The focus of the subject study was on the development of the closed-form solutions for displacement, velocity and acceleration, based upon the utilization of the Laplace transform, experienced by each collision partner involved in a collinear collision under the constraints of linearity in the force-deflection response during closure and separation and while subject to any number of net externally applied loads for which an analytic Laplace transform was determinable. Starting from the basic expression of Newton’s Second Law, the coupled equations of motion were developed in standard matrix-vector form by the introduction of the definitions of structural deflections in terms of the displacements of the center of mass of each collision partners and the massless common collision interface. The solution for the equations of motion was determined by applying the Laplace Transform and determining the solution for the dynamic stiffness matrix and transfer function by means of an Eigendecomposition. The closed-form analytic for the Laplace domain displacement was readily amenable to the inverse Laplace Transform and thereby provided a closed-form analytic solution for displacement in the time domain. The first and second time derivatives of this solution provided the closed form solution for velocity and acceleration, respectively. The reduced forms of each of these three equations, addressing the specific limits of the achievement of common velocity at the terminus of closure and the achievement of zero acceleration due to the collision force at the terminus of separation, were then developed.

The use of the residue theorem, instead of a partial fraction expansion, for the evaluation of the transfer function, coupled with the reduction in the complexity of the general problem secondary to both the complex conjugate nature of the solutions for the Eigenvalue problem and the reduction of the same to only complex roots for the solution to the characteristic polynomial of the dynamic stiffness matrix for the undamped problem, revealed a solution set comprised of a rigid body mode and a solution based upon the circular frequency of the effective system mass and stiffness. Depending on the nature of the net externally applied forces, the solution for the time of terminus of closure could be determined from basic trigonometric relationships or from equating the velocities of the collision partners in the Laplace domain, solving for the Laplace variable and then performing an inverse Laplace transform to obtain the solution in the time domain. The terminus of the separation could be solved for in a similar manner through the use of the acceleration of either collision partner.