# Mathematical model validated by a crash test for studying the occupant’s kinematics and dynamics in a cars’ frontal collision

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## Abstract

The aim of the paper was to determine the kinematic parameters that influence the occupant injury risk through a mathematical model. The developed model is a 2D model composed of 4 bodies (2 vehicles, thorax and head). The head and thorax are interconnected with a rotation joint and a torsion spring meant to stiffen the relative movement between the bodies. The thorax is connected with the vehicle body by a linear spring meant to simulate the seatbelt stiffness. The model was solved using Lagrange principle and the validation of the model was made through a crash test performed using the same initial conditions and comparing the obtained values of the displacement, velocity and acceleration parameters with the ones obtained with the mathematical model. The head and torso were chosen due to the fact that they are the common parts of the body that get injured, especially the head with the change of 80 % to cause fatal injury in car’s frontal collision. Once the model was validated, the stiffness of the seatbelt was modified in order to determine the behavior of the occupant in case of car frontal collisions. When the seatbelt stiffness was reduced, the occupant displacement and velocity increased, while by increasing the stiffness, these parameters decreased. The values of the developed model presented a high degree of similarity with the results obtained from the crash test with an error of 10 %. This model can be used by engineers to easily asses the occupant injury risk in case of vehicle frontal collisions.

## Key words

Safety Collision Occupant Mathematical model Kinetic energy Potential energy Crash test Dummy## Nomenclature

- m
mass of body, kg

- v
velocity of the body, m/s

- h
height from the point the object will fall, m

- k
rigidity of spring, N/m

- X
lengh of the spring, m

- Δx
spring deformation, m

- T
kinetic energy, J

- V
potential energy, J

- q
generalized coordinate system, -

- V
_{1t} velocity of the first moving body in the simple vehicle collision model, m/s

- V
_{2t} velocity of the second moving body in the simple vehicle collision model, m/s

- k
_{1t} rigidity of the fisrt moving body, N/m

- k
_{2t} rigidity of the second moving body, N/m

- m
_{1t} mass of the first body in the simple model, kg

- m
_{2t} mass of the second body in the simple model, kg

- X
_{1t} position of the first body relative to the model coordinate system xOy, m

- X
_{2t} position of the second body relative to the model coordinate system xOy, m

- ΔX
_{1} deformation of the first body, m

- ΔX
_{2} deformation of the second body, m

- m
_{1} mass of the first vehicle (VEH 1), kg

- m
_{2} mass of the second vehicle (VEH 2), kg

- m
_{3} mass of the thorax and pelvis of the occupant, kg

- m
_{4} mass of the neck and head of the occupant, kg

*l*_{3}lengh of the thorax and pelvis, m

*l*_{4}lengh of the neck and head, m

- x
_{1} displacement of the first vehicle’s center of mass on the X axis relative to the coordinate system xOy, m

- x
_{2} displacement of the second vehicle’s center of mass on the X axis relative to the coordinate system xOy, m

- x
_{3} displacement of the occupant thorax and pelvis on the X axis relative to the vehicle’s center of mass, m

- y
_{4} displacement of the head and neck on the Y axis, m

- α
_{4} angular displacement of the neck and head relative to the thorax, rad

- k
_{1} vehicle VEH 1 frontal chassis structure rigidity, N/m

- k
_{2} vehicle VEH 2 frontal chassis structure rigidity, N/m

- k
_{3} seatbelt rigidity between body m

^{3}and the vehicle VEH 1 centre of mass- k
_{4} neck-thorax articulation rigidity, Nm/rad

*l*_{1}maximum deformation coordinate for vehicle VEH 1, m

*l*_{2}maximum deformation coordinate for vehicle VEH 2, m

*l*_{t}maximum displacement of the thorax m

^{3}, m- F
_{s} friction force between the thorax and pelvis and the seat, N

- F
_{f} friction force between the vehicle tire and road, N

- F
_{c} collision forces of the vehicles, N

- v
_{1} vehicle VEH 1 velocity, m/s

- v
_{2} vehicle VEH 2 velocity, m/s

- g
gravitational acceleration, m/s

^{2}

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