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.


Bifurcation Diagram Material Point Periodic Motion Velocity Discontinuity Unilateral Constraint 
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  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

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© Springer Science + Business Media B.V. 2009

Authors and Affiliations

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

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