Singh, J. and Carpenter, N., "Work-Energy Relationships for the Collinear Single Degree of Freedom Impact Model under the Case of Net Unbalanced Externally Applied Forces," SAE Technical Paper 2013-01-0794, 2013, doi:10.4271/2013-01-0794.
Externally applied unbalanced forces and their corresponding impulses are generally excluded from consideration in regards to the evaluation of the collision phase events for a system comprised of two motor vehicles undergoing collinear impact. This exclusion is generally warranted secondary to the fact that the collision force and its corresponding impulse are dominant during the collision phase. Conceptually, two exclusions exist to this approach. The first is the situation in which significant physical restraints are present to the displacement of one or both collision partners and are of sufficient magnitude as to require inclusion. Generally, this represents the exceptional case and includes, but is not limited to, situations in which one vehicle is snagged, in a non-eccentric manner, by a rigid narrow-width object such as a pole or other similar restraint, prior to the occurrence of the subsequent vehicle-to-vehicle collision under evaluation. The second is the situation of collisions that are of exceedingly low collision severity. In such collisions, the effects of externally applied forces and their impulses are non-negligible when compared with the collision force and its impulse. The inclusion of externally applied forces and their collision phase impulses in the evaluation of such collisions is normative.The focus of the subject study, therefore, is the development of the applicable work-energy relationships for modeling collinear collisions in cases in which the effects of unbalanced forces, relative to the center of mass of the two-vehicle system undergoing impact, are non-negligible. The theoretical development presented is based upon the direct application of Newton's Second Law of motion. The dynamic center of mass reference frame is employed to develop the analytic closed-form solutions for the change in velocity experienced by each collision partner as a function of both the closing velocity and the impulses generated by the external impedances. The work-energy relationships are first derived in general form. The dynamic center of mass reference frame is utilized to solve for the common velocity reached by the collision partners at the terminus of closure. The resultant closed-form relationships derived for the terminus of closure and the terminus of separation, relating the closing velocity and separation velocity, respectively, to both the internal and external work, are applicable irrespective of the form used for the force-deflection response of either collision partner. The linear load-linear unload force-deflection model is implemented, secondarily, as a specific application of this modeling methodology.