Stability and bifurcation analysis of super- and sub-harmonic responses in a torsional system with piecewise-type nonlinearities

The nonlinear dynamic behaviors induced by piecewise-type nonlinearities generally reflect super- and sub-harmonic responses. Various inferences can be drawn from the stability conditions observed in nonlinear dynamic behaviors, especially when they are projected in physical motions. This study aimed to investigate nonlinear dynamic characteristics with respect to variational stability conditions. To this end, the harmonic balance method was first implemented by employing Hill’s method, and the time histories under stable and unstable conditions were examined using a numerical simulation. Second, the super- and sub-harmonic responses were investigated according to frequency upsweeping based on the arc-length continuation method. While the stability conditions vary along the arc length, the bifurcation phenomena also show various characteristics depending on their stable or unstable status. Thus, the study findings indicate that, to determine the various stability conditions along the direction of the arc length, it is fairly reasonable to determine nonlinear dynamic behaviors such as period-doubling, period-doubling cascade, and quasi-periodic (or chaotic) responses. Overall, this study suggests analytical and numerical methods to understand the super- and sub-harmonic responses by comparing the arc length of solutions with the variational stability conditions.