An alternating frequency-time harmonic balance method for fast-slow dynamical systems

The Alternating Frequency-Time (AFT) Harmonic Balance method has been widely applied in the analysis of non-linear mechanical systems under periodic excitation. Customarily, a periodic displacement is considered as ansatz in a harmonic balance analysis. In the present work, a deviation from the latter ansatz is realized and the periodicity is assumed in the velocity, leading to a linear term in the displacement of the system. The latter approach aims to facilitate the analysis of a certain class of systems, which are characterized by a fast periodic motion and a slow non-periodic motion. The motivation of this study originates in the area of offshore engineering and more specifically in the topic of monopile installation. During vibratory pile installation, the pile is forced into the soil under the combined action of a periodic excitation at the pile top and the self-weight of the pile and the vibratory device. As a result, the pile simultaneously penetrates into the soil as a rigid body (slow motion) and vibrates in the driving frequency and its super-harmonics both as a rigid and a flexible body (fast motion). In this study, the AFT harmonic balance with the ansatz of periodic velocity is implemented in different problem cases. A set of non-linear mechanical systems are analysed, ranging from a single-degree-of-freedom to a continuum, to showcase the potential application of the method and to verify its accuracy.