Two mathematical models are developed and analyzed in this paper to predict the dynamic response in vehicle crashes. The first model is developed to capture the front-end intrusion of the vehicle structure in frontal collision. The second model is proposed to define the interaction between the child occupant and vehicle passenger compartment and to predict the acceleration injuries during a sudden impulse load. In these mathematical models, the bumper and vehicle body are defined by lumped masses and longitudinal rails of the front-end structure are defined by plastic springs. Moreover, the child occupants are considered as lumped masses, connected to the child seat and vehicle body masses by means of restraint systems. The occupant restraint characteristics of seat belt are represented by stiffness and damping elements. To obtain the dynamic response of the occupant, the equations of motion of the vehicle impact system in both full and offset scenarios are developed and analytically solved using Incremental Harmonic Balance Method (IHBM). The injury criteria, child’s acceleration and vehicle’s font-end deformation, are used to interpret the results. It is demonstrated from the simulations that the dynamic response and injury criteria are easily captured and analyzed. It is also shown that the mathematical models are flexible, useful in optimization studies and it can be used at initial design stage.

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