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The limiting performance of ballistically loaded helmets designed to reduce head injuries is studied analytically. The projectile does not penetrate the helmet. This analysis evaluates the absolute minimum of the peak displacement of the helmet shell relative to the head, provided that criteria measuring the severity of head injuries lie within prescribed limits. Rather than optimize a specific design configuration, e.g. a viscoelastic foam liner, characteristics of a time-dependent force representing the helmet liner are calculated. The formulation reduces the limiting performance analysis to an optimal control problem

Capabilities of Helmets for Preventing Head Injuries Induced by Ballistic Impacts