# On the Mutual Coefficient of Restitution in Two Car Collinear Collisions

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1 //006 On the Mutual Coefficient of Restitution in Two Car Collinear Collisions Milan Batista Uniersity of Ljubljana, Faculty of Maritie Studies and Transportation Pot poorscako 4, Sloenia, EU (Jan. 006) Abstract In the paper two car collinear collisions are discussed using Newton's law of echanics, conseration of energy and linear constitutie law connecting ipact force and crush. Two ways of calculating the utual restitution coefficient are gien: one already discussed by other authors that does not include the car's stiffness and a new one based on car stiffness. A nuerical exaple of an actual test is proided.. Introduction For the odeling of the collinear car collision two ethods are usually used. The first is the so-called ipulse-oentu ethod based on classical Poisson ipact theory, which replaces the forces with the ipulses ([3], [9]). The second ethod treats a car as a deforable body; so the constitutie law connecting contact force with crush is necessary. For the copression phase of ipact the linear odel of force is usually adopted and the odels differ in the way the restitution phase of collision is treated ([5], [], [], [4]). The purpose of this paper is to extend the linear force odel discussed in [] to the collinear ipact of two cars. In the quoted article it is proposed that a car is characterized by its ass, stiffness and liit elocity for peranent crush. The latter properties can be established by a fixed barrier crush test. Also, the proposed restitution odel is siple: rebound elocity is constant. The question arises as to how these

2 //006 characteristics can be incorporated into the two car collision odel since it is well known that the utual coefficient of restitution is the characteristic of ipact; i.e., it is a two car syste and not the property of an indiidual car ([], [4]). To answer the aboe question, first the well-known theory of central ipact is specialized for collinear car collisions. The kinetic energy losses are then discussed and the restitution coefficient is related to the. The third section of the paper discusses two odels for calculating the utual restitution coefficient based on indiidual car characteristics. The last section is deoted to a description of the use of the present theory in accident reconstruction practice. The section ends with a nuerical exaple.. Two car collinear collision Consider a collinear ipact between two cars where collinear ipact refers to rear-end and head-on collisions. Before ipact the cars hae elocities and respectiely and after ipact they hae elocities u and u (Figure ). Figure. The two car ipact: (a) pre-ipact elocities, (b) end of copression elocity, (c) post-ipact elocities

3 //006 3 In the collision phase the oeent of cars is goerned by Newton's nd and 3rd laws (Figure ). On the basis of these laws equations of otion of the cars can be written as follows d F d dt = and dt = F () where and are the asses of the cars and F is contact force. Figure. Newton's 3rd law applied to collinear ipact of two cars Following Poisson's hypothesis ([3]), the ipact is diided into two phases: copression and restitution. In the copression phase the contact force F raises and the cars are defored. The copression phase terinates when the relatie elocity of cars anishes; i.e., when cars hae equal elocity (Figure ). The copression phase () thus integrates the changes fro initial elocities to coon elocity u. This leads to the following syste of equations ( ) ( ) u = P u = P () c c where Pc τ c Fdt is copression ipulse and τ c copression tie. Fro () one 0 obtains the elocity after copression + u = + (3)

4 //006 4 and the copression ipulse ( ) Pc = + (4) In the restitution phase the elastic part of internal energy is released. Equations () are integrated fro u to the end elocities, which gies two equations for three unknowns ( ) ( ) u u = P u u = P (5) r r where Pr τ c Fdt is restitution ipulse and τ r is restitution tie. In order to sole 0 syste (5) for an unknown's post-ipact elocity and restitution ipulse the constitutie equation is needed. According to the Poisson hypothesis the restitution ipulse is proportional to copression ipulse P r = ep (6) c where e is the restitution coefficient. Because contact force is non-negatie, so are copression and restitution ipulse. Fro (6) this iplies that e 0. Note. Instead of (6), one can use Newton's kineatical definition of restitution coefficient u e = u which is in the case of centric ipact without friction equialent to Poisson s definition. Howeer in the case of non-centric ipact with friction Newton's odel could lead to oerall energy increase ([0]).

5 //006 5 The total ipulse is P= Pc + Pr so by using (4) and (6) P= ( + e) Δ + (7) Soling (5) and (6) and taking into account (4) gies the well known forulas (see for exaple [3], [9]) for the cars post-ipact elocities ( + ) e = Δ = Δ u u e + + ( + ) e = + Δ = + Δ u u e + + (8) where Δ =. The aboe equations can be used for calculation of post-ipact elocities if pre-ipact elocities are known, asses of cars are known and, in addition, the restitution coefficient is known. 3. Energy consideration At car ipact the kinetic energy is dissipated. Applying the principle of conseration of energy one obtains, after copression, ( + ) u + = +Δ E (9) where Δ E is axial kinetic energy lost (or axial energy absorbed by crush). By using (3) one has E Δ = + Δ (0)

6 //006 6 Siilarly, by applying the principle of conseration of energy to the oerall ipact process u u + = + +Δ E () one finds the well known forula for total kinetic energy lost (see for exaple [9]) Δ E = ( e ) + Δ () Since, by the law of therodynaics, ΔE 0, it follows fro () that e. Now, fro (0) and () one has ( ) gien by ([9]) Δ E = e Δ E, so the utual restitution coefficient is ΔE e = = ΔE a ΔE ΔE 0 (3) where ΔE0 ΔE Δ E is the rebound energy. The forula obtained is the basis for relating the utual coefficient of restitution e with the restitution coefficients obtained for indiidual cars in the fixed barrier test. 4. The utual coefficient of restitution Let T be a barrier test elocity of a first car and T a barrier test elocity of a second car. Let these elocities be such that the axial kinetic energy lost can be written as T T Δ E = + (4) and in addition the rebound energy can be written as (see [7])

7 //006 7 e T e T Δ E0 = + (5) The utual restitution coefficient is therefore fro (3), (4) and (5), by using (0), e = e + e T T T+ T (6) For the odel of the barrier test proposed in [] the restitution coefficients of cars are e in, 0 = T 0 and e = in, T (7) where 0 and 0 are liited ipact elocities where all the crush is recoerable ([]). The task is now to deterine appropriate test elocities of cars which satisfy (4). 4. Model A. Let T be the barrier test elocity (or barrier equialent elocity [6]) of the first car for the sae crush as in a two car ipact and T the barrier test elocity for the sae crush for the second car. Then the test elocities for the sae crush ust satisfy relations ([], [6]) T kδ = and T kδ = (8) where k and k are stiffness of the cars and δ and δ are actual axial dynaics crush of the cars. Fro (8) one has

8 //006 8 = k δ and k = T δ (9) T On the other hand, fro (0), (4) and (8) it follows that kδ k δ Δ E = Δ = + + (0) Defining oerall axial crush δ δ + δ and taking into account the law of action and reaction kδ = kδ one obtains δ k δ δ k = = δ k+ k k+ k () Substituting () into (0) yields Δ kδ Δ E = = () where is syste ass and k is syste stiffness, gien by kk k + k + k (3) Fro () one has δ = Δ and therefore fro (9) the required test elocities are k (see also [6]) k k = Δ and = Δ (4) T T k k

9 //006 9 Substituting (4) into (4) leads to identity proides the required utual restitution coefficient = + and substituting it into (6) k k k e = ke k + ke + k (5) This equation for the calculation of e has (to the author s knowledge) not yet been published. Knowing the ass and stiffness of the cars and Δ one can calculate test elocities fro (4), restitution of indiidual cars fro (7), the utual restitution coefficient fro (5) and post-ipact elocities fro (8). 4. Model B. This odel does not include cars stiffness and it's based on (0) and (4) only. Equating (0) and (4) results in the equation Δ = + (6) T T for two unknowns. To sole it one could set = = (7) T 0 T where 0 is a new unknown elocity. Substituting (7) into (4) one obtains after siplification ( ) ( ) + = 0 0 0, so 0 = + + (8) This is in fact the elocity of the centre of the ass of colliding cars. Substituting (8) into (7) yields unknown test elocities

10 //006 0 ( ) ( ) = = T T + + (9) Note that in calculation of restitution coefficients (7) the absolute alues of test elocities should be used. Substituting (9) into (6) gies the utual restitution coefficient e = e + e + (30) This forula was deried by different arguents of Howard et al ([7]) and is also quoted by Watts et al ([5]). 4.3 Copartent of the odels Coparing (4) and (5) one finds that test elocities of both odels are the sae if stiffness is proportional to the ass; i.e., k = k 0 and k = k 0 where k 0 is a constant. While the test elocities of the odels differ, the utual restitution coefficient differs only in the case when just one car is crushed peranently, since when T 0 and T 0 then both e = e = so by (5) or (30) it follows e = and when T > 0 and T > 0 then substituting (7) and appropriate test elocities into (5) or (30), and taking (0) into account, yields e = + Δ 0 0 (3)

11 //006 Note that (3) can not be used directly for calculating the utual restitution coefficient in adance since the classification of ipact--fully elastic, fully plastic or ixed-- depends on test elocities. At last the question arises as to which odel is ore physically justified. While Model A has a sound physical base connecting test elocities with crushes, Model B requires soe additional analysis. It turns out that it can be interpreted as follows. The copression ipulse (4), can be written by using (3) as Pc = Δ. Using () one could define test elocities of indiidual cars as elocities resulting at the end of the copression phase in a fixed barrier test as the sae ipulse as in an actual two car collision; i.e., P = Δ = = (3) c T T Fro this equation, test elocities gien already by (9) result. Now by (6) restitution ipulse is Pr = epc = e Δ, so by (5) and (3) one ust hae e Δ = e = e. But this can be fulfilled only in the special case when e T T = e, and consequently, by (30), when e= e. This consequence raises a doubt about Model B s adequacy for general use. 4.4 Exaples The aboe forulas were ipleented into the spreadsheet progra (Table ).As the exaple, a full scale test (test no. 7) reported by Cipriani et al ([4]) was executed. In this test the bullet car ade ipact with the rear of the target car at a elocity of 5 /s or 8 k/h. The ass of the cars and their stiffness was taken fro the report; howeer, the liit speed was taken to be 4 k/h for both cars ([]). The result of the calculation is shown in Table. The calculated elocity difference for the target car is 4.8 k/h, which differs fro that easured (3.9 /s or 4.0 k/h) by about 5%. The calculated elocity change for the bullet car is.3 k/h and the easured one was.9 /s or 0.4 k/h. The discrepancy is thus about 7%. If one takes the liit speed to be 3 k/h,

12 //006 then the calculated alue of elocity change for the bullet car is 3.6 k/h, differing fro that easured by about %, and the calculated alue of elocity change for the target car is 0.4, which actually atches the easured alue.. Table. Spreadsheet progra for calculation of post-ipact elocities Full scale test 7 of Cipriani et al ([4]) Vehicle Vehicle ass kg stiffness kn/ liit elocity k/h 4 4 ipact elocity k/h 8 0 Delta V k/h 8.00 elocity after copression k/h 7.8 syste ass kg syste stiffness kn/ test elocity k/h test restitution restitution 0.45 post ipact elocity k/h Delta V k/h Maxial crush Residual crush Accident Reconstruction In a real car accident the proble is not to deterine post-ipact elocities but usually the opposite; i.e., to calculate the pre-ipact elocities. For deterining pre-ipact elocities, howeer, the post-ipact elocities deterined fro skid-arks should be known. If only the peranent crushes of cars are known then only the elocity changes for indiidual cars in an accident can be calculated. If the characteristics of cars are known--i.e., ass, stiffness and liit elocity--then the proble is soled as follows. Let δ r be residual crush of the first ehicle. The axial crush, then, is ([])

13 //006 3 δ = δ + δ (33) r 0 where the recoerable part of crush is calculated as δ 0 = 0. The axial crush of k the second car can be calculated in the sae way or fro Newton s 3rd law as δ k = δ (34) k The axial energy lost at ipact is then calculated fro Δ E =Δ E +Δ E (35) kδ where Δ E = and fro (), kδ Δ E =. The pre-ipact elocity difference is thus, ΔE Δ = (36) To calculate elocity changes of indiidual ehicles the first test elocities are calculated by (8) ΔE ΔE = = (37) T T Fro (7) the restitution coefficient for indiidual cars are calculated and fro (5) the utual coefficient of restitution. Fro (8) the elocity differences of indiidual cars at ipact are ( + e) ( + ) e Δ = u = Δ Δ = u = Δ + + (38)

14 //006 4 The aboe forulas were prograed into a spreadsheet progra (Table ). As the exaple, the car to car test described by Kerkhoff et al ([8]) is considered. In this test the test car (bullet) struck the rear of the stationary car (target) at a speed of 40.6 ph or 65 k/h. The actual easured Δ was.6 ph or 36. k/h. As can be seen fro Table, the calculated alue Δ for the bullet car is 36. k/h; i.e., the discrepancy between actual and calculated alue is 0.% and the calculated ipact elocity 64.4 k/h differs fro the actual by.3 %. Note that the deforation of the stationary car was not reported, so (34) is used for calculation of its axial dynaic crush. The liit speed for both cars was taken to be 4 k/h ([]). The discrepancy of calculated alues in the preious case is so inial because the actual low ipact elocity tests were used for deterination of stiffness. If one used for the calculation the default alues of CRASH stiffness and appropriate calculated liit elocity for class cars the discrepancy would increase (Table 4). Thus, in this case the calculated elocity change of the bullet car is 38.5 k/h, which differs fro the actual change by about 6% and the calculated Δ is 5. k/h, differing by about 0%. Table. Spreadsheet progra for calculation of elocity differences at ipact. Car to car test no by Kerkhoff et al ([8]) Vehicle Vehicle ass kg Data stiffness kn/ liit speed k/h crush 0.6? recoerable crush axial crush syste ass kg syste stiffness kn/ ax energy lost kj test elocity k/h test restitution restitution 0. Delta V k/h

15 //006 5 References [] M. Batista, A Note on Linear Force Model in Car Accident Reconstruction [] R.M.Brach. Friction, Restitution, and Energy Loss in Planar Collisions Trans ASME, Journal of Applied Mechanics, 5, 64-70, 984 [3] R.M.Brach, R.M.Brach. A Reiew of Ipact Models for Vehicle Collision. SAE Paper [4] A. L. Cipriani. F. P. Bayan, M. L. Woodhouse, A. D. Cornetto, A. P. Dalton, C. B. Tanner, T. A. Tibario, E. S. Deyerl. Low-speed Collinear Ipact Seerity: A Coparison between Full-Scale Testing and Analytical Prediction Tools with Restitution Analysis, SAE Papers [5] R.I.Eori. Analytical Approach to Autoobile Collisions. SAE Papers [6] P.V.Hight, D.B.Lent-Koop, R.A.Hight. Barrier Equialent Velocity, Delta V and CRASH3 Stiffness in Autoobile Collisions. SAE Papers [7] R.P.Howard, J.Boar, C.Bare. Vehicle Restitution Response in Low Velocity Collisions. SAE Paper 9384 [8] J.F.Kerkhoff, S.E.Hisher, M.S.Varat, A.M.Busenga, K.Hailton. An Inestigation into Vehicle Frontal Ipact Stiffness, BEV and Repeated Testing for Reconstruction. SAE Paper [9] R.H.Macillan, Dynaics of Vehicle Collision, Inderscience Enterprise Ltd. 983 [0] M.T.Manson. Mechanics of Robotic Manipulation. MIT Press,00, pp.4 [] R.R.McHenry. A Coparison of Results Obtained with Different Analytical Techniques for Reconstruction of Highway Accidents. SAE Papers [] R.R.McHenry, B.G.McHenry, Effects of Restitution in the Application of Crush Coefficients, SAE [3] E.W.Routh. The Eleentary Part of A Treatise on the Dynaics of a Syste of Rigid Bodies. Doer Publications, 960 [4] S.Tany, The Linear Elastic-Plastic Vehicle Collision, SAE 9073 [5] A.J.Watts, D.R.Atkinson, C.J.Hennessy. Low Speed Autoobile Accidents. Lawyers & Judges Publishing Copany, Tuscon, AZ, 999

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