APPLICATION OF AN ASYMPTOTIC METHOD TO TRANSIENT DYNAMIC PROBLEMS

Abstract

A new method to solve linear dynamics problems using an asymptotic method is presented. Asymptotic methods have been efficiently used for many decades to solve non-linear quasistatic structural problems. Generally, structural dynamics problems are solved using finite elements for the discretization of the space domain of the differential equations, and explicit or implicit schemes for the time domain. With the asymptotic method, time schemes are not necessary to solve the discretized (space) equations. Using the analytical solution of a single degree of freedom (DOF) problem, it is demonstrated, that the Dynamic Asymptotic Method (DAM) converges to the exact solution when an infinite series expansion is used. The stability of the method has been studied. DAM is conditionally stable for a finite series expansion and unconditionally stable for an infinite series expansion. This method is similar to the analytical method of undetermined coefficients or to power series method being used to solve ordinary differential equations. For a multi-degree-of-freedom (MDOF) problem with a lumped mass matrix, no factorization or explicit inversion of global matrices is necessary. It is shown that this conditionally stable method is more efficient than other conditionally stable explicit central difference integration techniques. The solution is continuous irrespective of the time segment (step) and the derivatives are continuous up to orderN-1 whereNis the order of the series expansion.