The process of fronttoside collision of motor vehicles in terms of energy balance
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Abstract
The reconstruction of a road accident can be treated as the resolution of an “inverse problem” in mechanics using analytical or numerical models. In the road accident reconstruction research, an assumption is often made that a predominant part of the energy lost during vehicle collisions is consumed by permanent deformation of vehicle components. Other parts of the dissipated energy can be ignored due to their insignificant amount. In this article, this assumption will be verified for the fronttoside collision of passenger cars. The main objective of this paper is to determine the important components of the energy balance dissipated during the collision. These components were determined on the basis of experimental results, which included three crash tests with a fronttoside collision of motor vehicles of the same make and model, with the rightangle impact of one car against the side of another. The results of experiments were used to construct the model of the dynamics of the motor vehicle collision. The model was then used as a basis for the determination of the forces, displacements and velocities during vehicle collision. The above made it possible to determine vehicle force/deformation curves and then the key components of the dissipated energy in function of the duration of the contact phase of the vehicle collision. Based on the results of the model and crash tests, conclusions were formulated that provide an important insight into the reconstruction of the fronttoside collisions of motor vehicles.
Keywords
Cars’ collision dynamics Collision energy balance1 Introduction
The road accident analysis carried out by forensic experts is usually based on simplified methods and procedures due to a limited access to the numeric data necessary for computations. The reconstruction of the collision between two motor vehicles consists of the resolution of an “inverse problem” in mechanics, where the course of an accident is reconstructed, based on the accident description using appropriate analytical or simulation models derived from the principles of mechanics. When accident analyses and reconstructions are carried out in a standard way, i.e. without using the finite element method (FEM) or multibody system dynamics (MBD), the calculations are based on discrete dynamic models of colliding objects. These models are supplemented by empirically determined force/deformation curves, vehicle parameters and empirical relations between the velocity of the impacting vehicle and the measured deformation of vehicle bodies [3, 11, 13, 14, 15, 17, 23], described with the use of energy rasters in a two or threedimensional meshed models.
Basic research addressed in [2, 5, 8, 26, 27] shows an immense amount of detailed scientific problems that are encountered when collisions are modelled and simulated even in relatively simple discrete systems. This is especially true when not only the impactrelated phenomena but also the processes of energy dissipation due to frictional effects are to be taken into account. The introduction of dry friction with its singularities (Painlevé paradox, stickslip processes, indeterminacy problems) fundamentally complicates the collision models and their analyses. As an example, if temporary bonds (nanoscale junctions) are to be taken into account, the Gauss principle must be used in the synthesis of the model, which will lead to extended mathematical descriptions even in seemingly simple cases [32]. Considering the experience from the analyses of nonlinear vibrations in discrete dynamic systems (“multirigid body systems” and “nonsmooth systems”), where bifurcations and chaos are very frequent phenomena, the extended collision models directly derived from the laws of physics may be reasonably expected to show significant sensitivity. As a consequence of this sensitivity, and combined with the uncertainty of the data, the researchers who deal with the analysis and reconstruction of collisions in real systems are inclined to use simplified empirical formulas adapted for the processes to be examined.
In the area of applied research, there are many publications directly dedicated to motor vehicle collisions. They report crash test results, describe the techniques of measurements and the processing of signals and images recorded, and present simulation results obtained using software of the Multibody System Dynamics (MBD) type, e.g. MADYMO or the FEM type, e.g. LSDYNA [9, 25]. A separate group of publications describes the accident reconstruction carried out using simplified analytical methods or specialized supporting software (e.g. CRASH) [4, 22, 28, 29]. In some research procedures, advanced signal processing and identification techniques are also used, as for example those based on the wavelet transformation and the NARMA method [18]. An extensive review of those approaches, together with examples of the application of experimental, analytical and simulation results to the analysis and reconstruction of motor vehicle collisions can be found, for example, in [1, 10, 21].
One of the basic methods in the analysis and reconstruction of motor vehicle collisions in the road traffic is the method of energy balance. The energy balance is compiled for all vehicles involved in the collision. The shares of individual components in the energy balance may vary and depend on the types and characteristics of the objects involved in the collision, and even on the specific scenario of the accident. In most cases, the raster methods used by experts (i.e. the methods based on the empirical relations between the velocity of the impacting vehicle and the vehicle body deformations along a simplified mesh model) reduce the energy balance to two main components, namely the preimpact kinetic energy of the vehicles involved and the work carried out on the deformation of their bodies. The development of efficient methods to define also the remaining components of the energy balance, based on simple models and data recorded after the accident, is an objective of scientific research carried out at many research centres.
The methods and computational procedures concerning headon vehicle collisions are relatively well explored, and the reconstruction of such accidents is often based on the dynamic 1D models or on the use of raster methods. Usually, an assumption is made that in the headon collisions no lateral movement of the impacting vehicle takes place and, consequently, taking into account only two components of the energy balance is a correct hypothesis. These two components are the energy of the preimpact translational motion and the energy of vehicle body deformation. Such hypothesis has been used to develop the methods of determining the impact velocity based on the vehicle body deformation depth (the Campbell method and the McHenry method). These methods are often used for accident analysis and reconstruction. As an example, they were used in [33] for the reconstruction of headon collisions of several passenger cars with a barrier, and the accuracy of determining the preimpact vehicle velocity was within twenty per cent. It was experimentally shown [19, 20] that even if the vehicle body deformation is of the order of 50 centimeters, the energy dissipated during a headon collision can be roughly computed from the kinetic energy and the deformation raster by means of a linear equation, having the coefficient of restitution determined experimentally. An estimation of the energy dissipated for the deformation of colliding vehicles with significantly different masses has been presented in [12]. For the frontal impact against the side of a stationary vehicle, the dissipated energy depends to a significant degree on the properties of the vehicle side body structure and on its previous repairs, if any [24]. The energy dissipation takes place both during the normal deformation and as a result of the relative movement of the vehicles, as will be further elaborated in this paper.
The fronttoside collisions, i.e. those taking place when a motor vehicle hits its front against the side of another vehicle, occur in a more complex way than the headon collisions. In the fronttoside collisions, the process of destruction of vehicle bodies is accompanied by at least a 2D movement of the vehicles in relation to each other, and due to that the model becomes more complex. Limiting the energy balance to only two mentioned earlier components becomes unacceptable. Although the fronttoside collisions are quite frequent (in Poland, they constitute about 25% of all road accidents), their computer models are still relatively unexplored and, in consequence, it is challenging to analyse and reconstruct such scenarios.
The problems of energy balance and fronttoside collisions have been separately addressed in many publications dedicated to motor vehicle collisions. In the available literature, however, there is a lack of comprehensive and sufficiently indepth analyses of these problems. Therefore, extending the area of energy analysis to the fronttoside collisions requires further experimental and modelling research. A justifiable question arises whether the description of these problems in a 2D space is adequate or rather a 3D model have to be build. As an example, even in the case of an almost completely planar (2D) but nonsteady state motion, certain interactions take place as a result of the relative movements of the vehicles in the contact area during the collision. Does this disturbance affect considerably the accuracy of the accident reconstruction computations? The analysis of the available literature does not answer this question unequivocally due to the lack of comparative studies.
This article reports on the progress in the search for an efficient method of analysis and reconstruction of the fronttoside collision of motor vehicles taking into account the process of energy dissipation resulting not only from the deformation of vehicle bodies but also from the fact that the cars slide on the road surface and move in relation to each other in the deformation zone (i.e. in the area of contact during the collision). The main objective of this paper is to determine the important components of the balance of the energy dissipated as a result of the collision and to show the variations of these components during the vehicle collision contact phase. In the road accident reconstruction research, an assumption is made that a predominant part of the energy lost during vehicle collisions is consumed by the permanent deformation of vehicle bodies, and the remaining parts of the dissipated energy can be ignored due to their insignificant value. In this paper, an attempt will be made to verify this assumption for the fronttoside collision of passenger cars, based on the results of crash tests and computer modelling. This article is an extension of the paper presented at the 14th International Conference “Dynamical Systems – Theory and Applications” DSTA 2017.
2 Experimental studies
Three crash tests were carried out at the Automotive Industry Institute (PIMOT) in Warsaw, involving six passenger cars of the same make and model. In each test, the front of car A was impacting the left side of car B, close to the B pillar (Fig. 1).
The moments of inertia of vehicle wheels were determined from the generally available data for passenger car wheels of the specific class. The relevant vehicle characteristics were as specified in Table 1.
Vehicle characteristics
Description  AB1  AB2  AB3 

Mass of car A, \(m_\mathrm{A}\) (kg)  1569  1532  1545 
Mass of car B, \(m_\mathrm{B}\) (kg)  1568  1594  1629 
Moment of inertia of car A, \(I_\mathrm{A}\) (\(\hbox {kgm}^{2}\))  2722  2636  2661 
Moment of inertia of car B, \(I_\mathrm{B}\) (\(\hbox {kgm}^{2}\))  2710  2777  2827 
\(L_\mathrm{AB}\) (m)  1.25  1.34  1.33 
Velocity \(V_\mathrm{A}\) (m/s)  15.1  12.7  15.3 
The crash tests were carried out at the PIMOT facility on a dry concrete surface. During the tests, the steering wheels of both cars were left free and their road wheels had no brakes applied. The tests were performed on Honda Accord cars manufactured between 2000 and 2002. The cars were in good technical condition and had undamaged and noncorroded bodies, which had not been previously repaired. At the centre of mass of each car, a threeaxial accelerometer was installed, together with sensors for measuring the angular velocity components of the car body with respect to the three coordinate axes.
The data acquisition system used to record the acceleration and angular velocity components was placed in the car trunk, in the housing designed for vehicle crash tests. The vibroisolating performance of such a housing was previously verified in separate tests. The sensor signals were subjected to a lowpass filtering. The signals were recorded with the use of a 10kHz filter, and an additional filter with the cutoff frequency of 50 Hz was used at the final stage. That way the measured signals were reduced to the frequency band that can be reproduced in the modelling and equationsolving process. The filter cutoff frequency was selected after comparing with 100 and 25Hz filters, to compromise between the expected calculation accuracy, interpretability of the calculation results, and the benefit of reaching conclusions that would be relevant and suitable for the modelling and reconstruction of road accidents of this type.
Highspeed cameras were installed above the crash test location to record car positions with a frequency of 1000 frames per second.
3 Modelling of the collision

Local coordinate systems, fixed to the bodies of each of the “i” cars involved in the collision. The local coordinate system \(O_{i}x_{i}y_{i}z_{i}\) has its origin \(O_{i}\) situated at the centre of mass of the ith car and its \(O_{i}x_{i}\) axis is parallel to the car longitudinal centreline. The signals recorded in the local coordinate systems were the components of the acceleration vector of the car mass centre (\(a_{{\textit{xi}}}\), \(a_{{\textit{yi}}}\), \(a_{{\textit{zi}}})\) and of the angular velocity component (\(P_{i}\), \(Q_{i}\), \(R_{i})\) of the car.

Local levelled coordinate systems, attached to the mass centres of each of the “i” cars. The local coordinate system \(O_{i}x_{{\textit{Pi}}}y_{{\textit{Pi}}}z_{{\textit{Pi}}}\) has its origin \(O_{i}\) situated at the mass centre of the ith car, the \(O_{i}x_{{\textit{Pi}}}y_{{\textit{Pi}}}\) plane is parallel to the road surface, and the \(O_{i}x_{{\textit{Pi}}}\) axis is parallel to the car longitudinal symmetry plane. The local levelled coordinate systems were used to formulate the equations of motion of the planar model of the collision.

Global coordinate system \(O_\mathrm{G}X_\mathrm{G}Y_\mathrm{G}Z_\mathrm{G}\), attached to the road. The \(O_\mathrm{G}X_\mathrm{G}Y_\mathrm{G}\) plane of this system is situated at the road surface level and the \(O_\mathrm{G}Z_\mathrm{G}\) axis is pointing vertically upwards. The \(O_\mathrm{G}X_\mathrm{G}\) axis is parallel to the vector of preimpact velocity of car A and the \(O_\mathrm{G}Y_\mathrm{G}\) axis is parallel to the vector of preimpact velocity of car B. In the global coordinate system, the time histories of the translational and rotational velocities, and the trajectories of the involved cars were determined.
 \(B_{i}\,(B_{{\textit{ix}}}, B_{{\textit{iy}}})\)

inertial force acting on the ith car, expressed in the levelled coordinate system attached to the mass centre of the car;
 \(M_{{\textit{zi}}}\)

moment of the inertial resistance acting on the car during its accelerated rotation around the vertical axis;
 \(m_{i}\), \(I_{i}\)

vehicle mass and mass moment of inertia of the car relative to the vertical axis;
 \(a_{{\textit{xPi}}}, a_{{\textit{yPi}}}\)

longitudinal and lateral components of the acceleration vector of the centre of car mass, calculated from (3);
 \(F_{i1}(F_{xi1}, F_{yi1})\)

tangent road reaction force acting on the wheels of the front car axle and its components; and similarly
 \(F_{i2}(F_{xi2}, F_{yi2})\)

tangent road reaction force acting on the wheels of the rear car axle and its components, expressed in both cases in the levelled coordinate system attached to the centre of mass of the ith car;
 \(F_{i}(F_{{\textit{xi}}}, F_{{\textit{yi}}})\)

impact force acting on the vehicle and the longitudinal (x) and lateral (y) components of this force, expressed in the levelled coordinate system attached to the mass centre of the ith car;
 \(x_{i}\), \(y_{i}\)

coordinates of point Z, i.e. the point of application of the impact force, in the levelled coordinate system attached to the mass centre of the ith car (Fig. 2);
 \(\gamma _{{\text {AB}}}\)

angle of rotation of vehicle B around the vertical axis (car B yaw angle), measured from the longitudinal symmetry plane of vehicle A (\(\gamma _{{\text {AB}}}=\Psi _\mathrm{A}\Psi _\mathrm{B}\)).
This way, a system of six differential equations with respect to the coordinates defining the positions of the mass centres and rotations of cars A and B was obtained (6), comprising also two algebraic equations. This system will be treated as a system of solely algebraic equations, because the time histories of components of the vector of translational acceleration of the mass centres and of the vectors of angular acceleration of the cars are known. They were calculated from the experimental results with the time step of \(\Delta t=0.0001~\hbox {s}\), by converting the measurement results to the levelled coordinate systems using relation (3). Therefore, the unknowns in Eq. (6) are components \(F_{x\mathrm{{A}}}\), \(F_{y\mathrm{{A}}}\), \(F_{x\mathrm{B}}\), \(F_{y\mathrm{{B}}}\) of the impact force and the lateral reaction forces \(F_{y\mathrm{A}1}\), \(F_{y\mathrm{A}2}\), \(F_{y\mathrm{{B}}1}\), \(F_{y\mathrm{{B}}2}\) acting on car wheels.
4 Experimental studies and modelling results
Selected measurements of accelerations are shown in Fig. 3. They represent the longitudinal \(a_{{\textit{xi}}}\) and lateral \(a_{{\textit{yi}}}\) components of the accelerations of the mass centres of cars A and B in the local coordinate systems after applying the 50Hz filter.

Collision of the cars and their temporary contact (with the compression phase followed by the restitution phase);

Separation of the cars;

Start of the independent postseparation movements of the cars.
It is worth noticing that the maximum value of the longitudinal component of the impact force \(F_{x\mathrm{{A}}}\) in test AB2 was markedly lower than in the other two tests. The reason for this is the lower value of the impact velocity recorded in test AB2. Another factor having an influence on the impact force curves is the distance between the impact point (point Z in Fig. 2) and the mass centre of car B (Table 1).
 \(O_\mathrm{A}C^{0}\)

length of the line segment \(O_\mathrm{A}C\) at the beginning of the collision phase (\(t=0~\hbox {s}\)).
 K

average stiffness coefficient of the front of car A approximated from the deformation characteristics [16].
 \({\text {ZD}}^{0}\)

length of the segment ZD at the time instant \(t=0\).
5 Basic energy components involved in the energy dissipation process

Initial kinetic energy of the vehicles, determined by their preimpact translational motion and rotational motion of vehicle wheels and rotating components of the powertrain;

Deformation work of vehicle bodies in their contact zone;

Friction work done when the vehicle bodies slide against each other in the contact area between them;

Work related to the displacement of vehicles in their translational and rotational motion;

Energy dissipated in the vehicles suspensions and tyres;

Thermal and vibrational energy released as a result of deformations and processes of destruction of vehicle components.
 \(I_{{\textit{kp}}}\)

moment of inertia of the front wheels and rotating components of the powertrain;
 \(I_{{\textit{kt}}}\)

moment of inertia of the rear wheels;
 \(\omega _{{\textit{ki}}}\)

angular velocity of the wheels of the ith vehicle.
 \(s_{i1}\)

lateral displacement of the front wheels of the ith vehicle during the contact phase of the collision;
 \(s_{i2}\)

lateral displacement of the rear wheels of the ith vehicle during the contact phase of the collision.
Values of individual components of the energy balance
Test AB1  Test AB2  Test AB3  Test AB1  Test AB2  Test AB3  

\(E_{k\mathrm{{AB}}}(t=0)\) (kJ)  226.9  160.5  185.5  \(E_{k\mathrm{{AB}}}(t=0)\) (%)  100  100  100 
\(E_{k\mathrm{{AB}}}(t=t_{k)}\) (kJ)  128.0  95.9  92.4  \(E_{k\mathrm{{AB}}}(t=t_{k)}\) (%)  56.4  59.8  49.8 
\(E_\mathrm{D}(t=t_{k})\) (kJ)  68.9  48.3  67.4  \(E_\mathrm{D}(t=t_{k})\) (%)  30.3  30.1  36.3 
\(E_\mathrm{T}(t=t_{k})\) (kJ)  11.9  5.5  3.4  \(E_\mathrm{T}(t=t_{k})\) (%)  5.2  3.4  1.8 
\(E_{t\mathrm{{AB}}}(t=t_{k})\) (kJ)  13.8  7.4  17.3  \(E_{t\mathrm{{AB}}}(t=t_{k})\) (%)  6.1  4.6  9.4 
\(E_{p}(t=t_{k})\) (kJ)  4.4  3.4  4.9  \(E_{p}(t=t_{k})\) (kJ)  2.0  2.1  2.7 
The values of the considered energy components at characteristic time instants during the contact phase of the collision are listed in Table 2.
Preimpact velocity of car A, obtained from the experiment and as computed
Velocity of car A (m/s)  AB1  AB2  AB3 

\(V_\mathrm{A}\) (experiment)  15.1  12.7  15.3 
\(V_\mathrm{A}\)(calculated for \(\Delta E=E_\mathrm{D}+E_\mathrm{T}+E_{t\mathrm{{AB}}}+E_{p}\))  15.9  13.0  15.4 
\(V_\mathrm{A}\)(calculated for \(\Delta E=E_\mathrm{D}\))  13.4  11.3  13.2 
The calculation results presented in Table 3 are an example of using the energy balance of the fronttoside motor car collision. They are showing that using the car body deformation work during the collisions of this type, which often happens for the accident reconstruction purposes, results in underestimating the velocity of the impacting car by 11–14%.
6 Summary and conclusions

Measurements carried out during motor vehicle crash tests;

Calculations of the forces, velocities and displacements in the car contact zone, based on the model of the motor vehicle collision dynamics;

Verification of car positions by means of a framebyframe analysis of the video recordings of the experiments.
 1.
It has been confirmed that the Eq. (29) can be useful for the calculation of the impacting car’s velocity in the fronttoside collisions of passenger cars if some necessary initial assumptions are made. Also, it has been found that the value of the lost energy \(\Delta E\) should be taken higher by about 25–30% than the energy calculated from the postimpact vehicle deformation.
 2.
It has been pointed out that the work of deformation of the front and side parts of the cars makes about 70–75% of the total energy lost (Table 2) by the vehicles during the contact phase of the collision and that this percentage is definitely lower than that observed in headon car collisions.
 3.
During the rightangle collision, the vehicles involved remain in a practically unchanged position relative to each other for the entire duration of the contact phase of the collision (\(t_{k}=0.090.12~\hbox {s}\), Fig. 1), in spite of their translational and rotational motion during that period (Figs. 6, 7). Identical findings have been presented in [30]. This fact considerably facilitates the interpretation of the measurement results obtained from the experiments, at least for the central impact locations of certain range.
 4.
The impact against a motionless vehicle differs from a similar impact against a vehicle in motion in that far less energy is lost in the former case due to friction between the vehicle bodies sliding on each other and, on the other hand, much more energy is used for the translation and rotation of the vehicles on the road surface during the contact phase of the collision.
Notes
Compliance with ethical standards
Conflicts of interest
The authors declare that they have no conflict of interest.
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