A polynomial extrapolation-based wavelet-Galerkin method for dynamic response reconstruction

Abstract

In this work we present a single scale wavelet-Galerkin method (WGM) for obtaining the linear dynamic response of single degree of freedom (SDOF) and multi-degree of freedom (MDOF) systems, the equation of motion is discretized using a polynomial extrapolation-based time stepping scheme. The aim of the study is to investigate the accuracy of the dynamic response resulting from the application of the proposed method and to explore the Daubechies wavelet types that can be used for unconditional stability. In our work we solve only for the scaling function coefficients of the displacement response, the scaling function coefficients of the velocity and the acceleration responses are obtained from the solved scaling function coefficients of the displacement function and the pertaining connection coefficients. The classical central differences method (CCDM) is used to generate consistent values of the velocity and the acceleration response to replace the distorted ones at the edges of the time interval. Stability analysis shows that the method using D6 wavelets is unconditionally stable for the practical range of the product of the time step and the frequency of vibration. An application of a SDOF system with initial conditions shows that for D6 wavelets the calculated responses at an adequate approximation level correspond well with those obtained using Newmark’s average acceleration method (AAM). The method is also applied to a MDOF system without performing a modal analysis, the resulting dynamic response is found to be accurate enough to be compared to that of the mode superposition method used by the STAAD pro program.