out of phase with the table. This indicates that the table (or the body) would deflect in one direction while the head and neck deflect in the opposite direction. The response for every dominant frequency for each impact location was not analyzed because this would not be an effective way to compare the dynamic performance of two helmets. Instead, an overall (aggregate) response of the head-neck system was analyzed that consisted of the Complex Mode Indicator Function (CMIF). The CMIF estimates the eigenvalues and eigenvectors of the frequency response function matrix using the Singular Value Decomposition (SVD) in Equation (5). In this set of measurements, the frequency response matrix consists of the frequency response, H1, of each sensor (x, y, z) for all impacts. The [S] matrix consists of the singular values of [H]. The [U] and [V] matrices are the left and right singular value vectors, respectively. These matrices contain phase information and are orthogonal matrices with unity length. The CMIF can be used to estimate the global resonant frequencies as well as the global deflection vectors: ( ) [ ]( ) [ ]( )[ ]( )[ ] ( ) i i i i i N N N N N N i N N S V U H × − × × × = 0 0 0 1 ω (5) The dominant modes for the head, neck, and table obtained from using the CMIF (Fig 4) indicate several strong modes of vibration for each component. In addition, the CMIF indicates common (heavily coupled) modes (20~30 Hz) as well as frequency ranges where the responses of the components are relatively decoupled (~ 10 Hz). The modal impact tests that were conducted on the system with the helmet strapped to the head were similar to the modal tests on the head-neck system. The main difference in these two tests is that two rounds of tests were conducted with the helmet using two different modal impact hammers: small and large hammers. The small impact hammer (PCB 086D05) excites a large frequency range (0-2,500 Hz) with a sharp impulse; the large impact hammer (PCB 086C01) excites a smaller frequency range (0-250 Hz) with a wider broadband loading. All future Fig 4 Dominant CMIF Modes for the head, neck, and table based on impact tests without a helmet 560
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