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

  • * Corresponding author: Stéphane Junca

    * Corresponding author: Stéphane Junca 
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  • 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.


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  • 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

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