A robust and efficient stability analysis of periodic solutions based on harmonic balance method and Floquet-Hill formulation

Abstract

Harmonic balance method (HBM) incorporated with arc-length continuation and Floquet-Hill formulation now becomes an essential and strong tool for computing periodic solution branches and performing stability analysis, which can provide great insight into the responses of nonlinear dynamic systems. However, due to the redundancy of eigenvalues of Hill’s matrix over the actual Floquet exponents of periodic solutions and the distortion on the redundancy relation of the eigenvalues induced by order truncation of harmonics, it is not routine work to correctly select the actual Floquet exponents from the eigenvalues of Hill’s matrix in order to achieve correct stability and bifurcation analysis, especially the order of harmonics in HBM is low. In this work, a new robust strategy for Floquet exponents filtering is introduced, which is chosen based on the character of the real part (FEF-RP) of the Hill’s eigenvalues. Furthermore, an adaptive arc-length control for the arc-length continuation (ALC) is proposed, which is continuously governed by the minimum parameterized eigenvalue of the Jacobian matrix (ALC-MPE) and can provide smooth and efficient continuation of the turning points of periodic solutions. The study on a 4DOFs rotor–stator rubbing system and a 3DOFs damped Duffing oscillator system shows the feasibility of the proposed methods for arc-length control (ALC-MPE) and Floquet exponents filtering (FEF-RP).