Dynamics of pseudo-rigid ball impact on rigid foundation

This paper concerns the dynamics induced by the ideally elastic normal impact of a linearly elastic pseudo-rigid sphere on a rigid, stationary foundation. An impact map is derived and studied by numerical and analytical means. Periodic, quasi-periodic, and chaotic response is observed consistently with the symplectic nature of the map.

[1]  Robert G. Muncaster,et al.  Invariant manifolds in mechanics II: Zero-dimensional elastic bodies with directors , 1984 .

[2]  D. Saari,et al.  Stable and Random Motions in Dynamical Systems , 1975 .

[3]  D. Steigmann On pseudo-rigid bodies , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[4]  Invariant manifolds in mechanics I: The general construction of coarse theories from fine theories , 1984 .

[5]  Impact of an elastic pseudo-rigid body on a rigid foundation , 2000 .

[6]  Enrico Fermi,et al.  On the Origin of the Cosmic Radiation , 1949 .

[7]  John M. Greene,et al.  A method for determining a stochastic transition , 1979, Hamiltonian Dynamical Systems.

[8]  A. Lichtenberg,et al.  Regular and Chaotic Dynamics , 1992 .

[9]  B. Chirikov A universal instability of many-dimensional oscillator systems , 1979 .

[10]  Robert Dewar,et al.  Non-linear dynamics , 2000 .

[11]  Robert G. Muncaster,et al.  The theory of pseudo-rigid bodies , 1988 .

[12]  Vladimir Igorevich Arnolʹd,et al.  Problèmes ergodiques de la mécanique classique , 1967 .

[13]  Ralph Abraham,et al.  Foundations Of Mechanics , 2019 .

[14]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory , 1980 .

[15]  C. Oliveira,et al.  Bifurcations and chaos for the quasiperiodic bouncing ball , 1997 .

[16]  Jerrold E. Marsden,et al.  Nonsmooth Lagrangian Mechanics and Variational Collision Integrators , 2003, SIAM J. Appl. Dyn. Syst..

[17]  A. Kolmogorov,et al.  Preservation of conditionally periodic movements with small change in the Hamilton function , 1979 .

[18]  N. MacDonald Nonlinear dynamics , 1980, Nature.

[19]  Nicholas B. Tufillaro,et al.  Experimental approach to nonlinear dynamics and chaos , 1992, Studies in nonlinearity.