Development of lumped-parameter mathematical models for a vehicle localized impact

Transcription

1 Journal of Mechanical Science and Technology 5 (7) () 737~747 DOI.7/s6--55-x Developent of luped-paraeter atheatical odels for a vehicle localized ipact Witold Pawlus, Haid Reza Karii * and Kjell Gunnar Robbersyr Departent of Engineering, Faculty of Engineering and Science, University of Agder, N-4898 Gristad, Norway (Manuscript Received October 5, ; Revised March 5, ; Accepted March 5, ) Abstract In this paper, we propose a ethod of odeling for vehicle crash systes based on viscous and elastic properties of the aterials. This paper covers an influence of different arrangeent of spring and daper on the odels response. Differences in siulating vehicle-torigid barrier collision and vehicle-to-pole collision are explained. Coparison of the odels obtained fro wideband (unfiltered) acceleration and filtered acceleration is done. At the end we propose a odel which is suitable for localized collisions siulation. Keywords: Data analysis; Filtering; Modeling; Vehicle crash Introduction This paper deals with establishing an appropriate atheatical odel representing vehicle soft ipacts such as localized pole collisions. In siulation of the vehicle collision, eleents which exhibit viscous and elastic properties are used. Models utilized by us consist of energy absorbing eleents (EA) and asses connected to their both ends. We focus on finding a odel with such an arrangeent of springs, dapers and asses, which siulated, will give a response siilar to the car's behavior during the real crash. Due to the fact that real crash tests are coplex and coplicated events, their odeling is justified and advisable. Every car which is going to appear on the roads has to confor to the worldwide safety standards. However, crash tests consue a lot of effort, tie and oney. The appropriate equipent and qualified staff is needed as well. Therefore our goal is to ae possible siulation of a vehicle crash on a personal coputer. When it coes to odeling the vehicle crash we can distinguish two ain approaches. The first one utilizes CAE (Coputer Aided Engineering) software including FEA (Finite Eleent Analysis) while the second one bases on the analytical ethod presented in this paper. Much research has been done so far in both of those areas. Refs. [-3] provide a brief overview of different types of vehicle collisions. This paper was recoended for publication in revised for by Associate Editor Mohaad Abdul Aziz Irfan * Corresponding author. Tel.: , Fax.: E-ail address: haid.r.arii@uia.no KSME & Springer Approach presented here - atheatical odeling of a crash event with the equations of otion which can be solved explicitly with closed for solutions - is different that the ethods which have been shown in Refs. [4-7]. In order to siulate the collision of a car the software based on FEM (Finite Eleent Method) was utilized. After the creation of 3D CAD and FE odels, the crash siulations were perfored. Results obtained showed good correlation between the test and odel responses. When it coes to deterining crush stiffness coefficients, in Ref. [8] it is presented a ethod which eploys CRASH3 coputer progra. Vehicle structure was odeled as a hoogenous body and then the coparative analysis of the crash response of vehicles tested in both: full-overlap and partial-overlap collisions, was done. A luped paraeter odeling (LPM) is another way of approxiation of the vehicle crash. It is an analytical ethod of forulating a odel which can be further used for siulation of a real event. It allows us to establish dynaic equations of the syste - differential equations - which give the coplete description of the odel's behavior, see Refs. [9] and []. To be able to create a atheatical odel of a vehicle collision, it is often enough (and ore efficient) not to analyze the coplicated crash pulse recorded during the full-scale experient but just to study an approxiation of the easured acceleration signal. Those approxiated functions were copared to experiental pulses in Ref. []. Subsequently they were tested to obtain different odels' responses which were copared to the original pulse. Results confired that the crash pulse approxiation is a reasonable ethod to siplify the collision analysis. Recently, the Haar wavelet-based perforance analysis of the safety barrier for use in a full-scale

2 738 W. Pawlus et al. / Journal of Mechanical Science and Technology 5 (7) () 737~747 test was proposed in Ref. []. Refs. [3-7] tal over coonly used ways of describing a collision - e.g. investigation of tire ars or the crash energy approach. Vehicle crash investigation is an area of up-to-date technologies application. Refs. [8-] discuss usefulness of such developents as neural networs or fuzzy logic in the field of odeling of crash events. It is extreely iportant to assess what factors have an influence on the crash severity for an occupant. As in the case of a vehicle crash siulation, also here we can distinguish two ain ways of exaining the occupant behavior during an ipact. Ref. [3] focuses on finding the relationship between the car's daage and occupant injuries. On the other hand, Ref. [4] eploys FEM software to closely study the crash severity of particular body parts. In Refs. [5-8], basic atheatical odels are proposed to represent a collision. The ain part of this research is devoted to ethods of establishing paraeters of the vehicle crash odel and to real crash data investigation, e.g. creation of a Kelvin odel (spring and daper connected in parallel with ass) for a real experient, its analysis and validation. After odel's paraeters extraction a quic assessent of an occupant crash severity is done. Finally, the dynaic response of such a syste was siilar to the car's real behavior in the tie interval which corresponds to the collision's duration. Paraeters of this assebly (spring stiffness and daping coefficient) were obtained analytically with closed-for solutions according to Ref. [9]. In this paper, we present a process of iproving the accuracy of the vehicle crash odel. We start with siulation of the vehicle to pole ipact by using the Kelvin odel (spring and daper in parallel connected to ass). Afterwards, by filtering the crash pulse data, ore accurate response of the syste is obtained. Model establishent is done one ore tie. This allows us to copare what the crash odels are for both: raw and filtered data, and to decide which of the is ore suitable to represent vehicle to pole collision. In total, four different odels created for the filtered data are elaborated here and it is being assessed which of the gives the ost exact description of the car's behavior in the pole collision. The ain contribution of this paper is the evaluation of the proposed odeling ethodology results with the full-scale experiental data. When copared to the previous wor which concerns the siilar area of research [5], the current study presents ore detailed insight into vehicle localized ipact odeling. By the coparative analysis of different viscoelastic odels responses, it is decided which of the is the ost suitable to siulate vehicle-to-pole collision. To assess odel's fidelity, its structural properties and dynaic responses are exained. Fig.. Schee of the test collision [3]. Fig.. The car is undergoing deforation. Fig. 3. Kelvin odel.. Experiental setup In the experient conducted by UiA [4] the test vehicle, a standard Ford Fiesta. L 987 odel was subjected to a central ipact with a vertical, rigid cylinder at the initial ipact velocity v = 35 /h. Mass of the vehicle (together with the easuring equipent and duy) was 873 g. Experient's schee is shown in Fig.. Vehicle accelerations in three directions (longitudinal, lateral and vertical) together with the yaw rate at the center of gravity were easured. Using noral-speed and high-speed video caeras, the behavior of the obstruction and the test vehicle during the collision was recorded. Fig. shows one of the crash stages. 3. Raw data analysis - Kelvin odel According to Ref. [5], the Kelvin odel shown in Fig. 3 has been proposed to represent the vehicle to pole collision. Sybols used: - spring stiffness, c - daping coefficient, - ass, V - initial ipact velocity. Known paraeters of the odel are: = 873 g - ass V =.8 /s - initial ipact velocity. Paraeters which we obtain fro the crash pulse analysis (acceleration of the car in the x-direction - longitudinal) shown

3 W. Pawlus et al. / Journal of Mechanical Science and Technology 5 (7) () 737~ Table. Coparison between car s and Kelvin odel s responses - raw data. 6 4 Wideband data Filtered acceleration Paraeter Crash pulse analysis Kelvin odel Dynaic crush C [c] Tie of dynaic crush t [s] Acceleration [g]; velocity [/h]; crush [c] C=57c 8 t Acceleration =8s Velocity Crush 5 5 Tie [s] Fig. 4. Raw data analysis. Acceleration [g]; velocity [/h]; displaceent [c] Model acceleration 6 Model velocity Model displaceent Car acceleration 8 Car velocity Car displaceent 5 5 Tie [s] Fig. 5. Kelvin odel s response - raw data. in Fig. 4 are listed in Table. At the tie when the relative approach velocity is zero, the axiu dynaic crush occurs. The relative velocity in the rebound phase then increases negatively up to the final separation (or rebound) velocity, at which tie a vehicle rebounds fro an obstacle. The contact duration of the two asses includes both contact ties in deforation and restitution phases. When the relative acceleration becoes zero and relative separation velocity reaches its axiu recoverable value, separation of the two asses occurs. By following Ref. [9] (ethod of calculating daping factor ζ and natural frequency f is covered in Ref. [5]), spring stiffness and daping coefficient c of the Kelvin odel are deterined to be: = 4π f = 4 π (.9375 Hz) 873g = 9739 N/ c= 4π fζ= 4π.9375Hz. 873g = 33 N s/. Acceleration in x direction [g] Tie [s] Fig. 6. Butterworth 3 rd order filtering results. Aplitude x Wideband data Filtered acceleration Frequency [Hz] Frequency [Hz] Validation of the odel has been done in Matlab Siulin software - the response of the Kelvin odel with above estiated paraeters is shown in Fig. 5. Coparison of dynaic crush and tie of dynaic crush fro the crash pulse analysis and Kelvin odel response is done in Table. Rear. Since the raw data has been used above, the discrepancy between the real initial ipact velocity (which is V = 9.86 /s = 35 /h) and initial ipact velocity obtained fro the raw data analysis (which is V =.8 /s = 39 /h) is visible. Therefore, to eliinate inaccuracies in odeling caused by this velocity difference we need to filter the acceleration easureents. 4. Acceleration easureents filtering Digital filtering ethod has been used here [3]. Frequency response corridors for an appropriate channel class are specified in this standard. Since our goal is to analyze the crash pulse (i.e. integration for velocity and displaceent) we select the channel class CFC 8. Filter utilized by us was Butterworth 3rd order lowpass digital filter with cut-off frequency f N = 3 Hz. Coparison between the wideband data and data filtered with this ethod is shown in Fig. 6. In Fig. 7, the coparison in the frequency doain between the raw and filtered acceleration is presented. Since the scale is linear, we clearly see that the filtering helped us to get rid of the high frequency coponents of the crash pulse. This aes its analysis ore efficient and gives us results which better correspond to the reality than the ones obtained fro wideband data (velocity and displaceent). Aplitude x 4 Wideband data Filtered acceleration Fig. 7. Frequency analysis of crash pulses in linear scale (left - whole spectru, right - cut-off frequency region).

4 74 W. Pawlus et al. / Journal of Mechanical Science and Technology 5 (7) () 737~747 Table. Coparison between car s and Kelvin odel s responses - filtered data. Paraeter Crash pulse analysis Kelvin odel Dynaic crush C [c] Tie of dynaic crush t [s] Acceleration [g]; velocity [/h]; crush [c] Tie [s] Fig. 8. Filtered data analysis. Acceleration [g]; velocity [/h]; displaceent [c] 6 4 This has crucial influence on our further considerations because in order to develop a good odel, we need to have at our disposal real paraeters of the crash test (e.g. initial velocity). 5. Filtered data analysis 5. Kelvin odel Let us deterine what the axiu dynaic crush and the tie at which it occurs are for the filtered data. Paraeters which we obtain fro the crash pulse analysis (acceleration of the car in the x-direction - longitudinal) shown in Fig. 8 are listed in Table. Proceeding in the sae anner as in Section 3, we obtain the following paraeters of the Kelvin odel: = 4π f = 3445 N/ c = 4π fζ = 47 N s/. d c =5c t =76s Acceleration Velocity Crush Model acceleration Model velocity Model displaceent 4 Car acceleration Car velocity Car displaceent Tie [s] Fig. 9. Kelvin odel s response - filtered data. Fig.. Spring-ass odel. Kelvin odel response for those paraeters is shown in Fig. 9. Coparison between the odel and reality for the filtered data is done in Table. Filtering the data has iproved our calculations - we have obtained the real value of the initial velocity V = 9.86 /s = 35 /h. However, we observe a larger discrepancy between the dynaic crush fro the acceleration's integration and odel's prediction than for the raw data. This allows us to clai that since the ethod utilized in both of those cases reains the sae and accuracy of our calculations has increased because of the data filtering, the Kelvin odel is not suitable for odeling the ipact exained by us. For that reason we investigate a sipler odel which consists of spring and ass only. 5. Spring-ass odel The otion of this syste is a non-decayed oscillatory one (sinusoidal) because there is no daping in it [9]. This arrangeent is shown in Fig.. Sybols: - spring stiffness, - ass, a - absolute displaceent of ass. Let us introduce the following notation: V - initial barrier ipact velocity [/s] f - structural natural frequency [Hz]. Response of this syste is characterized by the following equations: α() t = Vωesin( ωet) () α() t = Vcos( ωet) () V α() t = sin( ωet) ω (3) e which represent deceleration, velocity and displaceent, respectively. Furtherore we define: V C = ω e (4) π t = ω (5) e ω e = (6)

5 W. Pawlus et al. / Journal of Mechanical Science and Technology 5 (7) () 737~ Table 3. Coparison between car s and spring-ass odel s responses - filtered data. Paraeter Crash pulse analysis Spring-ass odel Dynaic crush C [c] Tie of dynaic crush t [s] Acceleration [g]; velocity [/h]; displaceent [c] 6 4 Model acceleration Model velocity Model displaceent 4 Car acceleration Car velocity Car displaceent Tie [s] Fig.. Spring-ass odel s response. as axiu dynaic crush, tie of axiu dynaic crush and syste's circular natural frequency, respectively. To investigate what the paraeters C and t of such a odel are, first we need to find the spring stiffness. By substituting Eq. (6) to Eq. (4) and rearranging one gets: V =. (7) C Fro Fig. 8 it is obtained C =.5 = 5 c and V = 9.86 /s = 35 /h for filtered data. Therefore (9.86 / s) / = g = N (.5) π π π t = = = ωe N/ 873g =.83 s. Spring-ass odel's response for above spring stiffness (initial velocity and ass of the car reain the sae) is shown in Fig.. Let us copare what the dynaic crush and the tie at which it occurs are for the car and odel see Table 3. Results obtained in this step are good. The dynaic crush estiated by the spring-ass odel is exactly the sae as the reference dynaic crush of a real car. When it coes to the tie when it occurs, the difference between the odel and reality is less than %. This odel gives us good approxiation of the car's behavior during the crash. It is a particular case of a Kelvin odel in which daping has been set to zero as well as of a Maxwell odel in which daping goes to infinity. Fig.. Maxwell odel - designates zero-ass. 6. Maxwell odel - introduction The arrangeent in which spring and daper are connected in series to ass is called Maxwell odel - Fig.. To derive its equation of otion it is proposed to place sall ass ' between spring and daper. By doing this, the inertia effect which occurs for the spring and daper is neglected and the syste becoes third order differential equation which can be solved explicitly [9]. According to Fig. we define d and d' as absolute displaceent of ass and absolute displaceent of ass ', respectively. We establish the following equations of otion (EOM): d = c( d d') (8) d ' ' = cd ( d') d'. (9) By differentiating Eq. (8) and Eq. (9) w.r.t. tie and setting ' = we obtain: d = c( d d') () = cd ( d') d'. () We su up both sides of Eqs. () and () and rearrange: d' = d. () We substitute Eq. () into Eq. (8) and finally obtain the EOM found below: d+ d+ d =. (3) c Therefore, characteristic equation of the Maxwell odel is: ss [ + s ]. c + = (4) In this syste, the rebound of the ass depends on the sign of discriinant Δ of the quadratic equation in bracets. For positive Δ there is no rebound, i.e.: > 4. c

6 74 W. Pawlus et al. / Journal of Mechanical Science and Technology 5 (7) () 737~747 In this case, roots of the characteristic equations (Eq. (4)) are, respectively: s = s = a+ b s = a b where: a = c b =. c On the other hand, for negative Δ the rebound occurs when: v ad d = b d = d. However, in a Maxwell odel, the ass ay not rebound fro the obstacle. It eans that its displaceent increases with tie to an asyptotic value. The paraeter, which deterines whether the rebound will occur or not, is daping coefficient. When it is less than a liiting one (naed transition daping coefficient c*), the ass will be constantly approaching an obstacle, whereas when it is higher, there will exist a dynaic crush at a finite tie. Another boundary situation is for daping coefficient c =. Then the Maxwell odel degenerates into spring-ass syste. To deterine the value of transition daping coefficient we assue that c =, or equivalently < 4. c c = * In this case, roots of the characteristic equation (Eq. (4)) are, respectively: s = s = a+ ib s = a ib where: a = c b = c. Since this case is of our greater interest than the previous one (due to the fact that in the experient rebound occurred) we will describe in details its response. Displaceent of a ass is given by the forula: α = de + e [ dsin( bt) + d cos( bt)]. (5) st at Initial conditions (t = ) are: α = α = v α =. where v is the initial ipact velocity. Constants are: av d = a + b and * c =. (6) Indeed, for c < c * we have Δ > - it eans no dynaic crush at a finite tie. We are able to assess what the inial daping should be, which we add to the siple spring-ass odel entioned above, which will produce the dynaic crush not extended in an infinite period of tie. According to Eq. (6), for the odel and crash test being analyzed in Section 5., we calculate the transition daping coefficient: * N/ 873g c = = 877 N s/. For every daping greater than this value, the Maxwell odel fored fro the spring-ass odel fro Section 5., will give us the response ore and ore siilar to the springass odel characteristics presented in Fig., as it is shown in Fig. 3. It is noting that the final displaceent (or asyptotic value - for transition daping coefficient) achieved by the ass in this odel is characterized by the equation (V - initial ipact velocity, - ass, c - daping coefficient): crush V c =. (7) This syste is appropriate for siulating soft ipacts or offset ipacts because the tie of dynaic crush is longer than for Kelvin odel. We assue the sae paraeters for

7 W. Pawlus et al. / Journal of Mechanical Science and Technology 5 (7) () 737~ Displaceent [] c * c *.4 4c * Infinite daping Tie [s] Fig. 3. Maxwell odel responses for different values of daping. 7. Maxwell odel analysis When it coes to Maxwell odel, we will discuss just two cases for which Δ <, i.e. when the rebound occurs, because that is what happens during the experient. We are going to start with the siplification of this situation, in which daping coefficient of this odel has a liiting, transitional value. Then we proceed to the full Maxwell odel's analysis. 7. Maxwell odel with transition daping coefficient This is the particular case of a Maxwell odel in which ass' displaceent reaches an asyptotic value given by Eq. (7). For * c = paraeters of Eq. (5) degenerate into: Displaceent [] Maxwell odel Kelvin odel a = ω b = v d = b d = v ω d = v ω.5 Kelvin t =.9s Maxwell t =.8s where Tie [s] Fig. 4. Maxwell and Kelvin odels responses coparison. both odels, e.g.: = N/, c = 5 N-s/, = 5 g, v = /s. In Fig. 4, it is seen that for the Maxwell odel the dynaic crush occurs later than for the Kelvin odel. This is an analog situation to the real crash: in a vehicle-torigid barrier collision (Kelvin odel) the whole ipact energy is being consued faster, therefore the crash is ore dynaic than the vehicle-to-pole collision (Maxwell odel) - under the assuption that we copare the sae cars with the sae initial ipact velocities - as in the exaple above. It is noting that we do not investigate here the agnitude of the displaceent of both odels - as we can see for the sae paraeters it is higher for the Maxwell odel. Above exaple just illustrates the dynaic responses of those two systes and in order to apply those two odels to the real crash one needs to assess what spring stiffness and daping coefficient of both of the are separately. ω =. We tae advantage of the following trigonoetric relationships: sin( bt) li = t b b licos( bt) =. b Finally we coe up with the following equation of ass' displaceent: v t α = [ ( ωt + ) e ω ]. (8) ω To establish the paraeters of the Maxwell odel (spring stiffness and daping coefficient c) we just substitute to Eq. (8) values of initial ipact velocity v, axiu dynaic crush α and tie of axiu dynaic crush t taen fro the acceleration easureents analysis shown in Fig. 8 - we obtain ω = 37.5 rad/s. Knowing circular natural frequency ω and ass of the whole vehicle = 873 g, we calculate spring stiffness and transition daping coefficient c*: = ω = = 8966 N/ * c = = = 6377 N s/ Response of the odel with above coputed paraeters

8 744 W. Pawlus et al. / Journal of Mechanical Science and Technology 5 (7) () 737~747 6 Acceleration [g]; velocity [/h]; displaceent [c] 4 Model acceleration Model velocity Model displaceent 4 Car acceleration Car velocity Car displaceent Tie [s] Displaceent [] Experient Fitting.5..5 Tie [s] Fig. 5. Maxwell odel s (transition daping coefficient) response. and initial ipact velocity v = 9.86 /s = 35 /h is shown in Fig. 5. As we can see the value of axiu dynaic crush is the sae as the one obtained fro the experient's data analysis (C =.5 ) and tie when it occurs is uch longer - approxiately t =. s (copared to experient's t =.76 s). 7. Maxwell odel - rebound Response of the Maxwell odel is described by Eq. (5). Having the car's displaceent curve fro the experient we can establish paraeters of the odel (spring stiffness and daping coefficient c) just by fitting the curve defined by Eq. (5) to that real graph. However paraeters which we obtain by fitting Eq. (5) are: a, b, d, d and d. Since d and d (we do not discuss d separately because d = -d ) are functions of v as well, it is not guaranteed that the odel's paraeters which we obtained would be correct - in another words, v wouldn't be fixed if we fit Eq. (5) to the experient's displaceent. Therefore we express Eq. (5) only in ters of a and b (which are just functions of, c and ass = 873 g) and set initial ipact velocity to v = 9.86 /s. The equation which we obtain has the following for: av α = + a + b av v at a b av e + sin( bt) + cos( bt). b a + b (9) Fitting Eq. (9) to the experient's results has been done in Matlab Curve Fitting Toolbox and is shown in Fig. 6. Fitting Eq. (9) not Eq. (5) resulted in loss of approxiation's accuracy but on the other hand, we are sure that the initial ipact velocity has the correct value. Fro the above operation we obtain paraeters a and b of Eq. (9) which are equal to: a = = 4.79 c s Fig. 6. Fitting the Maxwell odel's response to the real experient's displaceent. Acceleration [g]; velocity [/h]; displaceent [c] 6 4 Model acceleration Model velocity Model displaceent 4 Car acceleration Car velocity Car displaceent Tie [s] Fig. 7. Coplete Maxwell odel s response. b = =.6. c s Now we have all we need to calculate daping coefficient c and spring stiffness of the Maxwell odel, respectively: a ( + b) c= = 9546 N s/ a = ac = 5787 N/. Response of the odel is shown in Fig. 7. As we can see the value of axiu dynaic crush is exactly the sae as the one obtained fro the experient's data analysis (C = 5 c) and tie when it occurs is longer - t = 4 s (copared to the experient's t = 76 s). However, as it is going to be shown in the following Section, the overall response of the Maxwell odel is the ost siilar to the car's behavior during collision with a pole. 8. Models coparison To represent vehicle to pole collision we established in total four odels here (spring-ass odel, Kelvin odel, Maxwell odel with transition daping coefficient and coplete Maxwell odel). Let us copare their responses with the car's

9 W. Pawlus et al. / Journal of Mechanical Science and Technology 5 (7) () 737~ Displaceent [c] Real car Kelvin odel Spring ass odel Maxwell odel (transition daping) Maxwell odel 5 5 Tie [s] Fig. 8. Models responses coparison. behavior during the experient analyzed by us - see Fig. 8. Characteristics which the best represents the overall car's behavior during the crash period belongs to the Maxwell odel. Although Kelvin and spring-ass odels give good approxiation in the beginning of the crash (up to the tie of axiu dynaic crush), they copletely fail when it coes to the crash representation after the rebound. Maxwell odel with transition daping coefficient shows correctly just the axiu dynaic crush, not its tie at all. Therefore the Maxwell odel gives the best overall outcoe - there is no difference in axiu displaceent and about 7% of divergence for tie of axiu dynaic crush coparing to the reality. And the entire shape of the Maxwell odel's response resebles closely the real car's crush. 9. Conclusion and future wors In this paper, we studied a process of iproving the accuracy of the vehicle crash odel. First, we siulated the vehicle under a pole ipact by using the Kelvin odel. Afterwards, by filtering the crash pulse data, ore accurate response of the syste was obtained. Model establishent was done one ore tie. Finally, we copared the crash odels and it was concluded which of the is ore suitable to represent vehicle to pole collision. The obtained results indicate that the Kelvin odel is not appropriate for siulation of the collision which we deal with. Based on Section 6, for the data prepared in the proper way, we establish a proper odel. Results obtained fro studying Maxwell odel provided us with satisfactory results. Coparative analysis of the odel's and real car's responses turned out to be appropriate. Therefore if one wants to siulate a vehicle to pole collision it is advisable to use Maxwell odel. It is desirable to verify whether the other viscoelastic odels which were not discussed in this paper are capable of vehicle crash siulation. In particular, so called hybrid odels (systes coposed of two springs, one daper, and a ass) ay be proising for this application. Furtherore, a twoass-spring-daper odel can be used to represent interactions between fore- and aft-frae of a vehicle. On top of that, it is advisable to exaine ethods for nonlinear syste paraeters identification. Since all the odels presented in the current study are luped paraeter ones which are valid only for the data which were used for their creation, they cannot be used to siulate e.g. a high-speed vehicle collision. However, the capabilities of atheatical odels with nonlinear paraeters (stiffness and daping) to siulate a variety of crash events are required to be assessed. Noenclature : Spring stiffness c : Daping coefficient : Mass V ; V : Initial ipact velocity ζ : Daping factor f : Structural natural frequency C : Maxiu dynaic crush t : Tie of axiu dynaic crush f N : Cut-off frequency a; d : Absolute displaceent of ass α : Model displaceent ω e ; ω : Circular natural frequency : Zero-ass d : Absolute displaceent of ass c * : Transition daping coefficient Δ : Discriinant References [] M. Borovinse, M. Vesenja, M. Ulbin and Z. Ren, Siulation of crash tests for high containent levels of road safety barriers, Engineering Failure Analysis, 4 (8) (7) [] R. Harb, E. Radwan, X. Yan and M. Abdel-Aty, Light truc vehicles (LTVs) contribution to rear-end collisions, Accident Analysis & Prevention, 39 (5) (7) [3] A. Soica and S. Lache, Theoretical and experiental approaches to otor vehicle: pedestrian collision, 3rd WSEAS International Conference on Applied and Theoretcial Mechanics, Spain, Deceber (7). [4] O. Klyavin, A. Michailov and A. Borovov, Finite eleent odeling of the crash-tests for energy absorbing lighting coluns, ENOC-8, Saint Petersburg, Russia, June (8). [5] C. -H. Lin, Modeling and siulation of van for side ipact sensing tests, General Motors R&D Center, USA, Paper Nuber 7-6 (7). [6] A. Bižal, K. Jernej, R. Uroš and F. Matija, Nuerical siulation of crash test for the vehicle student roadster, FISITA 8 (8). [7] R. J. Yang, L. Tseng, L. Nagy, and J. Cheng, Feasibility study of crash optiization, advances in design autoation, Proceedings of the 994 ASME Design Technical Conference, th Design Autoation Conference, Minneapolis, Minnesota, USA, 69 () (994) [8] J. A. Neptune and J. E. Flynn, A ethod for deterining

10 746 W. Pawlus et al. / Journal of Mechanical Science and Technology 5 (7) () 737~747 crush stiffness coefficients fro offset frontal and side crash tests, SAE Paper 984 (998). [9] A. Deb and K. C. Srinivas, Developent of a new lupedparaeter odel for vehicle side-ipact safety siulation, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Autoobile Engineering, () (8) [] P. Jonsén, E. Isasson, K. G. Sundin and M. Oldenburg, Identification of luped paraeter autootive crash odels for buper syste developent, International Journal of Crashworthiness, 4 (6) (9) [] M. S. Varat and S. E. Husher: Crash pulse odeling for vehicle safety research, 8th ESV Paper (). [] H. R. Karii and K. G. Robbersyr, Signal analysis and perforance evaluation of a vehicle crash test with a fixed safety barrier based on Haar wavelets, International Journal of Wavelets, Multiresoloution and Iage Processing, 9 () () [3] G. Šušteršič, I. Grabec and I. Prebil, Statistical odel of a vehicle-to-barrier collision, International Journal of Ipact Engineering, 34 () (7) [4] X. Y. Zhang, X. L. Jin and J. Shen, Virtual reconstruction of two types of traffic accident by the tire ars, Lecture Notes Coput. Sci. 48, (6) [5] D. Trusca, A. Soica, B. Benea, and S. Tarulescu, Coputer Siulation and Experiental Research of the Vehicle Ipact, WSEAS Transactions on Coputers, 8 () (9) [6] D. Vangi, Energy loss in vehicle to vehicle oblique ipact, International Journal of Ipact Engineering, 36 (3) (9) 5-5. [7] W. Pawlus, H. R. Karii and K. G. Robbersyr, Matheatical odeling of a vehicle crash test based on elastoplastic unloading scenarios of spring-ass odels, International Journal of Advanced Manufacturing Technology, DOI:.7/s x,. [8] T. Oar, A. Esandarian and N. Bedewi, Vehicle crash odelling using recurrent neural networs, Matheatical and Coputer Modelling, 8 (9) (998) 3-4. [9] I. A. Harati, A. Rovid, L. Szeidl and P. Varlai, Energy distribution odeling of car body deforation using LPV representations and fuzzy reasoning, WSEAS Transactions on Systes, 7 () (8) [] V. Giavotto, L. Puccinelli, M. Borri, A. Edelan and T. Heijer, Vehicle dynaics and crash dynaics with inicoputer, Coputers & Structures, 6 (-4) (983) [] W. Pawlus, J. E. Nielsen, H. R. Karii and K. G. Robbersyr. Coparative analysis of vehicle to pole collision odels established using analytical ethods and neural networs. The 5th IET International Syste Safety Conference, Manchester, UK, October (). [] W. Pawlus, H.R. Karii and K. G. Robbersyr, Analysis of vehicle to pole collision odels: analytical ethods and neural networs. International Journal of Control Theory and Applications, 3 () () [3] C. Conroy, G. T. Toinaga, S. Erwin, S. Pacyna, T. Vely, F. Kennedy, M. Sise and R. Coibra, The influence of vehicle daage on injury severity of drivers in head-on otor vehicle crashes, Accident Analysis & Prevention, 4 (4) (8) [4] Y. Niu, W. Shen and J. H. Stuhiller, Finite eleent odels of rib as an inhoogeneous bea structure under highspeed ipacts, Medical Engineering & Physics, 9 (7) (7) [5] W. Pawlus, J. E. Nielsen, H. R. Karii and K. G. Robbersyr, Matheatical odeling and analysis of a vehicle crash, 4th European Coputing Conference, Bucarest, Roania, April (). [6] W. Pawlus, J. E. Nielsen, H.R. Karii and K.G. Robbersyr, Developent of atheatical odels for analysis of a vehicle crash, WSEAS Transactions on Applied and Theoretical Mechanics, 5 () () [7] W. Pawlus, J. E. Nielsen, H. R. Karii and K. G. Robbersyr, Further results on atheatical odels of vehicle localized ipact, The 3rd International Syposiu on Systes and Control in Aeronautics and Astronautics, Harbin, China, June (). [8] W. Pawlus and J. E. Nielsen, Developent of atheatical odels for vehicle to pole collision, Shaer Publishing, Maastricht, The Netherlands (). [9] M. Huang, Vehicle Crash Mechanics, CRC Press, Boca Raton, USA (). [3] K. G. Robbersyr, Calibration test of a standard Ford Fiesta.L, odel year 987, according to NS-EN 767, Project Report 43/4, Gristad: Agder University College (4). [3] ISO 6487:, Road vehicles - easureent techniques in ipact tests instruentation. world phenoena. Witold Pawlus is a Research Assistant at the University of Agder in Norway. He received his B.Sc. in in Mechatronics fro AGH University of Science and Technology in Kraów, Poland. His research interests are within the area of nonlinear systes paraeters identification and atheatical odeling of real

11 W. Pawlus et al. / Journal of Mechanical Science and Technology 5 (7) () 737~ Haid Reza Karii is a Professor in Control Systes at the University of Agder in Norway. His research interests are in the areas of control theory with an ephasis on applications in engineering. Dr. Karii is a senior eber of IEEE and serves as a Chairan of the IEEE Chapter on Control Systes in Norway. He is also serving as an editorial board eber for soe international journals, such as Mechatronics, Journal of the Franlin Institute, etc. He is a eber of IEEE Technical Coittee on Systes with Uncertainty, IFAC Technical Coittees on Robust Control and on Autootive Control. Kjell G. Robbersyr is a Professor in Machine Design at both: the University of Agder and Hogsolen in Bergen in Norway. Machine design (coponent and syste design, product developent) and vehicle crash siulations (FEA) are his ain research interests. Fro 997 to 5 he was the leader of a scientific group woring in the area of full-scale crash testing, including e.g. perforance evaluation of safety barriers.