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

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