Simple Model of Bouncing Ball Dynamics
- 338 Downloads
Nonlinear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered and the Poincaré map, describing evolution from an impact to the next impact, is described. Displacement of the limiter is assumed as periodic, cubic function of time. Due to simplicity of this function analytical computations are possible. Several dynamical modes, such as fixed points, two cycles and chaotic bands are studied analytically and numerically. It is shown that chaotic bands are created from fixed points after first period doubling in a corner-type bifurcation. Equation for the time of the next impact is solved exactly for the case of two subsequent impacts occurring in the same period of limiter’s motion making analysis of chattering possible.
KeywordsBouncing ball Simple models Exact solutions
Unable to display preview. Download preview PDF.
- 1.di Bernardo M., Budd C.J., Champneys A.R., Kowalczyk P.: Piecewise-Smooth Dynamical Systems. Theory and Applications. Series: Applied Mathematical Sciences. Springer, Berlin (2008)Google Scholar
- 2.Luo A.C.J.: Singularity and Dynamics on Discontinuous Vector Fields. Monograph Series on Nonlinear Science and Complexity. Elsevier, Amsterdam (2006)Google Scholar
- 3.Awrejcewicz J., Lamarque C.-H.: Bifurcation and Chaos in Nonsmooth Mechanical Systems. World Scientific Series on Nonlinear Science: Series A. World Scientific Publishing, Singapore City (2003)Google Scholar
- 6.Mehta, A. (ed.): Granular Matter: An Interdisciplinary Approach. Springer, Berlin (1994)Google Scholar
- 8.Wiercigroch M., Krivtsov A.M., Wojewoda J.: Vibrational energy transfer via modulated impacts for percussive drilling. J. Theor. Appl. Mech. 46, 715–726 (2008)Google Scholar
- 10.Luo, A.C.J., Guo, Y.: Motion switching and chaos of a particle in a generalized Fermi-acceleration oscillator. Math. Probl. Eng. 2009, Article ID 298906 (2009). doi: 10.1155/2009/298906
- 14.Okniński, A., Radziszewski, B.: Chaotic dynamics in a simple bouncing ball model. Acta Mech. Sin. 27, 130–134 (2011). arXiv:1002.2448 [nlin.CD] (2010)Google Scholar
- 16.Okniński, A., Radziszewski, B.: Simple models of bouncing ball dynamics and their comparison. arXiv:1002.2448 [nlin.CD] (2010)Google Scholar
- 17.Okniński, A., Radziszewski, B.: Grazing dynamics and dependence on initial conditions in certain systems with impacts. arXiv:0706.0257 [nlin.CD] (2007)Google Scholar