Convergence estimates for a series approximation of dynamic response of a perturbed system

Abstract

This paper presents a new method for efficiently computing the response of a perturbed dynamic system. The method is based on predicting the convergence of a Neumann series expansion that approximates the displacement response of a perturbed system. The Neumann series is often faster than a direct solve, but only converges when the spectral radius of the system is less than unity. However, direct computation of the spectral radius is often expensive. The proposed method avoids this expense by predicting convergence using a ratio of adjacent terms early in the evaluation of the series. Stochastic simulations are presented for determining the convergence bound on this ratio as a function of term index. These simulations demonstrate that the resulting convergence predictions are very accurate after only 5–20 terms, and that the proposed method is significantly faster than the computation of the spectral radius. The proposed method is also inherently faster than other published methods, as it requires fewer terms in the series to assess convergence. Numerical examples involving the steady-state vibrations of an elastic bar and an elastic plate illustrate the accuracy and efficiency of the method.