An investigation into the interaction between a nonlinear resonance and a fixed anti-resonance in a harmonically excited system

Abstract

The study of linear mechanical structures with discrete nonlinear attachments is a widespread field in structural dynamics as it is important in many areas of engineering. A common type of nonlinearity is a hardening spring, which can be described by a cubic polynomial representing the force-deflection relationship. The frequency response of a system consisting of a hardening spring and a mass is characterized by an amplitude-dependent resonance frequency, that increases as the displacement of the mass increases. The way in which the resonance of a nonlinear attachment interacts with the resonance of a linear host structure has already been discussed in the literature, however little attention has been paid to the interaction of the nonlinear resonance with a fixed anti-resonance of the host structure. This paper aims to fill this gap by addressing the physical aspects of the interaction between an anti-resonance and a resonance due to a nonlinear resonant attachment. A two-degrees-of-freedom system representing a general structure with two modes of vibration is used as a host structure to which a nonlinear resonant device is attached. The host structure has a fixed anti-resonance prior to the nonlinear device being attached to the structure, and the influence of this device on the anti-resonance is investigated. This is achieved by using the harmonic balance method and a numerical continuation algorithm. In the study, it was found that the anti-resonance interacts with the nonlinear resonance, where some of the peaks bend toward higher frequencies because of the hardening stiffness. Further increasing the nonlinearity, results in an isola being formed inside the main frequency response curve. When the motion is extremely high, quasi-periodic or chaotic-like responses are observed in specific frequency regions.