A technique for overcoming load discontinuity in using Newmark method

Abstract

Although a closed form solution can be obtained for the shock response to an impulse by using the Duhamel integral method for a single degree-of-freedom system, it is analytically verified herein that an extra amplitude distortion will be introduced at the end of the impulse by using the Newmark method. In fact, the relative amplitude error in the displacement response due to the discontinuity at the end of the impulse can be reliably estimated by the ratio of the area of extra impulse over the area of input impulse, where the area of extra impulse is equal to a half of the product of the time step and load discontinuity. It seems that the momentum equation of motion can be used to easily overcome the discontinuity problem at the end of an impulse. This is because the external force is replaced by its corresponding external momentum, which is obtained from the time integration of external force, and thus the discontinuity problem automatically disappears by using the momentum equation of motion. Numerical studies reveal that this technique is applicable to nonlinear systems. Meanwhile, it is also numerically illustrated that the discontinuity problem can be overcome by performing a very small time step immediately upon termination of the applied impulse.