THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT

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1 THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT Leon Bogdanoić B. Eng., Milan Baisa D. Sc. Uniersi of Ljubljana Facul of Mariime Sudies and Transporaion Po pomorščako 4, SI- 63 Pororož, Sloenia ABSTRACT This paper discusses hrow models for ehicle/pedesrian collisions. Mos of he models proposed in lieraure neglec he road gradien. For ha reason a model including his parameer is deried and analsed wih daa from a real acciden. The resuls (compared o hose produced b oher models) are showing ha he road gradien has a significan influence on he deerminaion of he iniial ehicle speed, which is he main objecie of echnical analses of ehicle/pedesrian collisions. 1 INTRODUCTION Seeral models describing pos-impac moion of pedesrians inoled in accidens wih moor ehicles can be found in lieraure ([1], [], [3], [4] and [5]). These models are eiher heoreical or empirical. Theoreical soluions gie reliable resuls, howeer considerable daa from real world collisions is needed o sole mahemaical equaions. On he oher hand empirical soluions need no paricular daa, howeer heir applicaion is limied onl o well defined scenarios [1]. This paper deals mainl wih heoreical models. An oeriew of heoreical hrow models shows ha he are mainl based on mechanics of a mass poin. This assumpion is accepable onl for he pos-impac phase. The pedesrian should be reaed oherwise as an ariculaed, fleible and sof bod wih limied rigidi (due o skeleal srucure) [], since ha is fundamenal for he deerminaion of he ehicle/pedesrian ineracion. The analsis of he ehicle/pedesrian ineracion is he basis for he deerminaion of he launch heigh of pedesrians. Generall wo pes of scenarios can occur. For low ehicle fronal shapes, i is common o hae a primar collision beween he fronal area of he ehicle and he lower par of he pedesrian followed b a secondar collision of an upper par of he pedesrian wih he hood, windshield or roof. In low speed collisions, he secondar collision ma be noneisen or minor []. If a secondar collision occurs, he ehicle suffers a loss of momenum, which has o be considered in a calculaion. Afer he impac, he pedesrian is hrown and begins he fligh phase, which is followed b he sliding phase. These wo phases can be modelled eiher ogeher or separael. 1

2 AN OVERVIEW OF THROW MODELS An oeriew of he mos relean heoreical hrow models will be presened. Also an empirical model widel used b epers will be described. Mos of he heoreical hrow models rea he pedesrian as a projecile (mass poin). In his wa he oal hrow disance S as a funcion of he iniial speed and all of he oher quaniies (such as pedesrian launch angle θ, iniial launch heigh H, coefficien of fricion µ beween he pedesrian and he ground, road gradienα, ec.) or he iniial speed as a funcion of he oal hrow disance S and all of he oher quaniies is obained. I has o be sressed ou ha he oal hrow disance is referred as he disance he pedesrian undergoes from impac o his res posiion on he ground (see Figure 1). g V θ H α S R Figure 1: Pos-impac moion of he pedesrian Collins [3] deried he following equaion for he oal hrow disance S : S H g + µ g (1) where g 9,81 m/ s is he erical acceleraion due o grai. If S is known, he iniial speed from equaion (1) can be epressed: ( ) µ g µ H+S µ gh () The main disadanage of his model is ha i neglecs he pedesrian launch angle (and also he road gradien).

3 Searle [4] considered he launch angle θ and obained: ( θ + µ θ) cos sin S + µ H (3) µ g Soling he aboe equaion for produces he following epression: ( H) µ g S µ cosθ + µ sinθ (4) Because he alue for he pedesrian launch angle is ofen no aailable, Searle obained an epression for he minimum pedesrian iniial speed b equaing he firs deriaie o zero ( ): θ ( H) µ g S µ min 1+ µ (5) This model also neglecs he road gradien. Han and Brach [] spli he pedesrian moion ino 3 phases. Upon his model he oal hrow disance S can be wrien as: S L + R+ s (6) where L is he disance he pedesrian undergoes in he ehicle/pedesrian conac phase, R is he disance coered in he fligh phase and s is he disance beween he ground impac of he pedesrian and he res posiion. The parameer L is dependen on ehicle s fronal geomer, pedesrian cener-of-grai heigh a impac and ehicle speed afer firs impac wih he pedesrian. The disance R can be epressed as: where 1 R Rcosθ grsinα (7) R sinθ sin θ + ghcosα + (8) gcosα gcosα is he fligh ime from he launch o he conac wih he ground and α is he road gradien. The disance s has he form: 3

4 ( + µ ) s g cos sin ( µ α + α) (9) where cosθ g sinα (1) R and sinθ g cosα (11) R are he eloci componens of he pedesrian a he ime of he impac wih he ground. B neglecing he road gradien (i.e. α ) he following equaion for he iniial speed of he pedesrian can be obained: µ g( S ( L + µ H) ) ( µ sinθ + cosθ) (1) Because he ehicle speed ' c and he pedesrian iniial speed (a he ime he pedesrian is launched) usuall differ, Han and Brach adoped a coefficien η o relae hem: η' c (13) The iniial conac speed of he ehicle c can be calculaed due o he conseraion of momenum: where mc is he mass of he ehicle and m + m ' (14) c p c c mc mp is he mass of he pedesrian. An empirical model widel used b epers can be found in [5]. The iniial ehicle speed (in km/h) can be calculaed due o he following formula: S (15) c 1 where S is he oal hrow disance (in m). The reliabili of he calculaed resul is in he range of ± 1%. 4

5 3 A THROW MODEL INCLUDING ROAD GRADIENT The real world accidens ofen occur on he roads wih a gradien. Mos of he heoreical hrow models neglec his parameer, howeer such simplificaions ma lead o erroneous resuls. Therefore a model including his parameer will be deried and analzed. Basic equaions The noaion of ariables and parameers is he same as before (Figure 1). From he Newon s nd Law he equaions of moion are obained: ma mg sinα µ N ma mg cosα + N (16) where a and a are he acceleraion componens, m is he mass of he pedesrian, µ is he fricion coefficien, g is he erical acceleraion due o he grai, α is he road grade and N is he erical reacion (which is zero when he pedesrian is in he air). Combining he wo equaions (16), he following epression is deried: ( sin cos ) a + µ a g α + µ α (17) The inegraion of equaion (17) wih respec o ime gies he relaionship beween speed componens: ( sin cos ) 1 + µ g α + µ α + C (18) The inegraion of equaion (18) wih respec o ime gies he relaionship beween posiion componens: ( ) + µ g sinα + µ cosα + C 1 + C (19) The iniial condiions for he deerminaion of inegraion consans C 1 and C are: ( ) + µ ( ) µ ( ) + µ ( ) ( cosθ + µ sinθ) H () From equaions (18), (19) and () he alues for inegraion consans are obained: C C 1 ( cosθ µ sinθ) + µ H (1) The equaions (18) and (19) can now be wrien in he following form: 5

6 ( cos sin ) ( sin cos ) + µ θ + µ θ g α + µ α + µ µ H + ( cosθ + µ sinθ) g( sinα + µ cosα) () These equaions coer he sliding phase of pedesrian s moion. In he fligh phase he erm N in (16) anishes: ma ma mg sinα mg cosα (3) When hese equaions are inegraed wice wih respec o ime and considering he iniial condiions: (), () H, () cosθ and () sinθ he speed componens cosθ gsinα sinθ gcosα (4) and posiion componens 1 cosθ g sinα 1 H + sinθ g cosα (5) in he fligh phase are obained. Toal hrow disance When he pedesrian sops, he following condiions are fulfilled:,,, S (oal hrow disance) and sop (sopping ime of he pedesrian). From equaions () follows: and furhermore: ( θ µ θ) ( α µ α) cos + sin g sin + cos S µ H + ( cosθ + µ sinθ) sop g( sinα + µ cosα) sop g ( cosθ + µ sinθ) ( sinα + µ cosα) ( cosθ + µ sinθ) ( α + µ α) S µ H + g sin cos sop sop (6) (7) 6

7 Fligh disance The fligh disance R and he fligh ime R can be deried from (5). The epressions are he same as in (7) and (8). Iniial speed The iniial speed is obained from he second equaion in (7): B using rigonomeric ideniies ( α + µ α )( µ ) g sin cos S H cosθ + µ sinθ (8) sinθ anθ 1+ an θ and cosθ 1 1+ an θ his epression can be pu in he following form: ( α + µ α )( µ )( + θ) g sin cos S H 1 an 1+ µ anθ (9) B equaing he firs deriaie o zero: ( α + µ α )( µ ) ( ) ( cos + sin ) d g sin cos S H sinθ + µ cosθ anθ min µ dθ θ µ θ (3) he minimum iniial speed min is deried: ( α + µ α )( µ ) g sin cos S H min 1+ µ (31) 7

8 4 VALIDATION OF THE MODEL The deried hrow model was alidaed wih he daa from a real acciden. All parameers perinen o he acciden are shown in Table 1. Table 1: Daa from a real ehicle/pedesrian collision Quani Smbol Uni Value grai acceleraion g m/ s 9.8 road gradien p % iniial heigh H m 1. mass of he pedesrian m kg 6 mass of he ehicle m kg 94 coefficien of resiuion η 1. conac phase disance m 1. sum of conac phase disance and fligh disance (due o phsical eidence) sum of conac phase disance and oal hrow disance (due o phsical eidence) p L R + R m 18.5 * m L m S + S m 6.8 * m L m The main goal of his echnical analsis is o deermine he iniial ehicle speed c, which can be calculaed, if he iniial pedesrian speed is known (equaions (13) and (14)). The speed in he deried model is obained from equaion (9). Because no all of he parameers in his equaion are known (i.e. he pedesrian launch angle θ and he coefficien of fricion µ ) a direc calculaion of is impossible. This problem can be oercome b using he leas square mehod. In his wa he alues for, θ and µ can be calculaed, which saisf he known parameers (i.e. * R m and * S m ). The leas square mehod formulaion for his problem is he following: The parameers, θ and µ hae o be chosen in such a wa ha he funcion reaches is minimum. The necessar condiions are: ( ) ( ) ( ) F,, θ µ R Rm + S Sm (3) F F, θ F and µ (33) 8

9 From (33) a ssem of hree nonlinear equaions is obained and hae o be soled for, θ and µ. For R and S he epressions from (7) and (7) are used, while R m and S m are obained from phsical eidence (see Table 1). Using Maple 1, he numerical soluion shown in Table is calculaed. Because he ssem (33) has muliple soluions, soling was resriced o inerals ( 36, 7 ) km/ h, θ (, 45) and µ (.,1.), which suis phsical consisenc. L Table : The leas square mehod soluion for, θ and µ + R L + S c θ [m] [m] [km/h] [km/h] [ µ R sop ] [s] [s] Figure shows ha he funcion ( ) for a range of iniial speeds. F,, θ µ has indeed a minimum for he calculaed θ and Figure : Graphical represenaion of he funcion ( ) speeds F,, θ µ for a range of iniial 1 Maple 7., Waerloo Maple Inc. 9

10 The calculaed coefficien of fricion µ is in he range of recommended alues (from.45 o.55) [1], howeer here is no concordance in lieraure abou ha and he alues ma ar from.45 o 1. [5]. In he discussed acciden a higher alue for µ has o be adoped, because here is srong eidence ha he pedesrian suffered a collision wih he edge of he paemen. The coefficien of fricion µ is now a known parameer and he reformulaed problem has he following form: ( F( ) ( ) ( ) θ R Rm + S Sm ) min, ; F F, θ The soluion for µ.7 is shown in Table 3. Table 3: The leas square mehod soluion for, θ ( µ.7 ) + R L + S c θ R [m] [m] [km/h] [km/h] [ sop ] [s] [s] L The resuls show ha he ehicle had an iniial speed c of 56 km/h and he pedesrian s iniial speed was 53 km/h. (34) 5 COMPARISON WITH OTHER MODELS A direc comparison beween he model presened in his paper and he Han-Brach s model (including road gradien) shows ha he resuls produced (appling he leas square mehod) are idenical. The main adanage of he proposed model is a more compac form of equaions for he deerminaion of he iniial pedesrian speed or he oal hrow disance. The iniial ehicle speeds c obained b indiidual models are gien in Table 4. Table 4: The iniial ehicle speeds c in km/h calculaed b indiidual models Roim (empirical) Collins Searle Han-Brach Han-Brach (incl. road gradien) This paper (incl. road gradien) c The resuls (Table 4) show ha he use of models neglecing road gradien wih accidens occurred on graded roads oeresimae he iniial ehicle speed. In such cases i is recommended o appl models which incorporae ha parameer. 1

11 Figure 3 shows he iniial ehicle speed c dependen on he road gradien and he coefficien of fricion for he analzed acciden. I is eiden ha his wo parameers hae a significan influence on he deerminaion of he iniial ehicle speed. Figure 3: The iniial ehicle speed c as a funcion of he road gradien α and he coefficien of fricion µ 6 CONCLUSION The deried hrow model considers all he parameers (including road gradien) relean o ehicle/pedesrian collisions. I is shown ha he road gradien has a considerable effec on soluions and mus be aken ino accoun in cases wih graded roads. The model (ogeher wih oher models) was alidaed wih daa from a real acciden and gies idenical resuls as Han-Brach s model, howeer he deried formulae are more concise and compac. The resuls are showing ha models neglecing road grade oeresimae he alue of he iniial ehicle speed. The applicaion of he leas square mehod was performed in his paper o oercome he problem of unknown parameers. Wih he use of his mehod a se of minimum alues of searched parameers is obained, which suis phsical eidence. 11

12 REFERENCES [1] A. Toor, M. Araszewski, "Theoreical s. empirical soluions for ehicle/pedesrian collisions", , Acciden Reconsrucion 3, Socie of Auomoie Engineers, Inc., Warrendale, PA, 3, pp [] I. Han, R. M. Brach, "Throw model for fronal pedesrian collisions", , Socie of Auomoie Engineers, Inc., Warrendale, PA, 1. [3] J. C. Collins, Acciden Reconsrucion, Charles C. Thomas Publisher, Springfield, Illinois, 1979, pp [4] J. A. Searle, "The phsics of hrow disance in acciden reconsrucion ", 93659, Socie of Auomoie Engineers, Inc., Warrendale, PA, [5] F. Roim, Elemeni sigurnosi cesonog promea, Sezak 1, Eksperize promenih nezgoda, Znanseni saje za prome JAZU, 1989, pp [6] J. J. Eubanks, W. R. Haigh, "Pedesrian inoled raffic collision reconsrucion mehodolog", 91591, Socie of Auomoie Engineers, Inc., Warrendale, PA,

1. VELOCITY AND ACCELERATION

1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under