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First Return Time to the contact hyperplane for $N$-degree-of-freedom vibro-impact systems

 1 Department of Mathematics and Informatics, Thang Long University, Hanoi, Vietnam 2 LJAD Mathematics, Inria & CNRS, Université Côte d'Azur, Nice, France 3 Department of Mechanical Engineering, McGill University, Montréal, Québec, Canada

* Corresponding author: Stéphane Junca

Received  May 2020 Revised  August 2020 Early access  February 2021

The paper deals with the dynamics of conservative $N$-degree-of-freedom vibro-impact systems involving one unilateral contact condition and a linear free flow. Among all possible trajectories, grazing orbits exhibit a contact occurrence with vanishing incoming velocity which generates mathematical difficulties. Such problems are commonly tackled through the definition of a Poincaré section and the attendant First Return Map. It is known that the First Return Time to the Poincaré section features a square-root singularity near grazing. In this work, a non-orthodox yet natural and intrinsic Poincaré section is chosen to revisit the square-root singularity. It is based on the unilateral condition but is not transverse to the grazing orbits. A detailed investigation of the proposed Poincaré section is provided. Higher-order singularities in the First Return Time are exhibited. Also, activation coefficients of the square-root singularity for the First Return Map are defined. For the linear and periodic grazing orbits from which bifurcate nonlinear modes, one of these coefficients is necessarily non-vanishing. The present work is a step towards the stability analysis of grazing orbits, which still stands as an open problem.

Citation: Huong Le Thi, Stéphane Junca, Mathias Legrand. First Return Time to the contact hyperplane for $N$-degree-of-freedom vibro-impact systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021031
References:
 [1] P. Ballard, The dynamics of discrete mechanical systems with perfect unilateral constraints, Archive for Rational Mechanics and Analysis, 154 (2000), 199-274.  doi: 10.1007/s002050000105.  Google Scholar [2] M. di Bernardo, C. Budd, A. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences, Springer Science & Business Media, London, 2008. Google Scholar [3] C. Budd and F. Dux, Intermittency in impact oscillators close to resonance, Nonlinearity, 7 (1994), 1191-1224.  doi: 10.1088/0951-7715/7/4/007.  Google Scholar [4] D. Chillingworth, Dynamics of an impact oscillator near a degenerate graze, Nonlinearity, 23 (2010), 2723-2748.  doi: 10.1088/0951-7715/23/11/001.  Google Scholar [5] C. Corduneanu, Almost Periodic Functions, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers, 1968. Google Scholar [6] J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, 2007. Google Scholar [7] M. Fredriksson and A. Nordmark, Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators, Proceedings of the Royal Society, 453 (1997), 1261-1276.  doi: 10.1098/rspa.1997.0069.  Google Scholar [8] S. Junca, H. Le Thi, M. Legrand and A. Thorin, Impact dynamics near unilaterally constrained grazing orbits, 9th European Nonlinear Dynamics Conference (ENOC), Budapest, Hungary, (2017), hal-01562154. Google Scholar [9] A.-N. Krylov, On the numerical solution of the equation by which, in technical questions, frequencies of small oscillations of material systems are determined, Izvestija AN SSSR (News of Academy of Sciences of the USSR), Otdel. Mat. I Estest. Nauk, VII (1931), 491-539.   Google Scholar [10] M. Legrand, S. Junca and S. Heng, Nonsmooth modal analysis of a N-degree-of-freedom system undergoing a purely elastic impact law, Commun. Nonlinear Sci. Numer. Simul., 45 (2017), 190-219.  doi: 10.1016/j.cnsns.2016.08.022.  Google Scholar [11] H. Le Thi, S. Junca and M. Legrand, Periodic solutions of a two-degree-of-freedom autonomous vibro-impact oscillator with sticking phases, Nonlinear Anal. Hybrid Syst., 28 (2018), 54-74.  doi: 10.1016/j.nahs.2017.10.009.  Google Scholar [12] J. Molenaar, J. G. de Weger and W. van de Water, Mappings of grazing-impact oscillators, Nonlinearity, 14 (2001), 301-321.  doi: 10.1088/0951-7715/14/2/307.  Google Scholar [13] A. Nordmark, Non-periodic motion caused by grazing incidence in an impact oscillator, Journal of Sound and Vibration, 145 (1991), 279-297.  doi: 10.1016/0022-460X(91)90592-8.  Google Scholar [14] A. Nordmark, Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators, Nonlinearity, 14 (2001), 1517-1542.  doi: 10.1088/0951-7715/14/6/306.  Google Scholar [15] M. Schatzmann, A class of nonlinear differential equations of second order in time, Nonlinear Analysis: Theory, Methods & Applications, 2 (1978), 355-373.  doi: 10.1016/0362-546X(78)90022-6.  Google Scholar [16] J. Sotomayor and M. A. Teixeira, Vector fields near the boundary of a 3-manifold, Dynamical systems Valparaiso 1986, Lecture Notes in Mathematics, 1331 (1988), 169–195. Google Scholar [17] A. Thorin, M. Legrand and S. Junca, Nonsmooth modal analysis: Investigation of a 2-dof spring-mass system subject to an elastic impact law, Proceedings of the ASME IDETC & CIEC: 11th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Boston, Massachusetts, 2015, hal-01185973. Google Scholar [18] A. Thorin and M. Legrand, Spectrum of an impact oscillator via nonsmooth modal analysis, 9th European Nonlinear Dynamics Conference (ENOC), Budapest, Hungary, 2017, hal-01509382. Google Scholar [19] A. Thorin, P. Delezoide and M. Legrand, Nonsmooth modal analysis of piecewise-linear impact oscillators, SIAM Journal on Applied Dynamical Systems, 16 (2017), 1710-1747.  doi: 10.1137/16M1081506.  Google Scholar [20] A. Thorin, P. Delezoide and M. Legrand, Periodic solutions of $n$-dof autonomous vibro-impact oscillators with one lasting contact phase, Nonlinear Dynamics, 90 (2017), 1771-1783.   Google Scholar [21] A. Thorin and M. Legrand, Nonsmooth modal analysis: From the discrete to the continuous settings, Advanced topics in nonsmooth dynamics, Springer, Cham, 2018,191–234. Google Scholar [22] P. Thota, Analytical and Computational Tools for the Study of Grazing Bifurcations of Periodic Orbits and Invariant Tori, Ph.D thesis, Engineering Sciences [physics], Virginia Polytechnic, Institute and State University, 2007, tel-01330429. Google Scholar

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References:
 [1] P. Ballard, The dynamics of discrete mechanical systems with perfect unilateral constraints, Archive for Rational Mechanics and Analysis, 154 (2000), 199-274.  doi: 10.1007/s002050000105.  Google Scholar [2] M. di Bernardo, C. Budd, A. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences, Springer Science & Business Media, London, 2008. Google Scholar [3] C. Budd and F. Dux, Intermittency in impact oscillators close to resonance, Nonlinearity, 7 (1994), 1191-1224.  doi: 10.1088/0951-7715/7/4/007.  Google Scholar [4] D. Chillingworth, Dynamics of an impact oscillator near a degenerate graze, Nonlinearity, 23 (2010), 2723-2748.  doi: 10.1088/0951-7715/23/11/001.  Google Scholar [5] C. Corduneanu, Almost Periodic Functions, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers, 1968. Google Scholar [6] J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, 2007. Google Scholar [7] M. Fredriksson and A. Nordmark, Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators, Proceedings of the Royal Society, 453 (1997), 1261-1276.  doi: 10.1098/rspa.1997.0069.  Google Scholar [8] S. Junca, H. Le Thi, M. Legrand and A. Thorin, Impact dynamics near unilaterally constrained grazing orbits, 9th European Nonlinear Dynamics Conference (ENOC), Budapest, Hungary, (2017), hal-01562154. Google Scholar [9] A.-N. Krylov, On the numerical solution of the equation by which, in technical questions, frequencies of small oscillations of material systems are determined, Izvestija AN SSSR (News of Academy of Sciences of the USSR), Otdel. Mat. I Estest. Nauk, VII (1931), 491-539.   Google Scholar [10] M. Legrand, S. Junca and S. Heng, Nonsmooth modal analysis of a N-degree-of-freedom system undergoing a purely elastic impact law, Commun. Nonlinear Sci. Numer. Simul., 45 (2017), 190-219.  doi: 10.1016/j.cnsns.2016.08.022.  Google Scholar [11] H. Le Thi, S. Junca and M. Legrand, Periodic solutions of a two-degree-of-freedom autonomous vibro-impact oscillator with sticking phases, Nonlinear Anal. Hybrid Syst., 28 (2018), 54-74.  doi: 10.1016/j.nahs.2017.10.009.  Google Scholar [12] J. Molenaar, J. G. de Weger and W. van de Water, Mappings of grazing-impact oscillators, Nonlinearity, 14 (2001), 301-321.  doi: 10.1088/0951-7715/14/2/307.  Google Scholar [13] A. Nordmark, Non-periodic motion caused by grazing incidence in an impact oscillator, Journal of Sound and Vibration, 145 (1991), 279-297.  doi: 10.1016/0022-460X(91)90592-8.  Google Scholar [14] A. Nordmark, Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators, Nonlinearity, 14 (2001), 1517-1542.  doi: 10.1088/0951-7715/14/6/306.  Google Scholar [15] M. Schatzmann, A class of nonlinear differential equations of second order in time, Nonlinear Analysis: Theory, Methods & Applications, 2 (1978), 355-373.  doi: 10.1016/0362-546X(78)90022-6.  Google Scholar [16] J. Sotomayor and M. A. Teixeira, Vector fields near the boundary of a 3-manifold, Dynamical systems Valparaiso 1986, Lecture Notes in Mathematics, 1331 (1988), 169–195. Google Scholar [17] A. Thorin, M. Legrand and S. Junca, Nonsmooth modal analysis: Investigation of a 2-dof spring-mass system subject to an elastic impact law, Proceedings of the ASME IDETC & CIEC: 11th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Boston, Massachusetts, 2015, hal-01185973. Google Scholar [18] A. Thorin and M. Legrand, Spectrum of an impact oscillator via nonsmooth modal analysis, 9th European Nonlinear Dynamics Conference (ENOC), Budapest, Hungary, 2017, hal-01509382. Google Scholar [19] A. Thorin, P. Delezoide and M. Legrand, Nonsmooth modal analysis of piecewise-linear impact oscillators, SIAM Journal on Applied Dynamical Systems, 16 (2017), 1710-1747.  doi: 10.1137/16M1081506.  Google Scholar [20] A. Thorin, P. Delezoide and M. Legrand, Periodic solutions of $n$-dof autonomous vibro-impact oscillators with one lasting contact phase, Nonlinear Dynamics, 90 (2017), 1771-1783.   Google Scholar [21] A. Thorin and M. Legrand, Nonsmooth modal analysis: From the discrete to the continuous settings, Advanced topics in nonsmooth dynamics, Springer, Cham, 2018,191–234. Google Scholar [22] P. Thota, Analytical and Computational Tools for the Study of Grazing Bifurcations of Periodic Orbits and Invariant Tori, Ph.D thesis, Engineering Sciences [physics], Virginia Polytechnic, Institute and State University, 2007, tel-01330429. Google Scholar
A unilaterally constrained $N$-degree-of-freedom chain with $d > 0$
Possible types of contacts in unilaterally constrained discrete dynamics
The $3D$ Poincaré section for the 2 dof case on $\{u_2 = d\}$. It consists in the upper part $\{ \dot{u}_2 >0\}$ and the part $\{\dot{u}_2 = 0\}$ minus the interval $]a,b[$
Two branches on the right of the line $y = y_0$ when $\gamma>0$
Square-root singularity in dimension $3$. Two graphs of $x(y,{\bf{z}})$ are shown in red and blue. The square-root singularity arises along the curve $y = \alpha({\bf{z}})$ in purple
Power-root singularity (1/2, 1/3 and 1/4) in the plane $(u_1(T_0),\dot{u}_1(T_0))$ which is isomorphic to the set $\smash{\mathcal{H}^0}$ since $u_2(T_0) = d$ and $\dot{u}_2(T_0) = 0$. The blue area $u_1< d$ corresponds to grazing. The red branch $u_1 = d$ and $\dot{u}_1>0$ corresponds to the beginning of a sticking phase. The green area $u_1> d$ corresponds to states within a sticking phase. The solid green line $u_1 = d$ and $\dot{u}_1<0$ corresponds to the end of a sticking phase. The dark red dot correspond to a unique orbit the worst power-root singularity
]). A cube-root singularity appears near $v_1(0) = \dot{u}_1(0) = 5.86$">Figure 7.  First Return Time $T$ with respect to $v_1 = \dot{u}_1$ (near a periodic solution with one sticking phase per period [11]). A cube-root singularity appears near $v_1(0) = \dot{u}_1(0) = 5.86$
Neighborhood $D_{\epsilon}$
Instability of the fixed-point $(0,0)$ for the map (104). (a) $a = 1$, $c = 2$, $b = d = 0$: the recurrence goes away from $(0,0)$ along the line $y = cx/a$; (b) $c = \alpha a$, $d = \alpha b$: the instability does not realize if the starting point sits on the curve $\mathcal{C}: y = -a\sqrt{|x|}/b$ since the next iterate is $(0,0)$. (c) $b = c = 1$, $a = d = 0$: iterates leave $(0,0)$. The gradient color scale shows initial iterates in blue to final iterates in red irrespective of the magnitude
Instability of the fixed-point $(0,0)$ for the map (104). (a) when $0<d = 0.5<1$ and $c = 1$; (b) when $-1<d = -0.5<0$ and $c = 1$; (c) when $d = 0$ and $c = 1$. The gradient color scale shows initial iterates in blue to final iterates in red irrespective of the magnitude
First Return Time (red lines) with respect to the initial displacement of the first mass: (a) near the first GLM, (b) near the second GLM
Simple unilaterally constrained one-dof system
One-dof system orbit
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