ON CONVEX COMBINATIONS OF GRAZING LINEAR MODES FOR 2-DEGREE-OF-FREEDOM VIBRO-IMPACT SYSTEMS WITH 1:2 RESONANCE

DOI: 10.18173/2354-1059.2026-0001

Abstract

Grazing periodic orbits in vibro-impact mechanical systems are known to generate nonlinear normal modes. In the absence of internal resonance, many of the essential dynamical features already appear in two-degree-of-freedom (2-dof) mass–spring chains. In this note, we focus on such a system under the simplest internal resonance condition. Two grazing linear modes, denoted as GLOM1 and GLOM2, associated with the first and the second linear modes, play a key role.We will show that the grazing linear orbit generating one-impact-per-period (1IPP) solutions is neither GLOM1 nor GLOM2, but rather a convex combination of the two. This leads to the introduction of the set GLOc = [GLOM1, GLOM2] of grazing linear orbits. For all GLOc except the endpoint modes, the free-flight time to impact admits a particularly simple structure, providing a convenient framework for the future study of the nonlinear behavior of such grazing linear orbits.