In the mid 1970s a group of 12 staged and instrumented automobile collisions was conducted for the National Highway Traffic Safety Administration. These were two-vehicle collisions with a variety of initial speeds, vehicle orientations, and vehicle size mixes. Initial speeds were controlled and velocity components including angular velocities at separation were measured. At about the same time, development of the classic theory of impact of rigid bodies to planar vehicle collisions was taking place. Users of classic theory heretofore had neglected the existence of a moment between impacting bodies. Inclusion of a moment and introduction of a moment coefficient of restitution allows the formulation of a planar collision model consisting of six algebraic equations relating the six initial velocity components of the two vehicles to their six final velocity components. The model contains collision geometry, vehicle geometry, vehicle inertial properties, and three coefficients. These coefficients are the classic coefficient of restitution, a friction coefficient, and the newly defined moment coefficient. This paper discusses the application of the theory of least squares to fit the experimentally determined velocity components to the six equations of the vehicle collision model. The usual approach using the theory of least squares is to set to zero the partial derivatives of the sum of squares taken with respect to the unknowns. The original model equations can be added as constraints through the use of Lagrange multipliers. A set of 15 nonlinear algebraic equations results. This approach was tried unsuccessfully. Direct numerical minimization of the sum of squares using gradient projection techniques proved to be far superior. Solutions are obtained for the staged collisions. Results provide insight into velocity changes and their relationships to energy dissipation, the coefficients of restitution and friction and other collision parameters. The capability of calculating velocity changes of colliding vehicles should prove complementary to detailed finite element studies of vehicle crush properties.

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