Mathematical model and numerical analysis of helmeted head subjected to high speed impact
A physical model with discrete mass rigid bodies of a helmeted head under high speed impact is developed and the corresponding mathematical model is presented in this paper. The dynamic lumped mass model is comprised of the projectile, the helmet, the skull, and the brain. Translational head injury model (THIM) model is employed to simulate the skull and the brain, and Herztian Contact Theory describes the interactive force between the skull and the projectile and deformation disc (P&DD). A sketch illustrating the simplified dynamic lumped-mass model with discrete mass rigid bodies is drawn in Figure 1. Mass mi is the mass of the skull, which is moving directly under the force from the helmet, and m2 is the mass of the brain. The masses mi and m2 will always add up to total head mass. The stiffness k and the damper C1 form the non-linear skull stiffness in a given direction. The damper C2 is found to be a constant for all directions and was believed to be primarily the damping of the brain. These parameters were determined by fitting the model driving-point mechanical impedance response to the experimental driving-point mechanical impedance response data obtained from cadaver tests conducted to the sides of head. Mass m4 is the combined mass of the P&DD. Numerical solutions, which allow for the displacements, the velocities, the acceleration or deceleration of the skull, the brain, and the projectile, the interactive force between the skull and helmet, and the brain-skull separation, are obtained in order to analyze the protective performances of the helmet. The dynamics of the skull, the brain, the P&DD and the characteristic time points of events are presented in Figure 2. The results are validated by the available cavaderic experimental data and the reasons for the discrepancy are analyzed. These analyses provide insight into the helmet performance under the high speed impact and can be generally applied in the design of the helmet.
