Periodic solutions of a two-degree-of-freedom autonomous vibro-impact oscillator with sticking phases

This contribution explores the free dynamics of a simple two-degree-of-freedom vibro-impact oscillator. One degree-of-freedom is unilaterally constrained by the presence of a rigid obstacle and periodic solutions involving one sticking phase per period (1-SPP) are targeted. A solution method to obtain such orbits is proposed: it provides conditions on the existence of 1-SPP as well as closed-form solutions. It is shown that 1-SPP might not exist for a given combination of masses and stiffnesses. The set of 1-SPP is at most a countable set of isolated periodic orbits. The construction of 1-SPP requires numerical developments that are illustrated on a few relevant examples. Comparison with nonlinear modes of vibration involving one impact per period (1-IPP) is also considered. Interestingly, an equivalence between 1-SPP and a special set of isolated 1-IPP is established. It is also demonstrated that the prestressed system features sticking phases of infinite duration.