Super- and sub-harmonic response calculations for a torsional system with clearance nonlinearity using the harmonic balance method

This article presents a new multi-term harmonic balance method (HBM) for nonlinear frequency response calculations of a torsional sub-system containing a clearance type nonlinearity. The ability of the simplified subsystem to capture the salient behavior of the larger system is verified by the comparison of results to experimental data. Unlike previous analytical and numerical methods, the proposed HBM includes adaptive arc-length continuation and stability calculation capabilities to find periodic solutions in multi-valued nonlinear frequency response regimes as well as to improve convergence. Essential steps of the proposed HBM calculations are introduced, and it is validated by comparing time and frequency domain predictions with those yielded by numerical solutions, experimental studies, or analog simulations for several examples. Then, nonlinear frequency response characteristics of an oscillator with clearance nonlinearity are examined with focus on super- and sub-harmonics. We also explore some issues that are not fully resolved in the literature. For instance, the effect of mean operating point is examined for α = 0 and α = 0.18 - 0.25 cases where α is the stiffness ratio of the piecewise-linear elastic function. In addition, the number of harmonic terms that must be included in the HBM response calculations, given sinusoidal excitation, has been investigated. Finally, some simple analytical predictions for super and sub-harmonic resonances are presented.