Modal interactions in self-excited oscillators under external primary resonance

Abstract

Abstract The dynamics of two-degree-of-freedom oscillators including Rayleigh and Duffing type non-linearities is investigated. The method of multiple scales is first applied and a set of averaged equations is derived for cases of primary external resonance. These equations admit two types of constant solutions. The first type involves the directly excited mode only, while for some parameter combinations the non-linear damping terms excite the second mode also, even when no internal resonance is present. Stability and bifurcation analyses are then presented. Emphasis is placed on deriving explicit conditions on the system parameters that will lead to forms of the evolution equations which can be studied in detail, by utilizing results from the area of dynamical systems. To illustrate the effectiveness and accuracy of the analysis, numerical results are presented for a dynamic model of a specific practical system: namely, a three-parameter study is first carried over in order to reveal basic response features of a metal cutting system. Codimension one, two and three bifurcations are determined and their effect on the interaction and transition between the response modes is investigated. A representative sample of response diagrams and results from direct integration are also presented, providing an overall picture of the dynamics.