RESOLUTION OF THE TRANSIENT DYNAMIC PROBLEM WITH ARBITRARY LOADING USING THE ASYMPTOTIC METHOD

Abstract

Analysis of dynamic systems is more time consuming than of static ones due to the presence of inertia forces which vary in time. Equations of a dynamic system excited by arbitrary loads result in partial differential equations. The spatial part is discretized by the finite element method and the temporal part by implicit or explicit integration scheme. The time integration methods have already proved their effectiveness. However, in order to improve computing time for the resolution and quality of results, we present in this paper, a semi-analytical method based on an asymptotic method which allows to obtain a continuous solution for all time. In this method, the displacement field is expressed in power series. From this series, velocity and acceleration are easily computed. The load must be expressed also in series in the same manner as displacement. To do so, we use the Fourier integral to obtain an analytical function of an arbitrary load and then, we develop this function in power series using Taylor series. The dynamic asymptotic method (DAM) belongs to the conditionally stable-explicit methods. We apply this method in modal space in order to eliminate higher modes which influence the critical time (time segment length). Through numerical examples, we show better effectiveness of the asymptotic method compared to the Newmark method when we applied those schemes in the modal space.