\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author: Stéphane Junca

    * Corresponding author: Stéphane Junca 
Abstract Full Text(HTML) Figure(13) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 34A38, 70K50, 70H14; Secondary: 70K75.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  A unilaterally constrained $ N $-degree-of-freedom chain with $ d > 0 $

    Figure 2.  Possible types of contacts in unilaterally constrained discrete dynamics

    Figure 3.  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[ $

    Figure 4.  Two branches on the right of the line $ y = y_0 $ when $ \gamma>0 $

    Figure 5.  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

    Figure 6.  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

    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 $

    Figure 8.  Neighborhood $ D_{\epsilon} $

    Figure 9.  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

    Figure 10.  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

    Figure 11.  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

    Figure 12.  Simple unilaterally constrained one-dof system

    Figure 13.  One-dof system orbit

  • [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.
    [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.
    [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.
    [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.
    [5] C. Corduneanu, Almost Periodic Functions, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers, 1968.
    [6] J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, 2007.
    [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.
    [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.
    [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. 
    [10] M. LegrandS. 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.
    [11] H. Le ThiS. 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.
    [12] J. MolenaarJ. 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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [19] A. ThorinP. 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.
    [20] A. ThorinP. Delezoide and M. Legrand, Periodic solutions of $n$-dof autonomous vibro-impact oscillators with one lasting contact phase, Nonlinear Dynamics, 90 (2017), 1771-1783. 
    [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.
    [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.
  • 加载中

Figures(13)

SHARE

Article Metrics

HTML views(502) PDF downloads(313) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return