Mathematical functions to estimate indirect head impact responses were developed in this paper. Experimental human cadaver sinusoidal head impact responses in the A-P (frontal), L-R (lateral), P-A (rearward) and S-l (superior-inferior) directions were used as reference experimental bases for the generation of the “off-axis” approximate impedance functions. A lumped parameter model consisting of two masses, two dampers and one spring was used as the physical model in this study. This model is called the Translational Head Injury Model (THIM) and was used earlier in the development of the Translational Energy Criteria for primate and human head impact analyses [1,2,3].

The approximate impedance functions were generated by adjusting the values of all the lumped parameters in the THIM model (K, C₁, C₂, M₁ and M₂) to obtain smooth variations between reference points (A-P, L-R, P-A or S-l direction). These approximate impedance curves at the “middle” point or 45 degree rotation from a reference point on the head surface were generated base on linear logarithmic variation. Next, the values of K, C₁, C₂, M₁ and M₂ for the reference and intermediate impedance curves were used to develop a quadratic function for estimating lumped parameter values of the THIM model at other impact locations on the head surface. Two independent sets of approximate functions were generated based on available experimental data. Finally, these two sets of mathematical equations were combined to produce a computer program which can be used to estimate impedance curves for any off-axis or indirect head impact.

The approximate impedance curves at 22½, 45, and 67% degree rotations which were generated by the functions developed in this study show smooth logarithmic variations between the reference points. In general, the approximate impedance functions may be used to analyze off-axis head impacts which are much more common than direct head impacts in real world accident environments.