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

Huong Le Thi (1, 2) , Stéphane Junca (1, 2) , Mathias Legrand (3)

1 COFFEE - COmplex Flows For Energy and Environment
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 Structural Dynamics and Vibration Laboratory [Montréal]

This contribution explores the free dynamics of a simple two-degree-of-freedom vibro-impact oscillator. One degree-of-freedom is limited by the presence of a rigid obstacle and periodic solutions involving one sticking phase per period (1-SPP) are targeted. A 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.

A few SPP orbits are animated below (click right -> view image for optimal quality).
Simplest SPP


SPP with distinct masses but same stiffnesses


SPP with same masses but distinct stiffnesses


SPP with initial negative gap: sticking phase of infinite duration

Mots clés: vibration analysis; sticking phase; periodic solutions; impact dynamics

Domaines: Systèmes dynamiques [math.DS]; Vibrations [physics.class-ph]; Mécanique des structures [physics.class-ph]

Huong Le Thi, Stéphane Junca, Mathias Legrand. Periodic solutions of a two-degree-of-freedom autonomous vibro-impact oscillator with sticking phases. Nonlinear Analysis: Hybrid Systems, 2018, 28, pp.54-74. ⟨10.1016/j.nahs.2017.10.009⟩. ⟨hal-01305719v2⟩