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

Okniński, Andrzej; Radziszewski, Bogusław

doi:10.1007/978-1-4020-8778-3_11

Cited by 3 publications

(8 citation statements)

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“…We note that previous work has been done by Okninski et al in [21,22] on a similar inelastic, constant restitution coefficient bouncer [23]. In our work we confirm some of the basic results found in these investigations, as well as analyze some more global aspects of the dynamics.…”

Section: Introductionsupporting

confidence: 81%

Regular and chaotic dynamics of a piecewise smooth bouncer

Langer

Miller

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The dynamical properties of a particle in a gravitational field colliding with a rigid wall moving with piecewise constant velocity are studied. The linear nature of the wall's motion permits further analytical investigation than is possible for the system's sinusoidal counterpart. We consider three distinct approaches to modeling collisions: (i) elastic, (ii) inelastic with constant restitution coefficient and (iii) inelastic with a velocity-dependent restitution function. We confirm the existence of distinct unbounded orbits (Fermi acceleration) in the elastic model, and investigate regular and chaotic behavior in the inelastic cases. We also examine in the constant restitution model trajectories wherein the particle experiences an infinite number of collisions in a finite time i.e., the phenomenon of inelastic collapse. We address these so-called "sticking solutions" and their relation to both the overall dynamics and the phenomenon of self-reanimating chaos. Additionally, we investigate the long-term behavior of the system as a function of both initial conditions and parameter values. We find the non-smooth nature of the system produces novel bifurcation phenomena not seen in the sinusoidal model, including border-collision bifurcations. The analytical and numerical investigations reveal that although our piecewise linear bouncer is a simplified version of the sinusoidal model, the former not only captures essential features of the latter but also exhibits behavior unique to the discontinuous dynamics.

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Section: Introductionsupporting

confidence: 81%

Regular and chaotic dynamics of a piecewise smooth bouncer

Langer

Miller

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Our model consists thus of (5), (6) with control parameters R, γ , h. Let us note here that for h → 1 − it follows that γẎ 2 → −∞ and we obtain the model investigated in Ref. [46]. Since the period of motion of the limiter is equal to one, the map (5), (6) is invariant under the translation T i → T i + 1.…”

Section: Bouncing Ball: a Simple Motion Of The Tablementioning

confidence: 96%

“…This problem was investigated in Ref. [46] for h = 1, γẎ 2 = −∞ where it was shown that the solution T i+1 may be a discontinuous function of initial conditions T i , V i and control parameter γ . We shall investigate dynamical consequence of this discontinuity in the future [48].…”

Section: Bouncing Ball: a Simple Motion Of The Tablementioning

confidence: 98%

“…Accordingly, we have decided to simplify the limiter's periodic motion to make (5a) solvable. Very recently we have adopted a very simple motion of the table with piecewise constant velocity Y (T ) = T mod 1 and we have performed numerical and analytical computations [46]. This motion of the table implies that while the table moves up with a constant velocity, it moves down infinitely fast and hence the model is not fully physical.…”

Section: Bouncing Ball: a Simple Motion Of The Tablementioning

confidence: 99%

“…Moreover, the dynamics of this model is relatively simple and corresponds only partially to the dynamics of the model with periodic motion of the table [45]. It follows that the analytical results obtained in [46] cannot illuminate very complex dynamics of the standard model with periodic motion of the limiter. Therefore, in the present paper, we decided to investigate dynamics of a ball with a more realistic motion of the table (a preliminary report has been published in the conference proceedings, see [47]).…”

Section: Bouncing Ball: a Simple Motion Of The Tablementioning

confidence: 99%

See 2 more Smart Citations

Dynamics of impacts with a table moving with piecewise constant velocity

Okniński

Radziszewski

2009

Nonlinear Dyn

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Dynamics of a ball moving in gravitational field and colliding with a moving table is considered. The motion of the limiter is assumed as periodic with piecewise constant velocity. It is assumed that the table moves up with a constant velocity and then goes down with another constant velocity. The Poincaré map describing evolution from an impact to the next impact is derived. Several classes of solutions are computed in analytical form.

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Simple model of bouncing ball dynamics: displacement of the table assumed as quadratic function of time

Okniński

Radziszewski

2011

Nonlinear Dyn

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Nonlinear dynamics of a bouncing ball moving in gravitational field and colliding with a moving limiter is considered. Displacement of the limiter is a quadratic function of time. Several dynamical modes, such as fixed points, 2-cycles, grazing and chaotic bands are studied analytically and numerically. It is shown that chaotic bands appear due to homoclinic structures created from unstable 2-cycles in a corner-type bifurcation.

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

Cited by 3 publications

References 1 publication

Regular and chaotic dynamics of a piecewise smooth bouncer

Regular and chaotic dynamics of a piecewise smooth bouncer

Dynamics of impacts with a table moving with piecewise constant velocity

Simple model of bouncing ball dynamics: displacement of the table assumed as quadratic function of time

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