Theoretical analysis of co-dimension-two grazing bifurcations in $$\varvec{n}$$ n -degree-of-freedom impact oscillator with symmetrical constrains

Abstract

The co-dimension-two grazing bifurcation in n-degree-of-freedom impact oscillator with bilateral constraints is investigated. Using a classical approach of discontinuity mappings, the existence conditions for co-dimension-two grazing bifurcation are obtained and simplified skillfully. For the impact oscillator has two discontinuity surfaces, the existence conditions become complex compared with the unilateral constraints and are discussed in four different cases. Furthermore, The deduced theoretical results are employed to identify and explore the distribution of co-dimension-two grazing bifurcation points for a two-degree-of-freedom impact oscillator with bilateral constraints as an example. The co-dimension-two grazing bifurcation points are presented, and the complex dynamic behaviors in the vicinity of the co-dimension-two grazing bifurcation point are displayed numerically by unfolding chart and phase diagrams.