Convergence rates of harmonic balance method for periodic solution of smooth and non-smooth systems

Abstract

This paper presents a systematic and rigorous analysis on the convergence rates of the Harmonic balance Method (HB) for general smooth and non-smooth systems. In doing so, the convergence rates of Fourier truncation are established at first for functions with different smoothness, and then, the errors of HB are estimated with the help of a coercive condition and the established results on Fourier truncation errors. As is found in this work, when the restoring forces are discontinuous or of some special low smoothness, the convergence rates of HB become different from those of Fourier truncation. Numerical examples are studied and the results well verify the present theoretic convergence rates.