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Dynamics of a Material Point Colliding with a Limiter Moving with Piecewise Constant Velocity

  • Andrzej Okniński
  • Boguslaw Radziszewski

Vibro-impacting systems are very interesting examples of non-linear dynamical systems with important technological applications [1]. Dynamics of such systems can be extremely complicated due to velocity discontinuity arising upon impacts. A very characteristic feature of impacting systems is presence of non-standard bifurcations such as border-collisions and grazing impacts appearing in the case of motion with low velocity after impact, which often leads to complex chaotic motion [1].

The main difficulty with investigating impacting systems is in gluing pre-impact and post-impact solutions. The Poincaré map, describing evolution from an impact to the next impact, is thus a natural tool to study such systems. In the present paper we investigate motion of a material point in a gravitational field colliding with a moving motion-limiting stop. Typical example of such dynamical system, related to the Fermi model, is a small ball bouncing vertically on a vibrating table. Since evolution between impacts is expressed by a very simple formula the motion in this system is easier to analyze than dynamics of impact oscillators. It is possible to simplify the problem further assuming a special motion of the limiter.

Keywords

Bifurcation Diagram Material Point Periodic Motion Velocity Discontinuity Unilateral Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    di Bernardo M, Budd CJ, Champneys AR, Kowalczyk P (2008) Piecewise-Smooth Dynamical Systems. Theory and Applications. Springer, London.MATHGoogle Scholar
  2. 2.
    Stronge WJ (2000) Impact Mechanics. Cambridge University Press, Cambridge.MATHCrossRefGoogle Scholar
  3. 3.
    Okniski A, Radziszewski B (2007) Grazing dynamics and dependence on initial conditions in certain systems with impacts, arXiv:0706.0257.Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2009

Authors and Affiliations

  1. 1.Politechnika świeȩtokrzyska, Wydzial Zarzaȩdzania i Modelowania KomputerowegoTysiaȩcleciaPoland

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