American Institute of Mathematical Sciences

Advanced
July  2008, 10(1): 43-72. doi: 10.3934/dcdsb.2008.10.43

On the stopping time of a bouncing ball

Anna Maria Cherubini 1, , Giorgio Metafune 1, and  Francesco Paparella 2,

1. 

Dipartimento di Matematica E. De Giorgi, Università del Salento, 73100, Lecce, Italy, Italy
2. 

Dipartimento di Matematica e Fisica "Ennio De Giorgi", Università del Salento, 73100, Lecce, Italy

Received  June 2007 Revised  January 2008 Published  April 2008

We study a simple model of a bouncing ball that takes explicitely into account the elastic deformability of the body and the energy dissipation due to internal friction. We show that this model is not subject to the problem of inelastic collapse, that is, it does not allow an infinite number of impacts in a finite time. We compute asymptotic expressions for the time of flight and for the impact velocity. We also prove that contacts with zero velocity of the lower end of the ball are possible, but non-generic. Finally, we compare our findings with other models and laboratory experiments.
Keywords: granular gases, impact oscillators., inelastic collapse, Dynamical systems.
Mathematics Subject Classification: Primary: 37; Secondary: 76T25, 74M2.
Citation: Anna Maria Cherubini, Giorgio Metafune, Francesco Paparella. On the stopping time of a bouncing ball. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 43-72. doi: 10.3934/dcdsb.2008.10.43
[1]

Ming Gao, Jonathan J. Wylie, Qiang Zhang. Inelastic Collapse in a Corner. Communications on Pure & Applied Analysis, 2009, 8 (1) : 275-293. doi: 10.3934/cpaa.2009.8.275
[2]

Gilberto M. Kremer, Wilson Marques Jr.. Fourteen moment theory for granular gases. Kinetic & Related Models, 2011, 4 (1) : 317-331. doi: 10.3934/krm.2011.4.317
[3]

Zhongyi Huang, Peter A. Markowich, Christof Sparber. Numerical simulation of trapped dipolar quantum gases: Collapse studies and vortex dynamics. Kinetic & Related Models, 2010, 3 (1) : 181-194. doi: 10.3934/krm.2010.3.181
[4]

Fei Meng, Xiao-Ping Yang. Elastic limit and vanishing external force for granular systems. Kinetic & Related Models, 2019, 12 (1) : 159-176. doi: 10.3934/krm.2019007
[5]

Daxiong Piao, Xiang Sun. Boundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials. Communications on Pure & Applied Analysis, 2014, 13 (2) : 645-655. doi: 10.3934/cpaa.2014.13.645
[6]

Chao Wang, Dingbian Qian, Qihuai Liu. Impact oscillators of Hill's type with indefinite weight: Periodic and chaotic dynamics. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2305-2328. doi: 10.3934/dcds.2016.36.2305
[7]

Flaviano Battelli, Michal Fe?kan. Chaos in forced impact systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 861-890. doi: 10.3934/dcdss.2013.6.861
[8]

Da-Peng Li. Phase transition of oscillators and travelling waves in a class of relaxation systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2601-2614. doi: 10.3934/dcdsb.2016063
[9]

Ming He, Xiaoyun Ma, Weijiang Zhang. Oscillation death in systems of oscillators with transferable coupling and time-delay. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 737-745. doi: 10.3934/dcds.2001.7.737
[10]

Wolf-Jürgen Beyn, Thorsten Hüls. Continuation and collapse of homoclinic tangles. Journal of Computational Dynamics, 2014, 1 (1) : 71-109. doi: 10.3934/jcd.2014.1.71
[11]

El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449
[12]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355
[13]

Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447
[14]

John Erik Fornæss. Sustainable dynamical systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1361-1386. doi: 10.3934/dcds.2003.9.1361
[15]

Vieri Benci, C. Bonanno, Stefano Galatolo, G. Menconi, M. Virgilio. Dynamical systems and computable information. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 935-960. doi: 10.3934/dcdsb.2004.4.935
[16]

Josiney A. Souza, Tiago A. Pacifico, Hélio V. M. Tozatti. A note on parallelizable dynamical systems. Electronic Research Announcements, 2017, 24: 64-67. doi: 10.3934/era.2017.24.007
[17]

Mădălina Roxana Buneci. Morphisms of discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 91-107. doi: 10.3934/dcds.2011.29.91
[18]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1
[19]

Tobias Wichtrey. Harmonic limits of dynamical systems. Conference Publications, 2011, 2011 (Special) : 1432-1439. doi: 10.3934/proc.2011.2011.1432
[20]

Panayotis G. Kevrekidis, Vakhtang Putkaradze, Zoi Rapti. Non-holonomic constraints and their impact on discretizations of Klein-Gordon lattice dynamical models. Conference Publications, 2015, 2015 (special) : 696-704. doi: 10.3934/proc.2015.0696

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences