1 Throw Model for Frontal Pedestrian Collisions Coyright 1 Society of Automotive Engineers Inhwan Han, Associate Professor Hong Ik University, Choongnam, South Korea Raymond M. Brach, Professor University of Notre Dame, Notre Dame, IN, USA ABSTRACT A lanar model for the mechanics of a vehicleedestrian collision is resented, analyzed and comared to exerimental data. It takes into account the significant hysical arameters of wra and forward rojection collisions and is suitable for solution using mathematics software or sreadsheets. Parameters related to the edestrian and taken into account include horizontal distance traveled between rimary and secondary imacts with the vehicle, launch angle, center-of-gravity height at launch, the relative forward seed of the edestrian to the car at launch, distance from launch to a ground imact, distance from ground imact to rest and edestrian-ground drag factor. Vehicle and roadway arameters include ostimact, constant-velocity vehicle travel distance, continued vehicle travel distance to rest with uniform deceleration and relative distance between rest ositions of vehicle and edestrian. The model is resented in two forms. The first relates the throw distance, s, to the initial vehicle seed, v c, that is, s = f(v c ). The second, intended for reconstruction, relates the vehicle seed, v c, to the edestrian throw distance, s, that is, v c = f(s ). The first form is used extensively in the aer as means of comarison of the model to over 14 selected sets of exerimental data taken from the current literature. The second form is fit to exerimental data, roviding values of two model arameters, A and B. INTRODUCTION Well over 4 technical aers have been written in the ast 3, or so, years on the toic of vehicle-edestrian accidents. A significant number of these articles resent and discuss statistics of edestrian accidents, tyes and trends of injuries, tyes, trends and categories of vehicle frontal geometries, tyical vehicle deformations, etc. Others resent and discuss hysical and mathematical models for the analysis and reconstruction of such edestrian accidents. The form of these models varies widely, however and ranges from statistical and/or emirical equations to relatively sohisticated mathematical mechanics models. Examles of the former are Evans & Smith (1999), Haer, et al. () and Limert (1984, 1998). Examles of the latter include the simulation models of van Wijk, et al., (1983), Moser, et al., () and the momentum models of Wood, (1988). A comilation of numerous vehicle-edestrian accident reconstruction formulas is given by Russell, (undated). A book by Eubanks (1994), gives broad coverage to edestrian accidents including related toics such as tabulated edestrian seeds. A comilation of technical aers reresenting basic studies and results is contained in a volume edited by Backaitis (199). This aer deals with throw distance, the distance along the road between the location of the initial edestrian-vehicle contact and the edestrian rest osition. It deals with modeling of the relationshi between throw distance and vehicle seed. Using mathematics and hysics to model the throw distance resents a challenge, for several reasons. First and foremost is the fact that relative to the vehicle, the human body is articulated, flexible and soft, though the skeletal structure rovides limited rigidity (but caable of fracture). For low vehicle frontal shaes, it is common to have a rimary collision between the frontal ortion of the vehicle (bumer, grill, etc.) and the lower
2 Figure 1. Diagram of rimary (to) and secondary (bottom) imacts between a edestrian and a vehicle. art of the edestrian followed by a secondary collision of an uer art of the edestrian with the hood, windshield or roof as shown in Fig. 1. In low seed collisions, the secondary imact may be nonexistent or minor. Wide variations occur in human size, weight and form; wide variations exist in the size and frontal geometry of vehicles and wide variations occur in the initial conditions of contact between the two. For a given human form, size and weight and a given vehicle frontal geometry and comliance, the resulting motion of the edestrian deends significantly on the initial seed of the vehicle. All of these effects coule since the location and severity of a secondary collision is deendent on the seed. In addition, a ortion of the edestrian s motion before coming to rest is with ground contact. Because of the articulated shae of the human body, this motion can consist of combinations of bouncing, tumbling, rolling and sliding. In fact, this tye of motion has received considerable attention in its own right; for examle see Bratten (1989) and Searle (1993). All of these variations are difficult to model. They also are difficult to control during exeriments. Limert (1998) slits a vehicle-edestrian collision into 3 hases, the imact hase, the flight hase and the sliding and/or rolling hase. Five categories of vehicleedestrian collisions are described by Ravani, et al., (1981), Brooks, et al. (1987) and others. These are wra, forward rojection, fender vault, roof vault and somersault collisions. A wra collision is one where a significant ortion of the edestrian s body extends above the forward or frontal ortion of the vehicle and the edestrian s body "wras" u onto another ortion of the vehicle such as the hood (bonnet), windshield and/or roof; such as in Fig 1. Wra contact is accomanied by a rotation and bending of the edestrian s body and often is followed by a secondary imact of the head and/or uer torso. In a fender vault, the edestrian falls or rolls off to the side of the moving vehicle. For a roof vault, the vehicle asses under the edestrian as it does for a somersault, where the edestrian goes over the roof and rotates, tyically to an inverted, head-down, osition. In this aer, wra collisions are considered to be those that have both rimary and secondary collisions and a forward rojection has only a rimary collision. Of course, aberrations occur. An examle of this category is where a edestrian rises u onto the hood, erhas suffers a secondary collision, and remains on the hood without being thrown forward. In cases such as these, the edestrian tyically slides from the hood either to the side or forward, deending on the braking and steering of the vehicle. In this case there is a "carry" hase between imact and flight. This is not considered in this aer. The model develoed in this aer is intended to handle wra, forward rojection, roof vault and somersault collisions. Unfortunately, there seems to be no exerimental data available to study the last categories, so these are not treated here in any deth. Following imact the edestrian then is thrown forward from the vehicle and begins a flight hase, articularly if the vehicle is decelerating. A forward rojection collision is usually defined for a tye of collision where the edestrian s body does not extend significantly above the forward ortion of the vehicle and the initial contact drives the edestrian directly forward into the flight hase after a single imact. In both categories, the edestrian s center of gravity is above the ground when it dearts contact with the car. The mechanics of the flight hase can be described by the equations of free flight under gravity. Ordinarily, air drag is neglected for the flight hase and is neglected here. The flight hase is followed by an imact with the ground. The size and frontal geometry of the vehicle are factors that influence the mechanics of throw distance. Many authors have studied these asects of collisions; for examle, see Ashton and Mackay (1979), Sturtz and Suren (1976) and Evans and Smith (1999). There aears to be some disagreement on the significance of frontal geometry. On the one hand, Limert (1998) claims that frontal geometry and the identification and measurement of the location of the secondary imact can be critical in reconstruction to the extent that small errors can result in significant differences in seed reconstruction. Yet, Evans and Smith (1999) say that the effect of frontal geometry is overrated. On balance, frontal geometry robably does lay a role in vehicleedestrian mechanics and reconstruction but has no secial significance or sensitivity. It is also likely that the vehicle frontal geometry, edestrian height, weight and seed factors interact, or coule. For a given frontal geometry, the location and severity of the secondary imact deends on the seed and edestrian geometry. In the model that is resented, a single variable is used to reresent the distance
3 v c y x L R t t = s h θ v s 1 s t t c1 t 1 s d Figure. Coordinates, variables and events corresonding to a vehicle-edestrian collision. t s ϕ x t c between the rimary imact and the secondary imact. A main goal of this aer is to develo a model of a vehicle-edestrian collision that is somewhat more comrehensive than others, yet is not overly comlex and remains ractical and realistic. A comarison is made of the model with exerimental results. This comarison is rather extensive yet not fully satisfying, for several reasons. One reason is that measurements of some of the variables included in the model (such as the range of the flight hase and slide distance) tyically are not made or reorted from exeriments and so direct comarisons cannot be made. Other factors lay a role such as the common use of dummies for exeriments (for obvious reasons). The differences between results from exeriments using dummies and humans often is investigated through the use of actual accidents. Actual accidents, of course, require reconstruction, a rocess that introduces another set of associated unknowns and variations. Wood and Simms (1999) discuss some of these issues. Another aroach is to use cadavers, but the number of such exeriments is limited; these results also have their own eculiarities. Part of the aer is devoted to a reliminary analysis of some of the data used in the comarisons. Caution must always be used in the alication of any mathematical model of the mechanics of vehicleedestrian collisions. Collisions are not ideal. A edestrian is not a rigid body, contact often occurs between the edestrian and vehicle during the early art of the flight hase, motion along the ground is combination of sliding, rolling & tumbling, not simly sliding, etc. Consequently, wide variations can be exected between theory and ractice. And, in fact, wide variations occur in exeriments. This must be taken into account. A concerted study of the case-tocase variations from the model is not made in this aer since many of the variables in the model remain unmeasured; more exerimental work needs to be done. Confidence limits have been studied by other authors such as Evans & Smith (1999) and Haer, et al. (). Tyically these limits are broad and may not aly directly to a articular data set or to a secific alication of a model. A simlified edestrian collision index model is resented. A urose for the collision index model is that it can be used to hel interret exerimental data. It also is comared to the aer s model in redicting edestrian collision mechanics. PEDESTRIAN IMPACT MODEL THROW DISTANCE MODEL Figure shows a diagram that illustrates coordinates, arameters and some of the events of a vehicleedestrian collision. An objective is to determine the total throw distance, s, as a function of the initial vehicle seed, v c, with all of the other quantities (such as launch angle,, launch height, h, road grade, n, etc.) treated as known arameters. Another objective is to model the motion of the vehicle and relate it to the edestrian throw distance.
4 x x L The throw distance can be written as: s = x + R+ s x 1 (1) v c where x L is the v 1 v distance the edestrian moves in Figure 3. Samle relationshi between x L and the initial vehicle seed. the time between the rimary and secondary imacts. The arameter x L is included to reresent the combined influence of the vehicle frontal geometry, edestrian center-of-gravity height at imact, and vehicle seed deendence of the secondary imact. R is the range, or ground distance, covered during the flight hase and s is the distance between the ground imact of the edestrian and the rest osition. Some analysts, such as Limert (1998), reresent the initial velocity of the edestrian, v, as a fraction or multile of the forward seed of the vehicle. For that reason (and to allow comarisons of this concet with exerimental results) the initial launch seed of the edestrian, v, is exressed as () v = αv c where " is a constant and v c is the seed of the car at the time the edestrian is launched. The edestrian is launched following the secondary collision, or the rimary, if no secondary collision takes lace. Between initial contact and launch, the edestrian is brought u to seed and the vehicle suffers a momentum loss. Based on conservation of momentum, v = mc m + m v c c c where v c is the seed of the vehicle at initial contact, m c is the mass of the vehicle and m is the mass of the edestrian. The range, R can be exressed as R= v cosθ t gsin ϕt / (4) R R The quantity t R is the flight time from launch to ground contact and is vsinθ v sinθ + ghcosϕ tr = + gcosϕ gcosϕ (5) The velocity comonents of the edestrian at the time of ground imact are: vrx = vcosθ gsinϕtr (6) and vry = v θ gcosϕtr (7) sin L (3) These last two velocity comonents are velocity comonents just as the edestrian imacts the ground. Following imact, vertical rebound is ossible, even bouncing is ossible. However, it can be shown that the horizontal travel distance is not highly deendent on multile bouncing. Consequently, it is assumed that there is a single imact normal to the ground and that it is erfectly inelastic, so the ostimact rebound velocity is (8) v Ry = Further, it is assumed that Coulomb friction acts during the imact with the ground so the tangential comonent of the ostimact velocity is given by: v Rx = vrx + µ vry (9) The quantity, :, is the ratio of the tangential imulse to the normal imulse during the imact. Throughout this aer, the edestrian sliding drag factor is used for :. Note that v Ry is negative so that v Rx < v Rx. The final art of the throw distance is the slide distance, s. This is calculated using a edestrian drag factor, f where: ( v Rx ) s = (1) g( f cosϕ + sinϕ) The model for edestrian throw distance now is comlete where s = f(v c ) is given by using Eq 4 and 1 with Eq 1. Modeling of x L It was mentioned above that x L reresents the distance the edestrian (center of gravity) travels in the time between the initial and secondary imacts with the vehicle. In tests with dummies, Kallieris and Schmidt (1988) found that the location of the secondary imact deends on the seed of the car. Similar results were found by van Wijk, et al., (1983) from reconstructed accidents. In the following, a bilinear function is used to reresent this distance as it varies with vehicle seed. Figure 3 shows the function where the values, x 1 =.5 m (1.6 ft), x = 1.5 m (4.9 ft), v 1 = 5 m/s (11 mh) and v = m/s (45 mh) are used in this aer. By definition, x L = for forward rojection collisions. To some extent, the nonlinear function in Fig 3 is arbitrary and requires exerimental verification but it does allow roof vaults and somersaults to be modeled. VEHICLE MOTION Vehicle motion exists simultaneously with the edestrian motion and is searated into 3 ortions. The first is a distance, s. As a rough aroximation, it is assumed that over the distance, x L, the edestrian
5 travels at half its launch velocity so, t. x L '(v c ') and s. v c t. Next, the vehicle can travel a distance, s 1, at constant velocity and finally a subsequent ortion of the motion, s, with constant deceleration, a. Of course, s 1 can be zero if deceleration exists from t to rest. Consequently, d = s + s1 + s s d = vct+ vc ( tc1 t) + a ( t t ) / s c c1 THROW DISTANCE INDEX (11) Evans & Smith (1999) and Haer, et al. (), use curve fitting techniques to arrive at a relationshi between the throw distance, s, and the initial vehicle seed, v c. A modified form of their relationshi is: s = c1( vc + c) (1) For wra collisions, Evans & Smith choose c = and find (by fitting Eq 1 to secific exerimental data) that c 1 = 1'3.58 s 'm. Haer, et al., find values of c 1 = 1'3.53 s 'm and c =.7 m's for wra collisions by fitting to a wide range of exerimental data. As a means for comarison of exerimental data sets later in this aer, a throw distance drag index is defined by an equation as Eq 1 with c =, reresenting equivalent constant acceleration motion: v c feq = (13) gs Note that f eq is an equivalent constant drag factor corresonding to the entire throw distance, s, and not just the distance, s, when the edestrian is in contact with the ground. The quantity f eq corresonds to a fictitious, average deceleration factor for the throw distance rocess from imact to rest. RECONSTRUCTION THROW MODEL If the road grade angle, n, is close to zero, n., the edestrian throw-distance model derived above (given by Eq 4, 1 and Eq 1) can be ut into a form more useful for reconstruction of the seed of the vehicle. Doing this, an equation for the initial launch seed of the edestrian, v, is: where v = A s B (14) and A = B= x + f h L f f g sin θ + f sinθ + cos θ (15) (16) Using Eq, 3 and 14, the velocity of the car can be exressed as: where vc = A s B A A m c + = m αm c (17) (18) With the excetion that n., this model is a true inverse of the above throw-distance model. Recall that Eq 1, solved for the velocity v c, also is an equation used for reconstruction. Equations 17 and 1 have subtle, but imortant differences. First, since it is derived analytically not emirically, the arameters contained in A and B have hysical interretation, making Eq 17 more meaningful as well as more flexible. Admittedly, the values of the arameters in A and B are not always easy to estimate for secific reconstructions. Another difference between Eq 17 and 1 concerns their form. Eq 1 has an offset in the velocity whereas Eq 17 has an offset in the throw distance. Later, it will be seen that a deficiency with the throw distance index, Eq 13, is that at low velocities, it tends to underestimate the throw distance. Allowing a shift in the s coordinate ermits better matching of exerimental results. The method of least squares was used to find values of A and B that fit the model to exerimental data. This will be discussed later in more detail. EXPERIMENTAL DATA PEDESTRIAN DRAG FACTORS Exerimental Values of Hill The above edestrianimact model involves a calculation using the imulse ratio, :, on the coefficient of edestrian-ground friction (edestrian drag factor) as it strikes and momentarily slides along the road surface. This arameter has not been measured for edestrians. Exerimental data are therefore needed. Hill erformed a number of staged exeriments designed to measure the coefficient of friction from imact to rest. He used an adult size dummy constructed using a leather suit stuffed with a mixture of sand and sawdust contained in bags. The dummy was dressed in 5 tyes of clothing, and in each
6 test, the dummy was held, face u, horizontal to and aroximately.1m above the ground. It was released from the rear of a vehicle. The seed at release and the distance traveled by the dummy along the road between first contact and rest were measured. The avement in all tests was a dry, airfield tarmac. Hill used the following equation to calculate the coefficient of friction from his test results. f v = gs (19) Hill found a coefficient for body to road surface of.8, on average, to adequately reresent his test results and he used that value for calculation of vehicle seed from edestrian throw in actual accidents. Correction for Imact However, Eq 19 does not take account of the effect caused by the vertical imact which occurs when the dummy strikes the road surface. The imact may be a significant event that must be accounted for when calculating the coefficient of friction. Assuming zero road grade, Eq can be obtained from Eq 8 through 1 (where : = f ), 1 s = f g v cosθ f v sin θ + gh [ ] () Solving for f gives the following equation that rovides the friction drag from the test results for a zero launchangle (Hill s conditions). f = gs+ v gh [ ( gs + v gh) v gh gh ]/ (1) Note that the measured distance in the tests is the sliding and/or rolling distance s, and does not include flight range R. From Hill s tests, Eq 1 yields an average value of friction drag of.74, less than.8 obtained by Hill. This shows that failure to account for the vertical imact with the ground results in a coefficient that is too high. The imact is a significant event that must be taken into account when calculating the coefficient of friction. Coefficient of Friction Dummy Release Velocity, m/s Figure 4. Coefficient of friction as a function of velocity; values calculated from Hill s data, corrected for imact. Figure 4 shows the coefficients of friction calculated using Eq 1 lotted as a function of the horizontal release velocity for Hill s tests. Though the scatter in the data aears to vary with velocity, there seems to be no discernable trend between the releasing velocity and the coefficient of friction. Dummy Clothing Deendence In Hill s tests, the dummy was dressed in 5 tyes of clothing, and a minimum of ten tests comleted with each. The friction drag values show an influence of the clothing tye. Through the statistical method called Analysis of Variance (ANOVA), it was concluded that the differences in values of friction drag from clothing to clothing are statistically significant. This suggests that the coefficient of friction deends on clothing. Values of f for the different tyes of clothing had a range of.73 # f #.78 excet for nylon which had an average value of f =.61. So although values are statistically different, only nylon showed a meaningful change. Wood and Simms () carried out a similar analysis of the data of Hill (and others). Accounting for the imact gave average values of.7 and.734 for the coefficient f using Hill s data. EXPERIMENTAL THROW DATA In their SAE aer, Haer, et al., resented an extensive comilation of exerimental data collected from vehicle-edestrian exeriments. Some of the collection is used in this aer after grouing into to 3 categories. The first corresonds to actual accidents with adult edestrians, where corresonding values of
7 Exerimental data Model (Monte Carlo Simulation) Probability Probability s Variation s Variation Figure 5. Normal robability lot of the deviations of Hill s measured throw distance values from the throw distance index curve. v c and s are obtained through reconstruction of the accidents.these include data from Dettinger (1997), Hill (1994), Kramer (1975) and Sturtz (1976). The next category corresonds to wra trajectory collisions with adult sized dummies where s and v c are measured directly. These data came from Kramer (1975), Kuhnel (1974), Lucchini, et al. (198), Schneider and Beier (1974), Severy (1966), Stcherbatcheff, et al. (1975) and Wood (1988). The final category used child sized dummies with relatively high vehicle fronts so that the collisions all were forward rojection tyes. These data are from Lucchini, et al. (198), Severy (1966) and Sturtz (1976). Preliminary Analysis of Exerimental Data A value of f eq corresonding to each data set was calculated by minimizing the sum of squares of deviations of each data oint from the index curve given by Eq 13. If Eq 13 were an exact model, then the deviations of the exerimental values from the curve (sometimes referred to as residuals) should behave as normally (Gaussian) distributed, random exerimental "error" and should lot as a straight line on normal robability aer. Furthermore, the deviations should have a mean of zero and a small standard deviation. A lot of the throw-distance deviations of Hill s data from Eq 13 is Figure 6. Normal robability lot of the deviations of Schneider s measured throw distance values from the throw distance index curve and deviations between the throw distance index curve and values from a Monte Carlo random samle of the model. shown in Fig 5. The oints lie roughly along a curve, antisymmetric about zero. This imlies a narrow, symmetric distribution with wide tails, but close to normal. The mean (5% robability oint) is.11 m and the standard deviation is 3.1 m. Unfortunately, not all data sets have such nice characteristics. Figure 6, using Schneider s data (oen squares), is more tyical of those showing ossible non-normal behavior. These deviations are biased with a mean of m and have a standard deviation of 3.4 m. Moreover, the oints do not have a straight line trend. For comarison, a random samle of 11 oints was generated using a Monte Carlo simulation of the model equation. These oints also are lotted in Figure 6. Two arameters were given normal distributions, N(<,F), where < is the mean and F is the standard deviation. The statistical distributions used in the Monte-Carlo simulation are f = N(.8,.) and = N(4.1,.8). The samle oints of the model more closely follow a straight line, reflecting the normality of the distributions. The aarent non-normality of the exerimental variations cannot be exlained but must be either from exerimental events not observed and/or reorted or from effects not included in the model.
8 Table 1. Default values of Model Parameters ", velocity ratio, 1., launch angle, / (forward rojection) n, road grade angle, / :, edestrian-ground imact imulse ratio, f a, deceleration of vehicle,.5 f, edestrian-ground drag factor,.7 # f #.8 h, edestrian cg height at launch, 1 m (adult),.4 m (child, forward rojection) m c, mass of vehicle, 115 kg m, mass of edestrian, 65 kg (adult) and 3 kg (child) x L, edestrian dislacement from rimary to secondary vehicle imacts, m (forward rojection), Fig 3, (wra) With no exerimental error and an exact model all deviations between them would be zero. So the better the model and the more tightly controlled the exeriments, the closer the oints ass through 5%, and the more vertically they will be aligned (small variations). Since one of the measured variables, v c, aears to a ower in Eq 13, deviations may not lie along a straight line. According to statistical theory, in such a case deviations may have a distribution other than normal. Deviations of all data sets were examined and most of the robability lots showed some bias; Fig 6 is one of the "least normal". Variations of few data exhibited normality and there seemed to be no single discernable common trend. COMPARISON OF MODEL WITH DATA The rocess of fitting the model to the data sets was done in a consistent fashion. Selected arameter values were found using the method of least squares. Excet for those arameter values found by curve fitting, default values were used. These are listed in Table 1. In all cases the imulse ratio, :, reresenting the friction at ground imact was chosen to equal the edestrian drag factor, f, that is, : / f. This was done since there is no information on which to establish any other value of :. Model arameters that could have been fit include h, x L, f, " and. When exerimental values of these were unknown, they were assigned "tyical" values and held fixed. So, for examle, h = 1 m for adult tests and.4 m for children (forward rojection). For wra collisions, x L was modeled as shown in Fig 3. In wra collisions, " was held fixed at " = 1 and f and were found by the fitting rocedure. For the forward rojection cases, the launch angle was held fixed at = / and values of f and " were found by the fitting rocedure. The method of least squares was used to fit the model. The quantity Q was minimized, where Quv (, ) = [ s uv s ] () m(, ) e n where u and v reresent the fitted arameters, n indicates summation over all n values in a articular data set, m indicates model value of s and e indicates an exerimental value of s. In the wra exeriments, Q(u,v) = Q(f,); in the forward rojection cases, Q(u,v) = Q(f,"). Initially the fitting rocess was found to yield surious values and so the Q(u,v) surfaces were examined. It turned out that each Q surface formed a trough in (u,v) sace and the minimum would jum from one lace to another along the trough. This is because a change in, for examle, can be comensated to some extent by a change in f, forming a very shallow minimum. The same sort of behavior held for forward rojection data. Because of this behavior, it was necessary to constrain f. In all cases f was constrained to lie between.7 and.8. These Pedestrian Throw Distance, m Brach-Han Model s = v c /gfeq Figure 7. Dettinger s measured throw distance, the throw distance index curve and the Brach-Han model curve.
9 Table : Results of Fitting * Brach-Han Model v c = f(s ) Source Exerimental Model of Data Conditions f eq f " h A B Dettinger adult/reconstruction / Hill adult/reconstruction / Kramer adult/reconstruction / Sturtz adult/reconstruction / Kramer adult/dummy/wra / Kuhnel adult/dummy/wra / Lucchini adult/dummy/wra / Schneider adult/dummy/wra / Severy adult/dummy/wra / Stcherbatcheff adult/dummy/wra / Wood adult/dummy/wra / Lucchini child/forward rojection.46.8./ Severy child/forward rojection.57.8./ Sturtz child/forward rojection.55.8./ * Values in bold tye are from fitting the models to the corresonding exerimental data; others are default values. articular values were chosen for two reasons. First, from a hysical standoint, because of the resence of the flight hase (with essentially no drag), f must be greater than f eq. With few excetions, values of f eq were found to be greater than.6 (see Table ). Secondly, Hill s measurements indicate a mean value of.74. Reconstructed Wra Collisions Figures 7 and 8 show of the 4 data sets of adult/reconstructed, wra collisions with both the least-square index and model curves. Data in Fig 7 are from Dettinger data and Fig 8 from Sturtz. The model curves fit the data slightly better. The exerimental oints in Fig 7 show a much smaller deviation (scatter) from both curves; Fig 8 is one of the data sets with high scatter Pedestrian Throw Distance, m s = v c /gfeq Brach-Han Model Pedestrian Throw DIstance, m Point A Point B Figure 8. Sturtz s measured throw distance, the throw distance index curve and the Brach-Han model curve. Figure 9. Sturtz s measured throw distance comared to different Brach-Han model results.
10 Pedestrian Throw Distance, m Wood s Data Stcherbatcheff s Data s = v c /gfeq Brach-Han Model Pedestrian Throw Distance, m Brach-Han Model s = v c /gfeq Figure 1. Measured throw distances of Wood and Stcherbatcheff, the throw distance index curves and the Brach-Han model curves. Table contains the results of fitting of all of the selected data sets. For adult'reconstructed wra collisions, the equivalent drag coefficients ranged between.64 and.7. In all reconstructed wra collisions cases, the fitting rocedure gave values of f at the constraint boundaries and gave (unconstrained) launch angles between 5./ and 7.3/. Figure 9 shows 3 model curves fit to the adult/reconstruction data of Sturtz. These illustrate the sensitivity of the model to the fitted values. The leastsquares model curve is the solid line. Uer and lower dashed curves reresent the result of arbitrary changes of and f. The uer dashed curve is for the default values and for = 1/ and f =. while the lower dashed curve is for = / and f = 1.. Not all of the exerimental oints are included between the dashed curves. This is not because of a lack of flexibility of the model, however. Each of the oints can be fit exactly. For examle, with all default values, the measured throw distance of Point A is given exactly for = -1/ and f = 1.6. Point B is matched by = -15/ and f =.15. Unfortunately, there is no way to know if these arameter values are realistic for these exeriments. Points A and B seem unusually high and low, resectively Figure 11. Schneider s measured throw distance, the throw distance index curve and the Brach-Han model curve. Adult Dummy Wra Collisions This tye of collision had many more data sets available; seven were used in the fitting rocedure. Table shows the results. In all cases, fitted values of f were at a constraint boundary. Stcherbatcheff s and Wood s exeriments gave noticeably low values of the drag index. In addition, fitting to the model roduced high values of launch angle, namely = 35./ and = 4.9/, resectively. Their data and the fitted curves are shown in Fig 1. Underlying reasons for the differences of the launch angles of Stcherbatcheff and Wood and the rest of the cases are not known, excet to say that, for given vehicle velocities, the corresonding throw distances were larger than the other data sets. Arbitrarily omitting Stcherbatcheff and Wood, the average value of launch angle for wra collisions is 7.3/. This comares reasonably well with the average value of = 5.9/ from the adult/reconstructed exeriments. Figure 11 shows Schneider s data. This is the data whose statistical distribution of deviations is lotted in Fig 5. Neither the model nor the throw distance index curve could be forced u high enough by the fitting rocedure and so 9 of the 1 oints lie above the curve (note, oint have the same values). This is what causes the high bias in the mean value of the
11 35 4 Pedestrian Throw Distance, m Brach-Han Model _ s = v c /gfeq Pedestrian Throw Distance, m Brach-Han Model s = v c /gfeq Figure 1. Lucchini s measured forward rojection throw distances, the throw distance index curve and the Brach-Han model curve. deviations seen in Fig 6. Whether this reflects a roblem with the exeriments or a deficiency in the models is not known. Forward Projection Collisions Three sets of data were used in this comarison, all done with child-sized dummies and high-front vehicles. Data are from Lucchini (198), Severy (1966) and Sturtz (1986). Fitting was done for a fixed launch angle of = /. The dummies had an unknown center-of-gravity height so a nominal value of.4 m was used. In addition, the distance x L was fixed at. The least-square fitting rocedure was used to find values of " and f (where.7 # f #.8). Fitting of Eq 13 to find f eq rovided values of f eq =.46,.57 and.55 for the Lucchini, Severy and Sturtz data, resectively. The data sets all were quite different. Figures 1 and 13 show the fitted results and data of Lucchini and Sturtz, resectively. Lucchini s data is concentrated near velocity values with little scatter whereas the Sturtz data, with considerably more scatter, covers a much wider velocity range. Both the index curves and the model curves are very close for all 3 forward rojection data sets. A somewhat surrising and interesting result from these Figure 13. Sturtz s measured forward rojection throw distances, the throw distance index curve and the Brach-Han model curve. cases is that the fitted value of the velocity ratio, ", Table 3: Results of Fitting * the Reconstruction Model, Eq 13 Exerimental Data Conditions A B Source Tye Dettinger adult/reconstruction Hill adult/reconstruction Kramer adult/reconstruction Sturtz adult/reconstruction Kramer adult/dummy/wra Kuhnel adult/dummy/wra 3.5. Lucchini adult/dummy/wra Schneider adult/dummy/wra Severy adult/dummy/wra Stcherbatcheff adult/dummy/wra.81. Wood adult/dummy/wra Collision Tye Adult: reconstructions & dummies Child: reconstructions & dummies Adult/Reconstruction Adult/Dummy Child/Reconstruction Child/Dummy Reconstruction: adult & child Dummy: adult & child 3..5 * The coefficients A and B were found by fitting the data directly to Eq 17
12 Initial Vehicle Velocity m/s Pedestrian Throw Distance, m Pedestrian Throw Distance, m Figure 14. Model arameters A and B fitted to reconstructed collision data. ranges from 1. to 1.3. This means that the best fit to the data is for a edestrian forward launch velocity about 1. to 1.3 times greater than the forward velocity of the vehicle. The velocity loss of the vehicle due to the rimary imact can be ignored here (child-tovehicle mass ratio used here is 3/115), so the ratio, ", corresonds to a coefficient of restitution of the rimary (and only) imact. Therefore, the data indicate that in forward rojection collisions, the edestrian rebounds from the vehicle at a seed about % to 3% higher than the vehicle. All exeriments used dummies and it is not clear if this henomenon differs for humans, but the effect is consistent and significant. Roof Vault & Somersault Collisions These collisions are not discussed here to any great extent since there seems to be no exerimental data available. For these collisions it is exected that x L tyically would be larger than for corresonding wra collisions. The launch angle,, could be exected to take on values near or greater than 9/ and the factor " could very well be less than 1. Fitting of the Reconstruction Model The quantity Q (u, v) again was minimized, but where Quv (, ) = [ v uv v ] (3) cm(, ) ce n where n indicates summation over all n values in a articular data set, m indicates model value of v c and e indicates an exerimental value of v c and (u,v) = (A,B), Eq 17. This fitting was done without any constraints. Table 3 summarizes the results. Figure 14 shows a single curve with A = 3.5 and B =.94 lotted with collective adult/child/reconstruction data. Figure 15 shows a single curve with A = 3. and B =.5 lotted Figure 15. Model arameters A and B fitted to dummy collision data. with collective adult/dummy data. Comarison of the values of A and B in Table to those in Table 3 illustrates an extremely imortant difference between fitting of emirical and hysical models. The fitting rocess of EQ 17 done to reach the values listed in Table 3 was unconstrained and, in effect, no different than finding the constants c 1 and c in Eq 1. Consider the imlications of one of an examle air of values of A and B from Table 3. For the data of Kramer, adult/dummy/wra, A = 3.48 and B =.31. Also note that from Eq 16, f = ( B xl)/ h (4) If, for the exeriments, the "average" value of the distance, x L is about 1. m and the average cg height at beginning of launch is about 1. m, then the corresonding value of edestrian-ground frictional drag from Eq 4 is f = This certainly does not make hysical sense. This demonstrates why fitting of the throw model earlier in the aer was done to hysical constants. It also indicates why using a constraint was necessary when fitting the drag factor, f. A comrehensive comarison of the reconstruction model with constrained comarisons with exerimental data and with results of other mathematical models is lanned for a searate aer. DISCUSSION AND CONCLUSIONS An analysis, including a correction for imact effects, of Hill s direct measurements of edestrian-ground friction yields an average value of.74 and indicates that values between.7 and.8 are aroriate for f (for
13 an ashalt surface). Values of f greater than.7 also are exected since fitted values of the overall throw distance drag factor, f eq, are as high as.7 and.74. These results are in contrast to considerably lower values of f by direct measurements of others. For examle, Fricke (199) finds values between.45 and.65 and Bratten (1989), found values in the vicinity of.5. Secific reasons for these differences are not known. A model for edestrian throw distance has been develoed and resented. It contains what is believed to be the most imortant arameters and hysical variables that lay a role in lanar vehicle-edestrian collisions. The model is resented in two forms. One determines the throw distance as a function of the vehicle seed, s = f(v c ). The other is a reconstruction form where, v c = f(s ). Not all of the arameters in the model were measured, recorded and/or controlled in the exeriments and so not all of them could be evaluated or examined from the data. Fitted values of launch angles ranged from roughly from 5/ to 1/ for wra collisions, based both on reconstructed collisions and exeriments with dummies. In general, using leastsquare arameter values in the model and least-square fitted values of f eq in the index equation resulted in nearly identical curves. This imlies that the index equation (Eq 13) does nearly as good a job of matching exeriments as does the model. Although true, the model is based on hysical rinciles and measurable variables. Comared to the index equation, the model has greater flexibility and greater otential. If/when more detailed data becomes available, the model should rovide a more recise method for both analyzing and reconstructing edestrian collisions. It was shown that fitting emirical equations can resent the otential hazard of masking hysical reality. Finally, fitted values of the velocity ratio, ", of 1. and 1.3 resulted from the analysis of forward rojection collisions. Corresonding to the value of f =.8, this range of " indicates that the edestrian (based on exeriments using dummies) rebounds from the front of the vehicle at a seed higher than the vehicle by % to 3%. Of course, the edestrian s seed must be at least equal to the seed of the vehicle, otherwise the edestrian would "ass through" the vehicle. On the other hand, values of " between 1. and 1.3 seem rather high. This could be due to a difference in the restitution between dummies and human edestrians. Consequently, these results should be verified by future exeriments. ACKNOWLEDGMENT The authors graciously acknowledge the generosity of Andrew Haer, Michael Araszewski, Amrit Toor, Robert Overgaard, and Ravinder Johal for sulying most of the exerimental data used in this aer. Collecting and sulying the data allowed this aer s authors to accomlish much more toward the main goals of the aer. This research was conducted while the rincial author was on a sabbatical leave (1999-) from Hong Ik University, Choongnam, Korea. CONTACT Corresondence for this aer should be sent to R. M. Brach, Room 15 Hessert Center, Deartment of Aerosace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN , USA; REFERENCES Ashton, S. J. and Mackay, G. M., Car Design for Pedestrian Injury Minimization. Proc. of the Seventh International Technical Conference on Exerimental Safety Vehicles, U.S. DOT, Washington, D.C., 1979b,. 63; also SAE Paer Backaitis, Stanley H., Accident Reconstruction Technologies edestrians and Motorcycles in Automotive Collisions, SAE, PT-35, 199 Bratten, T.A., Develoment of a Tumble Number for Use in Accident Reconstruction SAE Paer 89859, 1989 Brooks, D., Wiechel, J., Sens, M. and Guenther, D. A., "A comrehensive review of edestrian imact reconstruction." SAE Paer 8765 Dettinger, J., Methods of imroving the reconstruction of edestrian accidents: develoment differential, imact factor, longitudinal forward trajectory, osition of glass slinters (in German). Verkehrsunfall und Fahrzeugtechnik, Dec. 1996, and Jan. 1997, 5-3 Eubanks, J. J., Pedestrian Accident Reconstruction, Lawyers & Judges Publishing Comany, Tucson, AZ, 1994 Evans, A. K. and Smith, R., Vehicle seed calculation
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