Using symmetry to understand nonlinear modal interactions

Abstract

With the drive for more efficent slender structures, nonlinear dynamic phenomena are increasingly being observed in, and sometimes designed into, engineering systems. The objective of this thesis is to develop a better theoretical understanding of nonlinear systems that manifest internal resonance and provides practical insights into the exploitation of such nonlinear behaviours in engineering practice. To achieve this, an analytical approach is employed, that is based on nonlinear normal mode, or backbone curve, analysis.
The geometry, i.e. synchronicity and asynchronicity, of internal resonances is investigated using conceptually simple, two-mode systems. Special dynamic behaviours that emerge from internal resonance are studied, including isolated backbone curves and backbone solutions where the phases of modal coordinates vary. The underpinning mechanisms that govern their existence are analytically derived and demonstrated using relevant engineering systems. These geometric features are generalised to account for arbitrary types of internal resonances for two-mode interactions with arbitrary eigenfrequency ratios; an analytical technique is proposed for the efficient and robust determination of internal resonances.
The research scope is then extended to forced-damped scenarios. By employing an energy-based method, the relationships between backbone curves and forced periodic responses are established. A semi-analytical, energy balancing method is formulated by combining the energy balancing principle across multiple harmonics with quadrature criteria. With known NNM solutions, it allows for efficient prediction of forced responses with the applicability and accuracy estimated via harmonic phase-shifts.
Based on the concept of resonant capture, backbone curves are used to interpret damped transient responses with applications of Targeted Energy Transfer (TET). The required backbone curves for realising TET are identified from a symmetry-breaking perspective. Using these insights, an analytical method is presented for parameter selection of a nonlinear energy sink; the effectiveness of this is demonstrated via a beam system.

Date of Award	27 Sept 2022
Original language	English
Awarding Institution	University of Bristol
Supervisor	Tom L Hill (Supervisor) & Simon A Neild (Supervisor)