rigid body impactmodel. This impact model employs restitution constraintsin the normal and tangential directions, so the uncertaintyinherent in identifying the normal and tangentialdirections is discussed. Guidance is given for theformulation of reasonable ranges for the coefficients ofrestitution.

ANALYTICAL UNCERTAINTY ANALYSIS

Analysis of crash severity should include analysis of theeffect that uncertainty has on the calculated change invelocity. Several methods are available to analyzeuncertainty in a dependent variable based onuncertainties in the independent variables. For theanalysis in this paper, the dependant variable is thechange in velocity and the independent variables consistof the coefficients of friction and restitution, post-impact

travel distances, approach and departure angles, andvehicle masses.

The most common method of uncertainty analysis is todetermine the upper and lower bounds of the dependentvariable based on the lowest and highest possible valuesof each input parameter [5, 20]. This method has theadvantage of being straightforward, even with non-lineaequations. However, this upper and lower bound methoddoes not result in information regarding the likelihoodthat any particular value within the obtained range for thechange in velocity corresponds to the actual value. Nobasis is provided to conclude that crash severity valuesnear the mean of the output range are any more likely tocorrespond to the actual value than those at theextremes. Further, the probability that the values of all ofthe independent variables fell at the extremes of theseranges in the actual crash is small. Therefore, the rangegenerated for the dependent variable the change invelocity will be unrealistic and wider than necessary.

Another method of uncertainty analysis uses differentia

calculus to relate variations in the independent variablesto the resulting variation in the dependent variable. Brach[5] and Tubergen [29] have expounded this methodwithin the context of crash reconstruction and thatdiscussion will not be repeated here. Suffice it to say thain the impact model described below, the linear andangular velocities are functions of ten independenvariables, the values of which are not known withcomplete certainty. Analysis of the uncertainty using thismethod will, therefore, include calculating partiaderivatives of the velocity equations with respect to eachof these ten independent variables. Further, analysis ouncertainty by this method should be limited to cases

where the variations are small, since the methodapproximates variations by linearizing the equationsaround some nominal value. And finally, as in the firstmethod, this method gives no basis to draw statisticaconclusions regarding individual values within the rangeof values obtained for the change in velocity.

An additional analytical method of uncertainty analysisaccounts for the statistical nature of the rangesformulated for the independent parameters and allowsfor statistical conclusions to be drawn regarding thevalue of the dependent variable. A probability distributionthat indicates the likelihood that any value in a range islikely to correspond to the actual value is specified for

each independent variable. This probability distributionwill be accounted for in analyzing the uncertaintyassociated with the final result. Statistical judgementscan then be made regarding which values are most likelyto occur. Brach [5] has detailed this method for a limitednumber of simple cases. However, this method becomesimpractical for more complex sets of equations, such asthe momentum equations considered below.


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MONTE CARLO SIMULATION

Monte Carlo simulation provides a method of uncertaintyanalysis that produces statistical data similar to thatproduced by the analytical statistical method above,while avoiding the limitations of that method for complexequations. The Monte Carlo technique is an approximatemethod of considering the effect of variations in theindependent variables on the uncertainty associated withthe dependent variable. The Monte Carlo techniqueutilizes the power of a personal computer to accomplishthousands of repeated calculations of the dependentvariable. These calculations are carried out withrandomly selected values for the independent variableswithin the confines of ranges and probability distributionsassigned to those ranges. The results are tracked andanalyzed statistically. Conclusions relating to the likelychange in velocity experienced by vehicle during a crashcan then be drawn in a statistically legitimate manner.

The Monte Carlo analysis performed below uses acommercially available Monte Carlo simulator called

Crystal Ball that works in tandem with Microsoft

Excel. To perform the analysis, the equations for theimpact model are configured in Excel. A range of valuesis then assigned to each independent parameter called

assumptions in Crystal Ball into which the actualvalue of that parameter should fall. Each range is thenassigned a probability distribution that indicates thelikelihood that any value in that range corresponds to theactual value of that parameter. Finally, the simulationruns a set number of calculations of the dependent

variable, called the forecast in Crystal Ball, with values

for each parameter chosen randomly within the confinesof the assigned ranges and probability distributions.

Results of these calculations are monitored and the finalresult is a probability distribution that indicates thelikelihood that any value in the resulting rangecorresponds to the actual value of the dependentvariable (the change in velocity). Figure 1 is an exampleof the output probability distribution obtained for avehicles change in velocity from the case study below.

FIGURE 1

CHOOSING PROBABILITY DISTRIBUTIONS

When choosing probability distributions for each

independent parameter, Crystal Ballallows the user toselect from a number of distributions including thosedepicted in Figure 2 below. Those distributions that mighbe applied within the context of crash reconstructioninclude the uniform, normal, and custom distributions.

FIGURE 2

Selection of a uniform distribution implies that any valuein the range is equally likely to occur. This distributionrepresents the most conservative assumption since thevalue of the parameter is allowed to vary more freelywithin the range than with any other distribution. Theresult is the widest possible variation in the outputparameter [30]. The uniform distribution should beselected for most independent parameters in crashseverity analysis since data that would justify choosingany other distribution is typically not available.

For instance, when a range and probability distribution isbeing assigned to the post-impact travel distance of avehicle, this range will often be based on a singlemeasurement of that distance. While we might expectrandom errors in measurement of the post-impact travedistance to vary normally, one measurement isinsufficient to tell us the mean and standard deviationproduced by those random errors. Further, uncertaintyassociated with the measurements of post-impact travedistances may go beyond random errors and includeerrors associated with identifying physical evidenceThere may be uncertainty associated with the exaclocation where tire marks terminate, for example. Using

a normal distribution with this independent variable wouldbe unjustifiable with the provided data.

An exception to the use of a uniform distribution is focoefficients of friction. Goudie, et al presented data from540 skid-to-stop tests under both wet and dry roadconditions and found that random variations in the valuesof the coefficients of friction approximated a normadistribution [13]. Still, the use of a normal distribution todescribe variations in the coefficient of friction for anyparticular reconstruction may be problematic. WhileGoudie tested three types of tires, his data is still for a


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single surface and a single vehicle. Random variations inthe coefficient of friction might be expected to exhibit anormal distribution for any vehicle-tire-surfacecombination. However, each vehicle-tire-surfacecombination would have a unique mean value and aunique standard deviation. In any particular crashreconstruction, the mean value for that particular vehicle-tire-surface combination would not be known. Anynumber of normal distributions would be possible (Figure

3). Even extensive post-accident testing at the site wouldnot necessarily yield the proper mean value andstandard deviation since that testing would likely involvea different vehicle with different tires and suspensionscharacteristics. Furthermore, the generation of astatistically significant amount of data for a particularreconstruction is typically not feasible.

FIGURE 3

We could perhaps place bounds on the possible meanvalues and standard deviations and construct a customdistribution for variations in the coefficient of friction. Forexample, given some roadway surface, an experiencedreconstructionist may be able to conclude that the meanvalue will likely occur between 0.7 and 0.85 (Figure 4).

Using the custom distribution feature in Crystal Ball

,the range between 0.7 and 0.85 could be assigned auniform distribution. The distribution could then tail offlinearly on either side to the extremes of the ranges(Figure 4). This distribution displayed in Figure 4 by thedashed line is wide enough to encompass the breadth ofvalues in the literature. Our intent is not to defend thisrange of values, but only the general shape of thedistribution. A tighter range of friction coefficients couldbe specified if there is justification for that tighter range,such as consideration of the actual roadway surfaceinvolved in the crash.

This distribution shape in Figure 4 recognizes that for agiven vehicle-tire-surface combination, variations in thecoefficient of friction are likely to exhibit a normaldistribution, but that in any particular reconstruction themean value of that normal distribution would beunknown. At the same time, this distribution recognizesthat limits can be placed on possible mean values, andvalues outside of these possible means need not beconsidered as likely as those within the range of possiblemeans. This distribution also allows for a wide range ofstandard deviations.

FIGURE 4

UNREALISTIC RESULTS

Some combinations of the independent variablesgenerated by the Monte Carlo simulator may produceunrealistic results, and the output of the Monte Carlosimulation should be filtered to eliminate thesecombinations [30]. Mathematically, these unrealisticresults occur because the equations of the impact modeonly constrain the output with respect to the independenvariables that are actually contained in the equations of

the impact model. For instance, there is no constraint inthe equations of the impact model placed on the lateraacceleration that a vehicle can achieve. If a crash occurswhile one vehicle is making a left turn, the analyst maybe able to place limits on the speed of that vehicle basedon the radius of the turn and the maximum lateraacceleration that the vehicle could achieve. The outpudata of the Monte Carlo simulation would have to befiltered based on this limit since the equations of theimpact model contain no inherent constraint that wouldlimit the speed of that vehicle based on maximum lateraacceleration.

The coefficients of restitution provide another criterion bywhich the result of each combination can be judged. Theimpact model formulated below does not require therestitution coefficients to be estimated in order to obtaina solution, and therefore the results are not constraintedwith respect to reasonable restitution values. Output fromthe Monte Carlo simulation should be filtered to includeonly combinations that produce reasonable restitutionvalues. The formulation of reasonable restitutioncoefficients is discussed below.

The most recent release of the Crystal Ball softwareallows for automatic filtering of the results of the Monte

Carlo simulation. The users manual should be consultedfor specific instruction for accomplishing this filtering.

WHAT DO THE RESULTS MEAN?

Application of the Monte Carlo technique is intended toimprove the legitimacy of crash severity analysis byenabling the reconstructionist to consider thousands ofpossible solutions within the bounds of the imperfecdata that is available. Legitimacy is not inherent inapplication of the Monte Carlo technique, though, sincethe Monte Carlo technique cannot establish the fidelity of


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the impact model in any particular case. Meaningfulapplication of the Monte Carlo technique must comeafter the assumptions of the impact model have beenadequately satisfied.

For example, a typical momentum-based impact modelassumes that tire forces can be neglected during theimpact. If a model with this assumption is applied toanalyze an impact of sufficient duration to makemomentum losses from tire forces significant, the error inthe solution induced by these momentum lossesundermines the fidelity of the model for that accident. Noamount of uncertainty analysis will solve this problem. Amore complex impact model is required that will accountfor the tire forces that are neglected in the first model.Monte Carlo simulation could then be applied with theimproved impact model.

Once the validity of the impact model has beenestablished, the validity of the ranges and probabilitydistributions assigned to the input parameters mustlikewise be established. In the same way that an

improperly applied impact model undermines theeffectiveness of the Monte Carlo technique, so do poorlychosen ranges for the input parameters. Test data fromthe literature should be used, the reliability of theavailable data should be assessed, and the range foreach parameter should be sufficiently wide toencompass the full range of possible values. Once theseconditions are met, then the results obtained from MonteCarlo simulation can be considered to give usmeaningful information about the likely value of thechange in velocity.

The results of the Monte Carlo analysis may be used in

various ways depending on the issues relevant to theparticular crash reconstruction. When the issue is crashseverity, a 51-percentile range, indicating the range ofvalues that are more probable than not, may be reported.When the impact speeds of the vehicles are also atissue, the 51-percentile range can likewise be reported[18]. If the overall distribution obtained straddles thespeed limit, the reconstructionist may report the percentprobability that the vehicle was exceeding the speedlimit.

THE IMPACT MODEL

The two-dimensional impact model employed fordemonstration of the Monte Carlo simulation techniquemakes use of the principle of impulse and momentum [6]and is derived with the following assumptions typical ofmomentum-based impact models [4, 7, 10, 16, 26, 27,31, 32]:

1. Tire forces and other external forces are

assumed to be negligible compared to the

collision forces. This assumption allows for theprinciple of conservation of linear momentum to be

applied and is generally considered valid for theanalysis of vehicular crashes. However, instancesexist when the duration of the impact is sufficientlylong to make external forces significant. Fonda [11has detailed a method for impact analysis thaincludes the momentum losses that result fromexternal forces. Fondas method can be appliedwhen the impact duration is too long to ignoreexternal forces that cause significant momentum

losses.

2. The resultant impulse applied to each vehicle is

concentrated at the impact center. A number opapers have discussed the identification of theimpact center [9, 14, 16, 31]. For the sake of brevitythis discussion will not be expanded here. Suffice itto say that the impact center is the point in the crushzone with a moment arm such that on the averagethe cross product of that moment arm with theimpulse produces the correct rotational impulse[31]. There is uncertainty associated with theidentification of the impact center that should be

accounted for in the analysis of crash severityFurther research should be done to explore theeffect of variations in the location of the impaccenter on the calculated change in velocity.

3. The interaction between the vehicles is assumed

to occur instantaneously.The collision forces aretransferred instantaneously and the pre-impact andpost-impact positions of the vehicles are assumed tocoincide. This instantaneous transfer of the collisionimpulse is assumed to occur when the maximumcrush to each vehicle has been reached. The validityof this assumption depends on the actual impac

duration. In a real crash the force is not transferredinstantaneously and movement of the vehicles doesoccur. The longer the impact duration, the moremovement there will be and the less valid thisassumption is.

4. Each vehicle is treated as a rigid body with two

translational degrees-of-freedom and one

rotational degree-of-freedom. Yaw rotations of thevehicles are considered, but roll and pitch rotationsare neglected. Vertical motion is also neglected.

5. Collision forces are treated as two-dimensiona

and assumed to act in the same plane in whichthe vehicles move.

6. The mass, center of gravity, and yaw moment of

inertia for each vehicle are assumed unchanged

by the impact.

The impact model employs a fixed X-Y coordinatesystem located, for convenience, at the impact center, asshown in Figure 5.


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FIGURE 5

The location and orientation of this coordinate systemare arbitrary. After the orientation of the fixed coordinateframe has been specified, the direction tangential to theimpact surface is identified in relationship to the X-axis.The normal direction is by definition orthogonal to thetangential direction, or normal to the impact surface. Thenormal and tangential directions form a coordinate frameuseful for setting up constraint equations. The orientation

of the tangential direction () along with the approach anddeparture angles of the vehicle velocities are measuredcounterclockwise from the positive X-axis.

Defining an arbitrarily oriented fixed coordinate systemseparate from the normal and tangential directionsallows the orientation of the tangential direction to beincluded as an angular parameter in the impact analysis.This allows the uncertainty inherent in the identification ofthe normal and tangential directions to be considered inthe analysis of crash severity.

Formulation of the impact equations begins withapplication of the principle of conservation of linearmomentum [6] in the X and Y directions. The followingequations are obtained:

The change in angular momentum of each vehicle iscalculated by application of the principle of impulse andmomentum. The following equations are obtained:

Pxand Pyare the components of the impulse applied toVehicle #1 in the X and Y directions, respectively. Anequal and opposite impulse is applied to Vehicle #2Since the impulse applied to each vehicle is equal to thechange in linear momentum experienced by that vehiclewe obtain the following equations:

Substitution of equations (5) and (6) into (3) and (4)yields four equations describing the impact. However, ageneral two-dimensional impact model includes 3degrees of freedom for each vehicle, therefore, sixequations are required to completely describe theimpact. Two constraint equations relating thecomponents of the closing speed to the components of

the separation speed at the impact center provide theadditional two equations necessary for a completeimpact model.

This impact model is similar to that derived by Ishikawa[16] in that restitution constraints in the normal andtangential directions provide the two additional equationsnecessary for a complete model. The restitutionequations are as follows:

where c1 and c2 are the tangential coordinates of thecenters of gravity of vehicles #1 and #2, respectivelyand d1and d2are the normal coordinates of the centersof gravity of vehicles #1 and #2, respectively. Thenumerators of equations (7) and (8) are the normal andtangential components of the separation speed. Thedenominators are the normal and tangential componentsof the closing speed. Transformation of restitutionequations (7) and (8) to the X-Y frame result in thefollowing equations:

where,

VmVmVmVm xfxfxixi 22112211 +=+ 1

VmVmVmVm yfyfyiyi 22112211 +=+ 2

) bPaP!!I xyif 11111 += (3)

) bPaP!!I xyif 22222 += (4)

) VVmVVmP xfxixixfx 222111 == 5

VVmVVmP yfyiyiyfy 222111 == (6)

( ) ( )

cossin

cossin

,,

,,

VVVV

eyclosexclose

ysepxsep

n +

+=

9

itiiti

ftfftf

tclose

tsep

t dVdV

dVdV

V

V

e 222111

111222

,

,

+

+==

8

( ) ( )

sincos

sincos

,,

,,

VVVV

eyclosexclose

ysepxsep

t +

+= 10

(7)

iniini

fnffnf

nclose

nsep

n

cVcVcVcV

VV

e222111

111222

,

,

+

+==

fxffxfxsep bVbVV 222111, += 11


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In these equations a1and a

2are the X-coordinates of the

centers of gravity of vehicles #1 and #2, respectively,and b1 and b2 are the Y-coordinates of the centers ofgravity of vehicles #1 and #2, respectively. Equations (1)through (4), in conjunction with equations (5) and (6),and equations (9) and (10) now provide a completedescription of the impact.

The values of the normal and tangential restitutioncoefficients are allowed to vary between values of -1 and1. Ishikawa introduced the idea of negative restitutionwithin the context of accident reconstruction [16]. Theresult of allowing restitution to fall below 0 is that theimpact model does not require a common velocity to be

achieved during the impact. This is advantageous sincethe impact model can be used to analyze sideswipe orbreak-through collisions where a common velocity isnever achieved. However, once the values of therestitution coefficients drop below 0, the classicaldefinition of the restitution constraints as a ratio ofdeformation and restitution impulses has beenabandoned.

If the orientation of the vehicle velocities before and afterimpact can be estimated, the number of unknowns isreduced to four and equations (1) through (4), with (5)and (6), are sufficient to produce a solution. Writing

equations (1) through (4) with approach and departureangles yields equations (15) through (18).

Application of the impact model demonstrated belowassumes that approach and departure angles can beestimated and equations (15) through (18) are utilized inthe Excel spreadsheet for the Monte Carlo analysis. Thecoefficients of restitution become unnecessary togenerate a solution, but they are used to monitorpossible solutions for reasonableness. The issue of

reasonable and unreasonable solutions is discussedbelow.

Finally, equations for the change in velocity experiencedby each vehicle during the impact are obtained by vectorsubtraction of the final velocities from the initiavelocities.

where,

The principle direction of force (PDOF) is calculatedgeometrically by the following equation:

RESTITUTION THEORETICAL ASPECTS

Traditionally, the constraint equations for twodimensional rigid body impact models have beenformulated by defining a coefficient of restitution in the

normal direction and an equivalent friction coefficient inthe tangential direction [4]. In classical mechanics, thenormal coefficient of restitution is defined at the impaccenter as the ratio of the normal impulses during therestitution and deformation phases [6].

Since the impulses during these phases are equal to thechange in momentum of each vehicle during that phase

substitution, algebraic manipulation, and consideration othe two-body system renders the familiar form of thenormal coefficient of restitution [6].

Definition of the coefficient of restitution as the ratio othe restitution and deformation impulses requires that the

fyffyfysep aVaVV 111222, += 12

)( ixiixixclose bVbVV 111222, += 13

) iyiiyiyclose aVaVV 222111, += (14)

( )

211

22222111

1 sin

sinsin

=

mVmVm

V ff

i (15)

( ) 2

22111111

22

2coscoscos

cos

1VmVmVm

mV ffii ++= (16)

( ) ( ) iiff abVIm

abVIm

111111

1

1

11111

1

1

1sincossincos += (17)

( ) ( ) iiff abVIm

abVIm

222222

2

2

22222

2

2

2sincossincos += (18)

22 VVV jyjxj += (19)

jjx

jy

Vj

VV

PDOF

+=

1tan180 (23)

jjijjfjx VVV coscos = (20)

jjijjfjy VVV sinsin = (21)

(24)dtF

dtF

impulsendeformationormal

impulsenrestitutionormal

Dn

Rn

ne

==

(25)VVVV

enini

nfnf

n

22

12

=

(22)2,1=j


10/17


11/17


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collision can only be determined by performing tests withthe actual vehicles [18]. Their tests exhibited restitutionvalues between 0.25 and 0.75 and they found noticeabledifferences between the different bumper types.

TANGENTIAL RESTITUTION

Estimating values for the coefficient of restitution in thetangential direction proves more problematic thanestimating corresponding values in the normal directions.Data from staged collisions is limited. The primary dataavailable for estimating coefficients of restitution in thetangential direction (and equivalent friction coefficients)again comes from Ishikawas staged collision data [16,17].

Side Impact Data - Figure 8 depicts tangential restitutionvalues reported by Ishikawa for 32 side impact testsplotted against tangential closing speed.

This group of data shows the greatest potential fordisplaying a general relationship between crash severity

and restitution. There is a trend of decreasing tangentialrestitution with increasing tangential closing speed. Thistrend makes intuitive sense in light of the physicalinterpretation of the tangential coefficient of restitution. Anegative restitution coefficient implies that a commonvelocity was never reached in the tangential direction.We would expect that the higher the closing speed in thetangential direction, the harder it would become forstructural engagement to cause a common velocity to bereached. Thus, in general, at lower closing speeds theengagement of the vehicle structures seem morecapable of bringing the vehicles to a common velocity inthe tangential direction.

FIGURE 8

Still, there is considerable variance in the relationshipbetween tangential closing speed and tangentialrestitution, indicating that there are other importantfactors. Intuitively, we would expect the normal closingspeed to affect tangential restitution values, as well,since normal closing speed effects the extent ofstructural engagement. A higher closing speed in thenormal direction would generally yield more significant

structural engagement and therefore a greater ability bythe vehicle structure to produce a common velocity in thetangential direction.

This intuition is confirmed by Ishikawas staged collisiondata. Figure 9 isolates the side impact tests withtangential closing speeds between 30 and 40 mph andplots their tangential restitution values against the normaclosing speed.

The structural engagement between the vehiclesappears to remain superficial up to around 15 mph, withthe normal closing speed playing little role in thetangential restitution value. Above a 15 mph normaclosing speed there appears to be a general trend ofincreasing tangential restitution values with increasingnormal closing speed.

FIGURE 9

Frontal Impact Data - Figure 10 depicts tangentia

restitution values reported by Ishikawa for 13 frontaimpact tests plotted against the closing speed in thetangential direction. Again, there is no clear correlationbetween the tangential restitution values and the crashseverity.

FIGURE 10

To estimate a reasonable range of tangential restitutionvalues for a particular crash, the reconstructionist shouldstart by establishing a correlation between that particular


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crash and the physical interpretation of the tangentialrestitution coefficient. For instance, for many sideswipecollisions a common velocity is never reached andtherefore the reconstructionist should expect a negativerestitution value. Likewise, the reconstructionist may beable to conclude that structural engagement between thevehicles was such that a common velocity in thetangential direction was reached and that the tangentialcoefficient of restitution should be zero or greater.

Beyond that general characterization, thereconstructionist should consider the tangentialrestitution values from collisions that most resemble thecrash in question. This data is limited and needsexpanding. Ishikawas data is helpful since he reports

values for both et and . Other researchers reportneither.

GENERAL CONSIDERATIONS

Selection of Normal and Tangential Directions -Restitution values reported for staged VTV tests, such as

those reported by Ishikawa, depend on thereconstructionists selection of the normal and tangentialdirections. Since there is uncertainty associated with theselection of these directions, analysis of the same databy another reconstructionist would likely lead to differentrestitution values. It would be worthwhile to investigatethe effect of the uncertainty in the normal and tangentialdirections. At any rate, reconstructionists shouldrecognize that restitution values published in theliterature have uncertainty associated with them.

Summary of Available Data - Reconstructionists shouldbe familiar with the staged collision data that is available

and use this data for estimating coefficients of restitution.References 2, 3, 8, 15, 16, 17, 21, and 23 are excellentsources of data. In the analysis of any particular crashthe reconstructionist should use the data from stagecollisions that best mimics that crash. The available datawill rarely be strictly analogous and the reconstructionistshould formulate ranges for the restitution values thatreflect the level of uncertainty associated with thosevalues. The ranges should be wide enough toencompass the full range of possible restitution values.

For the foreseeable future, frontal barrier impact data willcontinue to be the most readily available. While thenumber of staged VTV collisions in the literature atvarying impact configurations and severities continues togrow, there remain large gaps in the data for manyimpact configurations and for most vehicle combinations.Even if these gaps were filled for impact configuration,testing of every vehicle combination is not feasible andthis gap is permanent. Analysis of crash severity shouldgive an accurate picture of the uncertainty inherent inestimates of the coefficients of restitution.

CASE STUDY

BACKGROUND - The following case study is an attempto tie together the ideas that have been discussedincluding (1) the application of the impact model, (2) theformulation of ranges and probability distributions for theindependent parameters, (3) the filtering of the data forrealistic restitution values, and (4) the formulation ofconclusions based on the filtered data. Detailed

instructions for the use of Crystal Ball software are

contained in the user manual and are not repeated here.

FIGURE 11

The crash considered is an intersection collisioninvolving a Chevrolet pickup and a Nissan sedan. Thecrash occurred when the Nissan attempted to make aleft turn in front of the oncoming Chevrolet. Figure 11depicts the collision geometry, the point of impact, therest positions of the vehicles, and the coordinate system

used for application of the momentum model. Bothvehicles traveled approximately 22 feet after impact.

Parameter Low End High End

Weight of Chevy 6661 6861

Chevy Moment of Inertia 4712 5208

Chevy Approach Angle 181 191

Chevy Departure Angle 187 197

Chevy Post-Impact Travel Distance 20 22

Post-Impact Deceleration 0.60 0.95

Weight of Nissan 3410 3510

Nissan Moment of Inertia 1849 2043

Nissan Approach Angle 272 292

Nissan Departure Angle 192 202

Nissan Post-Impact Travel

Distance

20 24

Post-Impact Deceleration 0.6 0.95

Chevy X-Coordinate of Impact

Center

97.5 103.5

Chevy Y-Coordinate of Impact

Center

-16 -10

Nissan X-Coordinate of Impact

Center

-3.2 3.2

Nissan Y-Coordinate of Impact

Center

-28.6 -22.6

Orientation of Tangential Direction 96 106

TABLE 1


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MONTE CARLO SIMULATION - The equations of theimpact model were entered in an Excel spreadsheet.Each of the independent parameters was assigned arange of values and a uniform probability distribution.The range used for each variable is listed in Table 1.

A few of these ranges deserve comment. The Chevytruck was carrying a load, but details regarding theweight of the load were not available. Thus, the weight ofthe Chevrolet was allowed to vary more widely than theweight of the Nissan. Occupant weights were reasonablywell known. Vehicle weights were taken from publisheddata.

The point of impact and the points of rest weredocumented by the police and were considered wellestablished. The motion of both vehicles between thepoint of impact and the points of rest was relativelystraight. Therefore, the departure angles of both vehicleswere well established and were only allowed to varywithin a 10-degree range.

The approach angle of the Chevy was known reasonablywell. There were no pre-impact skid marks by the Chevybefore impact and the driver indicated that he had verylittle time to react before the accident. This wasconsistent with time-space analysis performed by theseengineers. The driver of the Chevy may have been ableto steer before impact, so the approach angle wasallowed to vary between 181 and 191 to allow for thepossibility of an evasive steer to the left before impact.

The ranges on the post-impact deceleration rates of thevehicles are wide enough to encompass the breadth ofvalues in the literature for dry asphalt. We do not intend

to prescribe the use of this range. The reconstructionistshould use data from the literature and knowledge of thespecific surface and formulate a range on a case bycase basis.

Finally, the impact center was located using the crushenergy analysis portion of EDCRASH. Here, again, weare not prescribing this as the preferred method forlocating the impact center.

Having assigned these ranges and a uniform distributionto each of the independent variables, the Monte Carlosimulation was run with 200,000 trials. The datagenerated by the simulation was filtered to eliminateunrealistic results based on conceptual considerations ofthe restitution. Since there was significant engagementbetween the vehicles, the authors concluded that thevehicles reached a common velocity in both the normaland tangential directions. The restitution values in boththe normal and tangential directions, therefore, had to begreater than or equal to zero. Restitution values werealso required to be less than 1, since these values wouldimply that energy was added to the collision.

Tangential restitution values were further restricted to arange of 0.0 to 0.25 since it was clear from physicaevidence that the vehicles did not depart far from thecommon velocity reached in the tangential directionFinally, impact speeds calculated for the Nissan wererequired to be positive. Of the original 200,000combinations of the parameters, 3,481 or approximately2% of these combinations survived these rejectioncriteria. This is similar to the percentage of accepted

values reported by other authors [30]. However, in theexperience of the authors, this percentage varies widelyfrom case to case.

Histograms were generated based on these remainingvalues and are displayed in Figures 12 through 15. Table3 lists the mean, standard deviation, and 51% range forthese histograms.

FIGURE 12

FIGURE 13


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FIGURE 14

FIGURE 15

Mean Standard

Deviation

51%

Range

Chevy Change in

Velocity

11.8 .7 11.3 to 12.4

Nissan Change in

Velocity

23.1 1.4 22.0 to 24.1

Chevy Impact

Speed

33.1 1.8 31.8 to 34.5

Nissan Impact

Speed

4.8 2.0 3.3 to 6.2

TABLE 2

CONCLUSIONS AND RECOMMENDATIONS

MONTE CARLO SIMULATION

1. Monte Carlo simulation provides a statistical analysistechnique to analyze the propagation of uncertainty

in crash severity analysis. The results of a MonteCarlo simulation allow the reconstructionist to drawconclusions about the probable crash severity withan understanding of which values are truly mostlikely.

2. The Monte Carlo technique cannot establish thefidelity of the impact model in any particular case.Meaningful application of the Monte Carlo techniquemust come after there has already been a strong linkestablished between the impact model and theaccident that the model mimics. In other words, the

assumptions of the model must fit well with theactual accident.

3. Poorly chosen ranges for the input parametersundermine the effectiveness and usefulness of theMonte Carlo simulation technique. The range foeach parameter should be wide enough toencompass the full range of possible values.

4. Selecting the uniform distribution to describevariations in the independent variables representsthe most conservative assumption since the value othe parameter is allowed to vary more widely thanwith any other distribution and will result in the widesvariation in the dependent variable. The uniformdistribution should be preferred for most independentparameters in crash severity analysis since data thatwould justify choosing any other distribution istypically unavailable.

5. A custom distribution can be constructed torepresent variations in the coefficients of friction tha

recognizes that random variations in the frictionvalues vary normally, but that also gives adequateconsideration to uncertainty in the mean andstandard deviation of that normal distribution.

6. Certain combinations of values of the independenparameters may produce unrealistic results, and theoutput of the Monte Carlo simulation should befiltered. Ranges formulated for the coefficients ofrestitution provide one criterion by which to judge thereasonableness of the result of each combinationand should be used to rule out certain combinationsHowever, the range set on normal and tangentia

restitution values should not be overly restrictivegiving adequate consideration to the uncertaintyassociated with restitution values.

THE IMPACT MODEL

1. The goal of this discussion has not been to defendthe fidelity and superiority of any particular impactmodel, or even to defend the use of certainconstraint parameters within the context omomentum impact models. The model utilized in thispaper was developed to give the reader insight intoissues surrounding the analysis of uncertainty inimpact modeling and crash severity analysis.

2. The impact model developed makes use ofcoefficients of restitution in both the normal andtangential directions. These coefficients of restitutionare allowed to vary between -1 and 1, with theimplication that there is no common velocityrequirement inherent in the model.

3. Parting with a common velocity requirement isadvantageous since the impact model can handlesideswipe and break-through collisions.


16/17


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CONTACT

The authors welcome comments, questions, osuggestions and can be contacted at the following

address:

Nathan A. RoseKnott Laboratory, Inc.7185 S. Tucson WayEnglewood, CO USA 80112(303) [email protected]

A copy of the Excel file used for the analysis in this paperwill be provided upon request.

Documents

Publications 2001-01-3347 Monte Carlo Momentum