The accuracy of the generalized-α method in the time integration of non-linear single- and two-DOF forced systems

Abstract

This paper deals with the accuracy of a numerical time integration scheme, the implicit Generalized-α method, when applied near resonant conditions in periodic steady-state vibrations of elastic linear and non-linear systems. In order to evaluate errors, analytical solutions of frequency response functions (FRFs) are determined by using the Harmonic Balance method in single- and two-degrees-of-freedom viscously damped systems, where the non-linearity is introduced through hardening Duffing oscillators. Successively, the Generalized-α method is implemented in conjunction with the Harmonic Balance method to trace numerical solutions of FRFs. It is shown that the effective resulting algorithm, the Algorithmic Harmonic Balance-ρ ∞ method, can define non-linear FRFs and allows the errors exhibited by the integration scheme near resonance in terms of frequency location and amplitude of the resonant peak to be quantified. The accuracy estimates demonstrate the robustness of the Generalized-α method also in the forced case and confirm its capability to reproduce amplitudes at resonance in the low frequency range and damp out them in the high frequency range.