ON CONVEX COMBINATIONS OF GRAZING LINEAR MODES FOR 2-DEGREE-OF-FREEDOM VIBRO-IMPACT SYSTEMS WITH 1:2 RESONANCE
DOI:
https://doi.org/10.18173/2354-1059.2026-0001Keywords:
unilateral contact, Grazing linear mode, free flight time, internal resonancesAbstract
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.
References
[1] Bernardo M di, Budd C, Champneys A & Kowalczyk P, 2008. Piecewise-smooth dynamical systems: Theory and applications. Applied Mathematical Sciences. Springer Science & Business Media.
[2] Nordmark A, (2001). Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators. Nonlinearity, 14(6), 1517-1542.
[3] Nordmark A, (1991). Non-periodic motion caused by grazing incidence in an impact oscillator. Journal of Sound and Vibration, 145(2), 279-297.
[4] Shaw SW & Holmes PJ, (1983). A periodically forced piecewise linear oscillator. Journal of Sound and Vibration, 90(1), 129-155.
[5] Bernardo M di, Budd CJ, Champneys AR, Kowalczyk P, Nordmark AB, Olivar Tost G & Piiroinen PT, (2008). Bifurcations in nonsmooth dynamical systems. SIAM Review, 50(4), 629-701.
[6] Shaw SW & Holmes P, (1983). Periodically forced linear oscillator with impacts: chaos and long-period motions. Physical Review Letters, 51(8), 623-626.
[7] Le TH, Junca S & Legrand M, (2023). First Return Time near grazing linear modes of discrete vibro-impact systems. Preprint.
[8] Le TH, Junca S & Legrand M, (2022). First return time to the contact hyperplane for N-degree-of-freedom vibro-impact systems. Discrete and Continuous Dynamical Systems – Series B, AIMS, 27, 1-44.
[9] Legrand M, Junca S & Heng S, (2017). Nonsmooth modal analysis of a N-degree-of-freedom system undergoing a purely elastic impact law. Communications in Nonlinear Science and Numerical Simulation, 45, 190-219.
[10] Thorin A, Delezoide P & Legrand M, (2017). Nonsmooth modal analysis of piecewise-linear impact oscillators. SIAM Journal on Applied Dynamical Systems, 16(3), 1710-1747.
[11] Thorin A & Legrand M, 2017. Spectrum of an impact oscillator via nonsmooth modal analysis. In: 9th European Nonlinear Dynamics Conference (ENOC). Budapest, Hungary.
[12] Thorin A, Legrand M & Junca S, 2015. Nonsmooth modal analysis: investigation of a 2-dof spring-mass system subject to an elastic impact law. In: Proceedings of the ASME IDETC & CIEC: 11th International Conference on Multibody Systems, Nonlinear Dynamics, and Control. Boston, Massachusetts.
[13] Shaw S & Pierre C, (1993). Normal modes for non-linear vibratory systems. Journal of Sound and Vibration, 164(1), 85-124.
[14] Yoong C & Legrand M. Nonsmooth modal analysis of a non-internally resonant finite bar subject to a unilateral contact constraint. In: Nonlinear structures and systems, Vol. 1. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham, 2020, 1-10.
[15] Le TH, Junca S & Legrand M, (2018). Periodic solutions of a two-degree-of-freedom autonomous vibro-impact oscillator with sticking phases. Nonlinear Analysis: Hybrid Systems, 28, 54-74.
[16] Thorin A & Legrand M, 2018. Nonsmooth modal analysis: from the discrete to the continuous settings. In: "Advanced topics in nonsmooth dynamics”. Springer, Cham, p. 191-234.
[17] Ballard P, (2000). The dynamics of discrete mechanical systems with perfect unilateral constraints. Archive for Rational Mechanics and Analysis, 154(3), 199–274.
