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Proceedings of the 9th Vienna International Conference on Proceedings ofModelling the 9th Vienna International Conference on Mathematical Proceedings of the 9th International Conference on Mathematical Vienna, Austria, February 21-23, 2018 Available online Proceedings ofModelling the 9th Vienna Vienna International Conference on at www.sciencedirect.com Mathematical Modelling Vienna, Austria, February 21-23, 2018 Mathematical Modelling Vienna, Vienna, Austria, Austria, February February 21-23, 21-23, 2018 2018

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IFAC PapersOnLine 51-2 (2018) 837–842

A vehicle model for crash stage simulation A vehicle model for crash stage simulation A vehicle model for crash stage ADario vehicle model crash stage simulation simulation Vangi*, Filippofor Begani*, Michelangelo-Santo Gulino*

Dario Vangi*, Filippo Begani*, Michelangelo-Santo Gulino* Florian Spitzhüttl** Dario Begani*, Michelangelo-Santo Dario Vangi*, Vangi*, Filippo Filippo Begani*, Michelangelo-Santo Gulino* Gulino* Florian Spitzhüttl** * Department of Industrial Engineering, University of Florence, 50139 Florence, Italy, (e-mail: [email protected]) Florian Spitzhüttl** Florian Spitzhüttl** * Department Industrial Engineering, University of Florence, 50139 Florence, Italy, (e-mail: [email protected]) **Instituteoffor Traffic Accident Research at Dresden University of Technology (VUFO), 01069 Dresden, Germany, (e* Industrial Engineering, University of 50139 Florence, (e-mail: [email protected]) * Department Department offor Industrial Engineering, University of Florence, Florence, 50139 Florence, Italy, Italy, (e-mail: [email protected]) **Instituteof Traffic Accident Research atmail:[email protected]) Dresden University of Technology (VUFO), 01069 Dresden, Germany, (e**Institute Dresden **Institute for for Traffic Traffic Accident Accident Research Research at atmail:[email protected]) Dresden University University of of Technology Technology (VUFO), (VUFO), 01069 01069 Dresden, Dresden, Germany, Germany, (e(email:[email protected]) mail:[email protected])

Abstract: Simulation of vehicle impact stages is gaining more importance as the years go by. The reasons Abstract: Simulation vehicle impact is gaininginmore importance theactive years go by. The reasons are an increase in the of requested vehiclesstages performances terms of passiveas and safety but also the Abstract: Simulation of vehicle impact stages is gaining more importance as the years go by. The reasons Abstract: Simulation of vehicle impact stages is gaining more importance as the years go by. The reasons are an increase in the requested vehicles performances in terms of passive and active safety but also the necessity to investigate causes which lead to car accidents. The paper describes a special purpose 2D are an in the vehicles performances in passive and safety but also the are an increase increase in time-efficient the requested requested vehicles performances in terms terms ofpaper passive and active active safety purpose but alsorule the necessity to investigate causes which lead to car accidents. Theof describes aequations special 2D vehicle model for crash stage simulation and therefore introduces the which necessity to causes which lead to car accidents. The describes aaequations special 2D necessity to investigate investigate which lead to car accidents. The paper paper describes special purpose purpose 2D vehicle for time-efficient crash stage simulation and therefore theanalytical which rule over themodel model first. Thecauses Finite Element Method (FEM) represents theintroduces basis for the formulation vehicle model for time-efficient crash stage simulation and therefore introduces the equations which rule vehicle model time-efficient crash stage simulation and therefore theanalytical equations whichofrule over the modelfor first. The Finite Element Method (FEM) represents theintroduces basis for the formulation of the problem. However, the different stiffness of the various vehicle areas involves a calibration the over model first. The Element Method (FEM) represents the for the formulation over the modelreal first. The Finite Finite Element Method (FEM) represents the basis basis for the analytical analytical formulation of thethe problem. However, the different stiffness of the various vehicle areas involves a calibration of the model, using vehicle-to-barrier crash tests as a reference (carried out by EuroNCAP, NHTSA, etc.). of the the stiffness various vehicle areas aa calibration of the of the problem. problem. However, the different different stiffness of the variousdemonstrate vehicle out areas involves calibration of the model, using realHowever, vehicle-to-barrier crash tests asof a the reference (carried byinvolves EuroNCAP, NHTSA, etc.). Based on the obtained stiffness value, performed simulations the applicability of the method model, using real vehicle-to-barrier crash tests as a reference (carried out by EuroNCAP, NHTSA, etc.). model, using real vehicle-to-barrier crash tests as a reference (carried out by EuroNCAP, NHTSA, etc.). Based the obtained stiffness value, performed simulations the applicability of the method to real on vehicle-to-vehicle impacts contained in databases likedemonstrate AREC, VERSUE, etc. Furthermore, realBased on the stiffness value, performed simulations the applicability of method Based on the obtained obtained stiffness value, performed simulations demonstrate thefor applicability of the the method to realcrashes vehicle-to-vehicle impacts contained in databases likedemonstrate AREC, VERSUE, etc.different Furthermore, realworld and results of the developed model simulations are compared four exemplary to real vehicle-to-vehicle impacts contained in like AREC, VERSUE, etc. Furthermore, realto realcrashes vehicle-to-vehicle impacts contained in databases databases like are AREC, VERSUE, etc. Furthermore, realworld and results of the developed model simulations compared different exemplary cases, highlighting the possibility to fully describe the events dynamics andfor thefour vehicles deformations. world crashes and results of the developed model simulations are compared for four different exemplary world crashes and results of the developed model simulations are compared for four different exemplary cases, highlighting the possibility to fully describe the events dynamics and the vehicles deformations. Therefore, the described model simulation times arethe evidently shortened in respect to moredeformations. complicated cases, highlighting the possibility to events dynamics and the vehicles cases, highlighting the possibility to fully fully describe describe eventsresources dynamics andrespect the vehicles deformations. Therefore, the described model simulation times arethe evidently shortened in to morethe complicated solution approaches, like FEM or Multi-Body models. These savings also imply possibility Therefore, the described model times are shortened in respect to more Therefore, the described model simulation times are evidently evidently shortened ini.e. respect to morethecomplicated complicated solution approaches, like FEM orsimulation Multi-Body models. These resources savings alsosimulation imply possibility to simulate activation of Advanced Driving Assistance Systems (ADAS), the of multiple solution approaches, FEM or models. resources savings also imply possibility solution approaches, like FEM or Multi-Body Multi-Body models. These These resources savings alsosimulation imply the theof possibility to simulate activationlike of the Advanced Driving vary. Assistance Systems (ADAS), i.e. the multiple impact configurations as ADAS features to simulate activation of Advanced Driving Assistance to simulate activation as of the Advanced Driving vary. Assistance Systems Systems (ADAS), (ADAS), i.e. i.e. the the simulation simulation of of multiple multiple impact configurations ADAS features Keywords: Crash simulation, Reduced order models,Control) Numerical simulation, Traffic impact as the ADAS features vary. © 2018,configurations IFAC (International Federation of Automatic Hosting by Elsevier Ltd.accidents All rights reserved. impact configurations as the ADAS features vary. Keywords: Crash simulation, Reduced order models, Numerical simulation, Traffic accidents Keywords:  Numerical Keywords: Crash Crash simulation, simulation, Reduced Reduced order order models, models, Numerical simulation, simulation, Traffic Traffic accidents accidents  1. Finite Element Models (FEM) requiring the vehicle to be  1. INTRODUCTION 1. Finite Element Models (FEM) of requiring theBecause vehicle of to the be discretized in a very large number elements. 1. INTRODUCTION 1. Finite Element Element Models (FEM) of requiring theBecause vehicle of to the be 1. Finite Models (FEM) requiring the vehicle to be discretized in a very large number elements. 1. INTRODUCTION high deformations and displacements implied in the crash, the Vehicles impact behaviour has always been an attractive topic discretized in a very large number of elements. Because of the 1. INTRODUCTION in a very large number elements. of the high deformations displacements implied inBecause the terms crash, ruling equations areand constituted of of many non-linear Vehicles impactfield. behaviour has alwaysabout beenstructural an attractive topic discretized in the research The knowledge response high deformations and displacements implied in the the terms crash,and the Vehicles impactfield. behaviour has always alwaysabout beenstructural an attractive attractive topic high deformations and displacements implied in crash, the ruling equations are constituted of many non-linear Vehicles impact behaviour has been an topic the approach is referred to as Non-Linear FEM (Pawlus, etand al., in the research The knowledge response to crashes for various means of transport can be decisive in ruling equations are constituted of many non-linear terms and in the research field. The The knowledge about structural structural response ruling equations are is constituted of many non-linear terms and the approach is referred to as Non-Linear FEM (Pawlus, et in the research field. knowledge about response 2011). The method accurate, but the calculation times are to crashes for various means of transport can be decisive in many different applications. The most advanced techniques to the approach is referred to as Non-Linear FEM (Pawlus, et al., al., to crashes for various means of transport can be decisive in approach is referred to as Non-Linear FEM (Pawlus, al., 2011). is generally accurate, but the calculation are to crashes for various means of most transport canoptimization be decisive of in high, soThe the method method is employed in the lasttimes partetof a many different applications. The advanced techniques to the simulate crashes allow for the structural 2011). The method is accurate, but the calculation times are many different applications. The most advanced techniques of to 2011). accurate, but the the calculation area high, soThe the method isphase, generally employed in the lasttimes part of many different applications. The most advanced techniques to new vehicle design when few vehicle models simulate crashes allow forto the structural optimization vehicles (crashworthiness) enhance occupants’ safety, as high, so the the method method isphase, generally employed in the the last part part of aa simulate crashes allow allow for forto the the structural optimization of so generally employed last of new vehicle designis when the few vehicle models simulate crashes structural optimization of available in libraries are analysed (Yildiz &inSolanki, 2012 ) or vehicles (crashworthiness) enhance occupants’ safety, In as high, well as for the reconstruction of road accidents dynamics. new vehicle design phase, when the few vehicle models vehicles (crashworthiness) to enhance occupants’ safety, as vehicle design phase, when the few vehicle models available in libraries are analysed (Yildiz & Solanki, 2012 ) or vehicles (crashworthiness) to of enhance occupants’ safety,the as new to investigate crashworthiness features of particular well as for the reconstruction road accidents dynamics. In the last few years, simulation algorithms also permitted available in libraries libraries are analysed analysed (Yildiz (Yildiz & Solanki, Solanki, 2012 )) or or well as for for the reconstruction ofalgorithms road accidents accidents dynamics.the In available in are & 2012 to investigate crashworthiness features of particular well as the reconstruction of road dynamics. In components (Wei, et al., 2016). Examples for commercial the last few years, simulation also permitted enhancement or optimization of specific also Advanced Driver to investigate crashworthiness features of particular the last few years, simulation algorithms permitted the © © © investigate featuresorfor of commercial particular components et al., 2016). Examples the last fewSystems years, simulation algorithms also permitted the to FEM software(Wei, are crashworthiness LS-DYNA , ABAQUS ANSYS . enhancement or optimization of specific Advanced Assistance (ADAS) features (Ming, et al., Driver 2016) components (Wei, et al., al., 2016). 2016). Examples for commercial © © © enhancement or optimization optimization of specific specific Advanced Driver components (Wei, et Examples for commercial FEM software are LS-DYNA , ABAQUS or ANSYS . the enhancement or of Advanced Driver 2. Multi-body (MB) models, in which different portions of Assistance Systems (ADAS) features (Ming, et al., 2016) © © © which actively change pre-crash conditions of et theal., vehicles. FEM software (MB) are LS-DYNA LS-DYNA ©, ABAQUS© or ANSYS©. Assistance Systems (ADAS) features (Ming, 2016) FEM , which ABAQUS or ANSYS . are software are 2. Multi-body models, in different portions of Assistance Systems (ADAS) features (Ming, et al., 2016) vehicle are connected through kinematic joints. Forces which actively change pre-crash conditions of the vehicles. Because of thesechange enhancements which involveofmany types of 2. Multi-body (MB) models, in which different portions of the the which actively pre-crash conditions the vehicles. 2. Multi-body (MB) models, in which different portions of vehicle are connected through kinematic joints. Forces are which actively pre-crash conditions ofmany the vehicles. exchanged by those constraints: the parts joints. are rigid and the the Because of from thesechange enhancements which involve types of road users, 2010 to 2015 the number of road fatalities in vehicle are connected through kinematic Forces are Because of these enhancements which involve many types of vehicle are connected through kinematic joints. Forces are exchanged by those constraints: the parts are rigid and the Because of from these enhancements which involve many types of variation depends on relative movements. However, roadEuropean users, 2010 to 2015 the number of road fatalities in shape the Union (EU) decreased of about 17 % (European exchanged by those those constraints: the parts parts are rigid rigidHowever, and the road users, from 2010 to 2015 the number of road fatalities in exchanged by constraints: the are and the shape variation depends on relative movements. road users, 2010 to 2015 the Nevertheless, number of road in Addition of FEM can be used to consider single parts the European Union (EU) decreased of about 17 still %fatalities (European Road Safetyfrom Observatory, 2017). 3 people shape variation depends on relative movements. However, the European Union (EU) decreased of about 17 % (European shape variation depends on relative movements. However, Addition of FEM can be used to consider single parts the European Union (EU) decreased of about 17 % (European (Hamza & Saitou, 2005). The Lagrange method Road Safety Observatory, 2017). Nevertheless, stillsimulative 3 people deformations die each hour, requiring steps forward also in the of FEM be used to consider single parts Road Safety Observatory, 2017). Nevertheless, stillsimulative people Addition Addition of applied FEM can can be type used to analysis, consider singlemethod deformations (Hamza & this Saitou, 2005). The Lagrange Road Safety Observatory, 2017). Nevertheless, still 33 people is the most in of based onparts the die each hour, requiring steps forward also in the approaches, to achieve near-zero fatalities in 2050 as deformations (Hamza & Saitou, 2005). The Lagrange method die each hour, requiring steps forward also in the simulative deformations (Hamza & Saitou, 2005). The Lagrange method is the most applied in this type of analysis, based on the die each hour, requiring steps forward also in the simulative D’Alembert’s principle. MB methods generally allow to approaches, to achieve near-zero fatalities in 2050 as prescribed byto the EU near-zero Transport fatalities White Paper. These applied in type of based on approaches, achieve in 2050 2050 as is is the the most most applied in this this type of analysis, analysis, based on the the D’Alembert’s principle. MB methods generally allow to approaches, to achieve near-zero fatalities in as analyse models’ kinematics and, regarding vehicle crashes, prescribed by the EU Transport White Paper. These enhancements in the need to generate more D’Alembert’s principle. MB methods generally allow to prescribed by also the result EU Transport Transport White Paper. These These D’Alembert’s principle. MB methods generally allowit to analyse models’ kinematics and, regarding vehicle crashes, prescribed by the EU White Paper. solve crash dynamics quicker than in the FEM case; is enhancements also result in the need to generate more synthetic data toalso evaluate ADAS performances, leadingmore to a analyse models’ kinematics and, regarding vehicle crashes, to to enhancements result in the need to generate analysecrash models’ kinematics and,than regarding vehicle crashes, to solve dynamics quicker in the FEM case; it is enhancements also result in the need to generate more commonly used in the early design stages of a vehicle, to study synthetic data to evaluate ADAS performances, leading to a significant increase in theADAS numberperformances, of simulationleading runs with solve crash dynamics quicker than in the FEM case; it is synthetic data to evaluate to a solve crash dynamics quicker than in the FEM case; it is commonly used in the early design stages of a vehicle, to study synthetic data to evaluate performances, leading toofa crashworthiness features, or for accident reconstruction significantcrash increase in theADAS number of simulation runs with different configurations. Considering the complexity commonly used in the early design stages of a vehicle, to study significant increase in the number of simulation runs with © commonly used in the early design stages of a vehicle, to study crashworthiness features, or for accident reconstruction significant increase in the number of the simulation runs with Some software examples are MADYMO (TASS different crash configurations. Considering the complexity of purposes. existing simulation methods, this raises necessity to reduce features, or accident reconstruction © different crash configurations. configurations. Considering the complexity complexity of crashworthiness crashworthiness features, or ©for for accident reconstruction purposes. Some ©software examples are MADYMO (TASS different crash Considering the of Int.), SIMPACK and MUSIAC . existing simulation methods, this raises the necessity to reduce © computation times significantly. Some examples are MADYMO © (TASS ©software © existing simulation methods, this this raises raises the the necessity necessity to to reduce reduce purposes. (TASS purposes. Some software examples are MADYMO Int.), SIMPACK and MUSIAC . existing simulation methods, 3. Impulsive models, based on momentum conservation and computation times significantly. © © Int.), SIMPACK © and MUSIAC©. computation times significantly. and based MUSIAC . velocities Int.), SIMPACK Impulsive models, on momentum conservation and Currently a wide range of numerical methods is in use for crash 3. computation times significantly. determining deformation energy and of the vehicles Impulsivedeformation models, based based on momentum momentum conservation and Currently aevaluation wide range of numerical methods is in2005): use for crash 3. 3. Impulsive models, on conservation and determining energy and velocities of the (forward vehicles dynamics purposes (Brach & Brach, after the impact starting from the initial conditions Currently a wide range of numerical methods is in use for crash determining deformation energy and velocities of the vehicles Currently wide range of numerical methods is in2005): use for crash after dynamics aevaluation purposes (Brach & Brach, determining deformation energythe and velocities of the (forward vehicles the impact from initial conditions reconstruction) orstarting vice versa This dynamics evaluation evaluation purposes purposes (Brach (Brach & & Brach, Brach, 2005): 2005): after starting from(backward the initial initialreconstruction). conditions (forward (forward dynamics after the the impact impactorstarting from the conditions reconstruction) vice versa (backward reconstruction). This reconstruction) or or vice vice versa versa (backward (backward reconstruction). reconstruction). This This reconstruction) Copyright © 2018, 2018 IFAC 1 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright 2018 responsibility IFAC 1 Control. Peer review©under of International Federation of Automatic Copyright © 1 Copyright © 2018 2018 IFAC IFAC 1 10.1016/j.ifacol.2018.04.018

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Dario Vangi et al. / IFAC PapersOnLine 51-2 (2018) 837–842

method is widely used because of the low calculation times, but it does not provide any information about vehicles' deformations nor accelerations (Brach, 1983; Ishikawa, 1993; Kolk, et al., 2016). PC-Crash©, Virtual Crash©, etc. are software packages which mainly use impulsive models. 4. Response Surface Models (RSM), appropriate for crashworthiness analysis (Simpson, et al., 2004). The vehicle impact behaviour is determined making use of a testing campaign: first, a full factorial Design Of Experiment (DOE) is created to consider all intended parameters, then data are acquired (from real tests or simulations) and eventually fitted to generate an analytical formulation describing the vehicle behaviour. The vehicle features are thus reconstructed making use of calculations, but no special purpose software is available to automate the process. 5. Reduced Order Dynamic Models (RODM) which are mainly based on FEM methods, with approximations of the problem to solve it more quickly. These methods have a lower accuracy in respect to the FEM. The most used category of RODM is the lumped-mass model (Jonsén, et al., 2009; Pahlavani & Marzbanrad, 2015) that substitute masses, dampers and springs to structural elements.

2. MODEL DEVELOPMENT Figure 1 shows two standard vehicle models penetrating each other due to an impact. The RODM algorithm discretizes only the perimeter of the vehicle or infrastructural element, at the height of the platform. The vehicle models’ perimeters are divided in 50 elements each which are sufficient to describe, in a satisfactory manner, their crash behaviour for accident reconstruction purposes. The elements transmit only tensile or compressive forces, and no bending state is allowed. The elements have only little possibility of changing their lengths in a predefined limit, to increase the solver calculation efficiency. The simulation starts at the impact instant. During the simulated impact and at each time step (some millisecond long), contact between the two vehicles' surface is detected by means of an algorithm determining which nodes of vehicle A are positioned inside vehicle B and vice versa. The vehicles are initially moved in the direction of initial speed 𝑣𝑣𝑖𝑖 . Forces between the vehicles’ nodes are assumed to act in the direction of relative motion, also referred to as PDOF (Vangi, 2008; Vangi, 2009), at the considered time step; such direction is obtained through the vector difference between the vehicles velocity. First attempt forces are imposed on each node; considering pairs of nodes, belonging to different vehicles, which are closest along the PDOF, iteration is repeated until the sum of their displacements reaches the distance at the time step beginning. Inertial properties of the vehicles are neglected and applied once the calculation is completed, to re-evaluate the vehicle’s velocity and displacements at the subsequent time-step. The process is then applied identically at the following time-step as long as an intrusion area exists.

While pure MB models (with rigid body) can accurately simulate the driving dynamics, no deformation is calculated. FEM and RSM model approaches calculate the deformations but provides for insufficient vehicle dynamics output. The paper thus describes a RODM routine providing both deformed shapes of vehicles and thorough information regarding accident kinematics and dynamics, as respectively obtainable from FEM/RSM and MB/impulsive model simulations. Road accidents can also be simulated trying out different impact configurations in a short time and can be combined with impulse models to get more detailed information on deformations and post-impact velocities and directions. Lower simulation time (ensuring a good accuracy) is a particularly desirable feature in all engineering problems because costs are reduced; this feature can be also decisive in ADAS control logic design: in real road conditions, the ADAS intervention minimizing damages to vehicles and occupants can be outlined almost in real time. The algorithm consists in a lumped-mass model, in which the vehicle discretization affects only its perimeter. The vehicle is treated in 2D, making the model suitable for crash analysis and reconstruction. The vehicle model considers only rods with little possibility to extend or shorten and that do not transmit bending moments. Rods are without mass and linked together by nodes as in the FEM. Deformation of the vehicle as a whole is the result of nodes displacements only, caused by the impact. Forces are transferred to nodes by springs, linked to nodes at one end and to a virtual point on the vehicle at the other end, coinciding with nodal position before vehicle's deformation (called nondeformed virtual vehicle). For the integration of motion equations, the inertial properties of the vehicle are applied on the centre of gravity. Elastic properties of springs can be determined from load-deformation curves slope obtained from crash tests or FEM simulations (McHenry, 1997). PC-Crash simulations will be used as reference to evaluate the proposed RODM performances in real-road accidents reconstruction.

Fig. 1. Two impacting vehicles with intruded nodes (light grey) and perimeters discretization in 50 elements each. To better understand the proposed method, figure 2 shows a n nodes simplified model: the rods lie on a straight line and the problem is in 1D. Nodes are connected to the non-deformable virtual vehicle by transversal x and longitudinal y springs applied on nodes. Springs follow Campbell model (Campbell, 1974), i.e. the vehicle is assumed to act as a homogeneous mean and to have a macroscopic linear behaviour. Springs stiffness varies from point to point, with different values in correspondence of side, front, corner and wheel nodes.

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𝑇𝑇𝑖𝑖𝑗𝑗 rod internal forces aligned with the axis of the element itself (n-1 equations) 𝑇𝑇(0_1)𝑥𝑥 𝑥𝑥1 − 𝑥𝑥0 = 𝑇𝑇(0_1)𝑦𝑦 𝑦𝑦1 − 𝑦𝑦0 𝑇𝑇(1_2)𝑥𝑥 𝑥𝑥2 − 𝑥𝑥1 = 𝑇𝑇(1_2)𝑦𝑦 𝑦𝑦2 − 𝑦𝑦1 ……………… 𝑇𝑇(𝑖𝑖−1_𝑖𝑖)𝑥𝑥 𝑥𝑥𝑖𝑖 − 𝑥𝑥𝑖𝑖−1 = 𝑇𝑇(𝑖𝑖−1_𝑖𝑖)𝑦𝑦 𝑦𝑦𝑖𝑖 − 𝑦𝑦𝑖𝑖−1 ……………… 𝑇𝑇(𝑗𝑗−1_𝑗𝑗)𝑥𝑥 𝑥𝑥𝑗𝑗 − 𝑥𝑥𝑗𝑗−1 = 𝑇𝑇(𝑗𝑗−1_𝑗𝑗)𝑦𝑦 𝑦𝑦𝑗𝑗 − 𝑦𝑦𝑗𝑗−1 ……………… 𝑇𝑇(𝑛𝑛−3_𝑛𝑛−2)𝑥𝑥 𝑥𝑥𝑛𝑛−2 − 𝑥𝑥𝑛𝑛−3 = 𝑇𝑇(𝑛𝑛−3_𝑛𝑛−2)𝑦𝑦 𝑦𝑦𝑛𝑛−2 − 𝑦𝑦𝑛𝑛−3 𝑇𝑇(𝑛𝑛−2_𝑛𝑛−1)𝑥𝑥 𝑥𝑥𝑛𝑛−1 − 𝑥𝑥𝑛𝑛−2 = 𝑇𝑇(𝑛𝑛−2_𝑛𝑛−1)𝑦𝑦 𝑦𝑦𝑛𝑛−1 − 𝑦𝑦𝑛𝑛−2

Fig. 2. 1D representation of a vehicle’s perimeter, with the nodes linked by springs to the virtual vehicle. Vehicles' motion is described by integrating motion equations in a single time step, also considering the forces transmitted by the wheels to the road; the latter are computed by the classical adherence circle model, like the one used in PC-Crash. By indicating with 𝑇𝑇(𝑖𝑖_𝑖𝑖+1) the forces transmitted through the rod linking nodes i and i+1, with 𝑘𝑘𝑖𝑖 the elastic constant of the springs relative to node i, with 𝑥𝑥𝑖𝑖 and 𝑦𝑦𝑖𝑖 node i coordinates, with 𝑥𝑥𝑖𝑖0 and 𝑦𝑦𝑖𝑖0 its coordinates at the time step beginning (nondeformed vehicle) and with 𝐹𝐹𝑗𝑗 the force applied to node j, the following equations (Eq. 1-4) for n nodes can be written:

The unknowns are the x and y nodal coordinates (2n) and the 𝑇𝑇𝑖𝑖_𝑗𝑗 forces components (2n-2), resulting in a total amount of 4n-2 equations in 4n-2 unknowns. The equations are not linear and must be solved with numerical methods or algorithms like the Newton’s or quasi-Newton (e.g. Newton-Raphson's) ones.

Equilibrium equations along the x axis (n equations) 𝑘𝑘0𝑥𝑥 (𝑥𝑥0 − 𝑥𝑥00 ) − 𝑇𝑇(01)𝑥𝑥 = 0 𝑘𝑘1𝑥𝑥 (𝑥𝑥1 − 𝑥𝑥10 ) + 𝑇𝑇(0_1)𝑥𝑥 − 𝑇𝑇(1_2)𝑥𝑥 = 0 ……………… 𝑘𝑘(𝑖𝑖)𝑥𝑥 (𝑥𝑥𝑖𝑖 − 𝑥𝑥𝑖𝑖0 ) + 𝑇𝑇(𝑖𝑖−1_𝑖𝑖)𝑥𝑥 − 𝑇𝑇(𝑖𝑖_𝑖𝑖+1)𝑥𝑥 = 0 ……………… (1) 0 𝑘𝑘𝑗𝑗𝑗𝑗 (𝑥𝑥𝑗𝑗 − 𝑥𝑥𝑗𝑗 ) + 𝑇𝑇(𝑗𝑗−1_𝑗𝑗)𝑥𝑥 − 𝑇𝑇(𝑗𝑗_𝑗𝑗+1)𝑥𝑥 + 𝐹𝐹𝑗𝑗𝑗𝑗 = 0 ……………… 0 ) + 𝑇𝑇(𝑛𝑛−3_𝑛𝑛−2)𝑦𝑦 − 𝑇𝑇(𝑛𝑛−2_𝑛𝑛−1)𝑥𝑥 = 0 𝑘𝑘(𝑛𝑛−2)𝑥𝑥 (𝑥𝑥𝑛𝑛−2 − 𝑥𝑥𝑛𝑛−2 0 ) 𝑘𝑘(𝑛𝑛−1)𝑥𝑥 (𝑥𝑥𝑛𝑛−1 − 𝑥𝑥𝑛𝑛−1 + 𝑇𝑇(𝑛𝑛−2_𝑛𝑛−1)𝑥𝑥 = 0

Once the displacements and tensions are obtained for both the vehicles, a check is performed to determine if the sum of the displacements is equal (within a certain tolerance) to the intrusion. If so, the force calculation process ends and the equations of motion are applied to the vehicles. If not, the first attempt forces values are changed based on how much the displacement of the related node is close to the intrusion. The algorithm is iterative, making it necessary to change first attempt forces and iterate until the criterion is fulfilled.

Equilibrium equations along the y axis (n equations)

𝑘𝑘0𝑦𝑦 (𝑦𝑦0 − 𝑦𝑦00 ) − 𝑇𝑇(0_1)𝑦𝑦 = 0 𝑘𝑘1𝑦𝑦 (𝑦𝑦1 − 𝑦𝑦10 ) + 𝑇𝑇(0_1)𝑦𝑦 − 𝑇𝑇(1_2)𝑦𝑦 = 0 ……………… 𝑘𝑘(𝑖𝑖)𝑦𝑦 (𝑦𝑦𝑖𝑖 − 𝑦𝑦𝑖𝑖0 ) + 𝑇𝑇(𝑖𝑖−1_𝑖𝑖)𝑦𝑦 − 𝑇𝑇(𝑖𝑖_𝑖𝑖+1)𝑦𝑦 = 0 ……………… (2) 𝑘𝑘𝑗𝑗𝑗𝑗 (𝑦𝑦𝑗𝑗 − 𝑦𝑦𝑗𝑗0 ) + 𝑇𝑇(𝑗𝑗−1_𝑗𝑗)𝑦𝑦 − 𝑇𝑇(𝑗𝑗_𝑗𝑗+1)𝑦𝑦 + 𝐹𝐹𝑗𝑗𝑗𝑗 = 0 ……………… 0 ) + 𝑇𝑇(𝑛𝑛−3_𝑛𝑛−2)𝑦𝑦 − 𝑇𝑇(𝑛𝑛−2_𝑛𝑛−1)𝑦𝑦 = 0 𝑘𝑘(𝑛𝑛−2)𝑦𝑦 (𝑦𝑦𝑛𝑛−2 − 𝑦𝑦𝑛𝑛−2 0 𝑘𝑘(𝑛𝑛−1)𝑦𝑦 (𝑦𝑦𝑛𝑛−1 − 𝑦𝑦𝑛𝑛−1 ) + 𝑇𝑇(𝑛𝑛−2_𝑛𝑛−1)𝑦𝑦 = 0

When the direction of vehicles' superimposition inverts, the restitution phase occurs (McHenry, 1997; Goldsmith, 2001) and different elastic constants k are applied, according to the desired load-crush law of the vehicle. When contact forces become null, the collision stage ends and the vehicles enter a post-collision phase, in which only road-tire forces are present. The springs' elastic constants are determined by comparing the vehicles deformation to the ones obtained in real crash tests. Assuming the system is linear (Campbell model), a linear relation also exists between stiffness of different vehicle’s areas. The characterization of the front area through a comparison with vehicle-to-barrier crash tests allows thus for the stiffness evaluation of all the vehicle’s different areas.

Constancy of distance 𝑑𝑑𝑖𝑖 between consecutive nodes within a certain tolerance ε (n-1 equations)

(𝑥𝑥1 − 𝑥𝑥0 )2 + (𝑦𝑦1 − 𝑦𝑦0 )2 < (𝑑𝑑0 + 𝜀𝜀)2 (𝑥𝑥2 − 𝑥𝑥1 )2 + (𝑦𝑦2 − 𝑦𝑦1 )2 < (𝑑𝑑1 + 𝜀𝜀)2 ……………… (𝑥𝑥𝑖𝑖 − 𝑥𝑥𝑖𝑖−1 )2 + (𝑦𝑦𝑖𝑖 − 𝑦𝑦𝑖𝑖−1 )2 < (𝑑𝑑𝑖𝑖−1 + 𝜀𝜀)2 ……………… 2 2 2 (𝑥𝑥𝑗𝑗 − 𝑥𝑥𝑗𝑗−1 ) + (𝑦𝑦𝑗𝑗 − 𝑦𝑦𝑗𝑗−1 ) < (𝑑𝑑𝑗𝑗−1 + 𝜀𝜀) ……………… (𝑥𝑥𝑛𝑛−2 − 𝑥𝑥𝑛𝑛−3 )2 + (𝑦𝑦𝑛𝑛−2 − 𝑦𝑦𝑛𝑛−3 )2 < (𝑑𝑑𝑛𝑛−3 + 𝜀𝜀)2 (𝑥𝑥𝑛𝑛−1 − 𝑥𝑥𝑛𝑛−2 )2 + (𝑦𝑦𝑛𝑛−1 − 𝑦𝑦𝑛𝑛−2 )2 < (𝑑𝑑𝑛𝑛−2 + 𝜀𝜀)2

(4)

More than 3000 vehicle-to-barrier crash tests have been analysed considering EuroNCAP, NHTSA and LaSIS (University of Florence) databases: the resulting stiffness allows to obtain the best-fitting post-impact motion, deformed shapes, pre-impact and post-impact velocities in respect to the real ones. Interestingly, the stiffness evaluation carried out in this study seems to point out that this parameter is peculiar to each class of vehicles (sedans, small cars, SUVs, etc.).

(3)

3



The possibility to efficiently simulate vehicle-to-barrier impacts is not sufficient by itself, because they represent a low amount of real-world cases. So, the ability to appropriately reconstruct vehicle-to-vehicle impacts through the RODM has been investigated, recreating crash tests gathered inside databases like AREC and VERSUE. The coherent results obtained imply only the necessity to further validate the model by comparison with real accidents data. 3. REAL CASES FOR MODEL VALIDATION

Fig. 5. Frontal impact with maximum intrusion (light grey) and final (dark grey) positions of the involved vehicles.

PC-Crash impulsive model simulations results are considered efficient indicators of the real accident kinematics: the comparison with the proposed RODM is based mainly on the 𝛥𝛥𝛥𝛥, representing the vector difference between the vehicles collision velocities 𝑣𝑣𝑖𝑖 and post-impact velocities 𝑣𝑣𝑓𝑓 ; also, the Equivalent Energy Speed (EES) is reported, because it helps in reducing the analysed crash to a barrier impact, simpler to treat and to visualize (Vangi, 2008). On the other hand, the vehicles’ deformed shapes obtained through the presented RODM are compared to pictures shot at the accident site: in fact, PC-Crash does not implement deformed shapes calculation.

3.3 Side impact In the intersection-located side impact shown in Figure 6, a Skoda Fabia and a BMW 550I were involved.

The reconstruction of 4 real road accidents included in the German In-Depth Accident Study (GIDAS) database are analysed in detail, to highlight the suitability of the proposed method for the solution of practical problems. In each of these cases, the areas interested by the impact are mostly the ones with different stiffness in a vehicle (front, rear, side, wheel). Numerical results of simulations are reported in Chapter 4, addressing also simulation times. Fig. 6. Side impact site with maximum intrusion condition (light grey) and final (dark grey) positions of the involved vehicles.

3.1 Rear-end impact Figure 4 shows a rear-end impact involving a FIAT Fiorino and an Audi A4 in an accident along a straight, one-way road before an intersection. The maximum intrusion condition is highlighted in light grey, while the final positions in dark grey (no intrusion). The Audi A4 was still, while the estimated impact speed 𝑣𝑣𝑖𝑖 of the FIAT Fiorino was 25 km/h.

3.4 Wheel engagement / Small overlap Figure 7 shows an accident where a wheel engagement and small overlap crash between two vehicles occurred. A Toyota Avensis hit a Renault Master and then a Mercedes A-Class as a result of the first crash. The Mercedes was considered for the determination of the Toyota rest position only.

Fig. 4. Rear-end collision site with maximum intrusion condition (light grey) and final (dark grey) positions of the involved vehicles. 3.2 Frontal impact Figure 5 shows the planimetry regarding the site of a frontal impact between a Toyota Corolla and a Chevrolet Kalos. The area of the road where the two vehicles collided is indicated by the presence of debris and depicted as a circle.

Fig. 7. Accident site for a small overlap crash: maximum intrusion condition (light grey) and final (dark grey) positions are reported. 4

Proceedings of the 9th MATHMOD Vienna, Austria, February 21-23, 2018


841

4. RESULTS AND DISCUSSION Comparing data between PC-Crash and the proposed RODM simulations summarized in Table 1 for the analysed impact scenarios, a high similarity in results can be assessed for what regards the post-impact velocity vf and the speed change Δv. In fact, the maximum calculated difference is about 3 km/h which cannot generate evident consequences on the real road accident dynamics. Only the very rare and special impact constellation of the small overlap crash presents a significant difference of the parameters. This is due to the special wheel engagement and the involved front suspension. Δv represents the main parameter to be considered because it is an index of both vehicles deformations (Iraeus & Lindquist, 2015) and injury risk for the occupants (Ranfagni, et al., 2017). On the other hand, the EES calculated values are slightly different (reaching more than 10 km/h) based on the used algorithm; while these differences are important, it is also worth noting that the EES is based on the dissipated deformation energy. If the deformations are available as in the RODM case, the check is carried out comparing the calculated to the real ones rather than considering the EES.

Fig. 10. Reconstructed deformed shapes of the Skoda Fabia (a) and the BMW 550I (b) and the real ones (c,d).

Deformed shapes of the vehicles are shown in Figures 8-11, both calculated – (a) and (b) – and real ones – (c) and (d). The vehicles deformations obtained through the reconstruction correspond to the real ones with a high accuracy. This demonstrates the suitability of the proposed algorithm not only for accident reconstruction purposes, but also for crashworthiness assessment of vehicles in various impact configurations.

Fig. 11. Comparison between Toyota Avensis (a) and Renault Master (b) reconstructed deformed shapes with real ones (c,d). Table 1. Initial conditions and results for the analysed scenario through PC-Crash and the proposed RODM. Speed (km/h) REAR-END 𝑣𝑣𝑖𝑖 𝑣𝑣𝑓𝑓 𝛥𝛥𝛥𝛥 EES FRONTAL 𝑣𝑣𝑖𝑖 𝑣𝑣𝑓𝑓 𝛥𝛥𝛥𝛥 EES SIDE 𝑣𝑣𝑖𝑖 𝑣𝑣𝑓𝑓 𝛥𝛥𝛥𝛥 EES WHEEL 𝑣𝑣𝑖𝑖 𝑣𝑣𝑓𝑓 𝛥𝛥𝛥𝛥 EES

Fig. 8. Reconstructed deformed shapes of the Fiat Fiorino (a) and the Audi A4 (b) and the real ones (c,d).

PC-Crash Fiorino A4 25 0 11 13 15 13 12 12 Kalos Corolla 58 58 12 3 67 57 59 54 Fabia 550I 20 37 20 22 23 15 24 19 Avensis Master 40 30 9 11 41 26 20 36

RODM Fiorino A4 25 0 11 13 15 13 12 12 Kalos Corolla 58 58 8 2 64 59 47 65 Fabia 550I 20 37 21 19 20 16 18 25 Avensis Master 40 30 10 13 31 20 25 36

Simulation times (W7x64, Intel Xeon 3.5 GHz, 32GB RAM) for the analysed impacts 𝑡𝑡 𝑠𝑠 are:

Fig. 9. Reconstructed deformed shapes of the Chevrolet Kalos (a) and the Toyota Corolla (b) and the real ones (c,d).

1. 2.

5

𝑠𝑠 = 7 𝑚𝑚𝑚𝑚𝑚𝑚; Rear-end crash 𝑡𝑡𝑅𝑅𝑅𝑅 𝑠𝑠 Frontal crash 𝑡𝑡𝐹𝐹 = 375 𝑚𝑚𝑚𝑚𝑚𝑚;


3. 4.


Side crash 𝑡𝑡𝑆𝑆𝑠𝑠 = 15 𝑚𝑚𝑚𝑚𝑚𝑚: 𝑠𝑠 = 32 𝑚𝑚𝑚𝑚𝑚𝑚. Wheel engagement 𝑡𝑡𝑊𝑊

GIDAS, 2017, http://gidas.org. [Online]. Goldsmith, W., 2001. Impact-The Theory and Physical Behaviour of Colliding Solids. Mineola, New York: Dover Publications, Inc.. Hamza, K. & Saitou, K., 2005. Design Optimization of Vehicle Structures for Crashworthiness Using Equivalent Mechanism Approximations. Transactions of ASME, Journal of Mechanical Design(127(3):), pp. 485-492. Iraeus, J. & Lindquist, M., 2015. Pulse shape analysis and data reduction of real-life frontal crashes with modern passenger cars. Int. J. Crashworth., 20(6). Ishikawa, H., 1993. Impact Model for Accident Reconstruction - Normal and Tangential Restitution Coefficients. SAE Paper, Issue 930654. Jonsén, P., Isaksson, E., Sundin, K. & Oldenburg, M., 2009. Identifcation of lumped parameter automotive crash models for bumper system development. Int. J. Crashworth., 14(6), pp. 533-541. Kolk, H. et al., 2016. Evaluation of a momentum based impact model and application in an effectivity study considering junction accidents. Hannover. LSTC, n.d. www.lstc.com/products/ls-dyna. [Online]. M. Pahlavani, J. M., 2015. Crashworthiness study of a full vehicle- lumped model using parameters optimisation. Int. J. Crashworth., 20(6), pp. 573-591. McHenry, R. a. M. B., 1997. Effects of Restitution in the Application of Crush Coefficients. SAE Technical Paper, Issue 970960. Milano, P., http://hdl.handle.net/10589/51382. [Online]. Ming, L., Jaewoo, Y. & Byeongwoo, K., 2016. Proposal and Validation of AEB System Algorithm for Various Slope Environments. Advanced Multimedia and Ubiquitous Engineering, Lecture Notes in Electrical Engineering (354). PC-Crash, 2017, www.pc-crash.it. [Online]. Pawlus, W., Karimi, H. R. & Robbersmyr, K. G., 2011. Application of viscoelastic hybrid models to vehicle crash simulation. Int. J. Crashworth., 16(2), pp. 195-205. Ranfagni, S., Vangi, D. & Fiorentino, A., 2017. Road Vehicles Passive Safety Rating Method. Detroit. Simpson, T. W. et al., 2004. Approximation Methods in Multidisciplinary Analysis and Optimization: A Panel Discussion. Structural and Multidisciplinary Optimization, 34(3), pp. 302-313. TASS, n.d. www.tassinternational.com/madymo. [Online]. Vangi, D., 2008. Ricostruzione della dinamica degli incidenti stradali – principi e applicazioni. Florence: Firenze University Press. Vangi, D., 2009. Energy loss in vehicle to vehicle oblique impact. Int. J. of Impact Engineering, 36(3), pp. 512-521. vCRASH, A. I., n.d. www.vcrashusa.com. [Online]. Wei, Z., Karimi, H. R. & Robbersmyr, K. G., 2016. Analysis of the relationship between energy absorbing components and vehicle crash response, Tech. Rep. 2016-01-1541. SAE Technical Paper, Issue 2016-01-154. Yildiz, A. R. & Solanki, K. N., 2012. Multi-objective optimization of vehicle crashworthiness using a new particle swarm based approach. Int. J. Adv. Manufact. Tech., 59(Issue 1–4), p. 367–376.

Calculation times for cases 1, 3 and 4 are extremely low in respect to traditional FEM and MB algorithms (of the order of days and hours respectively). This is less evident for case 2, in which they are comparable with the MB ones, probably due to high initial speeds and conditions superimposed for iterations. The accuracy is however the same for MB and RODM, making them interchangeable for this case reconstruction. 5. CONCLUSIONS The present work introduced a special purpose Reduced Order Dynamic Model (RODM) for the vehicles crash stage simulation. The problem of long simulation times, deriving from Finite Elements Models (FEM) or Multi-Body (MB) approaches use, was addressed. Discretization of vehicle’s perimeter only in a 2D environment reduces the number of analysed domains, simplifying equations to be solved inside them. The reconstruction accuracy can be assessed starting from a comparison between PC-Crash and the proposed RODM simulations regarding real road accidents. The RODM time for solution can be expressed in terms of minutes, while FEM and MB reconstruction times are of the order of days and hours respectively. 5 hours of RODM simulation were needed in the worst case, represented by a front impact at relatively high speed and involving high deformations. Time is however comparable to MB models’ solution ones. The developed method proved to be an efficient alternative to every crash dynamics reconstruction commercial software. In fact, it can be used for multiple purposes in the road safety research field: • road accidents reconstruction, for the investigation of their major causes; • crashworthiness analysis, for the determination of vehicles dynamic response to crashes; • ADAS simulations, for the study of new driving assistance systems and intervention verification in a specific event. Valuable features of the described algorithm lie in the reduction of simulation times, in the accuracy of solution but also in the possibility to subsequently improve its efficiency. In fact, the RODM uses non-linear equations which involve iterative calculations; the next steps will be taken, starting from the algorithm described in this work, towards a linearitybased method capable of further reducing the simulation time. REFERENCES Brach, R. M., 1983. Analysis of Planar Vehicle Collisions Using Equations of Impulse and Momentum. Accident Analysis and Prevention, 15(2), pp. 105-120. Brach, R. M. & Brach, M. R., 2005. Vehicle Accident Analysis and Reconstruction Methods. Warrendale, PA, USA: SAE International, ISBN 0-7680-0776-3. Campbell, K., 1974. Energy as a Basis for Accident Severity. SAE Paper, Issue 740565. European Road Safety Observatory, 2017. Annual Accident Report 2017. European Commission, 2010. Road Safety Programme 20112020: detailed measures, European Commision. 6