The optimal restraint for minimizing the peak thoracic compression of the SID-IIs side crash test dummy subjected to a prescribed impact is studied using a linear spring-mass model. This model consists of the thoracic and pelvic masses and three springs which connect the masses and interface them with an impacting surface through the restraint. The problem is posed as an optimal control problem, with the restraint, which could be any physical structure (e.g., an airbag) operating in a finite allowable space, treated as a displacement control element. Via an assumption of the linearity of the dummy model and a discretization scheme, the problem is approximated and transformed into a linear programming problem for numerical solution. Numerical solutions are obtained under different prescribed impacting surface motion histories, different restraint space values, and constraints on dummy responses. Results show that the general characteristics of the optimal restraint response is a rapid ramp up in velocity in the very beginning of the event, followed by a period of lower level of loading where the thoracic compression builds up, and then an approximately constant acceleration to maintain the compression. The corresponding theoretically minimum thoracic compression values under the various conditions studied are presented.

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