the throw model for vehicle/pedestrian collisions including road gradient
L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT Leon Bogdanović B. Eng., Milan Batista D. Sc. University of Ljubljana Faculty of Maritime Studies and Transportation Pot pomorščakov 4, SI- 6320 Portorož, Slovenia [email protected], [email protected] ABSTRACT This paper discusses throw models for vehicle/pedestrian collisions. Most of the models proposed in literature neglect the road gradient. For that reason a model including this parameter is derived and analysed with data from a real accident. The results (compared to those produced by other models) are showing that the road gradient has a significant influence on the determination of the initial vehicle speed, which is the main objective of technical analyses of vehicle/pedestrian collisions. 1
INTRODUCTION
Several models describing post-impact motion of pedestrians involved in accidents with motor vehicles can be found in literature ([1], [2], [3], [4] and [5]). These models are either theoretical or empirical. Theoretical solutions give reliable results, however considerable data from real world collisions is needed to solve mathematical equations. On the other hand empirical solutions need no particular data, however their application is limited only to well defined scenarios [1]. This paper deals mainly with theoretical models. An overview of theoretical throw models shows that they are mainly based on mechanics of a mass point. This assumption is acceptable only for the post-impact phase. The pedestrian should be treated otherwise as an articulated, flexible and soft body with limited rigidity (due to skeletal structure) [2], since that is fundamental for the determination of the vehicle/pedestrian interaction. The analysis of the vehicle/pedestrian interaction is the basis for the determination of the launch height of pedestrians. Generally two types of scenarios can occur. For low vehicle frontal shapes, it is common to have a primary collision between the frontal area of the vehicle and the lower part of the pedestrian followed by a secondary collision of an upper part of the pedestrian with the hood, windshield or roof. In low speed collisions, the secondary collision may be nonexistent or minor [2]. If a secondary collision occurs, the vehicle suffers a loss of momentum, which has to be considered in a calculation. After the impact, the pedestrian is thrown and begins the flight phase, which is followed by the sliding phase. These two phases can be modelled either together or separately.
1
L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
2
AN OVERVIEW OF THROW MODELS
An overview of the most relevant theoretical throw models will be presented. Also an empirical model widely used by experts will be described. Most of the theoretical throw models treat the pedestrian as a projectile (mass point). In this way the total throw distance S as a function of the initial speed v0 and all of the other quantities (such as pedestrian launch angle θ , initial launch height H , coefficient of friction µ between the pedestrian and the ground, road gradient α , etc.) or the initial speed v0 as a function of the total throw distance S and all of the other quantities is obtained. It has to be stressed out that the total throw distance is referred as the distance the pedestrian undergoes from impact to his rest position on the ground (see Figure 1).
g
y V0
x
θ
H
α
S
R
Figure 1: Post-impact motion of the pedestrian Collins [3] derived the following equation for the total throw distance S :
S = v0
v2 2H + 0 g 2µ g
(1)
where g = 9,81 m / s 2 is the vertical acceleration due to gravity. If S is known, the initial speed v0 from equation (1) can be expressed: v0 = 2 µ g ( µ H+S) − µ 2 gH
(2)
The main disadvantage of this model is that it neglects the pedestrian launch angle (and also the road gradient).
2
L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
Searle [4] considered the launch angle θ and obtained: v02 ( cos θ + µ sin θ ) S= + µH 2µ g 2
(3)
Solving the above equation for v0 produces the following expression: v0 =
2µ g ( S − µ H ) cos θ + µ sin θ
(4)
Because the value for the pedestrian launch angle is often not available, Searle obtained an expression for the minimum pedestrian initial speed by equating the first derivative to zero ∂v ( 0 = 0 ): ∂θ v0min =
2µ g ( S − µ H ) 1+ µ 2
(5)
This model also neglects the road gradient. Han and Brach [2] split the pedestrian motion into 3 phases. Upon this model the total throw distance S can be written as: S = xL + R + s
(6)
where xL is the distance the pedestrian undergoes in the vehicle/pedestrian contact phase, R is the distance covered in the flight phase and s is the distance between the ground impact of the pedestrian and the rest position. The parameter xL is dependent on vehicle’s frontal geometry, pedestrian center-of-gravity height at impact and vehicle speed after first impact with the pedestrian. The distance R can be expressed as: 1 R = v0t R cos θ − gt R2 sin α 2
(7)
v02 sin 2 θ + 2 gH cos α v0 sin θ tR = + g cos α g cos α
(8)
where
is the flight time from the launch to the contact with the ground and α is the road gradient. The distance s has the form:
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L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
s=
(v
x
+ µvy )
2
2 g ( µ cos α + sin α )
(9)
vx = v0 cos θ − gt R sin α
(10)
v y = v0 sin θ − gt R cos α
(11)
where
and
are the velocity components of the pedestrian at the time of the impact with the ground. By neglecting the road gradient (i.e. α = 0 ) the following equation for the initial speed of the pedestrian can be obtained: v0 =
2 µ g ( S − ( xL + µ H ) )
( µ sin θ + cos θ )
2
(12)
Because the vehicle speed v 'c 0 and the pedestrian initial speed v0 (at the time the pedestrian is launched) usually differ, Han and Brach adopted a coefficient η to relate them: v0 = η v 'c 0
(13)
The initial contact speed of the vehicle vc 0 can be calculated due to the conservation of momentum: vc 0 =
mc + m p mc
v 'c 0
(14)
where mc is the mass of the vehicle and m p is the mass of the pedestrian. An empirical model widely used by experts can be found in [5]. The initial vehicle speed (in km/h) can be calculated due to the following formula: vc 0 = 12 S
(15)
where S is the total throw distance (in m). The reliability of the calculated result is in the range of ± 10%.
4
L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
3
A THROW MODEL INCLUDING ROAD GRADIENT
The real world accidents often occur on the roads with a gradient. Most of the theoretical throw models neglect this parameter, however such simplifications may lead to erroneous results. Therefore a model including this parameter will be derived and analyzed. Basic equations
The notation of variables and parameters is the same as before (Figure 1). From the Newton’s 2nd Law the equations of motion are obtained: max = −mg sin α − µ N y ma y = −mg cos α + N y
(16)
where ax and a y are the acceleration components, m is the mass of the pedestrian, µ is the friction coefficient, g is the vertical acceleration due to the gravity, α is the road grade and N y is the vertical reaction (which is zero when the pedestrian is in the air). Combining the two equations (16), the following expression is derived: ax + µ a y = − g ( sin α + µ cos α )
(17)
The integration of equation (17) with respect to time gives the relationship between speed components: vx + µ v y = − g ( sin α + µ cos α ) t + C1
(18)
The integration of equation (18) with respect to time gives the relationship between position components: t2 x + µ y = − g ( sin α + µ cos α ) + C1t + C2 (19) 2 The initial conditions for the determination of integration constants C1 and C2 are: x ( 0) + µ y ( 0) = µ H vx ( 0 ) + µ v y ( 0 ) = v0 ( cos θ + µ sin θ )
(20)
From equations (18), (19) and (20) the values for integration constants are obtained: C1 = v0 ( cos θ + µ sin θ ) C2 = µ H
(21)
The equations (18) and (19) can now be written in the following form:
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L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
vx + µ v y = v0 ( cos θ + µ sin θ ) − g ( sin α + µ cos α ) t x + µ y = µ H + v0 ( cos θ + µ sin θ ) t − g ( sin α + µ cos α )
t2 2
(22)
These equations cover the sliding phase of pedestrian’s motion. In the flight phase the term N y in (16) vanishes: max = − mg sin α
(23)
ma y = − mg cos α
When these equations are integrated twice with respect to time and considering the initial conditions: x(0) = 0 , y (0) = H , vx (0) = v0 cos θ and v y (0) = v0 sin θ the speed components vx = v0 cos θ − gt sin α
(24)
v y = v0 sin θ − gt cos α
and position components 1 x = v0t cos θ − gt 2 sin α 2 1 y = H + v0t sin θ − gt 2 cos α 2
(25)
in the flight phase are obtained. Total throw distance
When the pedestrian stops, the following conditions are fulfilled: vx = 0 , v y = 0 , y = 0 , x = S (total throw distance) and t = tstop (stopping time of the pedestrian). From equations (22)
follows: 0 = v0 ( cos θ + µ sin θ ) − g ( sin α + µ cos α ) tstop
S = µ H + v0 ( cos θ + µ sin θ ) tstop − g ( sin α + µ cos α ) and furthermore:
tstop =
v0 ( cos θ + µ sin θ ) g ( sin α + µ cos α )
v 2 ( cos θ + µ sin θ ) S = µH + 0 2 g ( sin α + µ cos α ) 2
2 tstop
(26)
2
(27)
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L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
Flight distance
The flight distance R and the flight time t R can be derived from (25). The expressions are the same as in (7) and (8). Initial speed
The initial speed v0 is obtained from the second equation in (27):
v0 =
2 g ( sin α + µ cos α )( S − µ H ) cos θ + µ sin θ
(28)
By using trigonometric identities
sin θ =
tan θ 1 + tan 2 θ
and cos θ =
1 1 + tan 2 θ
this expression can be put in the following form:
v0 =
2 g ( sin α + µ cos α )( S − µ H ) (1 + tan 2 θ ) 1 + µ tan θ
(29)
By equating the first derivative to zero: 2 g ( sin α + µ cos α )( S − µ H ) dv0 = ( − sin θ + µ cos θ ) = 0 ⇒ tan θ min = µ 2 dθ ( cos θ + µ sin θ )
(30)
the minimum initial speed vmin is derived:
vmin =
2 g ( sin α + µ cos α )( S − µ H ) 1+ µ 2
(31)
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L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
4
VALIDATION OF THE MODEL
The derived throw model was validated with the data from a real accident. All parameters pertinent to the accident are shown in Table 1. Table 1: Data from a real vehicle/pedestrian collision Quantity gravity acceleration road gradient initial height mass of the pedestrian
Symbol g p H mp
Unit m / s2 % m kg
Value 9.8 - 8.8 1.0 60
mass of the vehicle
mv
kg
942
xL
− m
1.0 1.0
Rm* = xL + Rm
m
18.05
S m* = xL + Sm
m
26.8
η
coefficient of restitution contact phase distance sum of contact phase distance and flight distance (due to physical evidence) sum of contact phase distance and total throw distance (due to physical evidence)
The main goal of this technical analysis is to determine the initial vehicle speed vc 0 , which can be calculated, if the initial pedestrian speed v0 is known (equations (13) and (14)). The speed v0 in the derived model is obtained from equation (29). Because not all of the parameters in this equation are known (i.e. the pedestrian launch angle θ and the coefficient of friction µ ) a direct calculation of v0 is impossible. This problem can be overcome by using the least square method. In this way the values for v0 , θ and µ can be calculated, which satisfy the known parameters (i.e. Rm* and Sm* ). The least square method formulation for this problem is the following: The parameters v0 , θ and µ have to be chosen in such a way that the function
F ( v0 , θ , µ ) = ( R − Rm ) + ( S − S m ) 2
2
(32)
reaches its minimum. The necessary conditions are: ∂F ∂F ∂F = 0, = 0 and =0 ∂µ ∂v0 ∂θ
(33)
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L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
From (33) a system of three nonlinear equations is obtained and have to be solved for v0 , θ and µ . For R and S the expressions from (7) and (27) are used, while Rm and S m are obtained from physical evidence (see Table 1). Using Maple1, the numerical solution shown in Table 2 is calculated. Because the system (33) 0 has multiple solutions, solving was restricted to intervals v0 ∈ ( 36, 72 ) km / h , θ ∈ ( 0, 45 ) and
µ ∈ ( 0.2, 1.2 ) , which suits physical consistency. Table 2: The least square method solution for v0 , θ and µ
xL + R [m] 18.05
xL + S [m] 26.8
vc 0 [km/h] 51
v0 [km/h] 48
θ [0]
µ
26.0
0.56
tR [s] 1.36
tstop
[s] 3.31
Figure 2 shows that the function F ( v0 , θ , µ ) has indeed a minimum for the calculated θ and for a range of initial speeds v0 .
Figure 2: Graphical representation of the function F ( v0 , θ , µ ) for a range of initial speeds v0
1
Maple 7.00, Waterloo Maple Inc. 9
L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
The calculated coefficient of friction µ is in the range of recommended values (from 0.45 to 0.55) [1], however there is no concordance in literature about that and the values may vary from 0.45 to 1.2 [5]. In the discussed accident a higher value for µ has to be adopted, because there is strong evidence that the pedestrian suffered a collision with the edge of the pavement. The coefficient of friction µ is now a known parameter and the reformulated problem has the following form:
(
)
min F ( v0 , θ ) = ( R − Rm ) + ( S − S m ) ; 2
2
(34) ∂F ∂F = 0, =0 ∂v0 ∂θ
The solution for µ = 0.7 is shown in Table 3. Table 3: The least square method solution for v0 , θ ( µ = 0.7 )
xL + R [m] 18.05
xL + S [m] 26.8
vc 0 [km/h] 56
v0 [km/h] 53
θ
0
[] 20.0
tR [s] 1.19
tstop
[s] 2.90
The results show that the vehicle had an initial speed vc 0 of 56 km/h and the pedestrian’s initial speed was 53 km/h. 5
COMPARISON WITH OTHER MODELS
A direct comparison between the model presented in this paper and the Han-Brach’s model (including road gradient) shows that the results produced (applying the least square method) are identical. The main advantage of the proposed model is a more compact form of equations for the determination of the initial pedestrian speed or the total throw distance. The initial vehicle speeds vc 0 obtained by individual models are given in Table 4. Table 4: The initial vehicle speeds vc 0 in km/h calculated by individual models This paper Han-Brach Rotim (incl. road Collins Searle Han-Brach (incl. road (empirical) gradient) gradient) vc 0 62 62.5 61.5 60 56 56
The results (Table 4) show that the use of models neglecting road gradient with accidents occurred on graded roads overestimate the initial vehicle speed. In such cases it is recommended to apply models which incorporate that parameter.
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L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
Figure 3 shows the initial vehicle speed vc 0 dependent on the road gradient and the coefficient of friction for the analyzed accident. It is evident that this two parameters have a significant influence on the determination of the initial vehicle speed.
Figure 3: The initial vehicle speed vc 0 as a function of the road gradient α and the coefficient of friction µ 6
CONCLUSION
The derived throw model considers all the parameters (including road gradient) relevant to vehicle/pedestrian collisions. It is shown that the road gradient has a considerable effect on solutions and must be taken into account in cases with graded roads. The model (together with other models) was validated with data from a real accident and gives identical results as Han-Brach’s model, however the derived formulae are more concise and compact. The results are showing that models neglecting road grade overestimate the value of the initial vehicle speed. The application of the least square method was performed in this paper to overcome the problem of unknown parameters. With the use of this method a set of minimum values of searched parameters is obtained, which suits physical evidence.
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L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
REFERENCES
[1]
A. Toor, M. Araszewski, "Theoretical vs. empirical solutions for vehicle/pedestrian collisions", 2003-01-0883, Accident Reconstruction 2003, Society of Automotive Engineers, Inc., Warrendale, PA, 2003, pp. 117-126.
[2]
I. Han, R. M. Brach, "Throw model for frontal pedestrian collisions", 2001-01-0898, Society of Automotive Engineers, Inc., Warrendale, PA, 2001.
[3]
J. C. Collins, Accident Reconstruction, Charles C. Thomas Publisher, Springfield, Illinois, 1979, pp. 240-242.
[4]
J. A. Searle, "The physics of throw distance in accident reconstruction ", 930659, Society of Automotive Engineers, Inc., Warrendale, PA, 1993.
[5]
F. Rotim, “Elementi sigurnosti cestovnog prometa, Svezak 1, Ekspertize prometnih nezgoda”, Znanstveni savjet za promet JAZU, 1989, pp. 348-350.
[6]
J. J. Eubanks, W. R. Haight, "Pedestrian involved traffic collision reconstruction methodology", 921591, Society of Automotive Engineers, Inc., Warrendale, PA, 1992.
12
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT Leon Bogdanović B. Eng., Milan Batista D. Sc. University of Ljubljana Faculty of Maritime Studies and Transportation Pot pomorščakov 4, SI- 6320 Portorož, Slovenia [email protected], [email protected] ABSTRACT This paper discusses throw models for vehicle/pedestrian collisions. Most of the models proposed in literature neglect the road gradient. For that reason a model including this parameter is derived and analysed with data from a real accident. The results (compared to those produced by other models) are showing that the road gradient has a significant influence on the determination of the initial vehicle speed, which is the main objective of technical analyses of vehicle/pedestrian collisions. 1
INTRODUCTION
Several models describing post-impact motion of pedestrians involved in accidents with motor vehicles can be found in literature ([1], [2], [3], [4] and [5]). These models are either theoretical or empirical. Theoretical solutions give reliable results, however considerable data from real world collisions is needed to solve mathematical equations. On the other hand empirical solutions need no particular data, however their application is limited only to well defined scenarios [1]. This paper deals mainly with theoretical models. An overview of theoretical throw models shows that they are mainly based on mechanics of a mass point. This assumption is acceptable only for the post-impact phase. The pedestrian should be treated otherwise as an articulated, flexible and soft body with limited rigidity (due to skeletal structure) [2], since that is fundamental for the determination of the vehicle/pedestrian interaction. The analysis of the vehicle/pedestrian interaction is the basis for the determination of the launch height of pedestrians. Generally two types of scenarios can occur. For low vehicle frontal shapes, it is common to have a primary collision between the frontal area of the vehicle and the lower part of the pedestrian followed by a secondary collision of an upper part of the pedestrian with the hood, windshield or roof. In low speed collisions, the secondary collision may be nonexistent or minor [2]. If a secondary collision occurs, the vehicle suffers a loss of momentum, which has to be considered in a calculation. After the impact, the pedestrian is thrown and begins the flight phase, which is followed by the sliding phase. These two phases can be modelled either together or separately.
1
L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
2
AN OVERVIEW OF THROW MODELS
An overview of the most relevant theoretical throw models will be presented. Also an empirical model widely used by experts will be described. Most of the theoretical throw models treat the pedestrian as a projectile (mass point). In this way the total throw distance S as a function of the initial speed v0 and all of the other quantities (such as pedestrian launch angle θ , initial launch height H , coefficient of friction µ between the pedestrian and the ground, road gradient α , etc.) or the initial speed v0 as a function of the total throw distance S and all of the other quantities is obtained. It has to be stressed out that the total throw distance is referred as the distance the pedestrian undergoes from impact to his rest position on the ground (see Figure 1).
g
y V0
x
θ
H
α
S
R
Figure 1: Post-impact motion of the pedestrian Collins [3] derived the following equation for the total throw distance S :
S = v0
v2 2H + 0 g 2µ g
(1)
where g = 9,81 m / s 2 is the vertical acceleration due to gravity. If S is known, the initial speed v0 from equation (1) can be expressed: v0 = 2 µ g ( µ H+S) − µ 2 gH
(2)
The main disadvantage of this model is that it neglects the pedestrian launch angle (and also the road gradient).
2
L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
Searle [4] considered the launch angle θ and obtained: v02 ( cos θ + µ sin θ ) S= + µH 2µ g 2
(3)
Solving the above equation for v0 produces the following expression: v0 =
2µ g ( S − µ H ) cos θ + µ sin θ
(4)
Because the value for the pedestrian launch angle is often not available, Searle obtained an expression for the minimum pedestrian initial speed by equating the first derivative to zero ∂v ( 0 = 0 ): ∂θ v0min =
2µ g ( S − µ H ) 1+ µ 2
(5)
This model also neglects the road gradient. Han and Brach [2] split the pedestrian motion into 3 phases. Upon this model the total throw distance S can be written as: S = xL + R + s
(6)
where xL is the distance the pedestrian undergoes in the vehicle/pedestrian contact phase, R is the distance covered in the flight phase and s is the distance between the ground impact of the pedestrian and the rest position. The parameter xL is dependent on vehicle’s frontal geometry, pedestrian center-of-gravity height at impact and vehicle speed after first impact with the pedestrian. The distance R can be expressed as: 1 R = v0t R cos θ − gt R2 sin α 2
(7)
v02 sin 2 θ + 2 gH cos α v0 sin θ tR = + g cos α g cos α
(8)
where
is the flight time from the launch to the contact with the ground and α is the road gradient. The distance s has the form:
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L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
s=
(v
x
+ µvy )
2
2 g ( µ cos α + sin α )
(9)
vx = v0 cos θ − gt R sin α
(10)
v y = v0 sin θ − gt R cos α
(11)
where
and
are the velocity components of the pedestrian at the time of the impact with the ground. By neglecting the road gradient (i.e. α = 0 ) the following equation for the initial speed of the pedestrian can be obtained: v0 =
2 µ g ( S − ( xL + µ H ) )
( µ sin θ + cos θ )
2
(12)
Because the vehicle speed v 'c 0 and the pedestrian initial speed v0 (at the time the pedestrian is launched) usually differ, Han and Brach adopted a coefficient η to relate them: v0 = η v 'c 0
(13)
The initial contact speed of the vehicle vc 0 can be calculated due to the conservation of momentum: vc 0 =
mc + m p mc
v 'c 0
(14)
where mc is the mass of the vehicle and m p is the mass of the pedestrian. An empirical model widely used by experts can be found in [5]. The initial vehicle speed (in km/h) can be calculated due to the following formula: vc 0 = 12 S
(15)
where S is the total throw distance (in m). The reliability of the calculated result is in the range of ± 10%.
4
L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
3
A THROW MODEL INCLUDING ROAD GRADIENT
The real world accidents often occur on the roads with a gradient. Most of the theoretical throw models neglect this parameter, however such simplifications may lead to erroneous results. Therefore a model including this parameter will be derived and analyzed. Basic equations
The notation of variables and parameters is the same as before (Figure 1). From the Newton’s 2nd Law the equations of motion are obtained: max = −mg sin α − µ N y ma y = −mg cos α + N y
(16)
where ax and a y are the acceleration components, m is the mass of the pedestrian, µ is the friction coefficient, g is the vertical acceleration due to the gravity, α is the road grade and N y is the vertical reaction (which is zero when the pedestrian is in the air). Combining the two equations (16), the following expression is derived: ax + µ a y = − g ( sin α + µ cos α )
(17)
The integration of equation (17) with respect to time gives the relationship between speed components: vx + µ v y = − g ( sin α + µ cos α ) t + C1
(18)
The integration of equation (18) with respect to time gives the relationship between position components: t2 x + µ y = − g ( sin α + µ cos α ) + C1t + C2 (19) 2 The initial conditions for the determination of integration constants C1 and C2 are: x ( 0) + µ y ( 0) = µ H vx ( 0 ) + µ v y ( 0 ) = v0 ( cos θ + µ sin θ )
(20)
From equations (18), (19) and (20) the values for integration constants are obtained: C1 = v0 ( cos θ + µ sin θ ) C2 = µ H
(21)
The equations (18) and (19) can now be written in the following form:
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L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
vx + µ v y = v0 ( cos θ + µ sin θ ) − g ( sin α + µ cos α ) t x + µ y = µ H + v0 ( cos θ + µ sin θ ) t − g ( sin α + µ cos α )
t2 2
(22)
These equations cover the sliding phase of pedestrian’s motion. In the flight phase the term N y in (16) vanishes: max = − mg sin α
(23)
ma y = − mg cos α
When these equations are integrated twice with respect to time and considering the initial conditions: x(0) = 0 , y (0) = H , vx (0) = v0 cos θ and v y (0) = v0 sin θ the speed components vx = v0 cos θ − gt sin α
(24)
v y = v0 sin θ − gt cos α
and position components 1 x = v0t cos θ − gt 2 sin α 2 1 y = H + v0t sin θ − gt 2 cos α 2
(25)
in the flight phase are obtained. Total throw distance
When the pedestrian stops, the following conditions are fulfilled: vx = 0 , v y = 0 , y = 0 , x = S (total throw distance) and t = tstop (stopping time of the pedestrian). From equations (22)
follows: 0 = v0 ( cos θ + µ sin θ ) − g ( sin α + µ cos α ) tstop
S = µ H + v0 ( cos θ + µ sin θ ) tstop − g ( sin α + µ cos α ) and furthermore:
tstop =
v0 ( cos θ + µ sin θ ) g ( sin α + µ cos α )
v 2 ( cos θ + µ sin θ ) S = µH + 0 2 g ( sin α + µ cos α ) 2
2 tstop
(26)
2
(27)
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L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
Flight distance
The flight distance R and the flight time t R can be derived from (25). The expressions are the same as in (7) and (8). Initial speed
The initial speed v0 is obtained from the second equation in (27):
v0 =
2 g ( sin α + µ cos α )( S − µ H ) cos θ + µ sin θ
(28)
By using trigonometric identities
sin θ =
tan θ 1 + tan 2 θ
and cos θ =
1 1 + tan 2 θ
this expression can be put in the following form:
v0 =
2 g ( sin α + µ cos α )( S − µ H ) (1 + tan 2 θ ) 1 + µ tan θ
(29)
By equating the first derivative to zero: 2 g ( sin α + µ cos α )( S − µ H ) dv0 = ( − sin θ + µ cos θ ) = 0 ⇒ tan θ min = µ 2 dθ ( cos θ + µ sin θ )
(30)
the minimum initial speed vmin is derived:
vmin =
2 g ( sin α + µ cos α )( S − µ H ) 1+ µ 2
(31)
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L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
4
VALIDATION OF THE MODEL
The derived throw model was validated with the data from a real accident. All parameters pertinent to the accident are shown in Table 1. Table 1: Data from a real vehicle/pedestrian collision Quantity gravity acceleration road gradient initial height mass of the pedestrian
Symbol g p H mp
Unit m / s2 % m kg
Value 9.8 - 8.8 1.0 60
mass of the vehicle
mv
kg
942
xL
− m
1.0 1.0
Rm* = xL + Rm
m
18.05
S m* = xL + Sm
m
26.8
η
coefficient of restitution contact phase distance sum of contact phase distance and flight distance (due to physical evidence) sum of contact phase distance and total throw distance (due to physical evidence)
The main goal of this technical analysis is to determine the initial vehicle speed vc 0 , which can be calculated, if the initial pedestrian speed v0 is known (equations (13) and (14)). The speed v0 in the derived model is obtained from equation (29). Because not all of the parameters in this equation are known (i.e. the pedestrian launch angle θ and the coefficient of friction µ ) a direct calculation of v0 is impossible. This problem can be overcome by using the least square method. In this way the values for v0 , θ and µ can be calculated, which satisfy the known parameters (i.e. Rm* and Sm* ). The least square method formulation for this problem is the following: The parameters v0 , θ and µ have to be chosen in such a way that the function
F ( v0 , θ , µ ) = ( R − Rm ) + ( S − S m ) 2
2
(32)
reaches its minimum. The necessary conditions are: ∂F ∂F ∂F = 0, = 0 and =0 ∂µ ∂v0 ∂θ
(33)
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L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
From (33) a system of three nonlinear equations is obtained and have to be solved for v0 , θ and µ . For R and S the expressions from (7) and (27) are used, while Rm and S m are obtained from physical evidence (see Table 1). Using Maple1, the numerical solution shown in Table 2 is calculated. Because the system (33) 0 has multiple solutions, solving was restricted to intervals v0 ∈ ( 36, 72 ) km / h , θ ∈ ( 0, 45 ) and
µ ∈ ( 0.2, 1.2 ) , which suits physical consistency. Table 2: The least square method solution for v0 , θ and µ
xL + R [m] 18.05
xL + S [m] 26.8
vc 0 [km/h] 51
v0 [km/h] 48
θ [0]
µ
26.0
0.56
tR [s] 1.36
tstop
[s] 3.31
Figure 2 shows that the function F ( v0 , θ , µ ) has indeed a minimum for the calculated θ and for a range of initial speeds v0 .
Figure 2: Graphical representation of the function F ( v0 , θ , µ ) for a range of initial speeds v0
1
Maple 7.00, Waterloo Maple Inc. 9
L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
The calculated coefficient of friction µ is in the range of recommended values (from 0.45 to 0.55) [1], however there is no concordance in literature about that and the values may vary from 0.45 to 1.2 [5]. In the discussed accident a higher value for µ has to be adopted, because there is strong evidence that the pedestrian suffered a collision with the edge of the pavement. The coefficient of friction µ is now a known parameter and the reformulated problem has the following form:
(
)
min F ( v0 , θ ) = ( R − Rm ) + ( S − S m ) ; 2
2
(34) ∂F ∂F = 0, =0 ∂v0 ∂θ
The solution for µ = 0.7 is shown in Table 3. Table 3: The least square method solution for v0 , θ ( µ = 0.7 )
xL + R [m] 18.05
xL + S [m] 26.8
vc 0 [km/h] 56
v0 [km/h] 53
θ
0
[] 20.0
tR [s] 1.19
tstop
[s] 2.90
The results show that the vehicle had an initial speed vc 0 of 56 km/h and the pedestrian’s initial speed was 53 km/h. 5
COMPARISON WITH OTHER MODELS
A direct comparison between the model presented in this paper and the Han-Brach’s model (including road gradient) shows that the results produced (applying the least square method) are identical. The main advantage of the proposed model is a more compact form of equations for the determination of the initial pedestrian speed or the total throw distance. The initial vehicle speeds vc 0 obtained by individual models are given in Table 4. Table 4: The initial vehicle speeds vc 0 in km/h calculated by individual models This paper Han-Brach Rotim (incl. road Collins Searle Han-Brach (incl. road (empirical) gradient) gradient) vc 0 62 62.5 61.5 60 56 56
The results (Table 4) show that the use of models neglecting road gradient with accidents occurred on graded roads overestimate the initial vehicle speed. In such cases it is recommended to apply models which incorporate that parameter.
10
L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
Figure 3 shows the initial vehicle speed vc 0 dependent on the road gradient and the coefficient of friction for the analyzed accident. It is evident that this two parameters have a significant influence on the determination of the initial vehicle speed.
Figure 3: The initial vehicle speed vc 0 as a function of the road gradient α and the coefficient of friction µ 6
CONCLUSION
The derived throw model considers all the parameters (including road gradient) relevant to vehicle/pedestrian collisions. It is shown that the road gradient has a considerable effect on solutions and must be taken into account in cases with graded roads. The model (together with other models) was validated with data from a real accident and gives identical results as Han-Brach’s model, however the derived formulae are more concise and compact. The results are showing that models neglecting road grade overestimate the value of the initial vehicle speed. The application of the least square method was performed in this paper to overcome the problem of unknown parameters. With the use of this method a set of minimum values of searched parameters is obtained, which suits physical evidence.
11
L. Bogdanović, M. Batista
THE THROW MODEL FOR VEHICLE/PEDESTRIAN COLLISIONS INCLUDING ROAD GRADIENT
REFERENCES
[1]
A. Toor, M. Araszewski, "Theoretical vs. empirical solutions for vehicle/pedestrian collisions", 2003-01-0883, Accident Reconstruction 2003, Society of Automotive Engineers, Inc., Warrendale, PA, 2003, pp. 117-126.
[2]
I. Han, R. M. Brach, "Throw model for frontal pedestrian collisions", 2001-01-0898, Society of Automotive Engineers, Inc., Warrendale, PA, 2001.
[3]
J. C. Collins, Accident Reconstruction, Charles C. Thomas Publisher, Springfield, Illinois, 1979, pp. 240-242.
[4]
J. A. Searle, "The physics of throw distance in accident reconstruction ", 930659, Society of Automotive Engineers, Inc., Warrendale, PA, 1993.
[5]
F. Rotim, “Elementi sigurnosti cestovnog prometa, Svezak 1, Ekspertize prometnih nezgoda”, Znanstveni savjet za promet JAZU, 1989, pp. 348-350.
[6]
J. J. Eubanks, W. R. Haight, "Pedestrian involved traffic collision reconstruction methodology", 921591, Society of Automotive Engineers, Inc., Warrendale, PA, 1992.
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