A Solution of the Alekseevski-Tate Penetration Equations

Abstract

Abstract : The Alekseevski-Tate equations have been used for five decades to predict the penetration, penetration velocity, rod velocity, and rod length of long-rod penetrators and similar projectiles. These nonlinear equations were originally solved numerically and more recently by the exact analytical solution of Walters and Segletes. However, due to the nonlinear nature of the equations, penetration was obtained implicitly as a function of time. The current report obtains the velocities, length, and penetration as an explicit function of time by employing a perturbation solution of the nondimensional Alekseevski-Tate equations. Explicit analytical solutions are advantageous in that they clearly reveal the interplay of the various parameters on the solution of the equations. Perturbation solutions of these equations were first undertaken by Forrestal et al., up to the first order, and good agreement with the exact solutions was shown for relatively short times. The current study obtains a third-order perturbation solution and includes both penetrator and target strength terms. This report compares the exact solution to the perturbation solution, and a typical comparison between the exact and approximate solution for a tungsten rod impacting a steel armor target is shown. Also, alternate ways are investigated to normalize the governing equations in order to obtain an optimum perturbation parameter. In most cases, the third-order perturbation solution shows near perfect agreement with the exact solutions of the Alekseevski-Tate equations. This report compares the exact solution to the perturbation solution, and comments are made regarding the range of validity of the explicit solution.