Accident Analysis and Prevention 79 (2015) 1–12
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Accident Analysis and Prevention journal homepage: www.elsevier.com/locate/aap
Impact characteristics of a vehicle population in low speed front to rear collisions Naoya Nishimura a,b , Ciaran K. Simms a, * , Denis P. Wood c a b c
Centre for Bioengineering, School of Engineering, Trinity College, Dublin, Ireland Department of Vehicle and Mechanical Engineering, Meijo University, Japan Denis Wood Associates, Dublin, Ireland
A R T I C L E I N F O
A B S T R A C T
Article history: Received 3 November 2014 Received in revised form 2 February 2015 Accepted 3 February 2015 Available online xxx
Rear impact collisions are mostly low severity, but carry a very high societal cost due to reported symptoms of whiplash and related soft tissue injuries. Given the difficulty in physiological measurement of damage in whiplash patients, there is a significant need to assess rear impact severity on the basis of vehicle damage. This paper presents fundamental impact equations on the basis of an equivalent single vehicle to rigid barrier collision in order to predict relationships between impact speed, maximum dynamic crush, mean and peak acceleration, time to common velocity and vehicle stiffness. These are then applied in regression analysis of published staged low speed rear impact tests. The equivalent mean and peak accelerations are linear functions of the collision closing speed, while the time to common velocity is independent of the collision closing speed. Furthermore, the time to common velocity can be used as a surrogate measure of the normalized vehicle stiffness, which provides opportunity for future accident reconstruction. ã 2015 Elsevier Ltd. All rights reserved.
Keywords: Rear end collision Accident reconstruction Regression analysis Impact modeling
1. Introduction The injuries arising from rear impact are mostly low severity since the collision speed is generally much lower than for frontal impact. Nonetheless, due to its frequency, this crash mode results in 30% of automotive-related trauma, and low-severity rear impact accounts for more long-term injury than any other crash mode (Viano, 2002). Accordingly, the annual cost of whiplash type injuries is billions of dollars in the US alone (Viano and Olsen, 2001). The most significant rear impact related injuries are a set of soft tissue injuries frequently called whiplash associated disorders (WAD), which remain surprisingly poorly understood, despite significant research efforts. Injury mechanisms proposed include a hyperextension mechanism, an eccentric contraction mechanism, a hydrodynamic mechanism, and combined mechanisms of axial loading, shear force and bending (Yoganandan et al., 2013). This plethora of proposed mechanisms debated in the literature indicates the lack of consensus in relation to the precise biomechanical causes of whiplash. Nonetheless, the primary cause is relative motion between the torso and the head, driven by a
* Corresponding author at: FTCD, Parsons Building, Trinity College, Dublin, Ireland. Tel.: +353 1 896 3768. E-mail address: [email protected] (C.K. Simms). http://dx.doi.org/10.1016/j.aap.2015.02.001 0001-4575/ ã 2015 Elsevier Ltd. All rights reserved.
combination of the vehicle acceleration time history and the interaction of the occupant with the seat and other vehicle components. A particular challenge for low severity rear impact injuries is the tangible measurement of injury, though recent findings suggest that MRI is capable of quantifying neuromuscular degeneration in chronic whiplash (Elliott et al., 2014). A recent review has concluded that there is growing evidence that a claimant’s physiological and psychological stress response is a very significant factor in persistent symptoms following whiplash injury (Worsfold, 2014). Similarly, given the legal context of whiplash, a toolkit for identifying cases with a crash severity so low that the chances of whiplash injury are remote has been proposed (Moser et al., 2011). Given the difficulty in physiological measurement of damage in whiplash patients, there is a significant need to assess rear impact severity on the basis of vehicle damage. In particular, fundamental principles of injury biomechanics dictate that it is strongly desirable to have methods to assess the magnitude of the acceleration pulse as well as the velocity change imposed on the struck vehicle. However, for very low speed cases, there is frequently no visible damage to the vehicle at all, despite the fact that whiplash symptoms are frequently reported. There have been previous modeling approaches to reconstructing low velocity rear impact collisions. As reviewed by Scott et al. (2010, 2012), these approaches have broadly followed either the
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differential equation approach (Thomson and Romilly, 1993; Ojalvo and Cohen, 1997; Ojalvo et al., 1998; Brach 2003; Scott et al., 2010, 2012) which require explicit definition of the effective stiffness of the two-vehicle system but which provide acceleration time histories, or a momentum energy restitution approach (Bailey et al., 1995; Cipriani et al., 2002; Happer et al., 2003) which does not require an estimate of vehicle stiffness, but consequently limits predictions to velocity change rather than acceleration time history. From an injury biomechanics perspective, the differential equation approach which provides acceleration time histories is far preferable, but the variety of bumper designs make the effective stiffness of a specific vehicle pair involved in a rear end collision difficult to predict. There is therefore the need for a generic reconstruction model which can account for mean and variation. The recent work by (Scott et al., 2010, 2012; Bonugli et al., 2014; Funk et al., 2014) has shown that, when experimental knowledge of the combined bumper deformation behavior is known for a specific vehicle pair, the impact response for a specific collision can be found. However, the tests they completed showed non-linear and variable bumper stiffnesses, and the approach yields nontrivial errors in the coefficient of restitution. Furthermore, for assessment of visible damage, a single quasi static test is not sufficient. In Switzerland, the AGU accident research group has performed a set of 45 unbraked staged full overlap and over-ride/under-ride rear impact collisions with an impact speed range of 8–27 km/h (AGU, 2014). Fig. 1 shows the non-linearity and variability of the resulting acceleration time curves, leading to the preliminary conclusion that there is so much variability in these kinds of collisions due to vehicle design variations and impact configuration that it is impractical to develop a generic model. However, further analysis shows that significant trends can be identified in this dataset. Accordingly, the goal of this paper is to present the fundamental impact equations to predict relationships between impact speed, maximum dynamic crush depth, mean and peak acceleration, time to common velocity and vehicle stiffness, and to use these as a basis for regression analysis for the 45 low speed rear impact cases published by AGU (2014) for full overlap and over-ride/under-ride cases. This approach provides considerable insight into the mechanics of low speed rear impact collisions.
2. Regression analysis of staged low speed front to rear collisions
published by AGU (2014).
2.1. Fundamental collision modeling to inform staged tests regression analysis Here fundamental impact modeling is used to develop relationships between impact speed, crush depth, mean and peak acceleration and vehicle stiffness. These form the basis for regression analysis for the staged low speed rear impact tests published by AGU (2014). A detailed multibody or finite element modeling approach is not necessary, since a considerable body of experimental data is freely available through the AGU. The fundamental modeling approaches show which functional forms the regression analysis of the measurable impact variables should take. 2.1.1. Front to rear collision of two vehicles When two vehicles (m1,m2) are subject to a collinear impact, the system can be regarded as equivalent to a single vehicle impacting a rigid barrier. The equivalent mass meq impacts the rigid barrier at the collision closing speed Vccs and rebounds at the separating speed Vsep given by the following equations from conservation of momentum and conservation of energy considerations: meq ¼
m1 m2 ; m1 þ m2
(1)
V ccs ¼ V eq ¼ V 1 V 2 ; V sep ¼ V 02 V 01 ;
(2)
where Veq is the pre-impact velocity of the equivalent mass and this is equal to the collision closing speed, Vccs, of the colliding pair where V1, V2 are the pre-impact velocities of the two vehicles. The equivalent acceleration aeq and each vehicle acceleration (a1, a2) are related as follows: aeq ¼ a2 a1 ;
a1 ¼
m2 aeq ; m1 þ m2
a2 ¼
m1 aeq : m1 þ m2
2. Methods There are two main components to the work performed:
(3)
1. Fundamental collision modeling to inform the regression
analysis of the staged test data.
(a)
30
(b) 30
Vccs < 10km/h
Vccs < 10km/h
10km/h < Vccs < 13km/h
25
13km/h < Vccs < 17km/h
17km/h < Vccs < 20km/h
20
20km/h < Vccs
a e q [G ]
a e q [G ]
17km/h < Vccs < 20km/h
20
20km/h < Vccs
15 10 5
15 10 5
0
0 0
0.05
0.1
0.15
-5 -10
10km/h < Vccs < 13km/h
25
13km/h < Vccs < 17km/h
0.2
0.25
0
0.05
0.1
0.15
0.2
0.25
-5
T ime [s ]
-10
T ime [s ]
Fig. 1. Equivalent acceleration–time histories of the two vehicles collisions (front to rear collisions) published by AGU (2014) for (a) full engagement test and (b) under-ride/ over-ride tests.
N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
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Fig. 2. (a) Barrier force F as a linear function of deformation x, and (b) sinusoidal acceleration time relationship for a linear spring–mass collision model. Peak acceleration â is also shown.
The total change in velocity (DVtotal) and each vehicle velocity change (DV1, DV2) are given as:
DV total ¼ DV 1 þ DV 2 ¼ DV ccs þ V sep DV 1 ¼
m2 DV total ; m1 þ m2
DV 2 ¼
m1 DV total : m1 þ m2
2.1.2. Linear acceleration–displacement model (spring–mass model) The equations of motion of a linear spring–mass model of a single vehicle (mass m) colliding with a rigid barrier, where the spring constant (bumper stiffness) during the loading and unloading phases are k and kR are given as follows: m€ x ¼ kx: loading m€ x ¼ kR ðx CÞ½x C res ;C res 0 : unloading
(4)
From the law of conservation of energy, the nominal linear normalised stiffness k/m of the colliding pair is obtained from the test parameters Vccs and the maximum dynamic crush (Cdmax) as follows: k V ccs 2 ¼ (5) m C dmax This relationship can also be derived from the dynamic acceleration versus displacement response. The acceleration time relationships from staged tests shown in Fig. 1 are highly complex, but a subjective review indicates for modeling purposes that two simplified approaches are reasonable simplifications to represent the experimental traces: a linear acceleration–displacement model (a classical linear spring mass system) or a linear acceleration time model. With either of these two assumptions, closed form solutions are available to provide relationships between the principal impact parameters such as mean acceleration, peak acceleration, collision duration etc. An overview of these two modeling approaches is provided here.
(6)
with initial conditions at t = 0: x ¼ 0; x ¼ 0; x_ ¼ v; € _ € where m; x; x; x and Cres are respectively the mass, displacement, velocity, acceleration and residual deformation of the vehicle.The linear loading and unloading stiffnesses are k and kR respectively, while the impact speed is n and the maximum dynamic crush is Cdmax. A schematic of the resulting barrier force displacement relationship and acceleration time relationships during the loading and unloading phases is shown in Fig. 2. Further details of this model are given in Appendix A. 2.1.3. Linear acceleration–time model Considering the acceleration behavior of the vehicle as a linear function of time, the acceleration, velocity and deformation of the vehicle at the loading (closing) phase (0 t Tcv) are derived from integration: 2 € x ¼ a ¼ bt ¼ C 0 vt ¼ 2 vt T cv
1 1 1 x_ ¼ v bt2 ¼ v C 0 vt2 ¼ v 2 vt2 2 2 T cv
Fig. 3. Vehicle acceleration a as (a) a linear function of time t, and (b) a power function of relative displacement x for the linear acceleration–time elementary model.
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Table 1 Relationship between principal collision parameters for the linear acceleration displacement and linear acceleration time models. Parameter
Linear acceleration–displacement model
Linear acceleration–time model
Cdmax
2
2 3V ccs T cv
at â
pV ccs T cv Vccs/Tcv pV
^ ^ t =a a ^ ^d =a a
2/p 1/2
2
ccs
qffiffiffiffiffiffiffiffiffiffi k=m ¼ p=2T cv : for the linear acceleration displacement model
v¼
C 0 ¼ 2=T 2cv : for the linear acceleration time model
Vccs/Tcv V ccs
^¼2 a 1/2 3/8
T cv
=T cv
;v ¼
1 1 1vt3 x ¼ vt bt3 ¼ vt C 0 vt3 ¼ vt 2 ; 6 6 3T cv
(7)
where Tcv is the time required for the two vehicles to achieve a common velocity and the impact speed is n. The initial conditions x ¼ 0, and b is the slope of the linear at t = 0 are x ¼ 0; x_ ¼ v; € acceleration time function, and C0 is the slope constant. A schematic of the resulting barrier force displacement relationship and acceleration time relationships during the loading and unloading phases is shown in Fig. 3. Further details of this model are given in Appendix A. 2.1.4. Mean acceleration, peak acceleration and maximum dynamic deformation When the collision closing speed Vccs and the time to common velocity Tcv of two vehicles are known collision phenomena, the mean acceleration a, peak acceleration â and maximum dynamic deformation Cdmax can be calculated in closed form for both the linear spring mass model and the linear acceleration time model. The mean acceleration at can be considered on a time basis from integration of the acceleration–time curve but also on a displacement basis ad from integration of the acceleration–displacement curve. These relations are summarized in Table 1. For both models, the maximum dynamic crush Cdmax is proportional to the collision closing speed Vccs and the time to common velocity Tcv, while the time-based mean acceleration at is proportional to Vccs and inversely proportional to Tcv. The relationship between the mean and the peak acceleration is however different for the two models, due to the different shape of the acceleration time pulse. The following relations are derived from the natural frequency v and C0 given by Tcv as related to both elementary models:
(a) 14
rffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ 1:232C 0 : m
These equations based on elementary models show that the normalised linear stiffness (k/m) is related to the inverse of the square of the time to common velocity ð1=T 2cv Þ. Thus, Table 1 shows that the theoretical dependencies of maximum dynamic crush, mean acceleration and peak acceleration on collision closing speed and time to common velocity are fixed if a theoretical choice for the acceleration time relationship is made. Accordingly, these derivations can be used to guide regression based interrogation of the AGU staged low speed rear end collision tests (AGU, 2014). 2.2. Regression analysis of AGU staged low speed front to rear collisions AGU (2014) have carried out 45 unbraked front-to-rear impacts, of which 23 can be categorised as full engagement and 22 as over-ride/under-ride. Tables B1 and B2 (see Appendix B) detail the test numbers, the vehicles involved, impact speeds and related data. In the full engagement tests, the impact speed range was 8–21.5 km/h. There are 30 different car types with bullet vehicle (vehicle 1) mass range from 843–1903 kg and target vehicle (vehicle 2) mass range 838.5–1860 kg. For the over-ride/under-ride tests, the bullet vehicle mass ranged from 930–1997 kg and the target vehicle (vehicle 2) mass range was 930–2075 kg. Fig. 1 shows the acceleration time history curves for all of these tests. In the following sections, regression analysis is performed on the principal impact parameters of the AGU staged tests. 2.2.1. Relation between mean and peak acceleration, maximum dynamic crush and impact speed Given the linear relationship between time-based mean acceleration at (up to the time of common velocity Tcv) and collision closing speed Vccs implicit in both the linear acceleration
(b) 14 A(0 -Tcv)
12
10
10
Mean a t [G ]
12
Mean a t [G ]
(8)
8 6 4
A(0 -Tcv)
8 6 4
y = 0.3924x
2
y = 0.3292x
2
R² = 0.7049
R² = 0.6564
0
0
0
5
10
15
V c c s [km/h]
20
25
30
0
5
10
15
20
25
30
V c c s [km/h]
Fig. 4. The relationship between equivalent mean acceleration at (up to the time of common velocity Tcv) and collision closing speed Vccs for AGU test data for (a) full engagement tests and (b) under-ride/over-ride tests. Forced-zero linear regression results are also shown.
N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
(a)
(b)
30
25
25
20
20
P eak a [G ]
P eak a [G ]
30
15
10
y = 0.7943x
5
5
15
10
y = 0.651x
5
R² = 0.512
R² = 0.5829
0
0 0
5
10
15
20
25
30
0
5
10
15
20
25
30
V c c s [km/h]
V c c s [km/h]
Fig. 5. The relationship between the peak acceleration â and collision closing speed Vccs for AGU test data for (a) full engagement tests and (b) under-ride/over-ride tests. Forced-zero linear regression results are also shown.
displacement and linear acceleration time models (Table 1). Fig. 4 shows the results of forced-zero linear regression of these parameters for the 23 full engagement and 22 over-ride/underride staged AGU tests. Fig. 5 shows the equivalent linear relationship between peak acceleration and collision closing speed Vccs. Fig. 6 shows the relationship between maximum dynamic crush Cdmax and collision closing speed Vccs. 2.2.2. Relationship between time to common velocity and collision closing speed Fig. 7 shows the relationship between the time to common velocity Tcv of the two vehicles for each test and the collision closing speed (Vccs). The parameter Tcv, is calculated from the velocity–time curve. Linear regression results are also shown. 2.2.3. Linear acceleration–displacement or linear acceleration–time models It is possible to assess the appropriateness of the spring–mass and linear acceleration–time modeling approaches by considering the predicted acceleration time history relationship for the two models in comparison to actual experimental cases for both full overlap and over-ride/under-ride cases, see Figs. 8 and 9. The overall predictive capacity of the models for all staged test cases can be assessed from the predicted relationship between mean and peak acceleration for the two modeling approaches (see Table 1) in comparison with the actual response for all the AGU tests, see Fig. 10. 0.4
(a)
2.2.4. Mean acceleration – comparison of model and experimental results Table 1 shows that the mean acceleration on a time basis (at ) for both the linear spring mass model and the linear acceleration time model is equal to the collision closing speed (Vccs) divided by the time to common velocity (Tcv). Fig. 11 compares the model predictions for at with the AGU mean acceleration values for the full engagement and over-ride/under-ride tests. Fig. 12 shows the experimental and model predictions of ad versus peak acceleration â for the full engagement and over-ride/under-ride tests. 2.2.5. Relationship between normalised vehicle stiffness and time to common velocity Fig. 13 shows the relation between the experimental normalised equivalent stiffness (k/m) obtained from Vccs and Cdmax and the inverse square of the experimental values of the Tcv. Regression analysis shows a linear relation between (k/m) and 1=T 2cv with high correlation which compares with the predictions from the elementary models, Eq. (8). 3. Discussion For low velocity rear impact collisions, a method is needed to assess the magnitude of the acceleration pulse as well as the velocity change imposed on the struck vehicle. This is difficult, since for very low speed cases there is frequently no visible damage to the vehicle at all, despite the fact that whiplash associated
(b) 0.4
0.35
0.35
0.3
0.3
C dmax [m]
C dmax [m]
0.25 0.2
0.25 0.2 0.15
0.15
y = 0.0133x
0.1
y = 0.0142x
0.1
R² = 0.1537
R² = 0.6784
0.05
0.05
0
0
0
5
10
15
V c c s [km/h]
20
25
30
0
5
10
15
20
25
30
V c c s [km/h]
Fig. 6. The relationship between maximum dynamic crush Cdmax and collision closing speed Vccs for AGU test data (AGU, 2014) for (a) full engagement tests and (b) underride/over-ride tests. Forced-zero linear regression results are also shown.
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0.14
0.14
0.12
0.12
0.1
0.1
T c v [s ]
(b) 0.16
T c v [s ]
(a)0.16
0.08
0.08 0.06
0.06
0.04
0.04
y = -0.0008x + 0.0869
0.02
R² = 0.0405
5
10
15
20
25
R² = 0.0233
0
0 0
y = -0.0007x + 0.0994
0.02
30
V c c s [km/h]
0
5
10
15
V c c s [km/h]
20
25
30
Fig. 7. Dependence of the time to common velocity of two vehicles Tcv on the collision closing speed Vccs for AGU test data (AGU, 2014): (a) full engagement tests (Tcv: mean = 0.075, standard deviation = 0.015) and (b) under-ride/over-ride test (Tcv: mean = 0.087, standard deviation = 0.017). Tcv is calculated from the velocity–time curve.
symptoms are frequently reported. A further complication in reconstructing individual cases is the large variety of bumper systems which leads to significant variability in acceleration time relationships, as shown in Fig. 1. The analysis presented in this paper is predicated on the assumption of an equivalent single vehicle model to represent a low speed two vehicle collision which is grounded in fundamental mechanics. Application of this approach to the staged AGU low velocity rear impact tests (AGU, 2014) shows a linear relationship between both mean and peak acceleration and the vehicle collision closing speed, with high correlation coefficients (R2 > 0.5) for both
full engagement and over-ride/under-ride cases, see Figs. 4 and 5. A similar relationship between maximum dynamic crush and collision closing speed is seen in Fig. 6, although the correlation is significantly reduced for the over-ride/under-ride cases (R2 > 0.15 due to two outliers). Overall, each parameter at, ad , â and Cdmax shows a linear relation to Vccs (Figs. 4–6). Two possible acceleration time relationships are considered: a classic spring mass system which has a linear acceleration displacement relationship, or a linear acceleration time model. Table 1 shows that the mean acceleration at and peak acceleration â are proportional to collision closing speed Vccs and inversely
Fig. 8. Model and experimental acceleration–time curves for (a-1:a-3) the linear acceleration–displacement model and (b-1:b-3) the linear acceleration–time model for selected full engagement tests from AGU (2014).
N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
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Fig. 9. Model and experimental acceleration–time curves for (a-1:a-3) the linear acceleration–displacement model and (b-1:b-3) the linear acceleration–time model for selected over-ride/under-ride tests from AGU (2014).
proportional to the time to common velocity Tcv for both the linear spring mass and the linear acceleration time model approaches. It follows that Tcv can be considered as a measure of the stiffness of the vehicle deformation, which, for a nominally linear spring characteristic should be independent of collision closing speed. Fig. 7 shows that this is effectively the case, both for the full engagement and over-ride/under-ride AGU cases, with R2 < 0.05 in
both cases. The low R2 shows that Tcv can be considered to be independent of impact velocity with a mean value of 0.075 s for the full engagement test and 0.087 s for the over-ride/under-ride tests. This also indicates that the mean values of normalized stiffness k/m for the population of cars examined can be considered to be independent of car mass and impact velocity.
^: Fig. 10. Dependence of the mean acceleration at to time to common velocity Tcv on the peak acceleration â during collision for (a) full engagement test (at =a ^ : mean = 0.52, standard deviation = 0.083) from AGU tests (AGU, 2014). mean = 0.53, standard deviation = 0.091) and (b) under-ride/over-ride test (at =a
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Fig. 11. Relationship between the predicted and experimental at for (a) full engagement and (b) over-ride/under-ride cases for the AGU tests (AGU, 2014).
Application of these models to predict the loading and unloading time history relationship of staged crash tests (Figs. 8 and 9) shows that both the classic spring mass model and the linear acceleration time model provide very good overall fits to the experimental data, with inputs based only on time to common velocity (Tcv) and collision closing speed Vccs and the rebound velocity. Overall, Figs. 10–12 combine to show that the linear acceleration time model provides better predictions of the vehicle acceleration responses. Regression analysis indicates that the normalised vehicle stiffness is proportional to the inverse square of the time to common velocity (Tcv) as predicted by the fundamental impact modeling, see Fig. 13. This relationship is highly linear for both the full engagement and over-ride/under-ride cases (R2 > 0.85), and it indicates that the vehicles tested behave in a substantially similar manner in low speed rear impacts and that Tcv can be used as a surrogate measure of vehicle stiffness. The high R2 values in Fig. 13 for both the full engagement and over-ride/under-ride cases shows that Tcv is an excellent surrogate measure of the normalised equivalent vehicle stiffness (k/m). This is further demonstrated from the slope of the acceleration displacement relationships i.e., (k/m) for each test shown in Fig. 14, which show that the inverse square of the time to common velocity ð1=T 2cv Þ is a highly linear function of the slope of the acceleration displacement curves (R2 > 0.97), with very little scatter. In very recent work (Bonugli et al., 2014) have presented linear stiffness corridors from quasi static testing of bumper pairs, but since the vehicle masses are not
known for their data, direct comparison with the normalised stiffness data in Fig. 13 of this paper is not possible. Table 1 shows that for the linear acceleration time model, the mean acceleration at can be found from the collision closing speed Vccs divided by the time to common velocity Tcv. This is confirmed by the correspondence between the predicted and measured values of mean acceleration, ad shown in Fig. 11. The above suggests that measurement of time to common velocity, Tcv and collision closing speed Vccs could provide a basis for a relatively straightforward experimental characterisation of low speed vehicle stiffness and response. Although it has recently been found that peak accelerations on the basis of the dynamic stiffness of a bumper pair are within 5% of those predicted on the basis of the static bumper stiffness (Funk et al., 2014), the very high correlations shown in Fig. 13 indicate that a single measure of Tcv in the dynamic test can give an excellent measure of effective bumper stiffness. The regression data for the full engagement and under-ride/ over-ride tests in Fig. 6 show that the mean relationships between Cdmax (m) and Vccs (m/s) are: V ccs ¼ 21:5C dmax ðfull engagementÞ
V ccs ¼ 20C dmax ðunder ride=over rideÞ:
(9)
(10)
These equations provide a general indication of collision speed where maximum dynamic crush can be estimated. Table 1 also
Fig. 12. Relationship between mean acceleration ad and peak acceleration for the experimental AGU cases (AGU, 2014) for (a) full engagement tests and (b) the over-ride/under-ride tests. The theoretical predictions from the linear acceleration–time and linear acceleration–displacement models are also shown, together with a forced zero linear regression of the experimental cases.
N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
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Fig. 13. Relationship between the equivalent normalized stiffness k=m ¼ V 2ccs =C 2dmax and the inverse square of the time to common velocity ð1=T 2cv Þ for the linear acceleration time model for (a) full engagement tests and (b) the over-ride/under-ride tests from the AGU (2014).
shows that for the linear acceleration time model the maximum dynamic crush Cdmax is related to Vccs and Tcv as follows: 2 C dmax ¼ V ccs T cv : 3
(11)
opposed to the overall pulse duration, is substantially independent of collision closing speed for a set of full-scale staged collisions, and that Tcv can therefore be used as an explicit surrogate measure of the normalised vehicle stiffness. 4. Limitations
Therefore, once a relationship between residual crush (Cres), which can be measured after a collision, and Cdmax is established, since Tcv can be estimated on the basis of normalized vehicle stiffness, then this approach might facilitate more exact estimation of collision closing speed in the future. In this regard (Bonugli et al., 2014) have established the mean relationship Cdmax = 1.18Cres, although this encompasses full engagement, under-ride/over-ride and concentrated impact (tow-bar) data and does not fully account for those impacts where no residual crush is present. This paper has analysed full engagement and over-ride/underride collisions, but only for cases where the principal direction of force is aligned with the vehicles’ longitudinal axes, and this is a limitation of the present work. Nonetheless, the analysis shows for the first time that linear acceleration time models applied to both the loading and unloading phases of low speed rear impact yield closed form solutions which provide a close match to the available experimental data. Furthermore, although (Scott et al., 2010) found from a parametric modeling study that the overall pulse duration is independent of impact speed for fixed restitution and stiffness, they also found significant variations in overall pulse duration when restitution and stiffness are varied. In this paper it has been shown for the first time that the time to common velocity (Tcv), as
This analysis applies to unbraked front to rear full engagement and under-ride/over-ride impacts over the collision speed range 8–22 km/h for car masses in the range 830–2075 kg, mass ratio range for target car/bullet car of 0.5–1.8 and nominal equivalent normalised stiffness range 100–900/s2. Future research will apply this approach to offset and tow-bar crashes and to crashes involving trucks, buses, etc. 5. Conclusions Regression analysis of the staged AGU rear end collision tests (AGU, 2014) performed on the basis of fundamental impact modeling of an equivalent single vehicle system shows that, despite the complexity of the experimental linear acceleration time curves, either a linear acceleration time or a linear spring mass model provide very strong insights into the mechanics of low speed rear impact cases. The equivalent mean and peak accelerations are linear functions of the collision closing speed, while the time to common velocity is independent of the collision closing speed. Furthermore, the time to common velocity can be used as a
Fig. 14. Dependence of the slope a/Cd (forced regression for actual acceleration–displacement curve) on 1=T 2cv for AGU (2014) tests (a) the full engagement test and (b) the over-ride/under-ride tests. The normalized stiffness values reported here for the colliding pairs equate to the normalized stiffness of each car when these are the same as each other.
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surrogate measure of the normalized vehicle stiffness, which provides opportunity for post-hoc accident reconstruction. Acknowledgment The authors gratefully acknowledge Markus Muser from AGU for generously sharing their data. Appendix A.
A1. Linear spring mass model derivations
Fig. A1. Vehicle acceleration a for the loading and unloading phase as a linear function of time t.
The spring constant k is given by the impact velocity n and the maximum deformation Cdmax from the law of conservation of energy: k¼
mv2 C dmax 2
:
(A.1)
x¼
Normalising with respect to mass yields the ‘normalised stiffness’ k/m:
k V ccs ¼ m C dmax
pffiffiffiffiffiffiffiffiffiffiffiffi (loading), vR ¼ kR =m (unloading) and the initial conditions, as follows: Loading (closing) phase 0 t 2pv v
sinvt
v
x_ ¼ vcosvt
(A.3)
2 (A.2)
The normalised stiffness k/m is also the slope of the normalised force (F/m), i.e., acceleration (a) versus displacement (Cd) characteristic of the impact. The deformation, velocity and acceleration of the vehicle are pffiffiffiffiffiffiffiffiffiffi derived from integration using the natural frequency v ¼ k=m
€ x ¼ vvsinvt Unloading (rebounding) phase ð2pv t 2pv þ 2p vR Þ o n vv p v 1 þ x ¼ 2 cosvR t 2v v vR
Table B1 AGU full engagement test data (N = 23). Vehicle 2 is stationary in these tests. Test no.
AGU test no.
1 2 3
AZT_02.50 BMW 318 ATZ_02.52 Opel Vectra HS_11 Renault Espace
4 5 6
HS_35 WH0217 HS_03
Citroen C2 Opel Vectra A Peugeot 306
7 8
HS_24 WH0215
Mazda 121 Renault Clio
9 10
HS_26 HS_18
Alfa Romeo 156 VW New Beetle
11
HS_12
Opel Corsa C
12 13 14 15 16 17
HS_06 ATZ_02.51 ATZ_03.18 ATZ_02.54 HS_DTC25 HS_15
Vehicle 1 (bullet) Vehicle 2 (target)
20 21
MCC Smart Audi A3 Opel Zafira VW T4 VW Golf 3 Mercedes-Benz A-Klasse ATZ_02.53 Audi A4 HS_36 Mercedes Benz E320 HS_22 Renault Megane HS_43 Renault Kangoo
22 23
HS_05 HS_38
18 19
Opel Astra F VW Polo
Impact velocity (km/h)
Mass 1 (m1) Mass 2 (m2) Meq (kg) (kg) (kg)
âactual (g)
adisp (g)
atime (g)
Tcv (s)
T0–1 g (s)
Cdmax (m)
8.6 8.8 12.4
1400 1275 1616.5
1255 1200 1028
661.78 4 618.18 1.9 628.38 2.8
7.38 4.57 5.32
2.24 2.43 3.28
2.27 2.46 3.50
0.107 0.102 0.101
0.152 0.123 0.131
0.165 0.159 0.214
15.3 14.5 17.7
1182.5 1271 1330
1183 1026 1341
591.38 567.72 667.74
3.8 3.1 4.3
10.00 19.92 9.80
4.33 4.06 3.80
4.67 4.71 5.66
0.092 0.124 0.089 0.122 0.088 0.177
0.241 0.219 0.275
18.4 20.1
940.5 1026
1259 1271
538.34 5.3 567.72 3.2
17.56 13.85
6.35 6.27
6.39 7.14
0.081 0.107 0.080 0.105
0.287 0.273
14.5 14.1
1423 1347.5
1170.5 995.5
642.23 4.7 572.53 5.6
11.49 15.63
4.40 4.53
5.15 5.18
0.079 0.128 0.077 0.125
0.209 0.201
Vrebound (Km/h)
Opel Vectra Audi A3 Opel Corsa C VW Polo Renault Clio Mercedes W201 VW Golf IV Opel Vectra A Fiat Brava Nissan Micra K11 Renault Espace Opel Astra F BMW 318 VW Golf Audi A4 Toyota Prius Renault Clio
19.8
1028
1616.5
628.38 4
16.95
6.16
7.40
0.076
0.109
0.247
21.5 10.1 11.5 8 20.3 15.2
843 1222 1530 1878 1125 1174
1278 1380 1325 1295 1442 1195.5
507.95 648.10 710.07 766.47 631.96 592.33
5.7 3.9 4.1 3 3.9 5.9
9.50 8.73 9.50 5.03 19.92 12.82
7.18 3.91 4.43 3.17 7.88 5.75
8.24 4.11 4.79 3.33 8.48 6.67
0.074 0.069 0.069 0.068 0.068 0.065
0.116 0.103 0.096 0.099 0.087 0.106
0.282 0.128 0.144 0.089 0.279 0.185
VW T4 Fiat Punto
11.1 13
1308 1903
1860 1243.5
767.95 3 752.07 4.6
6.89 13.34
4.17 5.38
4.96 6.08
0.063 0.095 0.062 0.093
0.110 0.150
1200 1323.5
972.5 1353.5
537.17 669.17
4.8 5.7
10.00 16.95
5.33 7.71
5.98 8.90
0.061 0.094 0.057 0.084
0.135 0.177
1279.5 1183
838.5 1193
506.54 3.7 593.99 4.5
12.20 25.77
5.86 8.38
6.64 8.69
0.055 0.089 0.051 0.073
0.134 0.166
Toyota Yaris 12.9 Alfa Romeo 17.3 147 MCC Smart 13.9 Citroen C2 16.8
^ = 4.11– Ranges: mass 1 = 843–1903 kg; mass 2 = 838.5–1860 kg; Vccs = 8–21.5 km/h; Vsep = 1.9–5.9 km/h; Tcv = 0.051–0.107 s; Ttotal = 0.073–0.177 s; Cdmax = 0.089–0.287 m; a 25.77 g; at = 2.27–8.90 g; ad = 2.87–7.93 g.
N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
x¼
vv p x_ ¼ sinvR t vR 2v
3
2v2 ^ a
x þ
2
11
!
x 1€ : 3a ^2
(A.6)
(A.4) x is â, the displacement x is: When acceleration €
p € : x ¼ vvcosvR t 2v
^: C dmax ¼ 4v2 =3a
When t ¼ T cv ¼ 2pv, the velocity x_ is zero and the acceleration and displacement are the peak values â and Cdmax. At the end of € rebound, when the time t ¼ T total ¼ 2pv þ 2p vR , the acceleration x is 0 _ zero, the velocity x is v and the displacement x is the residual deformation Cres. A2. Linear acceleration time model derivations When the time t = Tcv, the acceleration € x and displacement x are the peak values â and Cdmax and the velocity x_ is zero. Furthermore, Tcv is independent of the initial impact velocity. The slope constant term C0 is given by the velocity conditions and the Tcv relation as follows:
T cv ¼
2v 1=2 2 2 ;b ¼ 2 v ¼ C 0 v; C 0 ¼ 2 : b T cv T cv
(A.5)
The acceleration–displacement curve for the loading phase is found by eliminating the time t from Eq. (6) and using the relation ^ ¼ 2v=T cv to yield: a
(A.7)
The acceleration–displacement curve is normalized by eliminating the impact velocity n using Eq. (A.6) to yield: x C dmax
¼
3 x 1 € x 3€ : ^ 2 a ^ 2a
(A.8)
The acceleration behavior of the vehicle for the unloading (rebounding) phase ðT cv t T total ¼ T cv þ T r Þ, is also considered as a linear function of time, as shown in Fig. A1. The acceleration, velocity and deformation of the vehicle during the unloading phase are derived from integration using initial ^ at tr ¼ 0 : conditions x ¼ C dmax ; x_ ¼ 0; € x ¼ a € ^ þ bu t r ; x ¼ a ¼ a tr ¼ t T cv
1 ^tr þ bu t2r x_ ¼ a 2
Table B2 AGU over-ride/under-ride (N = 22). Vehicle 2 is stationary in these tests. Test no.
AGU test no.
Vehicle 1 (bullet)
24 25 26 27 28 29
ATZ_04.08 ATZ_04.09 ATZ_04.12 ATZ_04.13 HS_08 HS_14
VW Polo Audi A8 VW Bora Audi 100 BMW E46 Audi A4(B5)
30 31 32
HS_20 HS_31 HS_47
33 34 35
WH0103 WH0104 WH0611
36
WH0612
37
WH0613
38 39 40 41
WH0615 WH0616 WH0617 WH0618
42 43
WI0709 WI0711
44
WI0712
45 46
WI0809 WI0810
47
WI0811
Vehicle 2 (target)
Audi A8 VW Polo Audi 100 VW Bora Ford Focus Jeep Grand Cherokee Mercedes W124 Renault Espace Subaru Legacy opel Meriva Volvo 850 Mitsubishi Lanser Fiat Punto Renault Twingo Renault Twingo Fiat Punto Audi A6 Avant Mercedes CKlasse Mercedes CAudi A6 Avant Class MCC Smart Citroen Xsara Picasso Peugeot 307SW Opel Vectra B Opel Vectra B Peugeot 307SW Ford Galaxy Peugeot 206 Chrysler Grand Renault Laguna Voyager Audi A3 Ford Focus VW Golf IV Ford Focus Turnier Ford Focus VW Golf IV Turnier BMW E39 523i Toyota Avensis Ford Mondeo II BMW E39 523i Turnier Peugeot 206 Ford Mondeo II Turnier
Impact velocity Mass 1 (km/h) (m1) (kg)
Mass 2 (m2) (kg)
Meq (kg)
Vrebound (Km/h)
âactual (g)
adisp (g)
17.1 10 12.9 11.2 14.3 20.2
1186 1997 1350 1517 1510 1411.5
2075 1186 1517 1350 1208 1900
754.66 744.09 714.32 714.32 671.11 809.86
4 3 5 4.4 2.5 5
12.19 7.94 10.05 7.23 8.41 14.54
5.51 3.22 4.77 3.49 5.07 5.13
16.7 27.1 15.6
1604.5 1558 1523
1508 1442 1282.5
777.38 3.2 748.88 4.2 696.22 2.5
9.28 23.95 7.26
12.7 16.1 16.5
950 930 1840
930 950 1334
469.95 4.4 469.95 3.2 773.33 2.3
20
1334
1840
773.33 3
25.3
776
1386
497.47
–
18.2 19 14.6 15.4
1424 1470 1770 1916
1470 1424 988 1528
723.32 723.32 634.10 850.10
3 4.4 4.2 4.1
18.9 17.7
1205 1321
1234 1201
18.5
1201
18.2 18.1 23.6
atime (g)
Tcv (s)
T0–1 g (s)
Cdmax (m)
0.084 0.092 0.075 0.089 0.071 0.092
0.108 0.114 0.105 0.119 0.094 0.139
0.254 0.179 0.183 0.181 0.167 0.309
3.91 7.89 3.77
4.48 0.105 0.143 11.85 0.063 0.112 4.40 0.100 0.136
0.274 0.259 0.238
6.37 11.77 9.58
3.56 5.17 4.32
3.81 0.096 0.136 5.49 0.081 0.105 5.62 0.081 0.122
0.209 0.252 0.219
11.20
4.88
6.45 0.086 0.137
0.277
–
–
–
–
–
7.57 16.42 10.00 8.17
3.49 5.03 4.70 4.18
3.82 5.52 4.76 4.65
0.138 0.086 0.073 0.091
0.167 0.133 0.113 0.132
0.358 0.305 0.189 0.222
609.66 6 629.07 6.5
13.76 10.39
7.08 5.33
8.13 0.060 0.097 0.166 5.86 0.077 0.128 0.231
1321
629.07 6.4
13.53
5.25
6.31
0.071 0.134
0.225
1550 1630
1329 1550
715.51 4.1 794.50 3.9
11.15 7.21
5.02 3.57
5.01 4.61
0.099 0.127 0.108 0.172
0.311 0.309
1137
1630
669.80 3.4
9.53
2.72
2.97 0.217
–
5.79 3.08 4.89 3.57 5.72 6.21
0.272
0.548
^ = 6.37– Ranges: mass 1 = 930–1997 kg; mass 2 = 930–2075 kg; Vccs = 10–27.1 km/h; Vsep = 2.3–6.5 km/h; Tcv = 0.060–0.138 s; Ttotal = 0.094–0.172 s; Cdmax = 0.166–0.358 m; a 23.95 g; at = 3.08–11.85 g; ad = 3.06–11.14 g.
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N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
1 2 1 3 ^t þ bu t ; x ¼ C dmax a 2 r 6 r
(A.9)
where bu is the slope of linear acceleration time function for unloading and tr is the local time after t = Tcv. When the unloading x is zero, and the phase is finished, i.e., tr = Tr, the acceleration € velocity x_ and displacement x become the rebounding velocity vr and the residual deformation Cres. The slope bu rebounding time Tr and Cres, respectively: bu ¼
^ a 2vr 1 4v2 ^T r 2 ¼ C dmax r ; C ¼ C dmax a ; Tr ¼ ^ ^ Tr 3 3a a
(A.10)
The acceleration–displacement curve for the unloading phase is given by eliminating the local time tr from Eq. (A.9):
2 € € x x 1 1 ^ Tr 1 þ 1þ 1 : x ¼ C dmax a ^ ^ 2 3 a a
(A.11)
Appendix B. See Tables B1 and B2. References AGU, 2014. Crash Test Database. AGU. http://www.agu.ch/1.0/en/crashtestdatenbank/. Bailey, M., et al., 1995. Data and methods for estimating the severity of minor impacts. SAE Technical Paper 950352. Society of Automotive Engineers World Congress, pp. 139–175.
Bonugli, E., et al., 2014. Characterisation of force deflection properties for vehicular bumper-to-bumper interactions. SAE Technical Paper 2014-01-1991. SAE International. Brach, R., 2003. Modelling of low speed, front-to-rear vehicle impacts. SAE Technical Paper 2003-01-0491. SAE International. Cipriani, A., et al., 2002. Low speed collinear impact severity: a comparison between full scale testing and analytical prediction tools with restitution analysis. SAE Technical Paper 2002-01-0540. SAE International. Elliott, J.M., et al., 2014. Mechanisms underlying chronic whiplash: contributions from an incomplete spinal cord injury? Pain Med. 15, 1938–1944. Funk, J., et al., 2014. Comparison of quasistatic bumper testing and dynamic full vehicle testing for reconstructing low speed collisions. SAE Int. J. Passenger Cars Mech. Syst. 7 (3). Happer, A., et al., 2003. Practical analysis methodology for low speed vehicle collisions involving vehicles with modern bumper systems. SAE Technical Paper 2003-01-0492. SAE International. Moser, A., et al., 2011. WITkit – whiplash injury risk claims, methodology and theory of operation. IRCOBI Conference Proceedings – International Research Council on the Biomechanics of Injury. Ojalvo, I., Cohen, E., 1997. An efficient model for low speed impacts of vehicles. SAE Technical Paper 970779. SAE International. Ojalvo, I., et al., 1998. Low speed car impacts with different bumper systems: correlation of analytical model with tests. SAE Technical Paper 980365. SAE International. Scott, W.R., et al., 2010. Simulation model for low-speed bumper-to- bumper crashes. SAE Int. J. Passenger Cars Mech. Syst. 3 (1), 21–36. Scott, W.R., et al., 2012. Reconstruction of low-speed crashes using the quasi-static force vs. deformation characteristics of the bumpers involved in the crashes. SAE Int. J. Passenger Cars Mech. Syst. 5 (1), 592–611. Thomson, R., Romilly, D., 1993. Simulaiton of bumpers during low speed impacts. Proceeding of the Canadian Multidisciplinary Road Safety Conference III, Saskatoon, Saskatchewan, Canada. Viano, D.C., 2002. Role of the Seat in Rear Crash Safety. SAE International. Viano, D.C., Olsen, S., 2001. The effectiveness of active head restraint, in preventing whiplash. J. Trauma 51, 959–969. Worsfold, C., 2014. When range of motion is not enough: towards an evidence-based approach to medico-legal reporting in whiplash injury. J. Forensic Legal Med. 25, 95–99. Yoganandan, N., et al., 2013. Patient mechanisms of injury in whiplash – associated disorders. Semin. Spine Surg. 25 (1), 67–74.
Contents lists available at ScienceDirect
Accident Analysis and Prevention journal homepage: www.elsevier.com/locate/aap
Impact characteristics of a vehicle population in low speed front to rear collisions Naoya Nishimura a,b , Ciaran K. Simms a, * , Denis P. Wood c a b c
Centre for Bioengineering, School of Engineering, Trinity College, Dublin, Ireland Department of Vehicle and Mechanical Engineering, Meijo University, Japan Denis Wood Associates, Dublin, Ireland
A R T I C L E I N F O
A B S T R A C T
Article history: Received 3 November 2014 Received in revised form 2 February 2015 Accepted 3 February 2015 Available online xxx
Rear impact collisions are mostly low severity, but carry a very high societal cost due to reported symptoms of whiplash and related soft tissue injuries. Given the difficulty in physiological measurement of damage in whiplash patients, there is a significant need to assess rear impact severity on the basis of vehicle damage. This paper presents fundamental impact equations on the basis of an equivalent single vehicle to rigid barrier collision in order to predict relationships between impact speed, maximum dynamic crush, mean and peak acceleration, time to common velocity and vehicle stiffness. These are then applied in regression analysis of published staged low speed rear impact tests. The equivalent mean and peak accelerations are linear functions of the collision closing speed, while the time to common velocity is independent of the collision closing speed. Furthermore, the time to common velocity can be used as a surrogate measure of the normalized vehicle stiffness, which provides opportunity for future accident reconstruction. ã 2015 Elsevier Ltd. All rights reserved.
Keywords: Rear end collision Accident reconstruction Regression analysis Impact modeling
1. Introduction The injuries arising from rear impact are mostly low severity since the collision speed is generally much lower than for frontal impact. Nonetheless, due to its frequency, this crash mode results in 30% of automotive-related trauma, and low-severity rear impact accounts for more long-term injury than any other crash mode (Viano, 2002). Accordingly, the annual cost of whiplash type injuries is billions of dollars in the US alone (Viano and Olsen, 2001). The most significant rear impact related injuries are a set of soft tissue injuries frequently called whiplash associated disorders (WAD), which remain surprisingly poorly understood, despite significant research efforts. Injury mechanisms proposed include a hyperextension mechanism, an eccentric contraction mechanism, a hydrodynamic mechanism, and combined mechanisms of axial loading, shear force and bending (Yoganandan et al., 2013). This plethora of proposed mechanisms debated in the literature indicates the lack of consensus in relation to the precise biomechanical causes of whiplash. Nonetheless, the primary cause is relative motion between the torso and the head, driven by a
* Corresponding author at: FTCD, Parsons Building, Trinity College, Dublin, Ireland. Tel.: +353 1 896 3768. E-mail address: [email protected] (C.K. Simms). http://dx.doi.org/10.1016/j.aap.2015.02.001 0001-4575/ ã 2015 Elsevier Ltd. All rights reserved.
combination of the vehicle acceleration time history and the interaction of the occupant with the seat and other vehicle components. A particular challenge for low severity rear impact injuries is the tangible measurement of injury, though recent findings suggest that MRI is capable of quantifying neuromuscular degeneration in chronic whiplash (Elliott et al., 2014). A recent review has concluded that there is growing evidence that a claimant’s physiological and psychological stress response is a very significant factor in persistent symptoms following whiplash injury (Worsfold, 2014). Similarly, given the legal context of whiplash, a toolkit for identifying cases with a crash severity so low that the chances of whiplash injury are remote has been proposed (Moser et al., 2011). Given the difficulty in physiological measurement of damage in whiplash patients, there is a significant need to assess rear impact severity on the basis of vehicle damage. In particular, fundamental principles of injury biomechanics dictate that it is strongly desirable to have methods to assess the magnitude of the acceleration pulse as well as the velocity change imposed on the struck vehicle. However, for very low speed cases, there is frequently no visible damage to the vehicle at all, despite the fact that whiplash symptoms are frequently reported. There have been previous modeling approaches to reconstructing low velocity rear impact collisions. As reviewed by Scott et al. (2010, 2012), these approaches have broadly followed either the
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N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
differential equation approach (Thomson and Romilly, 1993; Ojalvo and Cohen, 1997; Ojalvo et al., 1998; Brach 2003; Scott et al., 2010, 2012) which require explicit definition of the effective stiffness of the two-vehicle system but which provide acceleration time histories, or a momentum energy restitution approach (Bailey et al., 1995; Cipriani et al., 2002; Happer et al., 2003) which does not require an estimate of vehicle stiffness, but consequently limits predictions to velocity change rather than acceleration time history. From an injury biomechanics perspective, the differential equation approach which provides acceleration time histories is far preferable, but the variety of bumper designs make the effective stiffness of a specific vehicle pair involved in a rear end collision difficult to predict. There is therefore the need for a generic reconstruction model which can account for mean and variation. The recent work by (Scott et al., 2010, 2012; Bonugli et al., 2014; Funk et al., 2014) has shown that, when experimental knowledge of the combined bumper deformation behavior is known for a specific vehicle pair, the impact response for a specific collision can be found. However, the tests they completed showed non-linear and variable bumper stiffnesses, and the approach yields nontrivial errors in the coefficient of restitution. Furthermore, for assessment of visible damage, a single quasi static test is not sufficient. In Switzerland, the AGU accident research group has performed a set of 45 unbraked staged full overlap and over-ride/under-ride rear impact collisions with an impact speed range of 8–27 km/h (AGU, 2014). Fig. 1 shows the non-linearity and variability of the resulting acceleration time curves, leading to the preliminary conclusion that there is so much variability in these kinds of collisions due to vehicle design variations and impact configuration that it is impractical to develop a generic model. However, further analysis shows that significant trends can be identified in this dataset. Accordingly, the goal of this paper is to present the fundamental impact equations to predict relationships between impact speed, maximum dynamic crush depth, mean and peak acceleration, time to common velocity and vehicle stiffness, and to use these as a basis for regression analysis for the 45 low speed rear impact cases published by AGU (2014) for full overlap and over-ride/under-ride cases. This approach provides considerable insight into the mechanics of low speed rear impact collisions.
2. Regression analysis of staged low speed front to rear collisions
published by AGU (2014).
2.1. Fundamental collision modeling to inform staged tests regression analysis Here fundamental impact modeling is used to develop relationships between impact speed, crush depth, mean and peak acceleration and vehicle stiffness. These form the basis for regression analysis for the staged low speed rear impact tests published by AGU (2014). A detailed multibody or finite element modeling approach is not necessary, since a considerable body of experimental data is freely available through the AGU. The fundamental modeling approaches show which functional forms the regression analysis of the measurable impact variables should take. 2.1.1. Front to rear collision of two vehicles When two vehicles (m1,m2) are subject to a collinear impact, the system can be regarded as equivalent to a single vehicle impacting a rigid barrier. The equivalent mass meq impacts the rigid barrier at the collision closing speed Vccs and rebounds at the separating speed Vsep given by the following equations from conservation of momentum and conservation of energy considerations: meq ¼
m1 m2 ; m1 þ m2
(1)
V ccs ¼ V eq ¼ V 1 V 2 ; V sep ¼ V 02 V 01 ;
(2)
where Veq is the pre-impact velocity of the equivalent mass and this is equal to the collision closing speed, Vccs, of the colliding pair where V1, V2 are the pre-impact velocities of the two vehicles. The equivalent acceleration aeq and each vehicle acceleration (a1, a2) are related as follows: aeq ¼ a2 a1 ;
a1 ¼
m2 aeq ; m1 þ m2
a2 ¼
m1 aeq : m1 þ m2
2. Methods There are two main components to the work performed:
(3)
1. Fundamental collision modeling to inform the regression
analysis of the staged test data.
(a)
30
(b) 30
Vccs < 10km/h
Vccs < 10km/h
10km/h < Vccs < 13km/h
25
13km/h < Vccs < 17km/h
17km/h < Vccs < 20km/h
20
20km/h < Vccs
a e q [G ]
a e q [G ]
17km/h < Vccs < 20km/h
20
20km/h < Vccs
15 10 5
15 10 5
0
0 0
0.05
0.1
0.15
-5 -10
10km/h < Vccs < 13km/h
25
13km/h < Vccs < 17km/h
0.2
0.25
0
0.05
0.1
0.15
0.2
0.25
-5
T ime [s ]
-10
T ime [s ]
Fig. 1. Equivalent acceleration–time histories of the two vehicles collisions (front to rear collisions) published by AGU (2014) for (a) full engagement test and (b) under-ride/ over-ride tests.
N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
3
Fig. 2. (a) Barrier force F as a linear function of deformation x, and (b) sinusoidal acceleration time relationship for a linear spring–mass collision model. Peak acceleration â is also shown.
The total change in velocity (DVtotal) and each vehicle velocity change (DV1, DV2) are given as:
DV total ¼ DV 1 þ DV 2 ¼ DV ccs þ V sep DV 1 ¼
m2 DV total ; m1 þ m2
DV 2 ¼
m1 DV total : m1 þ m2
2.1.2. Linear acceleration–displacement model (spring–mass model) The equations of motion of a linear spring–mass model of a single vehicle (mass m) colliding with a rigid barrier, where the spring constant (bumper stiffness) during the loading and unloading phases are k and kR are given as follows: m€ x ¼ kx: loading m€ x ¼ kR ðx CÞ½x C res ;C res 0 : unloading
(4)
From the law of conservation of energy, the nominal linear normalised stiffness k/m of the colliding pair is obtained from the test parameters Vccs and the maximum dynamic crush (Cdmax) as follows: k V ccs 2 ¼ (5) m C dmax This relationship can also be derived from the dynamic acceleration versus displacement response. The acceleration time relationships from staged tests shown in Fig. 1 are highly complex, but a subjective review indicates for modeling purposes that two simplified approaches are reasonable simplifications to represent the experimental traces: a linear acceleration–displacement model (a classical linear spring mass system) or a linear acceleration time model. With either of these two assumptions, closed form solutions are available to provide relationships between the principal impact parameters such as mean acceleration, peak acceleration, collision duration etc. An overview of these two modeling approaches is provided here.
(6)
with initial conditions at t = 0: x ¼ 0; x ¼ 0; x_ ¼ v; € _ € where m; x; x; x and Cres are respectively the mass, displacement, velocity, acceleration and residual deformation of the vehicle.The linear loading and unloading stiffnesses are k and kR respectively, while the impact speed is n and the maximum dynamic crush is Cdmax. A schematic of the resulting barrier force displacement relationship and acceleration time relationships during the loading and unloading phases is shown in Fig. 2. Further details of this model are given in Appendix A. 2.1.3. Linear acceleration–time model Considering the acceleration behavior of the vehicle as a linear function of time, the acceleration, velocity and deformation of the vehicle at the loading (closing) phase (0 t Tcv) are derived from integration: 2 € x ¼ a ¼ bt ¼ C 0 vt ¼ 2 vt T cv
1 1 1 x_ ¼ v bt2 ¼ v C 0 vt2 ¼ v 2 vt2 2 2 T cv
Fig. 3. Vehicle acceleration a as (a) a linear function of time t, and (b) a power function of relative displacement x for the linear acceleration–time elementary model.
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N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
Table 1 Relationship between principal collision parameters for the linear acceleration displacement and linear acceleration time models. Parameter
Linear acceleration–displacement model
Linear acceleration–time model
Cdmax
2
2 3V ccs T cv
at â
pV ccs T cv Vccs/Tcv pV
^ ^ t =a a ^ ^d =a a
2/p 1/2
2
ccs
qffiffiffiffiffiffiffiffiffiffi k=m ¼ p=2T cv : for the linear acceleration displacement model
v¼
C 0 ¼ 2=T 2cv : for the linear acceleration time model
Vccs/Tcv V ccs
^¼2 a 1/2 3/8
T cv
=T cv
;v ¼
1 1 1vt3 x ¼ vt bt3 ¼ vt C 0 vt3 ¼ vt 2 ; 6 6 3T cv
(7)
where Tcv is the time required for the two vehicles to achieve a common velocity and the impact speed is n. The initial conditions x ¼ 0, and b is the slope of the linear at t = 0 are x ¼ 0; x_ ¼ v; € acceleration time function, and C0 is the slope constant. A schematic of the resulting barrier force displacement relationship and acceleration time relationships during the loading and unloading phases is shown in Fig. 3. Further details of this model are given in Appendix A. 2.1.4. Mean acceleration, peak acceleration and maximum dynamic deformation When the collision closing speed Vccs and the time to common velocity Tcv of two vehicles are known collision phenomena, the mean acceleration a, peak acceleration â and maximum dynamic deformation Cdmax can be calculated in closed form for both the linear spring mass model and the linear acceleration time model. The mean acceleration at can be considered on a time basis from integration of the acceleration–time curve but also on a displacement basis ad from integration of the acceleration–displacement curve. These relations are summarized in Table 1. For both models, the maximum dynamic crush Cdmax is proportional to the collision closing speed Vccs and the time to common velocity Tcv, while the time-based mean acceleration at is proportional to Vccs and inversely proportional to Tcv. The relationship between the mean and the peak acceleration is however different for the two models, due to the different shape of the acceleration time pulse. The following relations are derived from the natural frequency v and C0 given by Tcv as related to both elementary models:
(a) 14
rffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ 1:232C 0 : m
These equations based on elementary models show that the normalised linear stiffness (k/m) is related to the inverse of the square of the time to common velocity ð1=T 2cv Þ. Thus, Table 1 shows that the theoretical dependencies of maximum dynamic crush, mean acceleration and peak acceleration on collision closing speed and time to common velocity are fixed if a theoretical choice for the acceleration time relationship is made. Accordingly, these derivations can be used to guide regression based interrogation of the AGU staged low speed rear end collision tests (AGU, 2014). 2.2. Regression analysis of AGU staged low speed front to rear collisions AGU (2014) have carried out 45 unbraked front-to-rear impacts, of which 23 can be categorised as full engagement and 22 as over-ride/under-ride. Tables B1 and B2 (see Appendix B) detail the test numbers, the vehicles involved, impact speeds and related data. In the full engagement tests, the impact speed range was 8–21.5 km/h. There are 30 different car types with bullet vehicle (vehicle 1) mass range from 843–1903 kg and target vehicle (vehicle 2) mass range 838.5–1860 kg. For the over-ride/under-ride tests, the bullet vehicle mass ranged from 930–1997 kg and the target vehicle (vehicle 2) mass range was 930–2075 kg. Fig. 1 shows the acceleration time history curves for all of these tests. In the following sections, regression analysis is performed on the principal impact parameters of the AGU staged tests. 2.2.1. Relation between mean and peak acceleration, maximum dynamic crush and impact speed Given the linear relationship between time-based mean acceleration at (up to the time of common velocity Tcv) and collision closing speed Vccs implicit in both the linear acceleration
(b) 14 A(0 -Tcv)
12
10
10
Mean a t [G ]
12
Mean a t [G ]
(8)
8 6 4
A(0 -Tcv)
8 6 4
y = 0.3924x
2
y = 0.3292x
2
R² = 0.7049
R² = 0.6564
0
0
0
5
10
15
V c c s [km/h]
20
25
30
0
5
10
15
20
25
30
V c c s [km/h]
Fig. 4. The relationship between equivalent mean acceleration at (up to the time of common velocity Tcv) and collision closing speed Vccs for AGU test data for (a) full engagement tests and (b) under-ride/over-ride tests. Forced-zero linear regression results are also shown.
N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
(a)
(b)
30
25
25
20
20
P eak a [G ]
P eak a [G ]
30
15
10
y = 0.7943x
5
5
15
10
y = 0.651x
5
R² = 0.512
R² = 0.5829
0
0 0
5
10
15
20
25
30
0
5
10
15
20
25
30
V c c s [km/h]
V c c s [km/h]
Fig. 5. The relationship between the peak acceleration â and collision closing speed Vccs for AGU test data for (a) full engagement tests and (b) under-ride/over-ride tests. Forced-zero linear regression results are also shown.
displacement and linear acceleration time models (Table 1). Fig. 4 shows the results of forced-zero linear regression of these parameters for the 23 full engagement and 22 over-ride/underride staged AGU tests. Fig. 5 shows the equivalent linear relationship between peak acceleration and collision closing speed Vccs. Fig. 6 shows the relationship between maximum dynamic crush Cdmax and collision closing speed Vccs. 2.2.2. Relationship between time to common velocity and collision closing speed Fig. 7 shows the relationship between the time to common velocity Tcv of the two vehicles for each test and the collision closing speed (Vccs). The parameter Tcv, is calculated from the velocity–time curve. Linear regression results are also shown. 2.2.3. Linear acceleration–displacement or linear acceleration–time models It is possible to assess the appropriateness of the spring–mass and linear acceleration–time modeling approaches by considering the predicted acceleration time history relationship for the two models in comparison to actual experimental cases for both full overlap and over-ride/under-ride cases, see Figs. 8 and 9. The overall predictive capacity of the models for all staged test cases can be assessed from the predicted relationship between mean and peak acceleration for the two modeling approaches (see Table 1) in comparison with the actual response for all the AGU tests, see Fig. 10. 0.4
(a)
2.2.4. Mean acceleration – comparison of model and experimental results Table 1 shows that the mean acceleration on a time basis (at ) for both the linear spring mass model and the linear acceleration time model is equal to the collision closing speed (Vccs) divided by the time to common velocity (Tcv). Fig. 11 compares the model predictions for at with the AGU mean acceleration values for the full engagement and over-ride/under-ride tests. Fig. 12 shows the experimental and model predictions of ad versus peak acceleration â for the full engagement and over-ride/under-ride tests. 2.2.5. Relationship between normalised vehicle stiffness and time to common velocity Fig. 13 shows the relation between the experimental normalised equivalent stiffness (k/m) obtained from Vccs and Cdmax and the inverse square of the experimental values of the Tcv. Regression analysis shows a linear relation between (k/m) and 1=T 2cv with high correlation which compares with the predictions from the elementary models, Eq. (8). 3. Discussion For low velocity rear impact collisions, a method is needed to assess the magnitude of the acceleration pulse as well as the velocity change imposed on the struck vehicle. This is difficult, since for very low speed cases there is frequently no visible damage to the vehicle at all, despite the fact that whiplash associated
(b) 0.4
0.35
0.35
0.3
0.3
C dmax [m]
C dmax [m]
0.25 0.2
0.25 0.2 0.15
0.15
y = 0.0133x
0.1
y = 0.0142x
0.1
R² = 0.1537
R² = 0.6784
0.05
0.05
0
0
0
5
10
15
V c c s [km/h]
20
25
30
0
5
10
15
20
25
30
V c c s [km/h]
Fig. 6. The relationship between maximum dynamic crush Cdmax and collision closing speed Vccs for AGU test data (AGU, 2014) for (a) full engagement tests and (b) underride/over-ride tests. Forced-zero linear regression results are also shown.
6
N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
0.14
0.14
0.12
0.12
0.1
0.1
T c v [s ]
(b) 0.16
T c v [s ]
(a)0.16
0.08
0.08 0.06
0.06
0.04
0.04
y = -0.0008x + 0.0869
0.02
R² = 0.0405
5
10
15
20
25
R² = 0.0233
0
0 0
y = -0.0007x + 0.0994
0.02
30
V c c s [km/h]
0
5
10
15
V c c s [km/h]
20
25
30
Fig. 7. Dependence of the time to common velocity of two vehicles Tcv on the collision closing speed Vccs for AGU test data (AGU, 2014): (a) full engagement tests (Tcv: mean = 0.075, standard deviation = 0.015) and (b) under-ride/over-ride test (Tcv: mean = 0.087, standard deviation = 0.017). Tcv is calculated from the velocity–time curve.
symptoms are frequently reported. A further complication in reconstructing individual cases is the large variety of bumper systems which leads to significant variability in acceleration time relationships, as shown in Fig. 1. The analysis presented in this paper is predicated on the assumption of an equivalent single vehicle model to represent a low speed two vehicle collision which is grounded in fundamental mechanics. Application of this approach to the staged AGU low velocity rear impact tests (AGU, 2014) shows a linear relationship between both mean and peak acceleration and the vehicle collision closing speed, with high correlation coefficients (R2 > 0.5) for both
full engagement and over-ride/under-ride cases, see Figs. 4 and 5. A similar relationship between maximum dynamic crush and collision closing speed is seen in Fig. 6, although the correlation is significantly reduced for the over-ride/under-ride cases (R2 > 0.15 due to two outliers). Overall, each parameter at, ad , â and Cdmax shows a linear relation to Vccs (Figs. 4–6). Two possible acceleration time relationships are considered: a classic spring mass system which has a linear acceleration displacement relationship, or a linear acceleration time model. Table 1 shows that the mean acceleration at and peak acceleration â are proportional to collision closing speed Vccs and inversely
Fig. 8. Model and experimental acceleration–time curves for (a-1:a-3) the linear acceleration–displacement model and (b-1:b-3) the linear acceleration–time model for selected full engagement tests from AGU (2014).
N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
7
Fig. 9. Model and experimental acceleration–time curves for (a-1:a-3) the linear acceleration–displacement model and (b-1:b-3) the linear acceleration–time model for selected over-ride/under-ride tests from AGU (2014).
proportional to the time to common velocity Tcv for both the linear spring mass and the linear acceleration time model approaches. It follows that Tcv can be considered as a measure of the stiffness of the vehicle deformation, which, for a nominally linear spring characteristic should be independent of collision closing speed. Fig. 7 shows that this is effectively the case, both for the full engagement and over-ride/under-ride AGU cases, with R2 < 0.05 in
both cases. The low R2 shows that Tcv can be considered to be independent of impact velocity with a mean value of 0.075 s for the full engagement test and 0.087 s for the over-ride/under-ride tests. This also indicates that the mean values of normalized stiffness k/m for the population of cars examined can be considered to be independent of car mass and impact velocity.
^: Fig. 10. Dependence of the mean acceleration at to time to common velocity Tcv on the peak acceleration â during collision for (a) full engagement test (at =a ^ : mean = 0.52, standard deviation = 0.083) from AGU tests (AGU, 2014). mean = 0.53, standard deviation = 0.091) and (b) under-ride/over-ride test (at =a
8
N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
Fig. 11. Relationship between the predicted and experimental at for (a) full engagement and (b) over-ride/under-ride cases for the AGU tests (AGU, 2014).
Application of these models to predict the loading and unloading time history relationship of staged crash tests (Figs. 8 and 9) shows that both the classic spring mass model and the linear acceleration time model provide very good overall fits to the experimental data, with inputs based only on time to common velocity (Tcv) and collision closing speed Vccs and the rebound velocity. Overall, Figs. 10–12 combine to show that the linear acceleration time model provides better predictions of the vehicle acceleration responses. Regression analysis indicates that the normalised vehicle stiffness is proportional to the inverse square of the time to common velocity (Tcv) as predicted by the fundamental impact modeling, see Fig. 13. This relationship is highly linear for both the full engagement and over-ride/under-ride cases (R2 > 0.85), and it indicates that the vehicles tested behave in a substantially similar manner in low speed rear impacts and that Tcv can be used as a surrogate measure of vehicle stiffness. The high R2 values in Fig. 13 for both the full engagement and over-ride/under-ride cases shows that Tcv is an excellent surrogate measure of the normalised equivalent vehicle stiffness (k/m). This is further demonstrated from the slope of the acceleration displacement relationships i.e., (k/m) for each test shown in Fig. 14, which show that the inverse square of the time to common velocity ð1=T 2cv Þ is a highly linear function of the slope of the acceleration displacement curves (R2 > 0.97), with very little scatter. In very recent work (Bonugli et al., 2014) have presented linear stiffness corridors from quasi static testing of bumper pairs, but since the vehicle masses are not
known for their data, direct comparison with the normalised stiffness data in Fig. 13 of this paper is not possible. Table 1 shows that for the linear acceleration time model, the mean acceleration at can be found from the collision closing speed Vccs divided by the time to common velocity Tcv. This is confirmed by the correspondence between the predicted and measured values of mean acceleration, ad shown in Fig. 11. The above suggests that measurement of time to common velocity, Tcv and collision closing speed Vccs could provide a basis for a relatively straightforward experimental characterisation of low speed vehicle stiffness and response. Although it has recently been found that peak accelerations on the basis of the dynamic stiffness of a bumper pair are within 5% of those predicted on the basis of the static bumper stiffness (Funk et al., 2014), the very high correlations shown in Fig. 13 indicate that a single measure of Tcv in the dynamic test can give an excellent measure of effective bumper stiffness. The regression data for the full engagement and under-ride/ over-ride tests in Fig. 6 show that the mean relationships between Cdmax (m) and Vccs (m/s) are: V ccs ¼ 21:5C dmax ðfull engagementÞ
V ccs ¼ 20C dmax ðunder ride=over rideÞ:
(9)
(10)
These equations provide a general indication of collision speed where maximum dynamic crush can be estimated. Table 1 also
Fig. 12. Relationship between mean acceleration ad and peak acceleration for the experimental AGU cases (AGU, 2014) for (a) full engagement tests and (b) the over-ride/under-ride tests. The theoretical predictions from the linear acceleration–time and linear acceleration–displacement models are also shown, together with a forced zero linear regression of the experimental cases.
N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
9
Fig. 13. Relationship between the equivalent normalized stiffness k=m ¼ V 2ccs =C 2dmax and the inverse square of the time to common velocity ð1=T 2cv Þ for the linear acceleration time model for (a) full engagement tests and (b) the over-ride/under-ride tests from the AGU (2014).
shows that for the linear acceleration time model the maximum dynamic crush Cdmax is related to Vccs and Tcv as follows: 2 C dmax ¼ V ccs T cv : 3
(11)
opposed to the overall pulse duration, is substantially independent of collision closing speed for a set of full-scale staged collisions, and that Tcv can therefore be used as an explicit surrogate measure of the normalised vehicle stiffness. 4. Limitations
Therefore, once a relationship between residual crush (Cres), which can be measured after a collision, and Cdmax is established, since Tcv can be estimated on the basis of normalized vehicle stiffness, then this approach might facilitate more exact estimation of collision closing speed in the future. In this regard (Bonugli et al., 2014) have established the mean relationship Cdmax = 1.18Cres, although this encompasses full engagement, under-ride/over-ride and concentrated impact (tow-bar) data and does not fully account for those impacts where no residual crush is present. This paper has analysed full engagement and over-ride/underride collisions, but only for cases where the principal direction of force is aligned with the vehicles’ longitudinal axes, and this is a limitation of the present work. Nonetheless, the analysis shows for the first time that linear acceleration time models applied to both the loading and unloading phases of low speed rear impact yield closed form solutions which provide a close match to the available experimental data. Furthermore, although (Scott et al., 2010) found from a parametric modeling study that the overall pulse duration is independent of impact speed for fixed restitution and stiffness, they also found significant variations in overall pulse duration when restitution and stiffness are varied. In this paper it has been shown for the first time that the time to common velocity (Tcv), as
This analysis applies to unbraked front to rear full engagement and under-ride/over-ride impacts over the collision speed range 8–22 km/h for car masses in the range 830–2075 kg, mass ratio range for target car/bullet car of 0.5–1.8 and nominal equivalent normalised stiffness range 100–900/s2. Future research will apply this approach to offset and tow-bar crashes and to crashes involving trucks, buses, etc. 5. Conclusions Regression analysis of the staged AGU rear end collision tests (AGU, 2014) performed on the basis of fundamental impact modeling of an equivalent single vehicle system shows that, despite the complexity of the experimental linear acceleration time curves, either a linear acceleration time or a linear spring mass model provide very strong insights into the mechanics of low speed rear impact cases. The equivalent mean and peak accelerations are linear functions of the collision closing speed, while the time to common velocity is independent of the collision closing speed. Furthermore, the time to common velocity can be used as a
Fig. 14. Dependence of the slope a/Cd (forced regression for actual acceleration–displacement curve) on 1=T 2cv for AGU (2014) tests (a) the full engagement test and (b) the over-ride/under-ride tests. The normalized stiffness values reported here for the colliding pairs equate to the normalized stiffness of each car when these are the same as each other.
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N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
surrogate measure of the normalized vehicle stiffness, which provides opportunity for post-hoc accident reconstruction. Acknowledgment The authors gratefully acknowledge Markus Muser from AGU for generously sharing their data. Appendix A.
A1. Linear spring mass model derivations
Fig. A1. Vehicle acceleration a for the loading and unloading phase as a linear function of time t.
The spring constant k is given by the impact velocity n and the maximum deformation Cdmax from the law of conservation of energy: k¼
mv2 C dmax 2
:
(A.1)
x¼
Normalising with respect to mass yields the ‘normalised stiffness’ k/m:
k V ccs ¼ m C dmax
pffiffiffiffiffiffiffiffiffiffiffiffi (loading), vR ¼ kR =m (unloading) and the initial conditions, as follows: Loading (closing) phase 0 t 2pv v
sinvt
v
x_ ¼ vcosvt
(A.3)
2 (A.2)
The normalised stiffness k/m is also the slope of the normalised force (F/m), i.e., acceleration (a) versus displacement (Cd) characteristic of the impact. The deformation, velocity and acceleration of the vehicle are pffiffiffiffiffiffiffiffiffiffi derived from integration using the natural frequency v ¼ k=m
€ x ¼ vvsinvt Unloading (rebounding) phase ð2pv t 2pv þ 2p vR Þ o n vv p v 1 þ x ¼ 2 cosvR t 2v v vR
Table B1 AGU full engagement test data (N = 23). Vehicle 2 is stationary in these tests. Test no.
AGU test no.
1 2 3
AZT_02.50 BMW 318 ATZ_02.52 Opel Vectra HS_11 Renault Espace
4 5 6
HS_35 WH0217 HS_03
Citroen C2 Opel Vectra A Peugeot 306
7 8
HS_24 WH0215
Mazda 121 Renault Clio
9 10
HS_26 HS_18
Alfa Romeo 156 VW New Beetle
11
HS_12
Opel Corsa C
12 13 14 15 16 17
HS_06 ATZ_02.51 ATZ_03.18 ATZ_02.54 HS_DTC25 HS_15
Vehicle 1 (bullet) Vehicle 2 (target)
20 21
MCC Smart Audi A3 Opel Zafira VW T4 VW Golf 3 Mercedes-Benz A-Klasse ATZ_02.53 Audi A4 HS_36 Mercedes Benz E320 HS_22 Renault Megane HS_43 Renault Kangoo
22 23
HS_05 HS_38
18 19
Opel Astra F VW Polo
Impact velocity (km/h)
Mass 1 (m1) Mass 2 (m2) Meq (kg) (kg) (kg)
âactual (g)
adisp (g)
atime (g)
Tcv (s)
T0–1 g (s)
Cdmax (m)
8.6 8.8 12.4
1400 1275 1616.5
1255 1200 1028
661.78 4 618.18 1.9 628.38 2.8
7.38 4.57 5.32
2.24 2.43 3.28
2.27 2.46 3.50
0.107 0.102 0.101
0.152 0.123 0.131
0.165 0.159 0.214
15.3 14.5 17.7
1182.5 1271 1330
1183 1026 1341
591.38 567.72 667.74
3.8 3.1 4.3
10.00 19.92 9.80
4.33 4.06 3.80
4.67 4.71 5.66
0.092 0.124 0.089 0.122 0.088 0.177
0.241 0.219 0.275
18.4 20.1
940.5 1026
1259 1271
538.34 5.3 567.72 3.2
17.56 13.85
6.35 6.27
6.39 7.14
0.081 0.107 0.080 0.105
0.287 0.273
14.5 14.1
1423 1347.5
1170.5 995.5
642.23 4.7 572.53 5.6
11.49 15.63
4.40 4.53
5.15 5.18
0.079 0.128 0.077 0.125
0.209 0.201
Vrebound (Km/h)
Opel Vectra Audi A3 Opel Corsa C VW Polo Renault Clio Mercedes W201 VW Golf IV Opel Vectra A Fiat Brava Nissan Micra K11 Renault Espace Opel Astra F BMW 318 VW Golf Audi A4 Toyota Prius Renault Clio
19.8
1028
1616.5
628.38 4
16.95
6.16
7.40
0.076
0.109
0.247
21.5 10.1 11.5 8 20.3 15.2
843 1222 1530 1878 1125 1174
1278 1380 1325 1295 1442 1195.5
507.95 648.10 710.07 766.47 631.96 592.33
5.7 3.9 4.1 3 3.9 5.9
9.50 8.73 9.50 5.03 19.92 12.82
7.18 3.91 4.43 3.17 7.88 5.75
8.24 4.11 4.79 3.33 8.48 6.67
0.074 0.069 0.069 0.068 0.068 0.065
0.116 0.103 0.096 0.099 0.087 0.106
0.282 0.128 0.144 0.089 0.279 0.185
VW T4 Fiat Punto
11.1 13
1308 1903
1860 1243.5
767.95 3 752.07 4.6
6.89 13.34
4.17 5.38
4.96 6.08
0.063 0.095 0.062 0.093
0.110 0.150
1200 1323.5
972.5 1353.5
537.17 669.17
4.8 5.7
10.00 16.95
5.33 7.71
5.98 8.90
0.061 0.094 0.057 0.084
0.135 0.177
1279.5 1183
838.5 1193
506.54 3.7 593.99 4.5
12.20 25.77
5.86 8.38
6.64 8.69
0.055 0.089 0.051 0.073
0.134 0.166
Toyota Yaris 12.9 Alfa Romeo 17.3 147 MCC Smart 13.9 Citroen C2 16.8
^ = 4.11– Ranges: mass 1 = 843–1903 kg; mass 2 = 838.5–1860 kg; Vccs = 8–21.5 km/h; Vsep = 1.9–5.9 km/h; Tcv = 0.051–0.107 s; Ttotal = 0.073–0.177 s; Cdmax = 0.089–0.287 m; a 25.77 g; at = 2.27–8.90 g; ad = 2.87–7.93 g.
N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
x¼
vv p x_ ¼ sinvR t vR 2v
3
2v2 ^ a
x þ
2
11
!
x 1€ : 3a ^2
(A.6)
(A.4) x is â, the displacement x is: When acceleration €
p € : x ¼ vvcosvR t 2v
^: C dmax ¼ 4v2 =3a
When t ¼ T cv ¼ 2pv, the velocity x_ is zero and the acceleration and displacement are the peak values â and Cdmax. At the end of € rebound, when the time t ¼ T total ¼ 2pv þ 2p vR , the acceleration x is 0 _ zero, the velocity x is v and the displacement x is the residual deformation Cres. A2. Linear acceleration time model derivations When the time t = Tcv, the acceleration € x and displacement x are the peak values â and Cdmax and the velocity x_ is zero. Furthermore, Tcv is independent of the initial impact velocity. The slope constant term C0 is given by the velocity conditions and the Tcv relation as follows:
T cv ¼
2v 1=2 2 2 ;b ¼ 2 v ¼ C 0 v; C 0 ¼ 2 : b T cv T cv
(A.5)
The acceleration–displacement curve for the loading phase is found by eliminating the time t from Eq. (6) and using the relation ^ ¼ 2v=T cv to yield: a
(A.7)
The acceleration–displacement curve is normalized by eliminating the impact velocity n using Eq. (A.6) to yield: x C dmax
¼
3 x 1 € x 3€ : ^ 2 a ^ 2a
(A.8)
The acceleration behavior of the vehicle for the unloading (rebounding) phase ðT cv t T total ¼ T cv þ T r Þ, is also considered as a linear function of time, as shown in Fig. A1. The acceleration, velocity and deformation of the vehicle during the unloading phase are derived from integration using initial ^ at tr ¼ 0 : conditions x ¼ C dmax ; x_ ¼ 0; € x ¼ a € ^ þ bu t r ; x ¼ a ¼ a tr ¼ t T cv
1 ^tr þ bu t2r x_ ¼ a 2
Table B2 AGU over-ride/under-ride (N = 22). Vehicle 2 is stationary in these tests. Test no.
AGU test no.
Vehicle 1 (bullet)
24 25 26 27 28 29
ATZ_04.08 ATZ_04.09 ATZ_04.12 ATZ_04.13 HS_08 HS_14
VW Polo Audi A8 VW Bora Audi 100 BMW E46 Audi A4(B5)
30 31 32
HS_20 HS_31 HS_47
33 34 35
WH0103 WH0104 WH0611
36
WH0612
37
WH0613
38 39 40 41
WH0615 WH0616 WH0617 WH0618
42 43
WI0709 WI0711
44
WI0712
45 46
WI0809 WI0810
47
WI0811
Vehicle 2 (target)
Audi A8 VW Polo Audi 100 VW Bora Ford Focus Jeep Grand Cherokee Mercedes W124 Renault Espace Subaru Legacy opel Meriva Volvo 850 Mitsubishi Lanser Fiat Punto Renault Twingo Renault Twingo Fiat Punto Audi A6 Avant Mercedes CKlasse Mercedes CAudi A6 Avant Class MCC Smart Citroen Xsara Picasso Peugeot 307SW Opel Vectra B Opel Vectra B Peugeot 307SW Ford Galaxy Peugeot 206 Chrysler Grand Renault Laguna Voyager Audi A3 Ford Focus VW Golf IV Ford Focus Turnier Ford Focus VW Golf IV Turnier BMW E39 523i Toyota Avensis Ford Mondeo II BMW E39 523i Turnier Peugeot 206 Ford Mondeo II Turnier
Impact velocity Mass 1 (km/h) (m1) (kg)
Mass 2 (m2) (kg)
Meq (kg)
Vrebound (Km/h)
âactual (g)
adisp (g)
17.1 10 12.9 11.2 14.3 20.2
1186 1997 1350 1517 1510 1411.5
2075 1186 1517 1350 1208 1900
754.66 744.09 714.32 714.32 671.11 809.86
4 3 5 4.4 2.5 5
12.19 7.94 10.05 7.23 8.41 14.54
5.51 3.22 4.77 3.49 5.07 5.13
16.7 27.1 15.6
1604.5 1558 1523
1508 1442 1282.5
777.38 3.2 748.88 4.2 696.22 2.5
9.28 23.95 7.26
12.7 16.1 16.5
950 930 1840
930 950 1334
469.95 4.4 469.95 3.2 773.33 2.3
20
1334
1840
773.33 3
25.3
776
1386
497.47
–
18.2 19 14.6 15.4
1424 1470 1770 1916
1470 1424 988 1528
723.32 723.32 634.10 850.10
3 4.4 4.2 4.1
18.9 17.7
1205 1321
1234 1201
18.5
1201
18.2 18.1 23.6
atime (g)
Tcv (s)
T0–1 g (s)
Cdmax (m)
0.084 0.092 0.075 0.089 0.071 0.092
0.108 0.114 0.105 0.119 0.094 0.139
0.254 0.179 0.183 0.181 0.167 0.309
3.91 7.89 3.77
4.48 0.105 0.143 11.85 0.063 0.112 4.40 0.100 0.136
0.274 0.259 0.238
6.37 11.77 9.58
3.56 5.17 4.32
3.81 0.096 0.136 5.49 0.081 0.105 5.62 0.081 0.122
0.209 0.252 0.219
11.20
4.88
6.45 0.086 0.137
0.277
–
–
–
–
–
7.57 16.42 10.00 8.17
3.49 5.03 4.70 4.18
3.82 5.52 4.76 4.65
0.138 0.086 0.073 0.091
0.167 0.133 0.113 0.132
0.358 0.305 0.189 0.222
609.66 6 629.07 6.5
13.76 10.39
7.08 5.33
8.13 0.060 0.097 0.166 5.86 0.077 0.128 0.231
1321
629.07 6.4
13.53
5.25
6.31
0.071 0.134
0.225
1550 1630
1329 1550
715.51 4.1 794.50 3.9
11.15 7.21
5.02 3.57
5.01 4.61
0.099 0.127 0.108 0.172
0.311 0.309
1137
1630
669.80 3.4
9.53
2.72
2.97 0.217
–
5.79 3.08 4.89 3.57 5.72 6.21
0.272
0.548
^ = 6.37– Ranges: mass 1 = 930–1997 kg; mass 2 = 930–2075 kg; Vccs = 10–27.1 km/h; Vsep = 2.3–6.5 km/h; Tcv = 0.060–0.138 s; Ttotal = 0.094–0.172 s; Cdmax = 0.166–0.358 m; a 23.95 g; at = 3.08–11.85 g; ad = 3.06–11.14 g.
12
N. Nishimura et al. / Accident Analysis and Prevention 79 (2015) 1–12
1 2 1 3 ^t þ bu t ; x ¼ C dmax a 2 r 6 r
(A.9)
where bu is the slope of linear acceleration time function for unloading and tr is the local time after t = Tcv. When the unloading x is zero, and the phase is finished, i.e., tr = Tr, the acceleration € velocity x_ and displacement x become the rebounding velocity vr and the residual deformation Cres. The slope bu rebounding time Tr and Cres, respectively: bu ¼
^ a 2vr 1 4v2 ^T r 2 ¼ C dmax r ; C ¼ C dmax a ; Tr ¼ ^ ^ Tr 3 3a a
(A.10)
The acceleration–displacement curve for the unloading phase is given by eliminating the local time tr from Eq. (A.9):
2 € € x x 1 1 ^ Tr 1 þ 1þ 1 : x ¼ C dmax a ^ ^ 2 3 a a
(A.11)
Appendix B. See Tables B1 and B2. References AGU, 2014. Crash Test Database. AGU. http://www.agu.ch/1.0/en/crashtestdatenbank/. Bailey, M., et al., 1995. Data and methods for estimating the severity of minor impacts. SAE Technical Paper 950352. Society of Automotive Engineers World Congress, pp. 139–175.
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