A Novel Technique for Modeling Vehicle Crash using Lumped Parameter

Models

Gulshan Noorsumar

, Svitlana Rogovchenko

, Kjell G. Robbersmyr

Dmitry Vysochinskiy

and Andreas Klausen

Department of Engineering Sciences, University of Agder, Grimstad, Norway

Keywords:

Mathematical Model, Collision Mitigation, Lumped Parameter Model, Piece-wise Linear Spring Stiffness,

Vehicle Pitching, Maximum Dynamic Crush.

Abstract:

This paper presents a novel technique for modeling a full frontal vehicle crash. The crash event is divided

into two phases; the ﬁrst until maximum crush and the second part when the vehicle starts pitching forward.

This novel technique will help develop a three degrees of freedom (DOF) lumped parameter model (LPM)

for crash and support in the vehicle development process. The paper also highlights the design process for

reducing vehicle pitching in occupant protection load cases. The model has been validated against a ﬁnite

element (FE) simulation of a full frontal crash of a Chevrolet Silverado developed by the National Highway

Trafﬁc Safety Administration (NHTSA), and the LPM shows good correlation with the FE test data.

1 INTRODUCTION

Vehicle crashes have been among the major causes of

mortality in recent times (Du Bois et al., 2004). In Oc-

tober 2015, the European Commission had launched

a study to identify the most common crash scenarios

leading to serious injuries in a vehicle crash. The re-

sults of this study point to the fact that a frontal crash

is the most common crash scenario, followed by a side

impact, where occupants are severely injured (Noor-

sumar et al., 2020). Euro NCAP is a voluntary car

safety assessment program introduced to ensure safer

cars for occupants and vulnerable road users. This

program has been instrumental in driving regulations

across the globe and improving vehicle safety stan-

dards. During the past decades, several crash miti-

gation and avoidance techniques have been employed

by vehicle design engineers to meet these stringent

regulations. The vehicle front-end and side structures

have been modiﬁed to improve energy absorption ca-

pability (Elkady and Elmarakbi, 2012). Vehicle de-

sign engineers have resorted to various methodologies

https://orcid.org/0000-0002-6718-4508

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https://orcid.org/0000-0002-5411-3655

to improve the vehicle structure to absorb energy in

case of a crash and prevent intrusions in the occupant

compartment. These methodologies have been par-

tially successful in replacing full-time physical tests

in the vehicle development process. The vehicle in-

dustry still conducts a lot of physical crash tests to

validate the crash response generated from mathemat-

ical models.

One of the recent approaches is using ﬁnite ele-

ment methods (FEM) to model the full vehicle im-

pact scenario and conduct simulations to predict the

vehicle and occupant injury values. (Benson et al.,

1986) presented the calculations of crashworthiness

design thereby, laying the foundations for applica-

tion of FEM in the automotive industry. This tech-

nique has high accuracy in predicting injury values,

however the process involves manual efforts and is

computationally intensive. Lumped parameter mod-

els (LPM) were ﬁrst used in modeling vehicle crash

by (Kamal, 1970). In this paper the vehicle was rep-

resented by three lumped mass components and eight

resistances representing the deformable parts in the

vehicle. Mentzer et al. (Mentzer et al., 1992) em-

ployed real time crash data to determine parameters

for LPM used to represent the crash scenario. The

force deformation curves derived from these models

helped determine predictive models aiding in vehicle

Noorsumar, G., Rogovchenko, S., Robbersmyr, K., Vysochinskiy, D. and Klausen, A.

A Novel Technique for Modeling Vehicle Crash using Lumped Parameter Models.

DOI: 10.5220/0010529200620070

In Proceedings of the 11th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2021), pages 62-70

ISBN: 978-989-758-528-9

development.

Recently, LPMs were used by Elkady et al. to de-

velop a multi-DOF mathematical model to simulate a

crash event with active vehicle dynamics control sys-

tems (VDCS) (Elkady and Elmarakbi, 2012; Elkady

et al., 2012). The model replicated a full frontal

and offset impact between two vehicles and compared

the performance of a baseline vehicle with a vehicle

equipped with VDCS features. It also includes a 3-

DOF occupant impact model using Lagrangian for-

mulation. Munyazikwiye et al. use a mass-spring-

damper model with two lumped mass components

representing a vehicle impacting a rigid barrier. Af-

ter identifying the parameters, the model in this study

shows good correlation with test data which demon-

strates that a simple LPM can be used to represent the

impact dynamics successfully (Munyazikwiye et al.,

2013). Multi body modeling has also been used in the

past for simulating vehicle dynamics model for real-

istic applications (Riegl and Gaull, 2018).

Occupant injury prediction is an area of research

where the vehicle-occupant interaction in a vehicle

impact scenario is studied and the injury patterns of

occupants in the car are determined with mathemati-

cal models. Large vehicle deceleration has been iden-

tiﬁed as one of the main causes of head and chest in-

juries, and vehicle rotational motions in different axis

also lead to occupant injuries (Chang et al., 2006).

In a full frontal impact, vehicle pitch and drop are

signiﬁcantly greater compared to rolling and yawing

motions. In the recent past, increasing focus on un-

belted occupants to meet FMVSS 208 (Federal Mo-

tor Vehicle Safety Standards) requirements has led re-

searchers to observe that vehicle pitch and drop con-

tributed to higher head and neck injury values. The

objective of a vehicle structure is not just to absorb en-

ergy and optimize crash pulses, but also to minimize

vehicle pitch and drop (Chang et al., 2006; Woitsch

and Sinz, 2013). Chang et al. have developed an

FE model to study vehicle pitch and drop in body-

on-frame vehicles. The model is correlated to barrier

tests and also tries to predict factors affecting vehicle

pitch and drop in a crash event (Chang et al., 2005).

The research from Chang et al. points to the fact that

design of vehicle rails plays an important role in the

load distribution during an impact scenario for body-

on-frame vehicles. The out-of-plane bending of the

vehicle rails increases the role of a vertical component

of the barrier force, causing an imbalance in the vehi-

cle, leading to forward pitching on the vehicle. Wei

et al. have estimated the relationship between energy

absorbing components and the crash pulse, establish-

ing the fact that the bumper and the front rails both

signiﬁcantly contributing to the energy absorption in

a full frontal crash event (Wei et al., 2016).

In this paper, we simplify the system by splitting

the vehicle motion into two phases corresponding to

• the horizontal linear motion, and

• the rotation of the vehicle body.

We have decided to replicate a full frontal vehicle

crash event at 56 kilometers per hour (kmph) employ-

ing an LPM with multiple DOFs to predict

• the maximum deformation in the vehicle to absorb

energy, and

• the pitch angle of the vehicle due to the crash re-

sponse.

2 METHODOLOGY

Literature documents that a crash event leads to pitch-

ing, rolling and yawing of the vehicle along with the

deceleration of the vehicle and movement in horizon-

tal and vertical directions. It is difﬁcult to model the

impact scenario in different axes and generating the

governing equations. It was also observed that the

time for the vehicle to attain minimum velocity after

impact also coincides with the maximum deformation

on the vehicle.

In this study, we separate the horizontal transla-

tional motion from the vertical motion during the im-

pact event. In a full frontal crash event the vehi-

cle is observed to be experiencing forward pitching;

whereas the effect of rolling and yawing can be ne-

glected. Taking into account these assumptions we

split the crash event into two phases:

• time till maximum deformation and minimum ve-

hicle velocity after start of crash event t

, and

• time after maximum deformation to the end of the

crash event t

2.0.1 FEM Simulations

In this study, ﬁnite element simulation for a 2014

Chevrolet Silverado (Administration et al., 2016) run-

ning at 56 kmph and hitting a frontal barrier at 0% off-

set was conducted. The FE model was developed by

National Crash Analysis Center (NCAC) in collabo-

ration with NHTSA (National Highway Trafﬁc Safety

Administration) through the reverse engineering pro-

cess (Administration et al., 2016). The FE model con-

sisting of 1476 parts, 2,741,848 nodes and 2,870,507

elements has been correlated to NHTSA Oblique Test

and Insurance Institute for Highway Safety (IIHS)

Small Overlap Front Test. The FE model weighs 2582

A Novel Technique for Modeling Vehicle Crash using Lumped Parameter Models

kg which is close to the physical test vehicle weigh-

ing 2624 kg. It replicates the material and geometrical

properties of the physical vehicle (Singh et al., 2018).

The FE model was run on LS-DYNA with 32

CPUs in an HPC environment and the correspond-

ing curves generated were used for the parameter

estimation and validation of the LPM in MATLAB

Simulink. In the FE simulation, the acceleration of

some nodes on the vehicle body are recorded by the

solver LS-DYNA. These nodes are selected by the

user at the preprocessing stage. This process was em-

ployed to determine the acceleration of the vehicle

CG as well as the barrier forces, to be used for val-

idation in this study. Figure 1 shows the FE model

used in the simulations.

Figure 1: FE Model of a 2014 Chevrolet Silverado devel-

oped by NCAC.

This FE model generated the piecewise linear

curve data for spring and damper coefﬁcients. The al-

gorithm uses Newton-Euler numerical integration to

achieve the computed values and predict the time for

maximum dynamic crush of the vehicle. The algo-

rithm developed is explained in the next section.

2.1 Lumped Parameter Model

The LPM developed is a single-mass system with a

spring and damper unit in the front, known as Kelvin

model, representing the bumper system and the de-

formable system. The front springs allow transla-

tional motion only in the direction of x-axis (Huang,

2002). The suspension of the vehicle is represented

by a pair of springs and dampers which allow trans-

lation in the vertical z-axis and rotation around the

y-axis. The center of mass of the vehicle has 3 DOFs

making this system fairly complex to solve in a single

system. The lumped mass body can move in the direc-

tion of horizontal (x) and vertical (z) axes along with

rotation around one (y) axis. The center of gravity

(CG) of the vehicle is located at a distance l

from the

front end and l

from the rear end suspension points.

The distance l

represents the distance of the CG from

the front occupant compartment zone.

2.2 Vehicle Crash Model: Phase I

First we model only the translational movement of

the vehicle along the horizontal axis and hitting the

barrier at 0% offset. The mathematical model is de-

veloped in Simulink which replicates the maximum

vehicle deformation till the time of maximum crush

. This value also corresponds to the time when the

vehicle attains zero or minimum velocity. It should

be noted that the vehicle may not achieve zero veloc-

ity by the time of maximum deformation if the vehi-

cle front end is not able to absorb energy to undergo

plastic deformation. The mathematical model uses a

single DOF equation with a front spring-damper unit.

The stiffness of the spring is tuned to represent the

maximum deformation of the vehicle at a particular

speed. For this problem we have assumed a speed of

56 kmph (NHTSA regulations for frontal crash). The

motion of suspension system in the model has been

neglected during this phase of the event scenario. Fig-

ure 2 represents the vehicle in a deformed state. The

Simulink model predicts the time till maximum defor-

mation of the vehicle and the maximum displacement

of the vehicle CG.

The prediction of the values of spring deforma-

tion coefﬁcient k and damper coefﬁcient c used in the

general equation of motion have been a challenge for

researchers in the past (Marzbanrad and Pahlavani,

2011), (Pawlus et al., 2011). There have been several

parameter estimation studies conducted in the past to

determine the stiffness of the vehicle front in a crash

event. The behaviour of the front end system is highly

nonlinear but it has been approximated by a piece-

wise linear relationship (Munyazikwiye et al., 2017;

B Munyazikwiye et al., 2018).

Figure 2: Vehicle representation in Phase 1 of the event:

Deformed front end.

Equation of Motion:

m ¨x + c ˙x + kx = Q

(1)

where, Q

= 0 (i.e. no force component is added here);

k is the spring coefﬁcient; c is the damper coefﬁcient

SIMULTECH 2021 - 11th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

for the bumper model.

In the model developed for the ﬁrst phase, the spring

and damper coefﬁcients are parameterized using a

gradient-descent optimization algorithm developed

in (Klausen et al., 2014) for a single mass-spring-

damper system. The code searches for a global min-

ima by performing 100 re-runs of gradient descent

optimization, each with randomly generated initial

parameter values. The algorithm was modiﬁed to

improve the correlation between the test and com-

puted values. The non-linear force-deformation curve

for spring-damper system has been assumed to be

piecewise-linear with six breakpoints in the curve.

The forces on the spring are calculated using the gen-

eral relationship between the force and deformation

for a spring-damper system (Elkady and Elmarakbi,

2012), see Figure 3. The stiffness of the spring k and

the spring force component F

vary according to the

deﬂection values in the spring and are deﬁned as fol-

lows.

Figure 3: General piecewise force-deformation characteris-

tics (Elkady and Elmarakbi, 2012).

The spring stiffness and damping coefﬁcients in

the model, are deﬁned as the piecewise-linear func-

tions of x and ˙x, respectively. These functions are

k(x) =











−k

)·| ˆx|

+ k

for, | ˆx| ≤ x

−k

)·(| ˆx|−x

)

−x

)

+ k

for, x

≤ | ˆx| ≤ x

−k

)·(| ˆx|−x

)

−x

)

+ k

for, x

≤ | ˆx| ≤ x

−k

)·(| ˆx|−x

)

−x

)

+ k

for, x

≤ | ˆx| ≤ x

−k

)·(| ˆx|−x

)

−x

)

+ k

for, x

≤ | ˆx| ≤ x

−k

)·(| ˆx|−x

)

(C−x

)

+ k

for, x

≤ | ˆx| ≤ C

The damper characteristics are deﬁned similar to

the spring characteristics in the model

c( ˙x) =











−c

)·|

˙x|

˙x

+ c

for, |

˙x| ≤ ˙x

−c

)·(|

˙x|− ˙x

)

( ˙x

− ˙x

)

+ c

for, ˙x

≤ |

˙x| ≤ ˙x

−c

)·(|

˙x|− ˙x

)

( ˙x

− ˙x

)

+ c

for, ˙x

≤ |

˙x| ≤ ˙x

−c

)·(|

˙x|− ˙x

)

( ˙x

− ˙x

)

+ c

for, ˙x

≤ |

˙x| ≤ ˙x

−c

)·(|

˙x|− ˙x

)

( ˙x

− ˙x

)

+ c

for, ˙x

≤ |

˙x| ≤ ˙x

−c

)·(|

˙x|− ˙x

)

− ˙x

)

+ c

for, ˙x

≤ |

˙x| ≤ v

where, k is the spring coefﬁcient; c is the damper

coefﬁcient; ˆx is the computed vehicle deformation; ˙x

is the vehicle velocity;

˙x is the computed vehicle ve-

locity; v

is the velocity at the time of maximum dy-

namic crush; C is the maximum dynamic crush;

and F

are the built-up spring and damping

forces deﬁned by the following equations

= k(x) · x, (2)

= c( ˙x) · ˙x (3)

The proposed algorithm uses an optimization ap-

proach to minimize an objective function. The ob-

jective function to be minimized is the error function

E(Θ, t) where Θ denotes the unknown variables in the

mode. The error function is deﬁned as follows:

E(Θ, t) = E

(Θ, t) + E

(Θ, t), where

(4a)

(Θ, t) = |(a

− a

LPM

)| (4b)

(Θ, t) = |(v

− v

LPM

)| (4c)

(Θ, t) = |(x

− x

LPM

)| (4d)

where, a is the acceleration; v is the vehicle veloc-

ity; and x is the displacement.

The error function E(Θ, t) determines the differ-

ence between the FE and computed values at every

point, and the optimization algorithm tries to mini-

mize these error values by altering Θ = [k

, c

]∀i ∈

[1, 7]. The corresponding spring and damper coefﬁ-

cient values developed from this minimization algo-

rithm have been discussed in the results section of the

paper.

A Novel Technique for Modeling Vehicle Crash using Lumped Parameter Models

2.3 Vehicle Crash Model: Phase II

The second phase for the model describes what hap-

pens after the instant the vehicle achieves maximum

dynamic crush and minimum velocity. The vehicle

starts to pitch forward at this instant. Several studies

were conducted to understand the reason behind the

vehicle pitching forward (Chang et al., 2005; Chang

et al., 2006), suggesting that for body-on-frame vehi-

cles one of the reasons is the out of plane bending in

vehicle rails leading to a vertical force component in

the moment balance equation. The vertical force com-

ponent is added to gravity force acting downwards

and creates an imbalance of loading which leads to

the vehicle pitching. The prediction of this pitching

angle is important for determining the injury to oc-

cupants and a low pitching angle inﬂuences occupant

protection design in a vehicle. In this phase of the

event as shown in Figure 4 and Figure 5, we consider

only vertical motion of the suspension springs and a

rotation about the y-axis with angle θ.

Figure 4: Vehicle representation in Phase II of the event:

Vehicle Pitching forward.

Figure 5: Vehicle representation in Phase II with forces act-

ing on the vehicle and suspension springs in play.

We use the Lagrangian Formulation (Goldstein

et al., 2002):

∂L

∂ ˙q

−

∂L

∂q

∂D

∂q

= Q

(5)

where, in the general case, L = T −V , T is the total

kinetic energy of the system equal to the sum of the

kinetic energies of the particles, q

i = 1..n are gen-

eralized coordinates and V is the potential energy of

the system. For dissipation forces a special function

D must be introduced alongside L, Q

is the external

force acting on the system, which in this case is the

vertical force component experienced by the vehicle

at the time of maximum dynamic crush.

Kinetic Energy:

T =

m ˙x

(6)

Potential Energy :

V =

(x − l

θ)

(x + l

θ)

(7)

Dissipation Energy :

D =

( ˙x − l

θ)

( ˙x + l

θ)

(8)

The values of k

, k

, c

, l

and l

are taken from

standard automotive parameters from literature data

(Savaresi et al., 2010). Table 1 shows the parameter

values adopted from this study.

Table 1: Automotive Parameters set (Savaresi et al., 2010).

The value of the vehicle mass m and the moment

of inertia J for the lumped mass system has been cal-

culated from the FE model of the vehicle. The gov-

erning equations of motion are

θ + (k

+ k

)θ + (c

+ c

)

+ (k

− k

)x + (c

− c

) ˙x (9)

−Q

=m ¨x + (k

+ k

)x + (c

+ c

)

+ (k

− k

)θ + (c

+ c

) ˙x (10)

SIMULTECH 2021 - 11th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

3 RESULTS AND DISCUSSION

In this section we compare the results of the LPM

model with FE data generated from LS-DYNA sim-

ulations for a Chevrolet Silverado vehicle at 56kmph

with a full frontal impact loadcase.

3.1 Phase I

As mentioned in the previous section, Part I of the

event simulates the time till maximum deformation

of the vehicle; the spring and damper coefﬁcients are

determined using the Gradient Descent Optimization

with an error function explained in the previous sec-

tion. The computed and test (FE) values are plotted in

Figure 6 and shows good correlation of results. The

algorithm predicts the stiffness and damping coefﬁ-

cient values as shown in Figures 7 and 8.

Figure 6: Plot of computed and test values for parameter

model.

Figure 7: Spring coefﬁcient obtained from the algorithm.

Figure 8: Damper Coefﬁcient obtained from the algorithm.

The output from the Gradient Descent Optimiza-

tion algorithm is used to predict the deformation and

vehicle velocity in a MATLAB Simulink model;

Figure 9 shows a plot of maximum vehicle defor-

mation vs test deformation and the plot shows good

correlation. A similar plot (Figure 10) was generated

to compare the velocity of the vehicle at the CG (in

the case of LPM at the CG of the lumped mass). The

time the vehicle attains zero velocity is similar in the

plots but there is a small difference after 0.04s. The

reason for this deviation can be attributed to the spring

and damper characteristics which are approximated

for this study using a piece-wise linear function. The

model can be improved using a non-linear function

for the spring stiffness and damping characteristics.

If the model is simulated beyond the time the vehi-

cle attains zero velocity, a rebound is observed in the

velocity. This velocity rebound could be due to the in-

ternal strain energy store in the springs, and it would

be interesting to investigate this further in future re-

search.

Figure 9: Displacement of the vehicle CG curves compari-

son for LPM vs FE model.

Figure 10: X-Velocity curve comparison for LPM vs FE

model.

A Novel Technique for Modeling Vehicle Crash using Lumped Parameter Models

3.2 Phase II

The prediction for the second part of the model us-

ing Simulink was conducted and plotted against the

data from FE model. The force Q

in the govern-

ing equations is the vertical component of the barrier

force experienced by the vehicle in the crash. The

force curve is derived from the FE model and inputted

into the Simulink model to improve prediction. How-

ever, it will be of interest to mathematically explain

this force component in terms of residual impact en-

ergy after absorption. The Simulink model is run with

numerical integration (variable timestep- ode 45) and

the velocity of the lumped mass in z-direction along

with the pitching angle is compared to data from FE

model.The comparison with other numerical integra-

tion methods was kept out of scope of this study.

Figure 11 compares the forward pitching angle for

the FE model and the LPM developed in this study.

The pitch angle comparison shows a similar trend ob-

served in both the curves; the vehicle starts to pitch

around the same time during the crash event which

is crucial to designers planning airbag deployment in

vehicles and other active features on cars. The pitch

angle curve for the simulation LPM peaks higher than

the FE data at the start of the vehicle rotation but

slowly follows the FE data curve showing comparable

maximum pitch angle values which is also an impor-

tant observation for a vehicle safety design team. The

linear approximation for the spring and damper coef-

ﬁcients can be a contributing factor to the difference

in the values between the curves along with the barrier

force deﬁnition in the model. There might be energy

losses in the model which have not been accounted

for in this study.

Figure 11: Forward Pitch Angle curve overlay for LPM vs

FE model.

Figure 12 compares the z-velocity (vertical veloc-

ity) in the body with the curves generated from FE

data. The trend in the curve is similar but the peak

values are not matching in this simulation model. One

of the contributing factors to this deviation is the use

of standard linear spring and damper coefﬁcient val-

ues for the model (used from literature data). The lin-

ear value for the spring and damper coefﬁcients can

lead to the difference in the values for this parameter

as well. The values of l

and l

can also be tuned

further to represent the Chevrolet Silverado (2015)

model used in this study. However, the authors have

intentionally avoided ﬁne tuning these values assum-

ing that this data may not be available to vehicle de-

velopment team at the start of the design process. This

makes it inevitable to use standard values for automo-

tive parameters.

Figure 12: Z-Velocity curve comparison for LPM vs FE

model.

4 CONCLUSION AND NEXT

STEPS

The novel technique developed in this paper for mod-

eling a full frontal vehicle crash event successfully

predicts the event kinematics. The study demonstrates

that the two phase simulation model can describe a

highly complex dynamical multiple DOF system with

few equations and parameters, making the process of

using LPMs very simple and reliable for safety design

engineers. The study also highlights that parameter

identiﬁcation is an important part of accident recon-

struction process and its correlation has an inﬂuence

on the deformation and velocity during vehicle crash.

One of the major implications from the model devel-

oped in this study is the design of vehicle rails, as a

contributing factor to vehicle pitching forward.

This assumption used to arrive at a simpler LPM

model providing reliable results include the follow-

ing:

• The spring and damper characteristics are as-

sumed to be piecewise-linear with six breakpoints

although they are non-linear in physical systems.

SIMULTECH 2021 - 11th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

• The vehicle acceleration is assumed to be zero at

the time pitching starts in the crash event.

• Energy losses like friction and heat losses in the

vehicle during the crash event are neglected to

simplify the problem.

• Only vehicle rotations about the y-axis (pitching)

are considered for modeling in the full frontal im-

pact scenario; rotations about other axes are con-

sidered negligible and not impacting the occupant

injuries.

The next steps in this study include creating a

mathematical expression for the force components in

the mathematical model for the second phase which

includes pitching of the vehicle. The mathematical

expression for the force would help vehicle design

teams to reduce pitching on the vehicle by changing

the force components acting on the vehicle during a

collision. The prediction for maximum deformation

can be improved by using a non-linear force defor-

mation curve for the spring stiffness curve with larger

breakpoints along with including energy losses in the

model. The model currently uses standard spring and

damper coefﬁcients for the suspension model which

can be tuned to match a particular vehicle being stud-

ied; also including the mass and the distance of the

CG from the vehicle suspension connections.

ACKNOWLEDGMENTS

The authors would like to thank University of Agder

for the support to conduct this research.

REFERENCES

Administration, N. H. T. S. et al. (2016). Crash sim-

ulation vehicle models. URL: https://www. nhtsa.

gov/research-data/databasesand-software.

B Munyazikwiye, B., Vysochinskiy, D., Khadyko, M., and

G Robbersmyr, K. (2018). Prediction of vehicle crash-

worthiness parameters using piecewise lumped pa-

rameters and ﬁnite element models. Designs, 2(4):43.

Benson, D., Hallquist, J., Igarashi, M., Shimomaki, K., and

Mizuno, M. (1986). Application of dyna3d in large

scale crashworthiness calculations. Technical report,

Lawrence Livermore National Lab.

Chang, J. M., Ali, M., Craig, R., Tyan, T., El-Bkaily, M.,

and Cheng, J. (2006). Important modeling practices in

cae simulation for vehicle pitch and drop. SAE Trans-

actions, pages 62–72.

Chang, J. M., Rahman, M., Ali, M., Tyan, T., El-Bkaily, M.,

and Cheng, J. (2005). Modeling and design for vehi-

cle pitch and drop of body-on-frame vehicles. SAE

transactions, pages 329–338.

Du Bois, P., Chou, C. C., Fileta, B. B., Khalil, T. B., King,

A. I., Mahmood, H. F., Mertz, H. J., Wismans, J.,

Prasad, P., and Belwafa, J. E. (2004). Vehicle crash-

worthiness and occupant protection.

Elkady, M. and Elmarakbi, A. (2012). Modelling and anal-

ysis of vehicle crash system integrated with different

vdcs under high speed impacts. Open Engineering,

2(4):585–602.

Elkady, M., Elmarakbi, A., and MacIntyre, J. (2012). En-

hancement of vehicle safety and improving vehicle

yaw behaviour due to offset collisions using vehicle

dynamics. International journal of vehicle safety,

6(2):110–133.

Goldstein, H., Poole, C., and Safko, J. (2002). Classical

mechanics.

Huang, M. (2002). Vehicle crash mechanics. CRC press.

Kamal, M. M. (1970). Analysis and simulation of vehicle to

barrier impact. SAE Transactions, pages 1498–1503.

Klausen, A., Tørdal, S. S., Karimi, H. R., Robbersmyr,

K. G., Je

cmenica, M., and Melteig, O. (2014). Mathe-

matical modeling and optimization of a vehicle crash

test based on a single-mass. In Proceeding of the 11th

World Congress on Intelligent Control and Automa-

tion, pages 3588–3593. IEEE.

Marzbanrad, J. and Pahlavani, M. (2011). A system identiﬁ-

cation algorithm for vehicle lumped parameter model

in crash analysis. International Journal of Modeling

and Optimization, 1(2):163.

Mentzer, S. G., Radwan, R. A., and Hollowell, W. T. (1992).

The sisame methodology for extraction of optimal

lumped parameter structural crash models. Technical

report, SAE Technical Paper.

Munyazikwiye, B. B., Karimi, H. R., and Robbersmyr,

K. G. (2013). Mathematical modeling of vehicle

frontal crash by a double spring-mass-damper model.

In 2013 XXIV International Conference on Informa-

tion, Communication and Automation Technologies

(ICAT), pages 1–6. IEEE.

Munyazikwiye, B. B., Karimi, H. R., and Robbersmyr,

K. G. (2017). Application of genetic algorithm on pa-

rameter optimization of three vehicle crash scenarios.

IFAC-PapersOnLine, 50(1):3697–3701.

Noorsumar, G., Robbersmyr, K., Rogovchenko, S., and

Vysochinskiy, D. (2020). Crash response of a repaired

vehicle-inﬂuence of welding uhss members. Techni-

cal report, SAE Technical Paper.

Pawlus, W., Karimi, H. R., and Robbersmyr, K. G.

(2011). Development of lumped-parameter mathe-

matical models for a vehicle localized impact. Journal

of mechanical science and technology, 25(7):1737–

1747.

Riegl, P. and Gaull, A. (2018). Modeling and validation of

a complex vehicle dynamics model for real-time ap-

plications. In Proceedings of 8th International Con-

ference on Simulation and Modeling Methodologies,

Technologies and Applications - SIMULTECH,, pages

403–413. INSTICC, SciTePress.

Savaresi, S. M., Poussot-Vassal, C., Spelta, C., Sename, O.,

and Dugard, L. (2010). Semi-active suspension con-

trol design for vehicles. Elsevier.

A Novel Technique for Modeling Vehicle Crash using Lumped Parameter Models

Singh, H., Ganesan, V., Davies, J., Paramasuwom, M.,

Gradischnig, L., Wood, P., and Mogal, V. (2018).

Structural countermeasure/research program mass and

cost increase due to oblique offset moving deformable

barrier impact test. Technical report.

Wei, Z., Karimi, H. R., and Robbersmyr, K. G. (2016).

Analysis of the relationship between energy absorb-

ing components and vehicle crash response. Technical

report, SAE Technical Paper.

Woitsch, G. and Sinz, W. (2013). Inﬂuence of pitching and

yawing during frontal passenger vehicle crash tests on

driver occupant’s kinematics and injury. International

journal of crashworthiness, 18(4):356–370.

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